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Discrete PID control .

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11. Discrete PID Control

This demo shows how to use MATLAB to design and simulate sampled-data control systems. The design uses

frequency domain considerations leading to a pole-cancellation PID control method.

)(zC)(

sPAD/

DA /

u[k]e[k] y(t)u(t)

y[k]

r[k]

-

( )( )

( )

Y zP z

U z=

U(z)

Y(z)

Y(s)

The continuous-time process is given by

( )( ) ( )

1( )1 10 1 5

sseP s P s e

s s

= =

+ +

Observe that the system has a delay term of 1 sec. The sampling time is 1 sec.

Design a series PID compensator to meet the following design specifications:

- Phase Margin60- Settling time should be minimal

- The closed-loop system should follow unit step reference signal without steady error (type one system).

Define the continuous-time process without these time delay.

s=zpk(s);

P1s=1/((1+10*s)*(1+5*s));

Find the discrete-time process model assuming zero-order hold (without delay):

( ) 1 11( )

(1 )P s

P z z Zs

=

By MATLAB: Ts=1;

P1z=c2d(P1s,Ts,zoh);

Note that it is not necessary to include the default zoh string.Add the time-delay to the process by multiplying 1( )P z with

1z , since { } 1se zZ = .The z variable can be defined similarly to the z variable.

z=zpk(z,Ts) Pz=P1z/z

( ) ( ) ( )( )

( ) ( )1

1 1

0.904810.00905

0.9048 0.8187

zP z z P z P z

z z z z

+= = =

The zeros and poles of the discrete system are [zd,pd,kd]=zpkdata(Pz,v)

PI term is necessary to achieve the steady state zero error requirement. PD term is used to accelerate the system

response. The PI and PD break frequencies can be calculated similarly to the continuous system. iT is the largesttime constant of the system and dT is the second largest time constant.

PI : break frequency at 0.1

1

0.110

s

i

T

Te e e

= = = 0.9048

PD : break frequency at 0.2

1

0.20.5

s

d

T

Te e e

= = = 0.8187The discrete controller is

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Discrete PID control .

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0.9048 0.8187( )

1c

z zC z k

z z

=

Thec

kparameter is calculated to set the 60 phase margin.

First assumec

k =1 :

kc=1;

Cz=((z-0.9048)*(z-0.8187))/((z-1)*z);or with the poles of the discrete plant transfer function

Cz=((z-pd(1))*(z- pd(2)))/((z-1)*z)

Calculate the discrete-time loop transfer function ( ) ( ) ( )L z C z P z= . Lz=Cz*Pz;

Lz=minreal(Lz, 0.001); % cancel the zero-pole pairs

Thec

kparameter can be calculated by the margin command or read from a table.

a/ Use the margin command. The bode command tests thes

T sampling time to see if the system continuous or

discrete. If 0s

T = the system is considered to be a continuous-time system, if 0s

T > positive than the system is

considered to be a discrete-time system. [mag,phase,w]=bode(Lz);

kc=margin(mag,phase-60,w);b/Use a table

[numz,denz]=tfdata(Lz,v); [mag,phase,w]=dbode(numz,denz,Ts);

T=[mag, phase, w] % setting up a table

T =

0.0781 -114.8936 0.2200

0.0697 -117.8611 0.2462

0.0622 -121.1822 0.2756

0.0555 -124.8991 0.3084

0.0495 -129.0589 0.3452

It is seen that the phase shift of about -120 (required by the specification of PM=60) is achieved at a magnitude of

0.062. Consequently, gain by 1/0.0622 defines the proper gainc

k as follows:

kc=1/0.0622

The gain of the discrete-time controller is 16c

k = . Notice that the margin has to forms; its input can be a transfer

function or the bode amplitude and phase values.

Verify now the system behaviour. Check the phase margin by the margin command. Cz=kc*((z-pd(1))*(z- pd(2)))/((z-1)*z);

Lz=Cz*Pz; Lz=minreal(Lz);

margin(Lz);

The closed loop performance can be investigated by a Simulink model.

simulink % starts SIMULINK

Create a new file and copy the various blocks. The parameters of the block should be set to the required value.SIMULINK uses the variables defined in the MATLAB workspace.

C(z), P1(s): Control System Toolbox >LTI system :Cz, P1s

Sum: Simulink>Math>Sum: +-

Dead time, delay: Simulink>Continuous>Transport Delay: 1

Step input: Simulink>Sources>Step, Change the Step time parameter to zero

Zero-Order-Hold: Simulink>Discrete> Zero-Order-Hold, Sampling time: Ts

Scope: Simulink>Sinks>Scope

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Discrete PID control .

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Zero-Order

Hold

Transport

Delay

Step ScopeY

ScopeU

P1s

LTI System1

Cz

LTI System

The scope blocks can also be used to transfer the results of the simulation to the Matlab workspace. Change the

parameters in the Scope graphic window properties menu.

Data history: Save data to workspace

Variable name: My for ScopeY and Mu for ScopeU

Matrix format

From the Menu change the Simulation>Parameters>Stop Time parameter to 20

After the simulation the time, the output and control vectors can be gained from the transferred matrix form. ty=My(:,1), y=My(:,2) tu=Mu(:,1), u=Mu(:,2)

Plot the output, y(t) subplot(211), plot(ty,y), grid

and the control signal u(t). The u(t) control signal is the output of the zero order hold. subplot(212), stairs(tu,u) , grid

The discrete u[k] signal can also be ploted. hold on, plot(tu,u,*)

The performance parameters (settling time, overshoot) of the system can be calculated from they and u vectors.

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

0 2 4 6 8 10 12 14 16 18 20-5

0

5

10

15

20