Sliding mode controller design for linear stochastic systems with unknown parameters

This paper presents the sliding mode mean-square and mean-module controllers for linear stochastic systems with unknown parameters. In both cases, the controller equations are obtained using the separation principle, whose applicability to the considered problem is substantiated. Performance of the obtained controllers is verified in the illustrative example against the sliding mode mean-square and sliding mode controllers that are optimal for linear systems with known parameters. Simulation graphs demonstrating overall performance and computational accuracy of the designed optimal controller are included.     

Sliding mode state and parameter identification for linear stochastic systems

This paper presents the sliding mode mean-square and mean-module state filtering and parameter identification problems for linear stochastic systems with unknown parameters over linear observations, where unknown parameters are considered Wiener processes. The original problems are reduced to the sliding mode mean-square and mean-module filtering problems for an extended state vector that incorporates parameters as additional states. The obtained sliding mode filters for the extended state vector also serve as the optimal identifiers for the unknown parameters. Performance of the designed sliding mode mean-square and mean-module state filters and parameter identifiers are verified for both, stable and unstable, linear uncertain systems.     

Sliding mode controller design for linear systems with unmeasured states

This paper addresses the optimal controller problem for a linear system over linear observations with respect to different Bolza–Meyer criteria, where (1) the integral control and state energy terms are quadratic and the non-integral term is of the first degree or (2) the control energy term is quadratic and the state energy terms are of the first degree. The optimal solutions are obtained as sliding mode controllers, each consisting of a sliding mode filter and a sliding mode regulator, whereas the conventional feedback LQG controller fails to provide a causal solution. Performance of the obtained optimal controllers is verified in the illustrative example against the conventional LQG controller that is optimal for the quadratic Bolza–Meyer criterion. The simulation results confirm an advantage in favor of the designed sliding mode controllers.     

Sliding mode optimal control for linear systems

This paper addresses the optimal control problem for a linear system with respect to a Bolza–Meyer criterion with a non-quadratic non-integral term. The optimal solution is obtained as a sliding mode control, whereas the conventional linear feedback control fails to provide a causal solution. Performance of the obtained optimal controller is verified in the illustrative example against the conventional LQ regulator that is optimal for the quadratic Bolza–Meyer criterion. The simulation results confirm an advantage in favor of the designed sliding mode control.     

Sliding manifold design for linear systems with unmatched disturbances

This paper offers an efficient sliding manifold design method that minimizes the impact of unmatched disturbances onto SM dynamics and system accuracy. System sensitivity upon unmatched constant or slowly varying external disturbance vector is evaluated by the steady-state dependent criterion function. An infinite set of sliding manifolds that minimize the chosen optimization criterion is determined and a way of selecting a manifold out of that set that provides adopted SM dynamics is suggested. The proposed approach has been demonstrated on numerical examples and verified by simulations.     

Sliding mode filtering for stochastic systems with polynomial state and observation equations

In this paper, the mean-square and mean-module filtering problems for polynomial system states over polynomial observations are studied proceeding from the general expression for the stochastic Ito differentials of the estimate and the error variance. The paper deals with the general case of nonlinear polynomial states and observations. As a result, the Ito differentials for the estimates and error variances corresponding to the stated filtering problems are first derived. The procedure for obtaining an approximate closed-form finite-dimensional system of the sliding mode filtering equations for any polynomial state over observations with any polynomial drift is then established. In the examples, the obtained sliding mode filters are applied to solve the third-order sensor filtering problems for a quadratic state, assuming a conditionally Gaussian initial condition for the extended second-order state vector. The simulation results show that the designed sliding mode filters yield reliable and rapidly converging estimates.     

Sliding mode control for stochastic Markovian jumping systems subject to successive packet losses

In this work, the problem of sliding mode control is considered for a class of Markovian jumping systems. The packet dropout may happen when the state information is transmitted from the sensor to the controller. By means of an estimator for lost signals, an integral-like sliding function is constructed. And then, a sliding mode controller involving in dropout probability is designed such that the effect of packet losses can be effectively attenuated. Besides, the analysis on both the stability of sliding mode dynamics and the reachability of sliding surface are made. Finally, the numerical simulation results are given.     

