Control System - Stability of a control system

Control Systems
L H E Wijesinghe
BEng (Hons) in Electronic Engineering (SHU-UK)
MSc in Applied Electronics (University of Colombo – SL )

Learning Outcome
i. Stability (Asymptotic Stability, BIBO (Bounded Input Bounded
Output)Stability)
ii. Methods of determining stability

Dynamic System
System System
x(t) y(t) x(t) y(t)
Multi Input Multi Output System Single Input Single Output System
Hence forth our attention is focused to single input single output
systems (SISO)

SISO : Linear and Non Leaner System
SISO
Linear System
Non Linear System
𝑑2
𝑦(𝑡)
𝑑 𝑡2
+
2𝑑𝑦(𝑡)
𝑑𝑡
+3 𝑦(𝑡)=𝑥(𝑡)
𝑑2
𝑦 (𝑡)
𝑑 𝑡
2
+𝑡 𝑦 (𝑡)=𝑥 (𝑡)
𝑑2
𝑦(𝑡)
𝑑 𝑡
2
∗
𝑑𝑦 (𝑡)
𝑑𝑡
+
2𝑑𝑦 (𝑡)
𝑑𝑡
+3 𝑦 (𝑡)=𝑥(𝑡)
𝑒
𝑦 (𝑡)
𝑑𝑦 (𝑡)
𝑑𝑡
+3 𝑦(𝑡)=𝑥(𝑡)
Under linear system addition
and scalar multiplications are
preserved

SISO : Time invariant and variant System
SISO
Time invariant System
Time variant System
𝑑2
𝑦(𝑡)
𝑑 𝑡2
+
2𝑑𝑦(𝑡)
𝑑𝑡
+3 𝑦(𝑡)=𝑥(𝑡) 𝑑2
𝑦 (𝑡)
𝑑 𝑡2
+
𝑡 𝑑𝑦 (𝑡)
𝑑𝑡
+3 𝑦 (𝑡)=𝑥(𝑡)
𝑑2
𝑦(𝑡)
𝑑 𝑡2
∗
𝑑𝑦 (𝑡)
𝑑𝑡
+
2𝑑𝑦 (𝑡)
𝑑𝑡
+3 𝑦 (𝑡)=𝑥 (𝑡)
𝑡2
𝑑2
𝑦(𝑡)
𝑑 𝑡2
+
𝑡 𝑑𝑦(𝑡)
𝑑𝑡
+4 𝑦 (𝑡)=𝑥 (𝑡)

Conclusion
Dynamic System
SISO System
None Linear
Time Variant
Time Invariant
(coefficient free of t)
Linear
(Coefficient free of y(t))
Differential equations
with constant coefficients

• In this context we consider .Single Input ,continues time ,
linear, Time Invariant Dynamical Systems
• That can be modelled by constant coefficient differential
equations

Transfer Function of a dynamical system
from the above class
The transfer function of a system is the ratio between the
Laplace Transform of the output to the input when the
system is relaxed ()
Eg : Obtain the Transfer Function of the system descripted
by
System
x(t) y(t)
2 𝑑2
𝑦 (𝑡)
𝑑 𝑡2
+
3 𝑑𝑦 (𝑡)
𝑑𝑡
+2 𝑦 (𝑡)=
𝑑 𝑥 (𝑡)
𝑑𝑡
+2 𝑥(𝑡)

}
}
2𝑆2
𝑌 (𝑠)−2𝑆𝑦 (𝑠)−2 𝑦′
(0)+3 𝑆𝑌 (𝑠)−3 𝑦(0)+2𝑌 (𝑠)=𝑆𝑋 (𝑠)−𝑥′
(0)+2 𝑋 (𝑠)
2 𝑆2
𝑌 (𝑠 )+3 𝑆𝑌 (𝑠 )+2 𝑌 ( 𝑠)=𝑆𝑋 (𝑠 )+2 𝑋 (𝑠 )
(
𝑌 ( 𝑠 )
X ( s )
=
s + 2
( 2 𝑆
2
+ 3 𝑆 +2 ¿
¿

MATLAB Code
>> sys = tf ([1 2],[2 3 2]);
sys =

Stability
Definition 1 :
A system is said to be absolutely stable if it is sustainable
under normal operation conditions (Roughly Speaking)
Definition 2 :BIBO Stability
A system is said to be absolutely stable if it produces
bounded output for every bounded input (BIBO).

Stability
Definition 3 :
System is said to be absolutely stable if its impulse goes to
zero as time goes to infinity

MATLAB :Impulse Response
G(s)=
When X(s)=δ(s)  Y(s)=H(s)=Impulse response
>>sys = tf ([8 18 32],[1 6 14 24])
>>impulse(sys)

Stability
Definition 4:Asymptotic Stability
System is said to be absolutely stable if all the system poles
Lie in the left half of the s plane.

Methods of Determining Stability

Example : Root Locus diagram
A system is said to be absolutely stable if all the poles lie in
the left half of a s-plane.
Ex-:
Poles = -1 , -8
Zeroes = -j
Y=jω
X = σ
stable Unstable
Marginally
stable
-8 -1
-1
S plane

Exercise
Check the stability of the system with following transfer
functions

MATLAB : Root Locus diagram
>> sys = tf ([1 4 3],[1 6 8])
>> rlocus (sys)

Feedback
Y(s) = G(s)[(X(s) – H(s)*Y(s)]
Y(s)/X(s) = G(s)[(1– H(s)*[Y(s)/X(s)]]

= Proportional Controller
= Proportional Integral Controller
= Proportional Integral Derivative Contro

Example: Routh Hurwitz Criteria
If the Open loop transfer function of a system
Obtain the range of for which the system is absolutely stable using
Routh Hurwitz Criteria

1 120
34 K
A1 0
A2
A1 = -((1*K) – (34*120))/34
= (4080-K)/34
A2 = -((34*0) – (K*A1))/ A1
= K

NOTE
Number of roots in the right half plane = Number of sign
changes in the 1st
column
Therefor
To place all the roots in the left half plane = number of sign
changes should be
So
and
system to be stable

Exercise
Find the close loop controller gain for the following open loop
systems .

Polar Plots
Imaginary Axis
r
x
θ
y Real Axis

Parameters – Gain and Phase
In logarithmic coordinates , gain/phase versus frequency plot
known as Bode plots.
Ex -:
We can define s = σ + jω assume σ =0

Parameters – Gain and Phase
• Find r and θ of the above complex function of the gain.
• plot with (log scale )will frequency response of the system
• plot (liner scale)with (log scale)will phase response of the
system

Exercise
Plot bode plots for the circuits in the 2 lecture

MATLAB : Bode Plot
>> sys = tf ([1 4 3],[1 6 8])
>> bode (sys)

Control System - Stability of a control system

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