Full‐block multipliers for repeated,slope‐restricted scalar nonlinearities

This paper provides a comprehensive treatment of full‐block multipliers within the integral quadratic constraints framework for stability analysis of feedback systems containing repeated, slope‐restricted scalar nonlinearities. We develop a novel stability result that offers more flexibility in its application because it allows for the inclusion of general Popov and Yakubovich criteria in combination with the well‐established Circle and Zames‐Falb stability tests within integral quadratic constraint theory. A particular focus lies on the formulation of stability criteria in terms of full‐block multipliers, some of which are new, and thus typically involve less conservatism than current methods. Furthermore, a new asymptotically exact parametrization of full‐block Zames‐Falb multipliers is given that allows to exploit the complete potential of this stability test. Copyright © 2017 John Wiley & Sons, Ltd.     

Tractable stability analysis for systems containing repeated scalar slope‐restricted nonlinearities

This paper proposes an LMI‐based approach for studying the stability of feedback interconnections of a finite dimensional LTI system and a nonlinear element that consists of several identical scalar nonlinearities that have restrictions on their sector and slope. The results are based on the integral quadratic constraint stability analysis framework and other recent results that give a sharp characterisation of stability multipliers for monotone, repeated scalar nonlinearities. Several examples show the effectiveness of the proposed approach; the lack of conservatism in the results is noteworthy. Copyright © 2013 John Wiley & Sons, Ltd.     

gain bounds for systems with sector bounded and slope‐restricted nonlinearities

This paper proposes a new method for calculating a bound on the gain of a system comprising a linear time invariant part and a static nonlinear part, which is odd, bounded, zero at zero and has a restriction on its slope. The nonlinear part is also assumed to be sector bounded, with the sector bound being (possibly) different from that implied by the slope restriction. The computation of the gain bound is found by solving a set of linear matrix inequalities, which arise from an integral quadratic constraint formulation of a multiplier problem involving both Zames‐Falb and Popov multipliers. Examples illustrate the effectiveness of the results, and comparisons are made against the state‐of‐the‐art. Copyright © 2011 John Wiley & Sons, Ltd.     

Modified static anti‐windup for saturated systems with sector‐bounded and slope‐restricted nonlinearities

This paper proposes modified static anti‐windup techniques for saturated systems with sector‐bounded and slope‐restricted nonlinearities by augmenting the pre‐designed controller with the so‐called differential compensator to process the slope restriction. By using a purely quadratic Lyapunov function and with a modified sector condition dealing with actuator saturation, LMI‐based synthesis conditions are presented to address the problems of the estimates of the region of attraction and performance analysis of the closed‐loop system. Numerical examples illustrate the effectiveness of the proposed approaches. Copyright © 2016 John Wiley & Sons, Ltd.     

Robust Analysis of Discrete‐Time Lur'e Systems with Slope Restrictions Using Convex Optimization

This paper considers robust stability and robust performance analysis for discrete‐time linear systems subject to nonlinear uncertainty. The uncertainty set is described by memoryless, time‐invariant, sector bounded, and slope restricted nonlinearities. We first give an overview of the absolute stability criterion based on the Lur'e‐Postkinov Lyapunov function, along with a frequency domain condition. Subsequently, we derive sufficient conditions to compute the upper bounds of the worst case H₂ and worst case H∞ performance. For both robust stability testing and robust performance computation, we show that these sufficient conditions can be readily and efficiently determined by performing convex optimization over linear matrix inequalities.     

Zames-Falb Multipliers for Quadratic Programming

In constrained linear model predictive control, a quadratic program must be solved on-line at each control step, and this constitutes a nonlinearity. If zero is a feasible point for this quadratic program then the resultant nonlinearity is sector bounded. We show that if the nonlinearity is static then it is also monotone and slope restricted; hence, we show the existence of Zames-Falb multipliers for such a nonlinearity. The multipliers may be used in a general and versatile analysis of the robust stability of input constrained model predictive control.     

Robust nonlinear sampled‐data system analysis based on Fridman's method and S‐procedure

This paper is devoted to the evaluation of sampling interval providing robust exponential stability of nonlinear system with sector‐bounded nonlinearities. It extends our previous results (R. E. Seifullaev, A. L. Fradkov. Sampled‐data control of nonlinear oscillations based on LMIs and Fridman's method. In 5th IFAC International Workshop on Periodic Control Systems, 95‐100. Caen, France. 2013). The proposed approach exploits E. Fridman's method for linear systems based on a general time‐dependent Lyapunov–Krasovskii functional. With classical results of V. A. Yakubovich about S‐procedure, the problem is reduced to feasibility analysis of linear matrix inequalities. The results are illustrated by example: the pendulum system with friction and sector‐bounded multiple nonlinearities. Copyright © 2015 John Wiley & Sons, Ltd.     

