Lipschitz-Based Observer

• Lipschitz-based observer is a state estimation architecture for nonlinear systems that uses Lipschitz and sector bounds to guarantee stability and robust performance.
• It employs mixed-monotonicity decomposition and interval analysis to derive tight bounds on state and fault estimates, often via LMI-based synthesis.
• Practical applications include fault diagnosis, robust estimation in PDEs and DAE systems, ensuring exponential convergence and minimal conservatism.

A Lipschitz-based observer is a state or state/input estimation architecture tailored for nonlinear dynamical systems whose nonlinearity satisfies a Lipschitz or sector-bounded condition in the state variable. Such observers exploit explicit a priori (global or local) bounds on the Jacobian or partial derivatives of the system’s nonlinear mappings. By leveraging these bounds, they provide constructive synthesis and rigorous certificates of stability, robustness, and boundedness, often via Linear Matrix Inequality (LMI) formulations or interval analysis. Lipschitz-based observer methodologies have become foundational for robust estimation in nonlinear control, interval analysis, sampled-data systems, PDEs, distributed-parameter systems, and practical applications such as fault diagnosis and data-driven state reconstruction.

1. Problem Formulation and Lipschitz Assumptions

A Lipschitz-based observer is constructed for systems of the form

$$x_{k+1} = f(x_k, u_k) + D w_k, \quad y_k = h(x_k, u_k) + v_k,$$

with $$f:\mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^n$$, $$h: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^p$$ continuously differentiable and satisfying the uniform Lipschitz property: $$\exists L_f > 0: \|f(x_1, u) - f(x_2, u)\| \le L_f \|x_1 - x_2\|,\, \forall x_1,x_2,u,$$ and similarly for $h$ with constant $L_h$. Sector-bound (partial-derivative) bounds,

$$a^f_{ij} \le \frac{\partial f_i}{\partial x_j}(x,u) \le b^f_{ij},$$

imply mixed-monotonicity and enable tight interval bounding. Such Lipschitz and sector bounds are verified or computed offline, possibly using Jacobian sign-stability (JSS) over a compact set relevant for system trajectories (Khajenejad et al., 2020, Khajenejad et al., 2022).

2. Nonlinear Decomposition and Mixed-Monotonicity

By mixed-monotonicity, $f$ and $h$ admit decomposition functions $f_d(x^1, x^2; u)$, $h_d(x^1, x^2; u)$ with the following crucial properties:

• $f_d(x,x;u) = f(x,u)$ (same for $h$)
• $f_d$ (resp. $h_d$) is monotone increasing in its first argument and decreasing in its second
• Construction is explicit; for example, $f_d(x^1,x^2;u) = f(z,u) + C_f(x^1 - x^2)$, where $z_j$ is chosen according to sign of partials [Yang–Ozay, 2019].

Upper/lower interval bounds for the nonlinear flows are then tightened via

$$\delta^{\min}_f = f_d(x_{\min}, x_{\max}; u),\quad \delta^{\max}_f = f_d(x_{\max}, x_{\min}; u)$$

and analogs for the output. This decomposition yields interval estimators that are correct by construction without artificial conservatism (Khajenejad et al., 2020, Khajenejad et al., 2022).

3. Lipschitz-Based Interval Observer Structures

The canonical Lipschitz-based interval observer recursively computes: $$\begin{aligned} \begin{pmatrix} x_{k+1}^+ \ x_{k+1}^- \end{pmatrix} &= M_f \begin{pmatrix} \delta_f^{\max} \ \delta_f^{\min} \end{pmatrix} + M_h \begin{pmatrix} \delta_h^{\max} \ \delta_h^{\min} \end{pmatrix} + M_u u_k + M_w \begin{pmatrix} w^+ \ w^- \end{pmatrix} + M_v \begin{pmatrix} v^+ \ v^- \end{pmatrix} \end{aligned}$$ where $x_{k+1}^{+}$ ($x_{k+1}^{-}$) denote upper (lower) bounding estimates for the state at time $k+1$. The observer gain matrices $M_f$, $M_h$, $M_u$, $M_w$, $M_v$ are designed so the resulting affine recursion preserves interval inclusion and supports tightness. Unknown input or fault intervals are recovered using interval residuals and right-inverse (typically Moore–Penrose pseudoinverse) transforms based on the direct feedthrough structure (Khajenejad et al., 2020).

