Discrete time second order sliding mode observer for uncertain linear multi-output system

This paper presents a second order sliding mode observer (SOSMO) design for discrete time uncertain linear multi-output system. The design procedure is effective for both matched and unmatched bounded uncertainties and/or disturbances. A second order sliding function and corresponding sliding manifold for discrete time system are defined similar to the lines of continuous time counterpart. A boundary layer concept is employed to avoid switching across the defined sliding manifold and the sliding trajectory is confined to a boundary layer once it converges to it. The condition for existence of convergent quasi-sliding mode (QSM) is derived. The observer estimation errors satisfying given stability conditions converge to an ultimate finite bound (within the specified boundary layer) with thickness O(T²)$O\left(T^2\right)$ where T is the sampling period. A relation between sliding mode gain and boundary layer is established for the existence of second order discrete sliding motion. The design strategy is very simple to apply and is demonstrated for three examples with different class of disturbances (matched and unmatched) to show the effectiveness of the design. Simulation results to show the robustness with respect to the measurement noise are given for SOSMO and the performance is compared with pseudo-linear Kalman filter (PLKF).     

Fractional-order adaptive neuro-fuzzy sliding mode H∞ control for fuzzy singularly perturbed systems

This paper investigates the fractional-order (FO) adaptive neuro-fuzzy sliding mode control issue for a class of fuzzy singularly perturbed systems subject to the matched uncertainties and external disturbances. Firstly, a novel FO fuzzy sliding mode surface is presented. Secondly, by introducing an appropriate ε-dependent Lyapunov function, some H_∞ performance analysis criteria are given, which also ensure the robust stability of the sliding mode dynamics. Furthermore, a hybrid neuro-fuzzy network system (HNFNS) is introduced to estimate the matched uncertainty. Moreover, an FO adaptive fuzzy sliding mode controller is designed to drive the state trajectories of fuzzy singularly perturbed systems to the predefined FO sliding mode surface within a finite-time. Finally, two verification examples are presented to illustrate the validity of the proposed FO control scheme.     

Adaptive SMC based on the dynamic containment of the sliding variable

Conventional Sliding Mode Controllers (SMCs) exhibit a robust performance against matched bounded uncertainties and disturbances by containing them under a fixed controller’s effort. Consequently, the controller is commonly found excessive, leading to chattering and straining the actuator. As a solution, the variable-gain SMCs adapt to the instantaneous system requirements, thus attenuating the aforesaid effects and keeping the SMC’s benefits. However, the reported adaptive laws underlying such behavior commonly require arbitrary design considerations and do not consider practical implementation. Unlikely, in this work, a hysteresis-based adaptability law to drive the sliding variable to a boundary layer around zero is proposed. The sliding boundary—hysteresis’ width—will consistently “bounce” over the sliding variable, trying to shrink against it while preserving the sliding mode. This behavior finds its steady-state once the sliding variable and the sliding boundary’s dynamics are synchronized, with no need of subjective or arbitrary adjustments. The close-loop tuning can be derived from the system’s parameters alone, and its steady-state performance can be quantitatively predicted. Furthermore, a method to adjust the sliding surface parameters according to the system’s desired behavior is provided, all in a closed, analytical way. Finally, the physical actuator limits are taken into account and never exceeded, and the discrete nature of the devices normally used for SMC implementation is incorporated throughout. Two examples are studied to portray the proposal’s advantages.     

Nested backward compensation of unmatched perturbations via HOSM observation

This paper tackles the compensation problem of linear time invariant systems affected by unmatched perturbations. The proposed methodology exploits a high order sliding mode observer, guaranteeing theoretically exact state and perturbation estimation. A compensation based strategy is proposed to cope with the unmatched perturbations. The compensation of the desired coordinate is carried through a nested backward sliding surface design, which compensates some of the non-actuated state components, while the remaining states are maintained to be bounded. The feasibility of the technique was tested in an active suspension vehicle system.¹     

Compound tracking control based on MPC for quadrotors with disturbances

In this paper, a compound control strategy is proposed to realize the trajectory tracking task of quadrotors under operating constraints and disturbances. Disturbances caused by model uncertainties, environmental noises, and measurement disturbances are divided into matched disturbances and unmatched ones, which are compensated and suppressed separately by using two control components. The integral sliding mode control component is designed to actively reject the matched disturbances, and the control system is then transformed into an equivalent control system subject to equivalent disturbances only related to the unmatched disturbances. The remaining equivalent disturbances are treated by a robust model predictive control component based on the idea of constraints tightening, which minimizes the tracking error in an optimization framework and takes both state and input constraints into account explicitly. The derived compound control strategy is based on these two control components. Conditions are provided to guarantee the robust constraint satisfaction, recursive feasibility and closed-loop stability of the tracking error system. An illustrative example on the quadrotors shows the efficiency and robustness of this compound tracking control algorithm.     

