Sliding Modes in Control and Optimization.pdf

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本书是滑模控制中的经典书目，内容丰富，易懂。
retrace results in a discontinuity of the righthand part of the diflerential equations described, thus leading to diverse sliding mode equations. An approach describing the system motions. If such discontinuities are deliberately intro- followed by this author implies regulariztion through an introduction of a duced on certain surfaces in the system state space, then motions in a sliding boundary layer which, on thc onc hand, allows the reason for ambiguity to be mode may occur in the system. This type of motion features some attractive revealed and, on the other hiand, outlines the class of systems whose sliding properties and has long since become applied in relay systems. Sliding modes equations can be written quite unambiguously. Various methods of dcsigning are the basic motions in variable structurc systems automatic control systems treated in this book are applied to exactly such Consider the properties of sliding moles in some greater detail. First, the class of system trajectories of the state vector belong to manifolds of lower dinension than As was already noted, the approach used in the book is oriented toward a that of the whole state space, therefore the order of differential equations deliberate introduction of sliding modes over the intersection of surfaces on describing sliding motions is also reduced. Second, in most of practical which the control vector components undergo discontinuity. Realization of systems the sliding notion is control-independcnt and is determined merely such approach obviously implies the knowledge of the conditions of the by the propcrtics of the control plant and the position(or equations) of the occurrence of sliding modes, rarely discussed in the literature. From the discontinuity surfaces. This allows thc initial problem to be decoupled into point of view of mathematics, the problem may be reduced to that of finding ndcpcndent lower dimension subproblems wherein the control is"spent che area of attraction to the manifold of the discontinuity surfaces intersec only on creating a sliding mode while the required character of motion ove tion. Thc solutions suggested arc formulated in terms of the stability theory the intersection of discontinuity surfaces is provided by an appropriate choice and are obtained via generalization of the classical Lyapunoy theorems to of their cquations. These properties turn to be quite essential for solving discontinuous systems featuring not merely individual motions but. rather a many application problems characterizcd by high order differential equations set of motion trajectories from a certain domain on the discontinuity which prohibits the use of efficient analytic techniques and computcr technol- ogy. A third specific feature of a sliding mode is that under certain conditions Part l of the book is concluded with problems of robustness of discontinuous it may become invariant to variations of dynamic charactcristics of the systems with respect to small dynamic imperfections disregarded in an control plant which poses a central problem dealt with in the theory of idealized model yet always present in a real-life system due to small iner automatic control It is essential that, unlike continuous systems with non tialities of measuring instruments, actuators, data processing devices, etc. In measurable disturbances in which, the conditions of invariancy require the the thcory of singularly perturbed differential equations, the control in a usc of infinitely high gains, the same effect in discontinuous systems Is continuous system may be regarded as a continuous function of small time attained by using finitc control action constants,provided the fast decaying motions arc ncglcctcd; consequently Finally, a purely technological aspect of using discontinuous control sys the same property is featured by solutions of the equations describing the tems should hc mentioned, To improve performance, electric inertialess motion in the system. This conclusion provides grounds for applying simplified actuators are incleasingly employed now, built around power electronic models to solve various control problems in the class of continuous systems. In elements which may operate in a switching mode only. Thereforc even if we a discontinuous control system operating in a sliding mode small additiona employ continuous control algorithms the control itself is shaped as a hi timc constants neglected in the idealized model may cause a shift in the frequency discontinuous signal whose mean value is equal to the desired switching times thus making the controls in the idealized and real systems ontinuous control. A morc natural way then will be to employ such