IMA Journal of Mathematical Control and Information - recent issues

Properties of a subalgebra of H{infty}(D) and stabilization

Sasane, A. — 2008-03-06

Let D denote the open unit disc in C. Let T denote the unit circle and let S T. We denote by A_S(D) the set of all functions f : D S -> C that are holomorphic in D and are bounded and continuous in D S. Equipped with the supremum norm, A_S(D) is a Banach algebra, and it lies between the extreme cases of the disc algebra A(D) and the Hardy space H(D). We show that A_S(D) has the following properties:

P1. The corona theorem holds for A_S(D).

P2. The integral domain A_S(D) is not a Bézout domain, but it is a Hermite ring.

P3. The stable rank of A_S(D) is 1.

P4. The Banach algebra A_S(D) has topological stable rank 2.

The classes A_S(D) serve as appropriate transfer function classes for infinite-dimensional systems that are not exponentially stable, but stable only in some weaker sense. Consequences of the above properties to stabilizing controller synthesis using a coprime factorization approach are discussed.

Gain reduction in switched sliding-mode control

Ferrara, A., Scattolini, R. — 2008-03-06

A switched sliding-mode control strategy for a class of nonlinear uncertain systems is presented in this paper. It is characterized by an event-driven gain reduction mechanism which relies on a decomposition of the system state into regions. By enforcing sliding-mode behaviours on a suitable set of sliding manifolds, while avoiding the generation of limit cycles, the proposed strategy proves to globally asymptotically stabilize the origin of the system state space.

Optimal control problems of parabolic equations with an infinite number of variables and with equality constraints

Bahaa, G. M. — 2008-03-06

Optimal control problems of systems governed by parabolic equations with an infinite number of variables and with additional equality constraints are considered. The extremum principle, as well as sufficient condition of optimality, is formulated for the Neumann problem by using certain extensions of Dubovitskii–Milyutin method.

Optimality conditions for n x n infinite-order parabolic coupled systems with control constraints and general performance index

Bahaa, G. M., Kotarski, W. — 2008-03-06

A distributed control problem for n x n parabolic coupled systems involving operators with infinite order is considered. The performance index is more general than the quadratic one and has an integral form. Constraints on controls are imposed. Making use of the Dubovitskii–Milyutin theorem, the necessary and sufficient conditions of optimality are derived for the Dirichlet problem. Yet, the problem considered here is more general than the problems in El-Saify & Bahaa (2002, Optimal control for n x n hyperbolic systems involving operators of infinite order. Math. Slovaca, 52, 409–424), El-Zahaby (2002, Optimal control of systems governed by infinite order operators. Proceeding (Abstracts) of the International Conference of Mathematics (Trends and Developments) of the Egyptian Mathematical Society, Cairo, Egypt, 28–31 December 2002. J. Egypt. Math. Soc. (submitted)), Gali & El-Saify (1983, Control of system governed by infinite order equation of hyperbolic type. Proceeding of the International Conference on Functional-Differential Systems and Related Topics, vol. III. Poland, pp. 99–103), Gali et al. (1983, Distributed control of a system governed by Dirichlet and Neumann problems for elliptic equations of infinite order. Proceeding of the International Conference on Functional-Differential Systems and Related Topics, vol. III. Poland, pp. 83–87) and Kotarski et al. (200b, Optimal control problem for a hyperbolic system with mixed control-state constraints involving operator of infinite order. Int. J. Pure Appl. Math., 1, 241–254).

Weakening the strengthened condition of Weierstrass for the isoperimetric problem in the calculus of variations

Licea, G. S. — 2008-03-06

An alternate sufficiency proof for the fixed end-point isoperimetric problem in the calculus of variations is presented. This technique not only shows how the problem need not be transformed into a problem of Lagrange but also shows how we can weaken the classical strengthened condition of Weierstrass. The usefulness of this sufficiency result is illustrated with an example which cannot be transformed into a problem of Lagrange and for which it is possible to apply the alternate sufficiency theorem in order to conclude that a given extremal affords a strict strong minimum. On the other hand, we show that the classical sufficiency theorem does not respond for this case.

