Concepts of optimum stability for linear feedback systems

International Journal of Electrical Engineering Education,  Jan 1999  by De Paor, Annraoi

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Abstract Based on undergraduate teaching experience in control theory, the concept of gain margin is revised, vector margin systematised and the rudiments of its calculation exposed, and the concept of phase margin is illustrated. Some two-parameter problems are tackled via the Routh array. Concepts of centroid design and generalised vector margin design are discussed and finally, an eigenvalue-based optimum stability idea suggested by root locus analysis is explored.

1 INTRODUCTION

The prime prerequisite for good control is that the system be asymptotically stable. This requires that the roots of P(s), which are known as the system eigenvalues, all lie within the left half of the complex s plane. Standard techniques, such as the Routh array, the Nyquist stability criterion and the root locus method are normally invoked to ensure that, given G(s), C(s) is designed to confer asymptotic stability, while satisfying as far as possible other side conditions on dynamical performance, related to considerations such as disturbance rejection, speed of response, limitation of overshoots or undershoots, input tracking with zero steady-state error, etc.1-3.

Moving on from consideration of a single design parameter k, some illustrations are given of the determination of stability domains in a parameter plane, using the Routh array. Ideas such as centroid and inscribed circle centre or generalised vector margin design are explored, in the quest for optimum stability.

Finally, a technique which has informed several studies by the author (e.g., Refs. [4] and [5]) is illustrated. This is the invocation of an eigenvalue-based criterion. Given constraints imposed by system structure, optimum stability follows when the rightmost eigenvalue is as deep in the left half plane as possible.

It is hoped that at least some of the ideas explored here may help reflective teachers guide their students to a deeper understanding of the concept of asymptotic stability.

Inspired by the Nyquist criterion, we now consider the negative inverse frequency response locus of T(s), i.e., the parametric plot of -1/T(jomega), drawn with its real axis on the k line and its imaginary axis orthogonal to this. Nyquist analysis could indeed by performed by considering encirclements of the point

(k, 0) by the negative inverse Nyquist diagram of T(s), but urging the case for this is a task for another occasion. The construction considered is shown on Fig. 3.

The frequency response locus of -1/T(s) is draped around the k axis on

Fig. 6. It is seen immediately that there is an optimum phase margin, which can be evaluated by drawing a tangent from the origin to the locus. The value obtained is phiM = 12.11 deg, corresponding to the gain k = 1.18438. The corresponding normalised closed loop step response is plotted as curve (b) on Fig. 5.

3 A ROUTH APPROACH TO PARAMETER PLANE OPTIMUM STABILITY

The author has found that the Routh array is a very illuminating tool in single and two-parameter design studies on linear feedback systems. A few examples are given below, along with a few incidental observations which tend to simplify and systematise the approach for undergraduates.

This actually arose in the author's studies of living systems, in connection with the skeletal muscle reflex arc shown on Fig. 11. Here it is assumed, neglecting gravity torque and the fall-off of muscle force with velocity of contraction, that the torque acting to rotate a limb about a joint is proportional to the firing rate on a muscle spindle sensory nerve6. This firing rate is the output of a dynamical element actuated by the error between the joint angle and the value set by firing rate on a neuron descending from the brain along the spinal column. The parameter k in equation (15) is given by k = k^sub 2^/J, where J is limb moment of inertia and k^sub 2^ is a gain constant characteristic of the spindle. The parameter a is dependent on inherent derivative action within the muscle spindle. The problem in which the author was interested was how a and k might be tuned in nature to give, in some sense, an optimum response. One of the ideas explored was optimum stability in the (k, a) parameter plane.

The question now arises: what constitutes an optimum stability point on Fig. 12? One suggestion is to locate the design point at the centroid. To find this, each vertex is joined to the centre point of the opposite side. This has an appealing interpretation. Choosing a as any value within its permissible range, the line joining the vertex (0,1) in the scaled domain to the opposite midpoint (0.5, 0) is the locus along which the value of k is midway in its permissible range. Similarly, for any given value of k in its permissible range, the line joining the vertex (1, 0) to the opposite midpoint (0, 0.5) is the locus along which a is midway in its permissible range. In terms of the discussion in the previous section, centroid design maximises simultaneously the gain margins with respect to a and k.