[Paper Review] Boundary control of cascaded ODE-Heat equations under actuator saturation

Wen Kang, Emilia Fridman|arXiv (Cornell University)|Aug 12, 2016

Stability and Controllability of Differential Equations16 references3 citations

TL;DR

This paper proposes a backstepping-based boundary control law for cascaded ODE-heat systems with time-varying state delays and actuator saturation. By extending the backstepping method to delayed systems and using Lyapunov-Krasovskii functionals with LMIs, it derives delay-independent exponential stability conditions and computes an estimate of the domain of attraction, ensuring global exponential stability within a guaranteed region of initial conditions under saturation constraints.

ABSTRACT

In this paper, we consider boundary stabilization for a cascade of ODE-heat system with state delay under actuator saturation. To stabilize the system, we design a state feedback controller via the backstepping method and find a bound on the domain of attraction. The latter bound is based on Lyapunov method, whereas the exponential stability of the delayed cascaded systems is proved by using Halanay's inequality. Numerical examples illustrate the efficiency of the method.

Motivation & Objective

• To address the lack of boundary control methods for PDE-ODE cascades under actuator saturation and time-varying delays.
• To extend the backstepping method to systems with state delays, enabling stabilization via Volterra integral transformations.
• To derive delay-independent stability conditions using Halanay’s inequality and Lyapunov methods.
• To estimate the largest possible domain of attraction for the closed-loop system under saturation constraints.
• To provide LMI-based conditions for controller design that ensure exponential stability within a computable region of initial states.

Proposed method

• A Volterra integral transformation is used to map the original ODE-heat system with delay into a target system with coupled dynamics.
• The backstepping method is extended to handle time-varying delays in both the ODE and PDE components.
• Lyapunov-Krasovskii functionals are constructed for the target system to derive delay-independent exponential stability conditions.
• Linear matrix inequalities (LMIs) are employed to design a state feedback controller that respects actuator saturation.
• The domain of attraction is estimated by solving LMIs to maximize the size of the initial condition set ensuring exponential convergence.
• Numerical simulations validate the method using finite difference schemes with specific initial conditions and delay values.

Experimental results

Research questions

• RQ1 How can the backstepping method be extended to stabilize cascaded ODE-heat systems with time-varying state delays under actuator saturation?
• RQ2 What is the largest set of initial conditions for which the closed-loop system remains exponentially stable under control saturation?
• RQ3 Can delay-independent stability conditions be derived for such systems using Lyapunov methods and Halanay’s inequality?
• RQ4 How can LMIs be used to design a boundary controller that respects input saturation while ensuring exponential stability?
• RQ5 What is the relationship between the controller design parameters and the size of the estimated domain of attraction?

Key findings

• For Dirichlet actuation, a domain of attraction estimate was computed as $\mathcal{X}_{u} = \{(X_0,u_0) \in W_1 : 1.34\max_{[-h,0]}|X_0|^2 + 2.24\max_{[-h,0]}\|u_0\|^2 \leq 1\}$, with $h=0.4$, ensuring exponential stability for initial states within this set.
• For Neumann actuation, the domain of attraction was estimated as $\mathcal{X}_{u} = \{(X_0,u_0) \in W_1 : 13.96\max_{[-h,0]}|X_0|^2 + 16.67\max_{[-h,0]}\|u_0\|^2 + 0.47\max_{[-h,0]}\|u_0'\|^2 \leq 1\}$, with $h=0.4$, confirming stability for initial conditions inside this region.
• The method successfully stabilized the system in simulations when initial conditions were within the estimated domain of attraction, while instability occurred when they were outside.
• The minimal value of $\beta$ was found to be 0.0739 for Dirichlet actuation and 0.1176 for Neumann actuation, corresponding to the largest ellipsoids contained in the domain of attraction.
• Numerical results confirmed that the controller maintains exponential stability for initial states within the computed ellipsoidal bounds, even under actuator saturation.
• The approach is extendable to nonlinear ODE-heat systems with globally Lipschitz nonlinearities and to observer-based control designs.