Abstract

Nonlinear observer design is a fundamental research area in control theory that is constantly attracting attention from researchers in the community. While general and systematic frameworks for state estimation of linear systems with global convergence guarantees are well-established in the literature [1, 2], nonlinear observer design still suffers from a lack of generality and global convergence guarantees.
The literature abounds with various nonlinear observer methods such as high gain observers [3], immersion and invariance-based observers [4], observers based on geometric methods [5], those based on Linear Matrix Inequalities (LMI)s [6], algebraic estimators [7], approaches relying on an injective transformation into a larger latent space [8], and the well-known Extended Kalman Filter [9]. However, most of the above-mentioned observer designs rely heavily on the class of nonlinearity of the system or provide only local convergence guarantees. On the other hand, observer design based on contraction analysis is slowly regaining popularity since its first introduction in the late nineties in [10]. They offer a generic design framework that is suitable for a general class of smooth nonlinear systems while providing global exponential convergence guarantees.
Contraction analysis was introduced in [11], drawing on concepts from continuum mechanics and differential geometry. It offers a distinct perspective on stability analysis compared to traditional methods like Lyapunov theory [12, 13, 14]. Rather than focusing on the convergence of a system to a target trajectory or equilibrium point, contraction analysis assesses stability by determining whether all trajectories of a system converge toward one another. In essence, a system is considered contracting if it eventually "forgets" its initial conditions or any temporary disturbances, making this approach particularly well-suited for observer design. Indeed, the initial motivation behind contraction analysis was primarily related to the design of observers for nonlinear systems [15]. Numerous contraction-based nonlinear controllers and observers have since been proposed in the literature such as the work in [10, 16, 17, 11, 18, 19, 15]. Contraction analysis provides a systematic framework for designing nonlinear observers, leveraging sophisticated mathematical developments that ensure global exponential convergence. The observer’s correction term is determined by solving a Matrix Partial Differential Inequality (MPDI). However, despite the theoretical elegance of this approach, solving the MPDI presents significant analytical and numerical challenges that limit the use and implementation of contraction-based observers in their original form [20]. Therefore, we aim in the present paper to overcome this challenge by proposing a learning-based approach to design the contraction-based nonlinear observer’s gain satisfying the MPDI.

Artificial Neural Networks (ANNs) have emerged as powerful tools in approximating solutions to ordinary and partial differential equations (ODEs and PDEs). By leveraging their ability to approximate complex nonlinear functions, neural networks can be trained to satisfy the differential equations governing physical phenomena directly. This approach, often referred to as Physics-Informed Neural Networks (PINNs), was pioneered by Raissi et al. [21], and incorporates the underlying physical laws into the learning process, allowing the network to learn solutions that are consistent with the known physics. The advent of automatic differentiation has further revolutionized this field by enabling the efficient computation of derivatives required in the training process. This capability facilitates the direct incorporation of differential equations into the neural network’s loss function, ensuring that the learned solutions not only fit the data but also satisfy the physical laws described by the differential equations.

Physics-Informed Neural Networks (PINNs) have been extensively applied in modeling and parameter estimation of nonlinear dynamical systems. For instance, in [22], a Runge–Kutta-based PINN framework was proposed for parameter estimation and modeling of nonlinear systems where the physical loss improves the integral involved in the Runge-Kutta method, reducing the error in the learnable trajectories. From the perspective of this work, PINNs have been utilized in observer design for discrete-time and continuous-time nonlinear systems. The work in [23] introduced a PINN-based approach for designing nonlinear discrete-time observers, showing improved performance in state estimation without the need for explicit system models. Moreover, learning-based methods have been employed to design the KKL observer for autonomous nonlinear systems, as presented in [24, 25], and nonlinear systems subject to measurement delay [26, 27], where neural networks were trained to approximate a forward and inverse map involved in KKL observer design. The work in [28, 29] has shown that PINNs are effective in model predictive control, addressing practical challenges in the oil and gas industry, and accurately identified nonlinear dynamical systems. These findings emphasize the versatility of PINNs in system identification and observer design by embedding physical laws directly within neural network architectures, resulting in models that are both data-efficient and physically consistent, achieving robust generalization even for time horizons far beyond the training or operating points and maintaining resilience against additive noise.

