IMA Journal of Mathematical Control and Information - recent issues

Approximate controllability for semi-linear retarded stochastic systems in Hilbert spaces

Muthukumar, P., Balasubramaniam, P. — 2009-06-01

In this paper, semi-linear retarded stochastic control systems together with conditions for approximate controllability of associated linear system is used to obtain sufficient conditions for approximate controllability results in Hilbert spaces. An example is also provided to illustrate the theory.

State estimation for non-linear discrete-time systems with input signals

Xiao, M. — 2009-06-01

One of the typical approaches to construct an observer for non-linear systems is to seek a change of coordinates such that the observer design can be implemented in a relatively easier way. Thus, for such an approach, the main challenge is how to find such a change of coordinates. In this paper, we will provide an explicit expression of a change of coordinates which can transform a non-linear discrete-time system with inputs into a linear system with output injections. This leads to a linearizable error dynamics and thus the estimated state converges to the actual state exponentially in the new coordinates. We also discuss the observability and controllability issues which are related to the proposed change of coordinates.

Optimal control of vibrations of an elastic beam

Sun, B. — 2009-06-01

This paper is concerned with the optimal control problem of the vibrations of an elastic beam, which is governed by a non-linear partial differential equation. The functional analytical approach of Dubovitskii and Milyutin is adopted in investigation of the Pontryagin's maximum principle of the system. The necessary condition is presented for the optimal control problem in fixed final horizon case.

Delay feedback control in exponential stabilization of linear time-varying systems with input delay

Hien, L. V., Phat, V. N. — 2009-06-01

In this paper, we investigate the memory controller design for the exponential stabilization of linear time-varying systems with control delay. Based on state transformation and an improved Lyapunov–Krasovskii functional, new sufficient conditions for the exponential stabilization of the system are derived to design memory feedback controller which makes the system exponentially stabilizable. The conditions are given in terms of the solution of appropriate Riccati differential equations, which allows to compute simultaneously the two bounds that characterize the exponential stability rate of the solution. An application to robust control of uncertain linear control systems with input delay as well as illustrative examples to show the effectiveness of the obtained results is given.

Adaptive stabilization of a sine-Gordon equation with input disturbances

Kobayashi, T. — 2009-06-01

This paper is concerned with adaptive stabilization of the system governed by the sine-Gordon equation with input disturbances. The adaptive boundary controller is constructed by the concept of high-gain adaptive feedback and the estimation mechanism for the unknown parameters of the disturbances. The well posedness of the closed-loop system is justified. After the boundedness of the solution is shown, the stability of the closed-loop system and the convergence of the system state to zero are guaranteed by the LaSalle's invariance principle. The system with output disturbances is also discussed.

EP-based robustness analysis of optimal load allocation for boiler systems

Tsai, J. S. H., Du, Y. Y., Dunn, A. C., Shieh, L. S. — 2009-06-01

Based on evolutionary programming, a universal approach to examine the non-analytic solution robustness of an optimal allocation methodology for industrial boilers that is designed based on the second-order gradient method is proposed in this paper. The allocation algorithm is easy to implement without any specialized software and the precise robustness analysis considers the worst-case scenario when uncertainty factors are introduced into some of the boiler system parameters.

Stability of abstract non-linear non-autonomous difference-delay equations

Medina, R. — 2009-06-01

We consider a class of functional–difference equations in a Banach space with causal operators, which have the local Lipschitz property. Sufficient conditions for the boundedness of solutions and the Lyapunov stability are given. We offer some explicit stability criteria when the equations are defined on separable Hilbert spaces. Our approach is based on the representation of solutions combined with new norm estimates for operator-valued functions. Applications of the main results of the paper to the stability of abstract discrete-time control systems are especially investigated.

Supplement to: 'Boundary stabilization of hyperbolic systems related to overhead cranes' [H. Sano, IMA J. Math. Control Inf. (2008) vol. 25, 353-366, doi:10.1093/imamci/dnm031]

Sano, H. — 2009-06-01

In the paper cited in the heading, we treated the problem of stabilizing a flexible cable with two rigid loads, which was described by two kinds of hyperbolic equations. To show the asymptotic stability of the closed-loop system with a controller derived there, we used the LaSalle's invariance principle. However, in that paper, we need to supplement the proof of Theorem 5.1 and to revise the proof of Theorem 5.2. Throughout this note, we use the same notation as in the paper cited in the heading.

