Finite Dimensional Compensator Research Articles

Compensator design for the monodomain equations with the FitzHugh−Nagumo model

The problem of finite-dimensional compensator design for the monodomain equations with the FitzHugh−Nagumo model is investigated. Exponential stabilizability and detectability of the linearized infinite-dimensional control system is studied. It is shown that the system is not exactly null-controllable but still can be exponentially stabilized by finite-rank input and output operators provided the desired stability margin is small enough. Based on existing results on model order reduction of infinite-dimensional systems, a finite-dimensional compensator is obtained by LQG-balanced truncation. Using partial measurements, the compensator produces a feedback control that is shown to be locally stabilizing for the infinite-dimensional nonlinear control system. Examples motivated by cardiophysiology are used to illustrate these results in a numerical setup.

Alternative algebraic approach to stabilization for linear parabolic boundary control systems

In this paper, we develop an alternative algebraic approach to feedback stabilization problems of linear parabolic boundary control systems. The control system is general in the sense that no Riesz basis associated with the elliptic operator is expected. The control system contains a finite dimensional dynamic compensator, the dimension of which has been so far determined only by the actuators on the boundary. One of the purposes of the paper is to show a new alternative scheme that the dimension of the compensator could be determined by the sensors on the boundary: it leads to a realization of stabilizing compensators of lower dimension by comparison. This is achieved by introducing a new operator equation connecting two states of the controlled plant and the compensator. The other purpose is a generalization of the condition posed on the sensors. When the elliptic operator admits generalized eigenfunctions, the sharpest criterion is proposed on the minimum choice of the necessary number of the sensors. In fact, only one sensor is enough for the stabilization in the best case (the algebraic multiplicities of the corresponding eigenvalues may, of course, be greater than 1).

Finite-dimensional dual state feedback control of linear boundary control systems

In this article, the finite-dimensional control of distributed-parameter systems with boundary control and unbounded observation is considered. This problem is solved by extending the concept of dual observer-based compensators to infinite dimensions. These compensators have the advantage that, on the one hand, the separation principle can be used to directly determine a finite-dimensional compensator and, on the other hand they can be directly applied to boundary control systems with a time-derivative of the input. In order to achieve a low controller order as well as a desired control performance, a parametric approach is proposed to parameterise the degrees of freedom contained in the controller. Consequently, desired design specifications can be achieved by using a parameter optimisation. The proposed design of finite-dimensional dual observer-based compensators is demonstrated by means of a heat-conducting system with Neumann boundary control and boundary measurements.

A parametric approach to finite-dimensional control of linear distributed-parameter systems

This article considers the design of finite-dimensional compensators for distributed-parameter systems using eigenvalue assignment. The proposed compensator consists of an observer estimating additional outputs and a static feedback of the measurable and the estimated outputs. Since the additional outputs can be asymptotically reconstructed, the compensator can be designed using the separation principle, i.e. the closed-loop eigenvalues are given by the observer eigenvalues and the eigenvalues resulting from the static output feedback control. In order to solve the corresponding eigenvalue assignment problem, the parametric approach for the design of static output feedback controllers in finite-dimensions is extended to distributed-parameter systems. By using a parameter optimisation it is possible to assign all closed-loop eigenvalues within specified regions of the complex plane in order to stabilise the system and to assure a desired control performance. A heat conductor is used to demonstrate the proposed design procedure.

Stabilization of a class of partially observed infinite-dimensional systems with control constraints

The stabilization by a finite-dimensional compensator for a class of infinite-dimensional linear systems with control inequality constraints is investigated. The main result shows that the corresponding state feedback results cannot be directly extended to the composite system including a full-state observer. However, we get conditions of asymptotic stability for a particular subclass of systems with control constraints by an appropriate use of the positive invariance concept.

A compensator design controlling neutron flux distribution via observer theory

To suppress the spatial xenon oscillations in a nuclear reactor, an implementable stabilization scheme is proposed based on the finite dimensional compensator theory in control theory for the distributed parameter systems. The method is applied to a one-dimensional reactor whose dynamics is governed by one-group diffusion equation with its associated iodine and xenon dynamics. The modal decomposition of the state variables enables us to use the pole assignment algorithms developed in finite dimensional systems to obtain the stabilizing compensator gains. This allows us to estimate the states of a reactor in a transient using output measurement data and arbitrary initial conditions, and control the states using the estimated values. The resulting compensator is tested by using model-based data for measurement output through numerical simulations. The results show that unstable spatial xenon oscillations initiated by perturbations can be controlled by the finite dimensional compensator.

