‫بسم هللا الرحمن الرحيم‬
Advanced Control
Discrete forms of PID controllers
Reference: A. Visioli, Practical PID Control, Springer 2006
Computer Control
1.
The computer requests a value from the A/D converter. The A/D
converter samples the process signal, converts it to a number, and stores it
in the computer memory or a register.
2.
The computer performs the control calculations on the sampled process
signal(s) and computes the output(s) to the process.
3.
The computer output is sent to the D/A converter, which converts it to an
electronic signal, updates the output, and holds it constant until the next
update.
Computer Control
A good rule of thumb is that the sample time should be about onetenth of the effective process time constant
Discrete Form of PID Controllers
 Position Form
u (t )  u s  K c e(t ) 
Kc
I

t
0
e(t )dt  K c D
de(t )
dt
Sampling Time : Ts , Number of Sampling : k , Time : t = kTs
Upper rectangula r approximat iom :
k
 e(t )dt   e(iT )T
t
0
i 1
s
de(t k ) e(kTs )  e((k  1)Ts )
Backward Finite Difference :

dt
Ts
s
Discrete Form of PID Controllers
 Position Form
u (t )  u s  K c e(t ) 
Kc
u(k )  us  K c e(k ) 
K cTs
I

t
0
I
e(t )dt  K c D
K
 e(i) 
i 1
de(t )
dt
K c D
e(k )  e(k 1)
Ts
 Velocity Form
u (k )  u (k  1)  K c e(k )  e(k  1) 
K cTs
I
K c D
e(k )  2e(k  1)  e(k  2)
e( k ) 
Ts
Discrete Form of PID Controllers
 Velocity Form
u(k )  u(k  1)  g 0e(k )  g1e(k  1)  g 2e(k  2)
Where:
 Ts  D 
g 0  K c 1   
  I Ts 
 2 D 

g1   K c 1 
Ts 

K c D
g2 
Ts
Discrete Form of PID Controllers
 Backward Shift Operator (q -1) : y(k-n)=q-ny(k)
u (k ) g 0  g1q 1  g 2 q 2

e(k )
1  q 1
Tuning of Digital PID : Moore et al. (1969)
Use the continuous tuning formula of PID controller with corrected
dead time
Ts
t0 c  t0 
2