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 letter we investigate data-driven predictive control of discrete-time linear descriptor systems. Specifically, we give a tailored variant of Willems’ fundamental lemma, which shows that for descriptor systems the non-parametric modeling via a Hankel matrix requires less data compared to linear time-invariant systems without algebraic constraints. Moreover, we use this description to propose a data-driven framework for optimal control and predictive control of discrete-time linear descriptor systems. For the latter, we provide a sufficient stability condition for receding-horizon control before we illustrate our findings with an example.

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Willems’ Fundamental Lemma for Linear Descriptor Systems and Its Use for Data-Driven Output-Feedback MPC

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.

—We consider the design of state feedback control laws for both the switching signal and the continuous input of an unknown switched linear system, given past noisy input-state trajectories measurements. Based on Lyapunov-Metzler inequalities, we derive data-dependent bilinear programs whose solution directly returns a prov- ably stabilizing controller and ensures H 2 or H ∞ performance. We further present relaxations that considerably reduce the computational cost, still without requiring stabilizability of any of the switching modes. Finally, we show- case the ﬂexibility of our approach on the constrained stabilization problem for a perturbed linear system.

Data-driven stabilization of switched and constrained linear systems

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 this paper, we present a data-driven controller design method for continuous-time nonlinear systems with rational system dynamics, using no model knowledge but only measured data affected by noise. We first extend recent results on data-driven control for linear time-invariant systems by presenting a purely data-driven representation 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 guarantees for all continuous-time systems with rational system dynamics which are consistent with the measured data and the assumed noise bound. Finally, we apply the developed approach to a numerical example.

Data-Driven Stabilization of Nonlinear Systems with Rational Dynamics.

Data-driven analysis and control of dynamical systems have gained a lot of interest in recent years. While the class of linear systems is well studied, theoretical results for nonlinear systems are still rare. In this paper, we present a data-driven controller design method for discrete-time control-affine nonlinear systems. Our approach relies on the Koopman operator, which is a linear but infinite-dimensional operator lifting the nonlinear system to a higher-dimensional space. Particularly, we derive a linear fractional representation of a lifted bilinear system representation based on measured data. Further, we restrict the lifting to finite dimensions, but account for the truncation error using a finite-gain argument. We derive a linear matrix inequality based design procedure to guarantee robust local stability for the resulting bilinear system for all error terms satisfying the finite-gain bound and, thus, also for the underlying nonlinear system. Finally, we apply the developed design method to the nonlinear Van der Pol oscillator.

Robust data-driven control for nonlinear systems using the Koopman operator

The security of public-key cryptosystems relies on computationally hard problems, that are classically analyzed by number theoretic methods. In this paper, we introduce a new perspective on cryptosystems by interpreting the Diffie-Hellman key exchange as a nonlinear dynamical system. Employing Koopman theory, we transfer this dynamical system into a higher-dimensional space to analytically derive a purely linear system that equivalently describes the underlying cryptosystem. In this form, analytic tools for linear systems allow us to reconstruct the secret integers of the key exchange by simple manipulations. Moreover, we provide an upper bound on the minimal required lifting dimension to obtain perfect accuracy. To demonstrate the potential of our method, we relate our findings to existing results on algorithmic complexity. Finally, we transfer this approach to a data-driven setting where the Koopman representation is learned from data samples of the cryptosystem.

Koopman interpretation and analysis of a public-key cryptosystem: Diffie-Hellman key exchange

In this paper, we present a state-feedback controller design method for bilinear systems. To this end, we write the bilinear system as a linear fractional representation by interpreting the state in the bilinearity as a structured uncertainty. Based on that, we derive convex conditions in terms of linear matrix inequalities for the controller design, which are efficiently solvable by semidefinite programming. Further, we prove asymptotic stability and quadratic performance of the resulting closed-loop system locally in a predefined region. The proposed design uses gain-scheduling techniques and results in a state feedback with rational dependence on the state, which can substantially reduce conservatism and improve performance in comparison to a simpler, linear state feedback. Moreover, the design method is easily adaptable to various scenarios due to its modular formulation in the robust control framework. Finally, we apply the developed approaches to numerical examples and illustrate the benefits of the approach.

Robin Strässer

Papers

Data-Driven Control of Nonlinear Systems: Beyond Polynomial Dynamics

Data-Driven Stabilization of Nonlinear Systems with Rational Dynamics.

Robust data-driven control for nonlinear systems using the Koopman operator

Koopman interpretation and analysis of a public-key cryptosystem: Diffie-Hellman key exchange

Control of bilinear systems using gain-scheduling: Stability and performance guarantees