Passivity‐based sliding mode controller/observer for second‐order nonlinear systems

A passivity‐based sliding mode control for a class of second‐order nonlinear systems with matched disturbances is proposed in this paper. Firstly, a nonlinear sliding surface is designed using feedback passification, in which the passivity is employed to guarantee the closed‐loop system's stability. The passivity‐based controller comprising a discontinuous term guarantees globally asymptotical convergence to the sliding surface. A sliding mode‐based control law that satisfies the reaching and sliding condition is also developed. Moreover, the passivity‐based sliding mode observer is also developed to effectively estimate the system states. Compared with conventional sliding mode control, the proposed control scheme has a shorter reaching time; and hence, the system performance is less affected by disturbances, thus eliminating the need to increase the control input gain. Finally, simulation results demonstrate the validity of the proposed method.     

On the multi‐input second‐order sliding mode control of nonlinear uncertain systems

This note addresses the multi‐input second‐order sliding mode control design for a class of nonlinear multivariable uncertain dynamics. Among the most important peculiarities of the considered control problem, the considered sliding vector variable has a uniform vector relative degree [2,2, … ,2] with respect to the vector control variable, and only the sign of the sliding vector and of its derivative are available for feedback. Additionally, the symmetric part of the state‐dependent control matrix is supposed to be positive definite. Under some further mild restrictions on the uncertain system's dynamics, a control algorithm that realizes a multi‐input version of the ‘twisting’ second‐order sliding mode control algorithm is suggested. Simple controller tuning conditions are derived by means of a constructive Lyapunov analysis, which demonstrates that the suggested control algorithm guarantees the semiglobal asymptotic convergence to the sliding manifold. Simulation results, which confirm the good performance of the proposed scheme and investigate the actual accuracy obtained under the discrete‐time implementation effects, are given. Copyright © 2011 John Wiley & Sons, Ltd.     

Output‐feedback tracking in causal nonminimum‐phase nonlinear systems using higher‐order sliding modes

Asymptotic output‐feedback tracking in a class of causal nonminimum phase uncertain nonlinear systems is addressed via sliding mode techniques. Sliding mode control is proposed for robust stabilization of the output tracking error in the presence of a bounded disturbance. The output reference profile and the unknown input/disturbance are supposed to be described by unknown linear exogenous systems of a given order. Local asymptotic stability of the output tracking error dynamics along with the boundedness of the internal states are proven. The unstable internal states are estimated asymptotically via the proposed multistage observer that is based on the method of extended system center. A higher‐order sliding mode observer/differentiator is used for the exact estimation of the input–output states in a finite time. The bounded disturbance is reconstructed asymptotically. A numerical example illustrates the efficiency of the proposed output‐feedback tracking approach developed for causal nonminimum phase nonlinear systems. Copyright © 2009 John Wiley & Sons, Ltd.     

Acceleration of finite‐time stable homogeneous systems

Stabilization rates of power‐integrator chains are easily regulated. It provides a framework for acceleration of uncertain multiple‐input–multiple‐output dynamic systems of known relative degrees (RDs). The desired rate of the output stabilization (sliding‐mode control) is ensured for an uncertain system if its RD is known, and a rough approximation of the high‐frequency gain matrix is available. The uniformly bounded convergence time (fixed‐time stability) is obtained as a particular case. The control can be kept continuous everywhere except the sliding‐mode set if the partial RDs are equal. Similarly, uncertain smooth systems of complete multiple‐input–multiple‐output RDs (ie, lacking zero dynamics) are stabilized by continuous control at their equilibria in finite time and are accelerated. Output‐feedback controllers are constructed. Computer simulation demonstrates the efficiency of the proposed approach.     

Sliding mode output‐feedback causal output tracking for a class of discrete‐time nonlinear systems

The output tracking (OT) of arbitrary references in discrete‐time (DT) nonlinear systems is addressed by designing an output‐feedback control. A set of difference‐algebraic equations is proposed as an exact solution of the problem. Using a novel technique of approximating DT functions, the system disturbance and steady states, characterized by tracking error identically zero, for both the system state and the control input, are represented as signals generated by a disturbed dynamic system. Using the mentioned dynamics, the control system is extended. Then, a state observer is proposed to estimate the resulting extended system state. Finally, a DT sliding mode controller is designed to achieve the approximate OT. Simulations show the effectiveness of the proposed control scheme.     

