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## Über dieses Buch

The book is devoted to systems with discontinuous control. The study of discontinuous dynamic systems is a multifacet problem which embraces mathematical, control theoretic and application aspects. Times and again, this problem has been approached by mathematicians, physicists and engineers, each profession treating it from its own positions. Interestingly, the results obtained by specialists in different disciplines have almost always had a significant effect upon the development of the control theory. It suffices to mention works on the theory of oscillations of discontinuous nonlinear systems, mathematical studies in ordinary differential equations with discontinuous righthand parts or variational problems in nonclassic statements. The unremitting interest to discontinuous control systems enhanced by their effective application to solution of problems most diverse in their physical nature and functional purpose is, in the author's opinion, a cogent argument in favour of the importance of this area of studies. It seems a useful effort to consider, from a control theoretic viewpoint, the mathematical and application aspects of the theory of discontinuous dynamic systems and determine their place within the scope of the present-day control theory. The first attempt was made by the author in 1975-1976 in his course on "The Theory of Discontinuous Dynamic Systems" and "The Theory of Variable Structure Systems" read to post-graduates at the University of Illinois, USA, and then presented in 1978-1979 at the seminars held in the Laboratory of Systems with Discontinous Control at the Institute of Control Sciences in Moscow.

## Inhaltsverzeichnis

### Chapter 1. Scope of the Theory of Sliding Modes

Abstract
A number of processes in mechanics, electrical engineering, and other areas, are characterized by the fact that the righthand sides of the differential equations describing their dynamics feature discontinuities with respect to the current process state. A typical example of such a system is a dry (Coulomb) friction mechanical system whose force of resistance may take 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 control purposes, restrict the range of possible variations of control parameters, etc. leads to controls in the form of discontinuous functions of the system state vector and the system input actions.

### Chapter 2. Mathematical Description of Motions on Discontinuity Boundaries

Abstract
As already noted, the behaviour of system (1.7), (1.8) on discontinuity boundaries cannot be adequately described in terms of the classical theory of differential equations. To solve this problem, various special ways are usually suggested, in order to reduce the original problem to a form which yields a solution close, in a sense, to that of the original problem, and which allows the use of classical analysis techniques. Such a substitution of the problem is usually called regularization. The physical ways described in Chap. 2, Part 1 are, in essence, the ways of regularization of the problem of discontinuous dynamic system behaviour.

### Chapter 3. The Uniqueness Problems

Abstract
A system nonlinear with respect to control (1.7), (1.8) also permits formal application of the equivalent control method procedure, yielding some differential Eq. (2.6) which is true along the intersection of its discontinuity surfaces and may be regarded as its sliding mode equation.

### Chapter 4. Stability of Sliding Modes

Abstract
Major attention in the previous chapters was paid to methods of describing the motion along a discontinuity surface or an intersection of such surfaces. The equations of sliding mode, or the set of such equations in ambiguous cases, found in those chapters only indicate the possibility for this type of motion to exist. Let us now proceed directly to finding the conditions for further motion to be a sliding mode, should the initial state be on the intersection of discontinuity surfaces.

### Chapter 5. Singularly Perturbed Discontinuous Systems

Abstract
One of the major obstacles in the use of efficient tools for designing control systems is the high order of equations that describe their behaviour. In many cases they may be reduced to a lower order model by neglecting small time constants or rejecting fast components of the system overall motion. Fast motions may, for instance, be caused by small impedances in equations of electromechanical energy converters [155], time constants of electric motors in systems controlling slow processes [68], nonrigidity of flying vehicles construction [74] and many other reasons. The design of control systems resting upon the use of low-order models may be carried out both by analytical and by various computational techniques. (Application of computational techniques to the design of control systems may be seriously hindered not only by their high dimension, but also by the fact that the computational problems in such systems are generally ill-posed and require ad-hoc methods to be developed).

### Chapter 6. Decoupling in Systems with Discontinuous Controls

Abstract
Since we are going to “unify” the design principles by introducing sliding modes, let us confine ourselves with only those discontinuous systems whose sliding equations may be written quite unambiguously. From the entire variety of nonlinear systems, this limitation isolates a subclass which, generally speaking, may be presented by equations linear in control
$$x = f(xj) + B(x,t)u,$$
(6.1)
where x∈ℝ n and u∈ℝm. Formally, a standard statement of the control theory problems is in choosing such a control u, functionally or operationally dependent upon the system state, time and disturbances, that brings an appropriate transformation of the solution to the initial system (6.1). The idea of the “appropriate” transformation is treated in such a broad sense by the present-day control theory authors that it will be useful to give the most important problems statements in this book.

### Chapter 7. Eigenvalue Allocation

Abstract
In this chapter, we make use of the decomposition method for finding the desired allocation of eigenvalues in time-invariant linear systems of the form
$$x = Ax + Bu,x \in {\mathbb{R}^n},u \in {\mathbb{R}^m},$$
(7.1)
with constant matrices A and B. The discontinuity surfaces are assumed to be linear, i. e.
$$s = Cx$$
(7.2)
with C being a (m x n)-dimensional matrix.