A distinguishing property is that the interval estimates for unknown inputs are the tightest possible, i.e., no broader than minimally needed to contain the set of consistent solutions, as proven by propagating the interval linear constraints with a correctly designed right-inverse (Khajenejad et al., 2020).

4. Stability, Boundedness, and LMI Synthesis

Uniform boundedness of the computed interval widths—interpreted as estimator stability—can be certified by conditions of several forms:

• Small-gain inequality: $$\mathcal{L} := L_{f,d}\|T_f\| + L_{h,d}\|T_h\| \le 1$$
• Block-matrix inequalities: e.g., $5n \times 5n$ LMI involving observer gains and Lipschitz constants
• Schur-complement LMIs: $3n \times 3n$ LMI with variables $P \succ 0$, $\Gamma \succeq 0$

Feasibility of any of these ensures all interval widths $\Delta^x_k = \|x_{\max,k} - x_{\min,k}\|$ remain uniformly bounded, and usually that they contract asymptotically to zero in the absence of disturbance (Khajenejad et al., 2020, Khajenejad et al., 2022). The LMI framework enables convex synthesis of observer gains, admits explicit optimization over convergence rate, and directly incorporates Lipschitz and sector-bounds.

5. Implementation Procedure and Algorithmic Recipe

The typical offline and online workflow for a Lipschitz-based observer is:

1. Offline:
   • Compute mixed-monotone decompositions using analytic or mesh-based Jacobian bounds.
   • Design observer gain matrices subject to algebraic or LMI constraints that guarantee interval invariance and boundedness.
2. Online (at each time-step):
   • Evaluate decomposition functions on current state intervals to compute $\delta^{\min/\max}_{f,h}$.
   • Propagate interval estimates via the affine observer update.
   • Generate interval residuals and apply the pseudoinverse or polyhedral optimizations to recover tight unknown input or fault intervals.

This structure lends itself to direct software implementation—and is demonstrated in detailed, constructive recipes in the references (Khajenejad et al., 2020, Khajenejad et al., 2022).

6. Applications, Scope, and Extensions

Lipschitz-based observer theory extends far beyond classical ODE systems:

• DAE Systems: Extension to algebraic-domain systems via subspace-restricted LMIs on the Wong-sequence subspace (Berger et al., 2019).
• Interval Observers: Correct-by-construction embedding and bounding of all admissible system trajectories, including systems with measurement noise, actuator/sensor faults, and unknown inputs (Khajenejad et al., 2020, Khajenejad et al., 2022).
• PDE and Infinite-Dimensional Systems: Nonlinear observer design for parabolic, hyperbolic, and boundary-coupled PDEs with guaranteed decay rates leveraging spectral geometric inequalities and Lyapunov PDE functionals (Katz et al., 2021, Ferrante et al., 2021, Liu et al., 17 Jan 2026).
• Switched and Sampled-Data Systems: Dwell-time and sampled-data stabilization via Lyapunov–Metzler and piecewise time-dependent LMI conditions with explicit ties to the underlying Lipschitz bounds (Katz et al., 3 Nov 2025).
• One-Sided Lipschitz and Relaxed Sector Constraints: Generalization to systems with only one-sided Lipschitz continuity, reducing conservativeness compared to standard global Lipschitz observers (Abbaszadeh et al., 2013).

Numerical and hardware-in-the-loop examples confirm practical effectiveness across domains including process control, fault detection, distributed parameter systems, and even neural observer design using Lipschitz-bounded NNs in data-driven frameworks (Khajenejad et al., 2020, Tang, 2023).

7. Significance and Theoretical Guarantees

Lipschitz-based observers address the fundamental challenge of state/input estimation for nonlinear, possibly non-contractive, and noise-perturbed systems. Rigorous theoretical guarantees provided in the literature include:

• Existence and uniqueness of interval observer solutions
• Necessary and sufficient conditions for boundedness and tightness of state/fault intervals
• LMI-based or small-gain sufficiency for exponential convergence
• Absence of artificial conservatism in the recovered intervals for unknown inputs
• Practical guidance for gain design, error analysis, and software realization (Khajenejad et al., 2020, Khajenejad et al., 2022, Berger et al., 2019)

Collectively, these properties establish Lipschitz-based observers as an essential and mature tool in modern nonlinear estimation, with ongoing developments in scalability, robustness, and application to high-dimensional and distributed settings.