Multivariable Control Systems

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Multivariable Control Systems

• Ali Karimpour
• Assistant Professor
• Ferdowsi University of Mashhad

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Chapter 5
Controllability, Observability and Realization
Topics to be covered include

• Controllability of Linear Dynamical Equations
• Observability of Linear Dynamical Equations
• Canonical Decomposition of a Linear
  Time-invariant
• Dynamical Equation
• Realization of Proper Rational Transfer
  Function Matrices
• Irreducible Realizations
• Irreducible realization of proper rational
  transfer functions
• Irreducible Realization of Proper Rational
  Transfer Function Vectors
• Irreducible Realization of Proper Rational
  Matrices

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Controllability and Observability of Linear
Dynamical Equations
Definition 5-1
Definition 5-2
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Controllability and Observability of Linear
Dynamical Equations
Theorem 5-1
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Controllability and Observability of Linear
Dynamical Equations
Theorem 5-2
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Controllability and Observability of Linear
Dynamical Equations
Theorem 5-2(continue)
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Canonical Decomposition of a Linear
Time-invariant Dynamical Equation
Theorem 5-3 The controllability and observability
of a linear time-invariant dynamical equation
are invariant under any equivalence
transformation.
Proof Let we first consider controllability
Similarly we can consider observability
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Canonical Decomposition of a Linear
Time-invariant Dynamical Equation
Theorem 5-4 Consider the n-dimensional linear
time invariant dynamical equation
If the controllability matrix of the dynamical
equation has rank n1 (where n1ltn ), then there
exists an equivalence transformation
which transform the dynamical equation to
and the n1-dimensional sub-equation
is controllable and has the same transfer
function matrix as the first system.
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Canonical Decomposition of a Linear
Time-invariant Dynamical Equation
Theorem 5-4 (Continue)
Furthermore Pq1 q2 qn1 qn-1 where q1,
q2, , qn1 be any n1 linearly independent column
of S (controllability matrix) and the last n-n1
column of P are entirely arbitrary so long as
the matrix q1 q2 qn1 qn is nonsingular.
Proof See Linear system theory and design
Chi-Tsong Chen
Hence, we derive the reduced order controllable
equation.
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Canonical Decomposition of a Linear
Time-invariant Dynamical Equation
Theorem 5-5 Consider the n-dimensional linear
time invariant dynamical equation
If the observability matrix of the dynamical
equation has rank n2 (where n2ltn ), then there
exists an equivalence transformation
which transform the dynamical equation to
and the n2-dimensional sub-equation
is observable and has the same transfer function
matrix as the first system.
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Canonical Decomposition of a Linear
Time-invariant Dynamical Equation
Theorem 5-5 (Continue)
Furthermore the first n2 row of P are any n2
linearly independent rows of V (observability
matrix) and the last and the last n-n2 row of P
is entirely arbitrary so long as the matrix P
is nonsingular.
Proof See Linear system theory and design
Chi-Tsong Chen
Hence, we derive the reduced order observable
equation.
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Canonical Decomposition of a Linear
Time-invariant Dynamical Equation
Theorem 5-6 (Canonical decomposition theorem)
Consider the n-dimensional linear time
invariant dynamical equation
There exists an equivalence transformation
which transform the dynamical equation to
and the reduced dimensional sub-equation
is observable and controllable and has the same
transfer function matrix as the first system.
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Canonical Decomposition of a Linear
Time-invariant Dynamical Equation
Definition 5-3 A linear time-invariant dynamical
equation is said to be reducible if and only if
there exist a linear time-invariant dynamical
equation of lesser dimension that has the same
transfer function matrix. Otherwise, the
equation is irreducible.
Theorem 5-7 A linear time invariant dynamical
equation is irreducible if and only if it is
controllable and observable.
Theorem 5-8
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Realization of Proper Rational Transfer Function
Matrices
This transformation is unique
Realization
This transformation is not unique

• Is it possible at all to obtain the state-space
  description from the transfer function
• matrix of a system?

2. If yes, how do we obtain the state space
description from the transfer function matrix?
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Realization of Proper Rational Transfer Function
Matrices
Theorem 5-9 A transfer function matrix G(s) is
realizable by a finite dimensional linear time
invariant dynamical equation if and only if G(s)
is a proper rational matrix.
Proof See Linear system theory and design
Chi-Tsong Chen
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Irreducible realizations
Definition 5-4
Theorem 5-10
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Irreducible realizations
Before considering the general case
(irreducible realization of proper rational
matrices) we start the following parts
1. Irreducible realization of Proper Rational
Transfer Functions 2. Irreducible
Realization of Proper Rational Transfer Function
Vectors 3. Irreducible Realization of Proper
Rational Matrices
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Irreducible realization of proper rational
transfer functions
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Irreducible realization of proper rational
transfer functions
There are different forms of realization

