#### Title

Design of linear control systems with distributed parameter elements by parameter plane techniques

#### Date of Award

Spring 1968

#### Document Type

Dissertation

#### Degree Name

Doctor of Engineering Science in Electrical Engineering

#### Department

Electrical Engineering

#### First Advisor

Lawrence Eisenberg

#### Second Advisor

Andrew Ulrich Meyer

#### Third Advisor

Henry B. Weisbecker

#### Fourth Advisor

George J. Moshos

#### Abstract

This dissertation extends the application of parameter plane techniques to a number of areas. Specifically, the parameter plane technique can now be used to determine absolute stability, relative stability, and the location of roots of the closed-loop characteristic equation of systems containing both lumped and distributed parameter elements. The advantages of utilizing this technique are that results can be obtained as two system parameters are varied simultaneously and higher order systems can be treated as easily as low order ones.

Investigation of the mapping of real roots resulted in the derivation of a theorem which relates the number of real roots of the characteristic equation when parameters specify a given point in the parameter plane with the number of tangents which can be drawn from the operating point to certain segments of the zeta equals f 1 contours.

The parameter plane is also used to treat systems containing a hypothetical element whose transfer function is a member of the set of functions exp(-(sT)^{p/q} ) where p and q are integers and p is less than q . The technique developed here involves two different mappings. The first mapping transforms the s-plane equation into a complex w-plane *in *which the equation is *single *valued and then parameter plane mapping techniques are applied . In addition, the parameter plane technique is also extended to systems containing elements with transfer functions of the form exp( -(s^{P}T) )

The predictor configuration described in the literature and used to compensate transport lag in the plant of a feedback system is investigated for systems with plants containing distributed parameter elements. The difficulty with this configuration of exactly synthesizing the distributed element in the auliliary predictor loop is overcome through use of one of a set of rational polynomials for the distributed parameter element. A system with distributed lag is analyzed applying the newly extended parameter plane method in order to determine which polynomial mpproximation is optimum for the system under consideration. Also, the method is used to show how parameters associated with the polynomial can be chosen to make the system stable with wide gain variations and also achieve a minimum of damping of the transient response compared with the uncompensated system.

Several methods are derived for estimating the transient response of feedback systems with distributed parameter elements. The concept of the response consisting of the sum of rational and irrational terms is introduced and attention is focused initially on the rational portion of the response. A geometric interpretation is presented for the time to first peak, T_{p}, and the amount of first overshoot, M, for the case of distributed lag in terms of roots and zeros of the closed loop transfer function transformed into a new complex plane. This interpretation assumes that one pair of roots are dominant. In addition, a set of curves are presented which show the relationship between T_{p}, M, and settling time T_{s} and the s-plane location of the dominant root for systems with distributed lag.

#### Recommended Citation

Weinberg, Murray, "Design of linear control systems with distributed parameter elements by parameter plane techniques" (1968). *Dissertations*. 1337.

https://digitalcommons.njit.edu/dissertations/1337