Advances in Variable Structure Systems and Sliding Mode Control—Theory and Applications

We provide a Lyapunov-based design of homogeneous High-Order Sliding Mode (HOSM) Control and Observation (Differentiation) algorithms of arbitrary order for a class of Single-Input-Single-Output uncertain nonlinear systems. First, we recall the standard problem of HOSM control, which corresponds to the design of a state feedback control and an observer for a particular Differential Inclusion (DI), which represents a family of dynamic systems, including bounded matched perturbations/uncertainties. Next we provide a large family of zero-degree homogeneous discontinuous controllers solving the state feedback problem based on a family of explicit and smooth homogeneous Lyapunov functions. We show the formal relationship between the control laws and the Lyapunov functions. This also gives a method for the calculation of controller gains ensuring the robust and finite time stability of the sliding set. The required unmeasured states can be estimated robustly and in finite time by means of an observer or differentiator, originally proposed by A. Levant. We give explicit and smooth Lyapunov functions for the design of gains ensuring the convergence of the estimated states to the actual ones in finite time, despite the non vanishing bounded perturbations or uncertainties acting on the system. Finally, it is shown that a kind of separation principle is valid for the interconnection of the HOSM controller and observer, and we illustrate the results by means of a simulation on an electro-mechanical system.

The chapter proposes constructive conditions for verifying the input-to-state stability property of discontinuous systems using geometric homogeneity. Two sets of conditions are developed for a class of homogeneous and homogenizable systems described by differential inclusions. The advantage of the proposed conditions is that they are not based on the Lyapunov function method, but more related to algebraic operations over the right-hand side of the system.

In this chapter we show that the Sliding Mode Control (SMC) technique can be successfully applied to stochastic systems governed by the stochastic differential equations of the Itô type which contain additive as well as multiplicative stochastic unbounded white noise perturbations. The existence of a strong solution to the corresponding stochastic differential inclusion is discussed. To do this approach workable the gain control parameter is suggested to be done state-dependent on norms of system states. It is demonstrated that under such modification of the conventional SMC we can guarantee the exponential convergence of the averaged squared norm of the sliding variable to a zone (around the sliding surface) which is proportional to the diffusion parameter in the model description and inversely depending on the gain parameter. Then the behavior of a standard super-twist controller under stochastic perturbations is studied. The suggested analysis is based on the Lyapunov functions, designed for the stability analysis of the deterministic version of super-twist controllers. The major finding is that under stochastic (in fact, unbounded) perturbations the special selection of a gain-parameter of such controller, making it depending on this Lyapunov function and its gradient, provides the controller with an adaptivity property and guarantees the means square convergence of this function into the prespecified zone around the origin. Finally, The chapter deals with the problem of state estimation of “two-component” systems where the second vector - component may have unknown nonlinear Lipschitz-type dynamics and is subjected to stochastic perturbations of both additive and multiplicative types. Both vector components governed by a system of stochastic differential equations with state dependent diffusion. The system is supposed to be quadratically stable in the mean-squared sense. We consider a sliding mode observer with the gain parameter linearly depending on the norm of the output estimation error which is available during the process. It has the same structure as deterministic observer based on “the Equivalent Control Method”. The workability of the suggested observer is guaranteed for the group of trajectories with the probabilistic measure closed to one. All theoretical results are supported by numerical simulations.

A new concept of chattering characterization for the systems driven by finite-time convergent controllers (FTCC) in terms of practical stability margins is presented. Unmodeled dynamics of order two or more incite chattering in FTCC driven systems. In order to analyze the FTCC robustness to unmodeled dynamics the novel paradigm of Tolerance Limits (TL) is introduced to characterize the acceptable emerging chattering. Following this paradigm a new notions of Practical Stability Phase Margin (PSPM) and Practical Stability Gain Margin (PSGM) as a measure of robustness to cascade unmodeled dynamics is introduced. Specifically, PSPM and PSGM are defined as the values that have to be added to the phase and gain of dynamically perturbed system driven by FTCC so that the characteristics of the emerging chattering reach TL. For practical calculation of PSPM and PSGM, the Harmonic Balance (HB) method is employed and a numerical algorithm to compute Describing Functions (DFs) for families of FTCC (specifically, for nested, and quasi-continuous Higher Order Sliding Mode (HOSM) controllers) was proposed. A database of adequate DFs was developed. A numerical algorithm for solving HB equation using the Newton–Raphson method was suggested to obtain predicted chattering parameters. Finally, computational algorithms that identify PSPM and PSGM for the systems driven by FTCC were proposed. The algorithm of a cascade linear compensator design that corrected the FTCC, making the values of PSPM and PSGM to fit the prescribed quantities, was presented. In order to design the flight-certified FTCC for attitude for the F-16 jet fighter, the proposed technique was employed in a case study. The prescribed robustness to cascade unmodeled dynamics was achieved.

