Sliding Modes after the First Decade of the 21st Century

The objective of this chapter is to try to analyze the main stages in the development of sliding mode enforcing control algorithms, starting from the first Variable Structure Systems workshop (VSS90). I would like to underline that this is my personal opinion, I am just trying to understand the steps we have made as a community during the last twenty years after VSS90 as well as which problems still remain open. Of course, generally I will concentrate the chapter on results in open problems I have discovered working with my group and coauthors.

Establishing and exactly keeping constraints of high relative degrees is a central problem of the modern sliding-mode control. Its solution in finite-time is based on so-called high-order sliding modes, and is reduced to finite-time stabilization of an auxiliary uncertain system. Such stabilization is mostly based on the homogeneity approach. Robust exact differentiators are also developed in this way and are used to produce robust output-feedback controllers. The resulting controllers feature high accuracy in the presence of sampling noises and delays, ultimate robustness to the presence of unaccounted-for fast stable dynamics of actuators and sensors, and to small model uncertainties affecting the relative degrees. The dangerous types of the chattering effect are removed artificially increasing the relative degree. Parameters of the controllers and differentiators can be adjusted to provide for the needed convergence rate, and can be also adapted in real time. Simulation results and applications are presented in the fields of control, signal and image processing.

Sliding mode control has become a mature theory and found a number of useful applications. The theory of sliding mode control is based on mostly state space models and Lyapunov approach to analysis of the convergence of the system states to the sliding surface. This approach often limits the analysis to the second-order systems. Frequency-domain methods could potentially overcome the above-mentioned limitation of the state-space approach. Yet they find limited applications in sliding mode control theory. The present article is aimed at giving an overview of some available and emerging frequency domain methods of analysis of systems having conventional and second-order sliding modes. The method of analysis of transient oscillations is given in detail. A frequency-domain criterion of finite-time convergence is presented.

A new sliding mode control technique for a class of SISO dynamic systems is presented in this chapter. It is seen that the stability status of the closed-loop system is first checked, based on the approximation of the most recent information of the first-order derivative of the Lyapunov function of the closed-loop system, an intelligent sliding mode controller can then be designed with the following intelligent features: (i) If the closed-loop system is stable, the correction term in the controller will continuously adjust control signal to drive the closed-loop trajectory to reach the sliding mode surface in a finite time and the desired closed-loop dynamics with the zero-error convergence can then be achieved on the sliding mode surface. (ii) If, however, the closed-loop system is unstable, the correction term is capable of modifying the control signal to continuously reduce the value of the derivative of the Lyapunov function from the positive to the negative and then drives the closed-loop trajectory to reach the sliding mode surface and ensures that the desired closed-loop dynamics can be obtained on the sliding mode surface. The main advantages of this new sliding mode control technique over the conventional one are that no chattering occurs in the sliding mode control system because of the recursive learning control structure; the system uncertainties are embedded in the Lipschitz-like condition and thus, no priori information on the upper and/or the lower bounds of the unknown system parameters and uncertain system dynamics is required for the controller design; the zero-error convergence can be achieved after the closed-loop dynamics reaches the sliding mode surface and remains on it. The performance for controlling a third-order linear system is evaluated in the simulation section to show the effectiveness and efficiency of the new sliding mode control technique.

This chapter outlines some results concerning the application of second-order sliding-mode techniques in the framework of control and estimation problems for some classes of fractional-order systems (FOS). Concerning the control problems, a second-order sliding mode control approach is developed to stabilize a class of linear uncertain multivariable fractional-order dynamics. Concerning estimation and observation problems, two main results are illustrated. A method for reconstructing in finite time an external disturbance acting on a known FOS is presented, and, as a second instance, a method for estimating the discrete state of a switched FOS is discussed. Both the schemes make use of second-order sliding mode observers. The method for discrete state reconstruction in switched FOS find useful application in the framework of fault detection, as shown in the experimental section part. Key point of all the approaches herein presented is the use of fractional-order sliding manifolds. Simple controller/observer tuning formulas are constructively developed along the paper by Lyapunov analysis. Simulation and experimental results confirm the expected performance.

This chapter is devoted to a discussion about the relations between first and high order sliding mode algorithms and both types of Zeno (Chattering and Genuinely) behaviors of switched dynamical systems. Firstly, the Henstock-Kurzweil integral is recalled in order to set up the problem of switched systems with Zeno phenomena, which enables to include Filippov solution and take into account some singularities. Then, observer designs based on the well-known super twisting algorithm are proposed. For this kind of problems, the importance of finite time convergence of the observation error is studied, and some simulations are given to highlight the discussion. Lastly, the two tanks example is given in order to point out the differences between both Zeno phenomena types, to show that there is life after Zeno and that a higher order sliding mode observer can be efficient before, during and after both Zeno phenomena types.

