Zhao-Yan Li

Gradient based iterative algorithm for solving coupled matrix equations

Abstract This paper is concerned with iterative methods for solving a class of coupled matrix equations including the well-known coupled Markovian jump Lyapunov matrix equations as special cases. The proposed method is developed from an optimization point of view and contains the well-known Jacobi iteration, Gauss–Seidel iteration and some recently reported iterative algorithms by using the hierarchical identification principle, as special cases. We have provided analytically the necessary and sufficient condition for the convergence of the proposed iterative algorithm. Simultaneously, the optimal step size such that the convergence rate of the algorithm is maximized is also established in explicit form. The proposed approach requires less computation and is numerically reliable as only matrix manipulation is required. Some other existing results require either matrix inversion or special matrix products. Numerical examples show the effectiveness of the proposed algorithm.

Observer based output feedback control of linear systems with input and output delays

Abstract This paper is concerned with observer based output feedback control of linear systems with both (multiple) input and output delays. Our recently developed truncated predictor feedback (TPF) approach for state feedback stabilization of time-delay systems is generalized to the design of observers. By imposing some restrictions on the open-loop system, two classes of observer based output feedback controllers, one being finite dimensional and the other infinite dimensional, are constructed. It is further shown that, the infinite dimensional observer based output feedback controllers can be generalized to linear systems with both time-varying input and output delays. It is also shown that the separation principle holds for the infinite dimensional observer based output feedback controllers, but does not hold for the finite dimensional ones. Numerical examples are worked out to illustrate the effectiveness of the proposed approaches.

Stabilization of some linear systems with both state and input delays

Abstract This paper studies the stabilization problem for some linear systems with both state and input time-varying delays. Three classes of systems are considered. The first class of systems consists of a chain of integrators with constant delays in both the state and the input. The second and the third classes of systems have respectively multiple point and distributed time-varying delays in both the state and the control. The parametric Lyapunov equation based approach is utilized to design stabilization controllers for these three classes of systems. Moreover, it is shown that the designed controllers can stabilize the systems semi-globally in the presence of input saturation. Two numerical examples are worked out to illustrate the effectiveness of the proposed approaches.

Brief paper: Detectability and observability of discrete-time stochastic systems and their applications

This paper studies detectability and observability of discrete-time stochastic linear systems. Based on the standard notions of detectability and observability for time-varying linear systems, corresponding definitions for discrete-time stochastic systems are proposed which unify some recently reported detectability and exact observability concepts for stochastic linear systems. The notion of observability leads to the stochastic version of the well-known rank criterion for observability of deterministic linear systems. By using these two concepts, the discrete-time stochastic Lyapunov equation and Riccati equations are studied. The results not only extend some of the existing results on these two types of equation but also indicate that the notions of detectability and observability studied in this paper take analogous functions as the usual concepts of detectability and observability in deterministic linear systems. It is expected that the results presented may play important roles in many design problems in stochastic linear systems.

Least squares solution with the minimum-norm to general matrix equations via iteration

Two iterative algorithms are presented in this paper to solve the minimal norm least squares solution to a general linear matrix equations including the well-known Sylvester matrix equation and Lyapunov matrix equation as special cases. The first algorithm is based on the gradient based searching principle and the other one can be viewed as its dual form. Necessary and sufficient conditions for the step sizes in these two algorithms are proposed to guarantee the convergence of the algorithms for arbitrary initial conditions. Sufficient condition that is easy to compute is also given. Moreover, two methods are proposed to choose the optimal step sizes such that the convergence speeds of the algorithms are maximized. Between these two methods, the first one is to minimize the spectral radius of the iteration matrix and explicit expression for the optimal step size is obtained. The second method is to minimize the square sum of the F-norm of the error matrices produced by the algorithm and it is shown that the optimal step size exits uniquely and lies in an interval. Several numerical examples are given to illustrate the efficiency of the proposed approach.

