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

Robust H-infinity filtering for 2-D systems with intermittent measurements

This paper is concerned with the problem of robust H∞ filtering for uncertain two-dimensional (2-D) systems with intermittent measurements. The parameter uncertainty is assumed to be of polytopic type, and the measurements transmission is assumed to be imperfect, which is modeled by a stochastic variable satisfying the Bernoulli random binary distribution. Our attention is focused on the design of an H∞ filter such that the filtering error system is stochastically stable and preserves a guaranteed H∞ performance. This problem is solved in the parameter-dependent framework, which is much less conservative than the quadratic approach. By introducing some slack matrix variables, the coupling between the positive definite matrices and the system matrices is eliminated, which greatly facilitates the filter design procedure. The corresponding results are established in terms of linear matrix inequalities, which can be easily tested by using standard numerical software. An example is provided to show the effectiveness of the proposed approac

H∞ State Feedback Switched Control For Discrete Time-varying Polytopic Systems

This article treats, in a new framework, a classical design problem, namely H∞ state feedback switched control for discrete time-varying polytopic systems. The main goal is to jointly design a set of state feedback gains and a switching rule in order to assure a pre-specified H ∞ guaranteed level. The main contribution is to generalise switched system results to design a switched controller that does not require online measurements of the time-varying parameters. This property has important consequences mainly because the performance quality is preserved even in the lack of parameter information. Our conditions are based on modified Riccati-Metzler inequalities and are described in terms of linear matrix inequalities. An academical example compares the proposed technique with others from the literature. © 2013 Taylor and Francis Group, LLC.864591598Amato, F., Mattei, M., Pironti, A., Gain scheduled control for discrete-time systems depending on bounded rate parameters (2005) International Journal of Robust and Nonlinear Control, 15, pp. 473-494. , doi: 10.1002/rnc.1001Apkarian, P., Gahinet, P., A convex characterisation of gain-scheduled H ∞ controllers (1995) IEEE Transactions on Automatic Control, 40, pp. 853-864. , doi: 10.1109/9.384219Blanchini, F., Miani, S., Stabilization of lpv systems: State feedback, state estimation, and duality (2003) SIAM Journal of Control Optimization, 42, pp. 76-97. , doi: 10.1137/S0363012900372283Blanchini, F., Miani, S., Savorgnan, C., Stability results for linear parameter varying and switching systems (2007) Automatica, 43, pp. 1817-1823. , doi: 10.1016/j.automatica.2007.03.002Daafouz, J., Bernussou, J., Parameter dependent lyapunov functions for discrete time systems with time varying parametric uncertainties (2001) Systems & Control Letters, 43, pp. 355-359. , doi: 10.1016/S0167-6911(01)00118-9Daafouz, J., Bernussou, J., (2001) Poly-quadratic Stability and H ∞ Performance for Discrete Systems with Time Varying Uncertainties, pp. 267-272. , Proceedings of the 40th IEEE Conference on Decision and Control. 