[Paper Review] Dissipativity and optimal control

Lars Grüne|arXiv (Cornell University)|Jan 29, 2021

Advanced Control Systems Optimization72 references4 citations

TL;DR

This paper establishes a foundational link between strict dissipativity and optimal control, demonstrating that strict dissipativity ensures the turnpike property—where optimal trajectories spend most time near a steady-state—enabling stability and performance guarantees in model predictive control (MPC). The key contribution is showing that strict dissipativity with a linear storage function enables Lyapunov-based stability analysis and optimal transient and average cost approximation in MPC schemes.

ABSTRACT

The close link between dissipativity and optimal control is already apparent in Jan C. Willems' first papers on the subject. In recent years, research on this link has been revived with a particular focus on nonlinear problems and applications in model predictive control (MPC). This paper surveys these recent developments and some of Willems' and other authors' earlier results.

Motivation & Objective

• To clarify the theoretical connection between dissipativity and optimal control, especially in the context of model predictive control (MPC).
• To demonstrate how strict dissipativity implies the turnpike property, where optimal trajectories remain near a steady-state for most of the time horizon.
• To extend stability and performance analysis of MPC schemes beyond terminal conditions by leveraging dissipativity and the turnpike phenomenon.
• To unify and survey recent advances connecting dissipativity, detectability, and optimal control in both discrete and continuous time.
• To identify open problems in extending these results to infinite-dimensional systems and stochastic optimal control.

Proposed method

• Uses the concept of strict dissipativity with a storage function λ(x) and supply rate s(x,u) to derive Lyapunov functions for MPC closed-loop stability.
• Applies strong duality in MPC to show that strict dissipativity with a linear storage function yields a Lyapunov function, ensuring asymptotic stability.
• Establishes that strict dissipativity plus a controllability condition implies the turnpike property, where trajectories remain close to an optimal equilibrium.
• Derives performance bounds for MPC without terminal conditions, showing that the closed-loop cost is bounded by the optimal transient cost plus error terms δ₁(T) and δ₂(S).
• Uses the turnpike property to prove that MPC achieves approximately optimal average cost and transient cost, even without terminal constraints.
• Extends the framework to time-varying and periodic optimal trajectories using overtaking optimality, broadening applicability beyond equilibrium settings.

Experimental results

Research questions

• RQ1 How does strict dissipativity relate to the stability and performance of model predictive control (MPC) schemes?
• RQ2 What conditions ensure the turnpike property in optimal control, and how is it connected to strict dissipativity?
• RQ3 Can performance bounds for MPC without terminal conditions be established using dissipativity and the turnpike property?
• RQ4 How does strict dissipativity relate to classical detectability notions in nonlinear systems?
• RQ5 To what extent can dissipativity-based analysis be extended to infinite-dimensional systems and stochastic optimal control problems?

Key findings

• Strict dissipativity with a linear storage function is equivalent to strong duality in MPC, enabling Lyapunov-based stability analysis.
• The turnpike property occurs when strict dissipativity and a controllability condition hold, ensuring optimal trajectories remain near an equilibrium for most of the time horizon.
• For MPC without terminal conditions, the average cost satisfies $\overline{J}^{MPC}_{\infty}(x_0) \leq \ell(x^e,u^e) + \delta_1(T)$, with $\delta_1(T)$ decaying as $T$ increases.
• The transient cost in MPC without terminal conditions is bounded by $J^{MPC}_S(x_0) \leq \inf_{u\in\widetilde{\mathbb{U}}^S} J_S(x_0) + S\delta_1(T) + \delta_2(S)$, where $\delta_1(T)$ and $\delta_2(S)$ are error terms.
• For positive definite stage costs, $J^{MPC}_\infty(x_0)$ remains finite even without terminal conditions, and performance bounds improve with larger horizon $T$.
• The factor $S\delta_1(T)$ in the transient cost bound reflects the accumulation of residual stage costs when the closed-loop does not converge to equilibrium, a natural consequence of practical asymptotic stability.