Performance improvement of robust controllers for polynomially uncertain systems

This paper is concerned with the high-performance robust control of discrete-time linear time-invariant (LTI) systems with semi-algebraic uncertainty regions. It is assumed that a robustly stabilizing static controller is given whose gain depends polynomially on the uncertain variables. The problem of tuning this parameter-dependent gain with respect to a prescribed quadratic cost function is formulated as a sum-of-squares (SOS) optimization. This method leads to a near-optimal controller whose performance is better than that of the initial controller. It is shown that the results derived in the present work encompass the ones obtained in a recent paper. The efficacy of the results is elucidated by an example.     

Robust controllability of linear stochastic uncertain systems

The paper extends the concept of robust controllability via linear state feedback to stochastic uncertain systems. We show that the controllability of a stochastic uncertain system can be characterized using solutions to a game-type differential Riccati equation.     

Robust observability for a class of time-varying discrete-time uncertain systems

This paper introduces a concept of robust observability for a class of uncertain discrete-time systems. This notion is an extension of the standard notion of observability for discrete-time, linear time-varying systems. The uncertainty and noise are modelled deterministically via a sum quadratic constraint. A necessary and sufficient condition for robust observability is presented in terms of existence of a suitable solution to a Riccati difference equation. The set of possible initial states given noisy measurements over a finite number of sampling periods is shown to be an ellipsoid.     

Robust analysis and synthesis for a class of uncertain nonlinear systems with multiple equilibria

This paper focuses on a class of uncertain nonlinear systems which are subject to norm-bounded parameter uncertainty in the forward path and a vector-valued periodic nonlinearity in the feedback path, and addresses robust analysis and synthesis problems for such systems. Sufficient conditions for global asymptotic stability are derived in terms of linear matrix inequalities (LMIs) and a technique for the estimation of the uncertainty bound is proposed by solving a generalized eigenvalue minimization problem. The problem of robust synthesis is concerned with designing a feedback controller such that the resulting closed-loop system is globally asymptotically stable for all admissible uncertainties. It is shown that a solution to the robust synthesis problem for the uncertain system can be obtained by solving a synthesis problem for an uncertainty free system. A concrete example is presented to demonstrate the applicability and validity of the proposed approach.     

On controllability and observability for a class of impulsive systems

Many dynamic systems in physics, chemistry, biology, engineering, and information science have impulsive dynamical behaviors due to abrupt jumps at certain instants during the dynamical processes. These complex dynamic behaviors can be modeled by impulsive differential systems. This paper studies the controllability and observability for a class of time-varying impulsive control systems. Several sufficient and necessary conditions for state controllability and state observability of such systems are established and the corresponding criteria for time-invariant impulsive control systems are also obtained. Meanwhile, several new results associated with variation of parameters for time-varying impulsive control systems are derived.     

Robust stability and instability of biochemical networks with parametric uncertainty

Parameter perturbations in dynamical models of biochemical networks affect the qualitative dynamical behaviour observed in the model. Since this qualitative behaviour is in many cases the key model output used to explain biological function, the robustness analysis of the model’s behaviour with respect to parametric uncertainty is a crucial step in systems biology research. In this paper, we develop a new method for robustness analysis of the dynamical behaviour. As a first step, we provide a characterization of non-robust perturbations as a system of polynomial equalities and inequalities. In the second step, we apply the Positivstellensatz and Handelman representation of polynomials to check for the non-existence of solutions to this system, which can be relaxed to solving a linear program. Thereby, a solution to the linear program yields a robustness certificate for the considered dynamical behaviour. With these robustness certificates, we propose an algorithm to compute a lower robustness bound corresponding to a level of parametric uncertainty up to which no local bifurcations can occur. The applicability of the proposed method to biochemical network models is illustrated by analysing the robustness of oscillations in a model of the NF-κB signalling pathway. The results may be used to define a level of confidence in the observed model behaviour under parametric uncertainty, making them valuable for evaluating dynamical models of biological networks.     

