Data-Driven Superstabilizing Control of Error-in-Variables Discrete-Time Linear Systems

Abstract

This paper presents a linear-programming based algorithm to perform data-driven stabilizing control of linear positive systems. A set of state-input-transition observations is collected up to magnitudebounded noise. A state feedback controller and dual linear copositive Lyapunov function are created such that the set of all data-consistent plants is contained within the set of all stabilized systems. This containment is certified through the use of the Extended Farkas Lemma and solved via Linear Programming. Sign patterns and sparsity structure for the controller may be imposed using linear constraints. The complexity of this algorithm scales in a polynomial manner with the number of states and inputs. The Linear-Programming based algorithm is extended to positive-stabilization of switched linear systems and Linear Parameter-Varying systems. Effectiveness is demonstrated on example systems.

Quantifying the Safety of Trajectories using Peak-Minimizing Control

“…Proof. Proposition 6.1 proves that v(0, x, z) ≤ V (x, z) from (24). Constraint (27b) imposes that q(x) ≤ v(0, x, z) ≤ V (x, z) for all x ∈ X, which implies that q(x) ≤ inf z v(0, x, z) for all x ∈ X.…”

“…Verification of p ∈ Σ[x] 2d at fixed degree can be performed by solving an SDP. The per-iteration scaling of Interior Point Methods for solving this SDP rises in a jointly polynomial manner with n and d (with O(n 6d ) and O(d 4n )) [23] [24].…”

Controller Design for Robust Invariance From Noisy Data

“…Although the actual parameters [A ⋆ B ⋆ ] from ( 7) are unknown, each of the previous two bounds allows us to characterize the set of parameters [A B] consistent with (13) and data in (10) or with (15) and data in (9). For (13), see also (10), the set is…”

“…In [2], sufficient conditions for controller design are given when the measurement error on the state satisfies an energy bound, whereas we provide necessary and sufficient conditions here. Measurement errors and a process disturbance, which satisfy an instantaneous bound (in ∞-norm), are considered in [13]. To handle bilinearity in the set of system parameters, [13] formulates the controller design as a polynomial optimization problem, which is approximated by a converging sequence of sum-of-squares programs.…”

“…Measurement errors and a process disturbance, which satisfy an instantaneous bound (in ∞-norm), are considered in [13]. To handle bilinearity in the set of system parameters, [13] formulates the controller design as a polynomial optimization problem, which is approximated by a converging sequence of sum-of-squares programs. Here, an instrumental result, Proposition 1, enables the controller design by solving a single LMI.…”