Control System -1 Final Exams Answers

1. Mathematical Models Of System

Mathematical models of control systems are mathematical expressions which describe the

relationships among system inputs, outputs and other inner variables.

The mathematical model should reflect the dynamics of a control system and be suitable for

analysis of the system. Thus, when we construct the model, we should simplify the problem to

obtain the approximate model which satisfies the requirements of accuracy.

The solution of the differential equation

describing the process may be obtained by

classical methods such as the use of

integrating factors and the method of

undetermined coefficients.

2. Differential Equations of Linear Systems

The differential equations describing the dynamic performance of a physical system are obtained by

utilizing the physical laws of the process.This approach applies equally well to mechanical , electrical, fluid, and thermodynamic systems

3. Linearization of Nonlinear Differential Equations

4. The Laplace Transform Theorems of Laplace Transform

The Laplace transform exists for linear differential equations for which the transformation

integral converges. Therefore, for f(t) to be transformable, it is sufficient that

5. Properties of Transfer Function

The transfer function of a linear system is defined as the ratio of the Laplace transform of the

output variable to the Laplace transform of the input variable, with all initial conditions

assumed to be zero. The transfer function of a system (or element) represents the relationship

describing the dynamics of the system under consideration. A transfer function may be

defined only for a linear, stationary (constant parameter) system. A nonstationary system,

often called a time-varying system, has one or more time-varying parameters, and the Laplace

transformation may not be utilized. Furthermore, a transfer function is an input-output

description of the behavior of a system. Thus, the transfer function description does not

include any information concerning the internal structure of the system and its behavior. The

transfer function of the spring-mass-damper system is obtained from the original Equation,

rewritten with zero initial conditions as follows:

6. Block Diagrams of Control Systems

The dynamic systems that comprise automatic control systems are represented mathematically

by a set of simultaneous differential equations. As we have noted in the previous sections, the

Laplace transformation reduces the problem to the solution of a set of linear algebraic

equations. Since control systems are concerned with the control of specific variables, the

controlled variables must relate to the controlling variables. This relationship is typically

represented by the transfer function of the subsystem relating the input and output variables.

Therefore, one can correctly assume that the transfer function is an important relation for

control engineering. The importance of this cause-and-effect relationship is evidenced by the

facility to represent the relationship of system variables by diagrammatic means. The block

diagram representation of the system relationships is prevalent in control system engineering.

Block diagrams consist of unidirectional, operational blocks that represent the transfer

function of the variables of interest.

7. Signal-Flow Graph Models. Mason’s Signal-Flow Gain Formula

The transition from a block diagram representation to a directed line segment representation is

easy to accomplish by reconsidering the systems of the previous section. A signal- flow graph

is a diagram consisting of nodes that are connected by several directed branches and is a

graphical representation of a set of linear relations. Signal- flow graphs are particularly useful

for feedback control systems because feedback theory is primarily concerned with the flow

and processing of signals in systems. The basic element of a signal-flow graph is a

unidirectional path segment called a branch, which relates the dependency of an input and an

output variable in.

8. Characteristics of Time-Domain Method Transient and Steady State

Analyzing a system in the time-domain can bring us a direct perception of the system,

providing all information of the time-response. However, it is not easy to find the correlation

between the parameters of a system and its performance in the time-domain, or to design a

control system. The time-domain method is the basic analysis method, of which the concepts

and theories are used in the root-locus method and the frequency-domain method.

Dynamic Characteristics

Usually, the performance characteristics of a control system are specified in terms of the

transient response to a unit-step input which is sufficiently drastic. If the transient response to

a step input fulfills the requirements, the transient response to other inputs should be decent.

In specifying the transient-response characteristics of a control system to a unit-step input, it

is common to specify the following.

Steady-State Characteristics

The steady-state error is the difference between the input signal and the output signal when

the time approaches infinity. The steady-state error has different definitions, it is subjected to

a typical test signal.

Note that different system requires different performance. For example, passenger airplanes

expect no overshoot in the transient response for comfort reason, while it is acceptable for

fighter aircrafts, where agility is more important.

9. State Space Models

10. Feedback Control Systems Characteristics. Error signal

11. Feedback Control Systems Characteristics Sensitivity of Control Systems

12. Disturbance Signals İn A Feedback Control System

An important effect of feedback in a control system is the control and partial elimination of

the effect of disturbance signals. A disturbance signal is an unwanted input signal that affects

the output signal. Many control systems are subject to extraneous disturbance signals that

cause the system to provide an inaccurate output. Electronic amplifiers have inherent noise

generated within the integrated circuits or transistors;

radar antennas are subjected to wind gusts; and many systems generate unwanted distortion signals

due to nonlinear elements. The benefit of feedback systems is that the effect of distortion, noise, and

unwanted disturbances can be effectively reduced.

13. Measurement Noise Attenuation

14. The stability analysis of linear feedback systems

The concept of stability can be illustrated by considering a right circular cone placed on a

plane horizontal surface. If the cone is resting on its base and is tipped slightly, it returns to its

original equilibrium position. This position and response are said to be stable. If the cone rests

on its side and is displaced slightly, it rolls with no tendency to leave the position on its side.

This position is designated as the neutral stability. On the other hand, if the cone is placed on

its tip and released, it falls onto its side. This position is said to be unstable.

15. The Routh-Hurwitz Stability Criterion

A simple method known as the Routh-Hurwitz criterion allows us to determine the number of

closed-loop poles that lie in the right half of the s-plane by examining the characteristic

equation of the system. In the Laplace domain the characteristic equation can be written as

where the coefficients an are real quantities and that a0 ≠ 0 (the system is of type-0). We can

also factor q(s)

where rn represent the n roots of q(s) and Re{rn}<0 for a stable system. If the factored form is

multiplied out

Since Re{rn}<0 all the coefficients of the characteristic equation should have the same sign

and, for a stable system, be non-zero. These are necessary conditions, but not sufficient. i.e. a

system is definitely not stable if the conditions are not satisfied, however the system not

guaranteed to be stable if they are satisfied.