IMA Journal of Mathematical Control and Information - recent issues

Adaptive stabilization of Kirchhoff's non-linear strings with output disturbances

Kobayashi, T. — Mon, 23 Nov 2009 04:05:14 PST

This paper is concerned with adaptive stabilization of an undamped non-linear string with disturbed outputs by boundary feedback control. The adaptive controller is constructed by the concept of high-gain adaptive feedback and the estimation mechanism for the unknown parameters of the measurement noise. The local existence and uniqueness of the solution of the closed-loop system are justified. Moreover, global existence and boundedness of the solution are shown for smooth and small initial data. The stability of the closed-loop system is proved such that the convergence of the system state to zero and the convergence of the estimated parameter to the unknown parameter are guaranteed for the smooth and small initial data.

An adaptive tracking controller design for non-linear singularly perturbed systems using fuzzy singularly perturbed model

Li, L., Sun, F. C. — Mon, 23 Nov 2009 04:05:14 PST

For a class of non-linear singularly perturbed systems with unknown dynamic and external disturbance, an adaptive tracking controller using fuzzy singularly perturbed model is developed. First, a series of dynamic Takagi-Sugeno fuzzy singularly perturbed subsystems are built to approximate a non-linear singularly perturbed system and a stable reference model with the same fuzzy premise is chosen to design both expected trajectory and dynamic performance. Then, a kind of controller including adaptive feedback term is developed to make the states of the closed-loop system follow those of the reference model. The linear feedback gain of the controller can be solved by the linear matrix inequality approach. The adaptive sliding term is used to compensate the uncertainty and alleviate the disturbance. Lyapunov constitute techniques can be used to prove the stability of the closed-loop systems. Finally, the simulations results illustrate the effectiveness of this approach.

Convex hull of two quadratic constraints is an LMI set

Yildiran, U. — Mon, 23 Nov 2009 04:05:14 PST

In this work, we are interested in the convex hull of the region determined by two quadratic polynomial constraints. The main result is that if this region is not empty, the convex hull is either Rⁿ or the feasible set of another pair of quadratic constraints which are, in fact, positive linear combinations of the original ones. Based on this result, a losslessness condition is also derived for the classical semidefinite programming relaxation. The characterization of the convex hull we found does not have to be composed of concave quadratic constraints. However, we propose an algorithm to convert them into linear matrix inequalities (LMIs) and explain how the LMI characterization can be employed to solve a certain class of non-convex optimization problems. It is shown that this approach may perform better than the available relaxation methods for the optimization problem considered. Lastly, we show how the results developed can be applied to a certain class of control problems.

Improved robust stability and feedback stabilization criteria for time-delay systems

Mahmoud, M. S. — Mon, 23 Nov 2009 04:05:14 PST

This paper develops novel robust stability and feedback stabilization criteria with guaranteed performance for a class of linear continuous time-delay systems with polytopic uncertainties. The time-varying delay function is unknown and differentiable within bounded interval and the input delay is constant. The criteria is derived based on the constructive use of a new Lyapunov–Krasovskii functional together with the integral inequality. The developed stability condition is expressed in terms of linear matrix inequality that manipulates fewer decision variables and requires reduced computational load. Through a comparison with other existing stability methods, it is established that the developed method retains some useful terms that are frequently dropped out and does not employ any free-weighting matrices to avoid redundancy. A state-feedback stabilizing controller is designed to ensure that the closed loop is robustly stable with guaranteed performance. Representative examples are simulated to illustrate the developed results.

Improved results for non-linear discrete-time systems with time-varying delays

Mahmoud, M. S., Xia, Y. — Mon, 23 Nov 2009 04:05:14 PST

In this paper, complete results for delay-dependent stability, feedback stabilization and linear filtering for a class of non-linear discrete-time systems are developed. The system under consideration has time-varying delays with Lipschitz-type non-linearities and subject to real convex bounded parametric uncertainties in all system matrices. A major thrust of the analysis is the constructive use of an appropriate Lyapunov functionals coupled with ‘Finsler's lemma’ and free-weighting parameter matrices. We establish a linear matrix inequality (LMI) characterization of delay-dependent conditions under which the non-linear discrete delay system is robustly asymptotically stable with an L₂ gain smaller than a prescribed constant level. Feedback stabilization schemes, based on state, static output or by using dynamic output feedback, are designed to guarantee that the corresponding closed-loop system enjoys the delay-dependent asymptotic stability with an L₂ gain smaller than a prescribed constant level. Finally, the developed approach is applied to linear filtering to design both H and L₂ – L filters. All the developed results are expressed in terms of convex optimization over LMIs and tested on several representative examples.

Stabilization and decay estimate of linear control systems in Hilbert space with non-linear feedback

Berrahmoune, L. — Mon, 23 Nov 2009 04:05:15 PST

We consider the problem of stabilization of linear control systems in Hilbert space with non-linear feedback. Extensions of known results in the linear feedback case to the non-linear one are given. Furthermore, under appropriate controllability assumptions, energy decay estimates are established. Applications to the saturating control case are treated.

