Infinite-dimensional Linear Systems Research Articles

Time-Varying and Adaptive Discrete-Time Low-Gain Control of Infinite-Dimensional Linear Systems with Input Nonlinearities

Discrete-time low-gain control strategies are presented for tracking constant reference signals for infinite-dimensional, discrete-time, power-stable, linear systems subject to input nonlinearities. Both non-adaptive (but time-varying) and adaptive controls are considered. The discrete-time results are applied in the development of sampled-data integral control for infinite-dimensional, continuous-time, exponentially stable, regular, linear systems with input nonlinearities.

On exponential stabilizability of linear neutral systems

In this paper, we deal with linear neutral functional differential systems. Using an extended state space and an extended control operator, we transform the initial neutral system in an infinite dimensional linear system. We give a sufficient condition for admissibility of the control operator B, conditions under which operator B can be acceptable in order to work with controllability and stabilizability. Necessary and sufficient conditions for exact controllability are provided; in terms of a gramian of controllability N(μ). Assuming admissibility and exact controllability, a feedback control law is defined from the inverse of the operator N(μ) in order to stabilize exponentially the closed loop system. In this case, the semigroup generated by the closed loop system has an arbitrary decay rate.

Contingent Solutions for the Bellman Equation in Infinite Dimensions

The present paper is concerned with the study of the Hamilton–Jacobi–Bellman equation for the time optimal control problem associated with infinite-dimensional linear control systems from the point of view of continuous contingent solutions.

Optimizability and Estimatability for Infinite-Dimensional Linear Systems

An infinite-dimensional linear system described by $\dot{x}(t)=Ax(t)+Bu(t)$ $(t\geq 0)$ is said to be optimizable if for every initial state x(0), an input $u\in L^2$ can be found such that $x\in L^2$. Here, A is the generator of a strongly continuous semigroup on a Hilbert space and B is an admissible control operator for this semigroup. In this paper we investigate optimizability (also known as the finite cost condition) and its dual, estimatability. We explore the connections with stabilizability and detectability. We give a very general theorem about the equivalence of input-output stability and exponential stability of well-posed linear systems: the two are equivalent if the system is optimizable and estimatable. We conclude that a well-posed system is exponentially stable if and only if it is dynamically stabilizable and input-output stable. We illustrate the theory by two examples based on PDEs in two or more space dimensions: the wave equation and a structural acoustics model.

Time-Varying and Adaptive Integral Control of Infinite-Dimensional Regular Linear Systems with Input Nonlinearities

Closing the loop around an exponentially stable, single-input, single-output, regular linear system---subject to a globally Lipschitz, nondecreasing actuator nonlinearity and compensated by an integral controller with time-dependent gain k(t)---is shown to ensure asymptotic tracking of a constant reference signal r, provided that (a) the steady-state gain of the linear part of the system is positive, (b) the reference value r, is feasible in an entirely natural sense, and (c) the function $t\mapsto k(t)$ monotonically decreases to zero at a sufficiently slow rate. This result forms the basis of a simple adaptive control strategy that ensures asymptotic tracking under conditions (a) and (b).

A generalization of the concept of similarity of infinite-dimensional linear systems with applications to stability, stabilizability, and detectability

A generalization of the concept of similarity of infinite-dimensional linear systems with applications to stability, stabilizability, and detectability

Infinite-Dimensional Linear Dynamical Systems with Chaoticity

The authors present two results on infinite-dimensional linear dynamical systems with chaoticity. One is about the chaoticity of the backward shift map in the space of infinite sequences on a general Fréchet space. The other is about the chaoticity of a translation map in the space of real continuous functions. The chaos is shown in the senses of both Li-Yorke and Wiggins. Treating dimensions as freedoms, the two results imply that in the case of an infinite number of freedoms, a system may exhibit complexity even when the action is linear. Finally, the authors discuss physical applications of infinite-dimensional linear chaotic dynamical systems.

Robust boundary control for linear time-varying infinite dimensional systems

The problem of robust boundary control for a class of infinite dimensional systems under mixed uncertainties is addressed. A strong solution of the Dirichlet boundary problem corresponding to the perturbed evolution operator is introduced. The Lyapunov function approach is used for proving that there is a controller which stabilizes this class of systems under the presence of smooth enough internal and external perturbations and guarantees some tolerance level for the joint cost functional. The derived control consists of two parts: a compensating one and a linear feedback controller with a gain operator which is a positive inverse solution of a corresponding operator Riccati equation. A heating boundary control process is given as an illustration of the suggested approach

Null-Controllability of Discrete-Time Systems with Restrained Controls in Asplund Spaces

This paper is concerned with the null-controllability of the infinite-dimensional discrete-time linear system described by $$(A, B, \Omega ) x_{n + 1} = Ax_n + Bu_n$$ where xn ∈ X, un ∈ Ω ⊂ U, X and U are Asplund spaces, A ∈ L(X, X), B ∈ L (U, X), Ω is a convex set, int Ω ≠ 0 and 0 ∈ Ω.

Stability theory for infinite-dimensional linear systems based on an extension

Stability theory for infinite-dimensional linear systems based on an extension

Adaptive λ-tracking for a class of infinite-dimensional systems

For a class of high-gain stabilizable multivariable linear infinite-dimensional systems we present an adaptive control law which achieves approximate asymptotic tracking in the sense that the tracking error tends asymptotically to a ball centred at 0 and of arbitrary prescribed radius λ>0. This control strategy, called λ-tracking, combines proportional error feedback with a simple nonlinear adaptation of the feedback gain. It does not involve any parameter estimation algorithms, nor is it based on the internal model principle. The class of reference signals is W 1,∞, the Sobolev space of absolutely continuous functions which are bounded and have essentially bounded derivative. The control strategy is robust with respect to output measurement noise in W 1,∞ and bounded input disturbances. We apply our results to retarded systems and integrodifferential systems.

