Robust state‐feedback control of stochastic state‐multiplicative discrete‐time linear switched systems with dwell time

Linear discrete‐time switched stochastic systems are considered, where the problems of mean square stability, stochastic l₂‐gain and state‐feedback control design are treated and solved. Solutions are obtained for both nominal and polytopic‐type uncertain systems. In all these problems, the switching obeys a dwell time constraint. In our solution, to each subsystem of the switched system, a Lyapunov function is assigned that is nonincreasing at the switching instants. The latter function is allowed to vary piecewise linearly, starting at the end of the previous switch instant, and it becomes time invariant after the dwell. In order to guarantee asymptotic stability, we require the Lyapunov function to be negative between consecutive switchings. We thus obtain Linear Matrix Inequalities conditions. Based on the solution of the stochastic l₂‐gain problem, we derive a solution to the state‐feedback control design, where we treat a variety of special cases. Being affine in the system matrices, all the aforementioned solutions are extended to the uncertain polytopic case. The proposed theory is demonstrated by a practical example taken from the field of flight control. Copyright © 2015 John Wiley & Sons, Ltd.     

Stability,L2‐gain and asynchronous H ∞ control for continuous‐time switched systems

This paper is concerned with the stability and L₂‐gain problems for a class of continuous‐time linear switched systems with the existed asynchronous behaviors, where ‘asynchronous’ means that the switching of the controllers to be designed has a lag to the switching of the system modes. Firstly, a new sufficient condition on the asymptotic stability and weighted L₂‐gain analysis is obtained by using multiple Lyapunov functions combined with the average dwell time technique. Moreover, a result that is formulated in form of linear matrix inequalities is derived for the problem of asynchronous H _∞ control. Based on the result, the mode‐dependent controllers can be designed. Finally, an illustrative numerical example is presented to show the effectiveness of the obtained results.Copyright © 2013 John Wiley & Sons, Ltd.     

L1‐Gain Analysis and Control for Switched Positive Systems with Dwell Time Constraint

This paper studies the problems of L₁‐gain analysis and control for switched positive systems with dwell time constraint. The state‐dependent switching satisfies a minimal dwell time constraint to avoid possible arbitrary fast switching. By constructing multiple linear co‐positive Lyapunov functions, sufficient conditions of stability and L₁‐gain property are derived under the proposed switching strategy. Then, an effective state feedback controller is designed to ensure the positivity and L₁‐gain property of the closed‐loop system. Finally, a simulation example is given to illustrate the effectiveness of the proposed method.     

Exponential H∞ filtering for uncertain discrete‐time switched linear systems with average dwell time: A µ‐dependent approach

In this paper, the problem of exponential H_∞ filter problem for a class of discrete‐time polytopic uncertain switched linear systems with average dwell time switching is investigated. The exponential stability result of the general discrete‐time switched systems using a discontinuous piecewise Lyapunov function approach is first explored. Then, a new µ‐dependent approach is proposed, which means the analysis or synthesis of the underlying systems is dependent on the increase degree µ of the piecewise Lyapunov function at the switching instants. A mode‐dependent full‐order filter is designed such that the developed filter error system is robustly exponentially stable and achieves an exponential H_∞ performance. Sufficient existence conditions for the desired filter are derived and formulated in terms of a set of linear matrix inequalities, and consequently the minimal average dwell time and the corresponding filter are obtained from such conditions for a given system decay degree. A numerical example is presented to demonstrate the potential and effectiveness of the developed theoretical results. Copyright © 2007 John Wiley & Sons, Ltd.     

Stability and l2‐gain of discrete‐time switched systems with unstable modes

This paper addresses stability and l₂‐gain for discrete‐time switched systems with unstable modes based on slow/fast mode‐dependent average dwell time (MDADT) switching strategies. Firstly, by employing a class of multiple discontinuous Lyapunov functions (MDLFs) and developing a kind of alternative switching signals, the sufficient conditions on stability are established for the system without external disturbances under a slow/fast MDADT switching scheme with a tighter bounds on the dwell time. Furthermore, by defining indicator functions and exploring the features of slow/fast MDADT switching, the weighted l₂‐gain conditions are achieved for the system with external disturbances. Particularly, the criteria of stability and l₂‐gain are also established for the corresponding discrete‐time switched linear systems with unstable modes via the MDLFs method and the slow/fast MDADT switching strategy. Finally, two numerical examples are presented to illustrate the advantages of the proposed methods.     

