Throughout my Ph.D., I have had the privilege of co-supervising several master’s students alongside my advisors, Prof. Pauline Bernard and Prof. Florent Di Meglio, during their research internships at Mines Paris - PSL. The students and their research projects, listed by the order in which their results appear in this dissertation, are as follows:

• Clément Boutaric, Kalman-Like Observer for Hybrid Systems with Linear Maps and Known Jump Times, November \( 2022\) - February \( 2023\) , related results included in Section 3.2;
• Aymeric Cardot, Observer Design for Hybrid Systems with Fully Nonlinear Maps and Known Jump Times, November \( 2023\) - February \( 2024\) , related results included in Chapters 3.3 and 3.4;
• Shang Liu, Hybrid Observer Design for a Three-Link Biped Robot, November \( 2023\) - February \( 2024\) , related results included in Section 3.4;
• Sergio Garcia, Gluing KKLKravaris-Kazantzis$\slash$Luenberger Observer for Hybrid Systems with Unknown Jump Times, November \( 2023\) - February \( 2024\) , related results included in Section 4.2;
• Zikang Zhu, NNNeural Network-Based Implementation of Gluing KKLKravaris-Kazantzis$\slash$Luenberger Observer for Hybrid Systems with Unknown Jump Times, June - September \( 2024\) , related results included in Section 4.3.

Close to the end of my Ph.D., I had the pleasure of collaborating with Valentin Alleaume, a new Ph.D. student working with my supervisors. Our collaboration is reflected in several joint papers.

During my Ph.D., I have been honored with the following awards and recognitions:

• Honda Y-E-S Award Plus from Honda Foundation, for a research period at Keio University, Tokyo, Japan, January - March \( 2025\) ;
• Student Travel Support and Student Workshop Support for the \( 63\) rd IEEE Conference on Decision and Control (CDC), Milan, Italy, December \( 2024\) ;
• Doctoral Student Mobility Scholarship from Paris Mathematical Sciences Foundation, for a research visit at the University of California Santa Cruz, California, USA, April - May \( 2023\) ;
• Mines Paris Foundation International Mobility Scholarship, for the same research visit;
• Participant of the \( 9\) th Heidelberg Laureate Forum, Heidelberg, Germany, September \( 2022\) ;
• Travel Award for Postdocs and Ph.D. Students from Electronics, to attend the \( 11\) th IFAC Symposium on Fault Detection, Supervision and Safety for Technical Processes (SAFEPROCESS), Paphos, Cyprus, June \( 2022\) .

During my Ph.D., I have participated in the following technical courses:

• \( 44\) th International Summer School of Automatic Control, Grenoble, France, August \( 2023\) ;
• Introduction to Nonlinear Systems and Control (Instructor: Prof. Hassan Khalil), International Graduate School on Control, European Embedded Control Institute, Paris-Saclay, France, May \( 2022\) .

Gia Quoc Bao Tran, Valentin Alleaume, Pauline Bernard, Ricardo G. Sanfelice. "Existence and Regularity of KKLKravaris-Kazantzis$\slash$Luenberger Gluing Change of Coordinates for Hybrid Systems". Will be submitted to a journal, content taken from Section 4.2.

Gia Quoc Bao Tran, Valentin Alleaume, Pauline Bernard, Florent Di Meglio. "Dealing with Indistinguishability in Gluing KKLKravaris-Kazantzis$\slash$Luenberger Observer Design for Hybrid Systems with Unknown Jump Times". Will be submitted to a journal, content taken from Section 4.2 and Section 4.3.

Gia Quoc Bao Tran, Pauline Bernard, Ricardo G. Sanfelice. "Observer Design for Hybrid Systems with Known Jump Times". Will be submitted to a journal, content taken from Section 3.3 and Section 3.4.

Gia Quoc Bao Tran, Pauline Bernard, Ricardo G. Sanfelice. "Observer Design for Hybrid Systems with Partially Affine Forms and Known Jump Times: Applications a Walking Robots". Submitted to Automatica.
Link: https://hal.science/hal-04885099

Gia Quoc Bao Tran, Pauline Bernard. "Arbitrarily Fast Robust KKLKravaris-Kazantzis$\slash$Luenberger Observer for Nonlinear Time-Varying Discrete Systems". IEEE Transactions on Automatic Control, Volume \( 69\) , Issue \( 3\) , March \( 2024\) .
DOI: 10.1109/TAC.2023.3328833

Gia Quoc Bao Tran, Pauline Bernard, Lorenzo Marconi. "Observer Design for Hybrid Systems with Linear Maps and Known Jump Times". Hybrid and Networked Dynamical Systems (Editors: Romain Postoyan, Paolo Frasca, Elena Panteley, Luca Zaccarian), pp. \( 115\) -\( 154\) , Lecture Notes in Control and Information Sciences, Volume \( 493\) , Springer, March \( 2024\) .
DOI: 10.1007/978-3-031-49555-7_6

Gia Quoc Bao Tran, Sergio Garcia, Pauline Bernard, Florent Di Meglio, Ricardo G. Sanfelice. "Towards Gluing KKLKravaris-Kazantzis$\slash$Luenberger Observer for Hybrid Systems with Unknown Jump Times". \( 63\) rd IEEE Conference on Decision and Control (CDC), Milan, Italy, December \( 2024\) .
Link: https://hal.science/hal-04685195

Gia Quoc Bao Tran, Pauline Bernard, Vincent Andrieu, Daniele Astolfi. "Constructible Canonical Form and High-Gain Observer in Discrete Time". \( 63\) rd IEEE Conference on Decision and Control (CDC), Milan, Italy, December \( 2024\) .
Link: https://hal.science/hal-04685007

Gia Quoc Bao Tran, Pauline Bernard, Florent Di Meglio, Ricardo G. Sanfelice. "Observer Design for Hybrid Systems: Applications to Robotics and Hybrid Dynamics". Extended abstract at \( 11\) th European Nonlinear Dynamics Conference (ENOC), Delft, Netherlands, July \( 2024\) .

Gia Quoc Bao Tran, Pauline Bernard. "Kalman-Like Observer for Hybrid Systems with Linear Maps and Known Jump Times". \( 62\) nd IEEE Conference on Decision and Control (CDC), Singapore, December \( 2023\) .
DOI: 10.1109/CDC49753.2023.10383629

Gia Quoc Bao Tran, Pauline Bernard, Ricardo G. Sanfelice. "Coupling Flow and Jump Observers for Hybrid Systems with Known Jump Times". \( 22\) nd IFAC World Congress, Yokohama, Japan, July \( 2023\) .
DOI: 10.1016/j.ifacol.2023.10.522

Gia Quoc Bao Tran, Pauline Bernard, Florent Di Meglio, Lorenzo Marconi. "Observer Design based on Observability Decomposition for Hybrid Systems with Linear Maps and Known Jump Times". \( 61\) st IEEE Conference on Decision and Control (CDC), Cancun, Mexico, December \( 2022\) .
DOI: 10.1109/CDC51059.2022.9993225

Gia Quoc Bao Tran, Thanh-Phong Pham, Olivier Sename. "Reduced-Order Observer for a Class of Uncertain Descriptor NLPV Systems: Application to the Vehicle Suspension". Submitted to a journal.

Thach Ngoc Dinh\( ^\star\) , Gia Quoc Bao Tran\( ^\star\) . "Systematic Interval Observer Design for Linear Systems". Submitted to a journal.
DOI: 10.48550/arXiv.2405.06445

Thach Ngoc Dinh\( ^\star\) , Gia Quoc Bao Tran\( ^\star\) . "A Unified KKLKravaris-Kazantzis$\slash$Luenberger-Based Interval Observer for Nonlinear Discrete-Time Systems". IEEE Control Systems Letter (jointly presented at \( 63\) rd IEEE CDC), Volume \( 8\) , April \( 2024\) .
DOI: 10.1109/LCSYS.2024.3394324

Gia Quoc Bao Tran, Thanh-Phong Pham, Olivier Sename. "Fault Estimation Observers for the Vehicle Suspension with a Varying Chassis Mass". \( 12\) th IFAC Symposium on Fault Detection, Supervision and Safety for Technical Processes (SAFEPROCESS), Ferrara, Italy, June \( 2024\) .
DOI: 10.1016/j.ifacol.2024.07.207

Poster at \( 63\) rd IEEE Conference on Decision and Control. "Observer Design for Hybrid Systems". Milan, Italy, December \( 2024\) .

Seminar at Division of Systems and Control, Uppsala University. "Observer Design for Hybrid Systems". Uppsala, Sweden, October \( 2024\) .

Seminar at Division of Decision and Control Systems, KTH Royal Institute of Technology. "Observer Design for Hybrid Systems". Stockholm, Sweden, October \( 2024\) .

Poster at Mines Paris Ph.D. Student Day. "Observer Design for Hybrid Systems". Paris, France, June \( 2024\) .

Seminar at LAGEPP (CNRS and Université Lyon \( 1\) ). "Observer Design for Hybrid Systems". Villeurbanne, France, February \( 2024\) .

Poster at \( 62\) nd IEEE Conference on Decision and Control. "Observer Design for Hybrid Systems". Singapore, December \( 2023\) .

Talk at French Observer Group Day (Journée du GT SynObs). "Kalman-Like Observer for Hybrid Systems with Linear Maps and Known Jump Times". Paris, France, December \( 2023\) .

Student talk at \( 44\) th International Summer School of Automatic Control. "Observer Design for Hybrid Systems with Known Jump Times". Grenoble, France, August \( 2023\) .

Seminar at Hybrid Systems Laboratory, University of California Santa Cruz. "Observer Design for Hybrid Systems with Known Jump Times". Santa Cruz, California, USA, May \( 2023\) .

Poster at \( 5\) th NorCal Control Workshop, University of California Berkeley. "Observer Design for Hybrid Systems". Berkeley, California, USA, April \( 2023\) .

Talk at French Hybrid System Technical Committee Reunion (Réunion du CT SDH). "Coupling Observers for Hybrid Systems with Known Jump Times". Paris, France, February \( 2023\) .

Poster at \( 9\) th Heidelberg Laureate Forum. "Observer Design for Hybrid Systems by Observability Decomposition". Heidelberg, Germany, September \( 2022\) .

Seminar at Hanoi University of Science and Technology (HUST). "Arbitrarily Fast Robust KKLKravaris-Kazantzis$\slash$Luenberger Observer for Nonlinear Time-Varying Discrete Systems". Hanoi, Vietnam, August \( 2022\) .

Talk at French Observer Group Day (Journée du GT SynObs). "Arbitrarily Fast Robust KKLKravaris-Kazantzis$\slash$Luenberger Observer for Nonlinear Time-Varying Discrete Systems". Paris, France, June \( 2022\) .

Notations. Denote \( \emptyset\) as the empty set. Let \( \mathbb{R}\) (resp., \( \mathbb{Z}\) ) denote the set of real numbers (resp., integers, i.e., \( \{…,-2,-1,0, 1, 2, …\}\) ). Let \( \mathbb{N}\) denote the set of natural numbers, i.e., \( \{0, 1, 2, …\}\) . Let \( \mathbb{R}_{\geq a}\) (resp., \( \mathbb{R}_{> a}\) ) denote the interval \( [a,+\infty)\) (resp., \( (a,+\infty)\) ), and similarly for \( \mathbb{Z}_{\geq a}\) and \( \mathbb{N}_{\geq a}\) . Similarly, we denote \( \mathbb{R}_{\leq a}\) (resp., \( \mathbb{R}_{< a}\) ), as well as \( \mathbb{Z}_{\leq a}\) and \( \mathbb{N}_{\leq a}\) . Denote \( \mathbb{C}\) as the set of complex numbers. Let \( \Re(z)\) and \( \Im(z)\) be the real and imaginary parts of \( z \in \mathbb{C}\) , respectively. Given \( \rho > 0\) , define \( \mathbb{C}_\rho = \{\lambda \in \mathbb{C}: \Re(\lambda) < -\rho\}\) . Denote \( \mathbb{R}^{m \times n}\) (resp., \( §_{> 0}^{n}\) ) as the set of real \( (m \times n)\) - (resp., symmetric positive definite real \( (n \times n)\) -) dimensional matrices. The sequence \( a_1,a_2,…\) (of numbers, vectors, sets, functions, etc.) is denoted as \( (a_n)_{n\in \mathbb{N}}\) . For some set \( S\) , let \( S^{\mathbb{N}}\) be the set of sequences whose elements belong to \( S\) . The closure of a set \( S\) , denoted as \( \rm{cl}(S)\) , is the smallest closed superset of \( S\) , while the interior of \( S\) , denoted as \( \rm{int}(S)\) , is the largest open subset contained within \( S\) . Denote \( \mathbb{B}\) (resp., \( \mathring{\mathbb{B}}\) ) as the closed (resp., open) unit ball and \( § + \delta\mathbb{B}\) (resp., \( § + \delta\mathring{\mathbb{B}}\) ) as the set of points within (resp., strictly smaller than) a distance \( \delta > 0\) from any point in \( §\) . For \( x\in \mathbb{R}^n\) , \( B_{r}(x)\) denotes the open ball of radius \( r > 0\) centered at \( x\) . For a non-empty set \( S \subseteq \mathbb{R}^n\) , let \( d_S(x):= \inf_{y\in S}|x-y|\) be the distance from \( x \in \mathbb{R}^n\) to \( S\) .

A metric space is an ordered pair \( (M, d)\) where \( M\) is a set and \( d\) is a metric on \( M\) , satisfying the following conditions: i) \( d(x,y) \geq 0\) for all \( (x,y) \in M \times M\) and \( d(x,y) = 0 \Leftrightarrow x = y\) , ii) \( d(x,y) = d(y,x)\) for all \( (x,y) \in M \times M\) , and iii) \( d(x,z) \leq d(x,y) + d(y,z)\) for all \( (x,y,z) \in M \times M \times M\) (the triangle inequality). If \( M\) is a normed vector space, a possible metric is \( d(x,y):= |x-y|\) for \( (x,y) \in M \times M\) , where \( |\cdot|\) denotes the corresponding norm.

A subset \( S\) of a metric space \( (M,d)\) is said to be open if, for each \( x \in S\) , there exists \( \epsilon > 0\) such that any \( y \in M\) with \( d(x,y) < \epsilon\) is also in \( S\) . A closed subset of \( (M,d)\) is one whose complement in \( (M,d)\) is open. A set \( S\) in a metric space \( (M, d)\) is bounded if there exists \( M \geq 0\) such that \( d(x, y) \leq M\) for all \( (x,y) \in S \times S\) . In a finite-dimensional Euclidean space, a set is compact if it is both closed and bounded. This result is known as the Heine–Borel theorem. While the concept of compactness is more complex in general, this dissertation focuses on the finite-dimensional case, where compactness is defined simply as closedness and boundedness.

The Lebesgue measure is a function \( \mu\) that assigns a non-negative number to each measurable set, generalizing the notions of length, area, and volume. A property is said to hold for almost any (or almost every) point in a set \( S\) if the set of points where the property does not hold has Lebesgue measure zero, i.e., this property holds for all points in \( S\) except for a subset of \( S\) with Lebesgue measure zero.