2001b. pp. FloridaDaafouz, J., Bernussou, J., Geromel, J.C., On inexact lpv control design of continuous-time polytopic systems (2008) IEEE Transactions on Automatic Control, 53, pp. 1674-1678. , doi: 10.1109/TAC.2008.928119Deaecto, G.S., Geromel, J.C., H ∞ control for continuous-time switched linear systems (2010) ASME Journal of Dynamic Systems, Measurement, and Control, 132. , 041013, 1-7 doi: 10.1115/1.4001711Deaecto, G.S., Geromel, J.C., Daafouz, J., Dynamic output feedback H ∞ control of switched linear systems (2011) Automatica, 47, pp. 1713-1720. , doi: 10.1016/j.automatica.2011.02.046Deaecto, G.S., Geromel, J.C., Daafouz, J., Switched state-feedback control for continuous time-varying polytopic systems (2011) International Journal of Control, 84, pp. 1500-1508. , doi: 10.1080/00207179.2011.608134Decarlo, R.A., Branicky, M.S., Pettersson, S., Lennartson, B., Perpectives and results on the stability and stabilisability of hybrid systems (2000) Proceedings of the IEEE, 88, pp. 1069-1082. , doi: 10.1109/5.871309Garone, E., Casavola, A.C., Franz, G., Famularo, D., (2007) New Stabilisability Conditions for Discrete-time Linear Parameter Varying Systems, pp. 2755-2760. , Proceedings of the 46th IEEE Conference on Decision and Control. 2007. pp. New OrleansGeromel, J.C., Colaneri, P., Stability and stabilisation of discrete time switched systems (2006) International Journal of Control, 79, pp. 719-728. , doi: 10.1080/00207170600645974Geromel, J.C., Colaneri, P., Bolzern, P., Dynamic output feedback control of switched linear systems (2008) IEEE Transations on Automatic Control, 53, pp. 720-733. , doi: 10.1109/TAC.2008.919860Golub, G.H., Van Loan, C.F., (1996) Matrix Computations, , Baltimore, MD: Johns Hopkins University PressHeemels, W., Daafouz, J., Millerioux, G., Observer-based control of discrete-time lpv systems with uncertain parameters (2010) IEEE Transactions on Automatic Control, 55, pp. 2130-2135. , doi: 10.1109/TAC.2010.2051072Ji, Z., Wang, L., Xie, G., Quadratic stabilisation of uncertain discrete-time switched systems via output feedback (2005) Circuits Systems, and Signal Processing, 24, pp. 733-751. , doi: 10.1007/s00034-005-0920-2Leith, D.J., Leithead, W.E., Survey of gain-scheduling analysis and design (2000) International Journal of Control, 73, pp. 1001-1025. , doi: 10.1080/002071700411304Liberzon, D., (2003) Switching in Systems and Control, , Boston, MA: BirkhäuserLiberzon, D., Morse, A.S., Basic problems in stability and design of switched systems (1999) IEEE Control Systems Magazine, 19, pp. 59-70. , doi: 10.1109/37.793443Lin, H., Antsaklis, P.J., Hybrid state feedback stabilzation with l 2 performance for discrete-time switched linear systems (2008) International Journal of Control, 81, pp. 1114-1124. , doi: 10.1080/00207170701654354Montagner, V.F., Oliveira, R., Leite, V.J.S., Peres, P.L.D., (2005) Gain Scheduled State Feedback Control of Discrete-time Systems with Time-varying Uncertainties: An LMI Approach, pp. 4305-4310. , Proceedings of the 44th IEEE Conference on Decision and Control. 2005. pp. SevilleRugh, W.J., Shamma, J.S., Research on gain scheduling (2000) Automatica, 36, pp. 1401-1425. , doi: 10.1016/S0005-1098(00)00058-3Skafidas, E., Evans, R.J., Savkin, A.V., Petersen, I.R., Stability results for switched controller systems (1999) Automatica, 35, pp. 553-564. , doi: 10.1016/S0005-1098(98)00167-8Sun, Z., Ge, S.S., (2005) Switched Linear Systems: Control and Design, , London: SpringerVarga, R.S., (2000) Matrix Iterative Analysis, , 2nd, Heidelberg: Springer-VerlagXie, W., H 2 gain scheduled state feedback for lpv system with new lmi formulation (2005) IEE Proceedings - Control Theory and Applications, 152, pp. 693-697. , doi: 10.1049/ip-cta:20050052Xie, W., New lmi-based conditions for quadratic stabilisation of lpv systems (2008) Journal of Inequalities and Applications, 2008, pp. 1-12Zhai, G., Lin, H., Antsaklis, P.J., Quadratic stabilisability of linear switched systems with polytopic uncertainties (2003) International Journal of Control, 76, pp. 747-753. , doi: 10.1080/0020717031000114968Yoon, M.G., Ugrinovskii, V.A., Pszczel, M., Gain-scheduling of minimax optimal state-feedback controllers for uncertain lpv systems (2007) IEEE Transactions on Automatic Control, 52, pp. 311-317. , doi: 10.1109/TAC.2006.88790

