1. Multivariable Control Systems Ali Karimpour Assistant Professor Ferdowsi University of Mashhad

2. Ali Karimpour Sep 2009 Chapter 1 2 v Vector Spaces v Norms v Unitary, Primitive and Hermitian Matrices v Positive (Negative) Definite Matrices v Inner Product v Singular Value Decomposition (SVD) v Relative Gain Array (RGA) v Matrix Perturbation Linear Algebra Topics to be covered include:

3. Ali Karimpour Sep 2009 Chapter 1 3 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:

4. Ali Karimpour Sep 2009 Chapter 1 4 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:

5. Ali Karimpour Sep 2009 Chapter 1 5 Norm of vectors For p=1 we have 1-norm or sum norm For p=2 we have 2-norm or euclidian norm For p=∞ we have ∞-norm or max norm p-norm is:

6. Ali Karimpour Sep 2009 Chapter 1 6 Norm of vectors

7. Ali Karimpour Sep 2009 Chapter 1 7 Norm of real functions 1-norm is defined as 2-norm is defined as

8. Ali Karimpour Sep 2009 Chapter 1 8 Norm of matrices We can extend norm of vectors to matrices Sum matrix norm (extension of 1-norm of vectors) is: Frobenius norm (extension of 2-norm of vectors) is: Max element norm (extension of max norm of vectors) is:

9. Ali Karimpour Sep 2009 Chapter 1 9 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

10. Ali Karimpour Sep 2009 Chapter 1 10 Matrix norm for matrices If we put p=1 so we have Maximum column sum If we put p=inf so we have Maximum row sum

11. Ali Karimpour Sep 2009 Chapter 1 11 Unitary and Hermitian Matrices A matrixis unitary if A matrixis Hermitian if For real matrices Hermitian matrix means symmetric matrix. 1- Show that for any matrix V, are Hermitian matrices 2- Show that for any matrix V, the eigenvalues of are real nonnegative.

12. Ali Karimpour Sep 2009 Chapter 1 12 Primitive Matrices A matrixis nonnegative if whose entries are nonnegative numbers. A matrixis positive if all of whose entries are strictly positive numbers. Definition 2.1 A primitive matrix is a square nonnegative matrix some power ( positive integer ) of which is positive.

13. Ali Karimpour Sep 2009 Chapter 1 13 Primitive Matrices

14. Ali Karimpour Sep 2009 Chapter 1 14 Positive (Negative) Definite Matrices A matrixis positive definite if for any is real and positive A matrixis negative definite if for any is real and negative A matrixis positive semi definite if for any is real and nonnegative Negative semi definite define similarly

15. Ali Karimpour Sep 2009 Chapter 1 15 Inner Product An inner product is a function of two vectors, usually denoted by Inner product is a function that maps x, y to a complex number An Inner product must satisfy following properties:

16. Ali Karimpour Sep 2009 Chapter 1 16 Singular Value Decomposition (SVD) ?

17. Ali Karimpour Sep 2009 Chapter 1 17 Singular Value Decomposition (SVD) Theorem 1-1 : Let. Then there existand unitary matrices and such that

18. Ali Karimpour Sep 2009 Chapter 1 18 Singular Value Decomposition (SVD) Example Has no affect on the output or

19. Ali Karimpour Sep 2009 Chapter 1 19 Singular Value Decomposition (SVD) Theorem 1-1 : Let. Then there existand unitary matrices and such that 3- Derive the SVD of

20. Ali Karimpour Sep 2009 Chapter 1 20 Matrix norm for matrices If we put p=1 so we have Maximum column sum If we put p=inf so we have Maximum row sum If we put p=2 so we have

21. Ali Karimpour Sep 2009 Chapter 1 21 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 †

22. Ali Karimpour Sep 2009 Chapter 1 22 Matrix Perturbation 1- Additive Perturbation 2- Multiplicative Perturbation 3- Element by Element Perturbation

23. Ali Karimpour Sep 2009 Chapter 1 23 Additive Perturbation has full column rank (n). Then Suppose Theorem 1-3

24. Ali Karimpour Sep 2009 Chapter 1 24 Multiplicative Perturbation. Then Suppose Theorem 1-4

25. Ali Karimpour Sep 2009 Chapter 1 25 Element by element Perturbation : Suppose is non-singular and suppose is the ij th element of the RGA of A. The matrix A will be singular if ij th element of A perturbed by Theorem 1-5

26. Ali Karimpour Sep 2009 Chapter 1 26 Element by element Perturbation Example 1-3 Now according to theorem 1-5 if multiplied by then the perturbed A is singular or