Robust estimation with dynamic integral quadratic constraints: the discrete‐time case

Abstract

The authors consider the problem of robust output estimation in the discrete-time setting. The uncertainty in the system is characterised by integral quadratic constraints with dynamic multipliers. The main technical contribution of this study is to derive two different synthesis conditions in terms of linear matrix inequalities (LMI) by two different approaches. The LMI condition derived by the first approach allows one to find the robust estimator and the performance certificate simultaneously, whereas the condition by the second approach has a smaller size but requires an additional step for recovering the estimator. The synthesis conditions are validated by numerical examples. The results verify the equivalence of the two approaches and show that the two-step approach is in fact computationally favourable when the system has large number of state variables. NomenclatureSymbols R, R n , R n×m and Z + are used to denote the sets of real numbers, n-dimensional real vectors, n × m real matrices and non-negative integers, respectively. Symbols I n and 0 n×m are used to denote n-dimensional identity matrix and n × m zero matrix, respectively. The subscripts are dropped when the dimension is evident from the text. Given a matrix M , the transposition and the conjugate transposition are denoted by M and M * , respectively. Given matrix M and N , we use diag(M , N ) to denote the block-diagonal matrix with M and N being the (1,1) block and the (2,2) block, respectively.

Robustness analysis of uncertain discrete‐time systems with dissipation inequalities and integral quadratic constraints

“…However, the extension to uncertain LPV and uncertain nonlinear systems follows along the lines of the continuous-time results in [13,16,21]. The standard IQC homotopy theory developed for both continuous and discrete-time systems [1,[22][23][24][25] can also be generalized for systems that do not have frequency domain interpretations [26]. The homotopy approach emphasizes input-output properties, while internal states are incorporated more transparently in the dissipativity approach.…”

Cycling-Based Synthesis of Robust Output Estimators for Uncertain LPTV Systems

“…for the value ofw at k. Thisw does not have the structure shown in (3), in general. However, we can decompose it aš w =w (0) +w (1)…”

“…In particular, since the system models inevitably include modeling errors viewed as uncertainties, output estimation should be achieved even in the presence of uncertainties. In the earlier study [1], the synthesis of robust output estimators based on the idea of integral quadratic constraint (IQC) [2], [3] was discussed for discrete-time uncertain time-invariant systems. This paper is concerned with robust output estimator synthesis for discrete-time uncertain linear periodically timevarying (LPTV) systems.…”

“…Although the separator-type robust stability theorem was originally derived for LTI systems as noticed above, it can be readily applied to LPTV systems if we use the lifting technique [5], [6]. Hence, one of the simplest ways of designing robust output estimators for uncertain LPTV systems is to exploit the result in [1] for using the separator-type robust stability theorem in the synthesis through the lifting-based treatment of systems. Nevertheless, this paper considers dealing with LPTV systems through the use of the technique called cycling [5], [6], instead of lifting; we call the cycling-based robustness analysis approach using the separator-type theorem the cycling-based LPTV scaling [7], [8].…”

“…Since we evaluate the performance of the estimators by using the l 2 -induced norm, we first show in this paper the equivalence between the l 2 -induced norm of the cycled system with the structured signals and that of the system obtained by viewing the cycled system as the usual time-invariant system (i.e., without the structural constraints on signals). Based on such equivalence and the ideas in [1], [7], [8], we develop a method of designing robust output estimators, which allow us to predetermine the estimator period regardless of the system period. Note that the same topic has been already dealt with in our conference paper [9].…”