An Upper Bound On Properly Efficient Solutions In Multiobjective Optimization

An upper bound on properly efficient solutions in multiobjective optimization is derived for the case of convex programs. The upper bound is derived through simple computations without solving any parametric problem. Further developments lead to a compact set that contains the whole set of properly efficient solutions with prescribed maximal rates of substitution. © 1991.1028386Benayoun, de MontGolfier, Tergny, Laritchev, Linear programming with multiple objective functions: Step method (STEM) (1971) Math. Programming, 1, pp. 366-375Chankong, Haimes, On the characterization of non-inferior solutions of the vector optimization problem (1982) Automatica, 18, pp. 697-707Geoffrion, Proper efficiency and the theory of vector optimization (1967) J. Math. Anal. Appl., 22, pp. 618-630Hwang, Masud, Multiple objective decision making methods and applications (1979) Lecture Notes in Economics and Mathematical Systems, , Springer, Berlin, No. 43Weistroffer, Careful usage of pessimistic values is needed in multiple objectives optimization (1985) Oper. Res. Lett., 4, pp. 23-25Yu, (1985) Multiple Criteria Decision Making: Concepts, Techniques and Extension, , Plenum, New Yor

H2 Control For Discrete-time Systems Optimality And Robustness

This paper proposes a new approach to determine H2 optimal control for discrete-time linear systems, based on convex programming. It is shown that all stabilizing state feedback control gains belong to a certain convex set, well-defined in a special parameter space. The Linear Quadratic Problem can be then formulated as the minimization of a linear objective over a convex set. The optimal solution of this convex problem furnishes, under certain conditions, the same feedback control gain which is obtained from the classical discrete-time Riccati equation solution. Furthermore, the method proposed can also handle additional constraints, for instance, the ones needed to assure asymptotical stability of discrete-time systems under actuators failure. Some examples illustrate the theory. © 1992.291225228Anderson, Moore, (1971) Linear Optimal Control, , Prentice Hall, Englewood Cliffs, NJBernussou, Peres, Geromel, A linear programming oriented procedure for quadratic stabilization of uncertain systems (1989) Systems and Control Letters, 13, pp. 65-72Dorato, Levis, Optimal linear regulators the discrete time case (1971) IEEE Transactions on Automatic Control, 16, pp. 613-620Geromel, Peres, Bernussou, On a convex parameter space method for linear control design of uncertain systems (1991) SIAM J. on Control and Optimiz., 29, pp. 381-402Kwakernaak, Sivan, (1972) Linear Optimal Control Systems, , John Wiley, New YorkLuenberger, (1973) Introduction to Linear Programming, , Addison-Wesley, Reading, M

Output-feedback Stabilization Of Discrete-time Switched Systems

This paper considers the closed-loop stabilization problem for discrete-time linear switched system. The state variables are assumed to be not accessible so that the feedback strategy hinges on given output variables. The solution of this problem is based on the solution of suitable matrix inequalities for the construction of a full order switched filter and the derivation of the stabilization rule. The main theoretical basis is constituted by the so-called Lyapunov-Metzler inequalities which play a prominent role in the state-feedback stabilization of linear switched systems. The theoretical results are illustrated by means of an academic example. ©2007 IEEE.687692Boyd, S.P., El Ghaoui, L., Feron, E., Balakrishnan, V., (1994) Linear Matrix Inequalities in System and Control Theory, , SIAM, PhiladelphiaBranicky, M.S., Multiple Lyapunov functions and other analysis tools for switched and hybrid systems (1998) IEEE Trans. Automat. Contr, 43, pp. 475-482Costa, O.L.V., Fragoso, M.D., Marques, R.P., (2005) Discrete-Time Markov Jump Linear Systems Series: Probability and its Application, , SpringerDaafouz, J., Bernussou, J., Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties (2001) Systems & Control Letters, 43, pp. 355-359Daafouz, J., Riedinger, P., lung, C., Observer-based switched control design for discrete-time switched systems (2003) European Control Conference, UK. , CambridgeDeCarlo, R.A., Branicky, M.S., Pettersson, S., Lennartson, B., Perspectves and results on the stability and stabilizability of hybrid systems (2000) Proceedings of the IEEE, 88 (7), pp. 1069-1082Geromel, J.C., Colaneri, P., Stabilization of continuous-time switched systems (2005) IFAC World Congress, , PragueGeromel, J.C., Colaneri, P., Stability and stabilization of discrete-time switched systems (2006) International Journal of Control, 79, pp. 719-728Hespanha, J.P., Uniform stability of switched linear systems : Extensions of LaSalle's principle (2004) IEEE Trans. Automat. Contr, 49, pp. 470-482Hockerman-Frommer, J., Kulkarni, S.R., Ramadge, P.J., Controller switching based on output predictions errors (1998) IEEE Trans. Automat. Contr, 43, pp. 596-607Johansson, M., Rantzer, A., Computation of piecewise quadratic Lyapunov functions for hybrid systems (1998) IEEE Trans. Automat. Contr, 43, pp. 555-559Liberzon, D., Morse, A.S., Basic problems in stability and design of switched systems (1999) IEEE Control Systems Magazine, 19, pp. 59-70Liberzon, D., (2003) Switching in Systems and Control, , BirkhauserRotea, M.A., Williamson, D., Optimal Realizations of Finite Wordlength of Digital Filters and Controllers (1995) IEEE Trans. Circuits and Systems -I:Fundamentals, Theory and Applications, 42 (2), pp. 61-72Ye, H., Michel, A.N., Hou, L., Stability theory for hybrid dynamical systems (1998) IEEE Trans. Automat. Contr, 43, pp. 461-474Xie, G.M., Wang, L., Reachability realization and stabilizability of switched linear discrete-time systems (2003) Journal of Math. Analysis and Applications, 280, pp. 209-220Zhai, G., Quadratic stability of discrete-time switched systems via state and output feedback (2001) Proceedings of the 40th IEEE CDC, pp. 2165-216

