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

H 2 And H ∞ Filtering Design Subject To Implementation Uncertainty

This paper presents new filtering design procedures for discrete-time linear systems. It provides a solution to the problem of linear filtering design, assuming that the filter is subject to parametric uncertainty. The problem is relevant, since the proposed filter design incorporates real world implementation constraints that are always present in practice. The transfer function and the state space realization of the filter are simultaneously computed. The design procedure can also handle plant parametric uncertainty. In this case, the plant parameters are assumed not to be exactly known but belonging to a given convex and closed polyhedron. Robust performance is measured by the H 2 and H ∞ norms of the transfer function from the noisy input to the filtering error. The results are based on the determination of an upper bound on the performance objectives. All optimization problems are linear with constraint sets given in the form of LMI (linear matrix inequalities). Global optimal solutions to these problems can be readily computed. Numerical examples illustrate the theory. © 2005 Society for Industrial and Applied Mathematics.442515530Gevers, M., Li, G., (1993) Parametrizations in Control, Estimation and Filtering Problems, , Springer-Verlag, LondonWilliamson, D., Finite wordlength design of digital Kalman filters for state estimation (1985) IEEE Trans. Automat. Control, 30, pp. 930-939Williamson, D., Kadiman, K., Optimal finite wordlength linear quadratic regulators (1989) IEEE Trans. Automat. Control, 34, pp. 1218-1228Liu, K., Skelton, R.E., Grigoriadis, K., Optimal controllers for finite wordlength implementation (1992) IEEE Trans. Automat. Control, 37, pp. 1294-1304Hwang, S.Y., Minimum uncorrelated unit noise in state-space digital filtering (1977) IEEE Trans. Acoustics Speech Signal Process, 25, pp. 273-281Amit, G., Shaked, U., Minimization of roundoff errors in digital realizations of Kalman filters (1989) IEEE Trans. Acoustics Speech Signal Process, 37, pp. 1980-1982De Oliveira, M.C., Skelton, R.E., Synthesis of controllers with finite precision considerations (2001) Digital Controller Implementation and Fragility: A Modern Perspective, pp. 229-251. , R. S. H. Istepanian and J. F. Whidborne eds., Springer-Verlag, New YorkKeel, L.H., Bhattacharyya, S.P., Robust, fragile or optimal (1997) IEEE Trans. Automat. Control, 42, pp. 1098-1105Keel, L.H., Bhattacharyya, S.P., Authors' reply to: "Comments on 'Robust, fragile or optimal' " by P. M. Mäkilä (1998) IEEE Trans. Automat. Control, 43, p. 1268Dorato, P., Non-fragile controller design: An overview (1998) Proceedings of the 1998 American Control Conference, 5, pp. 2829-2831. , Philadelphia, IEEE, Piscataway, NJFamularo, D., Dorato, P., Abdallah, C.T., Haddad, W.H., Jadbabaie, A., Robust non-fragile LQ controllers: The static state feedback case (2000) Internat. J. Control, 73, pp. 159-165Yang, G.H., Wang, J.L., Robust nonfragile Kalman filtering for uncertain linear systems with estimator gain uncertainty (2001) IEEE Trans. Automat. Control, 46, pp. 343-348Haddad, W.M., Corrado, J.R., Robust resilient dynamic controllers for systems with parametric uncertainty and controller gain variations (2000) Internat. J. Control, 73, pp. 1405-1423Keel, L.H., Bhattacharyya, S.P., Stability margins and digital implementation of controllers (1998) Proceedings of the 1998 American Control Conference, 5, pp. 2852-2856. , (Philadelphia), IEEE, Piscataway, NJGeromel, J.C., Optimal linear filtering under parameter uncertainty (1999) IEEE Trans. Signal Process, 47, pp. 168-175Nesterov, Y., Nemirovskii, A., (1994) Interior-Point Polynomial Algorithms in Convex Programming, , SIAM, PhiladelphiaGeromel, J.C., Bernussou, J., Garcia, G., De Oliveira, M.C., H 2 and H ∞ robust filtering for discrete-time linear systems (2000) SIAM J. Control Optim., 38, pp. 1353-1368Geromel, J.C., De Oliveira, M.C., Bernussou, J., Robust filtering of discrete-time linear systems with parameter dependent Lyapunov functions (2002) SIAM J. Control Optim., 41, pp. 700-711De Oliveira, M.C., Bernussou, J., Geromel, J.C., A new discrete-time robust stability condition (1999) Systems Control Lett., 37, pp. 261-265Sayed, A.H., A framework for state-space estimation with uncertain models (2001) IEEE Trans. Automat. Control, 46, pp. 998-1013Balakrishnan, V., Huang, Y., Packard, A., Doyle, J.C., Linear matrix inequalities in analysis with multipliers (1994) Proceedings of the 1994 American Control Conference, 2, pp. 1228-1232. , Baltimore, MD, IEEE, Piscataway, NJGeromel, J.C., Peres, P.L.D., Bernussou, J., On a convex parameter space method for linear control design of uncertain systems (1991) SIAM J. Control Optim., 29, pp. 381-40

