Average reachability of continuous-time Markov jump linear systems and linear Markovian state observers

Stability of state estimators for Markov jump linear systems featuring time-varying and correlated noise processes are studied in this paper. Three conditions for stability are presented, starting with a more general one requiring positiveness of the covariance of the error estimate, and is applicable to a class of filters that contains the well known linear minimum mean square estimators. It is then derived a more strict condition based on the plant parameters only, which may be interpreted as requiring that the state additive noise pervades every system dynamics. Finally, we consider a structural notion linked with the reachability gramian and we show it is a sufficient condition for the previous ones to be fulfilled, thus linking the filter stability with the structure of the plant, and present a simple rank test. Illustrative examples are included

Average reachability of continuous-time Markov jump linear systems and the linear minimum mean square estimator

. In this paper we study the average reachability gramian for continuous-time linear systems with additive noise and jump parameters driven by a general Markov chain. We define a rather natural reachability concept by requiring that the average reachability gramian be positive definite. Aiming at a testable condition, we introduce a set of reachability matrices for this class of systems and employ invariance properties of the null space of the noise coefficient matrices to show that the system is reachable if and only if these matrices are of full rank. We also show for reachable systems that the state second moment is positive definite. One consequence of this result in the context of linear minimum mean square state estimation for reachable systems is that the expectation of the error covariance matrix is positive definite. Moreover, the average boundedness of the error covariance matrix is invariant to a type of perturbation in the noise model, meaning that the estimates are not overly sensitive, which consists in a property that is desirable in applications and sometimes referred to as stability of the estimator

Average reachability and average controllability for continuous-time markov jum linear systems

Neste trabalho estudamos as noções de alcançabilidade e controlabilidade para sistemas lineares a tempo contínuo com perturbações aditivas e saltos nos parâmetros sujeitos a uma cadeia de Markov geral. Definimos conceitos de alcançabilidade e controlabilidade médios de maneira natural exigindo que os valores esperados dos gramianos correspondentes sejam definidos positivos. Visando obter uma condição testável para ambos os conceitos, introduzimos conjuntos de matrizes de alcançabilidade e de controlabilidade para esta classe de sistemas e usamos certas propriedades de invariância para mostrar que: o sistema é alcançável em média, e, analogamente, controlável em média, se e somente se as matrizes respectivas, de alcançabilidade e de controlabilidade, têm posto completo. Usamos alcançabilidade média de sistemas para mostrar que a matriz de segundo momento do estado é definida positiva com uma margem uniforme. Uma consequência deste resultado no problema de estimação linear do estado é que a matriz de covariância do erro de estimação é positiva definida em média, no sentido que existe um nível mínimo de ruído nas estimativas. Na sequência, para estimadores lineares markovianos, estudamos a limitação do valor esperado da matriz de covariância do erro para mostrar que o filtro é estável num certo sentido, sendo esta uma propriedade desejável em aplicações reais. Quanto às aplicações da controlabilidade média, usamos este conceito para estabelecer condições necessárias e suficientes que garantem a existência de um processo de controle que leva a componente contínua do estado do sistema para a origem em tempo finito e com probabilidade positiva.In this work we study the reachability and controllability notions for continuous-time linear systems with exogenous inputs and jump parameters driven by a quite general Markov chain. We define a rather natural average reachability and controllability concepts by requiring that the associated gramians are average positive definite, respectively. Aiming at testable conditions for each concept, we introduce certain sets of matrices linked with the gramians, and employ some invariance properties to find rank-based conditions. We show for average reachable systems that the state second moment is positive definite. One consequence of this result in the context of linear estimation for reachable systems is that the expectation of the error covariance matrix is positive definite. Moreover, for linear markovian filters we study the average boundedness of the error covariance matrix to show that the filter is stable in an appropriate sense, which consists in a property that is desirable in real applications. Regarding the average controllability concept, we show that it is a necessary and sufficient condition for the feasibility of the following control problem: find a control process that drives the continuous component of the state to zero in finite time with positive probability