Set-valued Observer Research Articles

Reference management using state observers

The reference management algorithm is an auxiliary system which modifies the reference signal to fulfil the state and control constraints. In the previous works, most of such algorithms require the assumption that the state vector of the closed-loop system including the plant state is directly measurable, which may be unrealistic for some applications. In this paper, we develop a new reference management algorithm for discrete-time constrained systems with unmeasurable plant states. The present algorithm assures the constraints fulfilment when an ellipsoidal set of possible plant state vectors is available instead of the plant state itself. It is shown that the set of possible plant states can be obtained from the measured outputs by using the Luenberger-type observer or the set-valued observer. The effectiveness of the present method is demonstrated by a numerical example.

非線形システムに対する出力フィードバック型リファレンスガバナの設計

This paper develops a new reference management algorithm for constrained nonlinear systems with unmeasurable states. Instead of the state itself, the present method utilizes an ellipsoidal region in which the state is guaranteed to lie. Such a region can be obtained by using the set-valued observer due to Scholte and Campbell [9]. The present method requires a solution of the optimization problem which may not be solved effectively. For ease of implementation, we introduce a relaxed problem which is always efficiently solvable. When neither noise nor disturbance are present and the reference is constant, we show sufficient conditions for the modified reference to be settled to the reference in a finite time, and consequently for the convergence of the state to the desired equilibrium. In the presence of noise and/or disturbance, we derive somewhat conservative conditions for the finite-time settling of the modified reference to the original one and for the convergence of the state to the neighborhood of the equilibrium. The effectiveness of the present method is demonstrated by a numerical example.

On a Statistical Framework for Estimation from Random Set Observations

Using the theory of random closed sets, we extend the statistical framework introduced by Schreiber(11) for inference based on set-valued observations from the case of finite sample spaces to compact metric spaces with continuous distributions.

Set-valued observer design for a class of uncertain linear systems with persistent disturbance and measurement noise

In this paper, a class of linear systems affected by parameter variations, additive noise and persistent disturbances is considered. The problem of designing a set-valued state observer, which estimates a region containing the real state for each time instant, is investigated. The techniques for designing the observer are based on positive invariant set theory. By constructing a set-induced Lyapunov function, it is shown that the estimation error converges exponentially to a given compact set with an assigned rate of convergence.

Set-valued observers for control systems

We extend the usual definition of observers with known inputs to nonlinear dynamic systems that are not completely observable. To achieve this, we use set-valued analysis and invariance tools. In particular, the set of indistinguishable states that plays a key role in the problem is characterized in terms of an invariance kernel. A numerical example illustrates the results.

Anti-windup via constrained regulation with observers

A method is presented for the output-feedback control of discrete-time linear systems with hard constraints on state and control variables. Prior work has shown that optimal controllers for constrained systems take the form of a nonlinear feedback law acting on a set-valued state estimate. In this paper, conventional state estimation schemes are used. A nonlinear control law is derived which views the state estimation error as a disturbance. The resulting control law is then used in conjunction with the conventional observer, rather than set-valued observer, to achieve the desired constrained regulation. The significantly reduced real-time computations come at the cost of restricting the controller structure and thereby introducing possible conservatism in the achievable performance. The results are specialized to the problem of anti-windup for systems with control saturations. A “measurement governor” scheme is introduced that alters plant measurements in such a way to improve performance in the presence of controller saturations.

Set-valued observers and optimal disturbance rejection

A set-valued observer (also called guaranteed state estimator) produces a set of possible states based on output measurements and models of exogenous signals. We consider the guaranteed state estimation problem for linear time-varying systems with a priori magnitude bounds on exogenous signals. We provide an algorithm to propagate the set of possible states based on output measurements and show that the centers of these sets provide optimal estimates in an l/sup /spl infin//-induced norm sense. We then consider the utility of set-valued observers for disturbance rejection with output feedback and derive the following general separation structure. An optimal controller can consist of a set-valued observer followed by a static nonlinear function on the observed set of possible states. A general construction of this function is provided in the scalar control case. Furthermore, in the special case of full-control, i.e., the number of control inputs equals the number of states, optimal output feedback controllers can take the form of an optimal estimate of the full-state feedback controller.

Output feedback control for systems with constraints and saturations: scalar control case

We consider the problem of regulating the state of a linear system within a prescribed region of the state-space in the presence of disturbances and control saturations. While prior work has focused on full state-feedback, we consider the case of noisy output feedback. For the scalar control case, we combine existing work on full state-feedback and set-valued observers to derive a computational procedure which determines a priori whether the desired constrained regulation is possible. Furthermore, we show that constrained regulation can always be achieved by a set-valued observer followed by static feedback.

Approximate set-valued observers for nonlinear systems

A set-valued observer (SVO) produces a set of possible states based on output measurements and a priori models of exogenous disturbances and noises. Previous work considered linear time-varying systems and unknown-but-bounded exogenous signals. In this case, the sets of possible state vectors take the form of polytopes whose centers are optimal state estimates. These polytopic sets can be computed by solving several small linear programs. An SVO can be constructed conceptually for nonlinear systems; however, the set of possible state vectors no longer takes the form of polytopes, which in turn inhibits their explicit computation. This paper considers an "extended SVO". As in the extended Kalman filter, the state equations are linearized about the state estimate, and a linear SVO is designed along the linearization trajectory. Under appropriate observability assumptions, it is shown that the extended SVO provides an exponentially convergent state estimate in the case of sufficiently small initial condition uncertainty and provides a nondivergent state estimate in the case of sufficiently small exogenous signals.

Least squares fitting of compact set-valued data

In contexts where set-valued data occur in a natural manner, it is reasonable to ask whether such data can be directly processed: for example, to develop numerical techniques for constructing multifunction models which are a best fit to a body of set-valued observations. This note proposes methods of least squares fitting of such data for the case where each datum is a nonempty compact interval of the real line. Two models will be considered: a) interval-valued output observed from interval-valued input and b) interval-valued output resulting from exact real input