#### Abstract

All digital control systems contain at least one signal which is sampled in time and quantized in amplitude. Design of these systems is often based on the assumption that a very large number of levels of quantization is available leading to the approximation of the basically nonlinear system by a linear one. If the actual system is constructed so that the linear assumption is satisfied, the performance may be excellent but other design factors such as reliability, cost, weight, and power consumption may be very unsatisfactory. On the other hand, if the actual system is constructed so that only a few levels of quantization are available, the other factors may be quite satisfactory but a previously well-behaved system may now possess limit cycle oscillations, large static errors, and an objectionable transient response. Thus, an important problem in the field of digital control is the development of analysis and design procedures such that the designer has the freedom to select quantization schemes not satisfying the linear approximation but producing an overall satisfactory design. Two techniques are presented as a partial solution to the above problem. The first is based on a study of certain properties unique to quantized sampled-data system and uses Laplace transforms to carry out the analysis. It leads to closed form solutions but appears to be somewhat restricted in the class of systems to which it can be applied. The second technique is a numerical procedure based on the state transition method and uses a digital computer to carry out the numerical calculations, It is not limited by input type, order of the plant, state variables having other than zero initial conditions, or quantizer complexity. Using the first method, some closed form solutions are obtained for first and second order systems and the results favorably compared with the results obtained by the more general second method. Where possible the results of the second method are compared with the results of other workers. In other cases, typical results are checked by comparison with results from simulation on an analog computer. In all cases favorable comparisons are obtained. Design charts prepared by the numerical procedure are presented and examples given demonstrating their use to satisfy specifications on static accuracy, response time, and presence or absence of either overshoot or limit cycle oscillations. A set of rules are derived describing certain properties of the system^ e.g. a final value rule similar to the final value theorem for linear systems is obtained. These rules are found to he useful in , both analysis and design by reducing the number of computations required to solve a given problem, by providing physical insight into system operation, and by furnishing a check on certain results

#### Date of this Version

8-1-1962

## Comments

Thesis. Financial support from the Rational Science Foundation under Grant G-lk609, Contract Ro. 2991-61-7805 is gratefully acknowledged.