Time-invariant Perturbations Research Articles

Robust stability bounds for multi-delay networked control systems

ABSTRACTIn this paper, the robust stability of a perturbed linear continuous-time system is examined when controlled using a sampled-data networked control system (NCS) framework. Three new robust stability bounds on the time-invariant perturbations to the original continuous-time plant matrix are presented guaranteeing stability for the corresponding discrete closed-loop augmented delay-free system (ADFS) with multiple time-varying sensor and actuator delays. The bounds are differentiated from previous work by accounting for the sampled-data nature of the NCS and for separate communication delays for each sensor and actuator, not a single delay. Therefore, this paper expands the knowledge base in multiple inputs multiple outputs (MIMO) sampled-data time delay systems. Bounds are presented for unstructured, semi-structured, and structured perturbations.

Fuzzy supervisor approach on logic-based switching H2/H∞

In this paper the synthesis of a logic-based switching H2/ H∞ state-feedback controller for continuous-time linear time-invariant (LTI) singular perturbation systems is considered which achieves a minimum limit on the H2 performance level, while satisfying the H∞ performance. The proposed hybrid control scheme is based on a fuzzy supervisor, which manages the combination of controllers. A convex linear matrix inequality (LMI)-based formulation of the fast and slow subsystem controllers leads to a ε-independent structure which ensures a good performance in both the transient phase and the steady-state phase. The stability analysis uses the Lyapunov technique, inspired from switching system theory, to prove that the system with the proposed controller remains globally stable, despite the configuration (controller) changing. Simulations show that the proposed non-linear scheme composed of two low-order controllers improves the performance in comparison with the linear optimal scheme with higher order designed for a full-order system.

On l ∞ and l 2 robustness of spatially invariant systems

We consider spatiotemporal systems and study their l∞ and l2 robustness properties in the presence of spatiotemporal perturbations. In particular, we consider spatially invariant nominal models and provide necessary and sufficient conditions for system robustness for the cases when the underlying perturbations are linear spatiotemporal varying, and nonlinear spatiotemporal invariant, unstructured or structured. It turns out that these conditions are analogous to the scaled small gain condition (which is equivalent to a spectral radius condition and a linear matrix inequality for the l∞ and l2 cases, respectively) derived for standard linear time-invariant models subject to time-varying linear and time-invariant nonlinear perturbations. Copyright © 2009 John Wiley & Sons, Ltd.

Composite Systems with Uncertain Couplings of Fixed Structure: Scaled Riccati Equations and the Problem of Quadratic Stability

We consider large scale systems consisting of a finite number of separate uncertain subsystems which interact via uncertain couplings of arbitrarily prescribed structure. The couplings are viewed as structured perturbations of the block-diagonal system representing the collection of the separate nominal subsystems (the “nominal system”). We define spectral value sets and stability radii for these time-invariant structured perturbations and derive formulas for their computation. Scaled Riccati equations are introduced to obtain explicit formulas for the stability radii with respect to time-varying and possibly nonlinear perturbations of the given structure. From these we derive necessary and sufficient conditions under which the stability radii with respect to time-invariant and time-varying perturbations are equal. These results are obtained by constructing joint quadratic Liapunov functions of optimal robustness. With their help we prove necessary and sufficient conditions for quadratic stability and sufficient conditions for the validity of a generalized Aizerman conjecture.

이산시스템에서 시간지연을 갖는 시변 상태 지연 섭동의 안정 범위에 관한 연구

The stability robustness problem of linear discrete-time systems with delayed time-varying perturbations is considered. Compared with continuous time system, little effort has been made for the discrete time system in this area. In the previous results, the bounds were derived for the case of non-delayed perturbations. There are few results for delayed perturbation. Although the results are for the delayed perturbation, they considered only the time-invariant perturbations. In this paper, the sufficient conditions for stability can be expressed as linear matrix inequalities(LMIs). The corresponding stability bounds are determined by LMI(Linear Matrix Inequality)-based algorithms. Numerical examples are given to compare with the previous results and show the effectiveness of the proposed results.

Robust stability of time-varying polytopic systems via parameter-dependent homogeneous Lyapunov functions

This paper deals with robust stability analysis of linear state space systems affected by time-varying uncertainties with bounded variation rate. A new class of parameter-dependent Lyapunov functions is introduced, whose main feature is that the dependence on the uncertain parameters and the state variables are both expressed as polynomial homogeneous forms. This class of Lyapunov functions generalizes those successfully employed in the special cases of unbounded variation rates and time-invariant perturbations. The main result of the paper is a sufficient condition to determine the sought Lyapunov function, which amounts to solving an LMI feasibility problem, derived via a suitable parameterization of polynomial homogeneous forms. Moreover, lower bounds on the maximum variation rate for which robust stability of the system is preserved, are shown to be computable in terms of generalized eigenvalue problems. Numerical examples are provided to illustrate how the proposed approach compares with other techniques available in the literature.

A multiple model approach to nonlinear internal model control design

A nonlinear internal model control (NIMC) design using multiple model is proposed. In this design approach, the process model is described via validity functions as the weighted sum of a set of local models. Subsequently, an approximate inverse of the process model is constructed in order to design NIMC controller. It is shown that the validity functions can be treated as time-varying uncertainties in the resultant NIMC structure, and a structured singular value test for mixed time-varying and time-invariant perturbations is used to assess the performance limitation of the proposed NIMC design. Simulation results of a continuous stirred tank reactor illustrate that the proposed NIMC design method is superior to the conventional IMC design.