Identification of non-parametric FIR non-linear systems with low-degree interactive terms

Bai, E.-W., Chan, K.-S., Erdahl, C. — 2009-06-01

In this paper, an interactive term identification approach is proposed for identification of non-parametric finite impulse response non-linear systems under i.i.d. random sequences. The idea is to make a high-dimensional non-linear identification problem into a number of low-dimensional problems and thus to effectively combat the problem of the curse of dimensionality. Convergence results are established in the paper and numerical results support the theoretical analysis and demonstrate that the proposed approach is an attractive alternative to existing non-linear identification methods.

Necessary conditions for infinite-horizon Volterra optimal control problems with time delay

De La Vega, C. — 2009-03-11

Necessary conditions are proved for optimal control problems involving an infinite horizon and terminal conditions at infinity whose states are governed by Volterra integral equations with non-linear time delay.

Robust H{infty} dynamic output feedback control for 2D linear parameter-varying systems

Wu, L., Lam, J., Wang, C. — 2009-03-11

This paper is concerned with the problem of robust H dynamic output feedback control for 2D discrete-time linear parameter-varying systems. Given a Fornasini–Marchesini local state-space system with linear varying parameters, our attention is focussed on the design of full-order H dynamic output feedback controller, which guarantees the closed-loop system to be asymptotically stable and has a prescribed H disturbance attenuation performance. A sufficient condition for the existence of a desired robust output feedback controller is established in terms of parameterized linear matrix inequalities, and the corresponding controller synthesis is cast into a convex optimization problem which can be efficiently handled by using standard numerical software. A numerical example is provided to illustrate the effectiveness of the proposed design method.

Delay-dependent dissipativity of singular time-delay systems

Mahmoud, M. S. — 2009-03-11

In this paper, new results are established for the delay-dependent problems of dissipative analysis and state-feedback synthesis of singular time-delay (STD) systems with polytopic uncertainties. The developed results for the nominal singular system encompass available results on H approach, passivity and positive realness for STD systems as special cases. All the sufficient stability conditions are cast as linear matrix inequality-based feasibility tests. Robust dissipativity results are also derived. Numerical examples are provided.

Parametric solutions to the generalized discrete Sylvester matrix equation MXN - X = TY and their applications

Zhou, B., Duan, G.-R. — 2009-03-11

In this paper, an explicit, analytical and complete solution to the generalized discrete Sylvester matrix equation MXN – X = TY which is closely related with several types of matrix equations in control theory is obtained. The proposed solution has a neat and elegant form in terms of the Krylov matrix, a block Hankel matrix and an observability matrix. Based on the proposed solution, an explicit solution to the general discrete Lyapunov matrix equation is also derived. As an application, the parametric pole assignment for descriptor linear systems by proportional-plus-derivative state feedback is considered. The results presented here are parallel to our earlier results on the generalized Sylvester matrix equation AX – XF = BY.

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

Abouzaid, B., Achhab, M. E., Wertz, V. — 2009-03-11

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 new algorithm for finding numerical solutions of optimal feedback control

Guo, B.-Z., Sun, B. — 2009-03-11

A new algorithm for finding numerical solutions of optimal feedback control based on dynamic programming is developed. The algorithm is based on two observations: (1) the value function of the optimal control problem considered is the viscosity solution of the associated Hamilton–Jacobi–Bellman (HJB) equation and (2) the appearance of the gradient of the value function in the HJB equation is in the form of directional derivative. The algorithm proposes a discretization method for seeking optimal control–trajectory pairs based on a finite-difference scheme in time through solving the HJB equation and state equation. We apply the algorithm to a simple optimal control problem, which can be solved analytically. The consistence of the numerical solution obtained to its analytical counterpart indicates the effectiveness of the algorithm.

Optimal control for linear discrete-time systems with Markov perturbations in Hilbert spaces

Ungureanu, V. M. — 2009-03-11

In this article, we discuss a quadratic control problem for linear discrete-time systems with Markov perturbations in Hilbert spaces, which is linked to a discrete-time Riccati equation defined on certain infinite-dimensional ordered Banach space. We prove that under stabilizability and stochastic uniform observability conditions, the Riccati equation has a unique, uniformly positive, bounded on N and stabilizing solution. Based on this result, we solve the proposed optimal control problem. An example illustrates the theory.

Pattern Theory: from Recognition to Inference by Ulf Grenander and Michael Miller

Banks, S. P. — 2009-03-11

On robust stability of linear neutral systems with time-varying delays

Fridman, E. — 2008-12-01

The application of the direct Lyapunov method to the stability analysis of neutral systems with time-varying delays usually encounters a restrictive assumption on the function in the right side of the differential equation. This function is supposed to satisfy the Lipschitz condition with respect to the delayed state derivative with a constant less than 1. In the present paper, we extend the input–output approach to consider the stability of neutral type systems with uncertain time-varying delays and norm-bounded uncertainties. The assumption on the Lipschitzian constant can then be avoided. Sufficient stability criteria are derived in the frequency domain and the time domain, where the descriptor discretized Lyapunov–Krasovskii functional is applied. As a by-product, new necessary conditions for neutral-delay-independent/retarded-delay-dependent stability criteria are obtained. The method can be easily extended to L₂-gain analysis and can be applied to design problems.

