Robust Estimation for Discrete-Time Markovian Jump Linear Systems

Transcription

1 Universidade de São Paulo Biblioteca Digital da Produção Intelectual - BDPI Departamento de Engenharia Elétrica - EESC/SEL Artigos e Materiais de Revistas Científicas - EESC/SEL Robust Estimation for Discrete-Time Markovian Jump Linear Systems IEEE Transactions on Automatic Control, Piscataway, v 58, n 8, p , Aug Downloaded from: Biblioteca Digital da Produção Intelectual - BDPI, Universidade de São Paulo

2 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 58, NO 8, AUGUST Robust Estimation for Discrete-Time MarkovianJumpLinearSystems Marco H Terra, JoãoY Ishihara, Gildson Jesus, and João P Cerri Abstract This technical note deals with recursive robust estimation of Markov jump linear systems subject to unobserved chain state It is developed an augmented system modeled via norm bounded uncertainties Upper and lower bounds of an uncertain quadratic cost function are computed in order to define the variances of the augmented system Numerical examples are provided to show the effectiveness of the proposed estimators Index Terms Discrete-time, Markovian systems, robust estimation I INTRODUCTION State estimation of discrete-time Markovian jump linear systems (DMJLS) has been considered in some important applications related with robotics, communication, finance (see, eg, [1] [7] and references therein) Nominal and robust estimators have been investigated by several authors in the literature (see, eg, [8] [15]) Optimal and suboptimal schemes based on maximum likelihood estimates have been proposed as extensions of the original Kalman filter proposed in the 1960s An interesting solution for state estimation of DMJLS is the recursive approach proposed in [11], since it is well suited for online implementations However, it does not present robustness properties The DMJLS parameters are not subject to uncertainties To overcome this problem, some approaches have been proposed in the literature A well established technique is based on criteria, which consider robustness against disturbances and parametric uncertainties The solutions provided by this class of robust estimators are based on linear matrix inequalities (LMIs) (see, eg, [16] [24]) Due to possible infeasible solutions inherent to LMIs, all computations should be performed offline before these estimators be implemented On the other hand, standard robust Kalman-type estimators developed for systems not subject to Markovian jumps (see, eg, [25] and [26]) cannot be directly applied to estimate uncertain DMJLS when the Markov chain is not known In this technical note, recursive robust state estimators (in the predicted and filtered forms) for DMJLS are developed based on discretetime Riccati recursions They are deduced through the minimization of the worst-possible regularized residual norm The main difficulty to solve this kind of problem is to find an appropriate cost function considering the Markov chain not available An advantage of the proposed robust filters for DMJLS is that the conditions for stability can be easily stated and proved An important contribution of this note resides in the variance computation of the disturbance related with the states of an augmented DMJLS The computation of this variance is possible thanks to the formulas proposed to calculate upper and lower bounds of an uncertain Manuscript received July 20, 2009; revised June 16, 2011, December 26, 2011, and August 06, 2012; accepted November 27, 2012 Date of publication February 11, 2013; date of current version July 19, 2013 Recommended by Associate Editor Z Wang M H Terra and J P Cerri are with the Department of Electrical Engineering, University of São Paulo at São Carlos, CP 359, São Carlos, SP, , Brazil ( terra@scuspbr; jpcerri@uspbr; jpcerri@yahoocombr) J Y Ishihara is with the Department of Electrical Engineering, University of Brasília, CP 4386, Brasília, DF, , Brazil ( ishihara@ene unbbr) G Jesus is with the Department of Sciences and Technology, University of Santa Cruz, Ilhéus, BA, , Brazil ( gildsonj@gmailcom) Digital Object Identifier /TAC quadratic term They are obtained through the solution of optimization problems Simulations are performed for the robust predictor developed, the stationary predictor proposed in [11] and the robust predictor subject to polytopic uncertainties given in [16], for comparison purposes It is emphasized in this comparative study the importance of the recursiveness for online applications For the case with no jumps, the predicted and the filtered robust estimates developed reduce to the recursive robust estimators given in [25] In addition, if the system is considered without uncertainties, both estimates reduce to the respective standard Kalman-type estimators in the predicted and filtered forms This technical note is organized as follows: in Section II, the uncertain DMJLS and its augmented version are presented; in Section III, robust filters for DMJLS are deduced; and in Section IV, numerical examples are shown II PRELIMINARIES Consider the following uncertain DMJLS:, for each time instant and jump parameter,,,,,and are nominal parameter matrices subject to the uncertainties with and arbitrary matrices;,,, known matrices of appropriate dimensions Assume that is a finite state discrete-time Markov chain with and state transition probability matrix whose entries are given by for all Asitisusual, is the -valued random state, is the -valued random state disturbance, is the -valued random output sequence, is the -valued random output disturbance, and are independent wide sense stationary sequences of mutually independent random variables with zero mean and covariance matrices and, respectively Suppose that is a random initial state with and, is the indicator function defined as,if and,if In addition, and are independent of and ;and,forall instant and In the uncertain DMJLS (1), the state sequence alone, without taking into account, is not a Markovian process [2] However, it can be rewritten in terms of a Markovian process with a new state variable depending on, through the following augmented system: (1) (2) (3) /$ IEEE