This chapter considers the development of sliding mode control strategies for linear, time delay systems with bounded disturbances that are not necessarily matched. The emphasis is on the development of frameworks that are constructive and applicable to real problems. For many systems it may not be practical to measure all the system states and therefore a static output feedback sliding mode control design paradigm is considered. The novel feature of the method is that Linear Matrix Inequalities (LMIs) are derived to compute solutions to both the existence problem and the finite time reachability problem that minimize the ultimate bound of the reduced-order sliding mode dynamics in the presence of time varying delay and unmatched disturbances. The methodology is therefore constructive and provides guarantees on the level of closed-loop performance that will be achieved by uncertain systems which experience delay. An uncertain model with both matched and unmatched disturbances from the literature provides a tutorial example of the proposed method. A case study involving the practical application of the design methodology in the area of liquid monopropellant rocket motor control is also presented.

In this chapter, we propose a new feedback controller design approach for the sliding mode control of a large class of linear switched systems. The method is devoid of state measurements, and it efficiently extends the sliding mode control methodology to traditional input-output descriptions of the plant. The approach is based on regarding the average Generalized Proportional Integral (GPI) output feedback controller design as a guide for defining the sliding mode features. Throughout, it is assumed that the available output signal coincides with the system’s flat output, an output capable of completely differentially parameterizing all the variables in the system (inputs, original outputs and state variables) and exhibits no zero dynamics. Encouraging simulation results are presented in connection with a tutorial example. Experimental results are also presented for the trajectory tracking problem on a popular DC-to-DC switched power converter of the “buck” type.

This chapter briefly describes the main results developed by the authors in the area of output feedback sliding mode control. For the sake of simplicity, the focus is maintained on uncertain single-input-single-output (SISO) nonlinear systems, although several results have been extended to the control of multi-input-multi-output (MIMO) systems. For the considered class of nonlinear systems, linear growth restriction on the unmeasured states is assumed, while less restrictive conditions are imposed to the growth of nonlinearities depending on the measured output. We present different tracking controllers for plants with arbitrary relative degree. We consider several approaches to overcome the relative degree obstacle: linear or variable structure lead filters, high-gain observers with constant or dynamic gain, global hybrid estimation schemes combining lead filters or observers with locally exact differentiators based on high-order sliding mode. Global or semi-global stability properties can be proved either for asymptotic exact tracking or for tracking within a small residual error. Some experimental results are presented to illustrate the applicability of the control schemes in real systems.

This chapter will describe the use of sliding mode ideas for fault detection leading to fault tolerant control. The fundamental purpose of a fault detection and isolation (FDI) scheme is to generate an alarm when a fault occurs and to pin-point the source. Fault tolerant control (FTC) systems seek to provide, at worst, a degraded level of performance (compared to the fault free situation) in the event of a fault or failure developing in the system. This chapter will discuss how sliding mode methods for control system design and observer design, can be advantageously used for such schemes. The sliding mode observer FDI schemes seek to robustly estimate any unknown fault signal existing within the system based on appropriate scaling of the equivalent output estimation error injection signal. Both actuator fault and sensor fault problems are considered. One advantage of these sliding mode methods over more traditional residual based observer schemes is that because the faults are reconstructed, both the ‘shape’ and size of the faults are preserved. In the absence of modelling discrepancies, the faults would be reconstructed perfectly. In the uncertain case, the thresholds set for the reconstruction signals for alarm purposes, correspond directly to the level of faults than can (or must) be tolerated. A further benefit of this approach is that because faults are reconstructed, these signals can be used to correct a faulty sensor for example, to maintain reasonable performance until appropriate maintenance could be undertaken. This ‘virtual sensor’ can be used in the control algorithm to form the output tracking error signal which is processed to generate the control signal. In particular the chapter discusses recent advances which seek to obviate the traditional relative degree one minimum phaseness conditions. Also the effects of unmatched uncertainty are discussed. In all the methods proposed, efficient Linear Matrix Inequality methods are employed to synthesis the required gains. A recent application of sliding mode controllers for fault tolerant control is also presented. Here the inherent robustness properties of sliding modes to matched uncertainty are exploited. Although sliding mode controllers can cope easily with faults, they are not able to directly deal with failures – i.e. the total loss of an actuator. In order to overcome this, the integration of a sliding mode scheme with a control allocation framework is considered whereby the effectiveness level of the actuators is used by the control allocation scheme to redistribute the control signals to the ‘healthy’ actuators when a fault occurs.

This paper addresses the mean-square and mean-module filtering problems for a linear system with Gaussian white noises. The obtained solutions contain a sliding mode term, signum of the innovations process. It is shown that the designed sliding mode mean-square filter generates the mean-square estimate, which has the same minimum estimation error variance as the best estimate given by the classical Kalman-Bucy filter, although the gain matrices of both filters are different. The designed sliding mode mean-module filter generates the mean-module estimate, which yields a better value of the mean-module criterion in comparison to the mean-square Kalman-Bucy filter. The theoretical result is complemented with an illustrative example verifying performance of the designed filters. It is demonstrated that the estimates produced by the designed sliding mode mean-square filter and the Kalman-Bucy filter yield the same estimation error variance, and there is an advantage in favor of the designed sliding mode mean-module filter. Then, the 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.