Robust sliding mode control for stochastic singular Markovian jump systems with unmatched one-sided Lipschitz nonlinearities

This paper investigates sliding mode control of stochastic singular Markovian jump systems with nonlinearity. The unmatched nonlinearity satisfies one-sided Lipschitz condition and quadratically inner-boundedness. In term of a new technical variable transformation, sufficient conditions are developed for nonlinear stochastic singular Markovian jump systems constrained on sliding manifold to guarantee stochastic admissibility and uniqueness of solution based on implicit function theorem. The sliding mode control law by which the trajectories of system can be compelled to the predefined sliding surface in finite time no matter what initial state value is, is synthesized. The derivative singular matrix is fully considered in the whole design process such that the derived conditions can be checked easily.The technical treatment of the nonlinear matrix term avoids the classification discussion of sliding mode controller design. Convex optimization problems subject to linear matrix inequalities are formulated to optimize the desired indexes of interest. Finally, the effectiveness of the proposed approach is illustrated by a numerical example and a practical example.     

Super twisting control of a parametrically excited overhead crane

The principal contribution of this note is to avoid the parametric resonance effect and attenuate the payload oscillations in an overhead-crane system subject to periodic variations in the base support. Considering an appropriate sliding output with relative degree one, we present the sliding mode control design based on the Super-Twisting Algorithm (STA), ensuring finite time convergence to the desired sliding surface for the linear periodic system with Lipschitz continuous matched and unmatched uncertainties bounded together by their gradients by known functions. The suggested approach also provides chattering phenomenon attenuation. Obtained results are verified experimentally.     

Harmonic oscillator utilizing double-fold integral, traditional and second-order sliding mode control

To stabilize both amplitude and frequency of the second-order harmonic oscillator double-fold sliding mode control is employed. The first, integral sliding mode control, is used to compensate for the disturbance/uncertainty, which is unmatched by the second control. The second sliding mode control is designed to achieve the stabilization of the harmonic oscillator system while the system is in the integral sliding mode. The first (integral) and second sliding mode controls are implemented in both formats: traditional sliding mode control that requires high-frequency oscillating control action and second-order sliding mode (super-twisting) control that is continuous and provides for the higher accuracy of stabilization. It is shown that the output of the double-fold sliding mode controlled second-order harmonic oscillator is robust to bounded disturbances and model parameter uncertainties. Computer simulations are performed to manifest the theoretical analysis.     

Design of sliding surface for mismatched uncertain systems to achieve asymptotical stability

The design of an adaptive sliding mode control (SMC) scheme is proposed in this paper for stabilizing a class of dynamic systems with matched and mismatched perturbations. Two methods for designing a novel sliding surface function are introduced first. By utilizing a pseudocontrol input in the sliding surface function, one cannot only suppress the mismatched perturbations in the sliding mode, but also obtain the property of asymptotical stability. Then a sliding mode controller is designed to drive the controlled systems to the designated sliding surface in a finite time. Adaptive mechanism is also embedded in the controller as well as in the sliding surface function designed from the second method to overcome the perturbations, so that the informations of upper bound of perturbations are not required. An application of flight control and experimental results of controlling a servomotor are also given for demonstrating the applicability of the proposed control scheme.     

Sliding mode control for uncertain discrete-time systems with Markovian jumping parameters and mixed delays

This paper is concerned with the robust sliding mode control (SMC) problem for a class of uncertain discrete-time Markovian jump systems with mixed delays. The mixed delays consist of both the discrete time-varying delays and the infinite distributed delays. The purpose of the addressed problem is to design a sliding mode controller such that, in the simultaneous presence of parameter uncertainties, Markovian jumping parameters and mixed time-delays, the state trajectories are driven onto the pre-defined sliding surface and the resulting sliding mode dynamics is stochastically stable in the mean-square sense. A discrete-time sliding surface is firstly constructed and an SMC law is synthesized to ensure the reaching condition. Moreover, by constructing a new Lyapunov–Krasovskii functional and employing the delay-fractioning approach, a sufficient condition is established to guarantee the stochastic stability of the sliding mode dynamics. Such a condition is characterized in terms of a set of matrix inequalities that can be easily solved by using the semi-definite programming method. A simulation example is given to illustrate the effectiveness and feasibility of the proposed design scheme.     