A vector-induced matrix norm is defined based on a vector norm. Given a vector norm \( |\cdot|\) on \( \mathbb{R}^n\) , the induced norm \( \|A\|\) of a matrix \( A \in \mathbb{R}^{m \times n}\) is defined as:

In this dissertation, the \( 2\) -norm (or spectral norm) is primarily used for matrices, which is induced by the Euclidean vector norm. It is defined as \( \|A\|_2 = \sqrt{\lambda_{\max}(A^H A)}\) where \( \lambda_{\max}(A^H A)\) is the largest eigenvalue of \( A^H A\) , and \( A^H\) is the conjugate transpose of \( A\) .

A matrix \( A \in \mathbb{R}^{m \times n}\) is termed left-invertible if there exists a matrix \( B \in \mathbb{R}^{n \times m}\) such that \( BA = \rm{Id}_n\) , where \( \rm{Id}_n\) denotes the \( n \times n\) identity matrix. The matrix \( B\) is referred to as the left inverse of \( A\) , denoted as \( A^*\) . For \( A\) to be left-invertible, it must satisfy the rank condition \( \rm{rank}(A) = n\) , indicating that \( A\) has full column rank and therefore \( m \geq n\) . In this dissertation, we often deal with matrices that vary with inputs, necessitating that this left invertibility is uniform across all inputs. Specifically, the matrix \( u \mapsto A(u) \in \mathbb{R}^{m \times n}\) , defined for an input \( u \in \mathcal{U}\) (where \( \mathcal{U}\) is a possibly unbounded set), is said to be uniformly left-invertible if there exists a constant \( c > 0\) such that:

A matrix \( A \in \mathbb{R}^{n \times n}\) is invertible if it is both left-invertible and right-invertible, meaning that there exists a matrix \( B \in \mathbb{R}^{n \times n}\) such that \( BA = AB = \rm{Id}_n\) . The matrix \( B\) is called the inverse of \( A\) , denoted as \( A^{-1}\) . The rank condition for \( A\) to be invertible is \( \rm{rank}(A) = n\) , implying that \( A\) has full rank. The notion of uniform invertibility is also defined using (2), with \( m = n\) .

The Kronecker product of two matrices \( A \in \mathbb{R}^{m \times n}\) and \( B \in \mathbb{R}^{p \times q}\) , denoted as \( A \otimes B\) , is defined as:

Notations. Let \( \sigma_{\min}(A):=\sqrt{\lambda_{\min}(A^\top A)}\) (resp., \( \sigma_{\max}(A):=\sqrt{\lambda_{\max}(A^\top A)}\) ) denote the smallest (resp., largest) singular value of matrix \( A\) , where \( \lambda_{\min}\) (resp., \( \lambda_{\max}\) ) represents the minimal (resp., maximal) eigenvalue. Let \( |\cdot|\) be the Euclidean norm and \( \|\cdot\|\) the induced matrix norm which coincides with \( \sigma_{\max}(\cdot)\) . Denote \( A^\top\) as the transpose of matrix \( A\) and \( A^H\) as its conjugate transpose. Let \( A^\bot\) be the orthogonal complement of matrix \( A\) satisfying \( A^\bot A = 0\) and \( A^\bot (A^\bot)^\top > 0\) , and \( A^\dagger\) be the Moore-Penrose inverse of \( A\)  [37]; if \( A\) is square and invertible, then \( A^\dagger = A^{-1}\) . Let \( \rm{diag}(\lambda_1, \lambda_2, …, \lambda_n)\) be the diagonal matrix with entries \( \lambda_i\) , \( i = 1,2,…,n\) . Denote \( \rm{Id}_n\) (resp., \( 0_{m\times n}\) ) as the \( (n\times n)\) -dimensional identity matrix (resp., \( (m \times n)\) -dimensional zero matrix), with dimensions omitted when contextually clear. Let \( A \otimes B\) denote the Kronecker product of matrices \( A\) and \( B\) . For \( x = (x_1,x_2,x_3) \in \mathbb{R}^3\) , denote

The symbol \( \star\) in the matrix inequalities denotes the symmetric block, while in some long derivations, \( \star^\top P x\) denotes \( x^\top P x\) . A square matrix is Hurwitz (resp., Schur) if all of its eigenvalues lie strictly on the left-half complex plan (resp., strictly inside the unit ball centered at the origin of the complex plan).

A function (or map\( /\) mapping) \( f: M \to N\) between metric spaces \( (M, d_M)\) and \( (N, d_N)\) is continuous at a point \( x \in M\) if, for every \( \epsilon > 0\) , there exists \( \delta > 0\) such that \( d_M(x, y) < \delta\) implies \( d_N(f(x), f(y)) < \epsilon\) . The function \( f\) is continuous on \( S \subseteq M\) if it is continuous at every point in \( S\) ; if \( S = M\) , it is simply called continuous. If the same \( \delta > 0\) works for every \( \epsilon > 0\) , then \( f\) is uniformly continuous at \( x\) .

A real-valued function \( f\) defined on an interval \( I \subseteq \mathbb{R}\) is absolutely continuous on \( I\) if, for each \( \epsilon > 0\) , there exists \( \delta > 0\) such that for any finite collection of disjoint sub-intervals \( (x_k,y_k)\) of \( I\) with \( x_k < y_k \in I\) , if

then

It is locally absolutely continuous on \( I\) if, for every closed sub-interval \( I^\prime \subseteq I\) , the restriction of \( f\) to \( I^\prime\) is absolutely continuous.

Let us now recall comparison functions [38, Section 4.4]. A function \( \alpha: \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}\) is of class-\( \mathcal{K}\) if it is continuous, strictly increasing, and \( \alpha(0) = 0\) . A class-\( \mathcal{K}\) function belongs to class-\( \mathcal{K}_\infty\) if it is also unbounded, i.e., \( \lim_{r \to +\infty}\alpha(r) = +\infty\) . A function \( \beta: \mathbb{R}_{\geq 0} \times \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}\) is of class-\( \mathcal{KL}\) if i) for each fixed \( s\) , \( r \mapsto \beta(r,s)\) is of class-\( \mathcal{K}\) , and ii) for each fixed \( r\) , \( s \mapsto \beta(r,s)\) is decreasing and \( \lim_{s \to +\infty}\beta(r,s) = 0\) .

With these comparison functions at hand, we uniformize the notion of continuity as follows. A function \( f\) is \( \mathcal{K}\) -continuous on a compact set \( \mathcal{X}\) if there exists a class-\( \mathcal{K}\) function \( \rho\) such that On compacts, \( \mathcal{K}\) -continuity coincides with uniform continuity.

If \( f\) is continuous on \( \mathcal{X}\) and \( \mathcal{X}\) is compact, then \( f\) is \( \mathcal{K}\) -continuous on this set [39, Lemmas A.6 and A.9]. If \( \rho\) takes the form \( \rho(s) = Ls\) for some \( L > 0\) , then \( f\) is said to be Lipschitz on \( \mathcal{X}\) with Lipschitz constant \( L\) . A function \( f\) is locally Lipschitz on \( \mathcal{X}\) if \( f\) is Lipschitz on each compact subset of \( \mathcal{X}\) , but with possibly different Lipschitz constants depending on the compact subset. In this dissertation, we often consider multi-variable functions that are uniformly continuous with respect to one of its arguments, while uniformity also holds in the rest. For example, if \( f\) also has \( u \in \mathcal{U}\) as its extra argument besides \( x\) , this corresponds to the fact that there exists a class-\( \mathcal{K}\) function \( \rho\) such that

Here, the function \( \rho\) is the same for all \( u \in \mathcal{U}\) , and we say that \( f\) is uniformly continuous in \( x\) on \( \mathcal{X}\) , uniformly in \( u\) on \( \mathcal{U}\) . The same holds for the Lipschitz case when \( f\) is Lipschitz in \( x\) on \( \mathcal{X}\) , uniformly in \( u\) on \( \mathcal{U}\) , and we simply say that \( f\) is uniformly Lipschitz when no confusion is possible.

The notion of injectivity can be seen as a counterpart to continuity, as explained next. A function \( f\) is injective on a set \( \mathcal{X}\) if \( f(x_a) = f(x_b)\) implies \( x_a = x_b\) for all \( (x_a,x_b) \in \mathcal{X} \times \mathcal{X}\) . A function \( f\) is uniformly injective on a set \( \mathcal{X}\) if there exists a class-\( \mathcal{K}\) function \( \rho\) such that

Similarly to the above, using [39, Lemmas A.6 and A.9], if \( f\) is injective on \( \mathcal{X}\) and \( \mathcal{X}\) is compact, then \( f\) is uniformly injective on this set. If \( \rho\) takes the form \( \rho(s) = Ls\) for some \( L > 0\) , then \( f\) is said to be Lipschitz injective on \( \mathcal{X}\) . If this Lipschitz injectivity is uniform with respect to the other arguments \( f\) may have, then \( f\) is Lipschitz injective in \( x\) on \( \mathcal{X}\) , uniformly in the other arguments on their respective sets, and we simply say that \( f\) is uniformly Lipschitz injective on \( \mathcal{X}\) when no confusion is possible.

A function \( f: \mathbb{R}^n \to \mathbb{R}^m\) where \( m \geq n\) , which is uniformly injective on \( \mathcal{X} \subseteq \mathbb{R}^n\) , is left-invertible on \( f(\mathcal{X})\) , i.e., there exists a function \( g: f(\mathcal{X}) \to \mathbb{R}^n\) such that \( g(f(x)) = x\) for all \( x \in \mathcal{X}\) . The function \( g\) is called a left inverse of \( f\) on \( f(\mathcal{X})\) , denoted as \( f^*\) . A function \( f: \mathbb{R}^n \to \mathbb{R}^n\) that is uniformly injective on \( \mathcal{X} \subseteq \mathbb{R}^n\) is invertible on \( \mathcal{X}\) , i.e., there exists a function \( g: f(\mathcal{X}) \to \mathbb{R}^n\) such that \( g(f(x)) = x\) for all \( x \in S\) and \( f(g(z)) = z\) for all \( z \in f(\mathcal{X})\) . The function \( g\) is called an inverse of \( f\) on \( f(\mathcal{X})\) , denoted as \( f^{-1}\) . These are more general notions of (left-)inverses than the ones defined using matrices above, which are linear.

A function \( f\) is bounded on a set \( \mathcal{X}\) if there exists \( M \geq 0\) such that

A function \( f\) is locally bounded on \( \mathcal{X}\) if \( f\) is bounded on each compact subset of \( \mathcal{X}\) , but with possibly different bounds depending on the compact subset. Similarly to the above, if this boundedness is uniform with respect to the other arguments \( f\) may have, then \( f\) is uniformly bounded on \( \mathcal{X}\) .

A function is continuously differentiable or \( C^1\) if its derivative exists and is continuous. More generally, a function is \( C^k\) if it is continuously differentiable of order \( k \in \mathbb{N}\) , i.e., its derivative up to order \( k\) is continuous. It may happen that \( k = +\infty\) , in which case the function is infinitely differentiable. The gradient of a continuously differentiable function \( V: \mathbb{R}^n \to \mathbb{R}\) at \( x = (x_1,x_2,…,x_n) \in \mathbb{R}^n\) is

where \( \frac{\partial V}{\partial x_i}(x)\) , for each \( i \in \{1,2,…,n\}\) , is the partial derivative of \( V\) with respect to \( x_i\) , evaluate at \( x\) .

A set-valued function assigns to each element in the domain a set of elements in the codomain. A set-valued function \( F: \mathbb{R}^n \rightrightarrows \mathbb{R}^m\) is outer semi-continuous (or in some literature, upper semi-continuous) at \( x \in \mathbb{R}^n\) if for any sequence \( (x_n)_{n \in \mathbb{N}}\) converging to \( x\) and any sequence \( (z_n)_{n \in \mathbb{N}}\) where \( z_n \in F(x_n)\) converging to \( z \in \mathbb{R}^m\) , we have \( z \in F(x)\) .

Notations. For two functions \( f\) and \( g\) , \( f \circ g\) is their composition, namely for all \( x\) in the domain of \( g\) , \( g(x)\) is in the domain of \( f\) and \( (f \circ g)(x) = f(g(x))\) . Given a function \( (x,u) \mapsto f(x,u)\) that is locally Lipschitz in \( x\) on \( \mathcal{X}\) , uniformly in \( u\) on \( \mathcal{U}\) , define \( f_\rm{{sat}}\) as a map that is globally Lipschitz in \( x\) , uniformly in \( u\) on \( \mathcal{U}\) , and agrees with \( f\) on \( \mathcal{X} \times \mathcal{U}\) ; such a map is guaranteed to exist by [40, Corollary 1]. Let \( \nabla V(x)\) denote the gradient of a continuously differentiable scalar function \( V\) at \( x\) . Let \( \|\cdot\|_{C^1}\) be the \( C^1\) -norm of \( C^1\) functions on the compact set \( \mathcal{X}\) , defined as \( \|f\|_{C^1} = \max_{x\in\mathcal{X}}|f(x)| + \max_{x\in\mathcal{X}}\left|\frac{\partial f}{\partial x}(x)\right|\) .

In continuous time, a dynamical system with input and (measured) output is described by the following equations:

where \( x(t) \in \mathbb{R}^{n_x}\) represents the state and \( \dot{x}(t) = \frac{dx}{dt}(t)\) is its time derivative, \( u(t) \in \mathbb{R}^{n_u}\) is the known input, \( y(t) \in \mathbb{R}^{n_y}\) is the measured output, and \( (v(t),w(t))\) are the unknown additive disturbances and measurement noise, respectively, all at continuous time \( t \in \mathbb{R}_{\geq 0}\) . While it is possible to consider a nonlinear dependence on these unknown inputs, doing so may render \( (f,h)\) as functions of \( x\) not well-defined. From now on, when unnecessary, the dependence on time \( t\) is not explicitly written.

Similarly, in discrete time, a dynamical system with input and output is described by the following equations:

where \( x_k \in \mathbb{R}^{n_x}\) , \( u_k \in \mathbb{R}^{n_u}\) , \( y_k \in \mathbb{R}^{n_y}\) , and \( (v_k,w_k)\) represents the state, known input, measured output, and unknown additive disturbances and measurement noise, respectively, all at discrete time \( k \in \mathbb{N}\) .

