IMA Journal of Mathematical Control and Information - Advance Access

Boundary stabilization of hyperbolic systems related to overhead cranes

Sano, H. — 2008-02-14

In this paper, we treat the problem of stabilizing a flexible cable with two rigid loads which is described by two kinds of hyperbolic equations. From the engineering point of view, the model can be regarded as a distributed parameter overhead crane which conveys two loads simultaneously. In this, the mass of the cart is assumed to be not zero. After deriving a control law which does not increase an energy defined for the model, we prove the asymptotic stability of the closed-loop system by using the LaSalle's invariance principle. The stabilization problem is also discussed for the case of the system with one rigid load.

Global stabilization of non-linear discrete-time systems by linear feedback

Medina, R. — 2008-02-08

The stabilization problem for a class of non-linear, non-autonomous discrete-time systems is discussed. Based on the ‘freezing’ method to discrete-time systems, we derive explicit conditions for global feedback exponential stabilizability. This approach will allow us to avoid the construction of Lyapunov functions in some situations.

Controllability of linear difference equations in Hilbert spaces and applications

Leiva, H., Uzcategui, J. — 2008-02-08

In this paper, we present necessary and sufficient conditions for the exact and approximate controllability of the following linear difference equation: where Z, U are Hilbert spaces, A(·) l( , L(Z)), B(·) l( , L(U, Z)), u l²( , U) and ^* = {0}. Moreover, in the case of exact controllability, the control u l²( , U) steering an initial state z₀ to a final state z₁ in time n₀ is given by the formula according to Lemma 2.1. As a particular case, we consider the discretization on flow of the following controlled evolution equation z' = Az + Bu, z Z, u U, t > 0, where B L (U, Z), u L²(0, ;U) and A is the infinitesimal generator of a strongly continuous semigroup {T(t)}_t _{≥ 0} in Z, given by according to Lemma 1.1. These results are applicable to a broad class of reaction–diffusion systems such as the heat equation, the wave equation, the equation modelling the damped flexible beam, the strongly damped wave equation, the thermoelastic plate equation, etc. In Section 4, these results are applied to a discrete version of the n-dimensional heat and n-dimensional wave equation.

Digital design of combined PI and state feedback controller for non-linear stochastic systems

Zhou, H.-Q., Shieh, L.-S., Liu, C. R., Wang, Q.-G. — 2007-11-17

In this paper, a combined Proportional-Integral (PI) and state feedback linear discrete control scheme is proposed for non-linear stochastic systems. The state-dependent optimal linear model of the original non-linear plant is first constructed from system state feedback at each sampling period, then the discrete linear quadratic regulator approach with pole placement is applied to issue and update controller settings. If the system is under stochastic process noise, the innovation form of Kalman gain can be employed for optimal states estimation without requiring prior knowledge of noise properties. Stability conditions for sampled-data non-linear systems are addressed in discrete-time analysis. The effectiveness of the proposed method will be demonstrated by the simulation examples of both single-input single-output and multi-input multi-output non-linear processes.

A note on observer for one-sided Lipschitz non-linear systems

Hu, G.-D. — 2007-09-14

In this note, observer design of a class of non-linear systems is considered. A quasi-one-sided Lipschitz condition is introduced to estimate the influence of non-linear vector functions on the observer. Based on the quasi-one-sided Lipschitz condition, sufficient conditions for existence of observers of the class of non-linear systems are presented which are less conservative than the results based on Lipschitz condition or one-sided Lipschitz condition in literature. Furthermore, a gain matrix of the observer is given by the linear matrix inequality.

Design of static and dynamic output feedback controllers through Euler approximate models: uncertain systems with norm-bounded uncertainties

Ibrir, S. — 2007-09-14

We propose new sufficient linear matrix inequality (LMI) conditions for the stability of uncertain linear systems with static and dynamic output feedbacks. The design of stabilizing controllers is carried out through Euler approximate models where the sampling period appears as a linear variable to be determined. Pole placement is also considered in LMI setting. Examples showing the efficacy of the developed results are presented.

Gain-scheduled H{infty}-output feedback control for parameter-varying systems with delays

Zhang, B., Zhou, S., Xu, S. — 2007-09-14

This paper deals with the problem of gain-scheduled H-output feedback controller design for a class of parameter-varying systems with time-varying delays. By introducing equations involving free-weighting matrices, a new condition for the stability and H-performance analysis of the resulting closed-loop system is obtained. Based on this, a sufficient condition for the existence of a desired gain-scheduled H-output feedback controller is derived and expressed in terms of linear matrix inequalities (LMIs). When these LMIs are feasible, an explicit expression of the desired controller is given. Numerical examples are provided to demonstrate the applicability of the proposed design method.

Generalized PI observer design for linear systems

Wu, A.-G., Duan, G.-R. — 2007-09-05

A parametric design approach for generalized proportional–integral (PI) observers of linear systems is proposed. Based on an explicit parametric general solution to a type of Sylvester matrix equations, complete parameterizations for all observer gains are presented. Moreover, the parametric expression of the left eigenvector matrix of the observer system is also established. The proposed design method can offer all the degrees of design freedom which can be utilized to achieve various desirable system specifications and performances. Also, a numerical example is given to show the design procedure and illustrate the effect of the proposed approach.

