A complete analysis and design framework for linear impulsive and related hybrid systems

We establish a framework for systematically analyzing and designing output-feedback controllers for linear impulsive and related hybrid systems that might even be affected by various types of uncertainties. In particular, the framework encompasses uncertain switched and sampled-data systems as well as networked systems with switching communication topologies. The framework is based on recently developed convex criteria involving a so-called clock for analyzing impulsive systems under dwell-time constraints. We elaborate on the extension of those criteria for dynamic output-feedback controller synthesis by means of convex optimization and generalize the so-called dual iteration to impulsive systems. The latter originally and still constitutes a promising heuristic procedure for the challenging and non-convex design of static output-feedback controllers for standard linear time-invariant systems. Moreover, for uncertain impulsive systems as modeled in terms of linear fractional representations, we generalize the nominal analysis criteria by providing novel robust analysis conditions based on a novel time-domain and clock-dependent formulation of integral quadratic constraints. Finally, by combining the insights on nominal synthesis and robust analysis, we are able to tackle challenging output-feedback designs of practical relevance, such as the design of gain-scheduled, robust or robust gain-scheduled controllers for impulsive systems. Most of the obtained analysis and synthesis conditions involve infinite-dimensional (differential) linear matrix inequalities which can be numerically solved by using relaxation methods based on, e.g., linear splines, B-splines or matrix sum-of-squares that we discuss as well

Output-Feedback Synthesis for a Class of Aperiodic Impulsive Systems

We derive novel criteria for designing stabilizing dynamic output-feedback controllers for a class of aperiodic impulsive systems subject to a range dwell-time condition. Our synthesis conditions are formulated as clock-dependent linear matrix inequalities (LMIs) which can be solved numerically, e.g., by using matrix sum-of-squares relaxation methods. We show that our results allow us to design dynamic output-feedback controllers for aperiodic sample-data systems and illustrate the proposed approach by means of a numerical example

Input-Output-Data-Enhanced Robust Analysis via Lifting

Starting from a linear fractional representation of a linear system affected by constant parametric uncertainties, we demonstrate how to enhance standard robust analysis tests by taking available (noisy) input-output data of the uncertain system into account. Our approach relies on a lifting of the system and on the construction of data-dependent multipliers. It leads to a test in terms of linear matrix inequalities which guarantees stability and performance for all systems compatible with the observed data if it is in the affirmative. In contrast to many other data-based approaches, prior physical knowledge is included at the outset due to the underlying linear fractional representation

Optimization Algorithm Synthesis based on Integral Quadratic Constraints: A Tutorial

We expose in a tutorial fashion the mechanisms which underly the synthesis of optimization algorithms based on dynamic integral quadratic constraints. We reveal how these tools from robust control allow to design accelerated gradient descent algorithms with optimal guaranteed convergence rates by solving small-sized convex semi-definite programs. It is shown that this extends to the design of extremum controllers, with the goal to regulate the output of a general linear closed-loop system to the minimum of an objective function. Numerical experiments illustrate that we can not only recover gradient decent and the triple momentum variant of Nesterov's accelerated first order algorithm, but also automatically synthesize optimal algorithms even if the gradient information is passed through non-trivial dynamics, such as time-delays.Comment: A short version of this paper has been submitted to the CDC 202

Controller Design via Experimental Exploration with Robustness Guarantees

For a partially unknown linear systems, we present a systematic control design approach based on generated data from measurements of closed-loop experiments with suitable test controllers. These experiments are used to improve the achieved performance and to reduce the uncertainty about the unknown parts of the system. This is achieved through a parametrization of auspicious controllers with convex relaxation techniques from robust control, which guarantees that their implementation on the unknown plant is safe. This approach permits to systematically incorporate available prior knowledge about the system by employing the framework of linear fractional representations

Revisiting and generalizing the dual iteration for static and robust output‐feedback synthesis

The dual iteration was introduced in a conference paper in 1997 by Iwasaki as an iterative and heuristic procedure for the challenging and non‐convex design of static output‐feedback controllers. We recall in detail its essential ingredients and go beyond the work of Iwasaki by demonstrating that the framework of linear fractional representations allows for a seamless extension of the dual iteration to output‐feedback designs of practical relevance, such as the design of robust or robust gain‐scheduled controllers. In the paper of Iwasaki, the dual iteration is solely based on, and motivated by algebraic manipulations resulting from the elimination lemma. We provide a novel control theoretic interpretation of the individual steps, which paves the way for further generalizations of the powerful scheme to situations where the elimination lemma is not applicable. As an illustration, we extend the dual iteration to a design of static output‐feedback controllers with multiple objectives. We demonstrate the approach with numerous numerical examples inspired from the literature.Deutsche Forschungsgemeinschaf

A structure exploiting SDP solver for robust controller synthesis

In this paper, we revisit structure exploiting SDP solvers dedicated to the solution of Kalman-Yakubovic-Popov semi-definite programs (KYP-SDPs). These SDPs inherit their name from the KYP Lemma and they play a crucial role in e.g. robustness analysis, robust state feedback synthesis, and robust estimator synthesis for uncertain dynamical systems. Off-the-shelve SDP solvers require $O(n^6)$ arithmetic operations per Newton step to solve this class of problems, where $n$ is the state dimension of the dynamical system under consideration. Specialized solvers reduce this complexity to $O(n^3)$. However, existing specialized solvers do not include semi-definite constraints on the Lyapunov matrix, which is necessary for controller synthesis. In this paper, we show how to include such constraints in structure exploiting KYP-SDP solvers.Comment: Submitted to Conference on Decision and Control, copyright owned by iee

The Non-Strict Projection Lemma

The projection lemma (often also referred to as the elimination lemma) is one of the most powerful and useful tools in the context of linear matrix inequalities for system analysis and control. In its traditional formulation, the projection lemma only applies to strict inequalities, however, in many applications we naturally encounter non-strict inequalities. As such, we present, in this note, a non-strict projection lemma that generalizes both its original strict formulation as well as an earlier non-strict version. We demonstrate several applications of our result in robust linear-matrix-inequality-based marginal stability analysis and stabilization, a matrix S-lemma, which is useful in (direct) data-driven control applications, and matrix dilation

A Dynamic S-Procedure for Dynamic Uncertainties

We show how to compose robust stability tests for uncertain systems modeled as linear fractional representations and affected by various types of dynamic uncertainties. Our results are formulated in terms of linear matrix inequalities and rest on the recently established notion of finite-horizon integral quadratic constraints with a terminal cost. The construction of such constraints is motivated by an unconventional application of the S-procedure in the frequency domain with dynamic Lagrange multipliers. Our technical contribution reveals how this construction can be performed by dissipativity arguments in the time-domain and in a lossless fashion. This opens the way for generalizations to time-varying or hybrid systems