The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

The Kalman filter is the minimum-variance state estimator for linear dynamic systems with Gaussian noise. Even if the noise is non-Gaussian, the Kalman filter is the best linear estimator. For nonlinear systems it is not possible, in general, to derive the optimal state estimator in closed form, but various modifications of the Kalman filter can be used to estimate the state. These modifications include the extended Kalman filter, the unscented Kalman filter, and the particle filter. Although the Kalman filter and its modifications are powerful tools for state estimation, we might have information about a system that the Kalman filter does not incorporate. For example, we may know that the states satisfy equality or inequality constraints. In this case we can modify the Kalman filter to exploit this additional information and get better filtering performance than the Kalman filter provides. This paper provides an overview of various ways to incorporate state constraints in the Kalman filter and its nonlinear modifications. If both the system and state constraints are linear, then all of these different approaches result in the same state estimate, which is the optimal constrained linear state estimate. If either the system or constraints are nonlinear, then constrained filtering is, in general, not optimal, and different approaches give different results.

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Kalman filtering with state constraints: a survey of linear and nonlinear algorithms

Considers the problems of robust stability and stabilization for uncertain continuous singular systems with state delay. The parametric uncertainty is assumed to be norm bounded. The purpose of the robust stability problem is to give conditions such that the uncertain singular system is regular, impulse free, and stable for all admissible uncertainties, while the purpose of the robust stabilization is to design a state feedback control law such that the resulting closed-loop system is robustly stable. These problems are solved via the notions of generalized quadratic stability and generalized quadratic stabilization, respectively. Necessary and sufficient conditions for generalized quadratic stability and generalized quadratic stabilization are derived. A strict linear matrix inequality (LMI) design approach is developed. An explicit expression for the desired robust state feedback control law is also given. Finally, a numerical example is provided to demonstrate the application of the proposed method.

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Robust stability and stabilization for singular systems with state delay and parameter uncertainty

Brief paper: Actuator fault robust estimation and fault-tolerant control for a class of nonlinear descriptor systems

This note deals with the problems of robust stability and stabilization for uncertain discrete-time singular systems. The parameter uncertainties are assumed to be time-invariant and norm-bounded appearing in both the state and input matrices. A new necessary and sufficient condition for a discrete-time singular system to be regular, causal and stable is proposed in terms of a strict linear matrix inequality (LMI). Based on this, the concepts of generalized quadratic stability and generalized quadratic stabilization for uncertain discrete-time singular systems are introduced. Necessary and sufficient conditions for generalized quadratic stability and generalized quadratic stabilization are obtained in terms of a strict LMI and a set of matrix inequalities, respectively. With these conditions, the problems of robust stability and robust stabilization are solved. An explicit expression of a desired state feedback controller is also given, which involves no matrix decomposition. Finally, an illustrative example is provided to demonstrate the applicability of the proposed approach.

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Robust stability and stabilization of discrete singular systems: an equivalent characterization

The purpose of this paper is to give a necessary and sufficient condition under which for a given plant of descriptor system model there exists a normal, internally stabilizing controller that has an upper limit on its order and that satisfies a closed-loop H/sub /spl infin// norm bound. The approach used in this paper is based on a generalized version of bounded real lemma, thus the proofs are simple.

Bounded real lemma and H/sub /spl infin// control for descriptor systems

In this paper, we study the H/sub /spl infin// control problem for nonlinear descriptor systems governed by a set of differential-algebraic equations (DAEs) of the form Ex/spl dot/ = F(x, w, u), z = Z(x, w, u), y = Y(x, w, u), where E is, in general, a singular matrix. Necessary and sufficient conditions are derived for the existence of a controller solving the problem. We first give various sufficient conditions for the solvability of H/sub /spl infin// control problem for DAEs. Both state-feedback and output-feedback cases are considered. Then, necessary conditions for the output feedback control problem to be solvable are obtained in terms of two Hamilton-Jacobi inequalities plus a weak coupling condition. Moreover, a parameterization of a family of output feedback controllers solving the problem is also provided.

H/sub /spl infin// control for nonlinear descriptor systems

Analysis of stability robustness for generalized state-space systems with structured perturbations

Robust control analysis and design for discrete-time singular systems

A constrained filtering method is proposed to deal with the filtering problems for nonlinear systems with constraints. The problem is converted to a sequence of recursive estimation problems in which the system equations and constraint conditions are treated as pseudo-measurements. To resolve the singularity problem arising from the constraints, a modified maximum-likelihood method for nonlinear systems is developed. The simulation results from the application of the proposed scheme to the target tracking problem shows that the constrained filtering method can enhance the performance of filter design significantly.

Fan-Ren Chang

Papers

Bounded real lemma and H/sub /spl infin// control for descriptor systems

H/sub /spl infin// control for nonlinear descriptor systems

Analysis of stability robustness for generalized state-space systems with structured perturbations

Robust control analysis and design for discrete-time singular systems

Filtering method for nonlinear systems with constraints