In this book chapter, analysis of first-order and second-order sliding-mode (SM) control systems from the perspective of the gain margins of the averaged dynamics is done. These averaged dynamics result from the fact of the existence of chattering, so that averaging on the period of chattering can be considered. Analysis is done via the describing function method and the locus of a perturbed relay system method. First-order SM, hysteresis relay control, twisting algorithm and sub-optimal algorithm are compared in terms of the gain margins of the averaged dynamics that can describe the propagation of external signals through a SM control system.

In this chapter, we propose an adaptive sliding mode approach based on the extended equivalent control concept to cope with disturbances of unknown bounds in nonlinear systems. Some advantages with respect to previous proposed methods are its simplicity and capability of rejecting non smooth disturbances. Unlike other adaptive approaches, overestimation of the controller gain and the loss of the sliding motion can be avoided. The developed method guarantees ideal sliding modes and alleviates the chattering phenomena. Theoretically, the sliding variable becomes identically null after some finite time in spite the disturbances. Global stability is proved. Simulation results illustrate the potential and limitations of this novel adaptation strategy.

A fundamental asset of sliding-mode control is their robustness against unstructured uncertainties. For a successful controller design very little information about the nature of the uncertainty is required. However, in practical control problems often some information about the uncertainty is available. In this case, indirect adaptive control methods may exploit the available information about the uncertainty structure, resulting in powerful adaptation schemes. In this contribution we present a design method that combines these two control concepts. The link is given by the certainty-equivalence approach that relies on the availability of a Lyapunov function for the stabilizing control law. We analyze how various choices of Lyapunov functions impact on our proposed design method. To this end, we briefly review classes of uncertainties that may be handled by various sliding-mode design algorithms. An example serves to illustrate that the violation of these assumptions may yield an unstable closed loop system. We provide an experimental case study that demonstrates the efficiency of the proposed control concept when compared with the conventional super-twisting algorithm.

In this chapter, a variable structure observer is designed for a class of nonlinear large-scale interconnected systems in the presence of uncertainties and nonlinear interconnections. The modern geometric approach is used to explore system structure and a transformation is employed to facilitate the observer design. Based on the Lyapunov direct method, a set of conditions are developed such that the proposed variable structure systems can be used to estimate the states of the original interconnected systems asymptotically. The internal dynamical structure of the isolated nominal subsystems as well as the structure of the uncertainties are employed to reduce conservatism. The bounds on the uncertainties are nonlinear and are employed in the observer design to enhance robustness. A numerical example is presented to illustrate the results and simulation studies show that the proposed approach is effective.

A unilaterally constrained perturbed double integrator system is studied in this work. The aim is to establish uniform finite time stability of the non-linear dynamics in the presence of impacts due to the constraints on the position variable. A non-smooth transformation is utilized to first transform the system into a variable structure system that can be studied within the framework of a conventional discontinuous paradigm. Then, a finite time stable continuous controller is used and stability of the closed-loop dynamics is proven by identifying a new set of Lyapunov functions. The results enable continuous and discontinuous cases to be unified using one parameter that defines the set of Lyapunov functions for each case.

In this chapter we show how the Integral Sliding-Mode Control design paradigm can be usefully applied in the framework of Multi-Agent Systems to allow the agents dynamics to be affected by unknown disturbances. Existing consensus-based algorithms for the distributed estimation of pre-specified quantities such as, e.g., the average or the median value of the agents initial conditions fail to converge when disturbances affect the agents dynamics. In the present chapter, is discussed how to redesign the original “non-robust” algorithms from an integral sliding mode perspective, such that restoration on the ideal unperturbed scenario (e.g., convergence to the average or median value) is guaranteed in spite of the unknown perturbations. The theoretical results are fully derived within a Lyapunov analysis approach. Finally, to corroborate the developed approaches, simulative results are also presented and discussed.

In this chapter, the finite-time output consensus problem is studied for a class of leader-follower higher-order multi-agent systems, which are subject to both unmeasured states and mismatched disturbances. Moreover, the disturbances are allowed to be fast time-varying types. By integrating a finite-time observer technique and the nonsingular terminal sliding-mode control technique together, a feedforward-feedback composite consensus control scheme is proposed and it solves the finite-time output consensus problem. Firstly, an observer is developed for each follower to get precise information of the unmeasured states and disturbances in finite time. Secondly, by distributedly utilizing the state and disturbance estimates, integral-type terminal sliding-mode surfaces are designed for the followers. Finally, on the basis of these surfaces, sliding-mode protocols are derived. The designed protocols achieve finite-time output consensus of all the leader-follower agents. Simulations substantiate the efficacy of the proposed composite consensus control scheme.

