Download - Lecture-Slides CHAPTER 06
ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Stability of Feedback Control Systems
Chapter 6
ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
6.1: The Concept of Stability
6.2: The Routh-Hurwitz Stability Criterion
Outline
ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
• Stability of a feedback control system
– Fundamentally important.
– An unstable closed-loop system is of no use in practice.
– Many physical systems are inherently open-loop unstable (e.g.
inverted pendulum, etc.).
– Among the performance specifications used in design, the
most important requirement is that the system must be stable.
– Classification:
• Relative stability
• Absolute stability
6.1: The Concept of Stability
ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
• Stable system:
– A dynamic system with a bounded response to a bounded
input.
– Bounded input bounded output (BIBO) stability
– A linear time-invariant control system is stable if the output
eventually comes back to its equilibrium state.
– It is marginally stable (neutral) if oscillations of the output
continue forever.
– It is unstable if the output diverges without bound from its
equilibrium state.
6.1: The Concept of Stability
ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
• The stability of a cone:
6.1: The Concept of Stability
ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
• Unit impulse response of a second order system for
various poles on s-plane
6.1: The Concept of Stability
ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
• Stability in the s-plane
6.1: The Concept of Stability
ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
• Mathematical description
– Consider the first-order differential equation:
– where 'a' is a constant
– Transfer function
– Unit impulse response:
– There are three cases:
i) a > 0, y(∞) = 0 → stable
ii) a = 0, y(∞) = 1 → neutral (marginally stable)
iii) a < 0, y(∞) = ∞ → unstable
– Stability is related to
• The location of poles of the system transfer function
6.1: The Concept of Stability
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
6.1: The Concept of Stability
ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
– A necessary and sufficient condition for a feedback system to
be stable is that ALL THE POLES of the system transfer
function have NEGATIVE REAL PARTS.
6.1: The Concept of Stability
ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
• Consider the transfer function of a feedback control
system:
– To check for stability, we have to find the poles of the
system.
– Poles of the system are the roots of the characteristic
equation
– The necessary conditions (but not sufficient) for the stability
are:
• All the coefficients have the same sign.
• Non of the coefficients are zero.
6.1: The Concept of Stability
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
• RH stability method is used to assess system’s stability
– This method allow us to compute the number of poles in the
right half plane without computing the values of poles.
– The RH criterion is a necessary and sufficient criterion for
the stability of linear systems.
6.2: The Routh-Hurwitz Stability Criterion
ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
• The Routh-Hurwitz Criterion
– Take the characteristic equation:
– Make the Routh array/table:
6.2: The Routh-Hurwitz Stability Criterion
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
• Investigate the signs of the coefficients in the fist
column of Routh array.
– The number of roots of the characteristic equation with
positive real parts is equal to the number of changes in sign
of the first column of the Routh array
6.2: The Routh-Hurwitz Stability Criterion
ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
• Application of Routh's stability criterion in control
system analysis
– The R-H criterion can be used to determine the range of the
parameter values which maintain a stable system.
• Example:
– Consider closed-loop system shown below. Using the R-H
criterion, determine the range of K over which the system is
stable.
6.2: The Routh-Hurwitz Stability Criterion
ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
• Example continued …
– Transfer function:
– Routh’s table
6.2: The Routh-Hurwitz Stability Criterion
ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
• For stability, all the coefficients in the first column of
the Routh's table must be positive, i.e. we require:
4K/5 - 6 > 0 leading to K > 15/2
and K > 0
Therefore we choose a value for K > 15/2 for stability
6.2: The Routh-Hurwitz Stability Criterion
ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
• Determine the range of (K) and (a) for which the
system is stable.
Example 6.5