Robust stability analysis of time-varying systems using time-varying quadratic forms

We consider the robust stability of time-varying linear systems subject to structured time-varying uncertainty. We provide an alternate proof of the result that robust stability holds if and only if a scaled small-gain condition holds asymptotically. The new proof employs an extension of the so-called S-procedure losslessness theorem to time-varying quadratic forms which is of independent interest.     

Stability analysis of systems with uncertain time-varying delays

Stability of systems in the presence of bounded uncertain time-varying delays in the feedback loop is studied. The delay parameter is assumed to be an unknown time-varying function for which the upper bounds on the magnitude and the variation are given. The stability problem is treated in the integral quadratic constraint (IQC) framework. Criteria for verifying robust stability are formulated as feasibility problems over a set of frequency-dependent linear matrix inequalities. The criteria can be equivalently formulated as semi-definite programs (SDP) using Kalman-Yakubovich-Popov lemma. As such, checking robust stability can be performed in a computationally efficient fashion.     

Stability and stabilisation for time-varying polytopic quadratic systems

This paper develops the problems of local stability and stabilisation for time-varying polytopic quadratic systems. The state-space data is assumed to be dependent on parameters that are measurable in real time and vary in a compact set with bounded variation rates. These problems are solved by utilising the parameter-dependent quadratic Lyapunov function and S-procedure approach. Sufficient conditions for local stability and stabilisation are first formulated as optimisation problems with ‘quasi’ linear matrix inequalities. Simulation examples are then provided to confirm the effectiveness of the given approach.     

Performance analysis of gradient neural network exploited for online time-varying quadratic minimization and equality-constrained quadratic programming

In this paper, the performance of a gradient neural network (GNN), which was designed intrinsically for solving static problems, is investigated, analyzed and simulated in the situation of time-varying coefficients. It is theoretically proved that the gradient neural network for online solution of time-varying quadratic minimization (QM) and quadratic programming (QP) problems could only approximately approach the time-varying theoretical solution, instead of converging exactly. That is, the steady-state error between the GNN solution and the theoretical solution can not decrease to zero. In order to understand the situation better, the upper bound of such an error is estimated firstly, and then the global exponential convergence rate is investigated for such a GNN when approaching an error bound. Computer-simulation results, including those based on a six-link robot manipulator, further substantiate the performance analysis of the GNN exploited to solve online time-varying QM and QP problems.     

Probabilistic output admissible set for systems with time-varying uncertainties

This paper considers time-varying uncertain constrained systems, and develops a method for computing a probabilistic output admissible (POA) set. This set consists of the initial states probabilistically assured to satisfy the constraint. The time-invariant counterpart has already been investigated in Hatanaka and Takaba [Computations of probabilistic output admissible set for uncertain constrained systems, Automatica 44 (2) (2008), to appear]. We first define the POA set for time-varying uncertainties with finite dimensional probability space. Then, we show that an algorithm similar to Hatanaka and Takaba [Computations of probabilistic output admissible set for uncertain constrained systems, Automatica 44 (2) (2008), to appear] provides the POA set also in the time-varying case, as long as an upper bound of a what we call future output admissibility (FOA) index is available. We moreover present two methods for computing the upper bound of the FOA index: probabilistic and deterministic methods. A numerical simulation demonstrates the effectiveness of our algorithm.     

Simple stability criteria for systems with time-varying delays

This paper considers the problem of checking stability of linear feedback systems with time-varying but bounded delays. Simple but powerful criteria of stability are presented for both continuous-time and discrete-time systems. Using these criteria, stability can be checked in a closed loop Bode plot. This makes it easy to design the system for robustness.     

Stability analysis for linear delayed systems via an optimally dividing delay interval approach

This paper addresses the problem of stability for linear systems with time-varying delay. A novel augmented Lyapunov–Krasovskii functional is constructed by using the idea of optimally dividing the delay interval [0,τ(t)] into some variable sub-intervals and line integral technology. Using the novel augmented functional, the new delay-dependent stability criteria are proposed for linear systems with time-varying delay. The gain is that this stability criterion can lead to much less conservative stability results compared to other methods for linear systems with delay. Two numerical examples are provided to verify the effectiveness of the proposed criteria.     

Stability analysis of nonlinear quadratic systems via polyhedral Lyapunov functions

Quadratic systems play an important role in the modeling of a wide class of nonlinear processes (electrical, robotic, biological, etc.). For such systems it is mandatory not only to determine whether the origin of the state space is locally asymptotically stable, but also to ensure that the operative range is included into the convergence region of the equilibrium. Based on this observation, this paper considers the following problem: given the zero equilibrium point of a nonlinear quadratic system, assumed to be locally asymptotically stable, and a certain polytope in the state space containing the origin, determine whether this polytope belongs to the domain of attraction of the equilibrium. The proposed approach is based on polyhedral Lyapunov functions, rather than on the classical quadratic Lyapunov functions. An example shows that our methodology may return less conservative results than those obtainable with previous approaches.     

On stabilization for discrete linear time-varying systems

In this paper, we consider the stabilization problem for discrete linear time-varying systems in an operator-theoretic framework. By using the complete finiteness of a certain discrete nest algebra, we show that a system is stabilizable if and only if it has one kind of strong representation and we also give a parametrization for all the stabilizing controllers in terms of this strong representation. This result extends the Youla parametrization theorem by only requiring a strong left or a strong right representation but not both. The strong stabilization problem is also discussed.     

