Frequency-domain $L_2$-stability conditions for time-varying linear and nonlinear MIMO systems

The paper deals with the g2-stability analysis of multi-input-multi-output （MIMO） systems, governed by integral equations, with a matrix of periodic/aperiodic time-varying gains and a vector of monotone, non-monotone and quasi-monotone nonlin- earities. For nonlinear MIMO systems that are described by differential equations, most of the literature on stability is based on an application of quadratic forms as Lyapunov-function candidates. In contrast, a non-Lyapunov framework is employed here to derive new and more general g2-stability conditions in the frequency domain. These conditions have the following features： i） They are expressed in terms of the positive definiteness of the real part of matrices involving the transfer function of the linear time-invariant block and a matrix multiplier function that incorporates the minimax properties of the time-varying linear/nonlinear block, ii） For certain cases of the periodic time-varying gain, they contain, depending on the multiplier function chosen, no restrictions on the normalized rate of variation of the time-varying gain, but, for other periodic/aperiodic time-varying gains, they do. Overall, even when specialized to periodic-coefficient linear and nonlinear MIMO systems, the stability conditions are distinct from and less restrictive than recent results in the literature. No comparable results exist in the literature for aperiodic time-varying gains. Furthermore, some new stability results concerning the dwell-time problem and time-varying gain switching in linear and nonlinear MIMO systems with periodic/aperiodic matrix gains are also presented. Examples are given to illustrate a few of the stability theorems.     

Frequency-domain stability criteria for SISO and MIMO nonlinear feedback systems with constant and variable time-delays

New frequency-domain criteria are proposed for the $L_2$-stability of both nonlinear single-input-single-output (SISO) and nonlinear multiple-input-multiple-output (MIMO) feedback systems, described by nonlinear integral equations. For SISO systems, the feedback block is a constant scalar gain in product with a linear combination of first-and-third-quadrant scalar nonlinearities (FATQNs) with time-delay argument functions; and, for MIMO systems, it is a constant matrix gain in product with a linear combination of vector FATQNs also with time-delay argument functions. In both the cases, the delay function in the arguments of the nonlinearities may be, in general, i) zero, ii) a constant, iii) variable-time and iv) fixed-history (only for SISO systems). The stability criteria are derived from certain recently introduced algebraic inequalities concerning the scalar and vector nonlinearities, and involve the causal+anticausal O''Shea-Zames-Falb multiplier function (scalar for SISO systems and matrix for MIMO systems). Its time-domain $L_1$-norm is constrained by the coefficients and characteristic parameters (CPs) of the nonlinearities and, in the case of the time-varying delay, by its rate of variation also. The stability criteria, which are independent of Lyapunov-Krasovskii or Lyapunov-Razumikhin functions and do not seem to be derivable by invoking linear matrix inequalities, seem to be the first of their kind. Two numerical examples for each of SISO and MIMO systems illustrate the criteria.     

H∞ fuzzy control of semi-Markov jump nonlinear systems under σ-error mean square stability

This paper is concerned with the problem of time-varying H_∞ fuzzy control for a class of semi-Markov jump nonlinear systems in the sense of σ-error mean square stability. The nonlinear plant is described via the Takagi–Sugeno fuzzy model. By defining a time-varying mode-dependent Lyapunov function, a set of sufficient stability and stabilisation criteria for non-disturbance case is first derived and then applied to the investigation of H_∞ performance analysis and H_∞ fuzzy controller design problems of semi-Markov jump nonlinear systems. Different from the traditional stochastic switching system framework, the probability density function of sojourn time is exploited to circumvent the complex computation of transition probabilities. The derived conditions can cover the time-invariant mode-dependent and time-invariant mode-independent H_∞ fuzzy control schemes as special cases. A classic cart-pendulum system is presented to demonstrate the effectiveness and advantages of the proposed theoretical results.     

