Multivariable Control Systems

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Multivariable Control Systems

• Ali Karimpour
• Assistant Professor
• Ferdowsi University of Mashhad

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Chapter 1
Linear Algebra
Topics to be covered include

• Vector Spaces
• Norms
• Unitary, Primitive and Hermitian Matrices
• Positive (Negative) Definite Matrices
• Inner Product
• Singular Value Decomposition (SVD)
• Relative Gain Array (RGA)
• Matrix Perturbation

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Vector Spaces
A set of vectors and a field of scalars with some
properties is called vector space.
To see the properties have a look on Linear
Algebra written by Hoffman.
Some important vector spaces are
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Norms
To meter the lengths of vectors in a vector
space we need the idea of a norm.
Norm is a function that maps x to a nonnegative
real number
A Norm must satisfy following properties
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Norm of vectors
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Norm of vectors
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Norm of real functions
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Norm of matrices
We can extend norm of vectors to matrices
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Matrix norm
A norm of a matrix is called matrix norm if it
satisfy
Define the induced-norm of a matrix A as follows
Any induced-norm of a matrix A is a matrix norm
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Matrix norm for matrices
If we put p1 so we have
Maximum column sum
If we put pinf so we have
Maximum row sum
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Unitary and Hermitian Matrices
For real matrices Hermitian matrix means
symmetric matrix.
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Primitive Matrices
Definition 2.1 A primitive matrix is a square
nonnegative matrix some power (positive integer)
of which is positive.
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Primitive Matrices
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Positive (Negative) Definite Matrices
Negative semi definite define similarly
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Inner Product
An Inner product must satisfy following
properties
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Singular Value Decomposition (SVD)
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Singular Value Decomposition (SVD)
Theorem 1-1
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Singular Value Decomposition (SVD)
Example
Has no affect on the output or
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Singular Value Decomposition (SVD)
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Matrix norm for matrices
If we put p2 so we have
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Relative Gain Array (RGA)
The relative gain array (RGA), was introduced by
Bristol (1966).
For a square matrix A
For a non square matrix A
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Matrix Perturbation
1- Additive Perturbation
2- Multiplicative Perturbation
3- Element by Element Perturbation
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Additive Perturbation
Theorem 1-3
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Multiplicative Perturbation
Theorem 1-4
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Element by element Perturbation
Theorem 1-5
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Element by element Perturbation
Example 1-3
then the perturbed A is singular or