essentially differcnt. As a rcsult, thc solution obtained may prove to be algorithms which are deliberately oriented toward the use of discontinuous nonrobust to some insignificant changes in the nmodel of our process which, in controls its turn, makes very doubtful the entire applicability of the control algorithms Let us describe the problem faced in an attempt to employ the properties do the considered class of discontinuous systems. In this connection, a study of of sliding modes for the design of automatic control systems. Consider first singularly perturbed discontinuous systems is carried out to show that despite the mathematical problems treatcd in Part 1. Discontinuous dynamic systems the lack of any continuous corrclation betwcen the control and the small time are outside the scope of the classical theory of differential equations and constants, the systems with unambiguous equations of their sliding mode duire the development of ad-hoc techniques to study their behaviour motions are robust with respect to small dynamic imperfections Various publicaLions onl the matter have shown a diversity of viewpoints of The major focus in the book is on control systems design methods their authors as to how the motion on a discontinuity boundary should be discussed in Part Il. Despite the diversity of the control goals in the problems Preface Pr eric considered in this part. and in their solution techniques, all the design mobile control subjected to uncontrollable disturbances. a deliberate intro- procedures are built around a common principle: the initial system 1s duction of sliding modes in the loops responsible for the control of the syster decoupled into indcpcndent subsystems of lower dimension and sliding motion and the intensity of the sourcc permits such a problem to be solved motions with required properties are designed at the final stage of a contro using the informati system, provider proccss. Thc methods of the analysis of discontinuous dynamic systeins magnitudes the spatially distributed disturbances and their velocities suggested in the first part of the book serve as a mathematical background for bounded. The same statement is used to solve the problem in the case of a thc realization of this principle distributed or lumped control both for the control plants with a single spatial Part il of the book is aimed not only at stating and developing the results variable and for a set of interconnected distributed plants obtained in the sphere of discontinuous control system design, but also at The problems of control and optimization under incomplete information presenting thesc rcsults in close correlation with the basic concepts, problems on the operator and the system statc as well as computational problem and methods of the present-clay control theory. This refers both to newly associatcd with the search for the optimal set of system parameters are obtained results and to those which have appeared earlier in the literature considered in the final chapters of Part II. In solving these problcms, continu- Some problems traditionally posed in the lincar control theory such as ous or discontinuous asymptotic observers and filters, nominal or adjustable eigenvalue allocation and quadratic and parametric optimization have been models, search procedures, etC. are used wherein the dynamic processcs of treated with the use of the fundamental control theory concepts of control- adaptation, identification or tending to an extremum are conducted lability, stabilizabiliTy, observability and detectability. The analogs of the sliding mode. well-known theorems on the uniqueness and existence of solutions to such A sufficiently great deal of experience has been acquired up to the present problems are given, and procedures of obtaining thc solutions for systems in using sliding modes in variouis application problems, It seened expedient with sliding modes are suggested to devote a separate section of the book(Part [lI)to applications and provide c The problem of invariance as applied to lincar systems with vector-valued some examples, most convincing in the author's opinion. The first exampl ntrols and disturbances is treated under the assumption that a dynamic dealing with the control of a robot arm is given to illustrate the wholc idca model of disturbances with unknown initia! conditions is available. However, The use of sliding modes for this multivariable high order noulinear design in contrast to the well-known methuds, the approach suggested does not problem has made it possible to realize motions invariant to load torques and require the knowledge of accuratc valucs of the model parameters, for it is mechanical parameters which could be described by independent linear quite sufficient to know just the ranige of their possible variations