Robust control of uncertain systems: H2/H∞ control and computation of invariant sets

This thesis is mainly concerned with robust analysis and control synthesis of linear time-invariant systems with polytopic uncertainties. This topic has received considerable attention during the past decades since it offers the possibility to analyze and design controllers to cope with uncertainties. The most common and simplest approach to establish convex optimization procedures for robust analysis and synthesis problems is based on quadratic stability results, which use a single (parameter-independent) Lyapunov function for the entire uncertainty polytope. In recent years, many researchers have used parameter-dependent Lyapunov functions to provide less conservative results than the quadratic stability condition by working with parameterized Linear Matrix Inequalities (LMIs), where auxiliary scalar parameters are introduced. However, treating the scalar parameters as optimization variables leads to large computational complexity since the scalar parameters belong to an unbounded domain in general. To address this problem, we propose three distinct iterative procedures for H2 and H∞state feedback control, which are all based on true LMIs (without any scalar parameter). The first and second procedures are proposed for continuous-time and discrete-time uncertain systems, respectively. In particular, quadratic stability results can be used as a starting point for these two iterative procedures. This property ensures that the solutions obtained by our iterative procedures with one step update are no more conservative than the quadratic stability results. It is important to emphasize that, to date, for continuous-time systems, all existing methods have to introduce extra scalar parameters into their conditions in order to include the quadratic stability conditions as a special case, while our proposed iterative procedure solves a convex/LMI problem at each update. The third approach deals with the design of robust controllers for both continuous-time and discrete-time cases. It is proved that the proposed conditions contain the many existing conditions as special cases. Therefore, the third iterative procedure can compute a solution, in one step, which is at least as good as the optimal solution obtained using existing methods. All three iterative procedures can compute a sequence of non-increasing upper bounds for H2-norm and H∞-norm. In addition, if no feasible initial solution for the iterative procedures is found for some uncertain systems, we also propose two algorithms based on iterative procedures that offer the possibility of obtaining a feasible initial solution for continuous-time and discrete-time systems, respectively. Furthermore, to address the problem of analysis of H∞-norm guaranteed cost computation, a generalized problem is firstly proposed that includes both the continuous-time and discrete-time problems as special cases. A novel description of polytopic uncertainties is then derived and used to develop a relaxation approach based on the S-procedure to lift the uncertainties, which yields an LMI approach to compute H∞-norm guaranteed cost by incorporating slack variables. In this thesis, one of the main contributions is to develop convex iterative procedures for the original non-convex H2 and H∞ synthesis problems based on the novel separation result. Nonlinear and non-convex problems are general in nature and occur in other control problems; for example, the computation of tightened invariant tubes for output feedback Model Predictive Control (MPC). We consider discrete-time linear time-invariant systems with bounded state and input constraints and subject to bounded disturbances. In contrast to existing approaches which either use pre-defined control and observer gains or optimize the volume of the invariant sets for the estimation and control errors separately, we consider the problem of optimizing the volume of these two sets simultaneously to give a less conservative design.Open Acces

<i>H</i>₂ and mixed <i>H</i>₂/<i>H</i>_∞ Stabilization and Disturbance Attenuation for Differential Linear Repetitive Processes

Repetitive processes are a distinct class of two-dimensional systems (i.e., information propagation in two independent directions) of both systems theoretic and applications interest. A systems theory for them cannot be obtained by direct extension of existing techniques from standard (termed 1-D here) or, in many cases, two-dimensional (2-D) systems theory. Here, we give new results towards the development of such a theory in H2 and mixed H2/H∞ settings. These results are for the sub-class of so-called differential linear repetitive processes and focus on the fundamental problems of stabilization and disturbance attenuation

POLYNOMIAL STATIC OUTPUT FEEDBACK H ∞ CONTROL FOR CONTINUOUS-TIME LINEAR SYSTEMS VIA DESCRIPTOR APPROACH

International audienceThis paper deals with the problem of the robust static output feedback H ∞ control (SOFC) for continuous linear systems with polytopic uncertainties. The controller has been gotten by the use of descriptor redundancy. Under this approach a sufficient condition is provided for the existence of a solution to the problem. Thus, the advantage of this method is to obtain more free matrices in the design condition, also the polynomial approach helps to have a less conservative result. In the end, the performance of the method is shown by several examples

Robust H∞ Control of Takagi–Sugeno Systems with Actuator Saturation

Producción CientíficaThe robust static output feedback control for continuous-time Takagi–Sugeno systems subject to actuator saturation is solved here, including H∞ performance guarantees. Based on a polytopic model of the saturation, sufficient conditions are proposed for designing these controllers in terms of Linear Matrix Inequalities. With the aid of some special derivations, bilinear matrix inequalities are converted into a set of linear matrix inequalities which can be solved easily without requiring iterative algorithms or equality constraints, moreover, the output matrix of the considered system does not require to be full row rank. Finally, some examples are presented to show the validity of the proposed methodology

Static output-feedback stabilization of discrete-time Markovian jump linear systems: a system augmentation approach

This paper studies the static output-feedback (SOF) stabilization problem for discrete-time Markovian jump systems from a novel perspective. The closed-loop system is represented in a system augmentation form, in which input and gain-output matrices are separated. By virtue of the system augmentation, a novel necessary and sufficient condition for the existence of desired controllers is established in terms of a set of nonlinear matrix inequalities, which possess a monotonic structure for a linearized computation, and a convergent iteration algorithm is given to solve such inequalities. In addition, a special property of the feasible solutions enables one to further improve the solvability via a simple D-K type optimization on the initial values. An extension to mode-independent SOF stabilization is provided as well. Compared with some existing approaches to SOF synthesis, the proposed one has several advantages that make it specific for Markovian jump systems. The effectiveness and merit of the theoretical results are shown through some numerical example

Asymptotically exact stabilisation for constrained discrete Takagi-Sugeno systems via set-invariance

[EN] Given a Takagi-Sugeno (TS) system, this paper proposes a novel methodology to obtain the state feedback controller guaranteeing, asymptotically as a Polya-related complexity parameter grows, the largest (membership-shape independent) possible domain-of-attraction with contraction-rate performance lambda, based on polyhedral lambda-contractive sets from constrained linear systems literature. The resulting controller is valid for any realisation of the memberships, as usual in most TS literature. For a finite complexity parameter, an inner estimate of such largest set is obtained; the frontier of such approximation can be understood as the level set of a polyhedral control-Lyapunov function. Convergence of a proposed iterative algorithm is asymptotically necessary and sufficient for TS system stabilisation: for a high-enough value of the complexity parameter, any conceivable shape-independent Lyapunov controller design procedure will yield a proven domain of attraction smaller or equal to the algorithm's output. (C) 2016 Elsevier B.V. All rights reserved.This work has been supported by grants DPI2015-70433- P and DPI2016-81002-R, from Spanish Government (MINECO) and grant PROMETEOII/2013/004 from Generalitat Valenciana.Ariño-Latorre, CV.; Sala, A.; Pérez Soler, E.; Bedate Boluda, F.; Querol-Ferrer, A. (2017). Asymptotically exact stabilisation for constrained discrete Takagi-Sugeno systems via set-invariance. Fuzzy Sets and Systems. 316:117-138. https://doi.org/10.1016/j.fss.2016.10.004S11713831