Conic sector analysis using integral quadratic constraints

This paper interprets the Conic Sector stability results of Zames, and the multivariable generalizations provided by Safonov and Athans, in an integral quadratic constraint context. The ideas are formulated for the case where one element of a feedback interconnection is linear and the other satisfies a conic sector condition. This scenario allows the main stability results to be formulated using integral quadratic constraints, with all finite‐sector cases being captured using this framework. The stability results are expressed using frequency domain inequalities or linear‐matrix inequalities. Some examples show how the main technical result can, in the multivariable case, be used to provide a reduction in conservatism over standard results.     

Nonlinear Consensus Algorithms with Uncertain Couplings

Distributed algorithms for synchronization and consensus in multi‐agent networks are considered. The agents are assumed to be linear of arbitrary order, the interaction topology may switch, and the couplings are uncertain, assumed only to satisfy certain quadratic constraints. Using the Kalman‐Yakubovich‐Popov lemma and absolute stability theory techniques, consensus criteria for the networks of this type are obtained. These criteria extend a number of known results for agents with special dynamics and are close in spirit to the celebrated circle criterion for the stability of Lurie systems.     

New absolute stability criteria for time‐delay Lur'e systems with sector‐bounded nonlinearity

This paper is concerned with the problem of absolute stability of time‐delay Lur'e systems with sector‐bounded nonlinearity. Several novel criteria are presented by using a Lur'e–Postnikov function. For a general Lur'e system with known time delay, the absolute stability of it is analyzed by solving a set of linear matrix inequalities (LMIs). The maximum upper bound of the allowable time delay for a general Lur'e system is derived by solving a convex optimization problem. The feasibility of the LMIs implies some frequency‐domain interpretations which are similar to the frequency‐domain inequalities in the circle criterion and the Popov criterion. Copyright © 2009 John Wiley & Sons, Ltd.     

IQC‐synthesis with general dynamic multipliers

Analogously to the existing μ‐synthesis tools, we propose an alternative algorithm for the systematic design of robust controllers based on an iteration of standard nominal controller synthesis and integral quadratic constraint (IQC) analysis with general dynamic multipliers. The suggested algorithm enables us to perform robust controller synthesis for a significantly larger class of uncertainties if compared with the existing methods. Indeed, while the classical approaches are restricted to the use of real/complex time invariant or arbitrarily fast time‐varying parametric uncertainties, the IQC framework also offers, for example, the possibility to efficiently handle sector‐bounded and slope restricted nonlinearities or time‐varying parametric uncertainties and uncertain time‐varying time‐delays, both with bounds on the rate‐of‐variation. Secondly, in contrast to the classical approaches, the proposed techniques completely avoid gridding and curve‐fitting. We present new insights that allow us to reformulate the robust IQC analysis LMIs into a standard quadratic performance problem. This enables us to generate suitable initial conditions for each subsequent iteration step. Depending on the size of the problem, this can significantly speed up the synthesis process. The results are illustrated by means of two numerical examples. Copyright © 2013 John Wiley & Sons, Ltd.     

Fault estimation observer design for discrete‐time systems in finite‐frequency domain

This paper proposes a framework of fault estimation observer design in finite‐frequency domain for discrete‐time systems. First, under the multiconstrained idea, a full‐order fault estimation observer in finite‐frequency domain is designed to achieve fault estimation by using the generalized Kalman–Yakubovich–Popov lemma to reduce conservatism generated by the entire frequency domain. Then, a reduced‐order fault estimation observer is constructed, which results in a new fault estimator to realize fault estimation using current output information. Furthermore, by introducing slack variables, improved results on full‐order fault estimation observer and reduced‐order fault estimation observer design with finite‐frequency specifications are obtained such that different Lyapunov matrices can be separately designed for each constraint. Simulation results are presented to illustrate the advantages of the theoretic results obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

Application of Yakubovich criterion for stability of nonlinear stochastic systems

Frequency domain criteria for stability of a class of multiloop nonlinear stochastic systems are presented in this paper. The Yakubovich criterion is used in the investigation of asymptotic stability with probability one. The obtained results are illustrated by the example.     

Parametric Solutions to the Generalized Discrete Yakubovich‐Transpose Matrix Equation

This paper is concerned with the complete parametric solutions to the generalized discrete Yakubovich‐transpose matrix equation X − AX^TB = CY. which is related with several types of matrix equations in control theory. One of the parametric solutions has a neat and elegant form in terms of the Krylov matrix, a block Hankel matrix and an observability matrix. In addition, the special case of the generalized discrete Yakubovich‐transpose matrix equation, which is called the Karm‐Yakubovich‐transpose matrix equation, is considered. The explicit solutions to the Karm‐Yakubovich‐transpose matrix equation are also presented by the so‐called generalized Leverrier algorithm. At the end of the paper, two examples are given to show the efficiency of the proposed algorithm.     