liform differential equations with respect to each of the controlled parame Since all design methods are based on the idea of decoupling the system seems interesting to compare discontinuous systems with continuous ones Perhaps the most suitable for the application of sliding modes proved to be realizing the same idea through the use of infinitely high gains, As is well one of the basic engineering problems, that of control of clectrical machines known, the increase of the gain in an open-loop systcm decouples its overall and, in particular, of clectrical motors. Increasingly dominating nowadays motion into fast and slow components which may be synthesized indepen- are AC motors built around power electronic elements operating in a dently. The comparison shows that: systcms with infinitely high gains are switching mode, which puts thc problem of the algorithmic supply in the articular cases of singularly perturbed systems and their slow motions may forefront. The technological aspect of this problem has already been discus be described by the same cquations as used for sliding motions in discontinu sed and the considerations speaking in favour of the control algorithms ous systems. The only difference is that in continuous systems these motions oriented toward introduction of sliding modes. The principles of designing may be attained only asymptotically in tending the gains to infiniTy, while in multivariable discontinuous control systems suggested in Part II may be discontinuous systems the same is attainable in a finite time and with a finite easily interpreted in terms of clectrical engineering: serving as the control control vector components in this case are discontinuous voltages at the output of Besides finite-dimensional control problems, a design method for a system power semiconductor converters which are then fed to the motor phases and with a mobile control in a distributed parameter control plant is discusscd i to thc excitation winding (if any ) the motion diflerential equations for all the book. Most probably, this may be regarded as the first attempt to apply types of motorS which are generally nonlinear turn to hc lincar with respect sliding modes to control plants of this class. The problem is in steering the to the control (in which case the sliding equations are written unambigu controlled parameter to the required spatial distribution in the system with a ously); only some components of the state vector are to be controlled (for Preface XI instance, angular position of the motor shaft, angular vclocity and torque by Dr D. Izosimov who helped the author to write Chapter 17. The control magnetic flux, power coefficient, currents, etc. Wide-scale experimenta principles developed in lhese studies have been extended by post-graduate tests have testified to the efficiency of sliding mode control for any types of student S. Ryvkin to synchronous clectrical motors(Section 4, Chapter 17) electrical motors, including induction motors, which are known to be the Unquestionably, it was a good luck Tor the author that the editorial board most reliable and economic vet the hardest to control of the Publishing House made a decision to send the hook for rcvicw to Prof Considercd in Chapter 17 of Part iii is a set of problems associated with A Filippov, who studied it with a mathematician's rigour and Imade a control algorithms for DC, induction and synchronous motors and with number of useful remarks and advices to improve the manuscript mcthods of acquiring informalion on the controlled process state The appcarance of this book could be hardly possible without a permanent The closing chapter of this part presents the results obLained in implement- help froin n. Kostyleva throughout all of its stages starting with the idea ta ing sliding mode control for electrical drives of mctal-cutting machine tools write it and finishing with the prcparation of the final version of the and transport vchicles utilizing both independent and external powcr manuscript, The seminars held in the laboratory of discontinuous Control sources. The potential of sliding mode systcms is demonstrated for physical Systems of the Institute of Control Sciences and discussions with his co processes most diverse in their nature such as chemical fibre production workers have significantly helped the author shape his ideas on the meial melting in electric arc furnaccs, processes in petroleum relining and methodology of presenting this material and its correlation with various parts petrochemical industries, automation in fishery stabili ation of the resonant of thc control theor frequency of an accelerator intended for physical experiments The author is indebted to L. Govorova whu has faultlessly accomplished a It is obvious that all feasible applications of systems