• Observable canonical form realization

• Controllable canonical form realization

• Realization from the Hankel matrix

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Observable canonical form realization of proper
rational transfer functions
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Observable canonical form realization of proper
rational transfer functions
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Observable canonical form realization of proper
rational transfer functions
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Observable canonical form realization of proper
rational transfer functions
The derived dynamical equation is observable.
Exersise 1 Why?
The derived dynamical equation controllable as
well if numerator and denominator of g(s) are
coprime.
Exersise 2 Why?
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Controllable canonical form realization of proper
rational transfer functions
Let us introduce a new variable
We may define the state variable as
Clearly
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Controllable canonical form realization of proper
rational transfer functions
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Controllable canonical form realization of proper
rational transfer functions
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Controllable canonical form realization of proper
rational transfer functions
The derived dynamical equation is controllable .
Exersise 3 Why?
The derived dynamical equation observable as well
if numerator and denominator of g(s) are
coprime.
Exersise 4 Why?
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Controllable and observable canonical form
realization of proper rational transfer
functions
Example 5-2 Derive controllable and observable
canonical realization for following system.
Observable canonical form realization is
Controllable canonical form realization is
It is not controllable.
Why?
Why?
It is not observable.
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Irreducible realization of proper rational
transfer functions
Example 5-3 Derive irreducible realization for
following transfer function.
Observable canonical form realization is
Controllable canonical form realization is
It is controllable too.
It is observable too.
Why?
Why?
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Irreducible realization of proper rational
transfer functions
Realization from the Hankel matrix
The coefficients h(i) will be called Markov
parameters.
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Irreducible realization of proper rational
transfer functions
Realization from the Hankel matrix
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Irreducible realization of proper rational
transfer functions
Realization from the Hankel matrix
Now consider the dynamical equation
Let the first s rows be linearly independent and
the (s1) th row of H(n1,n) be linearly
dependent on its previous rows. So
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Irreducible realization of proper rational
transfer functions
Realization from the Hankel matrix
We claim that the s-dimensional dynamical
equation
is a controllable and observable (irreducible
realization).
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Irreducible realization of proper rational
transfer functions
Example 5-4 Derive irreducible realization for
following transfer function.
We can show that the rank of H(4,3) is 2. So
Hence an irreducible realization of g(s) is
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Realization of Proper Rational Transfer Function
Vectors
Consider the rational function vector
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Realization of Proper Rational Transfer Function
Vectors
This is a controllable form realization of G(s).
We see that the transfer function from u to yi
is equal to
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Realization of Proper Rational Transfer Function
Vectors
Example 5-5 Derive a realization for following
transfer function vector.
Hence a minimal dimensional realization of G(s)
is given by
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Realization of Proper Rational Matrices
There are many approaches to find irreducible
realizations for proper rational matrices.

• One approach is to first find a reducible
  realization and then apply
• the reduction procedure to reduce it to an
  irreducible one.

Method I, Method II, Method III and Method IV
2. In the second approach irreducible
realization will yield directly.
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Realization of Proper Rational Matrices
Method I Given a proper rational matrix G(s), if
we first find an irreducible realization for
every element gij(s) of G(s) as
Clearly this equation is generally not
controllable and not observable.
To reduce this realization to irreducible one
requires the application of the
reduction procedure twice (theorems 5-4 and 5-5).

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Realization of Proper Rational Matrices
Proof
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Realization of Proper Rational Matrices
Method II Given a proper rational matrix G(s),
if we find the controllable canonical- form
realization for the ith column, Gi(s), of G(s)
say,
This realization is always controllable. It is
however generally not observable.
Proof
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Realization of Proper Rational Matrices
Then the following dynamic equation is a
realization of G(s).
Exercise 6 Show that the above dynamical
equation is a controllable realization of G(s)
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Realization of Proper Rational Matrices
Method IV It is possible to obtain observable
realization of a proper G(s). Let
Consider the monic least common denominator of
G(s) as
Then after deriving H(i) one can simply show
Exercise 7 Proof equation (I)
Let A, B, C and E be a realization of G(s) then
we have
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Realization of Proper Rational Matrices
Then A, B, C and D be a realization of G(s) if
and only if
We can readily verify that
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Realization of Proper Rational Matrices
Now we shall discuss in the following a method
which will yield directly irreducible
realizations. This method is based on the
Hankel matrices.


Consider the monic least common denominator of
G(s) as


Define
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Realization of Proper Rational Matrices


We also define the two following Hankel matrices
It can be readily verified that
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Realization of Proper Rational Matrices
It can be readily verified that
Note that the left-upper-corner of M iT TN i
is H(i1) so
It can be readily verified that
But we want Irreducible Realization of Proper
Rational Matrices
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Irreducible Realization of Proper Rational
Matrices
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Irreducible Realization of Proper Rational
Matrices
Least common denominator of G(s), is
Non-zero singular values of T are 10.23, 5.79,
0.90 and 0.23.
So, r 4.
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Irreducible Realization of Proper Rational
Matrices
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Irreducible Realization of Proper Rational
Matrices