Behavioral systems theory in data-driven analysis, signal processing, and control

We present a novel data-driven MPC approach to control unknown nonlinear systems using only measured input-output data with closed-loop stability guarantees. Our scheme relies on the data-driven system parametrization provided by the Fundamental Lemma of Willems et al. We use new input-output measurements online to update the data, exploiting local linear approximations of the underlying system. We prove that our MPC scheme, which only requires solving strictly convex quadratic programs online, ensures that the closed loop (practically) converges to the (unknown) optimal reachable equilibrium that tracks a desired output reference. As intermediate results of independent interest, we extend the Fundamental Lemma to affine systems and we propose a data-driven tracking MPC scheme with guaranteed robustness. The theoretical analysis of this MPC scheme relies on novel robustness bounds w.r.t. noisy data for the open-loop optimal control problem, which are directly transferable to other data-driven MPC schemes in the literature. The applicability of our approach is illustrated with a numerical application to a continuous stirred tank reactor.

Linear tracking MPC for nonlinear systems Part II: The data-driven case.

In this paper, we present a data-driven controller design method for continuous-time nonlinear systems, using no model knowledge but only measured data affected by noise. While most existing approaches focus on systems with polynomial dynamics, our approach allows to design controllers for unknown systems with rational or general non-polynomial dynamics. We first derive a data-driven parametrization of unknown nonlinear systems with rational dynamics. By applying robust control techniques to this parametrization, we obtain sum-of-squares based criteria for designing controllers with closed-loop robust stability and performance guarantees for all systems which are consistent with the measured data and the assumed noise bound. We then apply this approach to control systems whose dynamics are linear in general non-polynomial basis functions by transforming them into polynomial systems. Finally, we apply the developed approaches to numerical examples.

Data-Driven Control of Nonlinear Systems: Beyond Polynomial Dynamics

In data-driven analysis and control, the so-called Fundamental Lemma by Willems et al. has gained a lot of interest in recent years. Using behavioural system theory, the Fundamental Lemma shows that the full system behaviour of a Linear Time-Invariant (LTI) system can be characterised by a single sequence of data of the system, as long as the input is persistently exciting. In this work, we aim to generalize this LTI result to Linear Parameter-Varying (LPV) systems. Based on the behavioural framework for LPV systems, we prove that one can obtain a result similar to Willems'. This implies that the result is also applicable to nonlinear system behaviour that can be captured with an LPV representation. We show the applicability of our result by connecting it to earlier works on data-driven analysis and control for LPV systems.

Fundamental Lemma for Data-Driven Analysis of Linear Parameter-Varying Systems

In a recent paper it was shown how a matrix S-lemma can be applied to construct controllers from noisy data. The current paper complements these results by proving a matrix version of the classical Finsler's lemma. This matrix Finsler's lemma provides a tractable condition under which all matrix solutions to a quadratic equality also satisfy a quadratic inequality. We will apply this result to bridge known data-driven control design techniques for both exact and noisy data, thereby revealing a more general theory. The result is also applied to data-driven control of Lur'e systems.

A Matrix Finsler's Lemma with Applications to Data-Driven Control

Willems et al. showed that all input-output trajectories of a discrete-time linear time-invariant system can be obtained using linear combinations of time shifts of a single, persistently exciting, input-output trajectory of that system. In this paper, we extend this result to the class of discrete-time single-input single-output flat nonlinear systems. We propose a data-based parametrization of all trajectories using only input-output data. Further, we use this parametrization to solve the data-based simulation and output-matching control problems for the unknown system without explicitly identifying a model. Finally, we illustrate the main results with numerical examples.

Data-Based System Analysis and Control of Flat Nonlinear Systems.

This paper introduces and analyzes an improved Q-learning algorithm for discrete-time linear time-invariant systems. The proposed method does not require any knowledge of the system dynamics, and it enjoys significant efficiency advantages over other data-based optimal control methods in the literature. This algorithm can be fully executed off-line, as it does not require to apply the current estimate of the optimal input to the system as in on-policy algorithms. It is shown that a persistently exciting input, defined from an easily tested matrix rank condition, guarantees the convergence of the algorithm. Moreover, the method avoids the use of linear matrix inequalities (LMIs) for control design, decreasing the corresponding computational complexity. A data-based method is proposed to design the initial stabilizing feedback gain that the algorithm requires. Robustness of the algorithm in the presence of noisy measurements is analyzed. Both theoretical and simulation comparisons are performed to show the advantages of this algorithm against other model-free control design methods.

Mohammad Alsalti

Papers

Data-Based System Analysis and Control of Flat Nonlinear Systems.

Efficient Off-Policy Q-Learning for Data-Based Discrete-Time LQR Problems.