Consensus control for a class of linear multiagent systems using a distributed integral sliding mode strategy

In this paper, a consensus framework is proposed for a class of linear multiagent systems subject to matched and unmatched uncertainties in an undirected topology. A linear coordinate transformation is derived so that the consensus protocol design can be conveniently performed. The distributed consensus protocol is developed by using an integral sliding mode strategy. Consensus is achieved asymptotically and all subsystem states are bounded. By using an integral sliding mode control, the subsystems lie on the sliding surface from the initial time, which avoids any sensitivity to uncertainties during the reaching phase. By use of an appropriate projection matrix, the size of the equivalent control required to maintain sliding is reduced which reduces the conservatism of the design. MATLAB simulations validate the effectiveness and superiority of the proposed method.     

Observer-based discrete-time sliding mode control for systems with unmatched uncertainties

This paper addresses an observer-based sliding mode control (SMC) approach for discrete-time systems with unmatched uncertainties. A modified sliding surface based on disturbance estimation and a sliding mode controller are designed to counteract with the unmatched disturbance. The proposed method exhibits the following three features. First, the hyperplane matrix is designed in a simple way based on the discrete-time Riccati equation. Second, a chattering-free SMC method is utilized. Third, the proposed approach retains the nominal performance of the system. The stability of the overall system is achieved and simulation results are presented to verify the effectiveness of the proposed method.     

Fixed-time nonlinear homogeneous sliding mode approach for robust tracking control of multirotor aircraft: Experimental validation

This paper presents a robust scheme for fixed-time tracking control of a multirotor system. The aircraft is subjected to matched lumped disturbances, i.e., unmodeled dynamics, parameters uncertainties, and external perturbations besides measurement noise. Firstly, a novel Nonlinear Homogeneous Continuous Terminal Sliding Manifold (NHCTSM) based on the weighted homogeneity theory is presented. The sliding manifold is designed with prescribed dynamics featuring Global Asymptotic Stability (GAS) and fixed-time convergence. Then, a novel Fixed-time Non-switching Homogeneous Nonsingular Terminal Sliding Mode Control (FNHNTSMC) is proposed for the position and attitude loops by employing the developed NHCTSM and an appropriate reaching law. Moreover, the control framework incorporates a disturbance observer to feedforward and compensate for the disturbances. The designed control scheme can drive the states of the system to the desired references in fixed-time irrespective of the values of the Initial Conditions (ICs). Since the existing works on homogeneous controllers rely on the bi-limit homogeneity concept in the convergence proofs, the estimate of the settling-time or its upper-bound cannot be given explicitly. In contrast, this study employs Lyapunov Quadratic Function (LQF) and Algebraic Lyapunov Equation (ALE) in the stability analysis of both controller and observer. Following this method, an expression of the upper-bound of the settling-time is explicitly derived. Furthermore, to assure the Uniform Ultimate Boundedness (UUB) of all signals in the feedback system, the dynamics of the observer and controller are jointly analyzed. Simulations and experiments are conducted to quantify the control performance. The proposed approach achieves superior performance compared with recent literature on fixed-time/finite-time control and a commercially available PID controller. The comparative results witness that the developed control scheme improves the convergence-time, accuracy, and robustness while overcoming the singularity issue and mitigating the chattering effect of conventional SMC.     

Adaptive sliding mode tracking control for a flexible air-breathing hypersonic vehicle

This paper is concerned with the adaptive sliding mode control (ASMC) design problem for a flexible air-breathing hypersonic vehicle (FAHV). This problem is challenging because of the inherent couplings between the propulsion system, the airframe dynamics and the presence of strong flexibility effects. Due to the enormous complexity of the vehicle dynamics, only the longitudinal model is adopted for control design in the present paper. A linearized model is established around a trim point for a nonlinear, dynamically coupled simulation model of the FAHV, then a reference model is designed and a tracking error model is proposed with the aim of the ASMC problem. There exist the parameter uncertainties and external disturbance in the model, which are not necessary to satisfy the so-called matched condition. A robust sliding surface is designed, and then an adaptive sliding mode controller is designed based on the tracking error model. The proposed controller can drive the error dynamics onto the predefined sliding surface in a finite time, and guarantees the property of asymptotical stability without the information of upper bound of uncertainties as well as perturbations. Finally, simulations are given to show the effectiveness of the proposed control methods.     

