Analysis and design of the well-known higher-order sliding mode (HOSM) controllers are usually made by means of homogeneity and contraction properties. Up to now, HOSM algorithms have not been properly addressed within a Lyapunov framework, and there is no Lyapunov-based analysis and design, despite the fact that Lyapunov's methods are one of the most important analysis and design tools in modern control theory of nonlinear systems. In this chapter, we design new families of homogeneous HOSM controllers for a class of single-input-single-output uncertain systems. In contrast to the well-known quasi-continuous and nested HOSM control (HOSMC) families, the proposed families of HOSM controllers are obtained by using the concept of control Lyapunov functions (CLFs). The CLFs are constructed explicitly by a modification of the wellknown Backstepping technique and applying properties of homogeneous systems. These ingredients allow us to construct continuously differentiable CLFs recursively, avoiding the problem of using complex mathematical tools for non-continuously differentiable Lyapunov functions. A Lyapunov framework for the HOSM design leads to synthesize different families of homogeneous HOSM controllers and it will allow us to investigate important nonlinear properties of these algorithms which cannot be easily studied with the known techniques hitherto established in HOSM theory.

Two main concepts for sliding surfaces design procedure: pole placement and optimal stabilization are generalized for the case of arbitrary order sliding modes. For the pole placement case, the formula of Ackermann-Utkin is extended allowing the design of sliding surfaces with arbitrary relative degree. The natural connection between order of singularity for singular optimal stabilization problem and order of sliding mode controller is shown and used in the design of the sliding surface and the sliding mode controller of corresponding order.

The control of linear systems subjected to perturbations by means of high-order sliding modes observation and identification techniques is presented. First, the regulation control of a system affected by matched perturbations is addressed. Later on, the method is extended to deal with systems affected by matched and unmatched perturbations. The performance of the proposed controllers is estimated in terms of the deterministic noise upper bounds, sampling time and execution time. The feasibility of the approach is shown by means of experiments and numerical simulations.

High-order sliding modes (HOSM) are able to control highly uncertain systems effectively, providing very good properties to the closed loop system.They have been to date analyzed and designed using basically geometric and homogeneity tools. However, modern control theory for nonlinear systems use the Lyapunov and Lyapunov-like functions as a basic tool for analysis and control. Despite this fact, only recently the Lyapunov theory has been used for HOSM. The aim of this work is to present, in an informal manner, some recent methods to construct Lyapunov functions for HOSM. These constructive methods will be exemplified with some classical HOSM algorithms, such as twisting, terminal, and super-twisting.

In this chapter, the notion of geometric homogeneity is extended for differential inclusions. This kind of homogeneity provides the most advanced coordinate-free framework for analysis and synthesis of nonlinear discontinuous systems. The main qualitative properties of continuous homogeneous systems are extended to the discontinuous setting: the equivalence of the global asymptotic stability and the existence of a homogeneous Lyapunov function; the link between finite-time stability and negative degree of homogeneity; the equivalence between attractivity and asymptotic stability are among the proved results.

We consider the classical reaching problem of sliding mode control design, that is to find a control law which steers the state of a linear time-invariant (LTI) system toward a given hyperplane in a finite time. Since the LTI system is subject to unknown but bounded disturbances, we apply the minimax observer which provides the best possible estimate of the system's state. The reaching problem is then solved in observer's state space by constructing a feedback control law. The cases of discontinuous and continuous admissible feedbacks are studied. The theoretical results are illustrated by numerical simulations.

This chapter extends the L₂-gain analysis toward sliding mode dynamic systems. The developed analysis is applied to standard sliding mode algorithms of the first order and to the popular twisting and supertwisting algorithms of the second order. The above algorithms are shown to be capable of not only rejecting matched uniformly bounded disturbances but also attenuating unbounded ones, including unmatched disturbances. Numerical simulations are involved to support the theory. Experimental results on a DC motor with friction complement the chapter.

The conventional harmonic balance principle is a convenient tool for finding periodic solutions in variable-structure systems, which may occur as chattering in sliding mode control or as a normal operating mode in relay systems, limit cycling tests aimed at controller tuning through variable-structure algorithms, etc. In the present book chapter, the conventional harmonic balance principle is extended to transient oscillatory processes in systems. This principle is termed the dynamic harmonic balance principle. It is formulated for the system having one single-valued oddsymmetric nonlinearity and linear plant without zeros in the transfer function. Based on the dynamic harmonic balance, the equations for the amplitude, frequency rate of change, and amplitude rate of change are derived. This principle is then illustrated by analysis of transient motions in a variable-structure system and the decaying motions of a rocking block.

Output-feedback high-order sliding-mode (HOSM) controls include HOSM-based differentiators and, therefore, possess complicated discontinuous dynamics. Their practical application naturally involves discrete noisy output sampling and numeric integration of the internal variables. Resulting hybrid systems are shown to be stable, and the corresponding asymptotic sliding-mode accuracies are calculated in the presence of Euler integration and discrete sampling, whereas both might feature variable or constant time steps. Discrete differentiators are developed which restore the optimal accuracy of their continuous-time counterparts. Numeric criteria detect the end of the differentiator transient. Simulation confirms the presented results.

