Multivariable Control Systems

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

2. Chapter 1 LinearAlgebra 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

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. 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. Norm of vectors p-norm is: 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

6. Norm of vectors

7. Norm of real functions 1-norm is defined as 2-norm is defined as

8. Norm of 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: We can extend norm of vectors to matrices

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. 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. Unitary and Hermitian Matrices A matrix is unitary if A matrix is Hermitian if 1- Show that for any matrix V, are Hermitian matrices 2- Show that for any matrix V, the eigenvalues of are real nonnegative. For real matrices Hermitian matrix means symmetric matrix.

12. Primitive Matrices A matrix is nonnegative if whose entries are nonnegative numbers. A matrix is 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. Primitive Matrices

14. Positive (Negative) Definite Matrices A matrix A matrix is positive definite if for any is negative definite if for any A matrix is positive semi definite if for any is real and positive is real and negative is real and nonnegative Negative semi definite define similarly

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. Singular Value Decomposition (SVD) ?

17. Singular Value Decomposition (SVD) : Let . Then there exist and unitary matrices and such that Theorem 1-1

18. Singular Value Decomposition (SVD) Example Has no affect on the output or

19. Singular Value Decomposition (SVD) Theorem 1-1 : Let . Then there exist and unitary matrices and such that 3- Derive the SVD of

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. 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. Matrix Perturbation 1- Additive Perturbation 2- Multiplicative Perturbation 3- Element by Element Perturbation

23. Additive Perturbation Suppose has full column rank (n). Then Theorem 1-3

24. Multiplicative Perturbation Suppose . Then Theorem 1-4

25. Element by element Perturbation is non-singular and suppose : Suppose is the ijth element of the RGA of A. The matrix A will be singular if ijth element of A perturbed by Theorem 1-5

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