Properties of the weighted logarithmic matrix norms

Hu, G.-D., Liu, M. — 2008-03-06

In this paper, we are concerned with the properties of the weighted logarithmic matrix norms. A relation between the elliptic logarithmic matrix norm and the weighted logarithmic matrix norm is given. Based on Lyapunov equations, two weighted logarithmic matrix norms are constructed which are less than 1-logarithmic matrix norm and -logarithmic matrix norm, respectively. Then, an iterative scheme is presented to obtain the logarithmically -efficient matrix norm. Numerical examples are given to illustrate the results.

Feedback theory for time-varying regular linear systems with input and state delays

Hadd, S., Rhandi, A., Schnaubelt, R. — 2008-03-06

We show that the class of regular time-varying systems is invariant under perturbations by time-varying state and input delays. In particular, we give explicit formulas of the resulting input, output and input–output maps. This result is used to solve the feedback problem for the delayed system. The relationship between the open- and the closed-loop system is investigated. Our results are applied to a parabolic boundary control problem with input and state delays.

Optimal boundary feedback stabilization of a string with moving boundary

Gugat, M. — 2008-03-06

We consider a finite string that is fixed at one end and subject to a feedback control at the other end which is allowed to move. We show that the behaviour is similar to the situation where both ends are fixed: As long as the movement is not too fast, the energy decays exponentially and for a certain parameter in the feedback law it vanishes in finite time. We consider movements of the boundary that are continuously differentiable with a derivative whose absolute value is smaller than the wave speed. We solve a problem of worst-case optimal feedback control, where the parameter in the feedback law is chosen such that the worst-case L^p-norm of the space derivative at the fixed end of the string is minimized (p [1, )). We consider the worst case both with respect to the initial conditions and with respect to the boundary movement. It turns out that the parameter for which the energy vanishes in finite time is optimal in this sense for all p.

Open-loop linearization of non-linear discrete input-output systems through simplification algorithms

Kotsios, S. — 2008-03-06

The problem of linear equivalence for a general class of non-linear systems is examined throughout this paper. A relevant algorithm is developed based on a factorization procedure. This factorization is based on the star product, an operation corresponding to the cascade connection of systems.

Infinite horizon quadratic control of linear singularly perturbed systems with small state delays: an asymptotic solution of Riccati-type equations

Glizer, V. Y. — 2007-11-20

An infinite horizon linear-quadratic optimal control problem for a singularly perturbed system with multiple point-wise and distributed small delays in the state variable is considered. The set of Riccati-type equations, associated with this problem by the control optimality conditions, is studied. Since the system in the control problem is singularly perturbed, the equations of this set are also perturbed by a small parameter of the singular perturbations. The zero-order asymptotic solution to this set of equations is constructed and justified. Based on this asymptotic solution, parameter-free sufficient conditions for the existence and uniqueness of solution to the original optimal control problem are established.

Adaptive regulator design for a class of parabolic distributed parameter systems with disturbed outputs

Kobayashi, T., Sakamoto, T., Oya, M. — 2007-11-20

In this paper an adaptive regulator is designed for collocated parabolic distributed parameter systems with disturbed outputs in the case of unbounded input and output operators. The adaptive regulator is constructed by the concept of high-gain output feedback and a new type of the estimation mechanism of the unknown parameters for the noise. The existence and uniqueness of the solution of the closed-loop system are justified. In the controlled system, the convergence of the system state to zero will be guaranteed.

Numerical stabilization of polynomial and matrix

Han, J. D., Jiang, Z., Nie, Y. Y. — 2007-11-20

In this paper, the conception of numerical stabilization, which is related to mantissa digits of computer and dimensions of system, is described; and several strategies for the numerical stabilization of polynomial and matrix are presented.

Estimates of the region of attraction of continuous-time cascade systems

Bai, X., Yang, X.-S., Li, H. — 2007-11-20

In this paper, we first present some new sufficient conditions for global asymptotic stability of continuous-time cascade systems. Then, we give some detailed results on the region of attraction of continuous-time cascade systems. Some examples are presented to illustrate these results.