Recent advances on recursive filtering and sliding mode design for networked nonlinear stochastic systems: A survey

Copyright © 2013 Jun Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Some recent advances on the recursive filtering and sliding mode design problems for nonlinear stochastic systems with network-induced phenomena are surveyed. The network-induced phenomena under consideration mainly include missing measurements, fading measurements, signal quantization, probabilistic sensor delays, sensor saturations, randomly occurring nonlinearities, and randomly occurring uncertainties. With respect to these network-induced phenomena, the developments on filtering and sliding mode design problems are systematically reviewed. In particular, concerning the network-induced phenomena, some recent results on the recursive filtering for time-varying nonlinear stochastic systems and sliding mode design for time-invariant nonlinear stochastic systems are given, respectively. Finally, conclusions are proposed and some potential future research works are pointed out.This work was supported in part by the National Natural Science Foundation of China under Grant nos. 61134009, 61329301, 61333012, 61374127 and 11301118, the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grant no. GR/S27658/01, the Royal Society of the UK, and the Alexander von Humboldt Foundation of Germany

Robust H-infinity finite-horizon control for a class of stochastic nonlinear time-varying systems subject to sensor and actuator saturations

Copyright [2010] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.This technical note addresses the robust H∞ finite-horizon output feedback control problem for a class of uncertain discrete stochastic nonlinear time-varying systems with both sensor and actuator saturations. In the system under investigation, all the system parameters are allowed to be time-varying, the parameter uncertainties are assumed to be of the polytopic type, and the stochastic nonlinearities are described by statistical means which can cover several classes of well-studied nonlinearities. The purpose of the problem addressed is to design an output feedback controller, over a given finite-horizon, such that the H∞ disturbance attenuation level is guaranteed for the nonlinear stochastic polytopic system in the presence of saturated sensor and actuator outputs. Sufficient conditions are first established for the robust H∞ performance through intensive stochastic analysis, and then a recursive linear matrix inequality (RLMI) approach is employed to design the desired output feedback controller achieving the prescribed H∞ disturbance rejection level. Simulation results demonstrate the effectiveness of the developed controller design scheme.This work was supported under Australian Research Council’s Discovery Projects funding scheme (project DP0880494) and by the German Science Foundation (DFG) within the priority programme 1305: Control Theory of Digitally Networked Dynamical Systems. Recommended by Associate Editor H. Ito

A survey on gain-scheduled control and filtering for parameter-varying systems

Copyright © 2014 Guoliang Wei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.This paper presents an overview of the recent developments in the gain-scheduled control and filtering problems for the parameter-varying systems. First of all, we recall several important algorithms suitable for gain-scheduling method including gain-scheduled proportional-integral derivative (PID) control, H 2, H ∞ and mixed H 2 / H ∞ gain-scheduling methods as well as fuzzy gain-scheduling techniques. Secondly, various important parameter-varying system models are reviewed, for which gain-scheduled control and filtering issues are usually dealt with. In particular, in view of the randomly occurring phenomena with time-varying probability distributions, some results of our recent work based on the probability-dependent gain-scheduling methods are reviewed. Furthermore, some latest progress in this area is discussed. Finally, conclusions are drawn and several potential future research directions are outlined.The National Natural Science Foundation of China under Grants 61074016, 61374039, 61304010, and 61329301; the Natural Science Foundation of Jiangsu Province of China under Grant BK20130766; the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning; the Program for New Century Excellent Talents in University under Grant NCET-11-1051, the Leverhulme Trust of the U.K., the Alexander von Humboldt Foundation of Germany

A review on analysis and synthesis of nonlinear stochastic systems with randomly occurring incomplete information

