N-sigma stability of stochastic systems with sliding mode control

In this paper, sliding mode control for discrete time systems with stochastic noise in their input channel has been discussed. The idea of process control using control charts has influenced this new approach towards dealing with systems with stochastic noise. The new approach approximates the stochastic noise as a bounded uncertainty, similar to having bounds in the control charts for stochastic process control data. For discrete time systems, this results in a bounded stability in probability of the quasi sliding mode, which is referred to as the N-sigma bounded stability. The probability associated with the stability notions is not fixed and the control engineer may desire lower or higher degrees of stability in terms of this probability. Thus one has design flexibility while implementing the theory in practice, where one might have to adjust the desired degree of stability due to hardware limitations.     

The stochastic harmonic oscillator with finite correlation times

The moments of an undamped stochastic harmonic oscillator are examined using iteration procedures described in earlier papers. The noise is similar to white noise, but has finite correlation time which is taken to be less than the period of the oscillator. When the time of observation is finite, the first moment is explicitly given, attempting first-order approximation. For infinite time, we establish certain conditions which assure boundedness of the first and second moments. All the results obtained are applied to the white noise when its correlation time is considered finite.     

Stabilization of highly nonlinear hybrid systems driven by Lévy noise and delay feedback control based on discrete-time state observations

There are many hybrid stochastic differential equations (SDEs) in the real-world that don’t satisfy the linear growth condition (namely, SDEs are highly nonlinear), but they have highly nonlinear characteristics. Based on some existing results, the main difficulties here are to deal with those equations if they are driven by Lévy noise and delay terms, then to investigate their stability in this case. The present paper aims to show how to stabilize a given unstable nonlinear hybrid SDEs with Lévy noise by designing delay feedback controls in the both drift and diffusion parts of the given SDEs. The controllers are based on discrete-time state observations which are more realistic and make the cost less in practice. By using the Lyapunov functional method under a set of appropriate assumptions, stability results of the controlled hybrid SDEs are discussed in the sense of pth moment asymptotic stability and exponential stability. As an application, an illustrative example is provided to show the feasibility of our theorem. The results obtained in this paper can be considered as an extension of some conclusions in the stabilization theory.     

Discrete Razumikhin-type stability theorems for stochastic discrete-time delay systems

By using the Razumikhin-type technique, for stochastic discrete-time delay systems, this paper establishes the discrete Razumikhin-type theorems on the pth moment stability, the global pth moment stability and the pth moment exponential stability, respectively. The almost sure exponential stability is also investigated by using the pth moment exponential stability and the Borel–Cantelli lemma. As the applications of t he established theorems, stability of a special class of stochastic discrete-time delay systems, synchronization of the stochastic discrete-time delay dynamical networks and stabilization of a stochastic discrete-time linear delay time invariant system are examined.     

Modeling and stabilization of networked control systems with bounded packet dropouts and occasionally missing control inputs subject to multiple sampling periods

The paper is concerned with the modeling and stabilization problem of networked control systems under simultaneous consideration of bounded packet dropouts and occasionally missing control inputs. In particular, the focus of the paper is to capture the case where the packet dropouts and control inputs missing are subject to multiple sampling periods, and not periodic as in existing results. By input-delay approach and then fully considering the probability distribution characteristic of packet dropouts in the modeling, the original linear system is firstly transformed to a switched stochastic time-delay system. Meanwhile, the probability distribution values of stochastic delay taking values in m(m ≥ 2) given intervals can be explicitly obtained, which is of vital importance to analyse the stabilization problem of considered system. Secondly, by means of the average dwell time technique, some sufficient conditions in terms of linear matrix inequalities for the existence of desired stabilizing controller are derived. Finally, an illustrative example is given to illustrate the effectiveness of the proposed stabilizing controller and some less conservative results are obtained.     

Quantized control of event-triggered networked systems with time-varying delays

The paper is a study of quantized control for stochastic Markov jump systems with interval time-varying delays and bounded system noise under event-triggered mechanism. A new scheme of Lyapunov–Krasovskii functional which contains the quadratic terms and integral terms is presented. Then quadratic convex technology, the theory of stochastic switching system, and logarithmic quantizer are applied to this paper. The design of quantized controller is obtained with those methodologies. Different from previous results, our derivation applies the idea of second-order convex combination. The conservatism of stability criteria for systems is reduced by using this method. A numerical example under different conditions is given to demonstrate the effectiveness and validity of the new design techniques.     

A novel hybrid filter-based fault diagnosis algorithm for switched systems with a dual noise term

This study considers state and fault estimation for a switched system with a dual noise term. A zonotopic and Gaussian Kalman filter for state estimation is designed to obtain state estimation interval in the presence of both stochastic and unknown but bounded (UBB) uncertainties. The switching state and fault state of the system are distinguished by detecting whether the system measurement date is within the bounds of its predicted output. Once the switched time is detected in the system, the filter zonotopic and Gaussian Kalman functions are initialized. Once the fault time is detected, a zonotopic and Gaussian Kalman filter-based fault estimator is constructed to estimate the corresponding faults. Finally, a numerical simulation is presented to demonstrate the accuracy and effectiveness of the proposed algorithm.     