Bounded controls for discrete-time linear systems subject to input time delay

This paper investigates the global stabilization of discrete-time linear systems with input time delay by bounded controls. Based on some special canonical forms containing time delays both in its input and state, two special discrete-time linear systems---multiple integrators and oscillators are first considered. The global stabilizing controllers are respectively established, and moreover, explicit conditions are established to guarantee the stability of the closed-loop systems. Subsequently, a concise design method is proposed for globally stabilizing general discrete-time linear system by combining the design methods for multiple integrators and oscillators. The designed controller is in the explicit form with explicit stability conditions being given, and thus is easier to use than the existing results. Finally, numerical simulations illustrate the effectiveness of the proposed approaches.     

Quantitative mean square exponential stability and stabilization of stochastic systems with Markovian switching

This paper is concerned with the quantitative mean square exponential stability and stabilization for stochastic systems with Markovian switching. First, the concept of quantitative mean square exponential stability(QMSES) is introduced, and two stability criteria are derived. Then, based on an auxiliary definition of general finite-time mean square stability(GFTMSS), the relations among QMSES, GFTMSS and finite time stochastic stability (FTSS) are obtained. Subsequently, QMSE-stabilization is investigated and several new sufficient conditions for the existence of the state and observer-based controllers are provided by means of linear matrix inequalities. An algorithm is given to achieve the relation between the minimum states’ upper bound and the states’ decay velocity. Finally, a numerical example is utilized to show the merit of the proposed results.     

Optimal control for linear systems with time delay in control input

This paper presents the optimal regulator for a linear system with time delay in control input and a quadratic cost function. The optimal regulator equations are obtained using the duality principle, which is applied to the optimal filter for linear systems with time delay in observations, and then proved using the maximum principle. Performance of the obtained optimal regulator is verified in the illustrative example against the best linear regulator available for linear systems without delays. Simulation graphs and comparison tables demonstrating better performance of the obtained optimal regulator are included.     

Memoryless parameter-dependent control strategy of stochastic strict-feedback time delay systems

For a class of stochastic strict-feedback nonlinear systems subject to different time delay states, this paper mainly concerns the problem of global asymptotic stabilization. Two new control strategies that the memoryless parameter-dependent state feedback control and the memoryless parameter-dependent output feedback control are taken into consideration, respectively. By skillfully constructing the Lyapunov-Krasovskii (L-K) functional, taking the proper determined parameter and employing the stochastic nonlinear time delay system (SNTDS) stability theory, the global asymptotic stability of the stochastic closed-loop system can be achieved. The proposed output feedback control scheme is finally utilized for the control design of the one-link manipulator system and two-stage chemical reactor system, which can verify the availability of the control approach.     

Robust adaptive sliding mode control for uncertain discrete-time systems with time delay

This paper focuses on robust adaptive sliding mode control for discrete-time state-delay systems with mismatched uncertainties and external disturbances. The uncertainties and disturbances are assumed to be norm-bounded but the bound is not necessarily known. Sufficient conditions for the existence of linear sliding surfaces are derived within the linear matrix inequalities (LMIs) framework by employing the free weighting matrices proposed in He et al. (2008) [3], by which the corresponding adaptive controller is also designed to guarantee the state variables to converge into a residual set of the origin by estimating the unknown upper bound of the uncertainties and disturbances. Also, simulation results are presented to illustrate the effectiveness of the control strategy.     

Output feedback control for stochastic nonlinear time delay systems using dynamic gain technique

This paper investigates the output feedback control for a class of stochastic nonlinear time delay systems based on dynamic gain technique. The nonlinear terms of the stochastic system satisfy linear growth condition on unmeasured state variables with the output dependent incremental rate, which makes the studied time delay stochastic system more general than the exiting results. Firstly, the full order dynamic gain observer is constructed. Then, the linear-like controller is designed without using recursive design method. Next, the stability analysis is given and a useful corollary is obtained. Finally, a simulation is given to illustrate the effectiveness of the proposed method.     

