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.     

Control Lyapunov function optimal sliding mode controllers for attitude tracking of spacecraft

The attitude tracking control problem for a rigid spacecraft using two optimal sliding mode control laws is addressed. Integral sliding mode (ISM) control is applied to combine the first-order sliding mode with optimal control and is applied to quaternion-based spacecraft attitude tracking maneuvres with external disturbances and an uncertainty inertia matrix. For the optimal control part the control Lyapunov function (CLF) approach is used to solve the infinite-time nonlinear optimal control problem, whereas the Lyapunov optimizing control (LOC) method is applied to solve the finite-time nonlinear optimal control problem. The second method of Lyapunov is used to show that tracking is achieved globally. An example of multiaxial attitude tracking maneuvres is presented and simulation results are included to demonstrate and verify the usefulness of the proposed controllers.     

Sliding mode regulator as solution to optimal control problem for non-linear polynomial systems

This paper presents the optimal control problem for a non-linear polynomial 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 polynomial-quadratic regulator does not lead to a causal solution and, therefore, fails. Performance of the obtained optimal controller is verified in the illustrative example against the conventional polynomial-quadratic regulator that is optimal for the quadratic Bolza-Meyer criterion. The simulation results confirm an advantage in favor of the designed sliding mode control.     

Optimal and robust control for linear state-delay systems

This paper presents the optimal regulator for a linear system with state delay and a quadratic criterion. The optimal regulator equations are obtained 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 demonstrating better performance of the obtained optimal regulator are included. The paper then presents a robustification algorithm for the obtained optimal regulator based on integral sliding mode compensation of disturbances. The general principles of the integral sliding mode compensator design are modified to yield the basic control algorithm oriented to time-delay systems, which is then applied to robustify the optimal regulator. As a result, the sliding mode compensating control leading to suppression of the disturbances from the initial time moment is designed. The obtained robust control algorithm is verified by simulations in the illustrative example.     

Fuzzy sliding mode control design for a class of disturbed systems

This paper discusses the problem of the fuzzy sliding mode control for a class of disturbed systems. First, a fuzzy auxiliary controller is constructed based on a feedback signal not only to estimate the unknown control term, but also participates in the sliding mode control due to the fuzzy rule employed. Then, we extend our theory into the cases, where some kind of system information can not be obtained, for better use of our theoretical results in real engineering. Finally, some typical numerical examples are included to demonstrate the effectiveness and advantage of the designed sliding mode controller.     

Tracking the singular arc of a continuous bioreactor using sliding mode control

A sliding mode controller is developed for optimal transient operation of a continuous bioreactor. The sliding mode is the singular arc from the solution of an optimal control problem. The proposed controller is applied through simulations to an anaerobic digester and its performance is evaluated in terms of optimality and robustness properties.     

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.     

Global fast terminal sliding mode controller for hydraulic turbine regulating system with actuator dead zone

This paper investigates the frequency change problem of hydraulic turbine regulating system based on terminal sliding mode control method. By introducing a novel terminal sliding mode surface, a global fast terminal sliding mode controller is designed for the closed loop. This controller eliminates the slow convergence problem which arises in the terminal sliding mode control when the error signal is not near the equilibrium. Meanwhile, following consideration of the error caused by the actuator dead zone, an adaptive RBF estimator based on sliding mode surface is proposed. Through the dead zone error estimation for feed-forward compensation, the composite terminal sliding mode controller has been verified to possess an excellent performance without sacrificing disturbance rejection robustness and stability. Simulations have been carried out to validate the superiority of our proposed methods in comparison with other two other kinds of sliding mode control methods and the commonly used PID and FOPID controller. It is shown that the simulation results are in good agreement with the theoretical analysis.     

History-dependent modified sliding mode interception strategies with maximal capture zone

In order to construct the guidance strategy in a realistic nonlinear noise-corrupted interception endgame against a maneuverable target, a linearized zero-sum differential game is considered. Assuming perfect information in this game, sufficient conditions are established, which guarantee that a continuous interception strategy with memory (history-dependent) has the maximal capture zone. Two examples of such a strategy are analyzed: a modified super-twisting second-order sliding mode control and a modified integral sliding mode control. Simulation results of the original nonlinear interception endgame demonstrate that these strategies considerably reduce the chattering created by the classical game optimal bang-bang strategy without deteriorating the homing performance.     

