A survey of models, analysis tools and compensation methods for the control of machines with friction

The design of variable-structure control (VSC) systems for a class of multivariable, nonlinear, time-varying systems is presented. Using the Utkin-Drazenovic method of equivalent control and generalized Lyapunov stability concepts, the VSC design is described in a unified manner. Complications that arise due to multiple inputs are examined, and several approaches useful in overcoming them are developed. Recent developments are investigated, as is the kinship of VSC and the deterministic approach to the control of uncertain systems. All points are illustrated by numerical examples. The recent literature on VSC applications is surveyed. >

Variable structure control of nonlinear multivariable systems: a tutorial

cannot in general be stabilized using a continuous closed loop control U(X), even if each state separately can be driven asymptotically to the origin. (An example is analyzed in Section 2.) In this paper we examine the possibility of stabilizing such systems with a continuous closed loop relaxed control. We find, indeed, that the family of systems stabilizable with relaxed controls is larger than the family of those stabilizable with ordinary controls. An even larger class is obtained if the continuity of the closed loop at the origin is not required. The latter class includes all one dimensional systems for which states can be driven asymptotically to the origin. This result does not hold in two dimensional systems and we provide a counter-example. It should be pointed that relaxed control-type stabilization is used both in theory and in practice; the method is known as dither. We shall comment on the similarities. Lyapunov functions for the system (1) help us in the construction of the continuous closed loop stabilizers. In fact, we find that the existence of a smooth Lyapunov function is equivalent to the existence of a stabilizing closed loop which is continuous except possibly at the origin; an additional condition on the Lyapunov function implies the continuity at the origin as well. We present these results in Section 4, after a brief introduction of closed loop relaxed controls, notations and terminology in Section 3. Prior to that, in Section 2, we discuss an example illustrating the power of relaxed controls. In the particular case of systems linear in the controls, relaxed controls can be replaced by ordinary controls, this is discussed in Section 5. The role of Lyapunov functions in the stability and stabilization theories is of course well known. Examples of systems with Lyapunov functions are available in the literature. We display some in Section 6, along with general comments on the construction, applications and counterexamples, including one which cannot be continuously stabilized, yet possesses a nonsmooth Lyapunov function. Closed loop stabilization with ordinary controls is analyzed extensively in the literature, see Sontag [8], Sussmann [ll] and references therein. Lyapunov functions techniques in stabil-

Stabilization with relaxed controls

For linear time-invariant multivariable feedback systems, the feedback properties of plant disturbance attenuation, sensor noise response, stability margins, and sensitivity to plant and sensor variation are quantitatively related to the Bode magnitude versus frequency plots of the singular values of the return difference matrix I + L and of the associated inverse-return difference matrix I + L^{-1} . Implied fundamental limits of feedback performance are quantitatively described and design tradeoffs are discussed. The penalty function in the stochastic linear quadratic Gaussian (LQG) optimal control problem is found to be a weighted-sum of the singular values, with the weights determined by the quadratic cost and noise intensity matrices. This enables systematic "tuning" of LQG cost and noise matrices so that the resulting optimal return difference and inversereturn difference meet inequality constraints derived from design specifications on feedback properties. The theory has been used to synthesize a multivariable automatic controller for the longitudinal dynamics of an advanced fighter aircraft.

Feedback properties of multivariable systems: The role and use of the return difference matrix

A survey of nonlinear system identification algorithms and related topics is presented by extracting significant results from the literature and presenting these in an organised and systematic way. Algorithms based on the functional expansions of Wiener and Volterra, the identification of block-oriented and bilinear systems, the selection of input signals, structure detection, parameter estimation and recent results from catastrophe theory and included. The limitations, relationships and applicability of the methods are discussed throughout.

Identification of Nonlinear Systems- A Survey

A dither is a high-frequency signal introduced into a nonlinear system with the object of augmenting stability. In this paper,[1] it is shown that the effects of dither depend on its amplitude distribution function. The stability of a dithered system is related to that of an equivalent smoothed system, whose nonlinear element is the convolution of the dither distribution and the original nonlinearity. The ability of dithers to stabilize large classes of nonlinear systems is explained in terms of an effective narrowing of the nonlinear sector. A feature of the approach taken here is that a deterministic (i.e., strong) concept of stability is established under probabilistic (i.e., weak) assumptions on the dither.

Dither in nonlinear systems

A dither is a high frequency signal introduced into a nonlinear system for the purpose of augmenting stability or quenching undesirable jump-phenomena. In an earlier paper [2], it had been shown that the effects of a dither depend on its amplitude distribution function. The stability of a dithered system was related to that of an equivalent smoothed system, whose nonlinear element is the convolution of the dither distribution and the original nonlinearity. In this paper, which is a continuation of [2], the ability of dithers to quench jump-phenomena (i.e., to induce continuity) is demonstrated. A notion of structural-stability for feedback equations is introduced, and it is shown that dithers can structurally stabilize large classes of nonlinear systems subject to a second-order Lipschitz condition. The quenching properties of dithers are explained in terms of an effective narrowing of the nonlinear incremental sector.

N. Shneydor

Papers

Dither in nonlinear systems

Structural stabilization and quenching by dither in nonlinear systems