Robust state estimation for perturbed continuous systems with circular pole and error variance constraints

In this paper, the problem of circular pole and variance-constrained state estimator design is studied for perturbed linear continuous-time systems. The system under consideration is subject to unstructured time-invariant norm-bounded parameter perturbations in both the state and measurement matrices. The problem we address is the design of a state estimator such that, for all admissible time-invariant perturbations, the poles of the steady-state filtering matrix are assigned inside a prespecified circular region, and the variance of the estimation error of each state is not more that the individual prespecified value, simultaneously. An algebraic Riccati-like matrix inequality approach is proposed to solve the above problem. Specifically, the existence conditions and an explicit expression for the desired estimators are obtained. Furthermore, an illustrative example is presented to demonstrate the effectiveness of the proposed design procedure.

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.

Improved stability robustness bounds for digital control systems in state-space models

This paper presents stability robustness bounds for linear discrete-time systems with time-invariant perturbations. These bounds extend the previous investigations for independent perturbations to dependent perturbations and provide improved results. It is shown that the dependent perturbation bound includes an independent perturbation bound previously reported as a special case. These results are illustrated with simple examples.

Robustness of pole-clustering in a ring for structured perturbation systems

A simple approach is proposed to ensure that all poles of an uncertain system are clustered in a prescribed ring. The explicit bounds on linear time-invariant structured perturbations are obtained. Under these allowable highly structured perturbations, both stability robustness and certain performance robustness will thus be ensured. In the literature, as far as we are aware, little effort has been devoted to investigating pole-clustering robustness in such regions.

On ℓ ∞, robust stability and performance with ℓ 2 and ℓ ∞ perturbations

In this paper we will consider systems with linear time-invariant perturbations. We will analyze robust performance in the ℓ 2 and ℓ ∞ settings. The ℓ 2 setting gives rise to the familiar case of structured singular values, and a stability criterion is given by the “small μ” theorem. We show that although the necessary and sufficient criterion of robust stability for the ℓ ∞ case (ℓ ∞ stability with structured ℓ ∞-gain bounded perturbations) is the same “small μ” criterion, a system with ℓ 2-gain bounded perturbations is never ℓ ∞ stable.

Robust control with pole assignment in a specified region for systems under parameter perturbations; application to a nonlinear system

AbstractThis paper deals with the problem of determination of feedback gains in order to achieve required performances under linear time invariant perturbations. Based upon the Lagrange multiplier method, the proposed method determines feedback gains that ensure both robust stability and robust performance for a linear uncertain system. It is applied to a nonlinear crane system control design. The nonlinear system is first partitioned into some uncertain linear systems. Then the proposed method is applied to each uncertain linear system. Application is easily demonstrated and, consequently, a nonlinear crane control system can be easily designed. Finally, it is shown in a numerical example that the proposed method is effective.

On robust stability with structured time-invariant perturbations

We consider the problem of robust stability for uncertain systems described in terms of multiple scalar structured time-invariant perturbations, which are norm-bounded. The issue we address is the role that the type of perturbation norm plays in the robust stability conditions. To this end, the L p -induced norm (for 1 ≤ p ≤ ∞) on the perturbations is considered, and it is shown that the structured singular value is the necessary and sufficient condition for robust stability for all these norms. We conclude that for time-invariant perturbations, in contrast to the time-varying case, robust stability conditions are independent of the type of perturbation norms.

Numerical simulation of the effect of spatial disturbances on vortexasymmetry

The steady asymmetric vortex pattern observed on slender bodies of revolution at large angle of attack was investigated using fine-grid Navier-Stokes computations. The computed results demonstrate the marked asymmetry which has been observed in experiments. To obtain asymmetry, it was found essential to introduce a space-fixed time-invariant perturbation into the computation. The computational results suggest that vortex asymmetry is forced by amplification of small disturbances, such as those due to surface roughness, occurring within the body viscous boundary layer.

Pole-assignment robustness in a specified disk

A sufficient condition is proposed to guarantee robust pole location within a specified circular region for a linear uncertain system. The explicit bounds on linear time-invariant perturbations with highly structured information are obtained. Under these allowable highly structured perturbations, both stability robustness and certain performance robustness will thus be ensured. The merit of the proposed new approach is demonstrated by a given example where the results achieved are much better than the ones reported recently.

Robustness of perturbed large-scale systems with local constant state feedback

The robustness problem of perturbed large-scale systems is considered. The nominal large-scale system is stabilized by a local constant state feedback ui(t) = kixi (t) and the local optimal control law ui (t) = − R i −1 Bi TPixi (t), respectively. In the above two cases, bounds are obtained for an allowable non-linear time-invariant (or time-varying) perturbation such that the resulting closed-loop large-scale system remains stable. The special case of a linear perturbation is also treated here.

Robustness of pole-assignment in a specified region

An analysis of pole-assignment is given for systems under linear time-invariant perturbations. Based upon the Lyapunov approach, new techniques to calculate allowable element bounds for highly structured perturbations are presented. Under these allowable perturbations both stability robustness and a certain performance robustness are thus ensured. Examples are given to illustrate the proposed methods.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Lyapunov approach to robust pole-assignment analysis

Abstract The robust pole assignment of systems with linear time-invariant perturbations is considered. Based upon the Lyapunov approach, new techniques for calculating the robustness bounds for pole assignment in any chosen region are presented. These take account not only of stability robustness but also of certain types of performance robustness. Examples are presented to illustrate the merit of the proposed analysis methods.

Lyapunov approach to robust pole-assignment analysis

Abstract The robust pole assignment of systems with linear time-invariant perturbations is considered. Based upon the Lyapunov approach, new techniques for calculating the robustness bounds for pole assignment in any chosen region are presented. These take account not only of stability robustness but also of certain types of performance robustness. Examples are presented to illustrate the merit of the proposed analysis methods.