Analysis of Control System and Synthesis of Real Compensator

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Initial data for the term paper performance. Order of carrying out calculations. Analyze uncompensated system. Synthesize the real PD-compensator ( ) which would guarantee desired phase margin at gain crossover frequency . Analyze compensated system.

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Analysis of Control System and Synthesis of Real Compensator

The standard task

Variant 18

Topic:

Analysis of Control System and Synthesis of Real Compensator.

Task:

The transfer functions (table 1, 2, 3) and structure diagram (fig. 1, 2, 3) are given. It is necessary to

I. Analyze uncompensated system:

1) calculate the transfer function of the whole uncompensated system:

a) without MATLAB;

b) using MATLAB;

2) define zeros and poles of obtained transfer function;

3) plot the time responses for obtained transfer function; the responses should be produced in two different windows of one figure graphic object using command subplot;

4) determine settling time and overshoot of the uncompensated system;

5) check whether the system is stable or not using Hurwitz criterion;

6) check whether the system is stable or not using Nyquist criterion;

7) if system is stable, determine the stability margins using Bode diagram;

8) do the conversion from the transfer functions to state space;

9) calculate the eigenvalues of the uncompensated system;

10) calculate -norm of the uncompensated system.

II. Synthesize the real PD-compensator () which would guarantee desired phase margin at gain crossover frequency :

1) calculate the phase shift of compensator at given frequency ;

2) determine the gain of proportional part of compensator ;

3) determine the gain of differential part of compensator ;

4) plot Bode diagram of the PD-compensator.

III. Analyze compensated system:

1) build in Simulink the compensated system;

2) determine settling time and overshoot of the compensated system using transient process received in Simulink;

3) calculate the transfer function of the whole compensated system;

4) plot the step response for uncompensated and compensated system; the responses should be produced in one window of figure graphic object using command hold on;

5) check whether the system is stable or not using sufficient condition of stability;

6) if system is stable, determine the stability margins using Bode diagram;

7) do the conversion from the transfer functions to state space;

8) calculate the eigenvalues of the compensated system;

9) calculate -norm of the compensated system;

10) do the conclusions.

Introduction

This term paper represents the analysis of control system and synthesis of real compensator. So here we will examine the uncompensated and compensated system, check whether the system is stable or not using Niquist, Hurwitz criterions and sufficient condition of stability, calculate eigenvalues and-norms of the uncompensated and compensated system. We will also synthesize the real PD-compensator () which would guarantee desired phase margin at gain crossover frequency and then will draw the conclusions.

1. Initial data for the term paper performance

Fig. 1. Control system of nuclear reactor rods position

Table 1. Individual variants for control system of nuclear reactor rods position

No

,

rad/sec

,

deg

6

2

0.1

24.4

3

60

2. Calculations

I. Analyze of uncompensated system

1) Calculation of the transfer function of the whole uncompensated system:

a) without Matlab

Theoretical information:

A transfer function of the analog system is the ratio of the Laplace transform of the output signal to the Laplace transform of the input one under zero initial conditions. A continuous time SISO (single-input-single-output) transfer function is characterized by its numerator and denominator , both polynomials of the Laplace variable .

Calculations:

The control system of nuclear reactor rods position looks like:

Fig. 2

So, the transfer function of open-loop uncompensated system is:

And the transfer function of closed-loop uncompensated system is:

b) with Matlab

Program code:

disp ('Task 1. Analyze of uncompensated system')

W1=tf([5]);

W2=tf([6], [1 6 0]);

W3=tf([0.2]);

W4=tf([10.5]);

W5=tf([1], [0.1 1]);

W12=series (W1, W2);

W123=series (W12, W3);

disp ('Transfer function of open-loop uncompensated system is')

Wolun=series (W123, W4)

disp ('Transfer function of closed loop uncompensated system is')

Wclun=feedback (Wolun, W5, - 1)% task 1.1

Results of the program:

Task 1. Analyze of uncompensated system

Transfer function of open-loop uncompensated system is

Transfer function:

63

-

s^2 + 6 s

Transfer function of closed loop uncompensated system is

Transfer function:

6.3 s + 63

-

0.1 s^3 + 1.6 s^2 + 6 s + 63

2) define zeroes and poles of obtained transfer function

Theoretical information:

Poles of the transfer function are the roots of the system characteristic equation. Characteristic equation is the denominator of transfer function reduced to zero.

Zeros of the transfer function are the roots of equation which is obtained by reducing the numerator of the transfer function to zero.

Analytical method is based on using of the operators pole(sys) which calculates the poles of the transfer function and zero(sys) which calculates zeros.

Graphic method is based on using of the operator pzmap(sys), which plots the pole-zero map of the continuous-time LTI model sys on a complex plane. For SISO systems, pzmap plots poles and zeros of the transfer function.

Program code:

disp ('Poles and zeros of closed loop uncompensated system')

P = pole(Wclun), Z = zero(Wclun)% task 1.2

figure(1)

pzmap(Wclun), grid on

Results of the program:

Poles and zeros of closed loop uncompensated system

P =

-14.8199

-0.5901 + 6.4933i

-0.5901 - 6.4933i

Z =

-10.0000

Fig.3

3) plot the time responses for obtained transfer function; the responses should be produced in two different windows of one figure graphic object using command subplot;

Theoretical information:

There are two main time responses, namely: a step response and an impulse one.

The response of the dynamic system on the step (Heavyside) function at its input is called a step response. To plot the step response of an arbitrary LTI (Linear Time Invariant) model sys on the screen, operator step(sys) is used.

The response of the dynamic system on the -function at its input is called an impulse response. To plot the impulse response of an arbitrary LTI model sys on the screen, operator impulse(sys) is used. The duration of simulation is determined automatically to display the transient behavior of the response.

Program code:

figure(2)

subplot (2,1,1), step(Wclun),

grid on, legend ('Step of uncompensated system')% task 1.3

subplot (2,1,2), impulse(Wclun),

grid on, legend ('Impulse of uncomp...