In the present paper, we propose an unsupervised learning approach to design a nonlinear observer for a general class of nonlinear systems based on the contraction analysis. The proposed approach relies on a Physics Informed Neural Network to enforce the contraction condition in the learning process. Based on the MDPI involved in the gain design of contraction-based nonlinear observer, we formulate an optimization problem by strategically designing the cost function of the Physics Informed Neural Network. Furthermore, we establish the robustness of the proposed learning-based nonlinear observer to the neural network approximation error and measurement noise and derive conditions ensuring exponential input-to-state stability.
The present paper is organized as follows: Section 2 gives some background on contraction analysis and contraction-based nonlinear observers. The problem addressed in this paper is formulated in section 3. Subsequently, we present the proposed unsupervised learning approach to design the observer’s gain in section 4, and analyze the robustness of the designed observer to the neural network approximation error and measurement noise in section 5. The performance and robustness of the proposed observer is evaluated through two numerical examples in 6. Finally, a summary of the contributions and future work directions are provided in section 7.

Notation.
For a square matrix \( M\) , \( \operatorname{He}\{M\}=\frac{1}{2}(M+M^T)\) is the Hermitian Part of the matrix \( M\) . The class \( \mathcal{C}^1\) is the class of continuous functions with continuous first derivatives. The Euclidean norm of a vector \( u\) is denoted by \( \left\|{u}\right\|\) .

Inspired by fluid mechanics and differential geometry, Contraction Analysis offers an alternative way of studying stability. Usually, stability is studied with respect to a nominal trajectory or an equilibrium point. Instead, contraction analysis studies the convergence of the solutions of a dynamical system to each other. In other words, the system is considered stable if the system’s final behavior is independent of the initial condition [11]. A central result that was derived in [10, 16] is that if all neighboring trajectories converge to each other, then the system is globally exponentially stable and all trajectories converge to a single one.
We consider the following general autonomous nonlinear system

where \( x(t) \in \mathbb{R}^n\) is the state, \( y(t) \in \mathbb{R}^p\) is the output, \( f: \mathbb{R}^n \rightarrow \mathbb{R}^n\) and \( h: \mathbb{R}^n \rightarrow \mathbb{R}^p\) are smooth vector fields. The main result of [11] is that if there exists a bounded symmetric and positive definite matrix \( P \in \mathbb{R}^{n \times n}\) , such that

Then, system (1) is contracting with rate \( \lambda\) . Furthermore, this result is considered to be a generalization and strengthening of the Krasovskii’s global convergence theorem [30].

2.1 Contraction analysis in the context of nonlinear observer design

While Theorem 1 provides a general approach for designing nonlinear observers, determining the correction term involves solving a matrix partial differential inequality, which is highly challenging both analytically and numerically, and to the best of our knowledge, no existing numerical solvers are specifically equipped to handle this class of problems. To overcome this limitation, we exploit the universal approximation principle of neural networks to numerically approximate the observer’s correction term. We specifically rely on an unsupervised Physics-Informed Neural Network approach that enforces the contraction conditions of Theorem 1 to learn the observer’s gain.
Moreover, although the observer in (3) satisfying (4) is global, using a neural network requires training on a closed set of interests. Therefore, we consider in the remainder of the paper, systems in the form of (1) that satisfy the following assumption

In the present section, we present the proposed Physics-Informed Neural Network approach to learn the gain of the observer in (3). Since the goal is to learn the correction term of the observer, we do not have pre-calculated trajectories for the estimated state \( \hat{x}\) . Therefore, we propose an unsupervised PINN-based approach to learn the observer’s gain by leveraging the contraction analysis and enforcing the conditions in Theorem 1 to guarantee the exponential stability of the learned observer.

Let \( \hat{k}_\theta: \mathbb{R}^n \times \mathbb{R}^p \rightarrow \mathbb{R}^n \) be the learned observer gain given by

where \(T_{\theta}\) is the function parameterized by the neural networks and \( \theta\) represents the weights and biases. The block diagram of the proposed learning-based nonlinear observer is depicted in Fig. 1. By integrating this contraction condition into the neural network training, we effectively regularize the model using the physics knowledge and constraints of the observer, which allows the neural network to learn the observer’s correction term solely from the contraction condition, reducing, therefore, the need for large datasets. Therefore, we formulate the contraction condition of Theorem 1 into an optimization problem and derive a loss function that enforces this condition.

