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Linear quadratic optimal control problem for linear systems with unbounded input and output operators: Numerical approximations
... There does exist some theory, similar to that described in Section 2 for systems with unbounded control operators, e.g., [26,27]. However, most of this theory only applies to systems with analytic semigroups and the semigroup associated with this system is not known to be analytic [23]. ...
Calculation of the solutions to linear quadratic optimal control problems for infinite-dimensional systems is considered. The sequence of solutions to a sequence of approximating finite-dimensional problems converges to the optimal control for the infinite-dimensional system if certain assumptions such as uniform stabilizability are satisfied. We use this result to calculate controllers for the infinite-dimensional system that are arbitrarily close to optimal. A one-dimensional heat equation and a problem of acoustic noise control are used to illustrate the algorithm. The numerical results are discussed.
The present Part II is a continuation of Part I of the paper bearing the same title. Part I referred to the continuous case. The present Part II refers to the corresponding discrete problem. To emphasize the continuity of the two parts, we keep a progressive numbering of the sections.
We study an approximation of the boundary control problem for the heat equation over a finite horizon. Our goal is to obtain an approximation of the value function and of the corresponding “locally optimal” trajectories. We examine here a time discretization also proving some a priori estimates of convergence for the value function of the time-discrete problem. Some hints are also given for the construction of a fully discrete scheme.
In this paper we report on the results of our continuing efforts on using the averaging approximation scheme for retarded functional differential equations. The central focus of this paper is our numerical studies of constructing feedback soLutions to linear quadratic regulator (LQR) problems for retarded systems with delay in control. For completeness, we shall also give a brief summary and discussion of an abstract approximation framework and convergence theory developed previously by Ito and Tran in[9]. In [9]we presented an approximation framework for the numerical treatment of algebraic Riccati equations for a class of linear infinite dimensional systems with unbounded input and output operators studied by Pritchard and Salamon in[19] In this paper we will call it the Pritchard-Salamon clans. This approximation theory which yields convergence of the approximating Riccati operators as well as convergence of the approximating gain operators extends earlier results developed in[7][2][8]in which the input and output operators are assumed to be bounded to the unbounded cases. The main features which distinguish the work in [9]from other work existing in the literature, see e.g.[11][14][15] are the assumptions on the smoothness of the underlined semigroup and the observation map. Because of the smoothness assumptions, the algebraic Riccati soLution has a smoothing property which in turn implies boundedness of the feedback gain operator. Although the theory developed in[9]does not cover many important boundary control problems studied by Lasiecka and Triggiani in [13] and Flandoli, Lasiecka, and Triggiani in[6] for example, it does enable us to treat the control problem governed by delay differential equations with delays in control and observation.
The linear quadratic regulator problem (LQR-problem) for delay systems is a topic for intensive research efforts since the early sixties. The theory now has reached a degree of maturity which justifies a survey on the main results obtained so far. Because of limitations in space (for these proceedings) and in time (on the author’s side) it is not our intention to give a complete survey (and correspondingly a complete list of references). We rather concentrate on the main results being well aware that our choice is to a large degree motivated by personal taste.
In this paper we consider several control problems governed by Burgers’ equation. We use a linearized model to compute feedback control laws and apply these laws to the full nonlinear model. We investigate problems with unbounded inputs and observations. A finite element scheme is used to compute feedback gains and several numerical experiments are performed to illustrate the theoretical results.
In this paper, the averaging approximation scheme for linear retarded functional differential equations with delays in control and observation is considered in the context of the state space theory developed by Pritchard and Salamon [SIAM J. Control Optim., 25 (1987), pp.~121--144]. Using known results from linear semigroup theory, convergence and estimate of convergence rate of the approximating semigroups are established. These extend results due to Banks and Burns [SIAM J. Control Optim., 16 (1978), pp.~169--208] and Lasiecka and Manitius [SIAM J. Numer. Anal., 25 (1988), pp.~883--907] on hereditary systems with delays in state, to the case when delays in control and observation are included. The main difference from the case when delays in input and output are excluded is that unbounded input and output operators must be dealt with in the abstract formulation. Moreover, in the presence of the unboundedness of the input and output operators, new convergence results of the state solutions and the output are also obtained.
The parametric-mixed-sensitivity problem is posed for the class of irrational transfer matrices with a realization as a Pritchard-Salamon system. An exact solution is obtained in terms of two uncoupled, standard Riccati equations.
