# a control engineer’s guide to sliding mode control[精选推荐pdf]

328IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 7, NO. 3, MAY 1999A Control Engineer’s Guide to Sliding Mode ControlK. David Young,Senior Member, IEEE,Vadim I. Utkin,Senior Member, IEEE,and¨Umit¨Ozg¨ uner,Member, IEEEAbstract—This paper presents a guide to sliding mode control for practicing control engineers. It offers an accurate assessment of the so-called chattering phenomenon, catalogs implementable sliding mode control design solutions, and provides a frame of reference for future sliding mode control research.Index Terms— Discrete-time systems, multivariable systems, nonlinear systems, robustness, sampled data systems, singularly perturbed systems, uncertain systems, variable structure systems.I. INTRODUCTION DURING the last two decades since the publication of the survey paper in the IEEE TRANSACTIONS ONAUTOMATIC CONTROLin 1977 [1], significant interest on variable struc- ture systems (VSS) and sliding mode control (SMC) has been generated in the control research community worldwide. One of the most intriguing aspects of sliding mode is the discontinuous nature of the control action whose primary function of each of the feedback channels is to switch between two distinctively different system structures (or components) such that a new type of system motion, called sliding mode, exists in a manifold. This peculiar system characteristic is claimed to result in superb system performance which includes insensitivity to parameter variations, and complete rejection of disturbances. The reportedly superb system behavior of VSS and SMC naturally invites criticism and scepticism from within the research community, and from practicing control engineers alike [2]. The sliding mode control research commu- nity has risen to respond to some of these critical challenges, while at the same time, contributed to the confusions about the robustness of SMC by offering incomplete analyzes, anddesign fixes for the so-called chattering phenomenon [3]. Many analytical design methods were proposed to reduce the effects of chattering [4]–[8]—for it remains to be the only obstacle forsliding mode to become one of the most significant discoveries in modern control theory; and its potential seemingly limited by the imaginations of the control researchers [9]–[11]. In contrast to the published works since the 1977 article, which serve as a status overview [12], a tutorial [13] of design methods, or another more recent state of the art assessment [14], or yet another survey of sliding mode research [15], the purpose of this paper is to provide a comprehensive guide toManuscript received April 7, 1997. Recommended by Associate Editor, J. Hung. The work of¨U.¨Ozg¨ uner was supported by NSF, AFOSR, NASA LeRC and LaRC, Ford Motor Co., LLNL. Sandia Labs, NAHS, and Honda R • offer a catalog of implementable robust sliding mode con- trol design solutions for real-life engineering applications; • initiate a dialog with practicing control engineers on sliding mode control by threading the many analytical underpinnings of sliding mode analysis through a series of design exercises on a simple, yet illustrative control problem; • establish a frame of reference for future sliding mode control research.The flow of the presentation in this paper follows the chrono- logical order in the development of VSS and SMC: First we introduce issues within continuous-time sliding mode in Section II, then in Section III, we progress to discrete-time sliding mode, (DSM) followed with sampled data SMC design in Section IV.II. CONTINUOUS-TIMESLIDINGMODESliding mode is originally conceived as system motion for dynamic systems whose essential open-loop behavior can be modeled adequately with ordinary differential equations. The discontinuous control action, which is often referredto as variable structure control (VSC), is also defined in the continuous-time domain. The resulting feedback system,the so-called VSS, is also defined in the continuous-time domain, and it is governed by ordinary differential equations with discontinuous right-hand sides. The manifold of the state-space of the system on which sliding mode occurs is the sliding mode manifold, or simply, sliding manifold. For control engineers, the simplest, but vividly perceptible example is a double integrator plant, subject to time optimal control action. Due to imperfections in the implementations of the switching curve, which is derived from the Pontryagin maximum principle, sliding mode may occur. Sliding mode was studied in conjunction with relay control for double integrator plants, a problem motivated by the design of attitude control systems of missiles with jet thrusters in the 1950’s [16]. The chattering phenomenon is generally perceived as mo- tion which oscillates about the sliding manifold. There are two possible mechanisms which produce such a motion. First, in the absence of switching nonidealities such as delays, i.e., theswitching