Linear Distributed Parameter Systems Research Articles

Linear distributed parameter systems: Closed-loop exponential stability with a finite-dimensional controller

Many engineering systems exhibit dynamical behavior which must be described by partial, rather than ordinary, differential equations. These distributed parameter systems (DPS) have infinite-dimensional state-space descriptions, yet they must be controlled by finite-dimensional feedback systems implemented with a limited number of control actuators and sensors and an on-line digital computer. The most basic problem to be solved is whether such a finite-dimensional controller can produce a stable closed-loop system. In this paper, results are presented which characterize the linear DPS which can be (exponentially) stabilized by a finite-dimensional controller. This characterization makes use of stabilizing subspaces and the solvability of the nonlinear asymmetric Riccati equation. Connections with the usual model reduction approaches to finite-dimensional DPS controller design are also developed.

Optimal Control of Linear Distributed Parameter Systems by Shifted Legendre Polynomial Functions

The optimal control problem of a linear distributed parameter system is studied by employing the technique of shifted Legendre polynomial functions. A partial differential equation, which represents the linear distributed parameter system, is expanded into a set of ordinary differential equations for coefficients in the shifted Legendre polynomial expansion of the input and output signals. Expressing the performance index in terms of the expansion coefficients, we transformed an optimal control gain problem into a two point boundary value problem by applying the maximum principle. The two-point boundary value problem is reduced into an initial value problem, the solution of which can be easily obtained by the proposed computational algorithm. An illustrative example will be used to prove this point.

Feedback Control of Linear Distributed Parameter Systems: What can be Accomplished with a Finite-Dimensional Controller?

Feedback Control of Linear Distributed Parameter Systems: What can be Accomplished with a Finite-Dimensional Controller?

Optimal experiment design for identification of linear distributed-parameter systems: Frequency domain approach

In this paper optimality conditions for experiment design are derived. The experiment is planned for identification of linear time invariant distributed-parameter systems. The determinant of the averaged information matrix is expressed in the terms of input spectral density matrix and spatial density of measurements, and then used as a measure of estimation accuracy. Presented results are applied to find optimal sensors positions and input signals in several examples.

Optimal Saturating Control of Distributed Parameter Systems

Optimization of linear distributed parameter systems, with finite dimensional control input subject to component saturation constraints, is considered. An integral cost is introduced, with integrand comprised of quadratic and non-quadratic state penalty terms. The latter (non-quadratic) terms are chosen so as to regularize the problem in the sense that an optimal solution, of a simple feedback form, is obtained. Finite and infinite time horizon problem formulations are examined and two examples are presented.

Optimal smoothing in linear distributed parameter systems via boundary measurements containing coloured noise

Optimal smoothing in linear distributed parameter systems via boundary measurements containing coloured noise

Symmetry reduction of linear distributed parameter systems

The well-known symmetry principles and group theoretic methods of mathematical physics are applied, from a control point of view, to the analysis of linear distributed parameter systems (LDS) endowed with some degree of geometric symmetry. The connection between properties of symmetry groups and controllability, observability and decoupling properties of a class of LDS is discussed and illustrated by an example.

Existence and comparison theorems for partial differential equations of Riccati type

In this paper, we discuss the partial differential equation of Riccati type that describes the optimal filtering error covariance function for a linear distributed-parameter system with pointwise observations. Since this equation contains the Dirac delta function, it is impossible to apply directly the usual methods of functional analysis to prove existence and uniqueness of a bounded solution. By using properties of the fundamental solution and the classical technique of successive approximation, we prove the existence and uniqueness theorem. We then prove the comparison theorem for partial differential equations of Riccati type. Finally, we consider some applications of these theorems to the distributed-parameter optimal sensor location problem.

Optimal control of linear distributed parameter systems by finite‐element Galerkin's technique

AbstractThe paper considers a finite‐element discretization for solving two distributed parameter system optimal control problems. Linear and quadratic finite‐element approximations are used in conjunction with Galerkin's method. The first problem involves heating a massive body in a furnace and takes into account furnace dyanmics. The input (fuel flow rate) is magnitude constrained, and the performance index is the terminal cost. The second problem pertains to a plug‐flow heat exchanger and is solved for end‐point control and control along the entire heat exchanger. Uniform wall flux and wall temperature serve as the control variables. The results are compared with earlier results, and advantages of the present approach are discussed.

Controllability, observability and duality in infinite dimensional linear systems with multiple norm-bounded controllers

Controllability of infinite-dimensional linear systems with multiple independent controllers, each of which is norm-bounded, is studied. Necessary and sufficient conditions for systems to be such are presented. Both the usual (cooperative) and the game (non-cooperative) controllability are examined. The problem of sending an initial state to some target sets is investigated. Applications to differential delay linear systems and simple distributed parameter systems are given. For the cooperative mode, a duality theory is introduced for the constrained infinite dimensional case.

