Robust Stability Of Sampled-data Systems Research Articles

An IQC Approach to Robust Stability of Aperiodic Sampled-Data Systems

Conditions for robust stability of sampled-data systems with non-uniform sampling patterns and structural uncertainties are derived. The problem is tackled under the integral quadratic constraint (IQC) framework, where the “aperiodic sampling operation” is modelled by an “delay-integration” operator. Characterization based on integral quadratic constrains (IQC) is identified for this operator and the IQC theory is applied to derive convex stability criteria. Compared to the dominating Lyapunov approach where the candidate Lyapunov-Krasovskii functionals or looped functionals need to be tailored for the systems under consideration and therefore the stability conditions need to be re-derived whenever additional uncertainties are considered, the proposed approach has the advantage of avoiding such endeavor. Numerical examples are given to illustrate this main point and effectiveness of the proposed approach.

On the Robustness of Sampled-Data Systems to Uncertainty in Continuous-Time Delays

In this technical note, we consider the robust stability of sampled-data systems with respect to continuous-time delay. We argue that many results in the literature implicitly assume that the delay is a multiple of the sampling period. We demonstrate that this approach might be misleading. Namely, we show that there are systems, which are destabilized by small continuous-time delays despite being delay-independent stable with respect to “integer” delays (this phenomenon has no continuous-time counterpart). We also propose a robustness analysis procedure based on embedding continuous-time delays into unstructured analog uncertainty with the subsequent reduction of the problem to a standard sampled-data <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> problem. Toward this end, a novel nominal model for the uncertain delay is put forward. It yields a tighter unstructured uncertainty covering than in the existing approaches, thus having a potential to reduce the conservatism of the method. This might be advantageous in the pure continuous-time case as well.

Separator-type robust stability theorem of sampled-data systems allowing noncausal LPTV scaling

This paper derives a necessary and sufficient condition for robust stability of sampled-data systems, which is stated by using the notion of separators that are dealt with in an operator-theoretic framework. Such operator-theoretic treatment of separators provides a new perspective, which we call noncausal linear periodically time varying scaling and leads to reducing conservativeness in robust stability analysis. A numerical example is given to demonstrate the results.

Improved stability conditions for sampled data systems with jitter

This paper is concerned with the discrete time approach for the robust stability of sampled-data systems (Aström and Wittenmark [1997]) with sampling jitter. We assume that the sampling is unknown, time-varying and bounded in a given interval. Several discrete-time approaches exist in the literature. The intention of the paper is to clearly analyse the relations between them and to indicate the sources of conservatism. It will point out that the different existing models can be studied in the framework of difference inclusions with polytopic and norm bounded components. Furthermore, it will show how to reduce the conservatisms by using recent stability analysis tools based on Lyapunov functions with non-monotonic Lyapunov increment. The idea is to improve the stability conditions by considering α-samples variations, i.e. ΔαV(xk) = V(xk+α) — V(xk) < 0. Numerical examples illustrate the effectiveness of the approach.

Linear periodically time-varying scaling and its properties

Stimulated by a general necessary and sufficient condition for robust stability of sampled-data systems stated in an operator-theoretic framework, we introduce a novel technique called linear periodically time-varying (LPTV) scaling. We then give a simple example of sampled-data systems in which this new scaling allows exact robust stability analysis for static uncertainties while the conventional linear time-invariant (LTI) scaling fails to do so. This leads us to an interesting question whether LPTV scaling can be effective also in other situations, e.g., in the robust stability analysis of continuous-time feedback systems regarded as a special class of sampled-data systems, or for other classes of uncertainties. We thus study some basic properties of LPTV scaling by confining ourselves to the so-called D -scaling, and show that it provides no advantage over LTI scaling when it is applied to continuous-time LTI feedback systems, regardless of the class of uncertainties we take into consideration. This demonstrates that the LPTV scaling of the type we deal with in this paper is in some sense a special technique for sampled-data systems but is indeed an effective and more natural technique than the conventional LTI scaling as far as such systems are concerned. The technique can be further extended to include what is called noncausal LPTV scaling, and the implication of the present study in such a larger framework of LPTV scaling is also described.

Robust stability of sampled-data systems under possibly unstable additive/multiplicative perturbations

Applies the FR-operator technique to the robust stability problem of sampled-data systems against additive/multiplicative perturbations, where a reasonable class of perturbations consists of unstable as well as stable ones. Assuming that the number of unstable modes of the plant does not change, we show that a small-gain condition in terms of the FR-operator representation (which is actually equivalent to a small-gain condition in terms of the L/sub 2/-induced norm) is still necessary and sufficient for the sampled-data system to be robustly stable against h-periodic perturbations, in spite of their possible instability. The result is derived by a Nyquist-type of arguments. Next, a necessary and sufficient condition for robust stability against linear time-invariant (LTI) perturbations is also given. Furthermore, we show that if the plant is either single-input or single-output, the condition can be reduced to a readily testable form. Finally, we clarify when the small-gain condition becomes a particularly poor measure for robust stability.

Computation-oriented expression of a non-conservative condition for robust stability of sampled-data systems

A new expression of a non-conservative robust stability condition is derived for sampled-data systems taking into consideration its convenience in numerical computation. Our expression is described using a global maximum of a certain differentiable function, while the conventional expression includes a global maximum of a structured singular value, which is expensive to compute. When the sampling period is long, the advantages of our expression become clear. A possible problem is that the maximized function may be multimodal. However, the conventional expression has the same problem and even a local maximum is shown to give useful information. Our expression is applicable to a fairly large class of systems including the multi-input multi-output additive and multiplicative perturbation models. Using an example, we show how our expression is computed and make a comparison with the conventional expression.

Robust stability bounds for sampled-data systems

Abstract This paper is concerned with the robust stability of sampled-data systems whose underlying continuous-time system is subjected to additive parameter perturbations. In particular, the process by which parameter perturbations in the underlying continuous-time model are propagated into the discrete-time one is examined, and upper bounds on these induced perturbations are proposed. In addition robust stability bounds for discrete-time systems are derived. Combining the above findings yields new robust stability bounds on the underlying continuous-time perturbations. Owing to the exponential-like structure of the induced perturbations, relatively computationally demanding implicit bounds, are derived for structured perturbations. It is shown in the paper that the sampling rate is a crucial design parameter when it comes to reducing the effect of the perturbations. Examples are given to illustrate the proposed results.

Analysis of structured LTI uncertainty in sampled-data systems

This paper investigates robust stability of sampled-data systems that have structured LTI uncertainty in their model description. An analysis technique is developed that provides upper and lower bounds on the stability radius of such systems.