Event-triggered control is a novel control implementation strategy where control signal is updated in aperiodic manner such that the stability of the system is retained. Here, event condition is continuously monitored to generate the triggering instant. So, this minimizes resource utilization and control effort while achieving certain control objective. Recently, event-triggered sliding mode control (SMC) is proposed in [25, 31] to ensure the robust stability in the presence of disturbance. In this strategy also, the plant state is continuously monitored for generating possible triggering instant. In order to avoid this continuous measurement, we propose an event-triggering strategy which evaluates the event at periodic interval only known as discrete event-triggered control. This strategy is very appealing if the state measurements are available only at periodic intervals. Here, with this discrete state measurements, the discrete event-triggered SMC is designed to guarantee the system stability. The event is detected only at periodic intervals and the control signal is updated whenever it is violated at these periodic instants. So, there is always a guaranteed lower bound for inter event time which is the time between two sampled measurements. Simulation results are given to demonstrate the efficacy of the proposed technique.

The chapter considers so-called integral sliding modes (ISM) and how they can be employed in the context of fault tolerant control. Two distinct classes of problems are considered: firstly a fault tolerant ISM controller is designed for an over-actuated linear system; secondly an ISM scheme is retrofitted to an existing feedback control scheme for an over-actuated uncertain linear system with the objective of retaining the pre-existing nominal performance in the face of faults and failures. Aerospace examples are used throughout to demonstrate the theoretical developments based on simulations. The chapter concludes with a case study which describes the implementation of an LPV extension of one of the ISM schemes for fault tolerant control on a motion simulator configured to represent a Boeing 747 aircraft subject to several realistic fault scenarios.

The induction motor (IM) is one of the most common electrical motor used in most applications. This motor runs at a speed less than its synchronous speed, therefore it is also called as asynchronous motor. The synchronous speed is the speed of rotation of the magnetic field in a rotary machine and it depends upon the frequency and number poles of the IM. The IM has been extensively used in many practical applications due to its simply construction, lower repair and maintenance costs, high reliability and relatively low manufacturing cost. With the development of power electronics, electrical technique and control theories, IMs have been able to be used in high-performance servo systems, such as speed servo systems, even position servo systems. This chapter describes the speed control method and technique of IM servo drives using terminal sliding-mode controller.

Sliding Mode Control is a nonlinear control methodology based on the use of a discontinuous control input which forces the controlled system to switch from one continuous structure to another, evolving as a variable structure system. This structure variation makes the system state reach in a finite time a pre-specified subspace of the system state space where the desired dynamical properties are assigned to the controlled system. In the past years, an extensive literature has been devoted to the developments of Sliding Mode Control theory. This kind of methodology offers a number of benefits, the major of which is its robustness versus a significant class of uncertainties and disturbances. Yet, it presents a crucial drawback, the so-called chattering phenomenon, which may disrupt or damage the actuators and induce unacceptable vibrations throughout the controlled system, limiting the practical applicability of the methodology, especially in case of mechanical or electromechanical plants. This drawback has been better studied recently. Theoretical developments, oriented to increase the order of the sliding mode, thus producing efficient Second Order and Higher Order Sliding Mode Control algorithms, may be useful to attenuate the drawbacks caused by the use of a discontinuous control. Then, Sliding Mode Control can be profitably used to efficiently solve automotive control and observation problems, as testified by several recent publications and research projects. The aim of this chapter is to provide an overview of available examples of application of sliding mode control to the automotive field, focusing on recent developments at the University of Pavia.

This chapter deals with the development of a hysteresis band controller in charge of fixing the switching frequency of a sliding mode controller. The proposed control measures the switching period of the control signal and modifies the hysteresis band of the comparator in order to regulate the switching frequency of the sliding motion. The switching frequency control system is modelled and a design criterion for the control parameters is derived to guarantee closed loop stability. The technique is applied to power electronics. Specifically, DC-to-DC and DC-to-AC power converters are considered, a sliding mode controller providing robustness in the face of load disturbances and input voltage variations is designed and, finally, the stability analysis of these systems when the switching frequency is regulated is also presented. A set of numerical simulations will confirm the achievement of fixed switching frequency, robustness of the controller, and stability of the closed-loop system.