On ℓ∞ and L∞ gains for positive systems with bounded time-varying delays

This paper is devoted to the analysis of the ?_∞ and L_∞ gains for positive linear systems with interval time-varying delays. Through exploiting the monotonicity of the state trajectory, we first prove that for positive systems with constant delays, the ?_∞ and L_∞ gains are fully governed by the system matrices but independent of the delay size. Moreover, for positive systems with bounded time-varying delays, by comparing with the nominal systems with constant delays, it turns out that the ?_∞ and L_∞ gains are exactly the same as that of the constant delay systems. The results in this paper reveal that the ?_∞ and L_∞ gains of positive linear systems are not sensitive to the magnitude of time delays and hence the computation of ?_∞ and L_∞ gains of positive systems with bounded time-varying delays can be reduced to computing the ?_∞ and L_∞ gains of the corresponding delay-free systems. Both continuous-time and discrete-time cases are considered in this paper.     

Quadratic and H∞ switching control for discrete-time linear systems with multiplicative noises

The goal of this paper is to study the switched stochastic control problem of discrete-time linear systems with multiplicative noises. We consider both the quadratic and the H_∞ criteria for the performance evaluation. Initially we present a sufficient condition based on some Lyapunov–Metzler inequalities to guarantee the stochastic stability of the switching system. Moreover, we derive a sufficient condition for obtaining a Metzler matrix that will satisfy the Lyapunov–Metzler inequalities by directly solving a set of linear matrix inequalities, and not bilinear matrix inequalities as usual in the literature of switched systems. We believe that this result is an interesting contribution on its own. In the sequel we present sufficient conditions, again based on Lyapunov–Metzler inequalities, to obtain the state feedback gains and the switching rule so that the closed loop system is stochastically stable and the quadratic and H_∞ performance costs are bounded above by a constant value. These results are illustrated with some numerical examples.     

Finite-horizon ℋ 2/ℋ ∞ control for a class of nonlinear Markovian jump systems with probabilistic sensor failures

This article is concerned with the mixed ?₂/?_∞ control problem over a finite horizon for a class of nonlinear Markovian jump systems with both stochastic nonlinearities and probabilistic sensor failures. The stochastic nonlinearities described by statistical means could cover several types of well-studied nonlinearities, and the failure probability for each sensor is governed by an individual random variable satisfying a certain probability distribution over a given interval. The purpose of the addressed problem is to design state feedback controllers such that the closed-loop system achieves the expected ?₂ performance requirement with a guaranteed ?_∞ disturbance attenuation level. The solvability of the addressed control problem is expressed as the feasibility of certain coupled matrix equations. The controller gain at each time instant k can be obtained by solving the corresponding set of matrix equations. A numerical example is given to illustrate the effectiveness and applicability of the proposed algorithm.     

H∞ controller synthesis for pendulum-like systems

This paper focus on a stabilization problem for a class of nonlinear systems with periodic nonlinearities, called pendulum-like systems. A notion of Lagrange stabilizability is introduced, which extends the concept of Lagrange stability to the case of controller synthesis. Based on this concept, we address the problem of designing a linear dynamic output controller which stabilizes (in the Lagrange sense) a pendulum-like system within the framework of the H_∞ control theory. Lagrange stabilizability conditions for uncertainty-free systems and systems with norm-bounded uncertainty in the linear part are derived, respectively. When these conditions are satisfied, the desired stabilization output feedback controller can be constructed via feasible solutions of a certain set of linear matrix inequalities (LMIs).     

Asymptotic stability of systems operating on a closed hypercube

Sufficient conditions for the global asymptotic stability of the equilibrium x_e = 0 of dynamical systems which are characterized by linear ordinary differential equations with saturation nonlinearities are established. The class of systems considered herein arises in the modeling of control systems and neural networks.     

Design of robust H ∞ filters for a class of uncertain nonlinear neutral stochastic systems with time delays

This article investigates the problem of robust H _∞ filtering for a class of nonlinear neutral stochastic time-delay systems with norm-bounded parameter uncertainties. The nonlinearities are assumed to satisfy the global Lipschitz conditions. By solving a set of certain linear matrix inequalities, an H _∞ filter is designed, which ensures both the robust stochastic stability and a prescribed H _∞ performance of the filtering error system for all admissible uncertainties. A numerical example is given to show the effectiveness of the design method proposed in this article.     

