Example I
Consider the first example of
G(s)=k2s+1=bs+a
We shall try to control this system with a PI control.
G(t)=KI∫edt+KCe(t)=[KI,KC][∫edt,e(t)]T
Let x1=∫edt, this means that ˙x1=x2=e(t)
G(t)=[KI,KC][∫edt,e(t)]T=[KI,KC][x1,x2]T
As per the State Space formulation,
˙X=AX+BU
Note,
˙x2=˙e
e=r−y=−bus+a
˙e+ea=−bu
˙x2=−x2a−bu
and from the definition given above,
˙x1=x2
Thus we can form both A and B matrices.
A=010−a
B=0−b
Also U=−kX,
˙X=AX+BU
sX=AX+B(−kX)
X(sI−A+kB)=0
Thus the eigenvalues for the closed loop system will be the solutions to the equation,
|sI−A+kB|=0
Desired closed loop Transfer Function is,
D(s)=1λs+1
But as we have a second order system, we restate the desired closed loop T.F for comparison purposes,
D(s)=(zs+1)(λs+1)(zs+1)
Thus to find the k value we can now compare the desired to actual closed loop pole equations.
Example II
Given system T.F G(s)=1s,
we need to find the Proportional control for this system.
u(t)=Kce(t)
we define x1=e(t)
from the diagram we can see,
e=r−y
e=−us
˙e=−u
which means ˙x1=−u
Thus,
A1×1=0
B1×1=−1
The desired closed loop T.F is,
GCL(s)=1λs+1
Similarly as the first example find k by solving,
|sI−A+kB|=λs+1
Example III
This time consider a 2nd order system,
G(s)=k(z1s+1)(z2s+1)
While the desired closed loop system is,
GCL(s)=1λs+1
We restate the specifications because the system is a second order system.
GCL(s)=(z1s+1)(z2s+1)(z1s+1)(z2s+1)(λs+1)
We will now control this system using PID,
u=KI∫edt+KCe(t)+KDdedt
u=[KI,KC,KD][∫edt,e(t),dedt]T
Lets define the following terms,
x1=∫edt;˙x1=x2=e(t);˙x2=x3=dedt
So we can rewrite u as,
u=[KI,KC,KD][x1,x2,x3]T
As e=r−y=−ku(z1s+1)(z2s+1)
double differentiating,
¨e=f(x1,x2,x3)+g(u)
From these we can find A3×3 and B3×1
Using these we can find k by solving,
|sI−A+kB|=(λs+1)(z1s+1)(z2s+1)
Inference
- So we have used PI/PID control via State Space formulation for the above system specifications.
- IMC based tuning or synthesis may also be used for these problems
- The tuning procedure is straightforward and relatively simple. The solution of all systems take the same form.
- We prefer PI/PID control via State Space formulation when the desired closed loop system specifications are performance based.
that was very helpful. Keep up the good work
ReplyDeleteKeep up the good work!
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