3 2066 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 58, NO 8, AUGUST 2013 The state of the original system (1) is recovered as According to [2], when (3) is not subject to uncertainties, the secondorder moments and are calculated through the following equations: The covariance is easily computed through (6) However, it is difficult to use (4) and (5) in their present forms, they depend on and In the next section, this problem is dealt with From the augmented system (3), one can show that,,,,, For each, the second-order moments of the augmented system are described by the following variables defined for and III ROBUST FILTERS FOR DMJLS In order to develop robust filters for the DMJLS (3), it is needed to define an upper bound to the covariance given in (5) It is composed by the difference of positive and negative uncertain terms In [27] is provided a formula, which was deduced from inequality arguments, to compute upper bounds of uncertain quadratic terms On the other hand, it was not found in the literature an expression to calculate lower bounds, necessary to compute the lower influence of in (5) In the next proposition and corollary, lower and upper bounds based on optimizations of a suitable cost function are proposed Proposition 1: Let the following quadratic cost function: is updated through the equation (4) (7),, and are assumed known, and, are unknown variables For any fixed, consider the following optimization problems subject to given by: with initial condition is a nonnegative function Define and Then, for all such that, (5) with and (6) and

4 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 58, NO 8, AUGUST Proof: It follows the arguments used in [28], which are based on Lagrange multipliers The next corollary defines upper and lower bounds in a way useful to be applied to the estimators considered in this note Corollary 1: Consider the following quadratic term: with,,,,,and are assumed known and Then, for all, the following inequalities hold: with with The parameters and are given by and with ; and stand for the minimum and maximum singular values of, respectively It can be shown that for each, the covariances,,,and are related as for all such that ;forall such that and for all such In particular, if and then and Proof: It follows from Proposition 1, through some appropriate identifications For all arbitrarily chosen, one has Consider then the following identifications:,,,,, based on the quadratic cost function (7) Recalling that, it is easy to check that, with Considering Proposition 1, one obtains the result Based on the Corollary 1, one can calculate bounds to (4) and (5) For the uncertain covariance sequence generated by (4), the lower and upper bounds and, respectively, are generated by the following recursive equations: Now, one can calculate an upper bound for (5) based on (8) and (9), as shown in (10), at the bottom of the page, for any and such that One can always choose and The convergence of (10) depends on (8) and (9) Provide that (10) holds, one can compute recursively (8) whose stability property is similar to the standard Riccati equations [29] However, the stability of (9) is not guaranteed for any This issue is illustrated with the following numerical example 1) Example 1: Consider the System (3), for,with (8) (9) If it is computed in terms of, one obtains the graphic of Fig 1 whose is given by One can see in this figure that there exists a minimum for whose value is not easy to be found recursively, the convergence of the Riccati equation is not guaranteed for any The computation of upper bounds of uncertain systems has been the subject of several authors, see for instance [30] [33] The approaches considered in these references are based on only LMIs In the following, the computation of is performed in two steps In the first, is obtained through a minimization problem and in the second, (10)

5 2068 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 58, NO 8, AUGUST 2013 uncertainties on the second moment of in the System (3), it is proposed in the following a robust predictor for (3) based on the optimization problem (13) To obtain and from (13), this problem can be rewritten as (14) (15) Fig 1 Maximum singular value of versus is obtained recursively For the bottom of the page One can obtain, one has (11), as shown at (12) According to [26], the solution of the optimization problem (14) (15) is given by (16) and is given by the modified weighting matrices and are defined by with An advantage in solving lower and upper bounds based on optimization problems is that one can identify if there exists local minimum for (8) and (9), following studies performed in [34] At this point, it has completed the definition of the augmented system (3) Bearing in mind that (12) provides the maximum influence of the and is a nonnegative scalar parameter which satisfies With these definitions, one can propose an algorithm to compute, following the framework developed in [25] Recursive Robust Predictor for DMJLS Step 0: (Initial conditions): Step 1: Compute,, and through (6), (8), (9), and (10), respectively (11)