Finite-Dimensional Compensators for the $H^\infty$-Optimal Control of Infinite-Dimensional Systems via a Galerkin-type Approximation

We study the existence of general finite-dimensional compensators in connection with the $H^{\infty}$-optimal control of linear time-invariant systems on a Hilbert space with noisy output feedback. The approach adopted uses a Galerkin-type approximation, where there is no requirement for the system operator to have a complete set of eigenvectors. We show that if there exists an infinite-dimensional compensator delivering a specific level of attenuation, then a finite-dimensional compensator exists and achieves the same level of disturbance attenuation. In this connection, we provide a complete analysis of the approximation of infinite-dimensional generalized Riccati equations by a sequence of finite-dimensional Riccati equations. As an illustration of the theory developed here, we provide a general procedure for constructing finite-dimensional compensators for robust control of flexible structures.

Finite Element Compensators for Thermo-Elastic Systems with Boundary Control and Point Observation

We consider a thermo-elastic plate equation on a bounded domain Ω, subject to a boundary control as a ‘bending moment’, and with point observation in the interior of Ω The free dynamics generates a s.c. analytic semigroup on the natural energy space. Both control operator and observation operator are fully unbounded, with a combined degree of unboundedness > 1, the super-critical case. We construct an approximation theory, based on FEM, which leads to a finite-dimensional dynamic compensator, possessing all the expected desirable properties, once it is inserted into the original continuous system. In particular, it preserves the same margin of stability of the continuous problem, except, perhaps, for an arbitrary ε>0.

A Reduced Basis Approach to the Design of Low-Order Feedback Controllers for Nonlinear Continuous Systems

In this paper, the authors discuss a reduced basis approach to the development of low-order nonlinear feedback controllers for hybrid distributed parameter systems. This approach involves the use of distributed parameter control theory to design "optimal" infinite dimensional feedback control laws and approximation theory to design and compute low-order finite dimensional compensators. The resulting finite dimensional controller combines a nonlinear observer with a linear feedback law to produce a practical design. The authors concentrate on a weakly nonlinear distributed parameter system to illustrate the ideas.

A Compensator Design Controlling Neutron Flux Distribution via Observer Theory

To suppress the spatial xenon oscillations in a nuclear reactor, an implementable stabilization scheme is proposed based on the finite dimen- sional compensator theory in control theory for the distributed parameter systems. The method is applied to a one-dimensional reactor whose dynamics is governed by one-group diffusion equation with its associated iodine and xenon dynamics. The modal decomposition of the state variables enables us to use the pole assignment algorithms developed in finite dimensional systems to obtain the stabilizing compensator gains. This allows us to estimate the states of a reactor in a transient using output measurement data and arbitrary initial conditions, and control the states using the estimated values. The resulting compensator is tested by using model-based data for measurement output through numerical simulations. The results show that unstable spatial xenon oscillations initiated by perturbations can be controlled by the finite dimensional compensator.

A compenasator design controlling neutron flux distribution via observer theory

To suppress the spartial xenon oscillations in a nuclear reactor, an implementable stabilization scheme is proposed based on the finite dimensional compensator theory in control theory for the distributed parameter systems. The method is applied to a one-dimensional reactor whose dynamics is governed by one-group diffusion equation with its associated iodine and xenon dynamics. The modal decomposition of the state variables enables us to use the pole assignment algorithms developed in finite dimensional systems to obtain the stabilizing compensator gains. This allows us to estimate the states of a reactor in a transient using output measurement data and arbitrary initial conditions: and control the states using the estimated values. The resulting compensator is tested by using model-based data for measurement output through numerical simulations. The results show that unstable spatial xenon oscillations initiated by perturbations can be controlled by the finite dimensional compensator.ated by perturbations can be controlled by the finite

Minimax control of switching systems under sampling

We consider a general class of systems subject to two types of uncertainty: A continuous deterministic uncertainty that affects the system dynamics, and a discrete stochastic uncertainty that leads to jumps in the system structure at random times, with the latter described by a continuous-time finite state Markov chain. When only sampled values of the system state is available to the controller, along with perfect measurements on the state of the Markov chain, we obtain a characterization of minimax controllers, which involves the solutions of two finite sets of coupled PDEs, and a finite dimensional compensator. For the linear-quadratic case, a complete characterization is given in terms of coupled generalized Riccati equations, which also provides the solution to a particular H ∞ optimal control problem with randomly switching system structure and sampled state measurements.

Finite Element Approximations of Compensator Design for Analytic Generators with Fully Unbounded Controls/Observations

An approximation theory leading to a design of a finite-dimensional compensator for control systems generated by analytic semigroups is presented. The novelty of this paper with respect to other results available in the literature is threefold: (i) it treats fully unbounded control/observation operators; (ii) it does not require compactness property of the underlined generator (an assumption that is often violated in practice); and (iii) the design of a finite-dimensional compensator is based on finite element approximation of the original model rather than on modal (eigenfunctions) approximations which, in turn, require the a priori knowledge of the eigenvalues for the system. Applications of the theory to heat equations and plate equations are provided.