with sliding Inodes hard task of typing the manuscript. could not he possibly covered in one book, The mples given in PartIn In preparing the English version of the book, the author has gratefully have made a stress upon the algorithms not touching the problems of their used a chance given by springer-Verlag to revise the book and to include technical implementation, Consideration of such problems would somehow some results obtained in the theory of sliding modes and its applications after fall out of the general theoretical line of the book. Nonetheless, the author the publication of the book in Russian in 1981. These refer to stochastic believes that it is some specific examples that makes the "Mathematical regularization of discontinuous dynamic systems (Section 5, Chapter 2),con- Tools "and"Design"parts of the hook appropriately completed trol of distributed systems (Sections 2 through 4, Chapter 12), the method of The author takes a chance to express his deep gratitude to all of his system parameter identification employing discontinuous dynamic models colleagues who have contributcd, directly or indirectly, to the book. Most (Section 2, Chapter 13), robustness of sliding modes to dynamic discrepancies fruitful have been his discussions of the mathematical and technical aspects of between the system and its model (Scctions 3 and 4, Chapter 1 4 ). Part IiI was the theory of singularly perturbed systems with Prof. A. Vasilyeva and Prof enriched by a new Chapter 18 describing direct application of the sliding P Kokotovic The results of these discussions have formed a framework of mode control to a large class of technological system Chapters 5 and 11 where a comparison is made between singularly perturbed The project of publishing the book in the English translation could hardly systems, high gain systems and discontinuous control systems. The results ot ever be realized without constant friendly support and help from Prof. Section 3 in Chapter 4 are a"direct corollary"of a discussion with Prof. M. Thoma, to whom the author is sincerely indebted A, Filippov of the problem of existence of multidimensional sliding modes Prof. D. Siljak has drawn the author's attention to a possibility of construct- Moscow aV Vadim I utkin ing an upper bound for the Lyapunov function using the comparison prin ciple(Section 6, Chapter 4). The statement of thc problem of mobile control for a distributed system has be ed by Prof. A. Butkovsky with n the author have subsequently solved this problem in a joint effort The book has gained from the results published in rcccnt years by Dr. K. Young(Section I, Chapter 13 and Chapter 15)who had started enthusias- tic work in this field while a post-graduate stuclent In the works conducted at the Institute of Control Sciences in the design of control algorithms for electrical machines, the basic contribution was made contents PartI. Mathematical Tools Chapter 1 Scope of thc Thcory of sliding Modcs ......,. 1 1 Shaping the probler 2 Formalization of Sliding Mode Description 7 3 Sliding Modes in Control Systems Chaptcr 2 mathcmatical Description of Motions on discontinuity boundari 12 1 Regulari Problem 12 2 Ec dent control method Regularization of Systems Linear with Respect to Control 16 4 Ph f the Equivalent Control 5 Stochastic regularization . F Systcms with Aiubi Sliding ee ns with scalar 30 1.2 Systes Nonlinear with respect to vector-Valued control 35 1..3 Example of Ambiguity in a System Linear with respect to Control, 36 Sets 3 Ambiguity in Systems Linear with Respect to Control Chapter 4 Stability of Sliding Modes 1 Problem Statement, Definitions, Necessary Conditions for Stability 44 2 An Analog of Lyapunov's Theorem to Determine the sliding mode doma 416 3 Piece wise Smooth Lyapunoy Functions 4 Quadratic Forms Method 5 Systcms with a Vcctor-Valucd Control Hicrarchy 6 The Finiteness of Lyapunov Functions in Discontinuous Dynamic Systems Contents Contents XIV 3 Combined systeins 148 Chapter 5 Singularly Perturbed Discontinuous SystenIs 1 Separation of Motions in Singularly Perturbed Systems 4 Invariant Systems Without Disturbance Measurements 149 2 Problem Statement for Systems with Discontinuous control 5 Eigenvalue Allocation in Invariant System with G Sliding Modes in Singularly Perturbed Discontinuous Non-measurable disturbances 争甲 151 Control Systems 70 Chapter 11 Systeins with High Gains and Discontinuous Controls.... 155 led motion syst cms 155 Part II. Design 2 Linear Time-Invariant Syslems 3 Equivalent Control Method for the Study Chapter o decoupling in Systems with Discontinuous Controls Non-linear High-Gain Systems 1. Problem Statement 4 Concluding remarks ,166 2 Invariant Transformations 80 3 Design Procedure Chapter 12 Control of Distributed-Parameter plants 169 4 Reduction of the Control System Equations to a Regular Form 81 I Systems with 1 Mobilc Control 169 4. 