Robust stability and controller synthesis of discrete linear switched systems with dwell time

Sufficient conditions are derived for the robust stability of discrete-time, switched, linear systems with dwell time in the presence of polytopic type parameter uncertainty. A Lyapunov function, in quadratic form, is assigned to each of the subsystems. This function is allowed to be time-varying and piecewise linear during the dwell time and it becomes time invariant afterwards. Asymptotic stability conditions are obtained in terms of linear matrix inequalities for the nominal set of subsystems. These conditions are then extended to the case where the subsystems encounter polytopic type parameter uncertainties. The developed method is applied to l ₂-gain analysis where a bounded real lemma is derived, and to H _∞ control and estimation, both for the nominal and the uncertain cases.     

Finite‐time stabilization of Markovian jump delay systems – a switching control approach

In this paper, a finite‐time stabilization problem is considered for a class of continuous‐time Markovian jump delay systems (MJDSs). A switching controller, which only depends on the average dwell time (ADT) switching laws, is proposed to make a trade‐off between robustness and adaptiveness when the design complexity of mode‐independent, mode‐dependent, and mode‐dependent and variation‐dependent control strategies is considered. First, the stochastic finite‐time boundedness for an MJDS is analyzed by an ADT approach. Second, the disturbance attenuation capability of MJDS is studied via a finite‐time weighted L₂ gain, which depends on the switching numbers. The impacts of finite‐time interval and L₂ gain acting on the ADT are also thoroughly discussed. Then, a switching controller is designed such that the resulting closed‐loop MJDS is stochastically finite‐time bounded and has a guaranteed disturbance attenuation level. Finally, a numerical example is provided to verify the effectiveness of the developed results. Copyright © 2016 John Wiley & Sons, Ltd.     

Finite‐time boundedness and L2‐gain analysis for switched positive linear systems with multiple time delays

In this paper, we study the finite‐time boundedness, stabilization, and L₂‐gain for switched positive linear systems (SPLS) with multiple time delays. Using multiple linear copositive Lyapunov functions, sufficient conditions in terms of linear matrix inequalities are obtained for the problems of finite‐time boundedness and stabilization and the design of state feedback controllers for SPLS. Under asynchronous switching, L₂‐gain analysis is developed for SPLS under the constraint of average dwell time. Numerical examples are given to illustrate our theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.     

Delay‐dependent exponential H ∞ filtering for discrete‐time switched delay systems

In this paper, the problem of delay‐dependent exponential H _∞ filtering for discrete‐time switched delay systems is investigated under average dwell time switching signals. Time delay under consideration is interval time‐varying in the states. By introducing a proper factor to construct a novel Lyapunov‐Krasovskii function and using average dwell time approach, sufficient conditions for the solvability of this problem, dependent on the upper and lower bounds of time‐varying delay, are obtained in terms of linear matrix inequalities. A numerical example is presented to demonstrate the effectiveness of the developed results. Copyright © 2011 John Wiley & Sons, Ltd.     

L2‐Gain Analysis for Synchronization of Stochastic Complex Network with Switched Topology and Time‐Varying Delay

We investigate the exponential stability and L₂‐gain analysis for the synchronization of stochastic complex networks under average dwell time switched topology with consideration of external disturbance, internal noise and fast time‐varying delay in the synchronized process. Based on the proposed stochastic network, a new L₂‐gain synchronization is proposed to solve the mean‐square exponential stable under switched topology with an H_∞ performance from the extrinsic disturbances to the synchronization error. The obtained results are applicable for the fast time‐varying case with larger‐than‐1 delay derivative. Finally, numerical simulations are performed to demonstrate the effectiveness of our strategies.     