Systems (11) and system (12) are non-autonomous due to the inputs \( (u(t),v(t))\) (and \( (u_k,v_k)\) ) as functions of time. This representation covers time-varying systems, which is a special case when \( u(t) = t\) (resp., \( u_k = k\) ). When \( (u(t),v(t))\) (resp., \( (u_k,v_k)\) ) are absent, these systems are said to be autonomous. These systems, described by differential and difference equations, are dynamical because their current state depends on the entire history of its past values and inputs. Indeed, initialized as \( x(0)\) and with input trajectories \( t \mapsto (u(t),v(t))\) , any solution to system (11), evaluated at time \( t \in \mathbb{R}_{\geq 0}\) , if defined, verifies

Similarly, initialized as \( x_0\) and with input trajectories \( k \mapsto (u_k,v_k)\) , any solution to system (12), evaluated at time \( k \in \mathbb{N}\) , if defined, verifies

Nonlinear phenomena such as finite-time escape [38, Chapter 1] or solutions leaving the flow set of a state-constrained system [41] can make solutions, for some initial conditions and input trajectories, defined only up to a certain time. Then, a solution \( t \mapsto x(t)\) (resp., \( k \mapsto x_k\) ) is (forward) complete if its time domain is the whole \( \mathbb{R}_{\geq 0}\) (resp., \( \mathbb{N}\) ). In the context of Kravaris-Kazantzis$\slash$Luenberger[KKL] observers considered later in Chapters 2.3 and 4.2, we often need to consider solutions in backward or negative time, leading to the notion of backward completeness of solutions. Note that in this dissertation, exogenous inputs are defined at all times, even though solutions may not be. Next, we denote:

• \( \mathcal{X}_0 \subseteq \mathbb{R}^{n_x}\) as the set of initial conditions \( x(0)\) (resp., \( x_0\) ) of interest for system (11) (resp., system (12));
• \( \mathfrak{U}\) as the set of input trajectories \( t \mapsto u(t)\) (resp., \( k \mapsto u_k\) ) of interest for system (11) (resp., system (12));
• \( \mathcal{U} \subseteq \mathbb{R}^{n_u}\) as the set of the values that \( u(t)\) (resp., \( u_k\) ) can take along a trajectory in \( \mathfrak{U}\) , for \( t \in \mathbb{R}_{\geq 0}\) (resp., \( k \in \mathbb{N}\) );
• \( \mathcal{V}\) as the set of input trajectories \( t \mapsto v(t)\) (resp., \( k \mapsto v_k\) ) that system (11) (resp., system (12)) is considered to have;
• \( \mathcal{W}\) as the set of input trajectories \( t \mapsto w(t)\) (resp., \( k \mapsto w_k\) ) that system (11) (resp., system (12)) is considered to have.

The knowledge of \( \mathcal{X}_0\) is usually not needed for observer design but is used in the statement of theoretical results to describe the set of solutions of interest. Here, we clarify the difference between the sets \( \mathfrak{U}\) and \( \mathcal{U}\) for the following reason. Some of the properties of solutions are linked to the considered inputs or even their derivatives, up to a high enough order, as trajectories of time; these include completeness (e.g., only some controlled solutions can be complete), invariance (not escaping from some bounded set), and some observability conditions like instantaneous observability [42, Assumption 3] or Uniform Complete Observability[UCO] [43] (e.g., only solutions subject to some persistently exciting control inputs are observable). On the other hand, some properties such as regularity of the system’s maps or observability conditions such as quadratic detectability (typically used for observer design using a Linear Matrix Inequality[LMI]—see for instance [44]) are point-wise and hence not depending on the history or variation of the inputs. Thus, inputs are classified either based on trajectories using the set \( \mathfrak{U}\) , or values using \( \mathcal{U}\) . With these definitions at hand, we make the following basic assumptions on the solutions to systems (11) and (12) of interest.

Notations. For a given input \( t \mapsto u(t)\) to the dynamics \( \dot{x} = f(x,u)\) that admit unique solutions (e.g., when \( f\) is locally Lipschitz in \( x\) , uniformly in \( u\) ), let \( \Psi_{f(\cdot,u)}(x_0,t,\tau)\) denote the associated flow operator from initial value \( x_0\) at initial time \( t\) evaluated after \( \tau\) time unit(s); based on (13), it verifies

The duration \( \tau\) can be negative, in which case we integrate \( f\) backward. If \( f\) is autonomous, then this notation reduces to \( \Psi_f(x_0,\tau)\) . If \( f\) takes the form \( f(x,u) = A(u)x + B(u)\) , then \( \Psi_{f(\cdot,u)}(x_0,t,\tau)\) is linear in \( x_0\) , taking the form

where \( \Phi_{A(u)}(t,t^\prime) \in \mathbb{R}^{n_x \times n_x}\) is the state transition matrix from time \( t^\prime \in \mathbb{R}\) to time \( t \in \mathbb{R}\) of the system \( \dot{x} = A(u) x\) , such that solutions to this system verify \( x(t) = \Phi_{A(u)}(t,t^\prime) x(t^\prime)\) and such that \( \Phi(t,t) = \rm{Id}\) for all \( t \in \mathbb{R}\) . If moreover \( f\) is Linear Time-Invariant[LTI], i.e., \( f(x) = Ax + B\) for some constant matrices \( (A,B)\) , then

i.e., as (16) with \( \Phi_A(\tau) = e^{A\tau}\) , which is now time-invariant, where the matrix exponential is \( e^{A} = \displaystyle\sum_{k=0}^{+\infty} \frac{(A)^k}{k!} \in \mathbb{R}^{n_x \times n_x}\) for \( A \in \mathbb{R}^{n_x \times n_x}\) (this expression is still true if \( A\) has complex components).

In this section, we introduce observers for continuous- and discrete-time systems and discuss their properties. Observers, as illustrated in Figure 7, are algorithms designed to estimate the states of dynamical systems from their known inputs and measured outputs. In continuous time, an observer for system (11) takes the general form

where \( \hat{z} \in \mathbb{R}^{n_z}\) is the observer state. Similarly, for system (11) in discrete time, a general observer is

Designing an observer involves determining the functions \( (\mathcal{F},\Upsilon)\) and an initialization set \( \mathcal{Z}_0 \subseteq \mathbb{R}^{n_z}\) such that the estimation error satisfies certain desired properties. The types of observers thus depend on these properties, namely observer (18) (resp., observer (19)), is an

• Asymptotic observer, if in the absence of disturbance \( t \mapsto v(t)\) (resp., \( k \mapsto v_k\) ) and noise \( t \mapsto w(t)\) (resp., \( k \mapsto w_k\) ), for any solutions to system (11) (resp., system (12)) initialized in \( \mathcal{X}_0\) with \( t \mapsto u(t) \in \mathfrak{U}\) (resp., \( k \mapsto u_k \in \mathfrak{U}\) ) and to observer (18) (resp., observer (19)) initialized in \( \mathcal{Z}_0\) and fed with \( t \mapsto (u(t),y(t))\) (resp., \( k \mapsto (u_k,y_k)\) ), we have

  \[ \begin{equation} \lim_{t \to +\infty} |x(t) - \hat{x}(t)| = 0, \text{~ or resp., ~} \lim_{k \to +\infty} |x_k - \hat{x}_k| = 0. \end{equation} \]

  (20)

  Note that this property and the next ones require the system’s solutions of interest to be complete, which is guaranteed by Assumption 1;

• Asymptotically stable observer, if there exists a class-\( \mathcal{KL}\) function \( \beta\) such that in the absence of disturbance \( t \mapsto v(t)\) (resp., \( k \mapsto v_k\) ) and noise \( t \mapsto w(t)\) (resp., \( k \mapsto w_k\) ), for any solutions to system (11) (resp., system (12)) initialized in \( \mathcal{X}_0\) with \( t \mapsto u(t) \in \mathfrak{U}\) (resp., \( k \mapsto u_k \in \mathfrak{U}\) ) and to observer (18) (resp., observer (19)) initialized in \( \mathcal{Z}_0\) and fed with \( t \mapsto (u(t),y(t))\) (resp., \( k \mapsto (u_k,y_k)\) ), we have

  \[ \begin{multline} |x(t) - \hat{x}(t)| \leq \beta(|x(0) - \hat{x}(0)|,t), ~~ \forall t \geq 0, \\ \text{~ or resp., ~} |x_k - \hat{x}_k| \leq \beta(|x_0 - \hat{x}_0|,k), ~~ \forall k \in \mathbb{N}. \end{multline} \]

  (21)

  These properties ensure not only asymptotic convergence but also the stability of the estimation error with respect to its initial condition. Specifically, the estimation error remains arbitrarily small at all times if its initial value is small enough. In certain contexts, especially in discrete time, (21) can be stated with respect to the initial condition but only holds from a certain time, i.e.,

  \[ \begin{multline} |x(t) - \hat{x}(t)| \leq \beta(|x(0) - \hat{x}(0)|,t), ~~ \forall t \geq t_0, \\ \text{~ or resp., ~} |x_k - \hat{x}_k| \leq \beta(|x_0 - \hat{x}_0|,k), ~~ \forall k \in \mathbb{N}_{\geq k_0}, \end{multline} \]

  (22)

  for some certain \( t_0 \geq 0\) (resp., \( k_0 \in \mathbb{N}\) ), then we say that the observer is asymptotically stable (with respect to its initial condition) after a certain time. If the same map \( \beta\) can be used to express the stability of the estimation error with respect to its value at any time in addition to the one at the initial time, namely

  \[ \begin{multline} |x(t) - \hat{x}(t)| \leq \beta(|x(t_0) - \hat{x}(t_0)|,t-t_0), ~~ \forall t \geq t_0, \\ \text{~ or resp., ~} |x_k - \hat{x}_k| \leq \beta(|x_{k_0} - \hat{x}_{k_0}|,k-k_0), ~~ \forall k \in \mathbb{N}_{\geq k_0}. \end{multline} \]

  (23)

  for any \( t_0 \geq 0\) (resp., \( k_0 \in \mathbb{N}\) ), then this observer is uniformly asymptotically stable. If the function \( \beta\) takes the form \( \beta(r,s) = c r e^{-\lambda s}\) for some \( c \geq 0\) and \( \lambda > 0\) , then this observer is exponentially stable. In discrete time, we can equivalently write \( \beta(r,s) = c r \lambda^s\) for some \( c \geq 0\) and \( \lambda \in (0,1)\) . Similarly to the above, if the parameters \( (c,\lambda)\) are uniform with respect to treating the estimation error at any time as the new initial condition, then the observer is uniformly exponentially stable. In this dissertation, we frequently achieve high-gain results, where the observer can be designed to ensure any desired convergence rate \( \lambda > 0\) for the estimation error. However, a large \( \lambda\) (indicating fast convergence) may result in a larger \( c\) , which means that the estimation error is amplified during the transient; this is known as the peaking phenomenon. Such an observer is said to be arbitrarily fast;

• ISSInput-to-State Stability observer, if there exist a class-\( \mathcal{KL}\) function \( \beta\) and a class-\( \mathcal{K}_\infty\) function \( \alpha\) such that in the presence of disturbance \( t \mapsto v(t)\) (resp., \( k \mapsto v_k\) ) in \( \mathcal{V}\) and noise \( t \mapsto w(t)\) (resp., \( k \mapsto w_k\) ) in \( \mathcal{W}\) , for any solutions to system (11) (resp., system (12)) initialized in \( \mathcal{X}_0\) with \( t \mapsto u(t) \in \mathfrak{U}\) (resp., \( k \mapsto u_k \in \mathfrak{U}\) ) and to observer (18) (resp., observer (19)) initialized in \( \mathcal{Z}_0\) and fed with \( t \mapsto (u(t),y(t))\) (resp., \( k \mapsto (u_k,y_k)\) ), we have

  \[ \begin{multline} |x(t) - \hat{x}(t)| \leq \beta(|x(0) - \hat{x}(0)|,t) + \alpha\left(\sup_{s \in [0,t]}|v(s)| + \sup_{s \in [0,t]}|w(s)|\right), ~ \forall t \geq 0, \\ \text{or resp., } |x_k - \hat{x}_k| \leq \beta(|x_0 - \hat{x}_0|,k) + \alpha\left(\sup_{s \in \{0,1,\ldots,k\}}|v_s| + \sup_{s \in \{0,1,\ldots,k\}}|w_s|\right), ~ \forall k \in \mathbb{N}. \end{multline} \]

  (24)

  Condition (24) is based on the traditional definition of Input-to-State Stability[ISS] as presented in [45]. This condition, typically achieved when the estimation error is exponentially stable, implies that the estimation error converges asymptotically to a neighborhood around zero, where the size of this neighborhood is determined by the magnitude of the disturbance and noise. A more recent definition, referred to in this dissertation as robust stability, which introduces a forgetting factor to penalize past disturbances and noise, is proposed in [46]. This approach refines the original ISSInput-to-State Stability definition by replacing (24) with the following conditions:

  \[ \begin{multline} |x(t) - \hat{x}(t)| \leq \beta(|x(0) - \hat{x}(0)|,t) + \sup_{s\in[0,t]}\beta_v(|v(s)|,t-s)+\sup_{s\in[0,t]}\beta_w(|w(s)|,t-s), ~ \forall t \geq 0, \\ \text{or resp., } \\ |x_k - \hat{x}_k| \leq \beta(|x_0 - \hat{x}_0|,k) + \max_{q\in\{0,1,\ldots,k-1\}}\beta_v(|v_q|,k-q-1)+\max_{q\in\{0,1,\ldots,k-1\}}\beta_w(|w_q|,k-q-1), ~ \forall k \in \mathbb{N}, \end{multline} \]

  (25)

  for some class-\( \mathcal{KL}\) functions \( \beta_v\) and \( \beta_w\) . This property is utilized throughout this dissertation whenever it is achieved.

There are other types of observers such as finite-time observers (where the estimation error becomes zero within a finite time in the absence of disturbances and noise) of which a case is sliding mode observers [47], interval observers [48, 49], etc.

In general, observer design involves transforming the given system into a suitable target form in a new set of coordinates, known as the \( z\) -coordinates, for which we know how to design an observer (via the map \( \mathcal{F}\) of observers (18) and (19)), giving us some good properties in these new coordinates such as convergence or stability as discussed earlier, and then we recover the estimate in the original \( x\) -coordinates (via the map \( \Upsilon\) of observers (18) and (19)), relying on an injectivity property of the transformation. The stronger the injectivity of this transformation in terms of uniformity, the more robust the properties that can be transferred from the \( z\) -coordinates to the \( x\) -coordinates. For instance, to recover asymptotic convergence in the \( x\) -coordinates, the transformation only needs to be injective uniformly in time. However, to bring back (arbitrarily fast) exponential stability or robustness of the estimation error, the transformation must be Lipschitz injective uniformly with respect to time. This general idea of nonlinear observer design is recapped in [39] for continuous-time systems. Notice from observers (18) and (19) that the transformation from the \( x\) - to the \( z\) -coordinates is not used for observer implementation; it allows us to design the observer maps and thus plays a role in the theoretical analysis only. Therefore, in some contexts such as a discrete-time Moving Horizon Observer[MHO] [50], this transformation may be implicit and we only need to know a way to recover the estimate from an appropriately designed observer in the \( z\) -coordinates consisting of storing the past outputs; this idea is elaborated in Theorem 1. For nonlinear observers, the target \( z\) -coordinates satisfy \( n_z \geq n_x\) , with equality holding for linear observers. The case \( n_z < n_x\) corresponds to a special class called reduced-order observers that is not considered in this dissertation.

