Experiment No. 06
Stability
1. Objective(s):
This activity aims to equip the students with the skills and knowledge in connecting systems and reducing
a multiple sub-systems.
2. Intended Learning Outcomes (ILOs):
At the end of this activity, the students shall be able to
a. Connect systems in parallel, in series and in feedback.
b. Reduce a multiple subsytems.
c. Determine the performance of the reduced system.
3. Discussion
Stability and Frequency Response
Objective:
At the end of this laboratory exercise, you should be able to:
Determine the stability of a system by analyzing its step response, pole – zero map and
frequency response
Plot the frequency response of a system with the aid of MATLAB
Tools Required:
Computer Workstation
MATLAB Software
The total response of a system is the sum of the forced and natural responses or
c(t)= c_forced (t)+c_natural (t)
Using these concepts, we present the following definitions of stability, instability, and marginal
stability:
A linear, time-invariant system is stable if the natural response approaches zero as time
approaches infinity.
A linear, time-invariant system is unstable if the natural response grows without bound as time
approaches infinity.
A linear, time-invariant system is marginally stable if the natural response neither decays nor
grows, but remains constant or oscillates as time approaches infinity.
Thus the definition of stability implies that only the forced response remains as the natural
response approaches zero.
Mathematically, stability for linear, time-invariant systems can be determined from the location of the poles
in the pole – zero map. If the poles are only in the left half-plane, the system is stable. If any poles are in
the right half – plane, the system is unstable. If the poles are on the jω – axis and in the left half-plane,
the system is marginally stable as long as the poles on the jω – axis are of unit multiplicity; it is unstable if
there are any multiple jω poles.
MATLAB Commands Review:
Generation of LTI models
1. Transfer functions
SYS = tf (NUM, DEN) creates a continuous-time transfer function SYS with numerator(s) coefficients,
NUM and denominator(s) coefficients, DEN. The output SYS is a tf object.
2. Zero – pole – gain
SYS = zpk (Z, P, K) creates a continuous-time zero-pole-gain (ZPK) model SYS with zeroes vector Z;
poles vector P, and gain K. The output SYS is a zpk object.
3. Frequency Response Data (FRD) models are useful for storing frequency responses of LTI
systems, including experimental response data.
Creation:
SYS = frd (RESPONSE, FREQS) creates an frd model SYS with response data in the RESPONSE vector
at frequency points in FREQS vector. The output SYS is an frd model.
4. State-space
SYS = ss (A, B, C, D) creates a continuous-time state-space (SS) model SYS with matrices A, B, C, D.
The output SYS is anss object. You can set D=0 to mean the zero matrix of appropriate
dimensions.
Data Retrieval
[NUM, DEN]=tfdata (sys,’v’) returns the numerator(s) coefficients and denominator(s) coefficients of the
transfer function SYS. For a transfer function with NY outputs and NU inputs, NUM and DEN are NYby-NU cell arrays where the (I, J) entry specifies the transfer function from input J to output I. SYS is
first converted to transfer function, if necessary.
[Z, P, K] = zpkdata (sys,’v’) returns the zeroes, poles, and gain for each I/O channel of the LTI model SYS.
The cell arrays Z, P and the matrix K have as many rows as outputs and as many columns as inputs,
and their (I, J) entries specify the zeroes, poles, and gain of the transfer function from input J to output
I. SYS is first converted to zero – pole – gain format if necessary.
[A, B, C, D] = ssdata (sys) retrieves the matrix data A, B, C, and D for the state – space model SYS. If SYS
is not a state-space model, it is first converted to a state – space representation.
[RESPONSE, FREQ] = frdata (SYS) returns the response data and frequency samples of the frequency
response data (FRD) model SYS.
Note: applicable only to FRD data!!!
Connecting systems
SYS = series (SYS1, SYS2) connects two LTI models
P1 and P2 are LTI systems in series.
SYS = parallel (SYS1, SYS2) connects the two LTI models P1 and P2 in parallel.
SYS = feedback (P1, P2, SIGN) computes an LTI model SYS for the closed – loop feedback system.
Negative feedback is assumed and the resulting system SYS maps r to c. To apply positive feedback,
use the syntax SYS = feedback (P1, P2, +1).
4. Resources:
To perform this activity, a computer workstation with MATLAB R2012a or higher installed is required. For
MATLAB, the control systems toolbox is required.
5. Procedure:
1. Get the overall transfer function of the systems given below and use ltiview to see the step responses
and pole – zero maps.
a. System A
+
R(s)
C(s)
-
E(s)
b. System B
+
R(s)
C(s)
-
E(s)
c. System C
+
R(s)
C(s)
-
E(s)
d. System D
+
e. System E
+
R(s)
C(s)
Which of the system is stable, unstable or marginally stable? How would you relate the stability of a system
with the pole location?
Stability
Course Code/Section:
Kalalo – Experiment 5
Orosco – Experiment 6
Ramos – Experiment 6
Roble – Experiment 5
6. Data and Results:
a.
Experiment No.:
6
Date Performed:
Date Submitted:
Instructor:
b.
c.
d.
e.
Stability:
a.
b.
c.
d.
e.
Stable
Unstable
Unstable
Unstable
Unstable
Roots of this characteristic equation gives the eigenvalues of the system , if the eigenvalues are negative
in nature, then the system is stable, otherwise not stable. When we plot these poles on S plane, all poles
falls on the left side of S plane. If any pole lie in right half, it means eigenvalue is positive, due to this does
not converge at certain point. Which leads towards unstable nature of system. When you plot the phase
portrait of such system it is shown below, all the arrows moves away from the origin.
7. Conclusion:
For this experiment, We therefore conclude that reducing of sub system and determining its stability would
be very easy with use of Matlab a two or three liner codes would show you the location of its poles thus
you can determine its stability.