device is switching ideally at an infinite frequency, the presence of parasitic dynamics in series with the plant causes a small amplitude high-frequency oscillation to appear in the neighborhood of the sliding manifold. These parasitic dynamics represent the fast actuator and sensor dynamics1063–6536/99$10.00 1999 IEEEYOUNG et al.: CONTROL ENGINEER’S GUIDE TO SLIDING MODE CONTROL329which, according to control engineering practice, are often neglected in the open-loop model used for control design if the associated poles are well damped, and outside the desired bandwidth of the feedback control system. Generally, the motion of the real system is close to that of an ideal system in which the parasitic dynamics are neglected, and the difference between the ideal and the real motion, which is on the order of the neglected time constants, decays rapidly. The mathematical basis for the analysis of dynamic systems with fast and slow motion is the theory of singularly perturbed differential equations [17], and its extensions to control theory have been developed and applied in practice [18]. However, the theory is not applicable for VSS since they are governed by differential equations with discontinuous right hand sides. The interactions between the parasitic dynamics and VSC generatea nondecaying oscillatory component of finite amplitude and frequency, and this is generically referred to as chattering. Second, the switching nonidealities alone can cause such high-frequency oscillations. We shall focus only on the delay type of switching nonidealities since it is most relevant to any electronic implementation of the switching device, in- cluding both analog and digital circuits, and microprocessor code executions. Since the cause of the resulting chattering phenomenon is due to time delays, discrete-time control design techniques, such as the design of an extrapolator can be applied to mitigate the switching delays [19]. These design approaches are perhaps more familiar to control engineers. Unfortunately, in practice, both the parasitic dynamics and switching time delays exist. Since it is necessary to compensate for the switching delays by using a discrete-time control design approach, a practical SMC design may have to be unavoidably approached in discrete time. We shall return to the details of discrete-time SMC after we illustrate our earlier points on continuous-time SMC with a simple design example, and summarize the existing approaches to avoid chattering.A. Chattering Due to Parasitic Dynamics—A Simple ExampleThe effects of unmodeled dynamics on sliding mode can be illustrated with an extremely simple relay control system example: Let the nominal plant be an integrator(1)and assume that a relay controller has been designed(2)The sliding manifold is the origin of the state-space Given any nonzero initial condition, the state trajectory is driven toward the sliding manifold. Ideally, if the relaycontroller can switch infinitely fast, then whereis the first time instant that, i.e., once the state trajectory reaches the sliding manifold, it remains on it for good. However, even with such an ideal switching device, unmodeled dynamics can induce oscillations about the sliding manifold. Suppose we have ignored the existence of a “sensor” with second-order dynamics, and the true system dynamics are governed by(3)(4)whereandare the states of the sensor dynamics. Clearly, sliding mode cannot occur onsinceis continuous, however, sinceis bounded,where is the time constant of the sensor. Furthermore, reaching an boundary layer ofis guaranteed since(5)The system behavior inside thisboundary layer canbe analyzed by replacing the infinitely fast switching device with a linear feedback gain approximation whose gain tendsto infinity asymptotically(6)The root locus of this system, withas the scalar gain pa- rameter, has third-order asymptotes asTherefore, the high-frequency oscillation in the boundary layer is unstable. Instead of having parasitic sensor dynamics, we may have second-order parasitic actuator dynamics in series with the nominal plant, in which case, the closed-loop dynamics are given by(7) (8)The characteristic equation of this system is identical to that of the parasitic sensor case. This is not surprising since the forward transfer function is identical in both cases. Thus,similar instability also occur with infinitely fast switching.B. Boundary Layer ControlThe most commonly cited approach to reduce the effects of chattering has been the so called piecewise linear or smooth approximation of the switching element in a boundary layer of the sliding manifold [20]–[23]. Inside the boundary layer, the switching function is approximated by a linear feedback gain. In order for the system behavior to be close to that of the ideal sliding mode, particularly when an unknown disturbanceis to be rejected, sufficiently high gain is needed. Note that in the absence of disturbance, it is possible to enlarge the boundary layer thickness, and