Now, we briefly discuss a very important continuous-time observer called the high-gain observer. This design comes in two primary versions: the classical high-gain observer [26] and the Kalman-like high-gain observer [27]. Let us discuss the former. For brevity, we consider the case where the continuous-time system (11) is autonomous, (i.e., \( u\) is absent) and has a one-dimensional output (i.e., \( n_y = 1\) ). First, the case of inputs is tricky and is studied in works such as the one concerning the Kalman-like version [27]. Second, if the output is multi-dimensional, the observer design and its analysis can be done block-wise, i.e., for each output. These designs require this system to be instantaneously observable (or in some literature, differentially observable) [51, Definition 1.2], implying that there exists an injective immersion into new \( z\) -coordinates of the observable canonical form [52, 1]

where \( \nu\) is an image of \( v\) through the immersion that typically depends on \( z\) , \( \varphi\) is a locally Lipschitz function, and the matrices

System (26) with such an \( A\) matrix has a canonical form and is always observable. A classical high-gain observer for system (26) is given by

where \( \varphi_\rm{{sat}}\) is obtained from \( \varphi\) as defined in Section 1.5.1.3, \( K = (k_1,k_2,…,k_{n_z})\) is chosen such that \( A - KC\) is Hurwitz (defined in Section 1.5.1.2) and \( D(\ell) = \rm{diag}(\ell,\ell^2,…,\ell^{n_z})\) , with \( \ell > 0\) an observer parameter to be chosen. Because \( \varphi\) is locally Lipschitz, with such a \( \varphi_\rm{{sat}}\) , we deduce that there exists \( L > 0\) such that

where \( \mathcal{Z}\) is the image of \( \mathcal{X}\) in Assumption 1 via the immersion, which is compact when \( \mathcal{X}\) is compact. The following lemma, whose proof can be found in works such as [53], establishes the fundamental properties of the classical high-gain observer.

From Lemma 1, we deduce that if \( \ell\) in observer (27) is chosen sufficiently large, there exist rational positive functions \( c_\nu\) and \( c_w\) such that

Similarly, there is a Kalman-like version of the high-gain observer for systems of the form (26) whose \( A\) matrix depends on \( (u,y)\) still following a canonical form—see [27]. In short, it uses a time-varying \( \mathbb{R}^{n_z} \times \mathbb{R}^{n_z}\) covariance matrix and provides, under the same observability, the same results as in Lemma 1 and (29). Two crucial properties of high-gain observers can thus be highlighted:

1. In the absence of disturbance \( \nu\) and noise \( w\) , the estimation error is exponentially stable, with an arbitrarily fast convergence rate achievable by increasing \( \ell\) . This property is achieved thanks to system (26) being observable over any arbitrarily short time interval, which is the strongest observability condition that exists in continuous time and not discrete time. However, making \( \ell\) large also reduces \( b_o(\ell)\) , potentially leading to large transient errors, a phenomenon known as peaking that is common in high-gain designs;
2. In the presence of \( (\nu,w)\) , the estimation is ISSInput-to-State Stability and even robustly stable, with gains that can be amplified as \( \ell\) increases (for more details on how \( (c_\nu(\ell),c_w(\ell))\) vary with \( \ell\) , see [54]).

These properties, including at least asymptotic convergence, may then be brought back to the \( x\) -coordinates thanks to the “sufficient” injectivity of the transformation, as explained above. In this dissertation, for hybrid systems with known jump times, we use a combination of observers to estimate different parts of the state: a high-gain observer for the instantaneously observable part and another observer for the remaining components. This combination is facilitated by Lyapunov analysis, with the two properties of high-gain observers being critical for the analysis—see Section 3.3. Some other variations of high-gain designs include the low-power high-gain observers [52] or sliding mode high-gain observers [47].

Observers for nonlinear discrete-time systems are reviewed in greater detail in Section 2.1.

A hybrid (dynamical) system with input and output is described by

where \( x \in \mathbb{R}^{n_x}\) is the state, \( u_c \in \mathbb{R}^{n_{u,c}}\) (resp., \( u_d \in \mathbb{R}^{n_{u,d}}\) ) is the exogenous flow (resp., jump) input, \( y_c \in \mathbb{R}^{n_{y,c}}\) (resp., \( y_d \in \mathbb{R}^{n_{y,d}}\) ) is the flow (resp., jump) output, \( C \subseteq \mathbb{R}^{n_x}\times\mathbb{R}^{n_{u,c}}\) (resp., \( D \subseteq \mathbb{R}^{n_x}\times\mathbb{R}^{n_{u,d}}\) ) is the flow (resp., jump) set, \( F:\mathbb{R}^{n_x}\times\mathbb{R}^{n_{u,c}} \rightrightarrows \mathbb{R}^{n_x}\) (resp., \( G:\mathbb{R}^{n_x}\times\mathbb{R}^{n_{u,d}} \rightrightarrows \mathbb{R}^{n_x}\) ) is the set-valued flow (resp., jump) map (often called inclusions), and \( h_c:\mathbb{R}^{n_x}\times\mathbb{R}^{n_{u,c}} \rightrightarrows \mathbb{R}^{n_{y,c}}\) (resp., \( h_d:\mathbb{R}^{n_x}\times\mathbb{R}^{n_{u,c}} \rightrightarrows \mathbb{R}^{n_{y,d}}\) ) is the flow (resp., jump) output map. As further clarified in Definition 3, the flow input \( u_c\) is a continuous-time signal, while the jump input \( u_d\) is a sequence of values. The state \( x\) of system (30) evolves as a continuous function of time following \( F\) when itself and some flow input \( u_c\) remain in \( C\) , and it jumps following \( G\) when itself and some jump input \( u_d\) belong to the set \( D\) . This state \( x\) thus exhibits both continuous- and discrete-time behavior, resulting in the unique properties of hybrid systems described in Section 1.5.3.2. When the inclusions consist of exactly one element (i.e., singletons), (30) becomes

with flow and jump maps \( f:\mathbb{R}^{n_x}\times\mathbb{R}^{n_{u,c}} \to \mathbb{R}^{n_x}\) and \( g: \mathbb{R}^{n_x}\times\mathbb{R}^{n_{u,d}} \to \mathbb{R}^{n_x}\) , respectively. While some classes of systems require inclusions, e.g., those with timers (see Example 16) or those encountered when describing friction [55], the majority of hybrid systems encountered in this dissertation take the form of (31) and we henceforth lay special emphasis on it. An autonomous hybrid system (30) (resp., (31)), i.e., with \( (u_c,u_d)\) absent, is shortly called a hybrid system with data \( (C,F,D,G)\) (resp., \( (C,f,D,g)\) ). Assumption 2 lists the hybrid basic conditions that ensure a hybrid system is well-posed.

To better illustrate hybrid systems, we introduce the following classical academic example that is revisited throughout this dissertation in various configurations.

Hybrid systems, as described by (30), provide a framework capable of expressing both continuous- and discrete-time behavior. Besides systems that require a hybrid representation such as the one given by (30) or an equivalent formulation as described later (examples include mechanical systems with impacts such as the bouncing ball in Example 1 and walking robots as in Section 3.4, or the stick-slip behavior studied in Section 4.3), the versatility of hybrid systems allows them to encompass a wide variety of system classes, including:

• Ordinary continuous- and discrete-time systems (11)\( /\) (12), possibly with constraints: These are straightforward special cases of (30), occurring when \( D = \emptyset\) and \( C = \mathbb{R}^{n_x}\) , and vice-versa. The general formulation can also handle systems with constraints [41] by replacing \( \mathbb{R}^{n_x}\) with a bounded subset;
• Impulsive differential equations [56]: Impulses are modeled as jumps in the solution, occurring at specific times dictated either by an exogenous input or a state-dependent variable;
• Switched systems [57]\( /\) hybrid automata [13, 58]: These are continuous-time systems where the state evolves continuously but the derivatives change discontinuously across different modes. For the former, exogenous modes are incorporated into the flow input \( u_c\) . For the latter where the mode changes depending on the state, to fall into the scope of (30), the system adds a discrete state \( q\) with \( \dot{q} = 0\) , incorporating the mode-switching dynamics in \( G\) or \( g\) , and uses \( C\) and \( D\) for switching conditions;
• Continuous-time systems with sporadic (multi-rate) measurements: These systems are effectively modeled with \( G\) or \( g\) being an identity map and using a timer for each output, to account for the intermittent nature of the measurements. Each timer counts down to zero; when it hits zero, the corresponding output is taken and this timer resets to a certain value modeling the measurement rate. See Example 16 for a case study.

Given the broad range of systems that hybrid models can represent, observers designed for hybrid systems can be applied across these classes and this idea is illustrated using examples throughout this dissertation. Variations of the model (30) that are other formulations of general hybrid systems include:

• Switched systems with state jumps [59, 60]: Normally, solutions to switched systems, e.g., in [57] are continuous (but not their derivatives), but by allowing the state to jump at the switches triggered by an input, we may cover most of the systems covered by (30). This form is particularly suited to describing systems where jumps are triggered externally and time-dependently, rather than by the system state;
• Systems with modes, guard sets, and reset maps [61, 62, 63]: This representation differs from (30) in the sense that they separate the state into components that evolve continuously in time and those that switch (called the modes). It is thus more appropriate for systems where the states can be distinctly divided in this manner, as opposed to hybrid systems like the bouncing ball in Example 1, where the hybrid behavior is more intertwined.

Let us now return to the forms (30)\( /\) (31), and introduce the concept of hybrid time domains and solutions. While the time domain for solutions to continuous-time systems is typically \( \mathbb{R}_{\geq 0}\) and for discrete-time systems is \( \mathbb{N}\) , or subsets of these in the case of incomplete solutions, the hybrid nature of hybrid systems necessitates defining solutions on two-component hybrid time domains, as follows.

We see from Definition 1 that each jump of the solution is instantaneous, i.e., it lasts zero units of ordinary time. This definition is used throughout this dissertation. In Section 4.2, for analysis purposes, we consider hybrid solutions in backward time and define the backward version of hybrid time domains, thereby extending Definition 1.

Given an initial condition and input trajectories, a solution to a hybrid system, defined formally in Definition 3, is described as a hybrid arc \( x: \mathbb{R}_{\geq 0} \times \mathbb{N} \to \mathbb{R}^{n_x}\) defined on a hybrid time domain, as follows.

Notations. For a hybrid arc \( (t,j) \mapsto x(t,j)\) , we denote \( \rm{dom} x\) its domain [10, Definition 2.6], \( \rm{dom}_tx\) (resp., \( \rm{dom}_j x\) ) the domain’s projection on the ordinary time (resp., jump) component, for \( j \in \mathbb{N}\) , \( t_j(x)\) the unique time such that \( (t_j(x),j)\in \rm{dom} x\) and \( (t_j(x),j-1) \in \rm{dom} x\) , and \( \mathcal{T}_j(x):=\{t\in \rm{dom}_t(x):(t,j)\in \rm{dom} x\}\) the \( j\) -th flow interval. The mention of \( x\) is omitted when no confusion is possible. Given a hybrid arc \( \phi\) defined on \( \rm{dom} \phi\) , let \( \phi_{|_{\mathcal{D}}}\) be the restriction of \( \phi\) to \( \mathcal{D} \subset \rm{dom} \phi\) . Let \( X(x,t_{j}(x),j)\) denote the solution to an autonomous hybrid system initialized as \( x\) and at hybrid time \( (t_{j}(x),j)\) that can be negative (i.e., after its \( j\) -th jump)—see about backward hybrid systems in Section 4.2. The graph of the hybrid arc \( x\) is the set

With these tools at hand, we now define solutions to system (31). This definition extends [10, Definition 2.6], which only covers the autonomous case, by incorporating exogenous flow and jump inputs seen as continuous- and discrete-time signals respectively, and it is used throughout this dissertation. In the literature, definitions of solutions to hybrid systems with inputs are given in [64, 65, 66] for other classes of inputs.

Now, let us define the terms related to the types of hybrid solutions found in this dissertation.

Note that in Definition 3, similar to the case of continuous- and discrete-time systems in Section 1.5.2.1, the hybrid arc can stop being defined after some time (i.e., being incomplete), but the inputs are given and complete, i.e., they are defined at all times.

Now we introduce some key notions and tools that are useful for hybrid observer design later.

Using comparison functions as done when defining asymptotic stability in continuous and discrete time above, we deduce that Definition 5 is equivalent to the existence of a class-\( \mathcal{KL}\) function \( \beta\) such that for any solution \( x\) to system (30)\( /\) (31), we have

This latter definition is used throughout this dissertation. The consideration of sets generalizes the notions of asymptotic stability introduced in Section 1.5.2.2, which is particularly relevant for hybrid systems with unknown jump times considered in Section 4. The term pre-attractive, as opposed to attractive, highlights the possibility of a maximal solution that is not complete, though it may be bounded. This distinction separates conditions for completeness (closely related to existence) from those for stability and attractivity, and asymptotic convergence holds when \( t+j\) tends to infinity, i.e., a complete solution. If solutions to system (30)\( /\) (31) are complete, then we omit the prefix “pre-” and get Uniform Global Asymptotic Stability[UGAS] of the set \( S\) for system (30)\( /\) (31). Similarly to Section 1.5.2.2, if the function \( \beta\) takes the form \( \beta(r,s) = c r e^{-\lambda s}\) for some \( c \geq 0\) and \( \lambda > 0\) , then we achieve Uniform Global pre-Exponential Stability[UGpES]; if \( \lambda\) can be made arbitrarily large, then the convergence is said to be arbitrarily fast.

The following Lyapunov conditions for hybrid systems, inspired from [10], which are the sufficient conditions for the UGpASUniform Global pre-Asymptotic Stability of the set \( S\) for system (31), provide a valuable tool for stability analysis and are utilized throughout this dissertation, possibly in slightly different variations.

If the class-\( \mathcal{K}_\infty\) functions in Lemma 2 are linear, we can show UGpESUniform Global pre-Exponential Stability of \( S\) , which can result in robustness against uncertainties. The same property can also be shown with \( \underline{\alpha}\) , \( \overline{\alpha}\) , and only \( \alpha_c\) (resp., \( \alpha_d\) ) being linear, under a dwell-time (resp., reverse dwell-time) condition, as in [22] and defined properly later in this dissertation. As shown later in this dissertation, such conditions are useful for bringing stability and convergence thanks to flows to make up for the lack of these at jumps, and vice-versa, achieving pre-asymptotic stability from the flow-jump combination.

Notations. For a function \( V:\mathbb{R}^{n_\eta}\to \mathbb{R}\) and a hybrid system with state \( \eta \in \mathbb{R}^{n_\eta}\) and input \( u\) , flow dynamics \( \dot{\eta}=f(\eta,u)\) and jump dynamics \( \eta^+=g(\eta,u)\) , we denote \( \dot{V}(\eta,u) :=\langle \nabla V(\eta),f(\eta,u)\rangle\) (where \( \langle a,b \rangle\) denotes the cross product of vectors \( a\) and \( b\) ) the derivative of \( V\) along the flows and \( V^+(\eta,u) :=V(g(\eta,u))\) the value of \( V\) after a jump.