Finite‐time boundedness of switched systems with time‐varying delays via sampled‐data control

In the framework of sampled‐data control, finite‐time boundedness (FTB) of switched systems with time‐varying delays is investigated. Sufficient conditions for FTB of switched systems with time‐varying delays via sampled‐data control are proposed. Moreover, considering the relationship between the sampling period and the mode‐dependent average dwell time, switching signals are designed. In addition, finite‐time weighted L₂‐gain (FTW‐L₂‐gain) of switched systems with time‐varying delays is proposed to measure their disturbance tolerance capacity within a finite‐time interval. Multiple Lyapunov‐Krasovskii functionals are applied to complete subsequent proofs in detail. Simulation results are exemplified to verify the proposed method.     

Robust non‐fragile H ∞ ∕ L2 − L ∞ control of uncertain linear system with time‐delay and application to vehicle active suspension

This paper presents an approach to design robust non‐fragile H _∞ ∕ L₂ ? L _∞ static output feedback controller, considering actuator time‐delay and the controller gain variations, and it is applied to design vehicle active suspension. According to suspension design requirements, the H _∞ and L₂ ? L _∞ norms are used, respectively, to reflect ride comfort and time‐domain hard constraints. By employing a delay‐dependent Lyapunov function, existence conditions of delay‐dependent robust non‐fragile static output feedback H _∞ controller and L₂ ? L _∞ controller are derived, respectively, in terms of the feasibility of bilinear matrix inequalities. Then, a new procedure based on LMI optimization and a hybrid algorithm of the particle swarm optimization and differential evolution is used to solve an optimization problem with bilinear matrix inequality constraints. Simulation results show that the designed active suspension system still can guarantee their own performance in spite of the existence of the model uncertainties, the actuator time‐delay and the controller gain variations. Copyright © 2014 John Wiley & Sons, Ltd.     

Local input‐to‐state stabilization and ℓ ∞ ‐induced norm control of discrete‐time quadratic systems

This paper addresses the problems of local stabilization and control of open‐loop unstable discrete‐time quadratic systems subject to persistent magnitude bounded disturbances and actuator saturation. Firstly, for some polytopic region of the state‐space containing the origin, a method is derived to design a static nonlinear state feedback control law that achieves local input‐to‐state stabilization with a guaranteed stability region under nonzero initial conditions and persistent bounded disturbances. Secondly, the stabilization method is extended to deliver an optimized upper bound on the ? _∞ ‐induced norm of the closed‐loop system for a given set of persistent bounded disturbances. Thirdly, the stabilization and ? _∞ designs are adapted to cope with actuator saturation by means of a generalized sector bound constraint. The proposed controller designs are tailored via a finite set of state‐dependent linear matrix inequalities. Numerical examples are presented to illustrate the potentials of the proposed control design methods. Copyright © 2014 John Wiley & Sons, Ltd.     

L2 − L∞ Consensus Control for High‐Order Multi‐Agent Systems with Nonuniform Time‐Varying Delays

This paper considers consensus problem for high‐order multi‐agent systems with dynamically changing topologies and nonuniform time‐varying delays. By means of the tree‐type transformation approach, the model transformation is conducted and the consensus problem is converted into an L₂ ? L_∞ control problem of equivalent reduced‐order systems. Furthermore, a Lyapunov‐Krasovkii function is constructed for stability analysis and sufficient conditions in terms of linear matrix inequalities are derived to ensure the consensus with the prescribed L₂ ? L_∞ performance. A numerical simulation is provided to verify the correctness of the theoretical results.     

Exponential stability and L 2-gain analysis for a class of faulty systems

This article is concerned with the exponential stability and L ₂-gain problems for a class of systems with failures. Possible failures of sensors, actuators and/or controllers are considered simultaneously. Inspired by a switching control scheme, we develop an idea that is turning the class of faulty systems into a switched system composed of stable and unstable subsystems. Based on an average dwell‐time method, an activation time ratio scheme between stable and unstable subsystems is derived. It is shown that, under the proposed scheme, the faulty systems ensure exponential stability as well as a desire weighted L ₂-gain performance level. A numerical example and a practical simulation example on the flight control system are given to illustrate the practicability of the proposed method.     