The problem of causal output tracking and observation in non-minimum phase nonlinear systems is studied. The extended method of Stable System Center (ESSC) is used in two-fold manner: i) to generate reference profile for unstable internal states; ii) to estimate states of unstable internal dynamics. Two applications of the proposed technique are considered for illustration purposes: output voltage tracking in a nonminimum phase DC/DC electric power converter and output tracking in SISO systems with time-delayed output feedback. A variety of traditional and higher-order sliding mode (HOSM) control and observation methods is employed in the majority of algorithms. Most of the theoretical results are covered by numerical simulations.

In this chapter, a nonlinear sliding surface is discussed to improve the transient response for general discrete-time multiple input multiple output linear systems with matched perturbations. The nonlinear surface modulates the closed loop damping ratio from an initial low to final high value to achieve better transient performance. The control law is based on the discrete-time sliding mode equivalent control and thus eliminates chattering. The control law is proposed based on two approaches: (1) reaching law based approach which needs only disturbance bounds and (2) disturbance observer based approach. Multirate output feedback is used to relax the need of the entire state vector for implementation of the control law. A possible extension of the nonlinear surface to input-delay systems is also presented.

In this chapter, a scheme for real time motion planning and robust control of a swarm of nonholonomic mobile robots evolving in an uncertain environment is derived. This scheme consists of two main parts: (i) a real time collision-free motion planner; (ii) a trajectory tracking controller. In implementation, the motion planner dynamically generates the optimal trajectory while the robot runs. High precision motion tracking is achieved by the design of a higher order sliding mode controller based on geometric homogeneity properties. Experimental investigations have been conducted using several test benchmarks of mobile robots in order to demonstrate the effectiveness of the proposed strategy.

Power electronics is concerned with electromechanical systems that carry power. In a wide sense, power electronics includes the analysis, synthesis and implementation of electrical motors and generators, as well as power converters. Several controllers designed in the framework of Sliding Mode can be found in specialized literature; in particular it is worth to quote the book of V.I. Utkin, J. Guldner and J. Shi [8] where several motors, generators and power converters were studied in the SMC domain. The authors took benefit of these systems to explain the advantages of SM as a robust control methodology and to show most of its applications in dynamical systems: as estimators, observers, …This chapter is also devoted to electrical generators and power converters stressing implementation issues. We want to emphasize implementation procedures based on theory in front of the trial and error, a tuning method widely used even at universities. On the other way around, we are also interested in theoretical problems appeared when implementing algorithms.

The objective of this chapter is to present advanced control methodologies of uncertain nonlinear systems. Firstly, adaptive sliding mode controller that retains the system’s robustness in the presence of the bounded uncertainties/perturbations with unknown bounds is proposed. Due to the on-line adaptation, the proposed approach allows reducing control chattering. Secondly, a high order sliding mode control strategy that features a priory knowledge of the convergence time is presented. Finally, the output feedback second order sliding mode controller is presented and discussed. The control algorithms are applied to experimental set-up equipped by electrical or electropneumatic actuators.

Controllers and observers for electromechanical systems are widely used and implemented in the industry in order to improve its performance. Among different electromechanical systems we can find interesting domains of application such as power systems, UAVs, teleoperation. This paper intents to show the advantages of the control and observer design using sliding mode techniques. These domains are related with the research topics of the Mechatronics laboratory of the Nuevo Leon University, in the CIIDIT-UANL Research Institute.

Inducing sliding motions in appropriate converters allows a systematic design of the three canonical elements for power processing, i.e., DC transformer, power gyrator and loss-free resistor (LFR). A search of candidates is performed by studying a great number of converters with topological constraints imposed by the nature of each canonical element. Several examples ranging from DC impedance matching by means of a DC transformer to LFR-based power factor correction illustrate the application of the synthesis.

On the basis of classical studies in robotics, it seems that the conventional sliding mode approach is not a suitable technique to design robotic controllers, due to the presence of the so-called chattering effect. However, studies have shown that a good reduction of the chattering effect can be achieved by relying on higher order sliding modes. This chapter presents the application of the Second Order Sliding Mode (SOSM) design methodology to the control and supervision of industrial manipulators, by proposing a robust control scheme and a diagnostic scheme to detect and, possibly, isolate and identify faults acting on the components of the system. The proposed SOSM motion controller and the SOSM observers designed to construct the diagnostic scheme are theoretically developed, and their practical application is suitably described. Indeed, the proposed approaches are experimentally verified on a COMAU SMART3-S2 industrial robot manipulator, obtaining satisfactory results.

The dynamics of the electric motors and generators (synchronous and induction) are highly nonlinear and content uncertainties including plant parameters variations magnetic saturation and external disturbances (load torque). On the other hand, the electric machine models are described by a class of nonlinear minimum phase systems which include the strict-feedback form or the nonlinear block controllable form (NBC-form) and stable residual stable. Therefore, in this case, to design a stabilized controller it is naturally to applied some feedback linearization (FL) technique: input-output (IO) linearization [9], backstepping (BS) [11] or block control (BC) ( [13]- [15]). It is interesting to note that the BC approach has some advantage comparing with the IO and BS ones.