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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 uncompensated system')

Results of the program:

Fig.4

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

Theoretical information:

Settling time is the period of time from the beginning of transients to the moment of time, after which the inequality

takes place, where is given small constant value, which is usually equal to of the steady state value. Or in other words, it is the time for the system output to settle down to within a tolerance band of the final value, normally between 2 or 5%.

Overshoot is maximal deflection of the output value in the transient process of the transient process from the steady state value. It normally expressed as percentage determined as:

,

where is maximal deflection of the transient process.

In the most cases a system would has sufficient stability margins if overshoot is less or equal to .

Results of the program:

From the Fig.4:

Overshoot of closed loop compensated system is 81.8%.

Settling time of closed loop uncompensated system is 6.76 sec.

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

Theoretical information:

Hurwitz stability criterion

The characteristic equation of order system is

.

For this system to be stable it is necessary and sufficient that the determinant of Hurwitz matrix and the determinants of all its diagonal minors are positive.

Hurwitz matrix is

.

The main diagonal of Hurwitz matrix contains coefficients of characteristic equation beginning with to . Elements which are located above the main diagonal have increasing indices; elements which are located under the main diagonal have decreasing indices. So odd-numbered lines contain coefficients of characteristic equation with odd indexes, and even-numbered lines contain coefficients of characteristic equation with even indexes. Places, in which coefficients are absent, are filled by zeros.

Program code:

[n0, d0]=tfdata (Wclun, 'v')

if d0 (1)>0&d0 (2)>0&d0 (3)>0&d0 (4)>0&(d0 (2)*d0 (3) - d0 (1)*d0 (4))>0

disp ('System is stable by Hyrwitz criterion')

disp ('because all coeficients in Hurwitz matrix are positive')

disp ('and particular case for 3rd order s-m is true')

else disp ('system is unstable')

end% task 1.5

Results of the program:

n0 =

0 0 6.3000 63.0000

d0 =

0.1000 1.6000 6.0000 63.0000

System is stable by Hyrwitz criterion because all coeficients in Hurwitz matrix are positive and particular case for 3rd order s-m is true

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

Theoretical information:

There are two formulations of Nyquist criterion of stability depending on whether the open-loop system is stable or not stable.

If the open-loop system is stable, the closed loop system would be stable if the Nyquist plot of the open-loop system does not encircle the point with coordinates (-1, 0) at the complex plane.

If the open-loop system is unstable, the closed loop system would be stable if the Nyquist plot of the open-loop system encircles the point with coordinates (-1, 0) at the complex plane times, where is the number of roots in RHP, in the counter-clockwise direction when frequency changes from zero to infinity.

Operator nyquist(sys) plots the Nyquist response of an arbitrary LTI model sys.

Program code:

figure(3), nyquist(Wolun), grid on% task 1.6

Results of the program:

Fig. 5

System is stable because Nyquist plot for the open-loop system doesn`t encircle point (-1; 0).

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

Theoretical information:

Stability margins

There is a special operator margin for calculating the stability margins in MatLab. Its syntax is

[Gm, Pm, Wcg, Wcp]=margin(sys),

where is gain margin, is phase margin, is frequency under which , is frequency under which . and are crossover frequencies.

When invoked without left-hand arguments, margin(sys) plots the open-loop Bode response with the gain and phase margins marked by vertical lines.

Program code:

figure(4), margin(Wolun), grid on, legend ('Uncompensated system')% task 1.7

Results of the program:



Fig. 6

So, gain margin is equal to infinity and phase margin - 41.1 deg.

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

Theoretical information:

State space description

Complete state space description of a continuous system with constant coefficients is the following:

where is the input vector; is the output vector; is the state vector characterizing the system dynamics.

To create the continuous-time LTI (Linear Time Invariant) models in MatLab it is necessary to use operator ss.

For example,

sys=ss (A, B, C, D)

creates the state space of a continuous-time system.

Relationship between state space and transfer function description

The diagram explaining the relationship between the state space description and description with the transfer functions is represented in fig. 7.

Fig. 7

For the continuous case we have

.

In MatLab operator ss2tf carries out conversion from state space description to transfer functions. If output is a vector than numerator Num of the transfer function is a matrix, each row of which contains numerator polynomial coefficients of the corresponding transfer function (from input to the first, second and so on outputs). If it is necessary to define any concrete transfer function then it is necessary to take the corresponding row from the matrix Num, for example tf2=tf (Num(2,:), Den). Denominator would be common for all transfer functions.