The dataset used for training as well as the loss function used in the training process are detailed in the following subsections.

4.1 Collocation point generation

4.2 Construction of the loss function

4.3 Training Pseudo-Code

In this section, we study the effect of the neural network approximation error and the measurement noise on the convergence of the observer by considering the following structure

with \( y_e\) and \( \hat{k}_{\theta}\) representing respectively the noisy output and learned observer gain, which is given by

where \( v\) is the measurement noise and \( \varepsilon\) is the neural network approximation error.

To study the robustness of the proposed PINN-based contraction nonlinear observer, we consider the following assumptions:

Assumption 2 can be easily satisfied by selecting a Lipschitz activation function for the neural network \( T_\theta\) . Moreover, since the states and the output of the system are bounded as per Assumption 1, the approximation error \( \varepsilon\) is bounded.

Theorem 3 indicates that increasing the contraction rate \( \lambda\) can further mitigate the impact of the neural network approximation errors and measurement noise on the estimation. However, a higher contraction rate reduces the likelihood of finding a gain \( k\) that satisfies the MPDI.

To evaluate the performance and disturbance and noise rejection of the proposed learning-based contraction nonlinear observer, we perform numerical simulations on two nonlinear systems: nonlinear Van der Pol and reverse Duffing oscillators given by (19) and (20), respectively.

A training dataset of 4000 samples was generated by uniformly sampling the region of interest \( (\mathcal{X}, \mathcal{Y})\) , where \( (\mathcal{X}, \mathcal{Y})=([-2,2] \times[-3,3], [-2, 2])\) for the Van der Pol oscillator and \( (\mathcal{X}, \mathcal{Y})=([-1,1]^2, [-1, 1])\) for the reverse duffing. The considered architecture for both systems is a multi-layer perceptron of 5 hidden layers with 30 neurons each. The neural network was trained using the Adam optimizer [33] with a learning rate of \( \alpha= 10^{-3}\) for 500 iterations followed by the same amount of iterations for the L-BFGS optimization algorithm [34] with \( \beta=1\) , which aligns with the methodologies reported in the literature that show better performance for training physics-based deep learning models [21, 35, 36, 37, 38, 28, 29]. The training was performed on a single NVIDIA A40 GPU.

A contraction rate \( \lambda\) of 2.5 is considered for both systems and the loss functions’ weights for both systems are provided in Table 1.

The simulation results are depicted in Fig.2 and 3 under a white Gaussian noise of \( 0.15\) magnitude. The Van der Pol oscillator was initialized at [-1, 2.5], the reverse duffing was initialized at [-0.5; 0.5], and the observer for both systems was initialized at the origin. One can see that the proposed learning-based contraction nonlinear observer was able to accurately estimate the states of both systems (19) and (20) and demonstrated good noise rejection. Furthermore, one can see that the estimation error stays bounded within a neighborhood of the origin defined by the approximation error of the neural network and the noise level. To further assess the robustness of the proposed observer to measurement noise, the PINN was trained for several values of the contraction rate \( \lambda\) , and the observer was simulated under the same level of measurement noise for \( \hat{x}(0)=x(0)\) . The results in Table \( 2\) confirm the findings of Theorem 3 and show that increasing the contraction rate reduces the effect of the measurement noise on the estimation error.

The present paper addressed a long-lasting drawback of contraction-based nonlinear observer design, by proposing an unsupervised learning-based approach to design the nonlinear observer’s gain. The proposed approach relies on a physics-informed neural network that enforces the contraction condition to better approximate the gain of an exponentially stable observer for an autonomous nonlinear system. The effect of the neural network approximation error and measurement noise was studied, and conditions for ensuring exponential input-to-state stability were derived. The proposed learning-based contraction nonlinear observer demonstrated good performance and noise rejection in numerical simulation. Future work focuses on extending the proposed approach to nonlinear non-autonomous systems.

Research reported in this publication was supported by King Abdullah University of Science and Technology (KAUST) with the Base Research Fund (BAS/1/1627-01-01) and (BAS/1/1665-01-01), and the National Institute for Research in Digital Science and Technology (INRIA). The authors would also like to thank Ibrahima N’doye for the fruitful discussions that led to this paper.