Three-stage Kalman filter for state and fault estimation of linear stochastic systems with unknown inputs

The paper studies the problem of simultaneously estimating the state and the fault of linear stochastic discrete-time varying systems with unknown inputs. The fault and the unknown inputs affect both the state and the output. However, if the dynamical evolution models of the fault and the unknown inputs are available the filtering problem will be solved by the Optimal three-stage Kalman Filter (OThSKF). The OThSKF is obtained after decoupling the covariance matrices of the Augmented state Kalman Filter (ASKF) using a three-stage U–V transformation. Nevertheless, if the fault and the unknown inputs models are not perfectly known the Robust three-stage Kalman Filter (RThSKF) will be applied to give an unbiased minimum-variance estimation. Finally, a numerical example is given in order to illustrate the proposed filters.     

Controller design for discrete-time hybrid linear parameter-varying systems with semi-Markov mode switching

The paper is concerned with the stability and stabilization problems for a family of hybrid linear parameter-varying systems with stochastic mode switching. The switching phenomenon is modeled by a semi-Markov stochastic process which is more generalized than a Markov stochastic process. With the construction of a Lyapunov function that depends on both the parameter variation and operating mode, numerical testable stability and stabilization criteria are established in the sense of σ-error mean square stability with the aid of some mathematical techniques that can eliminate the terms containing products of matrices. To test the effectiveness of the designed stabilizing controller, we apply the developed theoretical results to a numerical example.     

Minimization of disturbances effects in time delay predictor-based sliding mode control systems

Stabilization problem for a linear plant with time delay control is considered. A new method of the sliding mode control design minimizing the effects of system disturbances is presented. It is based on a combination of the well-known predictor-based sliding mode control algorithm with the recently developed invariant ellipsoid method. The theoretical results are supported by numerical simulations.     

Further result on H∞ filter design for continuous-time Markovian jump systems with time-varying delay

This paper investigates the problem of _H∞$H_{\infty }$ filtering for Markovian jump linear systems with time-varying delay. The aim of this problem is to design an _H∞$H_{\infty }$ filter that ensures stochastic stability of the filtering error system and a prescribed L₂-induced gain from the noise signals to the estimation error, for all admissible uncertainties. For solving the problem, we transform the system under consideration into an interconnection system. Based on the system transformation and the stochastic scaled small gain theorem, stochastic stability of the original system is examined via the stochastic stability version of the bounded realness of the transformed forward system. The merit of the proposed approach lies in its reduced conservatism, which is made possible by a precise approximation of the time-varying delay and the stochastic scaled small gain theorem. The proposed _H∞$H_{\infty }$ filtering condition is demonstrated to be less conservative than most existing results. Moreover, the _H∞$H_{\infty }$ filter design condition is further presented via convex optimizations, whose effectiveness are also illustrated via numerical examples.     

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.     

Fuzzy sliding mode control design of Markovian jump systems with time-varying delay

This paper studies the robust stochastic stabilization problem for a class of fuzzy Markovian jump systems with time-varying delay and external disturbances via sliding mode control scheme. Based on the equivalent-input-disturbance (EID) approach, an online disturbance estimator is implemented to reject the unknown disturbance effect on the considered system. Specifically, to obtain exact EID estimation Luenberger fuzzy state observer and a low-pass filter incorporated to the closed-loop system. Moreover, novel fuzzy EID-based sliding mode control law is constructed to ensure the stability of the closed-loop system with satisfactory disturbance rejection performance. By employing Lyapunov stability theory and some integral inequalities, a new set of delay-dependent robust stability conditions is derived in terms of linear matrix inequalities (LMIs). The resulting LMI is used to find the gains of the state-feedback controller and the state observer a for the resulting closed-loop system. At last, numerical simulations based on the single-link arm robot model are provided to illustrate the proposed design technique.     

Existence, continuation, and uniqueness problems of stochastic impulsive systems with time delay

This paper studies stochastic impulsive systems with time delay, where the impulse times are state-dependent. Using Itô calculus, we develop the essential foundation of the theory of the mentioned system. In particular, we establish results on local and global existence, forward continuation, and uniqueness of adapted solutions.