Full Order Dynamic Output Feedback H∞ Control For Continuous-time Switched Linear Systems

This paper is devoted to the analysis and solution of the output feedback H∞ control design problem for continuous-time switched linear systems. More specifically, the main goal is to design a switching rule together with a full order dynamic linear controller to satisfy a pre-specified H ∞ level defined by the L2 gains from the input to the output signal. The results reported in this paper are based on the so called Lyapunov-Metzler inequalities which express a sufficient condition for switched linear systems global stability. The solution of the previously mentioned output feedback control design problem through an LMI-based method is the main contribution of the present paper. An academic example borrowed from the literature is used for illustration. ©2009 IEEE.63776382Branicky, M.S., Multiple Lyapunov functions and other analysis tools for switched and hybrid systems (1998) IEEE Trans. Automat. Contr., 43, pp. 475-482Deaecto, G.S., (2007) Control Sinthesis for Dynamic Switched Systems (in Portuguese), , Master's thesis, FEEC - UnicampDeCarlo, R.A., Branicky, M.S., Pettersson, S., Lennartson, B., Perspectives and results on the stability and stabilizability of hybrid systems (2000) Proceedings of the IEEE, 88, pp. 1069-1082Garg, K.M., (1998) Theory of Differentiation: A Unified Theory of Differentiation Via New Derivate Theorems and New Derivatives, , Wiley-InterscienceGeromel, J.C., Colaneri, P., Stability and stabilization of continuous-time switched linear systems (2006) SIAM J. Control Optim., 45, pp. 1915-1930Geromel, J.C., Colaneri, P., Bolzern, P., Dynamic output feedback control of switched linear systems (2008) IEEE Trans. on Automatic Contr., 53, pp. 720-733Geromel, J.C., Deaecto, G.S., Switched state feedback control for continuous-time uncertain systems (2009) Automatica, 45, pp. 593-597Geromel, J.C., Korogui, R.H., Bernussou, J., On H2 and H∞ robust output feedback control for continuous-time polytopic systems (2007) IET Control Theory and Applications, 1, pp. 1541-1549Hu, T., Ma, L., Lin, Z., On several composite quadratic Lyapunov functions for switched systems (2006) Proceedings of the 45th IEEE Conf. on Decision and Control, pp. 113-118. , San Diego, USAJi, Z., Wang, L., Xie, G., Quadratic stabilization of switched systems (2005) Inter. Jour. of Systems Science, 36, pp. 395-404Ji, Z., Guo, X., Wang, L., Xie, G., Robust H∞ control and stabilization of uncertain switched linear systems: A multiple Lyapunov function approach (2006) ASME Journal of Dynamic Systems, Measurement, and Control, 128, pp. 696-700Lasdon, L.S., (1970) Optimization Theory for Large Systems, , Macmillan, NYLiberzon, D., Morse, A.S., Basic problems in stability and design of switched systems (1999) IEEE Control Syst. Mag., 19, pp. 59-70Liberzon, D., (2003) Switching in Systems and Control, , Birkhäuser, BostonPeleties, P., DeCarlo, R.A., Asymptotic stability of m-switched systems using Lyapunov-like functions (1991) Proceedings American Control Conference, pp. 1679-1684. , Boston, MASavkin, A.V., Petersen, I.R., Skafidas, E., Evans, R.J., Hybrid dynamical systems: Robust control synthesis problems (1996) Systems & Control Letters, 29, pp. 81-90Savkin, A.V., Skafidas, E., Evans, R.J., Robust output feedback stabilizability via controller switching (1999) Automatica, 35, pp. 69-74Skafidas, E., Evans, R.J., Savkin, A.V., Petersen, I.R., Stability results for switched controller systems (1999) Automatica, 35, pp. 553-564Sun, Z., Ge, S.S., (2005) Switched Linear Systems: Control and Design, , Springer, LondonXie, G., Wang, L., Periodical stabilization of switched linear systems (2005) Journal of Computational and Applied Mathematics, 181, pp. 176-187Zhao, J., Hill, D.J., On stability L2 gain and H∞ control for switched systems (2008) Automatica, 44, pp. 1220-123