Componentwise ultimate bound and invariant set computation for switched linear systems

We present a novel ultimate bound and invariant set computation method for continuous-time switched linear systems with disturbances and arbitrary switching. The proposed method relies on the existence of a transformation that takes all matrices of the switched linear system into a convenient form satisfying certain properties. The method provides ultimate bounds and invariant sets in the form of polyhedral and/or mixed ellipsoidal/polyhedral sets, is completely systematic once the aforementioned transformation is obtained, and provides a new sufficient condition for practical stability. We show that the transformation required by our method can easily be found in the well-known case where the subsystem matrices generate a solvable Lie algebra, and we provide an algorithm to seek such transformation in the general case. An example comparing the bounds obtained by the proposed method with those obtained from a common quadratic Lyapunov function computed via linear matrix inequalities shows a clear advantage of the proposed method in some cases.     

Systematic ultimate bound computation for sampled-data systems with quantization

We present a novel systematic method to obtain componentwise ultimate bounds in perturbed sampled-data systems, especially when the perturbations arise due to quantization. The proposed method exploits the system geometry as well as the perturbation structure, and takes intersample behavior into account. The main features of the method are its systematic nature, whereby it can be readily computer coded, without requiring adjustment of parameters for its application, and its suitability for dealing with highly structured perturbation schemes, whereby the information on the perturbation structure is directly taken into account. The latter feature distinguishes the method from other approaches that require a bound on the norm of the perturbation and thus disregard information on the perturbation structure. We apply the method to a numerical example taken from the literature to illustrate its simplicity and potential.     

On ultimate boundedness around non-assignable equilibria of linear time-invariant systems

In this note we investigate the following questions: given a (finite-dimensional) linear time-invariant (LTI) multivariable system and a constant desired value for its output, say y_?. Assume there is no assignable equilibrium point corresponding to y_?. How “close” to y_? can we ultimately keep the output using LTI static state-feedback stabilizing controllers? Can this neighborhood of y_? be reduced with dynamic, nonlinear, time-varying controllers? Our main contributions are the proof that the optimal ultimate boundedness neighborhood is achieved with LTI static state-feedback, the explicit computation of the neighborhood's size and the proof, under some reasonable rank assumptions, that the system has non-assignable values for the output if and only if it has a transmission zero at zero. Interestingly, there is no connection between this problem and the more familiar concepts of controllability and observability.     

A new Lyapunov function for stability of time-varying nonlinear perturbed systems

In this paper, global and local uniform asymptotic stability of perturbed dynamical systems is studied by using Lyapunov techniques. The restriction about the perturbed term is that the perturbation is bounded by an integrable function under the assumption that the nominal system is globally uniformly asymptotically stable. We use a new Lyapunov function to obtain a global uniform asymptotical stability of some perturbed systems.     

Gain scheduling control of nonlinear singularly perturbed time-varying systems with derivative information

In this paper, we analyse the ultimate boundedness of nonlinear singularly perturbed time-varying systems and propose a control law using gain scheduling where the slow state and the exogenous signals are used as scheduling variables. In our control scheme, we have some flexibility in selecting the slow manifold of the system. Moreover, the derivative information can be properly engaged to manipulate the size of ultimate bound in tracking error of the controlled system.     

A new method to obtain ultimate bounds and convergence rates for perturbed time‐delay systems

A new method is proposed to determine the ultimate bounds and the convergence rates for perturbed time‐delay systems when the Lyapunov–Krasovskii functionals and their derivatives are available. Compared with existing methods, the proposed method is more concise, more widely applicable, and the obtained results are less conservative. To show the three features, the proposed method is applied to improve three existing results, respectively. Copyright © 2011 John Wiley & Sons, Ltd.     

Asymptotic stability of nonlinear multiparameter singularly perturbed systems

Sufficient conditions are derived to guarantee the asymptotic stability of a class of nonlinear singularly perturbed systems with several perturbation parameters of the same order. Estimates of the region of attraction and bounds on the small parameters are obtained.     

Asymptotic stability of nonlinear singularly perturbed systems using higher order corrections

This paper examines a methodology for investigating the asymptotic stability properties of nonlinear singularly perturbed systems. This is achieved by constructing the so-called zeroth order model (uncorrected model). Based on this model a weighted scalar sum of the Lyapunov functions for two lower order subsystems is obtained. The estimates of the region of attraction and the upper bound on the perturbed parameter are established. It is shown that by extending this methodology to higher order corrected models, less conservative results are obtained. These considerable improvements are facilitated by introducing a new fast variable in both uncorrected and corrected models.     