In this dissertation, our goal is to design observers that estimate the state of hybrid systems using their known inputs and outputs, similar to the approach taken for continuous- and discrete-time systems discussed in Section 1.5.2.2. To begin, we analyze some key properties of hybrid systems to understand how they influence the observer design problem:

• Non-uniqueness of system representations

In continuous- or discrete-time systems, changing the representation (or the maps) requires a change of state variables. However, in hybrid systems, the coupling of continuous- and discrete-time dynamics, along with the flow and jump conditions, can lead to non-uniqueness in the system maps even with the same state variables. Since observer design relies heavily on these system maps, this non-uniqueness can present challenges in designing accurate observers. In certain cases, however, it plays a valuable role in analysis, revealing that some conditions imposed on hybrid systems, such as the invertibility of the jump dynamics, may be sufficient but not strictly necessary. For an example of this, refer to Example 15;

• Non-uniqueness of solutions

In continuous-time systems (11), given an initial condition, the resulting solution is unique if \( f\) is locally Lipschitz around this initial condition, uniformly in the time \( t\) (as per the Cauchy-Lipschitz theorem). In discrete-time systems (12), the solution is unique as long as \( f\) is a singleton. In hybrid systems (31), different solutions may arise from the same initial condition due to inclusions. For example, the timer

can evolve at any rate within \( [1, 1.5]\) during flows and reset to any value in \( [0, 2]\) after each jump. Even when the flow map is Lipschitz and the jump map is a singleton, solutions may still be non-unique if the flow and jump set overlap (\( C \cap D \neq \emptyset\) ), i.e., the solution can choose either to flow or jump when in this intersection. For instance, the timer

can either reset to zero or continue counting to ten when it is in the interval \( [8, 10]\) . If solutions are non-unique, determining the initial condition is not equivalent to determining the current state, which is the objective of observer design. However, non-unique solutions are permissible as long as they remain distinguishable from the measured output. The following conditions ensure the uniqueness of solutions to an autonomous hybrid system;

• Dependence of jump times on initial conditions (and inputs)

In continuous-time systems (11), the time domain of solutions is (a subset of) \( \mathbb{R}_{\geq 0}\) , and for discrete-time systems (12), it is \( \mathbb{N}\) . These domains are generally the same for the solutions of interest and are also shared with the estimate, making it straightforward to define the estimation error by comparing the real solution with the estimate at the same time. However, in hybrid systems, because the flow and jump conditions depend on the state and inputs via the sets \( C\) and \( D\) , the jump times and thus the time domain of solutions—see Definition 1—depend greatly on the initial condition and the inputs. This domain \( \rm{dom} x\) thus becomes solution-exclusive.

Example 3 (Jump times of a bouncing ball)

Consider the bouncing ball system in Example 1 with some parameters. In two scenarios where the ball is initialized at the same height but with different starting velocities, it touches the ground at different times and rebounds at different speeds, resulting in distinct time domains as illustrated in Example 3.

Because the initial condition and the inputs are unknown when designing observers (only a set of initial conditions and sets of considered solutions are known), we cannot ensure that the estimate provided by the observer shares the same time domain as the system’s solution. Consequently, directly comparing them at the same time becomes difficult, complicating the usual definition of estimation error and observer convergence [67];

• Zeno solutions

Recall from Definition 4 that a (maximal) solution is Zeno if it is not \( t\) -complete but is \( j\) -complete. This phenomenon is illustrated in the following example.

The Zeno phenomenon is against physics in the sense that an infinite number of events (jumps) can happen in a finite amount of time. It arises here because, in the mathematical model (30), it is supposed that each jump takes zero time. In observer design, besides the jumps which may bring us observability, we need to have a sufficient amount of flow time for observability and convergence. For instance, the high-gain observer (27) which is used to estimate part of the system’s state, converges asymptotically and thus needs infinite flow time. This is why Zeno can be an obstruction and we may need to assume some conditions on the time domains of hybrid solutions of interest to ensure some uniform amount of flowing and jumping.

Summary

Due to the considerations mentioned above, this dissertation focuses on hybrid systems with well-defined maps, specifically omitting the case of (30) of inclusions to work only on observer designs for systems of the form (31). Uniqueness of solutions is assumed whenever needed, especially in Section 4.2 when this is supposed to hold in backward time. We then divide these hybrid systems into two main groups, namely those whose jump times are known as treated in Section 3 and those with unknown jump times, addressed in Section 4. The first case assumes some mechanisms that allow us to detect when the solution jumps, such as impact sensors in walking robots, or some conditions related to the jump that can be checked if satisfied using the measured output, for instance, the bouncing ball in Example 1 where a jump occurs when the measurement equals zero. While it is true that this detection can be imperfect or delayed, such imperfections could be managed by the observer’s robustness provided that the maximal delay is smaller than the smallest possible time between successive jumps of the system solution, resulting in practical convergence—convergence of the estimation error to a bounded neighborhood of zero outside of the delays—as studied in [22, Section 6]. With this in mind, in Section 3 of this dissertation, we assume that the jump times are exactly detected to focus ourselves on the observer design problem. Similarly to Section 1.5.2.1, we denote:

• \( \mathcal{X}_0 \subseteq \mathbb{R}^{n_x}\) as the set of initial conditions \( x(0,0)\) of interest for system (31);
• \( \mathfrak{U}_c\) (resp., \( \mathfrak{U}_d\) ) as the set of input trajectories \( t \mapsto u_c(t)\) , called \( \mathfrak{u}_c\) (resp., \( j \mapsto u_{d,j}\) , called \( \mathfrak{u}_d\) ) of interest for system (31);
• \( \mathcal{U}_c \subseteq \mathbb{R}^{n_{u,c}}\) (resp., \( \mathcal{U}_d \subseteq \mathbb{R}^{n_{u,d}}\) ) as the set of the values that \( u_c(t)\) (resp., \( u_{d,j}\) ) can take along a trajectory \( \mathfrak{u}_c \in \mathfrak{U}_c\) (resp., \( \mathfrak{u}_d \in \mathfrak{U}_d\) ), for \( t \in \mathbb{R}_{\geq 0}\) (resp., \( j \in \mathbb{N}\) ).

Because the jump times of solutions to system (31) are known, we build for this system a synchronized observer of the general form

where \( \hat{\zeta} \in \mathbb{R}^{n_\zeta}\) is the observer state; \( \mathcal{F}\) , \( \mathcal{G}\) , and \( \Upsilon\) are the observer dynamics and output maps designed such that the dependence of \( \Upsilon\) on time \( (t,j)\) is only through the inputs \( (\mathfrak{u}_c,\mathfrak{u}_d)\) and each maximal solution \( (x,\hat{\zeta})\) to the cascade (31)-(42) initialized in \( \mathcal{X}_0 \times \mathcal{Z}_0\) for some appropriate \( \mathcal{Z}_0 \subseteq \mathbb{R}^{n_\zeta}\) and with inputs \( (\mathfrak{u}_c,\mathfrak{u}_d)\in \mathfrak{U}_c\times \mathfrak{U}_d\) is complete and verifies at least asymptotic convergence

Indeed, (42.a) ensures that the estimate \( \hat{x}\) shares the same time domain as the system solution \( x\) , allowing them to be directly compared at the same hybrid time. Using the language of sets, we see that condition (43) is equivalent to the asymptotic convergence to a set where the estimate equals the true state, i.e.,

When jump times are known, we prefer writing (43) over (44) since it is more direct and easily related to similar properties in continuous and discrete time. In Section 3, we assume certain properties of the solutions of interest, including completeness, and design various observer structures (42) based on the form of system (31), to guarantee (43) and potentially additional properties such as robustness or arbitrarily fast convergence rate as introduced in Section 1.5.2.2.

In the second scenario of unknown jump times addressed in Section 4, synchronization between the system and observer as in (42.a) becomes impossible, making asymptotic convergence more challenging to define. Examples of such hybrid systems whose jump times cannot be detected include the stick-slip behavior encountered in oil drilling, as discussed in Section 4.3. In this dissertation, we focus on the autonomous form of (31)

with the output \( y\) assumed to be continuous through the jumps, i.e., \( h(x) = h(g(x))\) for all \( x \in D\) , preventing jump detection through measurement alone. Building on the concept of gluing the time domain initiated in [32], we transform system (45) into continuous-time dynamics of the form

of dimension \( n_z \geq n_x\) , for some \( A\) Hurwitz. In these new coordinates, the state \( z\) evolves as a continuous-time function, effectively “gluing” the hybrid time domain as the method’s name suggests. This eliminates the need for jump detection in observer design, unlike in the previous case. An exponentially stable observer for system (47) is then obtained by running the same dynamics from any initial condition in \( \mathbb{R}^{n_z}\) , specifically \( \dot{\hat{z}} = A\hat{z} + By\) , as the estimation error verifies \( \dot{z} - \dot{\hat{z}} = A(z - \hat{z})\) with \( A\) being Hurwitz. Then, by ensuring that the transformation is injective (and hence left-invertible) thanks to some distinguishability condition and a sufficiently large dimension \( n_z\) , we can recover the estimate in the original \( x\) -coordinates. Formally, an observer for system (45) would take the form

where \( \Upsilon\) is the left inverse of the transformation. In this case, the estimate \( \hat{x}\) is a continuous-time signal instead of a hybrid one, and so convergence is not defined in the sense of (43) by comparing the solution and the estimate at the same hybrid time, but as the convergence of \( (x(t,j),\hat{x}(t))\) in time to the indistinguishable set

The set \( S_\mathcal{I}\) consists of points that generate solutions that cannot be distinguished based on the output \( y\) . Therefore, convergence to this set is the best achievable outcome with observer (47). Because \( y\) is continuous at jumps, points that either are the same, are one jump apart, or jump to the same point, are indistinguishable from each other and hence they belong to this set. The formulations and results are made more precise in Section 4.2.

If the pair \( (F_k,H_k)\) is quadratically detectable [44] uniformly in time, namely if there exists \( k \mapsto L_k\) , some \( 0 \leq a < 1\) , and \( Q \in §_{>0}^{n_x}\) such that the matrix inequality

holds, then an asymptotic observer for system (54) would be

This takes the structure of observer (50) with \( \hat{z}_k = \hat{x}_k\) , usually referred to as the Luenberger observer for linear systems, which is the discrete-time version of [74]. Actually, (55) is not a Linear Matrix Inequality[LMI] since it is not linear in its variables, but it may be transformed into one thanks to Schur’s lemma. While detectability alone is not sufficient for observer design in nonlinear systems, it becomes adequate for linear systems due to the well-defined structure of the maps, allowing the construction of the observer gains. Note that while it allows achieving convergence, detectability is not sufficient for tuning the convergence rate, for which we need stronger properties such as observability as in the next section.

In practice, LMILinear Matrix Inequality-based observers are frequently extended to nonlinear systems, as seen in [75] or the work by the author in [35], leveraging the linearity in the dynamics to incorporate additional useful properties, such as disturbance decoupling in the estimation error [76]. Some LMILinear Matrix Inequality-based approaches have been developed for discrete-time normal forms with Lipschitz nonlinearities as in [77]. The main limitation of these methods is that, besides having to assume that the nonlinear part of the dynamics is Lipschitz, they require that detectability comes from the linear part of the dynamics, and so they do not work when it is the nonlinear part that allows us to estimate the state. Moreover, they typically do not guarantee the solvability of the LMILinear Matrix Inequalitys.

For the linear form (54), the celebrated Kalman observer was introduced in the early 60s by Kalman and Bucy [78] as an optimal filter. In a stochastic context and under an Uniform Complete Observability[UCO] assumption, it minimizes the covariance of the estimation error in the presence of Gaussian dynamics and measurement noise. Its appeal lies in its systematic design and easiness of tuning, which is linked to the (assumedly known) covariance of those disturbances. The Kalman filter was later extended to discrete-time systems [79] and various other contexts [80], becoming widely used in industrial applications. The discrete-time Kalman filter, consisting of prediction, namely proceeding from the estimate with the system’s dynamics, and correction of the estimate by comparing it with the measurement, has been shown to exhibit asymptotic stability “in the large”, namely convergence of the estimation error after several discrete steps instead of after each step, within the stochastic and deterministic frameworks in [79, 81], respectively.

On the other hand, in the early \( 90\) s, an alternative Kalman-like observer was developed [82, 83], optimizing in a deterministic setting the ability of the estimate to explain the past output history, with a certain forgetting factor and weighting, describing the confidence in the output measurement. This design was extended to discrete-time systems in [43], taking the form

where \( \gamma \in (0,1]\) is a forgetting factor and \( k\mapsto R_k\) is a uniformly bounded positive definite weighting matrix, both serving as design parameters. Observer (57) takes the structure (50) with \( \hat{z}_k = (\hat{x}_k,P_k)\) . This observer works globally but must be initialized with \( P_0\) positive definite, so \( \mathcal{Z}_0 = \mathbb{R}^{n_x} \times §_{>0}^{n_x}\) . The forgetting factor allows for speeding up the convergence rate when we push it close to zero, and a form of (57) without this is proposed in [81].

Both the Kalman and Kalman-like observers rely on the following observability condition.

Indeed, the analysis of both types relies on the Lyapunov function \( V(x_k,\hat{x}_k,P_k) = (x_k - \hat{x}_k)^\top P_k^{-1} (x_k - \hat{x}_k)\) , and UCOUniform Complete Observability ensures the uniform lower boundedness of the observability Gramian, which in turn leads to that of \( P_k^{-1}\) . The covariance matrix \( P_k\) here serves to gather observability from the outputs, and thus this boundedness signifies the uniform sufficiency of information needed to estimate the state, related to the notion of persistence of excitation [84]. A significant advantage of Kalman(-like) designs is that they are systematic, meaning that they can be implemented without having to check the system’s UCOUniform Complete Observability, unlike in LMILinear Matrix Inequality-based designs where we have to solve the LMILinear Matrix Inequality, i.e., check the detectability, in order to obtain the observer’s gain. The difference between the Kalman-like observer and the Kalman one mainly lies in the absence of noise on the dynamics which facilitates a Lyapunov stability analysis by linking the Lyapunov matrix directly to the observability Gramian. The Kalman-like observer in [43] thus provides exponential stability with an arbitrarily fast rate after a certain time, achievable with a (quadratic) strict Lyapunov function, unlike in [79, 81], where the Lyapunov function decreases over a certain finite number of steps. Note also that for these Kalman(-like) designs, while the invertibility of the system dynamics typically appears in the analysis such as (58), it is rather a sufficient condition and is not required for observer implementation.

These rely on the left inversion of the (forward) observability map comprising future outputs, such as with Newton algorithms [94, 95], providing instantaneous estimation as soon as enough output information is gathered, but lacking filtering effects against measurement noise. In fact, very few works in the literature propose global nonlinear discrete-time observer designs that exhibit robustness against disturbances and measurement noise, e.g., [96, 50] and in Section 2.2 and Section 2.3. Note also that though these dead-beat observers can work under a mild constructibility condition that allows determining the current state, the papers [94, 95] assume the stronger observability condition.

The MHOMoving Horizon Observer [97, 98, 99, 96, 50] can be viewed as a more computationally demanding but also more robust variant of dead-beat observers. These minimize the estimation error using an observability map made of the \( m\) current and future outputs, relying on the injectivity of this map with respect to the state from \( m\) steps ago, thus requiring observability instead of constructibility\( /\) backward distinguishability. Specifically, these algorithms involve solving an online optimization problem to find \( \hat{x}_{k - m | k}\) , the estimate made at time \( k\) of the state \( m\) steps ago, that minimizes a cost function of the form

that imposes the system’s dynamics on the values \( \hat{x}_{q|k}\) to compute the \( m\) past estimated outputs. Once \( \hat{x}_{k - m | k}\) is obtained from this optimization, the current state can be estimated by propagating with the system’s dynamics. This observer takes the form (50), with \( (\mathcal{F}_k)_{k \in \mathbb{N}}\) serving to store the past outputs in \( z_k\) and \( (\Upsilon_k)_{k \in \mathbb{N}}\) the optimizer. Some versions of the cost function [50] include terms that allow for estimating the system’s disturbances. These exhibit robustness against modeling uncertainties or numerical errors [100]. Recently, they have been shown to have some kind of Input-to-State Stability[ISS] against disturbances in [50, Corollary 1], resulting in one of the first robust designs for nonlinear discrete-time systems. Similar to dead-beat observers, while the MHOMoving Horizon Observer can function under constructibility, the works [97, 98, 99] all assume observability, which is a stronger condition.