6 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 58, NO 8, AUGUST Step 2: If and,then Otherwise,itis chosen as There exists a unique positive-semidefinite solution (with ) for (21), following the guidelines defined in [11], and (17) and the parameters are changed to the corrected parameters (18) denotes spectral radius of the predictor with stationary gain The stability of this estimator is easily checked thanks to the characteristic of the parameter, it depends on only known parameters of the DMJLS The robust recursive filter presented in the next algorithm, shown through (23) (26), is deduced based on the optimization problem Step 3: Actualize with through the following recursive equations (22) (19) (20) Step 4: From, compute If the uncertainties of the robust predictor are canceled ( ), it reduces to the nominal predictor developed in [11] For the case with no jumps, it resembles to the robust estimator developed in [25] (when the singular system of this reference is considered in the standard state-space form) Remark 1: In order to guarantee the stability of the stationary robust predictor proposed, some features are assumed: System (1) is mean square stable (MSS), the Markov chain is ergodic and (and with the condition (17) satisfied, the positiveness of is also assured) For,,,, fixed and for all model parameters constant, it is considered the following algebraic Riccati Equation: and (21) Following the guidelines aforementioned, (22) can be rewritten in the same way it was written (14), whose solution is given by (16) The robust filters proposed in this technical note, for and without uncertainties, reduce to the standard Kalman filter [35] Recursive Robust Filter for DMJLS Step 0: (Initial conditions):,, Step 1: Compute,, and through (6), (8), (9), and (10), respectively Step 2: If and,then Otherwise,itis chosen (23) and the parameters are changed to the corrected parameters (24) Step 3: Actualize with through the recursive equations shown in (25) (26), shown at the bottom of the page, with Step 4: From, compute (25) (26)

7 2070 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 58, NO 8, AUGUST 2013 Fig 2 SKF and Robust Predictor for DMJLS IV NUMERICAL EXAMPLE In the following, two comparative studies are presented to show the effectiveness of the robust predictor (17) (20) Both studies were obtained based on the System (1) (2) with two Markovian states, transition probability matrix, and parameter matrices defined as First, a comparative study between the standard Kalman filter (SKF) developed in [11] for DMJLS and the robust predictor proposed is shown in Fig 2 The square root of the mean square error (rms) estimator given in [11] was simulated for the system with and without uncertainties, as the robust predictor was simulated for the system with uncertainties The curves were obtained from with the values of generated randomly The initial condition was considered Gaussian with mean and variance,, and are independent sequences of noises, and The parameters and were settled as 2 and 01, respectively Notice that the parameter changes for each recursive step of 4000 Monte Carlo simulations It is important to point out also that if in (17), the stability of the robust predictor is always guaranteed The tuning of the variance in (12) influences only the robust predictor performance, the stability is always guaranteed Notice in Fig 2 the advantage of using the robust predictor when the DMJLS is subject to uncertainties Second, a comparative study between the robust Markovian estimator proposed in [16] and the robust predictor developed is shown in Fig 3 The predictor of [16] was deduced for polytopic uncertainties and calculated exclusively through LMIs According to Remark 71 of [36, p 265], polytopic and norm bounded uncertainties can be Fig 3 Comparison between the Predictor of [16] and the Recursive Robust Predictor equivalent in some cases One of these cases is when the norm bounded uncertainties are diagonal In this form of representation, both uncertainties result in a polyhedral convex set Four scalar numerical models considered in [16] were taken into account here Our estimator outperforms the estimator given in [16] in three cases: a, c, and d; and underperforms [16] in the case b Both estimators are optimal in some sense, however the approach proposed in this technical note is more useful for online applications V CONCLUSION In this technical note, robust Kalman-type filters were developed for DMJLS The main feature of these filters is the guarantee of stability in online applications Furthermore, provided that the variances of the augmented model of the DMJLS system are calculated, the optimal filters can be obtained following the guidelines proposed in [25] In resume, this note shows that after some algebra, it is possible to apply recursive robust techniques to estimate DMJLS, even when they are time-varying For future works, one intends to deduce recursive robust estimators for this class of systems which do not depend on any tuning parameter REFERENCES [1] MAnandandPRKumar, EstimatingthestateofaMarkovchain over a noisy communication channel: A bound and an encoder, in Proc 49th IEEE Conf Decision and Contr, Atlanta, GA, 2010, pp [2]OLVCosta,MDFragoso,andRPMarques, Discrete-Time Markov Jump Linear Systems Probability and its Applications London, UK: Springer-Verlag, 2005 [3] W Li, F Jiang, Z Wang, G Zhou, and Z Zhu, Fault detection of Markov jumping linear systems, Mathemat Prob in Eng, vol 2012, pp 1 27, 2012 [4] X Ma, S M Djouadi, and H Li, State estimation over a semi-markov model based cognitive radio system, IEEE Trans Wireless Commun, vol 11, no 7, pp , Jul 2012 [5] VJMathewsandJKTugnait, Detectionandestimationwithfixed lag for abruptly changing systems, IEEE Trans Aerosp Electron Syst, vol AES-19, no 5, pp , 1983 [6] A A G Siqueira, M H Terra, and M Bergerman, Robust Control of Robots Fault Tolerant Approaches London, UK: Springer-Verlag, 2011 [7] HZhuandLHao, Bayesianfiltering for Markov switching stochastic volatility model with heavy tails, in Proc Int Conf on Manage and Service Sci, Changsha, China, 2009, pp 1 4