Resilient control based on disturbance observer for nonlinear singular stochastic hybrid system with partly unknown Markovian jump parameters

In this paper, the composite anti-disturbance resilient control is considered for nonlinear singular stochastic hybrid system with partly unknown Markovian jump parameters under multiple disturbances. Three kinds of disturbances are included in the studied system. One is generated by an external system and it enters the hybrid system from the channel of the control input. The other one is stochastic white noise. And the third one is the external unknown time-varying disturbance and it is supposed to be H₂ norm bounded. By combining the disturbance-observer-based-control scheme, H_∞ control technique and resilient control method, a composite anti-disturbance resilient controller is constructed to attenuate and eliminate the affection of these disturbances, and ensures the whole closed-loop system regular, impulse free and stochastically stable with the corresponding control performance. Then, some sufficient conditions and the gains of the controller and observer are obtained by using Lyapunov function method and the linear matrix inequalities (LMIs) technique. Finally, two numerical examples are given to show the effectiveness of presented method.     

Further stability results for random nonlinear systems with stochastic impulses

In this paper, the global asymptotic stability in probability and the exponential stability in $m$th moment are investigated for random nonlinear systems with stochastic impulses, whose occurrence is determined by a Poisson process. The stochastic disturbances in the impulsive random nonlinear systems are driven by second-order processes, which have bounded mean power. Firstly, the improved Lyapunov approaches for the global asymptotic stability in probability and the exponential stability in $m$th moment are established for impulsive random nonlinear systems based on the uniformly asymptotically stable function. Secondly, the improved results are further extended to the impulsive random nonlinear systems with Markovian switching. Finally, two examples are provided to verify the feasibility and effectiveness of the obtained results.     

A control problem with structural choices

The case is considered in which, during the operation of an optimal control system, the optimizer, in addition to applying his usual control, may switch structures. Necessary and sufficient conditions are derived and emphasis is placed on the special characteristics of this problem. Continuous and discrete time set-ups are considered and the separation principle is shown not to hold for the linear quadratic case in the presence of noise.     

On robust detection of discrete-time stochastic signals

The problem of designing robust systems for the detection of stochastic signals in noise is considered for the large-sample-size, small-signal case. By applying two previously-established models for the detection of stochastic signals, known results for the robust detection of deterministic signals are extended on a limited basis to the stochastic- signal case. The proposed detectors are seen to be robust over a class of possible noise statistics, based on a Huber-Tukey mixture model, which contains noises characterized by heavy-tailed probability density functions. In addition, numerical results are presented which verify the robustness property of the proposed detectors over wider classes of noise mixtures.     

Stochastic fault tolerant control of networked control systems

A problem of stabilization about uncertain networked control systems (NCSs) with random but bounded delays is discussed in this paper. By using augmented state-space method, this class of problems can be modeled as discrete-time jump linear systems governed by finite-state Markov chains. A new switched model based on probability is proposed to research problems of reliable control when actuators become ageing or partially disabled. Using improved V-K iteration algorithm, a class of reliable controllers are designed to make systems asymptotically mean square stable under several stochastic disturbances such as random time-delay and stochastic actuator failure and the maximal redundancy degree is given through this method.     

Robust stable pole-placement-based adaptive control of continuous linear systems with two parametrical estimation schemes

This paper deals with the pole-placement-type robust adaptive control of continuous linear systems in the presence of bounded noise and a common class of unmodeled dynamics provided that two estimation schemes are used in parallel. Both estimation schemes are introduced in order to minimize the plant identification error by selecting, as plant parameter estimates, a convex combination of both parameter estimates which leads to the selection of one of the estimation schemes, via a switching rule, on time intervals of at least a minimum prefixed residence duration. The weights of the individual parameter vector estimates are provided at each time by an optimization or suboptimization scheme for a quadratic loss function of the possibly filtered tracking error and/or control input. The robust stability of the overall adaptive scheme is ensured by an adaptation relative dead zone which takes into account the contribution of the unmodeled dynamics and bounded noise. The basic results are derived for two different estimation strategies which have either a shared regressor with the plant or individual regressors for the input contribution and its contributed derivatives. In this second case, the plant input is obtained from a similar convex combination rule as the one used for the estimators in the first approach. An extension of the basic strategies is also pointed out including a combined use of the (sub) optimization scheme with a supervisor of past measures for the on-line calculation of the estimator weights in the convex combination. Finally, the extension of the scheme for the use of any number of parametrical estimators is focused on.     