Compensator design of semi-linear parabolic systems

The stabilization of semi-linear parabolic distributed systems by means of boundary controls is treated, formulating the partial differential equation describing the system as an evolution equation in a Hilbert space. Here, by using the properties of an analytic semigroup, the global existence of the solution under static output feedback is shown. Moreover, in the similar way, the global existence of the solution and the stabihzability for the system which installs a finite dimensional dynamic compensator is shown.

A compensator design controlling neutron flux distribution via observer theory

To suppress the spatial xenon oscillations in a nuclear reactor, an implementable stabilization scheme is proposed based on the finite dimensional compensator theory in control theory for the distributed parameter systems. The method is applied to a one-dimensional reactor whose dynamics is governed by one-gorup diffusion equation with its associated iodine and xenon dynamics. The modal decomposition of the state variables enables us to use the pole assignment algorithms developed in finite dimensional systems to obtain the stabilizing compensator gains. This allows us to estimate the states of a reactor in a transient using output measurement data and arbitrary initial conditions, and control the states using the estimated values. The resulting compensator is tested by using model-based data for measurement output through numerical simulations. The results show that unstable spatial xenon oscillations initiated by perturbations can be controlled by the finite dimensional compensator.

Approximation in Control of Thermoelastic Systems

This paper develops an abstract framework for analysis and approximation of linear thermoelastic control systems, and for design of finite-dimensional compensators. The thermoelastic systems in this paper consist of abstract wave and diffusion equations coupled in a skew self-adjoint fashion. Linear semigroup theory is used to establish that the abstract thermoelastic models are well posed and to prove convergence of generic approximation schemes. Open-loop uniform exponential stability for a subclass of thermoelastic systems is proved via a Lyapunov function. An example involving the design of an optimal linear-quadratic-Gaussian (LQG) compensator for a thermoelastic rod illustrates the application of the abstract theory. Results of an extensive numerical study, including a comparison of the closed-loop performance of different compensator designs, are presented and discussed.

Finite-dimensional compensators for linear distributed control systems with delays in outputs

In this paper we are concerned with linear control systems of infinite dimension with delays in outputs. An asymptotic compensator of finite order is proposed. The compensator provides a control law that stabilizes a wide class of distributed systems. Moreover, the compensator is invariant with respect to small bounded perturbations of the system parameters.

Boundary Stabilization of Semilinear Parabolic Distributed Systems

Abstract We treat, in this paper, the stabilization of semilinear parabolic distributed systems by means of boundary controls. We formulate the partial differential equation describing the system as an evolution equation in a Hilbert space. Hero, by using the properties of an analytic semigroup, we show the global existence of tho solution under static output feedback. Moreover, in the similar way. we show the global existence of the solution and the stabilizability for the system which installs a finite dimensional dynamic compensator.

Approximation Theory for Linear-Quadratic-Gaussian Control of Flexible Structures

This paper presents approximation theory for the linear-quadratic-Gaussian optimal control problem for flexible structures whose distributed models have bounded input and output operators. The main purpose of the theory is to guide the design of finite-dimensional compensators that approximate closely the optimal compensator, which is infinite-dimensional. Design of the optimal compensator separates into an optimal linear-quadratic control problem and a dual optimal state estimation problem; the solution to each problem lies in the solution to an infinite-dimensional Riccati operator equation. The approximation scheme in the paper approximates the infinite-dimensional LQG problem with a sequence of finite-dimensional LQG problems defined for a sequence of finite-dimensional, usually finite-element or modal, approximations of the distributed model of the structure. Two Riccati matrix equations determine the solution to each approximating problem. The finite-dimensional equations for numerical approximation are developed, including formulas for converting matrix control and estimator gains to their functional representation to allow comparison of gains based on different orders of approximation. Convergence of the approximating control and estimator gains and of the corresponding finite-dimensional compensators is studied. Also, convergence and stability of the closed-loop systems produced with the finite-dimensional compensators are discussed. The convergence theory is based on the convergence of the solutions of the finite-dimensional Riccati equations to the solutions of the infinite-dimensional Riccati equations. A numerical example with a flexible beam, a rotating rigid body, and a lumped mass is given.

Design of servo systems for distributed parameter systems by finite dimensional dynamic compensator

The design problem of servo systems for distributed parameter systems is investigated. The output regulator of integral type is designed in order to guarantee internal stability and output regulation. The design procedure based on a dynamic stabilizing compensator is discussed. The output regulation of a wide class of distributed parameter systems is proved under the condition that a closed-loop system is stabilized by a dynamic compensator of general type. Then a closed-loop system can be stabilized by a finite dimensional dynamic compensator under some additional conditions. The reducabitity of the design procedure to a purely finite dimensional one is discussed.