Model transformation based sliding mode control of discrete-time two-dimensional Fornasini–Marchesini systems

This paper investigates the problem of sliding mode control (SMC) for discrete-time two-dimensional (2-D) systems subject to external disturbances. Given a 2-D Fornasini–Marchesini (FM) local state space model, attention is focused on designing the 2-D sliding surface and sliding mode controller, which guarantees the resultant closed-loop system to be asymptotically stable. Particularly, this problem is solved using the model transformation based method. First of all, sufficient conditions are formulated for the existence of a linear sliding surface guaranteeing the asymptotic stability of the equivalent sliding mode dynamics. Based on this, a sliding mode controller is synthesized to ensure that the associated 2-D FM system satisfies the reaching condition. The efficiency of the proposed 2-D SMC law design is shown by a numerical example. This paper extends the idea of model transformation to the 2-D systems and solves the SMC problem of a more general 2-D model in FM type for the first time.     

Fractional power rate reaching law for augmented sliding mode performance

High-frequency control switching, chattering, limits the practical application of sliding mode controllers. This paper proposes an enhanced reaching law, viz., Fractional Power Rate Reaching Law (FPRRL), for the design of sliding mode controllers to mitigate the chattering problem. Controller gains are selected to accommodate variations in switching function dynamically to prevent over-actuation near the sliding surface, which is the primary reason for chattering. Chattering mitigation is achieved without compromising other attributes of sliding mode control. Performance enhancements in the attributes, viz., reaching time, robustness, and reduced chattering magnitude have been established through the methodical analysis of the new reaching law. The proposed control strategy is then bench-marked with the state of the art reaching law methods through simulations by considering a twin-rotor MIMO system control problem.     

Stabilization of discrete-time stochastic systems via sliding mode technique

This paper considers the problem of sliding mode control for discrete-time stochastic systems with parameter uncertainties and state-dependent noise perturbation. An integral-like sliding surface is chosen and a discrete-time sliding mode controller is designed. The key feature in this work is that both the reachability of the quasi-sliding mode and the stability of system states are simultaneously analyzed, due to the existence of state-dependent noise perturbation. By utilizing an Lyapunov function involving system states and sliding mode variables, the sufficient condition for reachability is obtained. Finally, numerical simulation results are provided.     

Second order sliding mode output feedback control with switching gains—Application to the control of a pneumatic actuator

This paper proposes a switching gains output feedback controller, which is a sample-based second order sliding mode one. First, constructive convergence conditions are established for the controller, in the unperturbed and perturbed cases. Then, a gain adjustment mechanism of the gain, based on the aforementioned convergence conditions, allows us to reduce gain magnitude, and then to reduce the chattering effect. This new algorithm is experimentally applied for the position control of an electropneumatic actuator.     

On homogeneity and its application in sliding mode control

The paper is reviewing the tools to handle high-order sliding mode design and robustness. The main ingredient is homogeneity which can be checked using an algebraic test and which helps us in obtaining one of the most desired properties in sliding mode control that is finite-time stability. This paper stresses some recently obtained results about homogeneity for differential inclusions and robustness with respect to perturbations in the context of input-to-state stability. Lastly within this framework, most of the popular high-order sliding mode control schemas are analysed.     

Observer-based sliding mode control for a class of discrete systems via delta operator approach

In this paper, an observer-based sliding mode control (SMC) problem is investigated for a class of uncertain delta operator systems with nonlinear exogenous disturbance. A novel robust stability condition is obtained for a sliding mode dynamics by using Lyapunov theory in delta domain. Based on a designed sliding mode observer, a sliding mode controller is synthesized by employing SMC theory combined with reaching law technique. The robust asymptotical stability problem is also discussed for the closed-loop system composed of the observer dynamics and the state estimation error dynamics. Furthermore, the reachability of sliding surfaces is also investigated in state-estimate space and estimation error space, respectively. Finally, a numerical example is given to illustrate the feasibility and effectiveness of the developed method.     

Asynchronous sliding mode control of semi-Markovian jump systems with state saturation

This paper deals with the sliding mode control problem for semi-Markovian jump systems with state saturation, in which the controller may not be synchronized with the considered systems. A mode-detector is introduced to estimate the unavailable system mode, based on which an asynchronous sliding mode controller is designed. Then, both the $\mu$-exponential mean-square stability and the reachability of sliding surface are analyzed. Furthermore, a solving algorithm is given to acquire the feasible controller gains. Finally, the proposed asynchronous sliding mode control approach under state-saturation is illustrated via simulation results.     

Flatness-based adaptive sliding mode tracking control for a quadrotor with disturbances

In this paper, a flatness-based adaptive sliding mode control strategy is presented to solve the trajectory tracking problem of a quadrotor. According to the differential flatness theory, the typical under-actuated quadrotor dynamics is transformed into a fully-actuated one. Based on this model, backstepping sliding mode controllers are designed to solve the trajectory tracking problem. To improve the robustness to disturbances, extended state observers are applied as a feedforward compensation of disturbances. Moreover, considering the high-order dynamics and possible instability caused by large observer gains, the adaptive method is applied to compensate for the estimation error. The effectiveness of the proposed control scheme is verified in simulations.