Self-tuning fault-tolerant digital PID controller for MIMO analogue systems with partial actuator and system component failures

Hong Tsai, J. S., Lin, J. Y., San Shieh, L., Chandra, J., Guo, S.-M. — 2007-09-05

A new methodology is presented to synthesize a digitally redesigned, active, self-tuning, fault-tolerant proportional–integral–derivative (PID) controller for multi-input–multi-output (MIMO) analogue systems to against partial actuator and system component failures. The fault-tolerant control (FTC) scheme possesses the ability to accommodate for system failures automatically and maintains the acceptable overall system performance in the event of partial actuator and system component failures. The theoretically well-designed analogue PID controller is refined using the continuous-time linear-quadratic regulator approach to have the high-gain property. Then, a predication-based digital redesign technique is utilized to discretize the cascaded MIMO analogue PID controller for finding a low-gain digital PID controller. Besides, a self-tuning FTC scheme with a modified Kalman filter algorithm is proposed, which is not only for the control system design but also for the faulty system recovery. The designed scheme can easily be implemented using digital processors. An illustrative example is presented to demonstrate the effectiveness of the proposed methodology.

Adaptive regulator design for undamped second-order hyperbolic systems with output disturbances

Kobayashi, T., Sakamoto, T., Oya, M. — 2007-09-05

In this paper, an adaptive regulator is designed for undamped second-order hyperbolic systems with output disturbances in the case where the input and output operators are collocated. The systems may have an infinite number of poles and zeros on the imaginary axis. The adaptive regulator is constructed by a high-gain adaptive feedback and a new type of the estimation mechanism for the unknown parameters of the disturbances. First, the well-posedness of the closed-loop system is justified. Next, the stability of the closed-loop system is analysed. An example is given to illustrate the theory.

Resilient feedback stabilization of discrete-time systems with delays

Mahmoud, M. S., Boukas, E.-K., Shi, P. — 2007-06-08

A class of linear, uncertain discrete-time systems with state delay is considered. We develop an linear matrix inequality (LMI)-based analysis and redesign procedures for improved robust stability of discrete-time systems with state delay and bounded nonlinearities. Then, we address the robust stabilization using nominal and resilient feedback designs. In both cases, the trade-off between the size of the controller gains and the bounding factors is illuminated and incorporated into the design formalism. Seeking computational convenience, all the developed results are cast in the format of LMIs and several numerical examples are presented throughout the paper.

Symplectic Runge-Kutta methods for the Kalman-Bucy filter

Hu, G.-D. — 2007-05-20

In this paper, numerical methods for the Kalman–Bucy filter are investigated from the viewpoint of geometry. The differential matrix Riccati equation for the Kalman–Bucy filter is transformed into a linear differential Hamiltonian system. We show that the linear differential Hamiltonian system with two different initial conditions is on symplectic group. The two different initial conditions relate to two different statistical assumptions about the initial state of a linear time-varying dynamical system. Then, symplectic Runge–Kutta methods can be applied to the linear differential Hamiltonian system, which keep the numerical solution on the symplectic group. Numerical examples are given to illustrate the performance of the numerical methods.

Exact controllability of nonselfadjoint Euler-Bernoulli beam model via spectral decomposition method

Shubov, M. A. — 2007-05-19

The zero controllability problem for the hyperbolic equation, which governs the vibrations of the Euler–Bernoulli beam model of a finite length, is studied in this paper. The equation of motion is supplied with a one-parameter family of physically meaningful boundary conditions containing damping terms. The control is introduced as a separable forcing term g(x)f(t) in the right-hand side of the equation. A force profile function, g(x), is assumed to be given. To construct the control, f(t), which brings a given initial state of the system to zero on the specific time interval [0, T], the spectral decomposition method is applied. The necessary and/or sufficient conditions for the exact controllability as well as the explicit formulas for the control laws are given. Approximate controllability is also discussed. The exact controllability results are generalized to the case of multiple eigenvalues of the main dynamics generator.

A necessary and sufficient condition for the controllability of linear systems in Hilbert spaces and applications

Iturriaga, E., Leiva, H. — 2007-05-15

As we have announced in the title of this work, we show that a broad class of linear evolution equations is exactly controllable. This class is represented by the following infinite-dimensional linear control system:$$\dot{z}=\mathcal{A}z+\mathcal{B}u(t),\hbox{ \hspace{1em} }t > 0,\hbox{ \hspace{1em} }z\in Z,\hbox{ \hspace{1em} }u(t)\in U,$$where Z, U are Hilbert spaces, the control function u belong to L²(0, t₁;U), t₁ > 0, $$\mathcal{B}\in L(U,Z)$$ and $$\mathcal{A}$$ generates a strongly continuous semigroup operator T(t) according to Pazy. We give a necessary and sufficient condition for the exact controllability of this system and apply this result to a linear controlled damped wave equation.

Controllability of neutral functional evolution integrodifferential systems with infinite delay

Balachandran, K, Leelamani, A, Kim, J-H — 2007-05-10

In this paper, we establish sufficient conditions for the controllability of neutral functional evolution integrodifferential systems. The results are obtained by using the analytic semigroup theory and the Nussbaum fixed point theorem.