Finite‐time H ∞ control for discrete‐time switched nonlinear systems with time delay

In this paper, the problem of finite‐time H _∞ control is addressed for a class of discrete‐time switched nonlinear systems with time delay. The concept of H _∞ finite‐time boundedness is first introduced for discrete‐time switched delay systems. Next, a set of switching signals are designed by using the average dwell time approach, under which some delay‐dependent sufficient conditions are derived to guarantee the H _∞ finite‐time boundedness of the closed‐loop system. Then, a finite‐time H _∞ state feedback controller is also designed by solving such conditions. Furthermore, the problem of uniform finite‐time H _∞ stabilization is also resolved. All the conditions are cast into linear matrix inequalities, which can be easily checked by using recently developed algorithms for solving linear matrix inequalities. A numerical example and a water‐quality control system are provided to demonstrate the effectiveness of the main results. Copyright © 2013 John Wiley & Sons, Ltd.     

Non‐fragile H ∞ control for switched stochastic delay systems with application to water quality process

In this paper, the problem of non‐fragile observer‐based H _∞ control for discrete‐time switched delay systems is investigated. Both data missing and time delays are taken into account in the links from sensors to observers and from controllers to actuators. Because data missing satisfies the Bernoulli distribution, such problem is transformed into an H _∞ control problem for stochastic switched delay systems. Average dwell time approach is used to obtain sufficient conditions on the solvability of such problems. A numerical example and a real example for water quality control are provided to illustrate the effectiveness and potential applications of the proposed techniques. Copyright © 2013 John Wiley & Sons, Ltd.     

Stability of neutral stochastic switched time delay systems: An average dwell time approach

This paper considers a class of stochastic systems referred to as stochastic switched systems of neutral type with time‐varying delay, which combines switched systems with neutral stochastic systems. The systems consist of subsystems of two forms: (i) only stable subsystems and (ii) both stable subsystems and unstable subsystems. By establishing an integral inequality, the exponential stability in pth(p≥1)‐moment for such systems with only stable subsystems is first considered. Then, by using an average dwell time approach, the exponential stability in pth(p≥1)‐moment for the second form is addressed. An important finding of this study is that when the average dwell time is chosen to be sufficiently large and the total activation time of unstable subsystems is relatively small compared with that of stable subsystems, the exponential stability in pth(p≥1)‐moment for such systems can be guaranteed. Two major advantages of these new results are that the differentiability or continuity of the delay function is not required compared with the existing results in the literature, and the proposed approaches can be used to consider the case when the neutral item and the stochastic perturbation are simultaneously presented. An example is provided to verify the effectiveness and potential of the theoretic results obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

Singular L2‐gain control for switched nonlinear control systems under arbitrary switching

This article concerns with the synthesis of L₂ ‐gain state feedback controllers, without the standard regular assumption, for multi‐input switched nonlinear control‐affine systems under arbitrary switching. A common control storage function approach is developed for deriving sufficient conditions for the existence of uniform L₂ ‐gain controllers. Moreover, an explicit formula for constructing L₂ ‐gain controllers is presented. A numerical example is given for illustration.     

State and input simultaneous estimation for discrete‐time switched singular delay systems with missing measurements

This paper presents an approach to simultaneously estimating the states and inputs of discrete‐time linear switched singular state‐delayed systems with unknown inputs, multiple missing measurements, and average dwell time (ADT) switching. In each output measurement channel of the system, the data loss incident is controlled by an individual stochastic variable obeying a certain probability distribution on the interval [01]. The proposed approach is based on the design of a switched, loss‐probability‐dependent proportional integral observer under the ? ₂ input attenuation framework. By using piecewise Lyapunov function technique, ADT scheme, stochastic analysis, and projection lemma, sufficient conditions for the existence of such an observer are established in terms of linear matrix inequalities, which guarantee that the resulting estimation error system is stochastically exponentially admissible and achieves an (non‐weighted) ? ₂ gain from the augmented unknown input to the state and unknown input estimation errors under ADT switching. Moreover, a method is provided to seek the minimum allowable ? ₂ gain level for a desired ADT of the switching signals. The effectiveness of the proposed approach is illustrated by a simulation example of direct current (DC) servomechanism control system. Copyright © 2016 John Wiley & Sons, Ltd.