8 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 58, NO 8, AUGUST [8] G A Ackerson and K S Fu, On the state estimation in switching environments, IEEE Trans Autom Control, vol AC-15, no 1, pp 10 17, Jan 1970 [9] CAndrieu,MDavy,andADoucet, Efficient particle filtering for jump Markov systems application to time-varying autoregressions, IEEE Trans Signal Process, vol 51, no 7, pp , Jul 2003 [10] C G Chang and M Athans, State estimation for discrete systems with switching parameters, IEEE Trans Aerosp Electron Syst, vol AES-14, no 3, pp , 1978 [11] O L V Costa and S Guerra, Stationary filter for linear minimum mean square error estimation for discrete-time Markovian jump linear systems, IEEE Trans Autom Control, vol 47, no 8, pp , 2002 [12] A Doucet and C Andrieu, Iterative algorithms for state estimation of jump Markov linear systems, IEEE Trans Signal Process, vol 49, no 6, pp , Jun 2001 [13] M S Mahmoud, P Shi, and A Ismail, Robust Kalman filtering for discrete-time Markovian jump systems with parameter uncertainty, J Comput and Appl Math, vol 169, pp 53 69, 2004 [14] P Shi, E K Boukas, and R K Agarwal, Kalman filtering for continuous-time uncertain systems with Markovian jumping parameter, IEEE Trans Autom Control, vol 44, no 8, pp , Aug 1999 [15] J K Tugnait, Adaptive estimation and identification for discrete systems with Markov jump parameters, IEEE Trans Autom Control, vol AC-27, no 5, pp , 1982 [16] O L V Costa and S Guerra, Robust linear filtering for discrete-time hybrid Markov linear systems, Int J Control, vol 75, no 10, pp , 2002 [17] C Lee and I Fong, Robust Kalman filter design for discrete-time systems with Markovian jumping parameters, in Proc SICE Annual Conf, Fukui, Japan, Aug 2003 [18] M S Mahmoud and P Shi, Robust Kalman filtering for continuous time-lag systems with Markovian jump parameters, IEEE Trans Circuits Syst, vol 50, no 1, pp , Jan 2003 [19] H Shao, Delay-range-dependent robust filtering for uncertain stochastic systems with mode-dependent time delays and Markovian jump parameters, J Math Anal and Applicat, vol342,no2,pp , 2008 [20] C E Souza and M D Fragoso, Robust filtering for uncertain Markovian jump linear systems, Int J Robust and Nonlin Control, vol 12, no 5, pp , 2002 [21] Z Wang, J Lam, and X Liu, Robust filtering for discrete-time Markovian jump delay systems, IEEE Signal Process Lett, vol 11, no 8, pp , Aug 2004 [22] J Xiong and J Lam, Fixed-order robust filter design for Markovian jump systems with uncertain switching probabilities, IEEE Trans Signal Process, vol 54, no 4, pp , Apr 2006 [23] S Xu, T Chen, and J Lam, Robust filtering for a class of nonlinear discrete-time Markovian jump systems, J Optimi Theory and Appl, vol 122, no 3, pp , 2004 [24] S Xu, J Lam, and X Mao, Delay-dependent control and filtering for uncertain Markovian jump systems with time-varying delays, IEEE Trans Circuits Syst, vol 54, no 9, pp , Sep 2007 [25] J Y Ishihara, M H Terra, and J C T Campos, Robust Kalman filter for descriptor systems, IEEE Trans Autom Control, vol 51, no 8, pp , Aug 2006 [26] A H Sayed, A framework for state-space estimation with uncertain models, IEEE Trans Autom Control, vol 46, no 7, pp , Jul 2001 [27] Z Wang, J Zhu, and H Unbehauen, Robust filter design with timevarying parameter uncertainty and error variance constraints, Int J Control, vol 72, no 1, pp 30 38, 1999 [28] A H Sayed and V H Nascimento, Design criteria for uncertain models with structured and unstructured uncertainties, in Robustness in Identification and Control, A Garulli, A Tesi, and A E Vicino, Eds London, UK: Springer-Verlag, 1999, vol 245, pp [29] P Lancaster and L