Stability of Linear Feedback Systems 2

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Stability of Linear Feedback Systems (2)

• Modern Control Systems
• Lecture 15

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Outline

• Stability of state variable systems
• Design example
• System stability using Matlab

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Stability of state variable systems

• Stability of a system modeled in state variable
  form, same as the TF model, can be ascertained by
  using the characteristic equation of the system.
  Having obtained the characteristic equation, we
  assess the system stability either through the
  Routh-Hurwitz criterion or by finding the roots
  of the characteristic equation. The roots of the
  characteristic equation must lie in the left half
  of s-plane for the system to be stable.
• We now investigate how to find the characteristic
  equation for state variable systems when the
  system is represented by 1) signal-flow graph, 2)
  block diagram, or 3) state equation.

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Characteristic equation of state variable systems
represented by signal-flow graph

• For a signal-flow graph state model, the
  characteristic equation is the flow graph
  determinant ?, acquired by Masons gain formula.
• Consider the signal-flow graph model in the
  figure.

Using Masons gain formula, we obtain the flow
graph determinant ?
Thus the characteristic equation is
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Characteristic equation of state variable systems
represented by block diagram

• For a block diagram model, the characteristic
  equation is obtained using block diagram
  reduction techniques.
• Consider the previous example but represented in
  block diagram.

Eliminating the two feedback loops yields G1(s)
and G2(s). The closed-loop TF is then
The characteristic equation is thus
which is same as before.
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Characteristic equation of state variable systems
represented by state equation

• For state variable systems in state equation, the
  characteristic equation can be obtained from the
  determinant of (?I-A). That is, the nth-order
  equation in ? resulting from the evaluation of
  this determinant is the characteristic equation,
  expressed by

Again consider the previous example but
represented in state equation form.
?1, ?2,?n are called characteristic roots or
eigenvalues of the system.
The characteristic equation is given by
which is same as before.
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Design example tracked vehicle turning control

• The block diagram of a tracked vehicle turning
  control system is shown in the figure. We want
  to select K and a so that the system is stable
  and the steady-state error for a ramp command is
  less than 24 of the magnitude of the command.

The closed-loop TF of the system is
The characteristic equation is
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Design example tracked vehicle turning control
(contd)

• The characteristic equation is

We establish the Routh array to determine
stability.
where
For the elements in the first column to be
positive, we require that Ka, b3 and c3 be
positive. Therefore,
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Design example tracked vehicle turning control
(contd)
Since the error signal where the ramp input
r(t)At for tgt0 and
For ess10A/Ka to be less than 24 of A, Ka
41.7. In addition,

• The region of stability for Kgt0 is shown in the
  figure. So if we choose Ka42, then the selected
  point in the stable region of K70 and a0.6
  satisfies design specifications.

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System stability using Matlab

• Routh-Hurwitz criterion finds how many poles lie
  in the right half plane, but not the specific
  location of the poles. With Matlab, we can
  calculate the poles explicitly, thus allowing us
  to comment on systems relative stability.
• The Matlab function to compute system poles is
  pole.

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System stability using Matlab (contd)

• When the characteristic equation is a function of
  a single system parameter, Routh-Hurwitz
  criterion can be used to determine the range of
  values that the parameter may take to ensure
  system stability. In this situation, we can
  resort to Matlab to verify the result
  graphically. In addition, we can use the roots
  function to calculate the characteristic roots.

The characteristic polynomial of the system in
the figure is
Using Routh-Hurwitz criterion, we require 0ltKlt8
for stability. We can use Matlab to verify it.
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System stability using Matlab (contd)

• For state variable systems represented in state
  equation form, the characteristic equation is
  det(?I-A)0. We can use the poly function to
  obtain the characteristic equation associated
  with A.

For example, system matrix A is
The associated characteristic polynomial is
If we use poly(A) in Matlab, we can obtain the
same result.