H∞ control of a class of 2-D continuous switched delayed systems via state-dependent switching

This paper addresses the problem of state feedback H_∞ stabilisation of 2-D (two-dimensional) continuous switched state delayed systems represented by the Roesser model using the multiple Lyapunov functional approach. First, an asymptotical stability condition of 2-D continuous switched systems with state-dependent switching is derived. Second, a sufficient condition for H_∞ performance of the underlying system is established. Third, a state feedback controller is proposed to ensure that the resulting closed-loop system has a prescribed H_∞ performance level under a state-dependent switching signal. All the results are developed in terms of linear matrix inequalities. Finally, three examples are provided to demonstrate the validity and effectiveness of the proposed method.     

H∞ output tracking control of uncertain and disturbed nonlinear systems based on neural network model

The work presented in this paper seeks to address the tracking problem for uncertain continuous nonlinear systems with external disturbances. The objective is to obtain a model that uses a reference-based output feedback tracking control law. The control scheme is based on neural networks and a linear difference inclusion (LDI) model, and a PDC structure and H_∞ performance criterion are used to attenuate external disturbances. The stability of the whole closed-loop model is investigated using the well-known quadratic Lyapunov function. The key principles of the proposed approach are as follows: neural networks are first used to approximate nonlinearities, to enable a nonlinear system to then be represented as a linearised LDI model. An LMI (linear matrix inequality) formula is obtained for uncertain and disturbed linear systems. This formula enables a solution to be obtained through an interior point optimisation method for some nonlinear output tracking control problems. Finally, simulations and comparisons are provided on two practical examples to illustrate the validity and effectiveness of the proposed method.     

Robust switching-type H∞ filter design for linear uncertain systems with time-varying delay

This paper is concerned with the problem of robust H_∞ filter design for a class of linear uncertain systems with time-varying delay. The uncertainty parameters are supposed to be time-varying, unknown, but bounded, which appear affinely in the matrices of the considered system model. Based on linear matrix inequality (LMI) method and switching laws, a new switching-type filter is designed to guarantee the asymptotic stability and H_∞ performance level of the filtering error systems. The key feature is that the new proposed filter parameters are switching between certain fixed gains automatically via the designed switching law. It is shown that the new filter design method is less conservative than the traditional fixed gain filter design method. An example is given to illustrate the validity of the proposed design.     

Event-triggered robust H∞ control for uncertain switched linear systems

In this paper, an event-triggering scheme is implemented in uncertain switched linear systems with time-varying delays and exogenous disturbance. Instead of standard periodically time-triggered, sampled-data control systems, the event-triggered control systems sample data only when an event, typically defined as some performance error exceeding a tolerant bound, occurs. Specifically, considering the disturbance existing in the system, the event-triggered robust H_∞ control problem is studied. In order to guarantee the robust H_∞ performance, the event-triggered full state feedback control, multiple Lyapunov functions method and state-dependent switching law are utilised to construct sufficient conditions in terms of linear matrix inequalities. In particular, since the event-triggered signals and switching signals may interlace with each other, the influence from them on the analysis of robust H_∞ performance is clarified. Subsequently, sufficient design conditions of the sub-controllers’ gains are further presented. Moreover, the Zeno problem is discussed to exclude continuously triggering and sampling. Finally, numerical simulations are provided to verify the feasibility of the proposed approach.     

Weighted H ∞ performance analysis of switched linear systems with mode-dependent average dwell time

This article is concerned with the disturbance attenuation properties of a class of switched linear systems by using a mode-dependent average dwell time (MDADT) approach. The proposed switching law is less strict than the average dwell time (ADT) switching in that each mode in the underlying system has its own ADT. By using the MDADT approach, a sufficient condition is obtained to guarantee the exponential stability with a weighted H _∞ performance for the underlying systems. A numerical example is given to show the validity and potential of the developed results on improving the disturbance attenuation performance.     

Absolute stability criteria with prescribed decay rate for finite-dimensional and delay systems

We provide here an extension of Popov criterion, permitting to check exponential stability with prescribed decay rate (otherwise called α-stability) of nonlinear delay systems with sector-bounded nonlinearities. As for the celebrated result, the main hypothesis is expressed under a frequency form. For the delay-free case, the latter is equivalent to a linear matrix inequality, whose solution may be found by widespread algorithms.     