We propose in Section 2.2 below a discrete-time observer whose structure resembles that of the classical continuous-time high-gain observer (26). This observer gives exponential stability of the estimation error when pushed sufficiently fast, enabling robustness to be stated, and moreover, the convergence rate can be arbitrary. Note that arbitrarily fast convergence from the initial time cannot be achieved in discrete time due to the absence of instantaneous observability. More importantly, our high-gain observer exploits constructibility or backward distinguishability, which are among the weakest observability conditions.

The Kravaris-Kazantzis$\slash$Luenberger[KKL] observer is of particular interest in the nonlinear setting thanks to its requiring a fairly mild backward distinguishability—see in Definition 13. This observer consists of transforming system (49) into a stable filter of the output

where \( A\) is Schur. An observer readily exists in these target \( z\) -coordinates by running system (62) fed with \( y_k\) from any initial condition, and inverting this transformation recovers the estimate in the original coordinates. This paradigm thus fits the general structure (50), but with a particular form in the \( z\) -coordinates and with \( (\Upsilon_k)_{k \in \mathbb{N}}\) being a left inverse of the transformation, which can be obtained if the latter is (uniformly) injective. In [102], for autonomous systems, this injectivity is established under a mild backward distinguishability condition, constituting a systematic observer that applies to a large class of systems. Unfortunately, while injectivity without uniformity in time is sufficient in the time-invariant setting, this is not the case for non-autonomous systems, which require a strong and uniform kind of injectivity. In Section 2.3, we propose the KKLKravaris-Kazantzis$\slash$Luenberger design for non-autonomous systems, which provides a systematic arbitrarily fast robust observer design under a stronger uniform Lipschitz backward distinguishability condition on the system.

Notably, although they both rely on transforming the given dynamics into linear dynamics with output injection, the crucial distinction between KKLKravaris-Kazantzis$\slash$Luenberger designs and linearization-based techniques reviewed in Section 2.1.3 is that the former does not require a linear output map in the new coordinates (62) (we do not even need its expression), leading to much more generic global results as the class of systems where the method is applicable is much wider. Compared to the MHOMoving Horizon Observer discussed in Section 2.1.4.2, the KKLKravaris-Kazantzis$\slash$Luenberger observer is more systematic but is more limited in the computation of the transformation (for which learning-based methods are studied, such as [103]) and the tuning of robustness.

This chapter is based on the following definition.

In the linear context, this property is well-known to be weaker than observability when the dynamics are not invertible [73]. Observability, as defined in Definition 10, would instead require \( x_k\) to be a function of the future outputs (or \( x_{k-m}\) function of \( (y_{k-1},…,y_{k-m})\) in (64)), which is easily checked with the Kalman criterion but is not necessary for observer design. In the nonlinear context, constructibility notions are studied in [71], but from the local point of view of differential geometry, and in [70, Proposition \( 1\) ] as a local property. It is also related to the notion of backward distinguishability exploited in KKLKravaris-Kazantzis$\slash$Luenberger designs in [102] or Definition 11. But otherwise, in general, observability notions are used instead, by exploiting the “observability map” gathering the outputs as a function of the initial state (instead of final), as reviewed in Section 2.1.

Lemma 5 below shows that the constructibility of system (63) is necessary and sufficient for it to be transformed into what we call a constructible canonical form (67).

Proof. First, if Item (II) holds, then by Definition 9, there exists \( (\Psi_k)_{k \in \mathbb{N}_{\geq m}}\) and we define for all \( k \in \mathbb{N}_{\geq m}\) the maps \( \mathcal{T}_k = \Psi_k\) and \( \theta_k = h_k \circ \mathcal{T}_k\) , and for all \( k \in \mathbb{N}\) the maps \( \varphi_{i,k}\) , \( i \in \{1,2,…,m\}\) as \( \varphi_{1,k} = \rm{Id}\) , \( \varphi_{2,k}(z_1,y) = z_1\) , \( \varphi_{3,k}(z_1,z_2,y) = z_2\) , etc., which means Item (II) implies Item (III). On the other hand, if Item (III) holds, then because we have for all \( k \in \mathbb{N}_{\geq m}\) , \( z_{1,k} = \varphi_{1,k-1}(y_{k-1})\) , \( z_{2,k} = \varphi_{2,k-1}(z_{1,k-1},y_{k-1}) = \varphi_{2,k-1}(\varphi_{1,k-2}(y_{k-2}),y_{k-1})\) , etc., we have \( x_k = \mathcal{T}_k(z_k) = \mathcal{T}_k(\varphi_{1,k-1}(y_{k-1}),\varphi_{2,k-1}(\varphi_{1,k-2}(y_{k-2}),y_{k-1}),…)\) for all \( k \in \mathbb{N}_{\geq m}\) , which is a function of only \( (y_{k-1},y_{k-2},…,y_{k-m})\) and that corresponds to \( \Psi_k\) in Definition 9, so Item (III) implies Item (II). \( \blacksquare\)

Compared to [88, Section 4] and [71] where the constructible forms are linear (modulo output injection), our form (67) comes with as much nonlinearity in the dynamics and output as possible, allowing us to consider a large class of systems.

The following examples illustrate that we can rely on constructibility to transform the system into the constructible canonical form (67).

However, such transformations are situational since it is not clear how they can be obtained from constructibility. We then propose in the next part a constructive way, when possible, to find this transformation from the system’s maps.

We introduce the following definition.

Proof. First, by definition of \( h_{-1,k}\) , along the solutions \( k \mapsto x_k\) to system (63), we have for all \( k \in \mathbb{N}_{\geq m}\) , \( z_{1,k+1} =h_{-1,k+1}(x_{k+1})= y_k\) . Then, for each \( i \in \{2,…,m\}\) , by definition of \( h_{-i,k}\) , along the solutions \( k \mapsto x_k\) to system (63), we have for all \( k \in \mathbb{N}_{\geq m}\) , \( z_{i,k+1} = h_{-i,k+1}(x_{k+1}) = h_{k-(i-1)}(x_{k-(i-1)}) = h_{-(i-1),k}(x_k) = z_{i-1,k}\) , concluding the proof. \( \blacksquare\)

This property is used in Section 2.4 to design an observer for a discretized PMSMPermanent Magnet Synchronous Motor. The form (73), obtained through backward distinguishability, is a particular case of the form (67) and was also obtained under constructibility in the proof of Lemma 5. The difference is that in (72), the past outputs are expressed as an injective function of \( x_k\) (and thus \( z_k\) is a function of \( x_k\) in (73) and vice-versa), while in (64), \( x_k\) is directly written as a function of the past outputs (and thus we only have \( x_k\) as a function of \( z_k\) in (67)). In other words, backward distinguishability is sufficient, but not necessary, for constructibility. This is shown in Lemma 7 and is illustrated in Example 7.

The relationships among the introduced notions and properties are captured in Lemma 7.

Proof. We consider system (63) and prove each implication individually.
For (I\( 1\) ): If system (63) is observable, from (65), we have \( x_k =F_k(x_{k-m}) = (F_k \circ \Phi_{k-m}) (y_{k-m},y_{k-(m-1)},…,y_{k-1})\) for all \( k \in \mathbb{N}_{\geq m}\) , where \( F_k = (f_{k-1} \circ f_{k-2} \circ … \circ f_{k-m})\) . From Definition 9, system (63) is constructible. Now for the converse, assume that the dynamics of system (63) are invertible, so each \( f_k\) is invertible as \( f_k^{-1}\) for all \( k \in \mathbb{N}\) . If system (63) is constructible, from (64), we have \( x_k =F_k^\prime(x_{k+(m-1)}) = (F_k^\prime \circ \Psi_{k+(m-1)}) (y_{k+(m-1)},y_{k+(m-2)},…,y_k)\) for all \( k \in \mathbb{N}\) , where \( F_k^\prime = (f_k^{-1} \circ f_{k+1}^{-1} \circ … \circ f_{k+(m-2)}^{-1})\) . From Definition 10, system (63) is observable.
For (I\( 2\) ): Assume that system (63) is backward distinguishable. First, we use Lemma 6 to show that this implies the existence of a transformation into a particular constructible form (73). Then, we use Lemma 5 to show that this constructible form is equivalent to system (63) being constructible.
For (I\( 3\) ): If system (63) is constructible, by Lemma 5, (67)-(66) is an asymptotic observer for it.
For (I\( 4\) ): See in the proof of Lemma 4. \( \blacksquare\)

Consider a system in the constructible canonical form (67) where \( z_k \in \mathbb{R}^{n_z}\) is the state, and \( y_k\) is the measured output in \( \mathbb{R}^{n_y}\) . For system (67), we propose the following observer:

where \( k_i \in \mathbb{R}\) , \( i \in \{1,2,…,m\}\) and \( \gamma \in [0,1]\) are design parameters to be selected, which may be \( 0\) , and the maps \( \bar\varphi_{i,k}\) , \( i \in \{2,…,m\}\) and \( \bar\theta_k\) are such that Item (IV) of Assumption 3 below holds.

Because system (67) is constructible for all \( k \in \mathbb{N}_{\geq m}\) , we can neglect the correction terms in observer (76), i.e., pick \( \gamma = 0\) for an instantaneous convergence after \( m\) time steps. However, the price to pay is the absence of filtering effects against disturbances and noise. On the other hand, the observer structure (76) resembles the famous high-gain design in continuous time (26) discussed in Section 1.5.2.2, the difference being that: i) the triangularity constraint is lower diagonal, ii) all the maps may be nonlinear, and iii) there are no constraints on the choice of the scalars \( k_i\) , \( i \in \{1,2,…,m\}\) .

Theorem 1 shows the exponential stability of the estimation error with an arbitrarily fast rate, achievable by pushing \( \gamma\) smaller.

Proof. This technical proof has been moved to Section 6.1.1.1 to facilitate reading. It involves two main steps: i) Obtain the estimation error dynamics and re-scale these, and ii) Bound nonlinear terms by their Lipschitzness and push \( \gamma\) to dominate these terms. \( \blacksquare\)

Consider system (67) with disturbance \( v_k \in \mathbb{R}^{n_z}\)

The disturbance \( v_{i,k}\) could also model the non-Lipschitzness of \( \varphi_{i,k}\) . Theorem 2 below shows the robustness of observer (76) under the considered disturbance and noise in system (80).

Proof. This technical proof is an adaptation of that of Theorem 1 taking into account the uncertainties. It has been moved to Section 6.1.1.2 to facilitate reading. \( \blacksquare\)

From Theorem 2, we see that with \( 0 < \gamma < \min\{1,\gamma^\star\}\) :

• The estimation error is robustly stable with respect to disturbance \( v_k\) and noise \( w_k\) , in the sense of [46, Definition \( 2.3\) ], which differs from the classical Input-to-State Stability[ISS] in [45] by the exponentially penalization of the past values of \( (v_k,w_k)\) thanks to the forgetting factor \( \left(\frac{\gamma}{\gamma^\star}\right)^{k-1-j}\) ;
• Similarly to the high-gain design in continuous time (see e.g., [53]), the effects of \( v_{q,j}\) (past disturbance on line \( q\) ) on \( \tilde{z}_{i,k}\) (current estimation error on line \( i\) ) can be either magnified or reduced depending on \( q\) with respect to \( i\) , because of the coefficient \( \gamma^{q-i}\) . Note that, however, the impact of the disturbance on the last line \( (v_{m,j})\) can only be reduced (for \( \gamma\) sufficiently small) on the lines \( i<m\) , which does not give practical convergence, contrary to continuous time;
• In the proof of Theorem 2, it is not evident that we can attenuate the disturbance and noise by choosing \( k_i\) in observer (76) since this proof is done conservatively for the general case of nonlinear maps \( \varphi_{i,k}\) . However, for the specific form (73) widely used in Moving Horizon Observer[MHO] schemes, by picking \( k_1 < 0\) and \( k_i=0\) for \( i\neq0\) , we get a penalization factor in front of the noise.

In Section 2.2.2, we have seen that constructible systems (63) can be linked to constructible form (67) via some map (66), at least after \( m\) discrete steps. Then, an observer (76) can be designed, and we now study conditions to recover the asymptotic convergence in the \( x\) -coordinates. For this, the following assumption is made.

The asymptotic convergence is then recovered in the \( x\) -coordinates as follows.

Proof. Based on (82) and using [40], we can extend \( (\mathcal{T}_k)_{k \in \mathbb{N}_{\geq m}}\) into \( (\bar{\mathcal{T}}_k)_{k \in \mathbb{N}_{\geq m}}\) such that there exists \( c_1 > 0\) such that for all \( (z_a,z_b) \in \mathbb{R}^{n_z} \times \mathbb{R}^{n_z}\) and for all \( k \in \mathbb{N}_{\geq m}\) ,

Along any solution \( k \mapsto x_k\) to system (63) and any solution \( k \mapsto \hat{z}_k\) to observer (76) with \( 0 < \gamma < \min\{1,\gamma^\star\}\) and fed with \( y_k\) in (63), we have

and we have \( \lim_{k\to+\infty}|z_k - \hat{z}_k| = 0\) for all \( k \in \mathbb{N}_{\geq m}\) thanks to Theorem 1 when \( 0 < \gamma < \min\{1,\gamma^\star\}\) , which concludes this proof. \( \blacksquare\)

Combining the ingredients in Lemma 6, Theorem 1, and Lemma 8, we arrive at a constructive observer design for general nonlinear systems of the form (63), under constructibility or backward distinguishability.

In this part, \( A\) is chosen of the form \( \gamma \tilde{A}\) with \( \tilde{A}\) Schur, and \( \gamma \in (0, 1]\) sufficiently small to ensure uniformly Lipschitz injectivity of \( (T_k)_{k\in \mathbb{N}}\) after a certain time. This is done under the following distinguishability conditions.

Intuitively, the concatenation of a sufficient number \( m_i\) of the past outputs determines uniquely and uniformly the current state (and equivalently the trajectory as well). Equivalent kinds of uniform observability are assumed in [106, Theorem \( 4.1\) ] and [42, Theorem \( 2\) ] for autonomous and time-varying continuous-time systems respectively, leading to similar results with arbitrarily fast convergence of the estimation error.

For our uniform Lipschitz injectivity result, we make the following assumptions.

The following theorem shows the uniform Lipschitz injectivity of \( (T_k)_{k \in \mathbb{N}}\) after a certain time.