at the same time reduce the effective linear gain such that the resulting system no longer exhibits any oscillatory behavior about the sliding manifold. However, this system no longer behaves as dictated by sliding mode, i.e., simply put, in order to reduce chattering, the proposed method of piecewise linear approximation reduces the feedback system to a system with no sliding mode. This proposed method has wide acceptance by many sliding mode researchers, but unfortunately it does not resolve the core problem of the robustness of sliding mode as exhibited in chattering. Many sliding mode researchers cited the work in [3] and [22] as the basis of their optimism that the imple- mentation issues of continuous-time sliding mode are solved with boundary layer control. Unfortunately, the optimism of these researchers was not shared by practicing engineers, and this may be rightly so. The effectiveness of boundary330IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 7, NO. 3, MAY 1999layer control is immediately challenged when realistic par- asitic dynamics are considered. An in-depth analysis of the interactions of parasitic actuator and sensor dynamics with the boundary layer control [24] revealed the shortcomings of this approach. Parasitics dynamics must be carefully modeled and considered in the feedback design in order to avoid instability inside the boundary layer which leads to chattering. Without such information of the parasitic dynamics, control engineers must adopt a worst case boundary layer control design in which the disturbance rejection properties of SMC are severely compromised. A Boundary Layer Controller: We shall continue with the simple relay control example, and consider the design of a boundary layer controller. We assume the same second-order parasitic sensor dynamics as before. The behavior inside the boundary layer is governed by a linear closed-loop system(9) (10)whererepresents a bounded, but unknown exogenous disturbance. Whereas discontinuous control action in VSC can reject bounded disturbances, by replacing the switching control with a boundary layer control, the additional assumption that be bounded is needed since according to singular perturbation analysis, the residue error is proportional toGiven afinite, we can compute the root locus of this system with respect to the scalar positive gainAn upper boundexists which specifies the crossover point of the root locus on the imaginary axis. Thus, forthe behavior of this system is asymptotically stable, i.e., for any initial point inside the boundary layer(11)the sliding manifoldis reached asymptotically as The transient response and disturbance rejection of this feedback system are two competing performance measures to be balanced by the choice of an optimum gain value. If we assumethe associated root locus is plotted in Fig. 1 forwith a step size of 0.001. The critical gain isThus from the linear analysis, a boundary layer control withresults in a stable sliding mode, whereas with, oscillatory behavior about the sliding manifold is predicted. Fig. 2 shows the simulated error responses of the closed-loop system for these two gain values which agree with the analysis. In this simulation, a unity reference command for the plant state and a constant disturbanceis introduced. The tradeoff between chattering reduction and disturbance rejection can be observed from, of which the steady-state value0.005 (for the stable response), or the average value0.0025 (for the oscillatory response) is inversely proportional to the gain We note that even with, the resulting response is only oscillatory, but still bounded. This is because the linear analysis is valid only inside the boundary layer, and the VSC always forces the state trajectory back into the boundaryFig. 1.Boundary layer control with sensor dynamics: root locus.Fig. 2.Boundary layer control with sensor dynamics: Time responses for ? ? ???(oscillatory), and? ? ???(stable).layer region. However, as the gain increases, the frequency of oscillation increases as the magnitude of the imaginary parts of the complex root increases. Increasingly smaller amplitudebut higher frequency oscillation as gain approaches infinity. This is the chattering behavior observed when the switching feedback control action interacts with the neglected resonant frequencies of the physical plant. This example illustrates the advantages of boundary layer control which lie primarily in the availability of familiar linear control design tools to reduce the potentially disastrous chattering. However, it should also be reminded that if theacceptable closed-loop gain has to be reduced sufficiently to avoid instability in the boundary layer, the resulting feedbacksystem performance may be significantly inferior to the nom- inal system with ideal sliding mode. Furthermore, the preciseYOUNG et al.: CONTROL ENGINEER’S GUIDE TO SLIDING MODE CONTROL331details of the parasitic dynamics must be known and used properly in the linear design.C. Ob