[Paper Review] Data-Driven Stabilizing and Robust Control of Discrete-Time Linear Systems with Error in Variables

Jared Miller, Tianyu Dai+1 more|arXiv (Cornell University)|Oct 24, 2022

Advanced Control Systems Optimization4 citations

TL;DR

This paper proposes a sum-of-squares (SOS)-based framework for data-driven stabilization and robust H2 control of discrete-time linear systems with L-infinity bounded process, measurement, and input noise. By formulating the consistency set of all possible plants as a polynomial optimization problem and applying a theorem of alternatives to eliminate noise variables, the method enables convergence to superstabilizing, quadratically stabilizing, or positively stabilizing controllers via successive SOS relaxations, with improved computational efficiency and guaranteed robustness under worst-case H2 performance.

ABSTRACT

This work presents a sum-of-squares (SOS) based framework to perform data-driven stabilization and robust control tasks on discrete-time linear systems where the full-state observations are corrupted by L-infinity bounded input, measurement, and process noise (error in variable setting). Certificates of state-feedback superstability, quadratic stability or positive stability of all plants in a consistency set are provided by solving a feasibility program formed by polynomial nonnegativity constraints. Under mild compactness and data-collection assumptions, SOS tightenings in rising degree will converge to recover the true superstabilizing or positive stabilizing controller, with some conservatism introduced for quadratic stabilizability. The performance of this SOS method is improved through the application of a theorem of alternatives while retaining tightness, in which the unknown noise variables are eliminated from the consistency set description. This SOS feasibility method is extended to provide worst-case-optimal robust controllers under H2 control costs. The consistency set description may be broadened to include cases where the data and process are affected by a combination of L-infinity bounded measurement, process, and input noise.

Motivation & Objective

• Address the challenge of stabilizing discrete-time linear systems when full-state observations are corrupted by L-infinity bounded process, measurement, and input noise.
• Formulate data-driven control as a polynomial optimization problem (POP) over a consistency set of all plants compatible with noisy data.
• Develop a sum-of-squares (SOS) relaxation framework that converges to true superstabilizing or positive stabilizing controllers as the polynomial degree increases.
• Improve computational tractability by applying a theorem of alternatives to eliminate unknown noise variables from the consistency set description.
• Extend the framework to worst-case H2-optimal robust control by minimizing the worst-case H2-norm across all consistent plants.

Proposed method

• Model the system as a discrete-time linear system with L-infinity bounded process noise $w_t$, measurement noise $\Delta x_t$, and input noise $\Delta u_t$, using observed data $\mathcal{D} = \{\hat{u}_t, \hat{x}_t\}_{t=1}^T$.
• Formulate the consistency set of all possible plants $(A,B)$ as a bilinear inequality system involving unknown noise variables, which is transformed into a polynomial optimization problem (POP).
• Apply sum-of-squares (SOS) relaxations to approximate the POP via a sequence of convex semidefinite programs (SDPs), ensuring convergence to the true stabilizing controller as the polynomial degree increases.
• Utilize a theorem of alternatives based on robust semidefinite programming to eliminate noise variables $\Delta x_t, \Delta u_t, w_t$ from the constraints, reducing problem complexity while preserving tightness for superstability and positive stability.
• Construct SOS-based certificates for state-feedback superstability, quadratic stability, and positive stability using nonnegativity constraints on polynomials, with explicit formulations via the Psatz and multiplier variables.
• Extend the framework to H2-robust control by formulating a worst-case H2-norm minimization problem over the consistency set, solved via SOS-based SDP relaxations.

Experimental results

Research questions

• RQ1 Can a data-driven controller be designed to stabilize all plants consistent with noisy input-state data under L-infinity bounded process, measurement, and input noise?
• RQ2 How can bilinear dependencies on unknown noise variables in the consistency set be eliminated to improve computational tractability?
• RQ3 To what extent do SOS relaxations converge to the true superstabilizing or positively stabilizing controller as the polynomial degree increases?
• RQ4 Can the framework be extended to provide H2-optimal robust controllers under worst-case performance over the consistency set?
• RQ5 What is the impact of using a theorem of alternatives on conservativeness and computational complexity in the context of data-driven control with error-in-variables?

Key findings

• The SOS-based feasibility program for superstability and positive stability converges to the true stabilizing controller as the polynomial degree increases, under mild compactness and data-collection assumptions.
• The application of the theorem of alternatives eliminates unknown noise variables from the consistency set description, reducing computational complexity while preserving tightness for superstability and positive stability.
• For quadratic stabilizability, the method introduces some conservatism due to the relaxation, but still ensures convergence to a stabilizing controller as the degree increases.
• The framework enables worst-case H2-optimal robust control by minimizing the worst-case H2-norm across all data-consistent plants, with the resulting controller obtained via SOS-based SDP relaxation.
• The method is extendable to switched systems, where a single controller can be designed to superstabilize all subsystems using labeled data and noise-eliminated consistency sets.
• The computational cost is reduced by exploiting sparse cliques in the polynomial constraints, particularly when using the theorem of alternatives, which results in fewer and smaller SDP constraints.