The Yakubovich–Kalman–Popov lemma and stability analysis of dynamic output feedback systems

This paper presents theory related to stability analysis and stability criteria relevant for observer‐based feedback control systems. To this purpose, a special formulation of the Yakubovich–Kalman–Popov (YKP) lemma is provided. We exploit that controllability is not necessary for existence of Lur'e–Lyapunov functions as used in stability criteria. Constructive means for dynamic output feedback stabilization, positivity, factorization and passivity are provided. Copyright © 2005 John Wiley & Sons, Ltd.     

A Reset‐Time Dependent Approach for Stability Analysis of Nonlinear Reset Control Systems

A reset mechanism in controller can affect the stability property of a closed loop control system. In a simple word, there are stable reset control systems with unstable base‐systems and also unstable reset systems with stable base‐systems. The Lyapunov stability theory is a strong tool to investigate the stability of a nonlinear system. In this paper, based on the well‐known Lyapunov stability concept, some stability conditions for nonlinear reset control systems are addressed. These conditions are dependent on the reset‐times and hence the reset‐time intervals are explicitly emerged in the stability conditions. Some applications of these results are used in numerical examples to show the effectiveness of the proposed approach.     

Robust stability analysis for fractional‐order systems with time delay based on finite spectrum assignment

In this paper, the robust stability of a fractional‐order time‐delay system is analyzed in the frequency domain based on finite spectrum assignment (FSA). The FSA algorithm is essentially an extension of the traditional pole assignment method, which can change the undesirable system characteristic equation into a desirable one. Therefore, the presented analysis scheme can also be used as an alternative time‐delay compensation method. However, it is superior to other time‐delay compensation schemes because it can be applied to open‐loop poorly damped or unstable systems. The FSA algorithm is extended to a fractional‐order version for time‐delay systems at first. Then, the robustness of the proposed algorithm for a fractional‐order delay system is analyzed, and the stability conditions are given. Finally, a simulation example is presented to show the superior robustness and delay compensation performance of the proposed algorithm. Moreover, the robust stability conditions and the time‐delay compensation scheme presented can be applied on both integer‐order and fractional‐order systems.     

Implications of dissipativity,inverse optimal control,and stability margins for nonlinear stochastic feedback regulators

In this paper, we derive stability margins for optimal and inverse optimal stochastic feedback regulators. Specifically, gain, sector, and disk margin guarantees are obtained for nonlinear stochastic dynamical systems controlled by nonlinear optimal and inverse optimal Hamilton‐Jacobi‐Bellman controllers that minimize a nonlinear‐nonquadratic performance criterion with cross‐weighting terms. Furthermore, using the newly developed notion of stochastic dissipativity, we derive a return difference inequality to provide connections between stochastic dissipativity and optimality of nonlinear controllers for stochastic dynamical systems. In particular, using extended Kalman‐Yakubovich‐Popov conditions characterizing stochastic dissipativity, we show that our optimal feedback control law satisfies a return difference inequality predicated on the infinitesimal generator of a controlled Markov diffusion process if and only if the controller is stochastically dissipative with respect to a specific quadratic supply rate.     

Criteria for robust absolute stability of time-varying nonlinear continuous-time systems

The present paper establishes results for the robust absolute stability of a class of nonlinear continuous-time systems with time-varying matrix uncertainties of polyhedral type and multiple time-varying sector nonlinearities. By using the variational method and the Lyapunov Second Method, criteria for robust absolute stability are obtained in different forms for the given class of systems. Specifically, the parametric classes of Lyapunov functions are determined which define the necessary and sufficient conditions of robust absolute stability. The piecewise linear Lyapunov functions of the infinity vector norm type are applied to derive an algebraic criterion for robust absolute stability in the form of solvability conditions of a set of matrix equations. Several simple sufficient conditions of robust absolute stability are given which become necessary and sufficient for special cases. Two examples are presented as applications of the present results to a particular second-order system and to a specific class of systems with time-varying interval matrices in the linear part.     

On the stability issues of switched singular time‐delay systems with slow switching based on average dwell‐time

The issue of exponential stability analysis of continuous‐time switched singular systems consisting of a family of stable and unstable subsystems with time‐varying delay is investigated in this paper. It is very difficult to analyze the stability of such systems because of the existence of time‐delay and unstable subsystems. In this regard, on the basis of the free‐weighting matrix approach, by constructing the new Lyapunov‐like Krasovskii functional, and using the average dwell‐time approach, delay‐dependent sufficient conditions are derived and formulated in terms of LMIs to check the exponential stability of such systems. This paper also highlights the relationship between the average dwell‐time of the switched singular time‐delay system, its stability, exponential convergence rate of differential states, and algebraic states. Finally, a numerical example is given to confirm the analytical results and illustrate the effectiveness of the proposed strategy. Copyright © 2012 John Wiley & Sons, Ltd.