An example in linear quadratic optimal control

We construct a simple example of a quadratic optimal control problem for an infinite-dimensional linear system based on a shift semigroup. This system has an unbounded control operator. The cost is quadratic in the input and the state, and the weighting operators are bounded. Despite its extreme simplicity, this example has all the unexpected features discovered recently by O. Staffans (and also by M. Weiss and G. Weiss). More precisely, in the formula linking the optimal feedback operator to the optimal cost operator, as well as in the Riccati equation, the weighting operator of the input has to be replaced by another operator, which can be derived from the spectral factorization of the Popov function.

An identification problem for partially observed infinite dimensional linear stochastic systems

In the paper an identification problem is considered for partially observed autonomous infinite dimensional stochastic systems with additive noise. This state equation is described by an infinite dimensional linear Itô equation with additive Gaussian noise.The observations of the states are finite dimensional We formulate the identification problem as an deterministic optimization problem with appropriate objective functionals for finite and infinite time horizons in such a way so that estimations of the system parameters and the covariance operators of the noise processes can be determined. The existence of these estimations is proved and a finite dimensional approximation is given

On the stability of infinite-dimensional linear inequality systems

The principal aim of this paper is to study the stability of the solution set mapping of a system composed by an arbitrary set of linear inequalities in an infinite-dimensional space. The unknowns space is assumed to be metrizable, which allows us to measure the size of any possible perturbation. Conditions guaranteeing the closedness, the lower semicontinuity and the upper semicontmuity of this mapping, at a particular nominal system, are given in the paper. The more significant differences with respect to the finite dimensional case, previously approached in the context of the so-called semi-infinite optimization, are illustrated by means of convenient examples.

On Robust Stability of Positive Infinite-Dimensional Linear Systems

It is shown that for linear infinite-dimensional continuous-time positive systems the complex and real stability radius with respect to the single perturbation structure coincide. Furthermore, a simple formula for the complex stability radius of positive systems is derived by using the associated transfer function. These results generalize the finite-dimensional counterparts due to Hinrichsen and Son. The results are illustrated with an example dealing with a certain type of differential-difference equations.

The Salamon-Weiss class of well-posed infinite-dimensional linear systems: a survey

The Salamon-Weiss class of well-posed infinite-dimensional linear systems: a survey

Analytic system problems and J-lossless coprime factorization for infinite-dimensional linear systems

This paper extends the coprime factorization approach to the synthesis of internally stabilizing controllers satisfying an H ∞-norm bound to a class of systems with irrational transfer matrices. Using the coprime factorization description, the H ∞-control problem can be reduced to two stable analytic system problems. Such problems have solutions if and only if a certain J-lossless factorization exists. The full H ∞-synthesis problem is shown to be equivalent to the solution of two nested J-lossless factorizations. If the irrational transfer matrix has a state-space realization, then the known state-space formulas for the H ∞-control problem may be recovered using the relationship between J-lossless factorizations and solutions of Riccati equations. However, the results derived here are valid for a larger class of infinite-dimensional systems.

On-Line Parameter Estimation for Infinite-Dimensional Dynamical Systems

The on-line or adaptive identification of parameters in abstract linear and nonlinear infinite-dimensional dynamical systems is considered. An estimator in the form of an infinite-dimensional linear evolution system having the state and parameter estimates as its states is defined. Convergence of the state estimator is established via a Lyapunov estimate. The finite-dimensional notion of a plant being sufficiently rich or persistently excited is extended to infinite dimensions. Convergence of the parameter estimates is established under the additional assumption that the plant is persistently excited. A finite-dimensional approximation theory is developed, and convergence results are established. Numerical results for examples involving the estimation of both constant and functional parameters in one-dimensional linear and nonlinear heat or diffusion equations and the estimation of stiffness and damping parameters in a one-dimensional wave equation with Kelvin--Voigt viscoelastic damping are presented.

Low-Gain Control of Uncertain Regular Linear Systems

It is well known that closing the loop around an exponentially stable, finite-dimensional, linear, time-invariant plant with square transfer-function matrix $\BG(s)$ compensated by a controller of the form $(k/s)\Gamma_0$, where $k\in {\Bbb R}$ and $\Gamma_0in {\Bbb R}\mm$, will result in an exponentially stable closed-loop system which achieves tracking of arbitrary constant reference signals, provided that (i) all the eigenvalues of $\BG(0)\Gamma_0$ have positive real parts and (ii) the gain parameter k is positive and sufficiently small. In this paper we consider a rather general class of infinite-dimensional linear systems, called regular systems, for which convenient representations are known to exist, both in time and in frequency domain. The purpose of the paper is twofold: (i) we extend the above result to the class of exponentially stable regular systems and (ii) we show how the parameters k and $\Gamma_0$ can be tuned adaptively. The resulting adaptive tracking controllers are not based on system identification or parameter estimation algorithms, nor is the injection of probing signals required.

Dynamic stabilization of regular linear systems

We consider a general class of infinite-dimensional linear systems, called regular linear systems, for which convenient representations are known to exist both in time and in frequency domain. For this class of systems, we investigate the concepts of stabilizability and detectability, in particular, their invariance under feedback and their relationship to exponential stability. We introduce two concepts of dynamic stabilization, the first formulated as usual, with the plant and the controller connected in feedback, and the second with two feedback loops. Even for finite-dimensional systems, the second concept, stabilization with an internal loop in the controller, is more general. We argue that the more general concept is the natural one, and we derive sufficient conditions under which an observer-based stabilizing controller with an internal loop can be constructed.