### Chapter 8. Systems with Scalar Control

Abstract
The methods of this chapter for allocation of the eigenvalues of the characteristic equation of sliding mode motion may be, of course, applied to the special case to systems with one-dimensional control
$$\dot x = Ax + bu,$$
(8.1)
where u∈ℝ1, b is (n x 1)-dimensional column vector, and the pair {A, b} is controllable. For the system (8.1) we present a design procedure that differs from the procedure of Chap. 7 and allows one to realize stable sliding modes with desirable eigenvalue allocation over all the discontinuity plane by means of control as a piecewise linear function only of a part of the state vector components rather than all of them.

### Chapter 9. Dynamic Optimization

Abstract
Optimization of linear systems of the type (7.1) usually relies upon minimization of the quadratic criterion
$$I = \mathop \smallint \limits_0^\infty ({x^T}Q + {u^T}Ru)dt,$$
(9.1)
where Q is a positive semi-definite symmetric matrix, and R is a positive definite symmetric matrix. If the controlled variable is of the form y = Dx where y∈ℝk, k<n, and D is constant kxn matrix, then taking
$$Q = {D^T}D\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{ \geqslant } 0$$
(9.2)
one obtains that the first term in the criterion (9.1) is yTy and characterizes the degree of deviation of the controlled variable from the zero state. The second term in the criterion defines the penalty for controll “expenses”. The relation between the weight matrices Q and R defines the tradeoff between the two contradictory desires such as to have a rapidly decaying control process and to reduce power consumption for its realization.

### Chapter 10. Control of Linear Plants in the Presence of Disturbances

Abstract
Consider a control system operating in the presence of disturbances, and given reference inputs which define the desired profiles of controlled variables. Accordingly, the behaviour of control system in the case of linear controlled plant is described by linear non-homogeneous differential equation
$$\dot X = Ax + Bu + Qf(t),$$
(10.1)
where f (t) ∈ℝ1 is the vector characterizing external actions. If the error coordinates are substituted in the state vector for the controlled variables, reference inputs will play a part of disturbances in the new space. Assume that (10.1) is written after such a substitution, vector f (t) then is just a disturbing action, and it is desirable to reduce its effect upon the system behaviour or eliminate it at all. This problem is pivotal in the invariance theory. Let us discuss in short the ideas that may underlie the design of invariant systems.

### Chapter 11. Systems with High Gains and Discontinuous Controls

Abstract
For complicated dynamic plants that are described by non-linear time-varying high-dimensional differential equations, control systems whose general motion may be decoupled with respect to some attribute into partial components having smaller subspaces are especially attractive. Independent investigation of smaller-dimensionality problems enables a significant simplification of analysis and design of control systems.

### Chapter 12. Control of Distributed-Parameter Plants

Abstract
All the design methods that have been considered in this Part are oriented towards the control of dynamic plants described by ordinary differential equations. Theoretical generalizations to the infinite-dimensional cases involve basic difficulties due to the need of constructing a special mathematical apparatus for the study of discontinuous partial differential equations or, stated more generally, discontinuous equations in a Banach space [111]. The method of equivalent control developed in Part I for finite-dimensional systems was shown [110] to be applicable to the distributed systems described by secondorder parabolic or hyperbolic equations with discontinuous distributed or lumped control. A wise range of processes (e.g. thermal and mechanical) is described by the equations of this kind.

### Chapter 13. Control Under Uncertainty Conditions

Abstract
Direct application of the methods of the linear optimal system theory to control of multi-variable plants meets most commonly with two practical problems. First, it is difficult to formulate the objectives in terms of the criterion to be minimized. Second, the wide range of unpredictable variations of the plant operator parameters prevents the optimal algorithms to be realized in all the modes.

### Chapter 14. State Observation and Filtering

Abstract
In the previous chapters, the discussion of problems (in particular, eigenvalue allocation in Chap. 7) was based on the assumption that the system state vector is known. In practice, however, only a part of its components or some of their functions may be measured directly. This gives rise to the problem of determination or observation of the state vector through the information on the measured variables. Below, consideration will be given to the problem formulated in this way for the linear time-invariant system
$$\dot x = Ax + Bu,x \in {\mathbb{R}^n},u \in {\mathbb{R}^m},A,B = const.$$
(14.1)
and it will be assumed that one can measure the vector y that is a linear combination of the system state vector components:
$$y = Kx,y \in {\mathbb{R}^1},1\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{ \leqslant } 1 < n,K - const.$$
(14.2)

### Chapter 15. Sliding Modes in Problems of Mathematical Programming

Abstract
This chapter addresses the methods for solution of mathematical programming problems that enable one to find the extremum point by generating sliding modes on the boundary of the domain of permissible values of the arguments of the function to be minimized [80–82].

### Chapter 16. Manipulator Control System

Abstract
The dynamical properties and control of manipulator or robot arm have been extensively studied [11,101,112,125]. The dynamics of a six-degrees-of-freedom (six joints) manipulator is described by six coupled second-order non-linear differential equations. Physically, the coupling terms represent gravitational torques that depend on link positions, reaction torques due to accelerations, Coriolis and centrifugal torques. The degree of this interaction depends on Manipulator’s physical parameters such as weight and size of the link, and the weight it carries.

### Chapter 17. Sliding Modes in Control of Electric Motors

Abstract
This chapter addresses one of the most challenging applications of sliding modes, the control of electric motors. Its attractiveness is due to the wide use of motors, the advances of electronics in the area of controlled power converters, the insufficience of linear control methodology for essentially non-linear high-order plants such as a.c. motors.