David P. Goodall

2009 Special Issue: Global stability analysis and robust design of multi-time-scale biological networks under parametric uncertainties

Biological networks are prone to internal parametric fluctuations and external noises. Robustness represents a crucial property of these networks, which militates the effects of internal fluctuations and external noises. In this paper biological networks are formulated as coupled nonlinear differential systems operating at different time-scales under vanishing perturbations. In contrast to previous work viewing biological parametric uncertain systems as perturbations to a known nominal linear system, the perturbed biological system is modeled as nonlinear perturbations to a known nonlinear idealized system and is represented by two time-scales (subsystems). In addition, conditions for the existence of a global uniform attractor of the perturbed biological system are presented. By using an appropriate Lyapunov function for the coupled system, a maximal upper bound for the fast time-scale associated with the fast state is derived. The proposed robust system design principles are potentially applicable to robust biosynthetic network design. Finally, two examples of two important biological networks, a neural network and a gene regulatory network, are presented to illustrate the applicability of the developed theoretical framework.

Output feedback stabilisation for uncertain nonlinear time-delay systems subject to input constraints

Robust stabilisation of a class of imperfectly known systems with time-varying time-delays via output feedback is investigated. The systems addressed are composed of a nonlinear nominal system influenced by nonlinear perturbations which may be time-, state-, delayed state- and/or input-dependent. The output of the system is modelled by a nonlinear function, which may depend on the delayed states, and inputs, together with a feed-through term. Using bounding information on the perturbations, in terms of specified growth conditions, classes of unconstrained and constrained output feedback controllers are designed in order to guarantee a prescribed stability property for the closed-loop systems, provided appropriate stability criteria hold. Two stability criteria are given: one in terms of a linear matrix inequality (LMI) and the other is algebraic in nature, obtained using a Geršgorin Theorem.

Extended global total least square approach to multiple-model identification

Abstract The paper proposes a novel global total least square procedure which has been appropriately extended for multiple-model identification of nonlinear systems. The resulting scheme which is a hybrid iterative procedure, makes repeated use of both the behavioural and classical approaches. Model parameters are optimised in order to minimise the distance between an observed time series and the simulated time series of the resulting optimised behavioural model. The developed procedure is demonstrated when applied to an arbitrary nonlinear system.

Stability criteria for feedback-controlled, imperfectly known, bilinear systems with time-varying delay

The problem of synthesizing a class of continuous, memoryless feedback controls in order to stabilize a class of imperfectly known homogeneous-in-the-state bilinear time-delay systems is considered. In particular, bilinear systems with state time-delay in the linear term are investigated. The time-delay is assumed to be an unknown time-varying function with known upper bound on its derivative. As well as considering both matched and residual uncertainty, the uncertainty in the class of systems can be state, delayed-state and input dependent, and time-varying. Prior information on the bound of the system uncertainty is required; such bounding information allows for quadratic growth with respect to the state. For this stabilizability problem, a stability criterion, involving the upper bound on the derivative of the time-varying time-delay is obtained.

DISTURBANCE ESTIMATION AND CANCELLATION FOR LINEAR UNCERTAIN SYSTEMS

Abstract This paper considers a class of linear uncertain systems in which the uncertainty is an additive perturbation of a known (nominal) linear model. It is supposed that the uncertainty and/or disturbance is known to be bounded, but its bound is unkown. A novel, easy to implement, adaptive feedback control law is designed to estimate the bounded disturbance on-line. This information is then used to cancel the effect of the disturbance in the system. The main advantage is that, if further design objectives are to be realized (for example, with respect to a tracking problem), the controls can be designed on the information from the nominal model only and not on the uncertain model.

Robust Controls for Implicit Hereditary Systems of the Neutral Type with Time-Varying Delays

Abstract A feedback stabilization problem is investigated for a class of imperfectly known implicit systems with discrete and distributed time-varying delays. The imperfections acting on the systems, which may be time-, state-, delayed state-, and/or input-dependent, are modelled as additive nonlinear perturbations influencing a known set of nonlinear functional differential equations of the neutral type. Sufficient conditions, which include a delay-dependent matrix inequality and a delay-dependent stability criterion involving some bounding parameters for the uncertainty in the system, are presented and a class of robust feedback controls are designed to guarantee a prescribed stability property for the class of implicit systems.

Constrained Feedback Control of Imperfectly Known, Linear, Time-Delay Systems of Neutral Type

Abstract A class of robust continuous feedback controls, with memory, is synthesized and a delay-free stability criterion is presented for a class of imperfectly known time-delay systems of the neutral type, when there are constraints on the control input. The uncertain systems are modelled as nonlinear perturbations to a known linear neutral time-delay control system and, as well as being time and state dependent, the nonlinear perturbations are assumed to be state delay-dependent and input dependent. Moreover, prior information on the bound of the system uncertainty is required. A deterministic approach is taken and, using a Lyapunov-Krasovskiĭ functional, a global uniform asymptotic stability property is investigated for the class of systems.

Robust stabilizers for uncertain singularly perturbed time-delay systems of the neutral type

Abstract A feedback stabilization problem is investigated for a class of imperfectly known singularly perturbed time-delay systems of the neutral type, with discrete delays. The imperfections acting on the systems, which may be time-, state-, delayed state-, and/ or input-dependent, are modelled as additive nonlinear perturbations influencing a known set of nonlinear functional differential equations. Sufficient conditions, which include delay-dependent matrix inequalities, are presented and a class of robust feedback controls are designed to guarantee a prescribed stability property for the class of singularly perturbed systems.

Constrained feedback control of a class of additively perturbed nonlinear neutral time-delay systems

Abstract Adopting a deterministic methodology based on Lyapunov stability theory and Lyapunov-Krasovskii functional delay-independent stability criteria are proposed which are sufficient to ensure that all global closed-loop trajectories for a class of perturbed nonlinear delay systems of the neutral-type, subject to constraints on the control input, uniformly asymptotically converge to the state origin. An application of the results to a specific sub-class is provided and a clays of robust stabilising controls, satisfying prescribed constraints, is presented.

Stabilization of Imperfectly Known Nonlinear Systems with Multiple Time-Delays

Abstract This paper proposes a delay-independent stability criterion that is sufficient to ensure the stabilization of a class of large-scale imperfectly known nonlinear systems containing multiple time-delays. In particular, large-scale nonlinear systems containing delay within the state are considered. The approach adopted is a deterministic one utilising memory less feedback controls, Lyapunov stability theory and Lyapunov-Krasovski??? functionals. Copyright