High‐order sliding mode control design based on adaptive terminal sliding mode

High‐order sliding mode control techniques are proposed for uncertain nonlinear SISO systems with bounded uncertainties based on two different terminal sliding mode approaches. The tracking error of the output converges to zero in finite time by designing a terminal sliding mode controller. In addition, the adaptive control method is employed to identify bounded uncertainties for eliminating the requirement of boundaries needed in the conventional design. The controllers are derived using Lyapunov theory, so the stability of the closed‐loop system is guaranteed. In the first technique, the developed procedure removes the reaching phase of sliding mode and realizes global robustness. The proposed algorithms ensure establishment of high‐order sliding mode. An illustrative example of a car control demonstrates effectiveness of the presented designs. Copyright © 2011 John Wiley & Sons, Ltd.     

Global tracking sliding mode control for a class of nonlinear systems via variable gain observer

A novel output‐feedback sliding mode control strategy is proposed for a class of single‐input single‐output (SISO) uncertain time‐varying nonlinear systems for which a norm state estimator can be implemented. Such a class encompasses minimum‐phase systems with nonlinearities affinely norm bounded by unmeasured states with growth rate depending nonlinearly on the measured system output and on the internal states related with the zero‐dynamics. The sliding surface is generated by using the state of a high gain observer (HGO) whereas a peaking free control amplitude is obtained via a norm observer. In contrast to the existing semi‐global sliding mode control solutions available in the literature for the class of plants considered here, the proposed scheme is free of peaking and achieves global tracking with respect to a small residual set. The key idea is to design a time‐varying HGO gain implementable from measurable signals. Copyright © 2010 John Wiley & Sons, Ltd.     

Impulsive second‐order sliding mode control in reduced information environment

The perturbed system with input‐output dynamics of arbitrary and well‐defined relative degree is considered in a reduced information environment. A novel impulsive second‐order sliding mode control in the reduced information environment is proposed. The almost instantaneous convergence to the origin is achieved via impulsive control acting in a concert with second‐order sliding mode control, specifically supertwisting and twisting algorithms. The impulsive actions are implemented in a piecewise constant format. Numerical examples illustrate the efficiency of the proposed control algorithms.     

UNIVERSAL OUTPUT‐FEEDBACK SISO CONTROLLERS

It is proved in the paper that practically all known higher‐order sliding controllers can be combined with recently developed 2‐sliding‐mode‐based differentiators yielding universal output‐feedback Single‐Input‐Single‐Output (SISO) controllers. These controllers can be applied at least locally, whenever the system relative degree is known. The convergence is global, provided the system relative degree is permanent and few boundedness restrictions hold. No detailed mathematical model of the system is needed. The proposed output‐feedback controller provides for the exact finite‐time‐convergent output tracking of real‐time‐given smooth signals if the output measurements are exact. Otherwise the tracking accuracy is proportional to the magnitude of the sampling noise. The control may be made arbitrarily smooth, thereby removing the chattering effect. The theoretical results are illustrated by computer simulation.     

Prespecifiable fixed‐time control for a class of uncertain nonlinear systems in strict‐feedback form

In this paper, we investigate the prespecifiable fixed‐time control problem for a class of uncertain nonlinear systems in strict‐feedback form, where the settling (convergence) time is not only bounded but also user‐assignable in advance. One of the salient features of the proposed method lies in the fact that it makes it possible to achieve any practically allowable settling time by using a simple and effective control parameter selection recipe. Both fixed‐time stabilization and fixed‐time tracking are considered for uncertain strict‐feedback systems. Firstly, by adding exponential state feedback and using fractional power integration as Lyapunov function candidate, a global stabilizing control strategy is developed. It is proved that all the system states converge to zero within prespecified fixed‐time with continuous and bounded control action. Secondly, under more general uncertain nonlinearities and external disturbances, an adaptive fixed‐time controller is derived such that the tracking error converges to a small neighborhood of zero within preassigned time. Theoretical results are also illustrated and supported by simulation studies.     

