تخمین ترکیبی حالت و ورودی خطی گسسته سیستم های متغیر با زمان: رویکرد تئوری بازی ها

The H∞ hybrid estimation problem for linear discrete time-varying systems is investigated in this paper, where estimated signals are linear combination of state and input. Design objective requires the worst-case energy gain from disturbance to estimation error to be less than a prescribed level. Optimal solution of the hybrid estimation problem is the saddle point of a two-player zero sum differential game. On the basis of the differential game approach, necessary and sufficient solvable conditions for the hybrid estimation problem are provided in terms of solutions to a Riccati differential equation. Moreover, one possible estimator is proposed if the solvable conditions are satisfied. The estimator is characterized by a gain matrix and an output mapping matrix, where the latter reflects the internal relations between unknown input and output estimation error. At last, a numerical example is provided to illustrate the proposed approach.

When estimated signal includes both state and unknown input of the system, the estimation problem is referred to as state and input hybrid estimation (in the following, only hybrid estimation will be used for brevity). Hybrid estima- tion is originated from practical application and theory[1]. One practical example is load current estimation of uninter- ruptible power supply (UPS), where load current is a linear function of capacitor voltage (state) and back electromotive force (unknown input)[2]. From a theoretical view point, either ¯ltering (state estimation) or deconvolution (input estimation) is just a special case of the hybrid estimation. Both of the former two can be treated in the framework of hybrid estimation. Therefore, research on hybrid estima- tion is more general. Fault diagnosis is another important related area of hybrid estimation. Scheme of fault diagnosis can be designed on the basis of hybrid estimation approach because fault signal can be treated as unknown input. In the past decade, H1 optimization-based estimation has been an active research area[3¡5]. Di®erential game- theory approach is one of main time-domain approaches, because H1 estimation is a min-max problem in essen- tial. Di®erential game-theory approach can directly deduce estimator0s design method from the performance speci¯- cation and therefore, is a constructive approach. More- over, the existence conditions of the proposed estimator are necessary and su±cient so that the least conservativeness might be achieved. Di®erential game-theory approach is also capable of dealing with time-varying problems, which makes it a powerful math tool. Banavar and Speyer[6] ¯rst investigated H1 ¯ltering and smoothing for contin- uous linear time-varying (LTV) systems using di®erential game-theory approach. Later, discrete di®erential game- theory approach was applied to H1 ¯ltering for discrete LTV systems[7]. In contrast, a new H1 deconvolution ¯l- ter was derived by using game-theory approach[8]. It should be noted that existing conditions for the deconvolution ¯l- ter are not provided explicitly. Moreover, the performing speci¯cation is de¯ned in an indirect manner which makes their results unnecessarily complicated. Other related research of H1 hybrid estimation is in- troduced in the following. Optimal performance was ¯rst presented for continuous LTV system by di®erential game-theory approach. However, construction of the estimator was not discussed there. Khargonekar et al. gave results on H2=H1 hybrid estimation for continuous linear time- invariant (LTI) systems[9]. In [10], H1 ¯ltering was ex- plored, where uncertain initial state is deemed as a ¯cti- tious external input and the H1 ¯ltering was converted to an equivalent hybrid estimation problem. At last, Cuzzola and Ferrante proposed LMI conditions for H2 estimation for discrete LTI systems[1]. They also illustrated explicitly the theoretical and practical sense of hybrid estimation. Above research on hybrid estimation mostly focused on LTI systems. There is however still a lack of results for H1 hybrid estimation problem for discrete LTV systems, which will be the subject of this paper. We will use a game theory approach that incorporates maximum principle ar- guments to study such a problem over a ¯nite horizon. The connection is ¯rst established between H1 hybrid estima- tion problem and a two-player zero sum di®erential game. On the basis of the di®erential game approach, necessary and su±cient solvable conditions for the hybrid estimation problem are then provided in terms of solutions to a Ric- cati di®erential equation. Moreover, one possible estimator is proposed if the solvable conditions are satis¯ed. The es- timator is characterized by a gain matrix and an output mapping matrix, where the latter re°ects the internal rela- tions between unknown input and output estimation error. At last, e®ectiveness of the proposed approach is shown through a numerical example. Notation. Rn and Rm£n denote n-dimensional and (m £ n)-dimensional Euclidean space, respectively. L2 denotes square summable real sequences and kk stands for inner product in Euclidean space. For any h 2 L2, khk = (hTh)1=2.

H1 state and input simultaneous estimation for discrete- time LTV{hybrid estimation are investigated in this paper. On the basis of di®erential game-theory approach, neces- sary and su±cient solvable conditions for H1 state and input simultaneous estimation for discrete-time LTV sys- tems are proposed, which are equivalent to solvability of a set di®erential Riccati recursion. An estimator is pre- sented in case that the H1 SISE is solvable. The estimator is parameterized by a gain matrix and a projector matrix. The work in this paper shows that innovation information can be used to provide state observation and input estima- tion simultaneously. Because fault signal can be treated as exogenous input, with input estimation ability, one im- mediate application area of the proposed estimator is fault diagnosis.