Set-based state estimation of nonlinear discrete-time systems using constrained zonotopes and polyhedral relaxations

TL;DR

This paper tackles guaranteed set-based state estimation for nonlinear discrete-time systems with bounded uncertainties. It introduces CZPR, a method that propagates constrained zonotopes through nonlinear dynamics using lifted polyhedral relaxations of factorable functions and refines enclosures with nonlinear measurements, avoiding global linearization. A key contribution is the construction of polyhedral relaxations for trigonometric functions, enabling accurate enclosure of systems with sine/cosine terms, along with an equivalent enclosure that reduces generator and constraint counts. Across three challenging numerical examples, CZPR achieves tighter enclosures than CZ methods based on the Mean Value Theorem or DC programming, at the cost of higher computational effort in complex trigonometric settings, illustrating a favorable accuracy–complexity trade-off for many nonlinear systems.

Abstract

This paper presents a new algorithm for set-based state estimation of nonlinear discrete-time systems with bounded uncertainties. The novel method builds upon essential properties and computational advantages of constrained zonotopes (CZs) and polyhedral relaxations of factorable representations of nonlinear functions to propagate CZs through nonlinear functions, which is usually done using conservative linearization in the literature. The new method also refines the propagated enclosure using nonlinear measurements. To achieve this, a lifted polyhedral relaxation is computed for the composite nonlinear function of the system dynamics and measurement equations, in addition to incorporating the measured output through equality constraints. Polyhedral relaxations of trigonometric functions are enabled for the first time, allowing to address a broader class of nonlinear systems than our previous works. Additionally, an approach to obtain an equivalent enclosure with fewer generators and constraints is developed. Thanks to the advantages of the polyhedral enclosures based on factorable representations, the new state estimation method provides better approximations than those resulting from linearization procedures. This led to significant improvements in the computation of convex sets enclosing the system states consistent with measured outputs. Numerical examples highlight the advantages of the novel algorithm in comparison to existing CZ methods based on the Mean Value Theorem and DC programming principles.

Figures (10)

• Figure 1: An example of halfspace enclosure $Q_j \supset \{(z_j,z_a) \in \mathbb{R} \times Z_a : z_j = \text{sin}(z_a)\}$ (yellow), where $Z_a = [-\frac{3 \pi}{4}, \pi]$, along with the exact set $\{(z_j,z_a) \in \mathbb{R} \times Z_a : z_j = \text{sin}(z_a)\}$ (solid blue line).
• Figure 2: The $n_x$th root of the volume of $\square \hat{X}_k$ obtained using CZMV ($+$), CZDC ($\square$), and CZPR ($\diamond$), for Example 1.
• Figure 3: The $n_x$th root of the volume of $\diamond \hat{X}_k$ obtained using CZMV ($+$), CZDC ($\square$), and CZPR ($\diamond$), for Example 1.
• Figure 4: The enclosures $\hat{X}_k$ for $k=40$ for Example 1, obtained using CZMV (blue) and CZPR (yellow).
• Figure 5: The $n_x$th root of the volume of $\square \hat{X}_k$ obtained using CZMV ($+$), CZDC ($\square$), and CZPR ($\diamond$), for Example 2

Theorems & Definitions (20)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Proposition 1
• Remark 1
• Remark 2
• Proposition 2
• Remark 3