This paper describes three theories for robust controller design for infinite-dimensional linear systems. They concern unstructured
perturbations in the transfer function of additive, multiplicative and coprime-factor types. Explicit formulas are given for
the maximal robustness margin, as well as for both infinite-dimensional and finite-dimensional robustly stabilizing controllers
which achieve an a priori robustness margin.
In this note, a H∞ controller for systems with state delay is presented. For a prechosen γ the controller is obtained by solving Riccati partial differential inequalities (RPDIs). For small time delays, an asymptotic approximation of the controller is achieved by expanding the solution in powers of the delay. It is shown that a higher-order accuracy controller improves the performance. An example is brought where the results of the new method provide a satisfactory solution for a delay length comparable with the system ‘bandwidth’. The performance of the system under the zero-order accuracy controller, which corresponds to systems without delay, is studied. Explicit formula for the guaranteed performance level is obtained for the delay lengths that preserve the internal stability of the system.
The weighted mixed-sensitivity H∞-control problem we consider is {\gamma ^w} = \begin{array}{*{20}{c}} {\inf } \\ {K \in \delta } \end{array}{\left\| {\begin{array}{*{20}{c}} {{W_1}{{(I + GK)}^{ - 1}}V} \\ {{W_2}K{{(I + GK)}^{ - 1}}V} \end{array}} \right\|_\infty }, (1.1) where G is the transfer matrix of the plant, K is the transfer matrix of the controller, W
1, W
2 and V are certain rational weighting transfer functions and δ the class of stabilizing controllers. The weighted mixed-sensitivity H
∞
-control problem is well understood for rational transfer matrices (see Kwakernaak [15] for a tutorial) and there are standard routines for its numerical solution in MATLAB [1]. It is the purpose of this paper to develop numerical solutions for a class of irrational transfer matrices. We suppose that G is irrational, but it has the decomposition (1.2)where G
f
is rational (possibly unstable) and G
S
has components in H
∞
. In Kwakernaak [15], it is argued convincingly that (1.1) is the most useful mixed-sensitivity design for applications, and we refer to this paper for the engineering background and for a discussion of the weights W
1W
2 and V.
A feedback control computational methodology for reducing acoustic
sound pressure levels in a two-dimensional cavity with a flexible
boundary (a beam) is investigated. The control is implemented in this
model through voltages to piezoceramic patches on the beam which are
excited in a manner (out-of-phase) so as to produce pure bending
moments. The incorporation of the output feedback control in this system
leads to a problem with unbounded input and output terms. By writing the
resulting system as an abstract Cauchy equation, the problem of reducing
interior pressure levels can be posed in the context of an H<sub>∞
</sub>/MinMax time-domain state-space formulation. A summary of
extensive computational efforts comparing output feedback to full state
feedback and investigating the effect of the number and location of
sensors on performance of the control is presented
We present a variational framework based on sesquilinear forms for Galerkin approximation techniques for state feedback control in problems governed by infinite dimensional dynamical systems. Both parabolic and second order in time, hyperbolic partial differential equations with unbounded input and unbounded observation operators are included as special cases of our treatment. 1 Introduction In this paper we discuss the linear quadratic regulator (LQR) problem for a class of (essentially parabolic) unbounded input or boundary control problems. A variational framework using sesquilinear forms is developed to treat Dirichlet and Neuman boundary control problems for parabolic equations and strongly damped elastic systems. Using such a framework, convergence of Galerkin approximations to solutions of Riccati equations is also established. The boundary control problem for parabolic systems has been studied extensively over the last two decades, inspired by the monograph of J.L. Lions [21] ...
A feedback control computational methodology for reducing acoustic sound pressure levels in a 2-D cavity with a flexible boundary (a beam) is investigated. The control is implemented in this model through voltages to piezoceramic patches on the beam which are excited in a manner (out-of-phase) so as to produce pure bending moments. The incorporation of the output feedback control in this system leads to a problem with unbounded input and output terms. By writing the resulting system as an abstract Cauchy equation, the problem of reducing interior pressure levels can be posed in the context of an H 1 /MinMax time domain state space formulation. A summary of extensive computational efforts comparing output feedback with full state feedback and investigating the effect of the number and location of sensors on performance of the control is presented. 1 Introduction In this paper we continue our earlier studies on the active control of a 2-D structural acoustic model through the use of p...
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