Sliding Mode ∑‐Δ Modulation Control of the Boost Converter

A switched implementation of average dynamic output feedback laws trough a ∑‐Δ‐modulator, widely known in the classic communications and analog signal encoding literature, not only frees the sliding mode control approach from state measurements and the corresponding synthesis of sliding surfaces in the plant's state space, but it also allows to effectively transfer all desired closed loop features of an uniformly bounded, continuous, average output feedback controller design into the more restrictive discrete‐valued (ON‐OFF) control framework of a switched system. The proposed approach is here used for the input‐output sliding mode stabilization of the “boost” DC‐to‐DC converter. This is achieved by means of a well known passivity based controller but any other output feedback design would have served our purposes. This emphasizes the flexibility of the proposed sliding mode control design implementation through ∑‐Δ‐modulators.     

Uniform bounded input bounded output stability of fractional‐order delay nonlinear systems with input

The bounded input bounded output (BIBO) stability for a nonlinear Caputo fractional system with time‐varying bounded delay and nonlinear output is studied. Utilizing the Razumikhin method, Lyapunov functions and appropriate fractional derivatives of Lyapunov functions some new bounded input bounded output stability criteria are derived. Also, explicit and independent on the initial time bounds of the output are provided. Uniform BIBO stability and uniform BIBO stability with input threshold are studied. A numerical simulation is carried out to show the system's dynamic response, and demonstrate the effectiveness of our theoretical results.     

Nonlinear controls for a class of discrete‐time bilinear systems

For a discrete‐time neutrally stable bilinear system, a nonlinear state feedback control based on the passivity design has been proposed to stabilize the system globally and asymptotically. This paper shows that the decay rate resulting from the passivity control is not exponential, and the system's response speed becomes very sluggish asymptotically. A ‘normalized’ nonlinear control is therefore proposed to achieve exponential stability. The new exponentially stabilizing control not only improves the system's response speed, but also enhances the system's robustness against small parametric perturbations. Copyright © 2003 John Wiley & Sons, Ltd.     

Peaking free output‐feedback exact tracking of uncertain nonlinear systems via dwell‐time and norm observers

The control of uncertain nonlinear systems by high‐gain observer based output feedback is addressed. Two tracking sliding mode controllers are designed for a broad class of uncertain nonlinear systems with arbitrary relative degree and unmatched polynomial nonlinearities in the unmeasured states. The proposed strategies are based either on dwell‐time for control activation or on simple norm state observers to remove the peaking phenomenon related with high‐gain observers, depending on the nonlinearity growth conditions. In contrast with previous works, exact tracking is also achieved by means of a switching strategy based on locally exact differentiators. Global or semi‐global stability is proved by using Lyapunov theory and on small‐gain analysis. Simulations show that the proposed methodologies provide better and uniform transient behavior, larger regions of attraction, performance recovery with significantly smaller observer gains and good robustness properties with respect to exogenous disturbances and measurement noise. Copyright © 2011 John Wiley & Sons, Ltd.     

Practical Output Regulation of Uncertain Strict‐Feedback Form Systems

In this paper, an output feedback control is proposed to solve the practical output regulation problem of a class of nonlinear systems. In the first step of the design procedure, two sets of coordinate transformations are used to convert the output regulation problem of the system in the strict‐feedback form into the regulation problem of the transformed system in the uncertain normal form. Then, for the resulting system, a state feedback in the cast of nested sliding mode control is designed. Finally, by using the nonlinear separation principle, the output feedback controller is achieved by substituting the estimated states, resulting from the high‐gain observer, instead of real states. It can be shown that the states of the closed‐loop system are ultimately bounded. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Cooperative observers and regulators for discrete‐time multiagent systems