Program code:

[A0, B0, C0, D0] = tf2ss (n0, d0)

Results of the program:

A0 =

-16 -60 -630

1 0 0

0 1 0

B0 =

1

0

0

C0 =

0 63.0000 630.0000

D0 =

0

9) calculate the eigenvalues of the uncompensated system;

Theoretical information:

Eigenvalues

The eigenvalues of the square matrix are the roots of the equation: , where is a unit matrix. The result of using the operator is a vector containing the eigenvalues of the square matrix .

Program code:

eigenvalues = eig(Wclun)% task 1.9

Results of the program:

eigenvalues =

-14.8199

-0.5901 + 6.4933i

-0.5901 - 6.4933i

10) calculate -norm of the uncompensated system.

Theoretical information:

Calculation of -norm

The -norm of a stable continuous system with a transfer function is a square root from average value of a square of impulse response of a system, and at conversion to Laplace transformation according to Parseval theorem, this norm is determined by the following way:

,

where is trace of matrix; is matrix conjugated to ; is the integral of the error squared under input signal which is equal to -function.

The index can be calculated with the help of the controllability or observability gramians. The function gram allows to calculate Gram function for an estimation of system controllability, which is called controllability gramian. Controllability gramian is applied to research controllability properties of the system models set in state space, and also for construction of their minimal realizations. It is more convenient for calculations, than controllability matrix.

Controllability gramian of this system is determined with the integral:

.

In MatLab: Gc=gram (A, B)

Controllability gramian is positive defined if and only if the pair of matrices is controlled. Controllability gramian is calculated with the solution of the continuous Lyapunov equation:

.

In MatLab: Gcc=lyap (A, B*B').

Observability gramian of the system is determined with the integral:

.

In MatLab: Go=gram (A, C).

Observability gramian is positive defined if and only if the pair of matrixes is observed. Observability gramian is calculated with the solution of the continuous Lyapunov equation:

.

In MatLab: Goo=lyap (A, C'*C).

Constraint is that the matrix should be stable, i.e. for continuous models all eigenvalues should have negative real parts.

It is illustrated in MATLAB using the following commands: gram (controllability gramian), lyap (the solution of Lyapunov equation), trace (a trace of matrix).

Gc=gram (A, B);%or Gc=lyap (A, B*B');

J=sqrt (trace(C*Gc*C'));

%or J=sqrt (trace(B'*Go*B)), where Go is observability gramian.

-norm can also be determined in MatLab as follow:

h2_n=normh2 (A, B, C, D)

-norm accepts the value which is equal to infinity in the following cases:

- the model is unstable or neutral;

- the continuous model has a nonzero matrix D.

Program code:

H2op_loop = normh2 (A0, B0, C0, D0)% task 1.10

Results of the program:

H2op_loop =

4.6137

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

Theoretical information:

Synthesis of PD-controllers

Transfer function of ideal PD-controller is the following:

,

where is the gain of proportional part; is the gain of differential one.

Differentiator gain increased unrestrictedly results in the infinite gain on high frequencies. So, in order to restrict the gain on high frequencies an additional pole is appended in the differential component of PD-controller. In this case the transfer function of real PD-controller can be written as:

,

where is a very small value (). If is known so it is necessary to determine two parameters in the synthesis procedure.

To synthesis of PD-controller, first of all, it is necessary to calculate the phase shift of controller at frequency as

and to determine

.

Then is determined as

.

In MatLab:

theta=-pi+Pm/57.296-phase/57.296

Kp=cos(theta)/mag

Kd=sin(theta)/(om*mag)

% Creation of the transfer function of real PD-controller

Wc=tf([Kp*tau+Kd Kp], [tau 1])

Program code:

disp ('Task 2. Synthesis of real PD-compensator')

t1=0.0001;

w=2.5;

Pm=60;

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

Program code:

disp ('Phase shift of compensator at given frequency w ')

[mag, phase] = bode (Wolun, w)% task 2.1

theta=-pi+Pm/57.296-phase/57.296;

Results of the program:

mag =

3.8769

phase =

-112.6199

2) determine the gain of proportional part of compensator ;

Program code:

disp ('Coefficient of proportional unit ')

Kp=cos(theta)/mag% task 2.2

Results of the program:

Coefficient of proportional unit

Kp =

0.2558

3) determine the gain of differential part of compensator ;

Program code:

disp ('Coefficient of differentiated unit ')

Kd=sin(theta)/(w*mag)% task 2.3

Results of the program:

Coefficient of differentiated unit

Kd =

-0.0133

4) plot Bode diagram of the PD-compensator.