Maximum principle for optimal control of sterilization of prepackaged food

Sun, B., Wu, M.-X. — 2007-11-20

This paper is concerned with an optimal control problem of the sterilization of prepackaged food. The Dubovitskii–Milyutin approach is adopted in investigation of the Pontryagin's maximum principle of the system. The necessary condition is presented for the problem with fixed final horizon and phase constraints.

Necessary and sufficient conditions for parameter insensitive disturbance-rejection problems with dynamic output feedback

Otsuka, N. — 2007-11-20

In this paper, necessary and sufficient conditions for the parameter insensitive disturbance-rejection problem with dynamic output feedback which was studied in Otsuka (1999) to be solvable are given for the structured uncertain linear systems. Further, necessary and sufficient conditions for the solvability of the problems with static output feedback and state feedback are also studied as special cases. The obtained results contain some extensions of the previous results to the structured uncertain linear systems.

Estimates of the attraction region for a class of nonlinear time-delay systems

Melchor-Aguilar, D., Niculescu, S.-I. — 2007-11-20

In this paper, we propose various estimates of the attraction region for a class of nonlinear time-delay systems of the form The approach is constructive and makes use of a Lyapunov–Krasovskii functional associated to the linear part. Several illustrative examples (delayed logistic equation, stabilizing nonlinear oscillations by delayed output feedback, congestion control in high-performance networks and hereditary phenomena in physics) complete the presentation.

Direct approximation of observer error linearization for nonlinear forced systems

Robenack, K. — 2007-11-20

In this paper, we discuss a generalization of the extended Luenberger observer. Our approach can be interpreted as an approximate error linearization. The design procedure which is formulated in terms of Lie derivatives and Lie brackets can easily be implemented using computer algebra software.

Statistical solution to the capacity problem in direct-sequence code-division multiple-access communication systems

Eshima, N., Kohda, T., Tabata, M. — 2007-09-24

Capacity in the direct-sequence code-division multiple-access (DS/CDMA) communication system was considered according to the code acquisition performance with the conventional serial-search method, because code acquisition needs a difficult operation. Since capacity in DS/CDMA systems is defined by the maximum number of users that can simultaneously transmit their signals with the same carrier frequencies, assuring a larger capacity is very important in DS/CDMA systems with respect to the economy of frequency source. This paper reconsiders the capacity problem through a statistical approach to code acquisition. First, a DS/CDMA system model is reviewed. Second, properties of the counting method for code acquisition are discussed. It is proved that the method can acquire the target user's code under any number of interference users that simultaneously transmit their signals, and that the method can guarantee a considerable precision in code acquisition. Third, an observation time necessary for code acquisition is given from a statistical discussion, and the original counting method is modified into a simpler one. It is concluded that the capacity of the DS/CDMA communication system can be set by the ‘bit error-rate-based capacity or signal-to-noise ratio-based capacity’, i.e. ‘post-acquisition-based criterion’, rather than the ‘acquisition-based capacity’.

Robust adaptive sliding-mode control for continuous time-delay systems

Nounou, M. N., Nounou, H. N., Mahmoud, M. S. — 2007-09-24

This paper focuses on the development of a robust adaptive sliding-mode scheme for the control of a class of uncertain time-delay systems with time-varying delays. The uncertainties are parametric and norm-bounded. Two main problems are investigated. In the first problem, a sliding-mode observer-based output feedback control scheme is developed. In the second problem, a robust redesign approach is developed yielding an adaptive-based controller against controller and sliding-mode observer gain perturbations. The objective is to derive stability conditions for the closed-loop system in the presence of gain perturbations. Simulation results are obtained through a numerical example.

Estimation of nonlinear systems without a priori structural information and its application in Hammerstein model identification with dynamic nonlinearities

Bai, E.-W., Liu, Y. — 2007-09-24

In the paper, we discuss identification of a nonlinear system without structural information and propose two methods, the kernel method and the orthonormal basis method. The convergence results are established for both methods without a priori structural information. We then apply the results to identification of Hammerstein models with an unknown dynamic nonlinearity. It is also shown that identification of the linear part in Hammerstein models is possible with no knowledge of the dynamic nonlinearity.