A distributed controller approach for delay-independent stability of networked control systems

This article introduces a novel distributed controller approach for networked control systems (NCS) to achieve finite gain L₂ stability independent of constant time delay. The proposed approach represents a generalization of the well-known scattering transformation which applies for passive systems only. The main results of this article are (a) a sufficient stability condition for general multi-input-multi-output (MIMO) input-feedforward-output-feedback-passive (IF-OFP) nonlinear systems and (b) a necessary and sufficient stability condition for linear time-invariant (LTI) single-input-single-output (SISO) systems. The performance advantages of the proposed approach are reduced sensitivity to time delay and improved steady state error compared to alternative known delay-independent small gain type approaches. Simulations validate the proposed approach.     

State feedback H ∞ control of commensurate fractional-order systems

This article focuses on the state feedback H _∞ control problem for commensurate fractional-order systems with a prescribed H _∞ performance. For linear time-invariant fractional-order systems, a sufficient condition to guarantee stability with H _∞ performance is firstly presented. Then, by introducing a new flexible real matrix variable, the feedback gain is decoupled with complex matrix variables and further parametrised by the new flexible matrix. Moreover, iterative linear matrix inequality algorithms with initial optimisation are developed to solve the state feedback H _∞ suboptimal control problem for fractional-order systems. Finally, illustrative examples are given to show the effectiveness of the proposed approaches.     

Stabilisation for positive switched T-S fuzzy delayed systems under standard L1 and L∞ performance

In this paper, the stabilisation problem for positive switched T-S fuzzy delayed systems under standard L₁ and L_∞ performance is addressed. First, by proposing a time-scheduled multiple linear copositive Lyapunov function which is time-varying during the mode-dependent minimum dwell-time interval, sufficient conditions of asymptotical stability for positive switched T-S fuzzy delayed system under dwell-time switching are derived for the first time. Then, sufficient conditions for the underlying system to be asymptotically stable with standard L₁ and L_∞ performance are obtained, respectively. Based on the obtained results, a fuzzy state-feedback controller scheme is designed to stabilise the positive switched T-S fuzzy delayed system, while guaranteeing the positivity and asymptotical stability with standard L₁ performance in closed loop. Furthermore, the stabilisation problem under standard L_∞ performance is also solved. All the results are presented in the linear programming form. Finally, two numerical examples are presented to demonstrate the effectiveness of the obtained theoretical results.     

LTI modelling of NFIR systems: near-linearity and control, LS estimation and linearization

Linear time-invariant (LTI) modelling of nonlinear finite impulse response (NFIR) systems is studied from a control point of view. Nearly linear NFIR systems and their control-relevant properties are analysed in detail. The main modelling interest is in the analysis of least squares (LS) LTI identification when the true system is an NFIR system, which is possibly nearly linear. Linearization is used for comparison purposes as the second LTI modelling technique. Nearly linear systems provide a natural generalization of LTI systems to include nonlinearities that allow globally good LTI approximations, while at the same time, such nonlinearities can have a very dramatic effect on the local characteristics of the system. Several control-oriented examples illustrate the possible weaknesses and strengths of the studied LTI modelling techniques. Linearization is found to be especially vulnerable to the presence of even very small, only locally significant, nonlinearities. LS estimation can largely avoid such difficulties, but input design becomes a more critical issue than in standard linear estimation theory. Certain counter-intuitive properties of commonly used input-output stability notions, such as ?₂ stability, are discussed via the concept of near-linearity.     

Robust stabilizing control laws for a class of second-order switched systems

For a class of second-order switched systems consisting of two linear time-invariant (LTI) subsystems, we show that the so-called conic switching law proposed previously by the present authors is robust, not only in the sense that the control law is flexible (to be explained further), but also in the sense that the Lyapunov stability (resp., Lagrange stability) properties of the switched system are preserved in the presence of certain kinds of vanishing perturbations (resp., nonvanishing perturbations). The analysis is possible since the conic switching laws always possess certain kinds of “quasi-periodic switching operations”. We also propose for a class of nonlinear second-order switched systems with time-invariant subsystems a switching control law which locally exponentially stabilizes the entire nonlinear switched system, provided that the conic switching law exponentially stabilizes the linearized switched systems (consisting of the linearization of each nonlinear subsystem). This switched control law is robust in the sense mentioned above.