1 Single-input Systems 2 Design based on the lyapunov method 4.2 Multiple-Input Systems 3 Modal control·∴ 4 Design of Distributed Control of Muiti-Variablc Hcat Proccsses 186 Chapter 7 Eigenvalue Allocation 91 Controllability of Stationary Linear Systems 91 Chapter 1.3 Control Under Uncertainty Conditions 2 Canonical Controllability Form 94 1. Design of Adaptive Systems wiLh Reference model 3 Eigenvalue Allocation in Linear Systems, Stabilizability 2 Identification with Piecewise-Contintous Dynamic modcls 194 4 Design of Discontinuity surfaces 3 Method of self- Optimization ■·◆ 5 Stability of Sliding Modes 14 6 Estimation of Convergence to Sliding Manifold 108 Chapter 14 State Observation and filtering 206 1 The Luicnhcrgcr Ohscrvcr 206 Chapter 8 Systems with Scalar Control 111 2 Observer with discontinuous parameters 1 Design of Locally Stable Sliding Modes ,,1l1 3 Sliding Modes in Systems with Asymptotic Observers 210 2 Conditions of Sliding Mode Stability"in the large 4 Quasi-Optimal Adaptive filterin 217 3 Design Procedurc: An Example 4 Systems in the Canonical Form l23 Chapter 15 Sliding Modes in Prohlems of Mathenatical Programming. 2 I Problem statement Chapter9 Dynamic Optimization∴… ·131 2 Motion Equations and Necessary Existence Conditions I Problem Statement 13l for Sliding mode 2 Observability, Detectability 132 3 Gradient procedures for Piecewise smooth Function 227 Optimal Control in Linear Systems with Quadratic Criterion 135 4 Conditions for Penalty function Existencc. Optimal Sliding Mo convergence of Gradient Parametric Optimiza. ion 139 Smooth penalty funct Optimization in Time-Varying Syst 6 Linearly Independent constraints Clapter 10 Control of Linear Plants in the Presence of Dist 145 Part Ill. Applications 237 1 Problem statement 14S ce conditions Cha 239 2 Sliding Mode i Contents I Model of robut arin Partl. mathematical tools 2 Problcm statement 240 3 Design of Control 4 Design of Control System for a Two-joint manipulator 243 5 Manipulator simulation 246 6 Path contro Chapter 7 Conclusions 249 Scope of the Theory of sliding Modes Chapter 17 Sliding Modes in Control of Electrie Motors ...... 250 1 Problem statenent 250 2 Control of d.c mott 3 Control of Induction Motor 4 Control of Synchronous motor 260 Chapter 1.8 examples I Electric Drives for Metal-cutting Machine Tools 265 2 Vehicle control 269 3 Process Control 271 4 Other applications 275 References 278 1 Shaping the Problem Subject index A number of processes in mechanics, electrical engineering, and other areas are characterized by the fact that thc righthand sides of the differential cquations describing their dynamics feature discontinuities with respect to thc current nccss statc. a typica. mechanical system whose force of resistance may lake up either of the two sign-opposite values depending on the direction of the motion. This situation is often the case in automatic control systems where the wish to improve the system performance, minimize power consumed for the contrul purposes, restrict ble variat the form of discontinuous functions of the system state vector and the system Input act Formally, such discontinuous dynamic systems may be described by the equalion where the system state vector is xeR", t is time, andf(x 1) has discontinuities at a certuin set within the (n+I)-d (R, t. Let us confine oursch onsideration of only th sct of, possibly Iimc-varying, discontinuity surfaces of nl-dimensianal state spacc cier thesc surfaces 1o is discontinuity bound Chapter 1 Scope of the Theory of Sliding Modes i Shaping the probler defined by the equations s(x,t)=0,s(x,t)∈R!,i=1,,m Discontinuity in the righthand parts of the motion equations is the reason of certain peculiarities in the system behaviour. These peculiarities are observed even in the simplest casc of the above Coulomb friction system(Fig. I)described by equations of the type(1.1 )and(1. 2) mx+Px*+kx=0 (1.3 Where x is displacement, m is mass, k is the spring rigidity lk0品 P w with x> P(x)= 1.4) Po with x<o and Po is a positive constant. It is quite obvious that the discontinuity sur (1.2)in this ca se is the x-axis s=北=0. 1.5) notice the fact that the Coulomb friction P(')is not delined in points where velocity i equals zero Consider the behaviour of the mechanical system (1.3)-(1. 