**5. Concluding Remarks**

In this chapter we have studied the state and mode estimation problems for linear discretetime hybrid systems over the fixed time interval. The systems aren''t restricted to the Marko‐ vian jump systems and the added noises aren''t restricted to be Gaussian. With regard to concrete examples of the systems considered in this chapter, refer to [24,25]. Those examples show that the systems and estimation algorithms presented in this chapter cover extreme broad classes of dynamical systems affected by the noises not to be restricted to be Gaussian. We have adopted the MPT approach. The state and mode estimation approach adopted in this paper guarantees the optimality of estimation performance in the meaning of MPT dif‐ ferent from the previous work ([4,7]).

In this chapter we have considered the problems that both system state and modes are esti‐ mated. However we have considered the problems that the distributions of the modes over the fixed time interval not the modes themselves are estimated to grasp the global behavior of the hybrid systems over the long time intervals. In order to estimate both the system state and distributions of the modes we have introduced the averaged performance indices with respect to the candidates of the mode distributions for the averaged systems. This introduc‐ tion of the averaged systems and performance indices prevents the computational complexi‐ ty from increasing exponentially with time passage. For these performance indices we have formulated the optimal filtering and smoothing problems based on the available observed information. The estimation problems have been reduced to the optimal control problems to find the noises minimizing the introduced performance indices. We have derived the for‐ ward and backward matrix difference equations and the forward and backward filter equa‐ tions with the initial and terminal conditions respectively, which give the necessary conditions for the solvability of the optimal estimation problems. Then we have presented the optimal hybrid smoothing algorithm by the two filters approach. Finally we have stud‐ ied the numerical examples to compare the estimation performances by filtering and smoothing. We have obtained the better estimation performance by the smoothing algo‐ rithm than the filtering algorithm from the point of view of both state and modes estimation.

With regard to continuous-time cases, refer to [16,17,25]. In particular, in [17,25], the cases that concerned systems are assumed to be Markovian jump systems is also considered. In these papers the concept of quasi-stationary distributions is introduced for the Markovian mode processes and near optimality of limiting estimators with the quasi-stationary distri‐ butions is shown. It is well known that the concept of quasi-stationary distribution is very important and highly practical to grasp behavior of stochastic processes over long run time. As a further research issue it is very significant that the quasi-stationary distributions of sto‐ chastic mode processes and estimator with these distributions are investigated for the dis‐ crete-time hybrid systems.