Copyright q 2012 Hongli Dong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.In the context of systems and control, incomplete information refers to a dynamical system in which knowledge about the system states is limited due to the difficulties in modeling complexity in a quantitative way. The well-known types of incomplete information include parameter uncertainties and norm-bounded nonlinearities. Recently, in response to the development of network technologies, the phenomenon of randomly occurring incomplete information has become more and more prevalent. Such a phenomenon typically appears in a networked environment. Examples include, but are not limited to, randomly occurring uncertainties, randomly occurring nonlinearities, randomly occurring saturation, randomly missing measurements and randomly occurring quantization. Randomly occurring incomplete information, if not properly handled, would seriously deteriorate the performance of a control system. In this paper, we aim to survey some recent advances on the analysis and synthesis problems for nonlinear stochastic systems with randomly occurring incomplete information. The developments of the filtering, control and fault detection problems are systematically reviewed. Latest results on analysis and synthesis of nonlinear stochastic systems are discussed in great detail. In addition, various distributed filtering technologies over sensor networks are highlighted. Finally, some concluding remarks are given and some possible future research directions are pointed out. © 2012 Hongli Dong et al.This work was supported in part by the National Natural Science Foundation of China under Grants 61273156, 61134009, 61273201, 61021002, and 61004067, the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grant GR/S27658/01, the Royal Society of the UK, the National Science Foundation of the USA under Grant No. HRD-1137732, and the Alexander von Humboldt Foundation of German

Robust variance-constrained H∞ control for stochastic systems with multiplicative noises

This is the post print version of the article. The official published version can be obtained from the link below - Copyright 2007 Elsevier Ltd.In this paper, the robust variance-constrained H∞ control problem is considered for uncertain stochastic systems with multiplicative noises. The norm-bounded parametric uncertainties enter into both the system and output matrices. The purpose of the problem is to design a state feedback controller such that, for all admissible parameter uncertainties, (1) the closed-loop system is exponentially mean-square quadratically stable; (2) the individual steady-state variance satisfies given upper bound constraints; and (3) the prescribed noise attenuation level is guaranteed in an H∞ sense with respect to the additive noise disturbances. A general framework is established to solve the addressed multiobjective problem by using a linear matrix inequality (LMI) approach, where the required stability, the H∞ characterization and variance constraints are all easily enforced. Within such a framework, two additional optimization problems are formulated: one is to optimize the H∞ performance, and the other is to minimize the weighted sum of the system state variances. A numerical example is provided to illustrate the effectiveness of the proposed design algorithm.This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grant GR/S27658/01, the Nuffield Foundation of the UK under Grant NAL/00630/G, and the Alexander von Humboldt Foundation of Germany

Robust constrained model predictive control based on parameter-dependent Lyapunov functions

The problem of robust constrained model predictive control (MPC) of systems with polytopic uncertainties is considered in this paper. New sufficient conditions for the existence of parameter-dependent Lyapunov functions are proposed in terms of linear matrix inequalities (LMIs), which will reduce the conservativeness resulting from using a single Lyapunov function. At each sampling instant, the corresponding parameter-dependent Lyapunov function is an upper bound for a worst-case objective function, which can be minimized using the LMI convex optimization approach. Based on the solution of optimization at each sampling instant, the corresponding state feedback controller is designed, which can guarantee that the resulting closed-loop system is robustly asymptotically stable. In addition, the feedback controller will meet the specifications for systems with input or output constraints, for all admissible time-varying parameter uncertainties. Numerical examples are presented to demonstrate the effectiveness of the proposed techniques

Robust H∞ filtering for a class of nonlinear networked systems with multiple stochastic communication delays and packet dropouts

Copyright [2010] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.In this paper, the robust H∞ filtering problem is studied for a class of uncertain nonlinear networked systems with both multiple stochastic time-varying communication delays and multiple packet dropouts. A sequence of random variables, all of which are mutually independent but obey Bernoulli distribution, are introduced to account for the randomly occurred communication delays. The packet dropout phenomenon occurs in a random way and the occurrence probability for each sensor is governed by an individual random variable satisfying a certain probabilistic distribution in the interval. The discrete-time system under consideration is also subject to parameter uncertainties, state-dependent stochastic disturbances and sector-bounded nonlinearities. We aim to design a linear full-order filter such that the estimation error converges to zero exponentially in the mean square while the disturbance rejection attenuation is constrained to a give level by means of the H∞ performance index. Intensive stochastic analysis is carried out to obtain sufficient conditions for ensuring the exponential stability as well as prescribed H∞ performance for the overall filtering error dynamics, in the presence of random delays, random dropouts, nonlinearities, and the parameter uncertainties. These conditions are characterized in terms of the feasibility of a set of linear matrix inequalities (LMIs), and then the explicit expression is given for the desired filter parameters. Simulation results are employed to demonstrate the effectiveness of the proposed filter design technique in this paper.This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) of the U.K. under Grant GR/S27658/01, the Royal Society of the U.K., the Alexander von Humboldt Foundation of Germany, National Natural Science Foundation of China under Grant 60825303, 60834003, 973 Project under Grant 2009CB320600, Fok Ying Tung Education Foundation under Grant 111064, and the Youth Science Fund of Heilongjiang Province under Grant QC2009C63