Proof. This highly technical proof has been moved to Section 6.1.2.1 to facilitate reading. It involves three main steps: i) Express \( T_k\) as a sum of three parts, the first one linked to the initial condition \( T_0\) (that may not be injective), the second one linked to all the outputs from the initial time up to \( \overline{m}\) steps ago, and the last one linked to the closest \( \overline{m}\) outputs (that gives us injectivity), ii) Upper-bound the first two parts thanks to Lipschitzness and lower-bound the last part thanks to uniform Lipschitz backward distinguishability, and iii) Push \( \gamma\) sufficiently small to make the last part dominate the first two parts, bringing injectivity. \( \blacksquare\)

According to the proof of Theorem 3, once \( (T_k)_{k \in \mathbb{N}}\) has become uniformly Lipschitz injective on \( \mathcal{X}\) following Theorem 5, there exists a sequence of left inverse maps \( (T^*_k)_{k \in \mathbb{N}}: \mathbb{R}^{n_z} \to \mathbb{R}^{n_x}\) and \( c^\prime > 0\) such that

Exploiting Lipschitzness, the result of Theorem 3 can thus be strengthened as follows, obtaining exponential asymptotic stability of the estimation error in the \( x\) -coordinates, and an arbitrarily fast discrete-time observer.

Corollary 1 shows that observer (90) can be made arbitrarily fast after \( (T_k)_{k \in \mathbb{N}}\) has become uniformly Lipschitz injective, by picking \( \gamma\) closer to zero. Indeed, compared with (97), the estimation error in the \( x\) -coordinates is exponentially stable with \( c_1 = \frac{\overline{c}}{ \gamma^{\overline{m}-1}}\) and \( c_2 = \gamma \|\tilde{A}\|<1\) (according to the proof of Theorem 5). For any desired convergence rate \( c_2^\star \in (0, 1)\) , by picking \( 0 < \gamma < \min\left\{\frac{c_2^\star}{\|\tilde{A}\|}, \gamma^\star\right\}\) , we achieve \( 0<c_2 \leq c_2^\star\) . Note that this typically increases \( c_1\) , because if \( c_2 \leq c_2^\star\) then \( c_1 \geq c_1^\star = \frac{\overline{c} \|\tilde{A}\|^{\overline{m}-1}}{ (c_2^\star)^{\overline{m}-1}}. \) We seem to recover a discrete-time version of the well-known peaking behavior in continuous-time high-gain designs [26]. Note though that since \( k^\star \in \mathbb{N}_{\geq \overline{m}}\) , (113) can actually be re-written as

indicating that the peaking is over after \( k^\star\) . This observer is illustrated in Section 2.3.6.

In this part, we study the robust stability (in the sense of [46]) and ISSInput-to-State Stability properties [45] of the observer given by Corollary 1. Suppose the system has dynamics (84) added with some disturbance\( /\) uncertainty \( v_k\) and a measurement with noise \( w_k\) :

Then, if the pair \( (f_k)_{k\in \mathbb{N}}\) , \( (h_k)_{k\in \mathbb{N}}\) verifies the conditions of Theorem 5, we know that there exists a sequence of left inverses \( (T_k^*)_{k\in \mathbb{N}}\) for \( k \in \mathbb{N}_{\geq k^\star}\) that verifies (112). However, in practice, following for instance [103], such maps are only approximately known. Theorem 6 then shows the robustness of the estimation error in the \( x\) -coordinates with respect to all those uncertainties (since \( \gamma \|\tilde{A}\|<1\) ).

Proof. This highly technical proof has been moved to Section 6.1.2.2 to facilitate reading. It involves two main steps: i) Show that \( (T_k)_{k \in \mathbb{N}}\) provided by Theorem 5 is uniformly Lipschitz, and ii) Exhibit bounds on the uncertainties in the \( z\) -coordinates then bring these back to the \( x\) -coordinates. \( \blacksquare\)

Note that it is the exponential stability (rather than asymptotic stability) of the estimation error that provides the ISSInput-to-State Stability with respect to disturbances and measurement noise. We also see from (118) that accelerating the convergence by pushing \( \gamma\) closer to zero will worsen the effect of the disturbances and noise, but not that of the approximation of the inverse transformation.

In Item (V) of Assumption 7, we require that the map sequences \( (f^{-1}_k)_{k \in \mathbb{N}}\) and \( (h_k)_{k \in \mathbb{N}}\) are globally uniformly Lipschitz, which is due to the fact that we do not have backward invariance of the sequence on \( \mathcal{X}\) . Here, we would like to study how to relax that into a local requirement on a certain bounded set, without losing Item (VI) of Assumption 7.

Let us assume that, given the \( m_i\) of Item (VI) of Assumption 7, there exists a large enough \( \sigma_d > 0\) such that for all \( x \in \mathcal{X}\) and for all \( k \in \mathbb{N}_{\geq \overline{m}}\) where \( \overline{m}: = \max_{i \in \{1, 2, …, n_y\}} m_i\) , all the pre-images \( f^{-1}_{k-1}(x)\) , \( (f^{-1}_{k-2} \circ f^{-1}_{k-1})(x)\) , up to \( (f^{-1}_{k-\overline{m}} \circ f^{-1}_{k-(\overline{m}-1)} \circ … \circ f^{-1}_{k-1})(x)\) are in \( \mathcal{X} + \sigma_d\mathbb{B}\) . This means that we can change \( (f_k^{-1})_{k \in \mathbb{N}}\) as we want outside of \( \mathcal{X}+\sigma_d\mathbb{B}\) without altering Item (VI) of Assumption 7 (and without altering the system dynamics on the set \( \mathcal{X}\) where the solutions of interest evolve).

Now, for any \( \sigma_c > \sigma_d\) , let us consider a saturating function \( \chi: \mathbb{R}^{n_x} \to \mathbb{R}\) defined as

where \( g\) is any locally Lipschitz function such that \( \chi\) is locally Lipschitz. We then define \( (f^{\dagger}_k)_{k \in \mathbb{N}}: \mathbb{R}^{n_x} \to \mathbb{R}^{n_x}\) as

The set

illustrated in Figure 12 is backward invariant with respect to \( (f^{\dagger}_k)_{k \in \mathbb{N}}\) . Indeed, pick any \( x\in \mathcal{I}\) and any \( k \in \mathbb{N}\) . Then, either \( x \in \mathcal{X} + \sigma_c\mathbb{B}\) and thus \( f^{\dagger}_{k}(x) \in \mathcal{I}\) , or \( x \notin \mathcal{X} + \sigma_c\mathbb{B}\) and then \( \chi(x) = 0\) and \( f^{\dagger}_{k}(x) = x \in \mathcal{I}\) . It follows that all the requirements of global uniform Lipschitzness of \( (f^{-1}_k)_{k \in \mathbb{N}}\) , \( (h_k)_{k \in \mathbb{N}}\) , and \( T_0\) as in Assumption 7 can be replaced by uniform Lipschitzness on this backward invariant set \( \mathcal{I}\) , by replacing \( (f^{-1}_k)_{k \in \mathbb{N}}\) with \( (f^{\dagger}_k)_{k \in \mathbb{N}}\) defined in (120) in all the equations. Similarly, in Remark 10, we can check uniform Lipschitz backward distinguishability using \( (\mathcal{O}^{fw}_k)_{k \in \mathbb{N}}\) instead of \( (\mathcal{O}^{bw}_k)_{k \in \mathbb{N}}\) if \( (f^{-1}_k)_{k \in \mathbb{N}}\) is uniformly Lipschitz injective on \( \mathcal{I}\) . Actually, even the invertibility of each \( f_k\) as in Assumption 6 may only be required on \( \mathcal{I}\) .

In particular, \( \mathcal{I}\) is bounded if and only if the sequence of sets \( (f^{\dagger}_k(\mathcal{X} + \sigma_c\mathbb{B}))_{k \in \mathbb{N}}\) is uniformly bounded, which is guaranteed if \( (f^{-1}_k)_{k \in \mathbb{N}}\) is uniformly bounded on \( \mathcal{X} + \sigma_c\mathbb{B}\) . In this case, all those assumptions become much more favorable.

where \( \Omega_k = \begin{pmatrix}\cos(\varphi_k) & -\sin(\varphi_k) \\ \sin(\varphi_k) & \cos(\varphi_k)\end{pmatrix}\) , and \( \varphi_k = \frac{\omega_k \tau}{2}\) where

is an approximation of the rotation speed of the motor, assuming that this speed does not vary too fast, and the map \( \rm{sinc}\) is defined as

First, we attempt to design for system (132) a continuous-time high-gain observer [26]. Note that this requires derivating the output, which includes the voltage\( /\) current inputs, up to a certain order, thus can terribly magnify the noise in these trajectories as well as noise due to non-saliency of the PMSMPermanent Magnet Synchronous Motor, limiting the observer’s performance. To see this, notice that the function describing the output and its next two time derivatives along the solutions to system (132), given by

where \( \eta = u - Ri + L\frac{di}{dt}\) , is uniformly Lipschitz injective if there exists \( c_\eta > 0\) such that

It can be shown that this property holds if the motor speed is uniformly bounded away from zero [8]. Exploiting this, we perform the uniformly injective transformation

which puts system (132) into an observable canonical form in the \( z\) -coordinates. A continuous-time high-gain observer for system (132) would then be of the form

where \( \phi\) is a globally Lipschitz left inverse of (137), which depends on \( u\) , \( i\) , and their derivatives (which may introduce noise); \( (k_1,k_2,k_3)\) and \( \ell\) are observer parameters, with \( \ell > 0\) having to be chosen sufficiently large. However, as pointed out in Section 2.4.2, we must implement a discrete-time observer. Without any specific physical guidelines about the way observer (138) should be discretized, we use a naive Euler discretization scheme with a given sampling period \( \tau = 10^{-3}\) (s), leading to

We then implement observer (139) for the PMSMPermanent Magnet Synchronous Motor. Figure 14 shows an ineffective estimation performance, especially at high speeds, due to the lack of precision of the observer discretization, a phenomenon also seen below with the KKLKravaris-Kazantzis$\slash$Luenberger observer, as well as noise amplification.

In this part, we propose strategically discretizing the PMSMPermanent Magnet Synchronous Motor model first, and then designing and implementing the discrete-time high-gain observer of Section 2.2. Note that system (134) does not follow a discrete-time triangular form (67) because \( x_{1,k+1}\) depends on \( x_{1,k}\) , so we deploy the transformation in Lemma 6. Based on the knowledge from continuous time in (137), we conjecture that the maps \( (\mathcal{O}^{bw}_k)_{k \in \mathbb{N}}\) of order \( 3\) should be uniformly Lipschitz injective if \( \tau\) is sufficiently small. We then perform the change of variables

with \( g_k\) defined in system (134), depending only on the inputs. Then, we implement observer (76) and obtain the results in Figure 15 with visibly better accuracy compared to Figure 14. From this, we draw an important lesson: instead of designing and then discretizing a continuous-time observer, we should properly discretize the system based on its physics and then build a discrete-time observer. Note also that storing the past samples of the inputs as done in this discrete-time design is finite-dimensional and no derivatives need to be computed.

Unfortunately, with \( k_1 = k_2 =k_3 = 1\) in observer (76), we need to select an exceedingly small value of \( 7 \cdot 10^{-5}\) for \( \gamma\) to make the observer work. This necessity arises from the large Lipschitz constant of the inverse map of (140). The reason behind this lies in the fact that with a low sampling rate of \( \tau = 10^{-3}\) (s), past outputs closely resemble each other, resulting in a poorly conditioned transformation. This does not allow exploring the filtering properties of the high-gain observer in this particular example. A way to improve this is to store more past outputs, namely take more dimensions in \( z_k\) like in MHOMoving Horizon Observers [97, 98, 99, 96, 50]. Another way could be to increase the sampling period to make the outputs different enough, risking the deviation of the discretized model from the real one hence requiring an even more accurate discretization scheme.

In Figure 16, we compare the estimation errors given by the discrete-time high-gain observer for model (134) but with different choices of \( \gamma\) . It is seen that the smaller \( \gamma\) , the faster the convergence, with \( \gamma = 0\) corresponding to instantaneous estimation after time \( m\) . However, a smaller \( \gamma\) results in higher sensitivity to numerical discretization noise. Moreover, the choice of \( \gamma\) seems to have little effect in the region of too high rotating speeds (around \( 12\) -\( 15\) seconds), because the discretized model becomes less accurate and deviates from the true dynamics of the PMSMPermanent Magnet Synchronous Motor. The high-gain observer, regardless of \( \gamma\) , is anyway unable to effectively address the model inaccuracies introduced by discretization.

We now focus on designing discrete-time KKLKravaris-Kazantzis$\slash$Luenberger observers for different discretized models of the PMSMPermanent Magnet Synchronous Motor, starting with the Euler one in (133). Let us verify the assumptions needed for observer design, more particularly those required by Theorem 5.

• Assumption 5: The solutions to system (133), when injected with sinusoidal inputs (see Figure 13), are also sine waves, so they remain in a compact set in positive time;
• Assumption 6: The dynamics map of system (133), with \( (u_k)_{k \in \mathbb{N}}\) and \( (i_k)_{k \in \mathbb{N}}\) known, is invertible. Note that this is not necessarily required for the high-gain design in Section 2.4.3;
• Assumption 7: First, uniform Lipschitzness of the inverse dynamics and output maps of system (133) holds since the inputs \( (u_k)_{k \in \mathbb{N}}\) and \( (i_k)_{k \in \mathbb{N}}\) are uniformly bounded and solutions remain in a compact set. Second, the uniform Lipschitz backward distinguishability is very hard to check analytically in discrete time because it involves the inversion of the dynamics. Similar to Section 2.4.3, we use its continuous-time version related to (136) to conjecture that the equivalent property holds in discrete time if the sampling period \( \tau\) is sufficiently small.

Guided by Example 9 and the knowledge that a KKLKravaris-Kazantzis$\slash$Luenberger observer of dimension \( 3\) exists in continuous time, we look for a transformation of the form

where

Note that each \( b_{i,k}\) , \( i = 1,2,3\) is a vector in \( \mathbb{R}^2\) . With \( A\in \mathbb{R}^{3\times 3}\) Schur and the pair \( (A, B)\) controllable, we get that \( z_k\) is solution to system (86) if \( (a_k,b_k,c_k)_{k\in\mathbb{N}}\) evolve following

Note that \( a_k\) can be picked constant equal to \( (\rm{Id}-A)^{-1}B\) . Because \( y_k = 0\) for all \( k\) , \( z_k\) converges to zero exponentially fast and it is straightforward to pick for the observer, e.g., the particular solution \( \hat{z}_k = 0\) for all \( k \in \mathbb{N}\) . Then, the estimate is obtained by solving \( T_k(\hat{x}_k)=\hat{z}_k=0\) , namely

We now design the KKLKravaris-Kazantzis$\slash$Luenberger observer in Section 2.3 but for the model (134), where the discretization scheme incorporates the system’s rotating dynamics. Note that system (134), with the inputs \( (u_k)_{k \in \mathbb{N}}\) and \( (i_k)_{k \in \mathbb{N}}\) being sinusoidal, satisfies all the assumptions required by Theorem 5. Keeping the same pair \( (A, B)\) , we get this time different dynamics of \( (a_k,b_k,c_k)_{k\in\mathbb{N}}\)

with \( a_k\) still possibly constant equal to \( (\rm{Id}-A)^{-1}B\) , and the estimate is still obtained with (143). In Figure 17, the estimation errors with respect to the continuous-time trajectory of system (132) are compared among: i) The continuous-time KKLKravaris-Kazantzis$\slash$Luenberger observer from [8] with \( A_c = -\rm{diag}(10, 44, 80)\) discretized at \( \tau = 10^{-3}\) (s) using Euler’s scheme; ii) The discrete-time KKLKravaris-Kazantzis$\slash$Luenberger observer (142)-(143); and iii) The discrete-time KKLKravaris-Kazantzis$\slash$Luenberger observer with rotation correction (144)-(143), for \( A = e^{\tau A_c}\) , \( B = (1,1,1)\) , and \( \gamma = 1\) .