5)on the plane Fig, 2 with coordinates x and x(Fig. 2). As evidenced from the figure, the description of this behaviour may be obtained quite easily if |x|> Pork,i. e. if the PiX) discontinuity points are isolated. In this case one may apply, for instance, the point-to-point transformation technique. If the slate vector appears to stay within the segment. Is Po/k of the discontinuity straight line(1.5)(stagnation zone, as termed by A.A. Andronov in [8]), it will not leave this segment, Since in this case the function P()is not defined on the discontinuity straighT line, an immediate problem of an appropriate description of this motion arises. A fixed value of function P()with i=U is unsatisfactory for the purpose because the set IxIs Pu/ k is a set ofeyuilibrium points feasible only when P(i)+kx=0 Therefore an additional assumption is needed to enable the description of the P behaviour of the system at the discontinuity boundary: P(a) should be a multivalued function of the system state ig, 3 Another way is yet possible. The motion equation may be changed by using a more accurate model (Fig 3)recognizing viscous friction(or deformability uf links)which eliminates ambiguity. Then for |xls Po/k and |. e<c and the solution to(1.6) tends to zero the slower the less is the linearity zone in Fig. 3. Ux-I-kx=0 (1.6) the behaviour of a system without introduct of an additional probably, the reason for an incrcased interest to Coulomb friction systems, which has manifested itself at the beginning of this century and has kept onl strong Ic I decades. sub Che problem of describing See, for instance acad Mundcl'shtam. CoLection of Works. Nauka, n 79, 21-125 Chapter 1 Scope of the Theory of Sliding Modes ∴1Sha ping the Proble discontinuous dynamic systems became vital for automatic control people as Another example which refers to a ith two discontinuity surfaces(Fig, 5) well illustrates the motion along both a single discontinuity surface(arcs ab and cb) a distinguished feature of differential equations dcscribing any contro and the intersection of the two surfaces(arc bd) system is known to be the presence of a scalar or vector parameter tt referred As evidenced by these cxamples, the motion trajcctorics which belong to to as contror. the set of discontinuity points are singular since in any combination of con- sinuous 文=f(x,t,1l),t∈R. (1 is u (x, n)and i ( t ) they differ fr An accepted term for the molion on discontinuity surfaces is sliding ode In early regulators, the controls have mostly been of relay type which was Incidentally, the moLion along the segment x Po/ l in the Coulomb riction dictated by the nced to make their implementation as simple as possible, As mechanical system is also a sliding mode motion sult, the righthand part of the differential on of the system motion The problem of the sliding mode existence will be treated in full later in proved to be a discontinuous function of the systcm statc vector. This has forced the book; here it will be apt to note that a sliding mode does exist on a the control theory specialists think of an adcquate description of the bchaviour discontinuity surface whenever the distances to this surface and the velocity of of systems at discontinuity boundaries. For systems with isolated discontinuity its change s are of opposite signs [9], i.c. when points, some analysis and synthesis methods have been designed based on the classical theory of differential equations with the use of point-LO-point lim sso and lim s<0 (1.9) transformations and averaging at the occurrence of high frequency switching (see, for instance, [42, 109] The mathematical description of sliding modes is quite a challenge. It However, in attempts to mathematically describe certain application requires the design of special techniques. The solution of Eq(1. 1)is known problems the samc case as in the Coulomb friction mechanical system were to exist and be unique if a Lipshitz constant L may be found such that for any often faced when the totality of discontinuity points proved to be a nonzero two vectors i and x, measure set in time. This fact is easily revealed for a sufficiently general class ‖∫(x1,)-f(x2,≤L非x1-x2‖ (1.10) of discontinuous controls defined by the relationships It is evident that in the dynamic system(1.7)with the discontinuous control hs:(x)>0 1.8) (1.8), condition (lIO) is violated Ef 1) ty surfaces. Indeed (a, t) with sA(x)<o, i=l, if paints nd x, are on different sides of the 1-x2-0, inequality (.O)is not true for any fixed value of L. Therefore where u'=(u1,., Hur)and all functions ui(x, t)and ui(r, t)are continuous at least formally, some additional effort is needed to find a solution to system The state vector of such systems may stay on one of thc discontinuity surfaces r their intersection within a finitc time. For examplc, the systcm state vector (1.7)and(1.)at an occurrence of a sliding mode. moreover. let a lunction x(t) pretend to be a solution lying on the set of discontinuity points. Even in this trajectories belong to some discontinuity surface s (x)=0 if in the vicinity of this surface the velocity vectors/(A, 4, u)are directed toward each other(Fig. 4) f(x.,u x)=0 F识g.4 Fig. 5 【实例截图】
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