A robust multivariable controller is introduced for a class of distributed parameter systems. The system to be controlled is given as \dot{x} = Ax + Bu, y = Cx in a Banach space. The purpose of the control, which is based on the measurement y , is to stabilize and regulate the system so that y(t) \rightarrow y_{r}, as t \rightarrow \infty , where y r is a constant reference vector. Under the assumptions that operator A generates a holomorphic stable semigroup, B is linear and bounded, C is linear and A -bounded, and the input and output spaces are of the same dimension; a necessary and sufficient condition is found for the existence of a robust multivariable controller. This controller appears to be a multivariable PI-controller. Also, a simple necessary criterion for the existence of a decentralized controller is derived. The tuning of the controller is discussed and it is shown that the I-part of the controller can be tuned on the basis of step responses, without exact knowledge of the system's parameters. The presented theory is then used as an example to control the temperature profile of a bar, with the Dirichlet boundary conditions.

Robust multivariable PI-controller for infinite dimensional systems

A compensator design is proposed for a large class of parabolic distributed systems. Basically, it is a modal approach utilizing known finite-dimensional algorithms for observer and stabilizer design. The observer is constructed from observations taken from point or averaging sensors and the compensator is a feedback of this observer and can be a "distributed" or "boundary" implementation. Explicit sufficient conditions are given for the convergence of this scheme.

Finite-dimensional compensator design for parabolic distributed systems with point sensors and boundary input

We consider the class of diffusion processes described by the generalized heat equation and obtain a practical feedback controller for N system modes. The remaining process modes are effected by the controller and this control spillover is shown to reduce the desired system response time by a predictable amount.

Feedback control of linear diffusion processes

It is proved that for a class of parabolic distributed parameter systems with unbounded control and observation there exists a finite dimensional compensator using dynamic output feedback. Finite dimensional means here that the dynamics relating the input to the output is finite dimensional. A constructive design algorithm is presented and several examples are considered.

Finite dimensional compensators for parabolic distributed systems with unbounded control and observation

This paper reviews different observer design methods for linear dynamic distributed-parameter systems. In such systems, the states, inputs, and outputs depend on some spatial variable. This dependence, along with additional aspects such as the boundary conditions, increase the complexity of the state estimation problem and of the design methods. The paper in particular surveys observers for first-order and second-order linear distributed-parameter systems based on their infinite-dimensional and finite-dimensional descriptions.

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Observers for linear distributed-parameter systems: A survey

In this paper, we study feedback stabilization of a class of distributed systems governed by partial differential equations of parabolic type and its application to constructing a state estimator for asymptotic state identification. It is proved that, when a controller (an observation) can be arbitrarily constructed, observability (controllability) of the system is necessary and sufficient for stabilizing the system so that it has an arbitrarily large damping constant. As an application of this result, it is shown that a state estimator can be constructed, the output of which approaches asymptotically the real state of the system with an arbitrary convergence rate.

T. Matsushita

Papers

Feedback stabilization of a class of distributed systems and construction of a state estimator