Similar to Section 2.4.3, we see that it is better not to discretize an observer already designed so as not to introduce numerical errors, and that the discretization scheme has a huge impact on the precision of the discretized model, thus determining the observer’s performance.

In Figure 18, we compare the estimation errors given by the discrete-time KKLKravaris-Kazantzis$\slash$Luenberger observer for model (134) but with different choices of \( \gamma\) . Similar to Section 2.4.3, it is observed that a smaller \( \gamma\) gives a faster convergence, but a more serious amplification of numerical noise, which is coherent with the robustness results in Theorem 6. However, in the region of too high rotating speeds, the three choices of \( \gamma\) tend to perform the same, since the discretized model becomes less appropriate, which is something the observers cannot deal with. Last, it is interesting to notice that in this application case, as we choose for the observer in the \( z\) -coordinates \( \hat{z}_k = 0\) for all \( k \in \mathbb{N}\) , it is indeed the transformation \( (T_k)_{k \in \mathbb{N}}\) that serves to provide the estimation.

with the estimate \( \hat{x}\) obtained from a solution \( (t,j) \mapsto \hat{z}(t,j)\) to dynamics (146.a) as

In the literature, impulsive systems take the form

for \( t_1 < t_2 < … < t_k < …\) a given sequence of times when impulses take place, making the state jump instantly from \( x(t_k^-)\) to \( x(t_k^+)\) . On the other hand, switched systems are modeled by

where the exogenous switching signal \( \sigma\) determines the active mode. These are thus continuous-time systems with discrete events such as switches or impulses occurring at specific time instants that are generally known and therefore constitute an important large class of hybrid systems with known jump times with a wide range of applications, e.g., power electronics [144]. Their observability\( /\) determinability is analyzed (possibly with algebraic certificates) in [11, 136, 137, 138, 134]. In [145], switching observers are designed using a Linear Matrix Inequality[LMI] for both continuous- and discrete-time linear systems assuming the full detectability of each mode, while the switching signal is completely known. In the linear context, [134] suggests a change of coordinates based on the Kalman decomposition to extract the observable components of each individual mode, which are then estimated during each interval using a Luenberger observer. [12] develops parallel observers estimating the observable part of each mode under pre-determined transitions of bounded varying intervals, relying on the determinability gained after a high enough number of switches. These two works thus exploit observability gained by accumulating information from individual non-observable sub-systems under persistent switching. However, so far, results for switched systems mostly concern those with linear maps.

Oftentimes, for various reasons such as bandwidth limitations, communication safety, or simply the availability of the measurements, the output of a continuous-time system is only taken at specific time instants, leading to systems of the form

for \( t_1 < t_2 < … < t_k < …\) a sequence of times when the measurement is performed. This is a particular class of hybrid systems (145) with an identity jump map and known jump times. Some observer designs consist of adapting existing continuous-time observers, usually with constraints on the maximal sampling period. For instance, [146] adapts the high-gain observer design into a continuous-discrete extended Kalman filter where exponential stability is achieved if the gain belongs to an interval that contracts when the constant sampling period increases; and [147] proposes an impulsive Luenberger observer for linear systems with quantized measurements that guarantees stability if the varying quantizer transition inter-arrival time is taken sufficiently small. LMILinear Matrix Inequality-based impulsive and sample-and-hold designs with varying sampling times are developed in [148, 149] respectively, taking into account the Lipschitzness of the system nonlinearity. Moreover, a sample-and-hold observer is designed in [150] with a time-varying adaptation gain that is reset to \( 1\) at each sampling instant and decreases in between sampling events. On the other hand, hybrid designs with correction at sampling times only have also been developed using LMILinear Matrix Inequalitys, with both a polytopic approach [151] and a grid-based one [152]. Similarly to Section 3.1.2.1, results for this class of systems are mostly restricted to the linear context.

In nonlinear settings, attempts have been made in [153], supposing that a continuous-time observer has been designed as if the output were continuous, by proposing a hybrid observer consisting of the said observer and an output predictor for the time interval between two consecutive measurements. The work [154] does this by letting the estimate evolve with the system’s dynamics and correcting it each time a measurement arrives, using gains obtained by LMILinear Matrix Inequalitys.

with

where \( \lambda \geq 0\) and \( \gamma \in (0,1]\) are design parameters, \( R_c \in §_{>0}^{n_{y,c}}\) and \( R_d \in §_{>0}^{n_{y,d}}\) are (possibly time-varying) weighting matrices such that there exist positive scalars \( \underline{c}_{R_c}\) , \( \overline{c}_{R_c}\) , \( \underline{c}_{R_d}\) , and \( \overline{c}_{R_d}\) such that for all \( (t,j) \in \rm{dom} x\) , we have

To define the required observability concepts for our hybrid Kalman-like observer, we begin by introducing the notions of the Gramian and observability needed for this observer. Let us assume the following; recall that \( F\) and \( J\) are allowed to be time-varying.

Under Assumption 10, solutions \( x \in §_{\mathcal{H}}(\mathcal{X}_0,\mathfrak{U})\) are unique in both forward and backward time, so we can define hybrid transition matrices of system \( \mathcal{H}\) in (155). More precisely, given a solution \( x\in §_{\mathcal{H}}(\mathcal{X}_0,\mathfrak{U})\) with \( u_c = 0\) and \( u_d = 0\) , for all hybrid times \( ((t^\prime,j^\prime),(t,j))\in \rm{dom} x \times \rm{dom} x\) , we have

if \( t \leq t^\prime\) and \( j \leq j^\prime\) , with the domain of \( F\) and \( J\) inherited from \( \mathcal{D}\) , where \( \phi_F\) denotes the continuous-time transition matrix associated with \( F\) , i.e., describing solutions to \( \dot{x}=Fx\) .

Now, consider a pair of solutions \( x_a\) and \( x_b\) in \( §_{\mathcal{H}}(\mathcal{X}_0,\mathfrak{U})\) with the same inputs \( (F,J,H_c,H_d,u_c,u_d)\) such that \( \rm{dom} x_a = \rm{dom} x_b:=\mathcal{D}\) . Then, for all hybrid times \( ((t^\prime,j^\prime),(t,j))\in \mathcal{D} \times \mathcal{D}\) , we have

By summing and integrating squares, it follows that the equality of the outputs \( (y_c,y_d)\) along \( x_a\) and \( x_b\) between time \( (t^\prime,j^\prime)\in\mathcal{D}\) and a later time \( (t,j)\in\mathcal{D}\) is equivalent to

or, assuming the invertibility of \( J\) at the jump times,

where \( \mathcal{G}^{fw}_{(F,J,H_c,H_d)}((t^\prime,j^\prime),(t,j))\) (resp., \( \mathcal{G}^{bw}_{(F,J,H_c,H_d)}((t^\prime,j^\prime),(t,j))\) ) is the observability Gramian (resp., backward observability Gramian) between those times as defined next.

According to (160), we deduce that the observability between times \( (t^\prime,j^\prime)\) and \( (t,j)\) , namely the ability to reconstruct the initial state \( x(t^\prime,j^\prime)\) from the knowledge of the future output until \( (t,j)\) , is equivalent to the positive definiteness of the observability Gramian over this period. On the other hand, when \( J\) is invertible at the jump times, the ability to reconstruct the current state \( x(t,j)\) from the knowledge of the past output until \( (t^\prime,j^\prime)\) , i.e. the backward distinguishability or constructibility, is characterized by the positive definiteness of the backward observability Gramian over this period according to (161). Actually, in that case, both notions are actually equivalent but the backward observability Gramian tends to appear more naturally in the analysis of observers.

For observer design, the invertibility of the backward observability Gramian is typically assumed to be uniform, leading to the following hybrid Uniform Complete Observability[UCO], extending the classical UCOUniform Complete Observability condition of the Kalman and Bucy’s filter [78].

In this section, we exploit these newly defined notions to show three main results: i) The estimation error converges asymptotically to zero for any choice of \( \lambda \geq 0\) , \( \gamma \in (0,1]\) , under boundedness of the matrices and UCOUniform Complete Observability along the considered solution only (Section 3.2.2.2); ii) It is exponentially stable with an arbitrarily fast rate for appropriate choices of \( \lambda\) and \( \gamma\) if these requirements hold uniformly with respect to solutions (Section 3.2.2.3); and iii) It is robustly stable (in the sense of [46]) with respect to flow\( /\) jump input disturbances and measurement noise (Section 3.2.2.4).

In this part, we provide sufficient conditions for asymptotic convergence of the synchronized hybrid Kalman-like observer for a given solution \( x \in §_{\mathcal{H}}(\mathcal{X}_0,\mathfrak{U})\) , thanks to some boundedness and UCOUniform Complete Observability assumptions, made along that particular solution only.

Theorem 8 below then shows that all solutions in \( §_{\mathcal{H}}(\mathcal{X}_0,\mathfrak{U})\) can be estimated by observer (156). Note that for asymptotic convergence only (without stability guarantees), no uniformity with respect to solutions or time domains is required, but only along the time domain of each particular solution.

Proof. Consider a solution \( x \in §_{\mathcal{H}}(\mathcal{X}_0,\mathfrak{U})\) . By Assumption 9, it is complete. In the rest of this proof, all variables are evolving on \( \rm{dom} x\) and so are complete. Consider \( (t,j)\mapsto \Pi(t,j)\) with \( \Pi(0,0) \in §_{>0}^{n_x}\) and dynamics

Because \( J\) is invertible at jumps from Assumption 10, \( \Pi\) is well defined. It can be proven using mathematical induction that the closed form of \( \Pi(t,j)\) for all \( (t,j) \in \rm{dom} x\) is

(with \( \Psi_c\) and \( \Psi_d\) defined in Definition 16). Now, we show that \( \Pi\) is uniformly lower-bounded along \( \rm{dom} x\) . First use Gronwall’s inequality to show that \( \|\phi_F(t,t^\prime)\| \leq e^{c_F |t-t^\prime|}\) , then it follows that for any \( ((t^\prime,j^\prime),(t,j)) \in \rm{dom} x \times \rm{dom} x\) with \( t^\prime\leq t\) and \( j^\prime\leq j\) , we have \( \|\Phi_{F,J}((t,j),(t^\prime,j^\prime))\| \leq e^{c_F(t-t^\prime)}c_J^{j-j^\prime}\) . Because

this implies that

Then, for any \( (t,j) \in \rm{dom} x\) such that \( t+j\leq \Delta\) , we have

for some \( c_{\Pi,1} > 0\) . Next, for any \( (t,j) \in \rm{dom} x\) such that \( t+j\geq \Delta\) , we can always pick \( (t^\prime,j^\prime) \in \rm{dom} x\) (before \( (t,j)\) ) such that \( \Delta \leq (t-t^\prime)+(j-j^\prime)\leq \Delta+1\) and from Assumption 11, we have

Therefore, for all \( (t,j) \in \rm{dom} x\) , we have

which means that \( \Pi\) is uniformly lower-bounded and thus uniformly invertible on \( \rm{dom} x\) . Let us now study the dynamics of \( W := \Pi^{-1}\) , which is well defined and belongs to \( §_{>0}^{n_x}\) . During flows, we have

At jumps, using Woodbury matrix identity, we have

Therefore, \( W\) follows the same dynamics as \( P\) in (156). So if \( W(0,0)=P(0,0)\) then \( W(t,j) = P(t,j)\) for all \( (t,j) \in \rm{dom} x\) . This means that \( P=\Pi^{-1}\) (with \( \Pi(0,0)=(P(0,0))^{-1}\) ) and that \( P\) is invertible at all times (but not necessarily uniformly along \( \rm{dom} x\) since \( P\) may go to \( 0\) asymptotically if \( \Pi\) is not uniformly upper-bounded). Therefore, the estimation error \( \tilde{x} := x - \hat{x}\) follows the dynamics

where \( K = \Pi^{-1}H_d^\top (H_d \Pi^{-1}H_d^\top+R_d)^{-1}\) . Consider the Lyapunov function \( V(\tilde{x},\Pi)=\tilde{x}^\top \Pi \tilde{x}\) . For all \( (t,j) \in \rm{dom} x\) , \( V(\tilde{x}(t,j),\Pi(t,j)) \geq c_\Pi |\tilde{x}(t,j)|^2 \) , so Theorem 8 is proven if we show that \( V\) asymptotically converges to \( 0\) . Let us study the dynamics of \( V\) along (169) and (166). During flows, we have

Using Woodbury matrix identity yields

At jumps, thanks to the newly obtained expression of \( K\) and Woodbury matrix identity, we have

We see that \( V\) decreases strictly and exponentially to \( 0\) if \( \lambda > 0\) and \( \gamma \in (0,1)\) . We next show that actually, thanks to UCOUniform Complete Observability, it converges in-the-large even for \( \lambda = 0\) and \( \gamma = 1\) . In this case, we have

and thus, for all \( ((t^\prime,j^\prime),(t,j)) \in \rm{dom} x \times \rm{dom} x\) , we have

Applying Lemma 22 in the Appendix with \( \Delta_m=\Delta+1\) , \( K_c = \Pi^{-1}H_c^\top R_c^{-1}\) , and \( K_d=JK= J\Pi^{-1}H_d^\top (H_d \Pi^{-1}H_d^\top+R_d)^{-1}\) , which are indeed upper-bounded by \( c_{H_c}(c_\Pi \underline{c}_{R_c})^{-1}\) and \( c_J c_{H_d}(c_\Pi \underline{c}_{R_d})^{-1}\) respectively, there exists \( c_\mathcal{G}>0\) such that for all \( ((t^\prime,j^\prime),(t,j)) \in \rm{dom} x \times \rm{dom} x\) such that \( \Delta \leq (t-t^\prime) + (j-j^\prime) \leq \Delta+1\) , we have

exploiting the UCOUniform Complete Observability property in Assumption 11. We finally conclude that there exists \( c_V>0\) such that for any \( ((t^\prime,j^\prime),(t,j)) \in \rm{dom} x \times \rm{dom} x\) verifying \( \Delta \leq (t-t^\prime) + (j-j^\prime) \leq \Delta+1\) , we have

It remains to show that \( \tilde{x}\) converges asymptotically to \( 0\) using contradiction, similar to [81]. Assume that \( \tilde{x}\) does not converge to \( 0\) . Then, there exists \( \epsilon > 0\) such that for any \( (t^\prime,j^\prime) \in \rm{dom} x\) , we can always find (exploiting the completeness of \( x\) ) \( (t,j) \in \rm{dom} x\) such that \( (t-t^\prime)+(j-j^\prime) \geq \Delta\) and \( |\tilde{x}(t,j)| \geq \epsilon\) . Hence we have \( V(t,j) \leq V(t^\prime,j^\prime) -c_V \epsilon^2\) . By repeating this process, still thanks to the completeness of \( x\) , \( V\) becomes negative after a finite amount of time, which contradicts its definition. Therefore, by contradiction, \( \tilde{x}\) converges asymptotically to \( 0\) . \( \blacksquare\)