Stabilization of Discrete-Time Systems With Multiple Actuator Delays and Saturations

This paper studies the problems of global and semi-global stabilization of discrete-time linear systems with multiple input saturations and arbitrarily large bounded delays. By developing the methodology of truncated predictor feedback (TPF), state feedback control laws using only the current states of the systems are constructed to solve these problems. The feedback gains are dependent on the delay information of the open-loop system and thus are referred to as delay-dependent feedback. A method for determining the exact condition such that the resulting closed-loop system is asymptotically stable is also presented. Moreover, if the delays in the system are time-varying or even unknown, a modified delay-independent TPF is also established to solve the concerned problems. Numerical example has been worked out to illustrate the effectiveness of the proposed approaches.

Discrete-time l∞ and l2 norm vanishment and low gain feedback with their applications in constrained control

The existing low gain feedback, which is a parameterized family of stabilizing state feedback gains whose magnitudes approach zero as the parameter decreases to zero, has been designed in very specific ways. In this paper, by recognizing the l_∞ and l₂ slow peaking phenomenon that exists in discrete-time systems under low gain feedback, more general notions of l_∞ and l₂ norm vanishment are considered so as to provide a full characterization of the nonexistence of slow peaking phenomenon in some measured signals. Low gain feedback that does not lead to l_∞ and l₂ slow peaking in the control input are respectively referred to as l_∞ and l₂ low gain feedback. Based on the notions of l_∞ and l₂ vanishment, not only can the existing low gain feedback been recognized as an l_∞ low gain feedback, but also a new design approach referred to as the l₂ low gain feedback approach is developed for discrete-time linear systems. Parallel to the effectiveness of l_∞ low gain feedback in magnitude constrained control, the l₂ low gain feedback is instrumental in the control of discrete-time systems with control energy constraints. The notions of l_∞ and l₂-vanishment also result in a systematic approach to the design of l_∞ and l₂ low gain feedback by providing a family of solutions including those resulting from the existing design methods.

Positive operator based iterative algorithms for solving Lyapunov equations for Itô stochastic systems with Markovian jumps

This paper studies the iterative solutions of Lyapunov matrix equations associated with Ito stochastic systems having Markovian jump parameters. For the discrete-time case, when the associated stochastic system is mean square stable, two iterative algorithms with one in direct form and the other one in implicit form are established. The convergence of the implicit iteration is proved by the properties of some positive operators associated with the stochastic system. For the continuous-time case, a transformation is first performed so that it is transformed into an equivalent discrete-time Lyapunov equation. Then the iterative solution can be obtained by applying the iterative algorithm developed for discrete-time Lyapunov equation. Similar to the discrete-time case, an implicit iteration is also proposed for the continuous case. For both discrete-time and continuous-time Lyapunov equations, the convergence rates of the established algorithms are analyzed and compared. Numerical examples are worked out to validate the effectiveness of the proposed algorithms.

On improving transient performance in global control of multiple integrators system by bounded feedback

Abstract The global stabilization problem for multiple integrators systems by bounded control is considered. Two classes of nonlinear feedback laws are proposed. The first one consisting of nested saturation functions is a modification and generalization of that in [A.R. Teel, Global stabilization and restricted tracking for multiple integrators with bounded controls, Systems & Control Letters 18 (3) (1992) 165–171] and the other one consists of parallel connections of saturation functions. Both these two types of nonlinear feedback laws need only n ˜ ( n ˜ = n 2 if n is even and n ˜ = n + 1 2 if n is odd, where n is the length of the multiple integrators) saturation elements. Furthermore, the poles of the closed-loop system can be placed on any location of the left real axis when none of the saturation elements in the control laws is saturated. Both of them exhibit simpler structure, can significantly improve the transient performance of the closed-loop system, and are very superior to the other existing methods. Simulation of a fourth-order system is used to illustrate the effectiveness of the proposed methods.

On robustness of predictor feedback control of linear systems with input delays

Abstract This paper is concerned with the robustness of the predictor feedback control of linear systems with input delays. By applying certain equivalent transformations on the characteristic equation associated with the closed-loop system, we first transform the robustness problem of a predictor feedback control system into the stability problem of a neutral time-delay system containing an integral operator in the derivative. The range of the allowable input delay for this neutral time-delay system can be computed by exploring its delay dependent stability conditions. In particular, delay dependent stability conditions for the neutral time-delay system are established by partitioning the delay into segments. The conservatism of this method can be reduced when the number of segments in the partition is increased. Numerical examples are worked out to illustrate the effectiveness of the proposed method.