Positive invariance of linear uncertain systems and applications

Sufficient and necessary conditions for nonlinear impulsive systems considering linear constraints with bounded additive disturbances are obtained from a characterization of positive invariance properties for these systems. As the application of results obtained, the sufficient conditions for discrete state feedback control and for asymptotic stabilization of the linear uncertain systems with the state positively asymmetrical invariant set are presented. Finally, a numerical example is given.

The control of systems with uncertain dynamics and unpredictable disturbances has raised some challenging problems. This is particularly important when high system performance is to be guaranteed at all times. Recently, Time Delay Control has been suggested as an alternative control scheme. The proposed control system does not require an explicit plant model nor does it depend on the estimation of specific plant parameters. Rather, it combines adaptation with past observations to directly estimate the effect of the plant dynamics. This paper outlines the Time Delay Control law for a class of linear dynamic systems and then presents a sufficient condition for stability of linear uncertain systems with time delay. The ideas of Nyquist and Kharitonov are used in the development of a sufficient condition, which does not resort to using approximations for time delay. Like Nyquist, the condition depends on maps of the Nyquist path and, like Kharitonov, stability depends on four functions each yielding a stable system. In this paper we combine these ideas to determine the stability of systems where the Time Delay Controller is applied to single input single output, linear time-invariant plants whose coefficients are known to vary within certain defined intervals. The development is carried out in the context of Time Delay Control but it can be applied in more general cases. Two examples will illustrate the approach and the usefulness of the technique.

Kharitonov's theorems are generalized to the problem of so-called weak Kharitonov regions for robust stability of linear uncertain systems. Given a polytope of (characteristic) polynomials P and a stability region D in the complex plane, P is called D-stable if the zeros of every polynomial in P are contained in D. It is of interest to know whether the D-stability of the vertices of P implies the D-stability of P. A simple approach is developed which unifies and generalizes many known results on this problem. >

Delay-independent robust stability of uncertain linear systems

Deals with the problem of robust stability of linear systems with uncertain real time-varying parameters with magnitudes and rates of change (or variations, in discrete-time) which are confined to a given convex region. A notion of robust stability, referred to as bi-quadratic stability, which is based on a Lyapunov function that depends quadratically on the uncertain parameters is proposed. LMI based sufficient conditions for bi-quadratic stability of continuous- and discrete-time systems are developed. The proposed stability analysis methods have the advantage that they can be used for solving problems of robust control synthesis, including robust stabilization, robust H/sub 2/ control, and robust H/sub /spl infin// control.

The problem of stabilizing a class of uncertain time-delay systems via memoryless linear feedback is examined. The systems under consideration are linear systems with time-varying state delays. They also contain uncertain parameters (possibly time-varying) whose values are known only to within a prescribed compact bounding set. The main contribution given is to enlarge the class of time-delay systems for which one can construct a stabilizing memoryless linear feedback controller. Within this framework, a novel notion of robust memoryless stabilizability is first introduced via the method of Lyapunov functionals. Then a sufficient condition for the stabilizability is proposed. It is shown that solvability of a parameterized Riccati equation can be used to determine whether the time-delay system satisfies the sufficient condition. If there exists a positive definite symmetric solution satisfying the Riccati equation, a suitable memoryless linear feedback law can be derived.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time-domain results on the quadratic stabilization of uncertain systems and the frequency-domain results on H/sup infinity / optimization. A complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given. Relations between these robust stabilization problems and H/sup infinity / control theory are explored. It is also shown that in a number of cases, if a robust stabilization problem can be solved via Lyapunov methods, then it can be also be solved via H/sup infinity / control theory-based methods. >

In this paper, a delay-independent sufficient condition for stability of linear uncertain systems with multiple delays is established. A new stability criterion is obtained based on the related theory of spectral radius and modulus matrix by the frequency domain method. The results are presented based on the assumption that system matrix A is a Hurwitz one and there is no requirement of a negative matrix measure. In comparison with the results in the literature, it is showed that the proposed method is less conservative.

The robust stability of linear uncertain systems with multiple uncertain parameters is considered. A sphere norm bound condition on the uncertainties, which assures stability of the systems, is derived. This bound is shown to be less conservative than existing ones. A necessary condition to maximize this bound is also given. >

In this paper we consider adaptive distributed stabilization for uncertain multivariable linear systems with a time-varying diagonal matrix gain. We show that an unknown system matrix being an M-matrix is a sufficient condition to ensure uncertain linear systems to be stabilizable by matrix high gains, and derive a threshold condition to ensure exponential stability of uncertain linear systems stabilized by a monotonically increasing diagonal gain matrix. When each individual gain function in the matrix gain is updated by state-dependent functions, the boundedness and convergence of both system states and adaptive matrix gains are guaranteed. We apply the matrix gain stabilization approach to adaptive synchronization control for complex networks with time-varying coupling weights.

This paper is concerned with the stabilization of uncertain linear systems via observer-based controllers. The motivation comes from earlier work (see [6] and [7]) where sufficient conditions are given for uniform asymptotic stability of the combined plant-observer-controller. In this paper, we describe an augmented dynamical system whose stability via an observer-based controller implies likewise for the original plant. While the given system may have uncertainties in the matrices describing the model, input connection, output distribution and direct transmission, the augmented system only contains uncertainties in its model matrix. The advantages of "lumping" the uncertainty into the model matrix are also discussed.

The effects of observers on robust linear feedback controllers are studied. Sufficient conditions are obtained that guarantee full recovery of the allowable uncertainty bounds attainable by full state feedback. The effects of the resulting high gain observers on the disturbance rejection bounds are also studied. It is shown that full recovery of the uncertainty bound leads to possible large degradation in disturbance rejection. However, if there is only additive plant disturbance and no measurement disturbance, this degradation can be prevented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

This paper investigates the stability of linear uncertain systems with time-varying delay. Criteria for guaranteed asymptotic stability are derived based on a discretized Lyapunov functional approach. Our results are the generalizations of results reported in Gu (1997) to the case with a time-varying delay. Numerical examples are also included to show the effectiveness of the method.

This paper presents a robust linear dynamic compensator design algorithm for linear uncertain systems whose parameters vary within given bounded sets. The algorithm explicitly incorporates the structure of the uncertainty into the design procedure and utilizes the elemental perturbation bounds developed recently. The special cases of linear state feedback and measurement feedback controllers are considered and the relative trade offs are discussed. The design algorithm is illustrated with the help of a simple example.

The paper provides new results on the flow (positive) invariance of the families of 0-symmetrical sets, which are defined by arbitrary Hölder norms and time-dependent diagonal matrices. Thus, we introduce the concept of “diagonal invariance” as a system property with local or global character. For this property, we formulate sufficient conditions in the case of time-variant or -invariant, nonlinear systems and necessary and sufficient conditions in the case of time-variant or -invariant, linear systems. We also derive a comparison method that allows exploring the diagonal invariance of time-variant or -invariant, nonlinear or linear systems, by using time-invariant, linear comparison systems. We illustrate the applicability of the diagonal invariance criteria to the important class of nonlinear systems described by Hopfield neural networks. These new results represent a meaningful generalization of some previous researches developed by the same authors.