Coordinated decentralized sliding mode MRAC with control cost optimization for a class of nonlinear systems

In this paper, we develop an approach for solving the problem of sliding mode decentralized adaptive state-feedback tracking with continuous control actions for a class of uncertain nonlinear dynamical systems. In addition to the traditional asymptotic zero error tracking specification in the sliding mode decentralized model reference adaptive control (MRAC) problem formulation, here an additional requirement is specified explicitly in the problem statement. The tracking objective is described by a set of admissible reference trajectories, called a performance tube. The input signal to the reference model, selected within specified bounds, is used as a design parameter. The best reference trajectory is found by solving an additional optimization problem whose criterion penalizes the variance of the control signal.     

Launch vehicle attitude control using sliding mode control and observation techniques

In determining flight controls for launch vehicle systems, several uncertain factors must be taken into account, including a variety of payloads, a wide range of flight conditions and different mission profiles, wind disturbances and plant uncertainties. Crewed vehicles must adhere to human rating requirements, which limit the angular rates. Sliding mode control algorithms that are inherently robust to external disturbances and plant uncertainties are very good candidates for improving the robustness and accuracy of the flight control systems. Recently emerging Higher Order Sliding Mode (HOSM) control is even more powerful than the classical Sliding Mode Controls (SMC), including the capability to handle systems with arbitrary relative degree. This paper proposes sliding mode launch vehicle flight controls using classical SMC driven by the sliding mode disturbance observer (SMDO) and higher-order multiple and single loop designs. A case study on the SLV-X Launch Vehicle studied under a joint DARPA/Air Force program called the Force Application and Launch from CONtinental United States (FALCON) program is shown. The intensive simulations demonstrate efficacy of the proposed HOSM and SMC-SMDO control algorithms for launch vehicle attitude control.     

Novel integral sliding mode control for small-scale unmanned helicopters

Integral sliding mode (ISM) control which consists of a nominal control and a sliding-mode motion control, provides a nice framework for high tracking performance and good disturbance reduction. Our work develops ISM to attenuate the adverse effect of mismatched perturbations. By properly choosing sliding-manifold surface, the elimination of disturbances on control outputs enables to be achieved. Additionally, the chattering of sliding-mode control part is attenuated based on second-order sliding mode idea. Then, the proposed novel ISM control scheme is applied to address trajectory tracking problem for helicopters under perturbations. Approximated input-output linearization is implemented, such that the obtained linearized model is suitable for applying the proposed ism control. The stability of the closed-loop system for helicopter and its convergence to zeros of tracking errors are demonstrated by Lyapunov theory analysis. Several comparison simulations illustrate the effectiveness and superiority of the proposed methods.     

Dynamic output feedback sliding mode control for spacecraft hovering without velocity measurements

In consideration of target angular velocity uncertainty and external disturbance, a modified dynamic output feedback sliding mode control (DOFSMC) method is proposed for spacecraft autonomous hovering system without velocity measurements. As a stepping-stone, an additional dynamic compensator is introduced into the design of sliding surface, then an augmented system is reconstructed with the system uncertainty and external disturbance. Based on the linear matrix inequality (LMI), a sufficient condition is given, which guarantees the disturbance attenuation performance of sliding mode dynamics. By introducing an auxiliary variable, a modified version of adaptive sliding mode control (ASMC) law is designed, and the finite-time stability of sliding variable is established by the Lyapunov stability theory. Compared with other results, the proposed method is less conservative and can decrease the generated control input force significantly. Finally, two simulation examples are performed to validate the effectiveness of the proposed method.     

Robust finite-time sliding mode control for discrete-time singular system with time-varying delays

This paper explores the finite-time bounded issue for discrete-time singular time-varying delay system via sliding mode control method. A suitable discrete-time sliding mode control law is constructed to drive the state trajectories onto the specified sliding surface in a given finite time interval. Meanwhile, sufficient conditions for finite-time bounded to the closed-loop delayed system are provided in both reaching phase and sliding motion phase. In addition, the finite-time sliding mode controller gain matrix can be solved by using the linear matrix inequalities approach. Finally, three numerical examples are illustrated to demonstrate the superiority and practicability of presented results.     

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.