Filtering of distributed parameter systems with pointwise disturbances

We consider in this article a filtering problem for a linear distributed parameter system where the input disturbances are located in points instead of being distributed. The observation process is assumed to be distributed inside the domain. Although formally one can derive equations for the Kalman filter and the Riccati equation, the complete justification of these equations leads to problems which cannot be solved by standard methods, in the sense that some new functional spaces need to be introduced. The objective of the paper is to give a rigorous treatment of this nonclassical filtering problem, which is relevant for applications.

Optimal Allocation of Controllers and Sensors in Linear Distributed Parameter Systems

Optimal Allocation of Controllers and Sensors in Linear Distributed Parameter Systems

Optimal filtering and smoothing algorithms for linear distributed-parameter systems with pointwise observation

Abstract Sequential algorithms for filtering, fixed-interval, fixed-point and fixed-lag smoothing are developed for a class of linear distributed-parameter systems. The class of systems concerned is that involving noisy measurement data which are obtained from a finite number subdomain in the spatial domain, i.e. pointwise sensor. The basic tools of the development are the Wiener-Hopf equation, innovation theory and the fundamental value theory for the differential operator. The fixed-lag smoother obtained is new. Finally, for the numerical solution, the expansion method of eigen-functions in L2(D) and L2(D × D) is asserted, and it is shown that the results obtained here are general forms, which contain the results of the finite state space reported earlier.

An alternative approach to the derivation of distributed-type partitioned filters

Abstract In a recent paper Watanabe et al. (1980) considered the filtering for linear distributed-parameter systems with non-gaussian initial state via the ‘ partition theorem ’. It is shown here that the partitioned filtering equations for such systems are obtainable in a simpler manner by using the concept of the fundamental solution matrix for the distributed-type Kalman filter.

The duality method for optimal control problems of distributed parameter systems with state constraints

In this paper we consider the optimal control problems of linear distributed parameter systems of parabolic type with inequality state constraints. We apply the duality methods and transform the original infinite-dimensional problems to the dual finite-dimensional problems. We solve the following three types of optimal control problem. (1) The determination of input-time functions with minimum energy. (2) The determination of input-distribution functions with minimum norm. (3) The determination of initial states with minimum norm. Several illustrative examples are given with numerical results.

On Optimal Location of Spatially-Fixed Sensors in a Class of Linear Distributed Parameter Systems

On Optimal Location of Spatially-Fixed Sensors in a Class of Linear Distributed Parameter Systems

Filter implementation for a linear stochastic distributed-parameter system with a moving boundary

This paper is concerned with the filter implementation for a linear stochastic distributed-parameter system with a moving boundary. In the system considered here, it is assumed that the states of the system are expressed by partial differential equations, and the location of the moving boundary is expressed by an ordinary differential equation. The eigenfunction expansion method to decrease computation volume and to stabilize numerical computation for the filter implementation is introduced as follows; firstly, the estimates and their covariances are approximately expressed as a finite sum of eigenfunction series, and secondly ordinary differential equations with respect to Fourier coefficients of eigenfunctions are derived. The simulation study compares the practical applicability of the proposed method with the finite difference scheme.

Parameter adaptive control of stochastic distributed systems

Abstract The parameter adaptive control problem of linear stochastic distributed-parameter systems is considered and treated by invoking the partitioning/non-linear separation approach. Both continuous-time and discrete-time systems are studied, with volume and/or boundary control and disturbance inputs. The following cases are considered: spatially-continuous and point-wise measurements/control actions, continuous and quantized parameter space, non-gaussian initial state, uncertainty in continuous and point-wise measurements, cost-function with spatial-derivative penalties, systems with pure time delays, composite distributed- and lumped-parameter systems, scanning-type control, and simultaneous adaptive system and measurement control. The uncertain parameters are assumed to be time-and-space independent, but the spatial independence can be alleviated as will be shown elsewhere.

Stochastic distributed systems with point observations and boundary control: an abstract theory

There already exists a fairly complete theory for the problems of estimation and stochastic optimal control for linear distributed parameter systems, with Gaussian or non Gaussian noise disturbance. In [8] and [12] generalizations of the familiar finite dimensional results of the Kalman-Bucy filter and the separation principle are obtained using an abstract input-output Hilbert space representation for the system. However, in [8] and [12] all the input operators are assumed to be bounded and so it does not cover the important practical cases of control and noise on submanifolds of the spatial domain or point observations. Here we introduce unbounded system operators in the abstract input-output Hilbert space representation and thus extend all the results of [8] and [12] to allow for point observations and noise and control on submanifolds including the boundary. The theory is illustrated by several examples.

Fixed-lag smoothing estimator for a linear distributed parameter system

The fixed-lag smoothing estimator for a linear distributed parameter system with a noisy observation at discrete points on the spatial domain or its boundary is derived from the Wiener-Hopf theoretical viewpoint. The new feature of this paper is that the derivation of the optimal smoothing estimator is based only on the Wiener-Hopf theory. Using the Wiener-Hopf equations both for the optimal smoothing problem and for the optimal filtering problem, the optimal fixed-lag smoothing estimator which corresponds to the results obtained by using Kalman's limiting procedure is derived.