Parabolic Distributed Parameter Systems Research Articles

Discrete-time servomechanism design of parabolic distributed-parameter systems

In this paper we investigate a discrete-time servomechanism design for a parabolic distributed-parameter system with constant disturbances. The problem is to design a discrete-time feedback controller such that the resulting continuous-time closed-loop system will be stable and the discrete-time controlled output will converge to the constant reference vector. We construct a servomechanism using the concepts of state feedback and output feedback through an observer. The design procedure is basically a modal approach which is realized in finite-dimensional theories and algorithms. Explicit sufficient conditions are given for the existence of a finite-dimensional servomechanism.

Finite-dimensional servomechanism design by discrete-time input-output data for parabolic distributed parameter systems

The paper investigates a finite-dimensional servomechanism design for a parabolic distributed parameter system with constant disturbances, by discrete-time input-output data. The problem is to design a discrete-time feedback controller such that the resulting continuous-time closed-loop system will be stable and the discrete-time controlled outputs will be regulated. With some assumptions with respect to a control operator and an output operator, the paper presents explicit sufficient conditions for the existence of a discrete-time feedback controller which stabilizes and regulates the system. A finite-dimensional design procedure of a servomechanism is given and it is shown that the discrete-time controlled outputs will converge to the constant reference vectors with an arbitrarily assignable decay rate. In the design procedure it is necessary to construct a finite-dimensional discrete-time state observer in order to estimate the system state at sampling times from observations

Feedback Stabilization of Parabolic Distributed Parameter Systems by Discrete-Time Input–Output Data

In this paper we investigate feedback stabilizability of continuous-time parabolic distributed parameter systems by discrete-time input-output data. We construct a stabilizer using the concepts of state feedback and output feedback through an observer. The design procedure is basically a modal approach which is realized in finite-dimensional theories and algorithms. It is shown that any initial state is reduced with an arbitrary decay rate. Explicit sufficient conditions are given for the convergence of the design scheme.

A digital adaptive control law for a parabolic distributed parameter system

This paper deals with the design of a digital adaptive controller for a parabolic distributed parameter system. The space of inputs and the space of outputs are both one dimensional. After introducing an input-output representation, a digital adaptive control law is presented and analyzed.

Regulator design for distributed parameter systems with constant disturbances

This paper investigates a regulator problem for a distributed parameter system with constant disturbances. The regulator problem considered here is to determine a feedback control law which stabilizes and regulates the system. From a practical point of view, a design procedure is proposed for a regulator which can be realized in finite-dimensional theories and techniques for an infinite-dimensional system. In the design procedure it is necessary to construct a state observer in order to estimate the system state from observations. Explicit sufficient conditions are presented for the convergence of the schemes and the theory is applied to a parabolic distributed parameter system with more general types of inputs and outputs, given in suitable Hilbert spaces.

Finite Dimensional Compensators for Parabolic Distributed Systems with Unbounded Control and Observation

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.

Controllability, Observability and Stabilizability of Distributed Systems with Infinite Domain

In the present paper the approximate controllability, observability and stabilizability of specific class of self-adjoint parabolic distributed parameter systems defined on infinite interval are investigated. The system under consideration has only discrete point spectrum and appropriate eigenfunctions form a complete orthon armal set in the state space. Using the Hennite’s polynomials, necessary and sufficient conditions for approximate controllability, observability and feedback stabilizability are presented. These conditions are easily computable The illustrative example is also given.

On parabolic distributed optimal control problems with restrictions on the gradient

Convex optimal control problems for parabolic distributed parameter systems with an integral constraint for the gradient of the state are considered. The objective function is a general convex integral functional in which the control constraints are incorporated. Necessary and sufficient conditions for optimality are derived.

Finite-dimensional servomechanism design for parabolic distributed-parameter systems

The paper investigates a finite-dimensional servomechanism design for a parabolic distributed-parameter system with constant disturbances. The problem is to design a feedback controller such that the resulting closed-loop system will be exponentially stable and the controlled output will be regulated. With some assumptions with respect to a control operator and an output operator, the paper presents explicit sufficient conditions for the existence of a feedback controller which exponentially stabilizes and regulates the system. A finite-dimensional design procedure of a servomechanism is given and it is shown that the controlled output will converge to the constant reference vector with an arbitrarily assignable exponential decay rate. In the design procedure it is necessary to construct a finite-dimensional state observer in order to estimate the system state from observations.

