Lecture 6: Frequency Response Analysis and Stability. Objectives. To study system stability in the frequency domain using the Bode stability method. To understand the concepts of gain and phase margins. Frequency Response: determines the response of systems variables to a sine input.
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Frequency Response Analysis and Stability
Frequency Response: determines the response of systems variables to a sine input.
Why do we study frequency response?
No!
Yes!
Yes!
Frequency Response : Sine in sine out
0.2
Y, outlet from system
0
0.2
0.4
0
1
2
3
4
5
6
time
1
0.5
X, inlet to system
0
0.5
1
0
1
2
3
4
5
6
time
Frequency Response : Sine in sine out
Amplitude ratio = Y’(t) max / X’(t) max
Phase angle = phase difference between input and output
P
B
output
P’
A
input
Frequency Response : Sine in sine out
Amplitude ratio = Y’(t) max / X’(t) max
Phase angle = phase difference between input and output
These calculations are tedious by hand but easily performed in standard programming languages.
In most programming languages, the absolute value gives the magnitude of a complex number.
Frequency response of mixing tank.
Timedomain
behavior.
0.6
0.4
0.2
20
0
v1
0
0.2
0
20
40
60
80
100
120
20
TC
100
40
0
20
40
60
80
120
v2
STABILITY
No!
or
Yes!
We influence stability when we implement control. How do we achieve the influence we want?
1
0.5
0.5
0
0
0.5
0.5
1
1
1.5
1.5
0
0
0.5
0.5
1
1
First, let’s define stability: A system is stable if all bounded inputs to the system result in bounded outputs.
Sample
Inputs
Sample
Outputs
Process
bounded
bounded
unbounded
unbounded
The denominator
determines
the stability of
the closedloop
feedback system!
Set point response
Bode Stability Method
Calculating the roots is easy with standard software. However, if the equation has a dead time, the term e s appears. Therefore, we need another method.
The method we will use next is the Bode Stability Method.
E(s)
CV(s)
MV(s)
+
+
GC(s)
Gv(s)
GP(s)
+

CVm(s)
Loop open
GS(s)
Bode Stability: To understand, let’s do a thought experiment
No forcing!!
SP(s)
SP(s)
E(s)
E(s)
CV(s)
CV(s)
MV(s)
MV(s)
+
+
+
+
GC(s)
GC(s)
Gv(s)
Gv(s)
GP(s)
GP(s)
+
+


CVm(s)
CVm(s)
Loop closed
Loop closed
GS(s)
GS(s)
Bode Stability: To understand, let’s do a thought experiment
Under what conditions is the system stable (unstable)?
Hint: think about the sine wave as it travels around the loop once.
E(s)
CV(s)
MV(s)
+
+
GC(s)
Gv(s)
GP(s)
+

CVm(s)
Loop closed
GS(s)
Bode Stability: To understand, let’s do a thought experiment
If the sine is larger in amplitude after one cycle; then it will increase each “time around” the loop. The system will be unstable.
Now: at what frequency does the sine most reinforce itself?
E(s)
CV(s)
MV(s)
+
+
GC(s)
Gv(s)
GP(s)
+

CVm(s)
Loop closed
GS(s)
Bode Stability: To understand, let’s do a thought experiment
When the sine has a lag of 180° due to element dynamics, the feedback will reinforce the oscillation (remember the  sign).
This is the critical or phase crossover frequency, pc.
Let’s put the results together. GOL(s) includes all elements in the closed loop.
At the critical frequency: GOL(pcj) = 180
The amplitude ratio: GOL(pcj)  < 1 for stability
GOL(pcj)  > 1 for instability
The gain margin (GM) is define as:
GM =
Hence, if the gain margin is less than 1 (negative in dB), the system is unstable.
Another relevant term is the phase margin. To calculate it, we need to find the gain crossover frequency(gc) whichis the frequency at which the openloop gain crosses unity.
The phase margin (PM) is the distance between the open loop phase and 180 at frequency gc
PM =
If the phase margin is negative, the system is unstable.
The best way to understand these two points is to try a numerical example!
Find the gain and phase margin for the following process under proportional control. Is the system stable?
.
Find the delay margin as well.
.
=
(system stable)
dB.
.
= 2tan1()
= 12tan1(1)
=  1 
= 2.57 rad
= 147.3.
= +
= 0.57 rad
= 32.7.
=
=
= 0.57 sec.
We can check our result using the command “margin” in Matlab.
s=tf(‘s’);
G=2/(s+1)^2*exp(s);
margin(G)
Let us now use Simulink to understand the interpretation of gain margin and phase margins.
The system is stable.
The system is unstable.
Remember that the gain margin is 1.36.
Remember that the phase margin tells us that additional delay must not exceed 0.57.
The system is unstable.