The average-case identifiability and controllability of large scale systems

Needs for increased product quality, reduced pollution, and reduced energy and material consumption are driving enhanced process integration. This increases the number of manipulated and measured variables required by the control system to achieve its objectives. This paper addresses the question of whether processes tend to become increasingly more difficult to identify and control as the process dimension increases. Tools and results of multivariable statistics are used to show that, under a variety of assumed distributions on the elements, square processes of higher dimension tend to be more difficult to identify and control, whereas the expected controllability and identifiability of nonsquare processes depends on the relative numbers of measured and manipulated variables. These results suggest that the procedure of simplifying the control problem so that only a square process is considered is a poor practice for large scale systems.     

On the controllability and observability of discrete-time linear time-delay systems

This article studies the controllability and observability of discrete-time linear time-delay systems, so that the two properties can play a more fundamental role in system analysis before controller and observer design is engaged. Complete definitions of controllability and observability, which imply the stabilisability and detectability, respectively, and determine the feasibility of eigenvalue assignment, are proposed for systems with delays in both state variables and input/output signals. Necessary and sufficient criteria are developed to check the controllability and observability efficiently. The proofs are based on the equivalent expanded system, but the criteria only involve the delays and matrices of the same dimension as the original system. Finally, the duality between the suggested controllability and observability is presented.     

Robust stability of MIMO nonlinear uncertain systems using operator-based right coprime factorisation

In this paper, a class of multi-input–multi-output (MIMO) nonlinear systems with uncertainties is considered by using operator-based right coprime factorisation. First, based on proposed quotient operators, coupling effect of the MIMO nonlinear systems is discussed. Second, based on the operator-based right coprime factorisation approach, sufficient conditions for guaranteeing robust stability of the MIMO nonlinear systems with uncertainties are proposed by using a new unimodular operator. Finally, a simulation example is shown to illustrate the proposed design scheme for the MIMO nonlinear systems with uncertainties.     

一类不确定非线性离散时滞系统的鲁棒H∞滤波设计

考虑一类同时带有非线性动态和参数不确定性的离散时滞系统的鲁棒H∞滤波设计问题．假设参数不确定性具有线性分式形式,而非线性动态满足Lipschitz条件,给出滤波器使误差系统鲁棒渐近稳定且达到指定的干扰抑制水平．对参数已知情形,先建立广义有界实引理,然后给出H∞滤波器的存在条件,证明了H∞滤波器的存在性可归结为线性矩阵不等式的可解性,基于线性矩阵不等式给出了H∞滤波器的综合方法和步骤．对参数不确定性情形,通过引进标度参数,将不确定非线性离散时滞系统的鲁棒H∞滤波问题转化为确定系统的H∞滤波设计．最后给出仿真例子验证所得结果的有效性．     

Resilient linear filtering of uncertain systems

The problem of resilient linear filtering for a class of linear continuous-time systems with norm-bounded uncertainties is investigated. We have considered additive filter gain variations to reflect the imprecision in filter implementation. The design problem of resilient linear filter is formulated as a convex optimization problem over linear matrix inequalities. As a limiting procedure, the case of resilient Kalman filter is derived. All the developed results are conveniently extended to the case of multiplicative filter gain variations. Simulation studies are carried out to support the theoretical findings.     

Robust stabilization of mimo nonlinear time-varying mismatched uncertain systems

This paper considers the problem of robust stabilization of multi-input-multi-output (MIMO) nonlinear systems in the presence of mismatched time-varying uncertainties. Combining the differential geometric feedback linearization with the deterministic approach, it derives two robust stabilizing controllers i.e. the high-gain feedback controller and the variable structure controller. These two controllers only require the knowledge of the nominal system and the bounds of the uncertainties. Therefore, they are more feasible to be implemented. Both controllers reduce the effects of arbitrarily large bounded mismatched uncertainties successfully to achieve uniform ultimate boundedness stabilization of the MIMO nonlinear timevarying uncertain systems.     