Optimal control for the thermistor problem with a current limiting device

Cimatti, G. — 2007-09-24

The problem of finding in the thermistor problem the optimal difference of applied potential in order to minimize a suitable cost functional involving the temperature is studied. We find the optimal system in the elementary case of constant electric conductivity. In addition, a theorem of existence of the optimal solution is given in the general case of conductivities depending on the temperature.

An LMI-based approach for robust stabilization of time delay systems containing saturating actuators

El Haoussi, F., Tissir, E. H. — 2007-09-24

This paper deals with the problem of robust stabilization for uncertain systems with input saturation and time delay in the state. The parameter uncertainties are time-varying and unknown but are norm bounded. Sufficient conditions obtained via a linear matrix inequality formulation are stated to guarantee the local stabilization. The method of synthesis consists in determining simultaneously a state feedback control law and an associated domain of safe admissible states for which the stability of the closed-loop system is guaranteed when control saturations effectively occur. Numerical examples are used to demonstrate the effectiveness of the proposed design technique.

Time optimal control of retarded parabolic systems

Krakowiak, A. — 2007-09-24

In this paper, the time-optimal control problem for parabolic systems in which time lags appear in the integral form both in the state equation and in the Neumann boundary condition is presented. The particular properties of the optimal control are discussed.

Optimal control for the forward backward heat coupled systems

Dorville, R., Omrane, A. — 2007-09-24

We give a singular optimality system for the control of the coupled (ill-posed) forward–backward heat problem using both the classical penalization method and the low-regret method of Lions (1992).

For the first method, we add the Slater hypothesis on the admissible closed convex set of controls U_ad, while with the low-regret control method, we do not need this hypothesis.

Delay-dependent conditions for guaranteed cost observer-based control of uncertain neutral systems with time-varying delays

Yu, K.-W., Lien, C.-H. — 2007-09-24

In this paper, delay-dependent guaranteed cost observer-based control for neutral systems with time-varying delays is considered. Control and observer gains will be given from the linear matrix inequality feasible solutions. Optimal guaranteed cost observer-based control which will minimize the guaranteed cost of the system is provided.

Decentralized dynamic pole assignment with low-order compensators

Leventides, J., Karcanias, N. — 2007-09-24

The problem of arbitrary pole placement via dynamic decentralized output feedback is studied for minimal systems described by a proper transfer function matrix P(s) R^m ^{x p}(s) (m = m_i and p = p_i), with McMillan degree n. The family of controllers to be used includes those decentralized controllers with channels whose ith channel has maximum observability index at most d_i. The method presented here is based on asymptotic linearization around a decentralized degenerate compensator of the pole placement map related to the problem. It is shown that the method works generically when m⁺p > n, where m⁺ = min{d_i(p_i + m_i – 1) + m_i}, i = 1, ..., , and the smallest d_i of the compensator of the ith channel is the integral part of (n – pm_i)/p(p_i + m_i – 1).

On the geometry of stability regions of Smith predictors subject to delay uncertainty

Morarescu, C.-I., Niculescu, S.-I., Gu, K. — 2007-09-24

In this paper, we present a geometric method for describing the effects of the ‘delay-induced uncertainty’ on the stability of a standard Smith predictor control scheme. The method consists of deriving the ‘stability crossing curves’ in the parameter space defined by the ‘nominal delay’ and ‘delay uncertainty’, respectively. More precisely, we start by computing the ‘crossing set’, which consists of all frequencies corresponding to all points on the stability crossing curve, and next we give their ‘complete classification’, including also the explicit characterization of the ‘directions’ in which the zeros cross the imaginary axis. This approach complements existing algebraic stability tests, and it allows some new insights in the stability analysis of such control schemes. Several illustrative examples are also included.

New conditions on absolute stability of uncertain Lur'e systems and the maximum admissible perturbed bound

Hao, F. — 2007-09-24

The robust absolute stability problem for structured uncertain Lur'e systems is considered in this paper by using Popov criterion and extended strictly positive real lemma. The conditions on robust absolute stability for Lur'e systems with structured uncertainties are established in terms of multilinear matrix inequalities. An estimate of the maximum bound of all admissible perturbations is given by a generalized eigenvalue problem. Finally, a numerical example is worked out to illustrate the efficiency of the main results.