Sufficiency by a direct method in the variable state problem of calculus of variations: singular extremals

Licea, G. S. — Tue, 01 Sep 2009 08:59:32 PDT

A direct sufficiency proof for singular and non-singular extremals for the parametric variable state problem of Bolza in the calculus of variations is presented. This technique is self-contained in the sense that it makes no use of the classical concepts of conjugate or focal points, fields of extremals or certain matrix Riccati inequalities. Moreover, we also study the non-parametric variable state problem of Bolza and derive sufficient conditions for singular and non-singular extremals for weak and strong local minima.

Robust state reconstruction of linear neutral-type delay systems with application to lossless transmission lines: a convex optimization approach

Ibrir, S., Diop, S. — Tue, 01 Sep 2009 08:59:32 PDT

This paper is concerned with the problem of robust observer design for linear systems with neutral-type time delays. Delay-independent and delay-dependent conditions are presented to solve the observation/filtering issue under noisy output measurements. Stated as linear matrix inequalities (LMIs) conditions, these sufficient conditions enable the determination of the observer gains that guarantee both asymptotic convergence of the observer in case of noiseless measurements and robust filtering in case of presence of measurements errors. The proposed LMI conditions are derived without any major approximation or assumption on the neutral-type time delay system, which make the observer design straightforward and less conservative.

Reduced-order observer design for one-sided Lipschitz non-linear systems

Xu, M., Hu, G.-D., Zhao, Y. — Tue, 01 Sep 2009 08:59:32 PDT

In this paper, the reduced-order observer design for a class of non-linear systems is investigated. Based on the one-sided Lipschitz condition, we present sufficient conditions for the existence of the reduced-order observers of the class of non-linear systems, these conditions are less conservative than those based on Lipschitz condition in literature. It should be noted that the one-sided Lipschitz condition is directly applicable to the important class of the Lipschitz non-linear systems. In considering many problems, the present paper obtains the one-sided Lipschitz constants that are significantly smaller than the classical Lipschitz constants (see Appendix). Some examples are given to illustrate effects of the proposed approach. The last example is introduced with the goal, to illustrate our proposed method to be effective for the important class of the Lipschitz non-linear systems.

A counter example to a recent result on the stability of non-linear systems

Muhammad, S., van der Woude, J. — Tue, 01 Sep 2009 08:59:32 PDT

Langson & Alleyne (2002, J Dyn. Syst. Meas. Control, 124, 452–456) proposed certain sufficient conditions that guarantee the global asymptotic stability of a class of non-linear systems. In this paper, we give a counter example to demonstrate that the conditions stated in Langson & Alleyne (2002) are not sufficient for the global asymptotic stability.

On the rapid convergence of a class of decentralized decision processes: quantized progressive second-price auctions

Jia, P., Qu, C. W., Caines, P. E. — Tue, 01 Sep 2009 08:59:32 PDT

A progressive second price (PSP) auction mechanism was proposed in Semret et al. (2000, IEEE J. Select. Areas Commun., 18, 2499–2513) for network bandwidth allocation. In this paper, a quantized version of this mechanism (Q-PSP) is analysed where the agents have similar demand functions and submit bids synchronously. It is shown that the nonlinear dynamics induced by this mechanism are such that the prices bid by the various agents and the quantities allocated to these agents converge in at most five iterations or oscillate indefinitely; this behaviour is not only independent of the number of agents involved but also independent of the number of quantization levels.

Robust HFormula state feedback control of discrete-time systems with state delay: an LMI approach

Leite, V. J. S., Tarbouriech, S., Peres, P. L. D. — Tue, 01 Sep 2009 08:59:32 PDT

In this paper, uncertain discrete-time systems with delayed states are investigated. The uncertainty is supposed to belong to a known convex polytope and can affect all system matrices. Sufficient linear matrix inequality conditions are given for the computation of H-guaranteed costs and for the design of robust state feedback gains assuring an H attenuation level. The conditions proposed here can assure robustness irrespective of the value of the delay and, differently from other approaches in the literature, are formulated as convex optimization problems. If the delay is known and the delayed states are available, a feedback gain depending on past values of the state can be used to improve the closed-loop performance of the system. As illustrated by numerical examples, including the model of an industrial electric heater, the proposed techniques are simple to be applied and can lead to less conservative results when compared with other conditions from the literature.

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

Muthukumar, P., Balasubramaniam, P. — Mon, 01 Jun 2009 06:33:31 PDT

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. — Mon, 01 Jun 2009 06:33:31 PDT

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. — Mon, 01 Jun 2009 06:33:31 PDT

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. — Mon, 01 Jun 2009 06:33:31 PDT

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. — Mon, 01 Jun 2009 06:33:31 PDT

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. — Mon, 01 Jun 2009 06:33:31 PDT

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. — Mon, 01 Jun 2009 06:33:31 PDT

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. — Mon, 01 Jun 2009 06:33:31 PDT

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. — Mon, 01 Jun 2009 06:33:31 PDT

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.