Razumikhin-type theorems on pth moment boundedness of neutral stochastic functional differential equations with Makovian switching

This paper investigates pth moment boundedness of neutral stochastic functional differential equations with Markovian switching (NSFDEsMS) based on Razumikhin technique and comparison principle. And pth moment stability is examined as a special case. Since the stochastic disturbances and neutral delays are incorporated, the considered system becomes more complex. Besides, the coefficients of the estimated upper bound for the diffusion operation associated with the underlying NSFDEsMS also may be chosen to be sign-changing functions instead of constant functions or negative definite functions, as a result, our results can work in general non-autonomous neutral stochastic systems. Finally, two examples are provided to show the effects of the proposed methods.     

Non-fragile control of positive Markovian jump systems

This paper investigates the non-fragile control for positive Markovian jump systems both in continuous-time and discrete-time cases with actuator uncertainty. It is assumed that the coefficient matrices of the non-fragile controller is unknown and bounded. The state-feedback controller gain consists of nominal controller gain and gain perturbation. First, a set of state-feedback controllers for the considered system are designed by using a stochastic co-positive Lyapunov function integrated with linear programming approach. Under the designed controllers, the resulting closed-loop systems are positive and stochastically stable. Then, the proposed controller design approach is extended to discrete-time systems. Through comparisons, it is shown that existing results are special cases of the presented ones in the paper. Finally, two examples are given to illustrate the effectiveness of the proposed design.     

Response and stability of SDOF viscoelastic system under wideband noise excitations

The response and stability of a single degree-of-freedom (SDOF) viscoelastic system with strongly nonlinear stiffness under the excitations of wideband noise are studied in this paper. Firstly, terms associated with the viscoelasticity are approximately equivalent to damping and stiffness forces; the viscoelastic system is approximately transformed to SDOF system without viscoelasticity. Then, with application of the method of stochastic averaging, the averaged Itô differential equation is obtained. The stationary response and the largest Lyapunov exponent can be analytically expressed. The effects of different system parameters on the response and stability of the system are discussed as well.     

Linear stochastic system identification using correlation techniques

The identification of linear, discrete time, scalar output systems which are driven exclusively by white, zero mean, inaccessible noise sequences is discussed. Two principal results are presented. First, two methods (least squares and an autocorrelation technique) for identifying the system characteristic equation coefficients are compared. The least squares approach is shown to be biased except for special cases. In general, the bias cannot be removed. If the state transition matrix is of the phase variable form, bias removal requires a knowledge of the measurement noise variance and all but one of the state driving noise variances. The autocorrelation technique is not biased asymptotically and does not require a knowledge of the noise variances.Secondly, it is shown that the m² elements of the state transition matrix cannot be identified uniquely from the scalar output sequence autocorrelation coefficients if the system order is higher than one. The implication of this uncertainty in the state transition matrix on optimal filtering of the output sequence is briefly discussed.     

Barrier Lyapunov function-based decentralized adaptive neural control for uncertain high-order stochastic nonlinear interconnected systems with output constraints

Decentralized adaptive neural backstepping control scheme is developed for uncertain high-order stochastic nonlinear systems with unknown interconnected nonlinearity and output constraints. For the control of high-order nonlinear interconnected systems, it is assumed that nonlinear system functions are unknown. It is for the first time to control stochastic nonlinear high-order systems with output constraints. Firstly, by constructing barrier Lyapunov functions, output constraints are handled. Secondly, at each recursive step, only one adaptive parameter is updated to overcome over-parameterization problems, and RBF neural networks are used to identify unknown nonlinear functions so that the difficulties caused by completely unknown system functions and stochastic disturbances are tackled. Finally, based on the Lyapunov stability method, the decentralized adaptive control scheme via neural networks approximator is proposed, ultimately reducing the number of learning parameters. It is shown that the designed controller can guarantee all the signals of the resulting closed-loop system to be semi-globally uniformly ultimately bounded (SGUUB), and the tracking errors for each subsystem are driven to a small neighborhood of zero. The simulation studies are performed to verify the effectiveness of the proposed control strategy.     

Finite-time stability and stabilization of linear discrete time-varying stochastic systems

This paper studies the finite-time stability and stabilization of linear discrete time-varying stochastic systems with multiplicative noise. Firstly, necessary and sufficient conditions for the finite-time stability are presented via a state transition matrix approach. Secondly, this paper also develops the Lyapunov function method to study the finite-time stability and stabilization of discrete time-varying stochastic systems based on matrix inequalities and linear matrix inequalities (LMIs) so as to Matlab LMI Toolbox can be used.The state transition matrix-based approach to study the finite-time stability of linear discrete time-varying stochastic systems is novel, and its advantage is that the state transition matrix can make full use of the system parameter informations, which can lead to less conservative results. We also use the Lyapunov function method to discuss the finite-time stability and stabilization, which is convenient to be used in practical computations. Finally, three numerical examples are given to illustrate the effectiveness of the proposed results.