Stability analysis of particle swarm optimization without Lipschitz constraint

There are some adjustable parameters which directly influence the performance and stability of Particle Swarm Optimization algorithm. In this paper, stabilities of PSO with constant parameters and time-varying parameters are analyzed without Lipschitz constraint. Necessary and sufficient stability conditions for acceleration factor φ and inertia weight w are presented. Experiments on benchmark functions show the good performance of PSO satisfying the stability condition, even without Lipschitz constraint . And the inertia weight w value is enhanced to ( - 1,1).     

Stability analysis with Popov multipliers and integral quadratic constraints

It is shown that a general form of Popov multipliers can be used in stability analysis based on integral quadratic constraints (IQC). The Popov multiplier is nonproper and a condition that the nominal plant is strictly proper will be imposed in order to ensure boundedness of the IQC corresponding to the Popov multiplier. A consequence of our main result is that the classical Popov criterion can be combined with a stability criterion for slope restricted nonlinearities developed by Zames and Falb. An example shows that the combination of these two criteria is useful in applications.     

Stability analysis of systems via a new double free-matrix-based integral inequality with interval time-varying delay

ABSTRACT

This paper investigates the problem of delay-dependent stability analysis for systems with interval time-varying delay. By means of a new double free-matrix-based integral inequality and augmented Lyapunov–Krasovskii functionals containing as much information of time-varying delay as possible, a new stability criterion for systems is established. Firstly, by a double integral term two-step estimation approach and combined with single free-matrix-based integral inequalities, a stability criteria is presented. Then, compared with the double integral term two-step estimation approach, the proposed new double free-matrix-based integral inequality with more related time delays has potential to lead to a criterion with less conservatism. Finally, the validity of the presented method is demonstrated by two numerical examples.     

Delay-range-dependent stability for systems with time-varying delay

This paper is concerned with the stability analysis for systems with time-varying delay in a range. An appropriate type of Lyapunov functionals is proposed to investigate the delay-range-dependent stability problem. The present results may improve the existing ones due to a method to estimate the upper bound of the derivative of Lyapunov functional without ignoring some useful terms and the introduction of additional terms into the proposed Lyapunov functional, which take into account the range of delay. Numerical examples are given to demonstrate the effectiveness and the benefits of the proposed method.     

Minimax control for discrete-time time-varying stochastic systems

This paper gives a self-contained presentation of minimax control for discrete-time time-varying stochastic systems under finite- and infinite-horizon expected total cost performance criteria. Suitable conditions for the existence of minimax strategies are proposed. Also, we prove that the values of the finite-horizon problem converge to the values of the infinite-horizon problems. Moreover, for finite-horizon problems an algorithm of calculation of minimax strategies is developed and tested by using time-varying stochastic systems.     

Stability analysis of systems with stochastically varying delays

A stability analysis of linear systems with stochastically varying delay is performed. It is assumed that the delay function has the form of a sawtooth with switches occurring at the arrival times of a homogeneous Poisson process. Several notions of stochastic stability are considered and corresponding stability criteria are derived. For two examples the different criteria are compared and the effect on stability of various deterministic approximations is examined.     

Lyapunov functions for time-varying systems satisfying generalized conditions of Matrosov theorem

The classical Matrosov theorem concludes uniform asymptotic stability of time-varying systems via a weak Lyapunov function (positive definite, decrescent, with negative semi-definite derivative along solutions) and another auxiliary function with derivative that is strictly nonzero where the derivative of the Lyapunov function is zero (Mastrosov in J Appl Math Mech 26:1337–1353, 1962). Recently, several generalizations of the classical Matrosov theorem have been reported in Loria et al. (IEEE Trans Autom Control 50:183–198, 2005). None of these results provides a construction of a strong Lyapunov function (positive definite, decrescent, with negative definite derivative along solutions) which is a very useful analysis and controller design tool for nonlinear systems. Inspired by generalized Matrosov conditions in Loria et al. (IEEE Trans Autom Control 50:183–198, 2005), we provide a construction of a strong Lyapunov function via an appropriate weak Lyapunov function and a set of Lyapunov-like functions whose derivatives along solutions of the system satisfy inequalities that have a particular triangular structure. Our results will be very useful in a range of situations where strong Lyapunov functions are needed, such as robustness analysis and Lyapunov function-based controller redesign. We illustrate our results by constructing a strong Lyapunov function for a simple Euler-Lagrange system controlled by an adaptive controller and use this result to determine an ISS controller.     

Stability analysis of systems with aperiodic sample-and-hold devices

Motivated by the widespread use of networked and embedded control systems, improved stability conditions are derived for sampled-data feedback control systems with uncertainly time-varying sampling intervals. The results are derived by exploiting the passivity-type property of the operator arising in the input-delay approach to the system in addition to the gain of the operator, and are hence less conservative than existing ones.     

Stability analysis for uncertain switched neutral systems with discrete time-varying delay: A delay-dependent method

This paper considers the robust stability of a class of switched neutral systems with discrete time-varying delay and time-varying structure. Sufficient conditions for exponential stability criteria are developed for arbitrary switching signal with average dwell time. The results are obtained based on Lyapunov’s stability analysis via Krasovsky–Lyapunov’s functionals and the related stability study is performed to obtain delay-dependent results. It is proved that the stabilizing switching rule is arbitrary if all the switched subsystems are exponentially stabile. These conditions are delay-dependent and are given in the form of linear matrix inequalities (LMIs). Some examples are worked out to illustrate the effectiveness of the result.