On the asymptotic stability of a particular class of distributed parameter feedback control systems†

In this paper the stability problem for a particular class of non-linear feedback control systems governed by a parabolic partial differential equation is considered. A sufficient condition for global asymptotic stability of the null equilibrium state is derived by the method of comparison functions.     

Optimally sensitive control for distributed parameter systems†

The objective of this paper is the extension of optimally sensitive control techniques to distributed parameter systems. In particular, these adaptive control techniques are applied to distributed systems in which either the initial states or a vector of constant plant parameters are unknown. The result is the synthesis of a feedback control structure that adapts the nominal open-loop control in such a manner as to compensate for the errors introduced by model uncertainties. This requires defining the sensitivity matrices and deriving a set of distributed equations whose solution yields the requisite sensitivity matrices.     

Feedback control of distributed parameter systems with spatially concentrated controls†

The problem of designing an optimal feedback controller is discussed for a distributed parameter system governed by a partial differential equation of parabolic type. A quadratic performance index is evaluated with spatially concentrated and time-discrete control, and controllability of this system is investigated. Also an estimator with multi-pointwise observation is constructed. Computational algorithms of these controller and estimator are derived, and to show their validities, a few numerical examples are given     

Optimal control of distributed parameter systems with discrete constrained inputs†

The optimal control of linear distributed parameter systems, which are represent able by a linear vector integral equation, is investigated. Restricting the control action to be discrete in time, the problem of minimizing the mean-square error, between specified desired final state functions and the actual state functions at a prescribed final time, subject to an energy constraint on the controlling functions, is treated. A necessary and sufficient condition from functional analysis is used to derive an equation whose solution yields the optimal control vector. Two convergence properties for the discrete problem are established which can be used to determine a good approximate solution to the corresponding measurable optimal control problem. An illustrative example is given.     

Synthesis of sub-optimal feedback controls for a class of distributed parameter systems†

Technique are derived for the synthesis of sub-optimal feedback controls for parabolic and first-order hyperbolic systems. An explicit result for the time-invariant gain of a specified controller is obtained by a least square approximation of the closed-loop control to the optimal open-loop control. If a least square approximation of the state trajectories is used, a parameter search is shown to give the time-invariant gain. A time-varying gain can be obtained by a re-definition of the original optimal control problem, again with the controller functionality specified. The only require-mont in the closed-loop synthesis is that an optimal open-loop solution exists and is computable.     

Weighted residual methods and the suboptimal control of distributed parameter systems†

A suboptimal control algorithm for distributed parameter systems is developed in a framework which synthesizes weighted residual methods and mathematical programming. The heat exchanger example of Koppel et al. (1968) is employed for introducing the algorithm. First, the Galerkin procedure with polynomial modes is applied to obtain a lumped ODE model for the distributed parameter system. Then the state and control variables of the lumped control problem are approximated by cubic splines on a uniform mesh. Through collocation at the knots, the ODE model is reduced to a sot of linear algebraic equations and the suboptimal control is determined from the solution of a quadratic programming problem with sparse matrices.

Numerical results for the heat exchanger example are presented and compared with those obtained by the authors (Neuman and Sen 1972) using the Ritz-Trefftz algorithm (Bosarge and Johnson 1970) for the lumped control problem. For this example, the two algorithms yield essentially identical results with comparable computational requirements. Application of the Ritz-Trefftz algorithm, however, is limited to lumped, linear-quadratic control problems without constraints on the state or control. The approach advocated in this paper, therefore, offers a viable approach to control problems in distributed parameter systems.     

On the optimal control of two classes of non-linear distributed parameter systems†

An optimal control problem, for two classes of non-linear distributed parameter systems, is formulated. Sufficient conditions on the optimal control are derived. These conditions form a two-point boundary value problem of partial differential equations.     

On the optimal control for a class of distributed parameter systems†

An optimal control problem for a class of distributed parameter systems is discussed. A non-linear integral equation of a signum type is derived as a necessary and sufficient condition for the optimal control. Under very weak conditions, existence and unique ness of the solution to this non-linear integral equation are proved.     

On the controllability of distributed parameter systems†

The reachable states of distributed parameter systems where the magnitude of the control energy is constrained are found. The systems considered are described by linear partial differential equations on a bounded domain. The controls may be either distributed or at the boundary and are required to satisfy a constraint in the magnitude of their norm. The results are then used to find conditions for null controllability.     

The bounded energy optimal control for a class of distributed parameter systems†

The bounded energy optimal control for one-dimensional linear stationary distributed parameter system is solved here. The criterion function is a quadratic functional of the output.

Obtaining the optimal control involves the computation of the solution of a certain non-linear integral equation. The method of solving this integral equation is approximating the kernel of the integral operator by a sequence of degenerate kernels. It is shown that the sequence of approximate solutions of the approximate integral equations converges to the optimal solution; and that the sequence of approximate values of the criterion, converges to the optimal value of the criterion.     