This paper investigates the design of distributed observers for agents with identical linear discrete‐time state‐space dynamics networked on a directed graph interaction topology. The digraph is assumed to have fixed topology and contain a spanning tree. Cooperative observer design guaranteeing convergence of the estimates of all agents to their actual states is proposed. The notion of convergence region for distributed observers on graphs is introduced. It is shown that the proposed cooperative observer design has a robustness property. Application of cooperative observers is made to the synchronization problem. A command trajectory generator and pinning control are employed for synchronizing all the agents to a desired trajectory. Complete knowledge about the agent's state is not assumed. A duality principle is shown for observers and state feedback for distributed discrete‐time systems on graph topologies. Three different observer/controller architectures are proposed for dynamic output feedback regulator design, and they are shown to guarantee convergence of the estimate to the true state and synchronization of all the agents' states to the command state trajectory. This provides design methods for cooperative regulators based on a separation principle. It is shown that the observer convergence region and feedback control synchronizing region for discrete‐time systems are inherently bounded, so that the conditions for observer convergence and state synchronization are stricter than the results for the continuous‐time counterparts. This is in part remedied by using weighting of different feedback coupling gains for every agent. Copyright © 2012 John Wiley & Sons, Ltd.     

Robust output‐based controller design for enlarging the region of attraction of input saturated linear systems

The aim of this study was to design a constrained robust output feedback control strategy for stabilizing linear systems affected by uncertainties and output perturbations. The input is constrained by a saturation function. A classical Luenberger observer reconstructed the states from output observations. The state feedback control employed the estimated states given by the observer. This work proposed a Barrier Lyapunov Function (BLF) to manage the saturation problem in the control input. Moreover, an extended version of the attractive ellipsoid method (AEM) characterized the zone of convergence due the presence of perturbations in the output. A convex optimization procedure, formulated as a set of matrix inequalities, yielded the control parameters and the region of attraction for the close‐loop system as well as a minimal ultimately bounded set for the system trajectories. Numerical simulations supported the theoretical results formulated in this study.     

Robust event‐triggered output feedback learning algorithm for voltage source inverters with unknown load and parameter variations

We consider the output feedback event‐triggered control of an off‐grid voltage source inverter (VSI) with unknown inductance‐capacitance (L ? C) filter dynamics and connected load in the presence of an input disturbance acting at the inverter. Due to uncertain dynamics and unmodeled parameters in the L ? C filter connected to the VSI, we use an adaptive observer to reconstruct the system's states by measuring only the voltage at the output. The control mechanism is constructed based on an impulsive actor/critic framework that approximates the cost, the event‐triggered controller, and the worst case disturbance and generates the desired AC output with the least energy dissipation. We provide rigorous stability proofs and illustrate the applicability of our results through a simulation example.     

Global second‐order sliding mode control for nonlinear uncertain systems

Second‐order sliding mode (SOSM) control is used to keep exactly a constraint σ of the second relative degree or to avoid chattering phenomenon. Yet, the traditional SOSM controllers are designed based upon the assumption that the uncertainties or their derivatives are bounded by positive constants. In this paper, a global SOSM controller is designed for a general class of single‐input–single‐output nonlinear systems with uncertainties bounded by positive functions. Moreover, a variable‐gain robust exact differentiator is developed such that the SOSM controllers with finite‐time convergence can also be implemented even when the derivative of the constraint σ is unavailable. Simulation results are given to show the effectiveness of the proposed method.     

A Robust Control Scheme Based on Sliding Mode Observer to Drive a Knee‐Exoskeleton

In this paper, a robust control scheme to drive a knee‐exoskeleton has been designed to assist people with limited knee movement. Dynamic modeling and parameter identification of the “subject's lower limb‐exoskeleton” has been performed and a sliding mode observer (SMO) developed and integrated in the system's closed loop to provide velocity. Lyapunov's theory allows demonstration of the stability of the system. Experimental tests have been performed with the aid of five voluntary subjects, in sitting position, and during flexion/extension of the knee joint. The performance of the proposed control scheme is compared with those obtained using the classical proportional integrator derivative (PID) controller. These tests track the desired position and velocity trajectories with small tracking errors and evaluate the system's stability and system's robustness against the subject's parameters, variations, and external disturbances.