Robust finite-time stabilisation of uncertain linear systems

This article deals with the problem of finite-time stability and stabilisation of uncertain linear systems. For linear time-varying systems subject to norm-bounded uncertainties some conditions for finite-time stability are provided. These conditions are expressed in terms of differential linear matrix inequalities. Then the problem of controller design is tackled, both for the state feedback and for the output feedback case; in both cases the controller can be found solving a suitable set of LMIs. A typical engineering case-study is included, to illustrate the applicability of the devised conditions.     

Observer-based direct adaptive fuzzy control of uncertain nonlinear systems and its applications

A direct adaptive fuzzy control algorithm is developed for a class of uncertain SISO nonlinear systems. In this algorithm, it doesn’t require to assume that the system states are measurable. Therefore, it is needed to design an observer to estimate the system states. Compared with the numerous alternative approaches with respect to the observer design, the main advantage of the developed algorithm is that on-line computation burden is alleviated. It is proven that the developed algorithm can guarantee that all the signals in the closed-loop system are uniformly ultimately bounded and the tracking error converges to a small neighborhood around zero. The simulation examples validate the feasibility of the developed algorithm. Recommended by Editorial Board member Zhong Li under the direction of Editor Young-Hoon Joo. This work is supported by National Natural Science Foundation of China under grant 60674056, 60874056, and the Foundation of Educational Department of Liaoning Province (2008312). Yan-Jun Liu received the B.S. degree in Applied Mathematics from Shenyang University of Technology in 2001. He received the M.S. degree in Control Theory and Control Engineering from Shenyang University of Technology in 2004 and the Ph.D. degree in Control Theory and Control Engineering from Dalian University of Technology, China, in 2007. His research interests include fuzzy control theory, nonlinear control and adaptive control. Shao-Cheng Tong received the B.S. degree in Department of Mathematics from Jinzhou Normal College, China, in 1982. He received the M.S. degree in Department of Mathematics from Dalian Marine University in 1988 and the Ph.D. degree in Control Theory and Control Engineering from Northeastern University, China, in 1997. His research interests include fuzzy control theory, nonlinear control, adaptive control, and system identification etc. Wei Wang received the B.S. degree in Department of Automation from Northeastern University, China, in 1982. He received the M. S. degree in Department of Automation from Northeastern University in 1984 and the Ph.D. degree in Department of Automation from Northeastern University, China, in 1988. His research interests include adaptive predictive control, intelligent control, and production scheduling method etc. Yong-Ming Li received the B.S. degree in Applied Mathematics from Liaoning University of Technology in 2004. He received the M.S. degree in Applied Mathematics from Liaoning University of Technology in 2007. His research interests include fuzzy control theory, nonlinear control and adaptive control.     

Robust control of a class of uncertain nonlinear systems

This paper considers the robust control of a class of nonlinear systems with real time-varying parameter uncertainty. Interest is focused on the design of linear dynamic output feedback control and two problems are addressed. The first one is the robust stabilization and the other is the problem of robust performance in an H_∞ sense. A technique is proposed for designing stabilizing controllers for both problems by converting them into ‘scaled’ H_∞ control problems which do not involve parameter uncertainty.     

Robust delayed-state-feedback stabilization of uncertain stochastic systems

Due to time spent in computation and transfer, control input is usually subject to delays. Problems of deterministic systems with input delay have received considerable attention. However, relatively few works are concerned with problems of stochastic system with input delay. This paper studies delayed-feedback stabilization of uncertain stochastic systems. Based on a new delay-dependent stability criterion established in this paper, a robust delayed-state-feedback controller that exponentially stabilizes the uncertain stochastic systems is proposed. Numerical examples are given to verify the effectiveness and less conservativeness of the proposed method.