Design for positivity of analytic functions

Zeheb, E., Dolgin, Y. — 2008-12-01

A large group of control problems can be casted as a problem of design for positivity. A new method is presented for choosing a parameter which ensures the robust positiveness of any real analytic function. The method provides an analytic solution as opposed to numerical sampling techniques. The method is especially useful in design for polynomial positivity due to existence of efficient techniques for solution of systems of polynomial equations and inequalities. Examples are provided, which illustrate the applicability of the proposed method to real-life problems.

New characterization of controllability via stabilizability and Riccati equation for LTV systems

Phat, V. N., Ha, Q. P. — 2008-12-01

This paper presents a new characterization of controllability via stabilizability and Riccati equation for linear time-varying systems. An equivalence is given between the global null controllability, complete stabilizability and the existence of the solution of some appropriate Riccati differential equation.

Reduction of parameters for stabilizing controllers without coprime factorizability

Mori, K. — 2008-12-01

The present paper introduces a parameterization method of stabilizing controllers in the framework of the factorization approach. This parameterization method is a generalization of the previous parameterization methods. The existence of the coprime factorizability of plants is not assumed herein. Under this condition, the number of parameters of the proposed method is less than or equal to those of the previous methods.

Generalized internal model architecture for gain-scheduled control

Xie, W., Eisaka, T. — 2008-12-01

A two-degree-of-freedom controller architecture and its design strategy for linear parameter-varying (LPV) systems, where the dependent parameters are assumed to be measurable, are proposed in the generalized internal model control (GIMC) framework. First, coprime factorization and Youla parameterization for LPV systems are introduced based on a parameter-dependent Lyapunov function. Then, the GIMC architecture for linear time-invariant systems is extended to LPV systems with these descriptions. Based on this architecture, good tracking performance and good robustness (disturbance rejection) are compatibly achieved by a nominal controller and a conditional controller, respectively. Furthermore, each controller design problem is formulated in terms of linear matrix inequalities related to each L2-gain performance. Finally, a simple design example is illustrated.

Gap phenomenon in the homogenization of parabolic optimal control problems

D'Apice, C., De Maio, U., Kogut, P. I. — 2008-12-01

In this paper, we study the asymptotic behaviour of a parabolic optimal control problem in a domain , whose boundary contains a highly oscillating part. We consider this problem with two different classes of Dirichlet boundary controls, and, as a result, we provide its asymptotic analysis with respect to the different topologies of homogenization. It is shown that the mathematical descriptions of the homogenized optimal control problems have different forms and these differences appear not only in the state equation and boundary conditions but also in the control constraints and the limit cost functional.

Boundary control for hyperbolic systems involving infinite-order operators with control constraints

Bahaa, G. M. — 2008-12-01

In this paper, a class of n x n hyperbolic boundary control problems with control constraints is investigated. We first establish the solvability of an n x n system with non-homogeneous mixed Neumann conditions involving hyperbolic operator of infinite order. Also a boundary control problem for this system is considered. The necessary and sufficient conditions for the control to be optimal are obtained. Two mathematical examples for derived optimality conditions are presented.

Kharitonov theorem with degree drop: the complex case

Toker, O. — 2008-12-01

In this paper, we study the complex version of the Kharitonov theorem without the constant-degree assumption. It is proved that, for a complex-interval polynomial with degree drop, robust Hurwitz stability is equivalent to the Hurwitz stability of the eight Kharitonov polynomials. Furthermore, it is also shown that for a robustly stable complex-interval polynomial, degree drop cannot exceed one and degree drop can only occur at one of the corners of the rectangular region in the complex plane corresponding to the leading coefficient of the uncertain polynomial.

Weakly coprime factorization and continuous-time systems

Mikkola, K. M. — 2008-12-01

We give many necessary and sufficient conditions for the existence of a weakly coprime or Bézout coprime factorization of a transfer function, possibly operator valued. Some of these conditions are given in terms of the output- or state-feedback stabilizability of realizations. Our realizations are well-posed linear systems—continuous-time linear time-invariant infinite-dimensional systems. We also study further properties of such factorizations, their relations to discrete-time weakly coprime factorizations, counterexamples and (weak) left invertibility. Moreover, analogous discrete-time results are obtained. Control-theoretic consequences are indicated.