A minimax problem for distributed parameter systems†

The problem considered in this paper is the infinite dimensional analogue of Ho. We use the results of Ho and show that, under suitable conditions, the optimal pair of controls for finite dimensional systems converges to the optimal pair for infinite dimensional systems.     

An optimization problem in distributed parameter systems†

The optimal control problem for a furnace heating a one-dimensional slab with a quadratic performance index is analysed. This system is a typical distributed parameter system. The Hamiltonian is defined and the canonical equations are obtained. A Riccati type matrix partial differential equation is obtained from the canonical equations. An approximate method to solve these equations is derived and an example is presented to illustrate this method.     

High-gain feedback in non-linear control systems†

High-gain state and output feedback are investigated for non-linear control systems with a single additive input by using singular perturbation techniques.

Classical approximation results (Tihonov-like theorems) in singular perturbation theory are extended to non-linear control systems by defining a composite additive control strategy, a control-dependent fast equilibrium manifold and non-linear change of coordinates.

Those tools and an appropriate change of coordinates show that high-gain state feedback and variable structure control systems can be equivalently used for approximate non-linearity compensation in feedback-linearizable systems.

Next the effect of high-gain output feedback is shown to be related to the strong invertibility property and the relative order of invertibility. For strongly invertible systems the slow reduced subsystem coincides with the dynamics of the inverse system when zero input is applied and with the unobservable dynamics when a certain input-output feedback-linearizable transformation is applied.     

Analysis of a class of non-linear and time-varying parameter feedback systems†

The paper presents some simple methods of analysis for a class of feedback systems. A method of feedback iteration is presented, directly applicable to non-linear and time-varying parameter feedback systems, without involving any intermediate formulations to obtain the response to a given input. Then, a numerical method closely following that of Naumov is described. Next, methods of predetermining the time-varying parameters for specified output are presented. One is a direct method of backward signal flow and the other is a numerical method based on that of Naumov (1957). These methods have the advantage of lending themselves to digital computation. The method of successive iteration round the feedback loop was first used in the classic paper by Nyquist (1932).     

Design of dominant-type control systems with large parameter variations†

This paper presents a design procedure for dominant-type systems with large plant parameter variations. The principal contribution is that a fourth-order approximation is used in the dominant region instead of a third-order, which up to now had been the most advanced method. The s-domain specifications of the system are assumed to be in the form of an acceptable dominant closed-loop pole region with bounds on the location of the ‘far-off’ closed-loop poles. The design philosophy is to place compensation zeros within the acceptable dominant closed-loop pole region such that the dominant closed-loop poles remain within their prescribed region despite the large variation in the plant parameters. The design procedure is for plants with simultaneous independent variation in the gain factor and a pair of poles. The design is such as to minimize the sensitivity of the system to internal noise.     

Stability analysis of a class of non-linear distributed parameter systems—tubular chemical reactors†

A sufficient condition for local stability of the spatially discretized model of a class of non-linear distributed parameter systems is derived using the circle criterion for stability and bounded-input, bounded-output stability. As a physical system the catalytic packed-bed tubular reactor is taken to illustrate the application of the theorem to process control systems. The stability condition in the large derived from non-linear system equations is compared with the stability condition in the small derived from linearized system equations. Interpretations of the result are given in terms of physical variables.     

On the design of optimal output feedback control systems†

For a linear control system with quadratic performance index the optimal control takes a feedback form of all state variables. However, if there are some states which are not fed in the control system, it is impossible to obtain the optimal feedback control by using the usual mathematical optimization technique such as dynamic programming or the maximum principle.

This paper presents the optimal control of output feedback systems for a quadratic performance index by using a new parameter optimization technique.

Since the optimal feedback gains depend on the initial states in the output feedback control system, two cases where (1) the initial states are known, and (2) the statistical properties of initial states such as mean and covariance matrices are known, are considered here. Furthermore, the proposed method for optimal output feedback control is also applied to sampled-data systems.     

A note ‘ On the controllability of distributed parameter systems ’†

The aim of this note is to give a simple proof of an improved anil equivalent result to the one derived in the article under consideration, Herget (1970).     

Optimization of a non-linear distributed parameter system using periodic boundary control†

Conditions for loeal optimality are worked out using simple calculus of variations. To find the optimum control, a two-point boundary value problem in space and time has to be solved, which involves the solution of the adjoint differential equation together with the prooess equation. The method is applied to the optimization of a periodic process, consisting of a tubular reactor where a second-order homogeneous reaction takes place and a periodicity oondition of the state is satisfied everywhere along the reactor. Tho plug flow and the diffusion model are assumed. In the first case an exact solution is carried out. The improvements in yield compared with steady-state conditions are obtained and shown in graphs.