Rodman, Algebraic Riccati Equations Oxford and New York New York: Oxford Science Publications, 1995, vol 1 [30] M Fu, C E Souza, and Z Lu, Finite-horizon robust Kalman filter design, IEEE Trans Signal Process, vol 49, no 9, pp , Sep 2001 [31] Y S Hung and F Yang, Robust filtering with error variance constraints for discrete time-varying systems with uncertainty, Automatica, vol 39, no 7, pp , 2003 [32] F Yang, Z Wang, G Feng, and X Liu, Robust filtering with randomly varying sensor delay: the finite-horizon case, IEEE Trans Circuits Syst, vol 56, no 3, pp , Mar 2009 [33] X Zhu, Y C Soh, and L Xie, Design and analysis of discrete-time robust Kalman filters, Automatica, vol 38, no 6, pp , 2002 [34] A H Sayed, V H Nascimento, and F A M Ciparrone, A regularized robust design criterion for uncertain data, SIAM J Matrix Anal and Appl, vol 23, no 4, pp , 2002 [35] TKailath,AHSayed,andBHassibi, Linear Estimation Upper Saddle River, NJ: Prentice-Hall, 2000 [36] P Colaneri, J C Geromel, and A Locatelli, Control Theory and Design An and Viewpoint London, UK: Academic, 1993 Extensions of Padé Discretization for Linear Systems With Polyhedral Lyapunov Functions for Generalized Jordan Structures Surya Shravan Kumar Sajja, Francesco Rossi, Patrizio Colaneri, and Robert Shorten Abstract Recently, we showed that certain types of polyhedral Lyapunov functions for linear time-invariant systems, are preserved by diagonal Padé approximations, under the assumption that the continuous-time system matrix has distinct eigenvalues In this technical note, we show that this result also holds true in the case that has non-trivial Jordan blocks Index Terms Discretization, nontrivial Jordan blocks, Padé approximations, polyhedral Lyapunov functions, preservation of Lyapunov functions I INTRODUCTION Recently, we showed that certain types of polyhedral Lyapunov functions for linear time-invariant systems, are preserved by diagonal Padé approximations, under the assumption that the continuous-time system matrix has distinct eigenvalues [1] This result follows by making explicit use of the fact that the diagonal Padé approximation preserves the Jordan structure of a matrix if the matrix has distinct eigenvalues Unfortunately, this fact no longer holds when has nontrivial Jordan blocks, and the purpose of this technical note is therefore to extend the results of [1] to the case of nontrivial Jordan blocks Polyhedral Lyapunov functions are known to be nonconservative in the analysis of stability under arbitrary switching for polytopic and switched systems, when compared to quadratic Lyapunov functions [2] The motivation for wondering whether there exists a polyhedral LF that is shared under discretization is discussed in [1] Recall that the Manuscript received January 30, 2012; revised July 13, 2012; accepted December 24, 2012 Date of publication February 08, 2013; date of current version July 19, 2013 Recommended by Associate Editor D Arzelier S S K Sajja is with the Technische Universität Berlin, Fachgebiet Regelungssysteme, Berlin, Germany, on leave from NUI Maynooth, Kildare, Kildare Maynooth, Ireland ( ssuryashravankumar@gmailcom; suryasajja2009@nuimie) F Rossi is with the Aix-Marseille Université, Domaine universitaire de Saint Jérôme, LSIS, 13013, Marseille, France ( francescorossi@lsisorg) P Colaneri is with the Dipartimento di Elettronica e Informazione, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milano, Italy ( colaneri@eletpolimiit) R Shorten was with the Hamilton Institute, NUI Maynooth, Ireland He is now with IBM Research, Dublin 15, Ireland ( robertshorten@nuimie) Digital Object Identifier /TAC /$ IEEE