Adaptive control of a class of nonlinearly perturbed linear systems of relative degree two

A simple adaptive output feedback strategy, incorporating gains of Nussbaum type, is described and is shown to be a universal stabilizer for a class of nonlinearly perturbed, single-input, linear systems of relative degree two.     

Reachable set estimation for switched positive systems

This paper focuses on the problem of reachable set estimation for discrete-time switched positive systems under two possible classes of exogenous disturbance. The multiple linear copositive Lyapunov function approach is applied to determine the bounding hyper-pyramids for the reachable set. Based on some Lyapunov-based inequalities and the linear version of the S-procedure technique, the bounding hyper-pyramids for the reachable set can be determined by solving a set of inequalities. Two optimisation methods are adopted to make the bounding hyper-pyramids as small as possible. Genetic Algorithm (GA) is utilised to search for the optimal value of the decision variables in the obtained inequalities. Finally, numerical examples are given to illustrate the effectiveness of the theoretical findings.     

Integral-based event triggering controller design for stochastic LTI systems via convex optimisation

The presence of measurement noise in the event-based systems can lower system efficiency both in terms of data exchange rate and performance. In this paper, an integral-based event triggering control system is proposed for LTI systems with stochastic measurement noise. We show that the new mechanism is robust against noise and effectively reduces the flow of communication between plant and controller, and also improves output performance. Using a Lyapunov approach, stability in the mean square sense is proved. A simulated example illustrates the properties of our approach.     

Upper bound covariance control of discrete perturbed systems

The purpose of this paper is to develop a methodology, which is based on the state covariance assignment theory, to design feedback controllers such that a specified state covariance upper bound will be achieved. Furthermore, an example is given to illustrate the effectiveness of the present approach.     

Exponential ultimate boundedness of impulsive stochastic delay difference systems

This paper is concerned with the exponential ultimate boundedness problems for the impulsive stochastic delay difference systems. Several sufficient conditions on the global pth moment exponential ultimate boundedness are presented by using the Lyapunov methods and the algebraic inequality techniques, and the estimated exponential convergence rate and the ultimate bound are provided as well. As an application, the boundedness criteria are applied to a class of discrete impulsive stochastic neural networks with delays. The obtained results show that the impulses not only can stabilize an unstable stochastic difference delay system but also can make an unbounded stochastic difference delay system into a bounded system. Examples and simulations are also provided to demonstrate the effectiveness of the derived theoretical results.     

Robust state estimation for singularly perturbed systems

This paper deals with the design of interval observers for singularly perturbed linear systems. The full-order system is first decoupled into slow and fast subsystems. Then, using the cooperativity theory, an interval observer is designed for the slow and fast subsystems assuming that the measurement noise and the disturbances are bounded and the singular perturbed parameter is uncertain. This decoupling leads to two observers that estimate the lower and upper bounds for the feasible state domain. A numerical example shows the efficiency of the proposed technique.     

Near optimal smoothing for singularly perturbed linear systems

A two time-scale lower order design is proposed as a near optimal solution to the fixed-interval smoothing problem for systems with slow and fast modes. The slow mode smoothing solution, in the limit as the perturbation parameter μ → 0, tends to that of the reduced-order problem; and the near optimal fast mode smoother is simply a weighted sum of lower order two time-scale filters. While the near optimal solution affords a significant reduction in computational complexity, the performance degradation, as illustrated in an example, is typically negligible.     

Exponential ultimate boundedness and stability of stochastic differential equations with impulese

The paper mainly studies globally pth moment exponentially ultimate boundedness and pth moment exponential stability of impulsive stochastic functional differential equations. By using the Lyapunov direct method of Razumikhin-type condition and the principle of comparison, some sufficient conditions for globally pth moment exponentially ultimate boundedness and globally pth moment exponential stability are presented. Theorems require the linear coefficients of the upper bound of Lyapunov differential operators are time-varying functions; this generalizes the previous results. When the time delay is not considered in the system, a unified criterion is given to achieve boundedness and stability when the system is disturbed by stabilizing impulse and destabilizing impulse. It shows that the stochastic differential equation may be unbounded or unstable, and it can be bounded or stable by adding appropriate impulsive perturbation. Finally, we use two examples to illustrate the validity of our results.     

Stability bound of discrete multiple time-delay singularly perturbed systems

In this paper, the exact stability bound for discrete multiple time-delay singularly perturbed systems is examined. The obtained results, based on state space approach for the critical stability criterion of discrete systems, are sufficient conditions for the original system and its reduced subsystems about stability problems. Besides, we show that the subsystems are, indeed, zero-order approximations to the original systems. An example is given to illustrate the proposed schemes.