Upload
maano1
View
221
Download
0
Embed Size (px)
Citation preview
7/27/2019 Lecture2 Methods of Determining Stability-rhc
1/30
Methods of Determining Stability
Eng R. L. Nkumbwa
Copperbelt University
2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
2/30
Determining Stability
For all practical purposes, there is no need to
compute the complete system response todetermine stability.
When the system parameters are all know the
roots of the characteristic equation can be found
using various methods including the AutomaticControl Systems Software (ACSYS) in
conjunction with MATLAB Toolbox.
10/3/20132 Eng. R. L. Nkumbwa @ CBU2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
3/30
Determining Stability
What are some of the methods for
determining stability in control engineering?
10/3/20133 Eng. R. L. Nkumbwa @ CBU2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
4/30
Methods for Determining Stability
RouthHurwitz Criterion this is an algebraic method
that provides information on the absolute stability of a
system that has as characteristic equation with constant
coefficients. The criterion tests whether any of the roots
of the characteristic equation lie in the right half s-plane.
Nyquist Criterion this is a semi graphical method that
gives information on the difference between the numberof poles and zeros of the closed loop transfer function
that are in the right half s-plane by observing the
behavior of the Nyquist plot of the loop transfer function.
10/3/20134 Eng. R. L. Nkumbwa @ CBU2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
5/30
Methods for Determining Stability
Bode Diagram this is the diagram plot ofthe magnitude of the loop transfer functionG(j) H (j) in dB and the phase of G (j)
H (j) in degrees, all verses frequency .
The stability of the closed loop system can bedetermined by observing the behavior ofthese plots.
10/3/20135 Eng. R. L. Nkumbwa @ CBU2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
6/30
Tip Off
It will be evident throughout this course, that
most of the design and analysis techniqueson control systems will represent alternative
methods of solving the same problem.
The designer simply has to choose the bestanalytical tool, depending on the particular
situation.
10/3/20136 Eng. R. L. Nkumbwa @ CBU2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
7/30
Tip Off
In the course of normal system design, it is not
merely necessary that the system be stable. It is essential that the system be sufficiently
stable that transient disturbances will decay
quickly enough to permit rapid recovery by the
controlled variable. In the next sections, however, the major
emphasis is on concepts and methods of
determining system stability.10/3/20137 Eng. R. L. Nkumbwa @ CBU2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
8/30
Routh-Hurwitz Criterion
What is RHC Analysis?
The RouthHurwitz Criterion (RHC) representsa method of determining the location of zeros
of a polynomial with constant real coefficients
with respect to the left half and right half of the
s-plane, without actually solving for the zeros.
10/3/20138 Eng. R. L. Nkumbwa @ CBU2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
9/30
Reminder
10/3/20139 Eng. R. L. Nkumbwa @ CBU2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
10/30
Output Signals Characteristic
for Stability, Neutral & Unstable
10/3/201310 Eng. R. L. Nkumbwa @ CBU2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
11/30
Stability Condition
A necessary and sufficient condition for a
feedback system to be stable is that allthe poles of the system transfer function
have negative real parts.
10/3/201311 Eng. R. L. Nkumbwa @ CBU2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
12/30
7/27/2019 Lecture2 Methods of Determining Stability-rhc
13/30
Routh-Hurwitz Criterion Background
The stability analysis has occupied the interest ofmany engineers including Maxwell andVishnegradsky.
In the late 1800s, A. Hurwitz and E.J. Routhindependently published a method ofinvestigating the stability of a linear system.
The Routh-Hurwitz stability method provides ananswer to the question of stability by consideringthe characteristic equation of the system.
10/3/201313 Eng. R. L. Nkumbwa @ CBU2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
14/30
Routh-Hurwitz Criterion
The Routh-Hurwitz criterion is based on
ordering the coefficients of the characteristicequation:
10/3/201314 Eng. R. L. Nkumbwa @ CBU2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
15/30
Routhian Array
Into an array as follows:
10/3/201315 Eng. R. L. Nkumbwa @ CBU2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
16/30
Routhian Rows
The rows of the schedule are then completed
as follows:
10/3/201316 Eng. R. L. Nkumbwa @ CBU2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
17/30
Routhian Rows
10/3/201317 Eng. R. L. Nkumbwa @ CBU2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
18/30
Hurwitz Determinants
Where,
10/3/201318 Eng. R. L. Nkumbwa @ CBU2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
19/30
RHC Principle
The Routh-Hurwitz criterion states that thenumber of roots of D(s) with positive real parts isequal to the number of changes in sign in thefirst column of the Routh Array.
This criterion requires that there be no changesin sign in the first column for a stable system.
This requirement is both necessary andsufficient.
10/3/201319 Eng. R. L. Nkumbwa @ CBU2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
20/30
Four Cases for First Column Array
No element in the first column is zero.
There is a zero in the first column, but otherelements of the row contain a zero in the first
column are non-zero.
There is a zero in the first column, the other
elements of the row contain a zero are also zero.
There is a zero in the first column, and the other
elements of the row contain the zero are also
zero, with repeated roots on the jw axis.10/3/201320 Eng. R. L. Nkumbwa @ CBU2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
21/30
Example: Third-Order System
The characteristic polynomial of a third-
order system is:
10/3/201321 Eng. R. L. Nkumbwa @ CBU2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
22/30
Example: Third-Order System
The polynomial satisfies all the necessary
conditions because all the coefficients existand are positive.
Utilizing the Routh array, we have:
10/3/201322Eng. R. L. Nkumbwa @ CBU
2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
23/30
Example: Third-Order System
10/3/201323Eng. R. L. Nkumbwa @ CBU
2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
24/30
Example: Third-Order System
Because there are two changes in sign in the
first column, we find that two roots of D(s) liein the right- hand plane, and our prior
knowledge is confirmed.
10/3/201324Eng. R. L. Nkumbwa @ CBU
2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
25/30
Exercise
To illustrate this method clearly, you are
required to do a research for each case.
10/3/201325Eng. R. L. Nkumbwa @ CBU
2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
26/30
Routh-Hurwitz Criterion Limitations
Routh-Hurwitz Criterion is only valid, if the
characteristic equation is algebraic with real
coefficients. If any one of the coefficients is
complex or if the equation is not algebraic, for
example, containing experimental functions or
sinusoidal functions of s, RHC simply cannot beapplied.
10/3/201326Eng. R. L. Nkumbwa @ CBU
2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
27/30
Routh-Hurwitz Criterion Limitations
The RHC cannot be applied to any other
stability boundaries in a complex plane, such
as the unit circle in the z-plane, which is the
stability boundary of discrete data systems.
10/3/201327Eng. R. L. Nkumbwa @ CBU
2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
28/30
MATLAB & ACSYS LAB
Spend time in the lab learning MATLAB and
ACSYS Software.
10/3/201328Eng. R. L. Nkumbwa @ CBU
2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
29/30
10/3/201329Eng. R. L. Nkumbwa @ CBU
2010
7/27/2019 Lecture2 Methods of Determining Stability-rhc
30/30
10/3/201330Eng. R. L. Nkumbwa @ CBU
2010