Functional reproducibility and decoupling using state feedback and dynamic compensation in parabolic distributed parameter systems

A sufficient condition for output functional reproducibility in a multivariate parabolic distributed parameter system is presented. A distributed prefilter (right-inverse system) is constructed. The output of the prefilter produces the required control input to the plant to generate a given real function as the output of the distributed system. Furthermore, a sufficient condition for the decoupling of the distributed system using state feedback and dynamic compensation is derived. It is shown that the compensator is a finite dimensional system. An example is presented to illustrate the application of the results

Identification of Noisy Distributed Parameter Systems Using Optimal Stochastic Approximation Procedure

The optimal stochastic approximation procedure (OSAP) is applied to the parameter identification problem of distributed parameter system (DPS) driven by random disturbances and observed through noisy measurements. This procedure is a stochastic approximation procedure (SAP) with an optimal gain sequence and an optimal transformation on the gradient of the objective function: these optimal values accelerate the convergence rate by minimizing the mean squared parameter estimation error, under the assumption that the density functions of the system and observation noises are known, or can be easily estimated. An example of parameter identification of a stochastic parabolic DPS is simulated on the digital computer. A comparison is made among the results of the optimal, the modified, the nominal first-order, and the nominal second-order SAP. It is shown that the OSAP gives higher accuracy and faster rate of convergence as compared to the nominal SAP

Computer simulation of a parametric optimization problem for an industrial process

Results in functional analysis are applied to the computation of the optimal control of a parabolic distributed parameter system. The control term is a coefficient of the state equation, and is computed by the steepest descent method. An a-priori feedback is determined in the case of a constant control on the optimization interval. Numerical results are discussed.

On identifiability of parameters in a class of parabolic distributed systems

In this paper we obtain a number of identifiability results for a particular class of parabolic distributed parameter systems. The parameters are assumed to be constant and the difficulties are due principally to the presence of the unknown parameters in the eigenfunctions. A lemma is established which permits us to obtain a number of identifiability results under certain assumptions on the boundary, conditions.

Identification of Parabolic Distributed-Parameter Systems by a Pattern-Recognition Approach

A pattern-recognition method is given for identification of distributed parameter systems. This approach takes care of the system characterization which is the first and critical step in the identification problem. Systems which can be represented by parabolic partial differential equations are considered. The pattern recognition system has been designed to classify the pattern measurements into identifiable classes. The inputs to this pattern recognition system are the data concerning the inputs and observation responses of the distributed systems and the output is the classification of the distributed parameter system.

A note on the application of the matrix Riccati equation to the optimal control of distributed parameter systems

Numerical results from an investigation of the Optimal control of a parabolic distributed parameter system are presented. The results from a matrix Riccati formulation are contrasted with those using a variable metric method. Spurious oscillations in the optimal control given by the Riccati algorithm are shown to stem from the use of Simpson's rule to approximate the integrals. It is concluded that the trapezoidal rule is to be preferred.

On the optimal pointwise control and parametric optimization of distributed parameter systems

The problem of determining the best pointwise location of the actuators in a parabolic distributed parameter system, having regard to the minimization of an energy criterion, is proposed in this study. The precise control problem is to find a minimal energy control law which enables one to reach a desired state, considering the location of the actuators as a parameter. Three different computation methods of the control law are recalled or developed. We present a critical study of their respective efficiency and difficulty of implementation. One example is solved by these three methods and numerical results are compared.

Controlability, observability, and duality in a distributed parameter system with continuous and point spectrum

The controllability of a specific system which is representative of the class of self-adjoint parabolic distributed parameter systems defined on a semi-infinite interval is established. Such systems possess nonempty continuous and/or point spectrum. Necessary and sufficient conditions for observability of the system are derived and, in the case where the observation is made at the boundary, a dual system is developed.

Computational aspects of the optimal control of distributed-parameter systems

Several techniques are examined for optimal control of hyperbolic and parabolic distributed-parameter systems. These include a direct search on the performance index and a method of steepest ascent based on a maximum principle analogous to that of Pontryagin for lumped-parameter systems. Extensive numerical results suggest that for the various physical systems investigated valid solutions can be obtained.