Program code:

disp ('Tf of compensation unit')

Wc=tf([Kp*t1+Kd Kp], [t1 1])% tf of compensation unit

disp ('Tf of compensator')

Wcomp=Wc% tf of compensator

figure(5)

bode(Wcomp), grid on,

legend('Compensator')% task 2.4

Results of the program:

Tf of compensation unit

Transfer function:

-0.01323 s + 0.2558

-

0.0001 s + 1

Tf of compensator

Transfer function:

-0.01323 s + 0.2558

-

0.0001 s + 1

Fig. 7

III. Analysis of compensated system:

1) build in Simulink the compensated system;



Fig. 8

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

Fig. 9

Settling time - 2.4; overshoot - 23.4%.

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

Program code:

disp ('Tf of open-loop compensated system')

Wolcomp = series (Wolun, Wcomp)

disp ('Tf of closed-loop compensated system')

Wclcomp=feedback (Wolcomp, W5, - 1)% task 3.3

Results of the program:

Tf of open-loop compensated system

Transfer function:

-0.8334 s + 16.12

-

0.0001 s^3 + 1.001 s^2 + 6 s

Tf of closed-loop compensated system

Transfer function:

-0.08334 s^2 + 0.7781 s + 16.12

-

1e-005 s^4 + 0.1002 s^3 + 1.601 s^2 + 5.167 s + 16.12

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;

Program code:

figure(7)

step(Wclcomp), grid on,

hold on, step(Wclun),

legend ('Closed-loop compensated', 'Closed-loop uncompensated')% task 3.4

Results of the program:

Fig. 10

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

Program code:

disp ('Poles of closed-loop compensated system')

pole(Wclcomp),

figure(8)

pzmap(Wclcomp), grid on% task 3.5

Results of the program:

Poles of closed-loop copmensated system

ans =

1.0e+003 *

-10.0000

-0.0130

-0.0015 + 0.0032i

-0.0015 - 0.0032i

Fig. 11

So the system is stable as all poles are negative.

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

Program code:

figure(9)

margin(Wolcomp), grid on,

legend ('Compensated system')% task 3.6

Results of the program:

Fig. 12

So gain margin is 17.1 dB and phase margin - 60 deg.

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

Program code:

[n1, d1]=tfdata (Wclcomp, 'v'),

[A1, B1, C1, D1]=tf2ss (n1, d1)% task 3.7

Results of the program:

n1 =

0 0 -0.0833 0.7781 16.1154

d1 =

0.0000 0.1002 1.6006 5.1666 16.1154

A1 =

1.0e+006 *

-0.0100 -0.1601 -0.5167 -1.6115

0.0000 0 0 0

0 0.0000 0 0

0 0 0.0000 0

B1 =

1

0

0

0

C1 =

1.0e+006 *

0 -0.0083 0.0778 1.6115

D1 =

0

8) calculate the eigenvalues of the compensated system;

Program code:

cl_loop_eigen=eig(A1)% task 3.8

Results of the program:

cl_loop_eigen =

1.0e+003 *

-10.0000

-0.0130

-0.0015 + 0.0032i

-0.0015 - 0.0032i

9) calculate -norm of the compensated system;

Program code:

H2cl_loop=normh2 (A1, B1, C1, D1)% task 3.9

Results of the program:

H2cl_loop =

1.4874

Conclusion

In this term paper we estimated properties of uncompensated dynamic system, synthesized real PD-compensator and as a result of including it in our system is changed. Also we estimated properties of compensated system and compared them with corresponding properties of uncompensated system. So we can say that overshoot is less and settling time of compensated system is more than in uncompensated system.

References

uncompensated system phase

1. Dorf R.C., Bishop R.H. Modern Control Systems. - Pearson Education, Inc., Pearson Prentice Hall, 2008. - 1046 p.

2. Бессекерский В.А., Попов Е.П. Теория систем автоматического управления. - СПб.: Профессия, 2003. - 752 с.

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