This chapter presents a set of experimental results concerning the sliding mode control of an electropneumatic system. The controller is implemented via a micro-processor as a discrete-time input. Three discrete-time control strategies are considered for the implementation of the discontinuous part of the sliding mode controller: explicit discretizations with and without saturation, and an implicit discretization (i.e., very easy to implement as a projection on the interval [-1, 1]). While the explicit implementation is known to generate numerical chattering, the implicit one is expected to significantly reduce chattering while keeping the accuracy. The experimental results reported in this work remarkably confirm that the implicit discrete-time sliding mode supersedes the explicit ones with several important features: chattering in the control input is almost eliminated (while the explicit and saturated controllers behave like high-frequency bang-bang inputs), the input magnitude depends only on the perturbation size and is “independent” of the controller gain and sampling time. On the contrary the explicit controller shows obvious chattering for all sampling times, its magnitude increases as the controller gain increases, and it does not reduce when the sampling period augments. The tracking errors are comparable for both methods, though the implicit method keeps the precision when the control gain increases, which is not the case for the explicit one. Introducing a saturation in the explicit controller does not allow to significantly improve the explicit controller behavior if one does not take care of the saturation width.

The discrete application of a continuous control designed using continuous time model of a plant is common practice. The work in this chapter evaluates new conditions required to be satisfied by the parameters of a sliding mode control so designed, such that the plant response remains bounded under such discrete application of the control. This work is taken forward to propose a generalized reaching law for discrete time sliding mode designs. This generalized reaching law is more flexible in its choice of functions and parameters, enabling it to deal with potentially new problem scenarios. A new problem in which the disturbance affecting the system is bounded by known functions, instead of a constant bound, is solved utilizing this generalized reaching law.

In this chapter, an adaptive second-order sliding mode technique is exploited to optimize the efficiency of a variable-speed wind turbine. A revisited form of a recently proposed adaptation algorithm is proposed to deal with the characteristics and control requirements of the wind energy conversion system (WECS), particularly model uncertainties and fastly varying disturbances due to gusty wind effects. The revisited algorithm enhances the reactivity of the adaptation strategy against fast varying uncertainties. The proposed approach is successfully used to control a doubly-fed induction generator (DFIG)-based wind turbine topology proving its suitability for this application area. The design and convergence analysis of the adaptive controller are developed considering a reduced-order model of the DFIG. However, the performance ofthe closed-loop system is extensively assessed through computer simulations made also over a full-order realistic model of the WECS under study.

Control of electric power system comprised proton exchange membrane fuel cell (PEMFC), along with shared and multiple load configurations of multiple-modular boost DC-DC power converters as power conditioners, is studied using sliding mode control techniques. The nonlinear coupling of the boost converters in a shared and multiple load configurations are overcome by conventional sliding mode controllers (SMCs) that facilitate the output tracking of the load voltage in both converter configurations. The nonminimum phase output tracking is alleviated by controlling the PEMFC current using the adaptive 2-SMC. The challenge of balancing the currents in the boost converters in the shared load configuration is addressed via conventional SMC. The efficiency and robustness of the three-fold controllers for the proposed electric power system for the both configurations has been confirmed via computer simulations.

In this chapter, the concept of model-based event-triggered control is revisited combining it with sliding mode control (SMC), so as to design robust networked SMC schemes. After some preliminaries on the conventional model-based event-triggered control, two alternative design frameworks are proposed along with the discussion of their stability properties. Finally, an example of application is presented to illustrate the two proposed strategies.

DC series motors produce very high torque at zero or low speed. It is for this reason and its simplicity these motors can still be found in oil drilling applications, among others. In oil drilling, the environment can be very harsh, leading to violent vibrations and shocks. Position sensors, needed to perform closed-loop speed or position control, can break easily under these conditions, causing production losses. Moreover, they increase the complexity and cost of the system. Therefore, it is desirable to avoid them. In this work, we propose the use of a step-by-step super twisting observer in order to determine the speed of the motor. These observers have the property of finite time convergence. However, two inherent properties of DC series motors could make the use of observers impossible. On one hand, there exists an observability singularity at zero current. On the other hand, there is the magnetic saturation which leads to incorrect speed observation if not taken into account. To overcome these limitations, we propose the use of an observer/estimator scheme and introduce the magnetic saturation into the observer model.An industrial application is conducted to highlight the performance of the proposed solution in the context of a sensorless speed control of a DC series motor.

This chapter presents a dynamic analysis and the control design for a family of unity power factor rectifiers with an inductive-capacitive-inductive filter. Both single-phase and three-phase topologies are considered, and the three-phase examples include a three-wire power converter and a four-wire configuration with neutral connection. In all cases, the control scheme consists in an inner current loop designed using the sliding mode control technique, and an external loop that regulates the output voltage. Differences among topologies are pointed out, such as the needed of using a decoupling matrix to solve the algebraic constraint on the grid currents (for the threewire case) or the use of two independent controllers that simultaneously regulate the DC bus voltage and keep the split bus balanced (in the four-wire power converter). Numerical simulations validate the proposed control schemes and show satisfactory performances of all closed-loop behaviors.

This chapter is focused on the design of robust controllers for electropneumatic actuators. This kind of systems is highly nonlinear, and their dynamics is uncertain due to frictions, uncertainties (especially in mass flow rate), and external perturbation. The use of robust controllers is then strongly recommended in order to get high performances for trajectories tracking or stabilization. This chapter presents very recent robust controllers based on sliding mode theory and adaptive gain. Through the application of control laws mixing high-order sliding mode and adaptive gain, the objective is to show their applicability to a real system and to compare their performances by using similar experimental benchmark.