1. Digital Control Systems
State Space Analysis(1)

2. INTRODUCTION
State :The state of a dynamic system is the smallest set of variables (called state variables) such that knowledge of these variables at t = t0, together with knowledge of the input for t ≥t0, completely determines the behavior of the system for any time t ≥ t0. State variables:The state variables of a dynamic system are the variables making up the smallest set of variables that determines the state of the dynamic system. If at least n variables x1,x2,… xn are needed to completely describe the behavior of a dynamic system (so that, once the input is given for t ≥ t0. and the initial state at t=t0 is specified, the future state of the system is completely determined), then those n variables are a set of state variables. State vector:If n state variables are needed to completely describe the behavior of a given system, then those state variables can be considered the n components of a vector x called a state vector. A state vector is thus a vector that uniquely determines the system state x(t) for any time t ≥ t0, once the state at t=t0 is given and the input u(t) for t ≥ t0 is specified.

3. INTRODUCTION
State space: The n-dimensional space whose coordinate axes consist of the x1-axis, x2-axis,..xn-axis is called a state space. State-space equations: In state-space analysis, we are concerned with three types of variables that are involved in the modeling of dynamic systems: input variables, output variables, and state variables. For Linear or Nonlinear discrete-time systems:

4. INTRODUCTION
For Linear Time-varying discrete-time systems:

5. INTRODUCTION
For Linear Time-invariant discrete-time systems:

6. INTRODUCTION
For Linear or Nonlinear continuous-time systems: For Linear Time-varying continuous time systems:

7. INTRODUCTION
For Linear Time Invariant continuous time systems:

8. STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMS
Canonical Forms for Discrete Time State Space Equations or There are many ways to realize state-space representations for the discrete time system represented by these equations:

9. STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMS
Controllable Canonial Form:

10. STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMS
Controllable Canonical Form: If we reverse the order of the state variables:

11. STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMS
Observable Canonical Form

12. STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMS
Observable Canonical Form: If we reverse the order of the state variables:

13. STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMS
Diagonal Canonical Form: If the poles of pulse transfer function are all distinct, then the state-space representation may be put in the diagonal canonical form as follows:

14. STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMS
Jordan Canonical Form: If the poles of pulse transfer function involves a multiple pole of orde m at z=p1 and all other poles are distinct:

15. STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMS
Example:

16. STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMS
Example:

17. EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Rank of a Matrix A matrix A is called of rank m if the maximum number of linearly independent rows (or columns) is m. Properties of Rank of a Matrix

18. EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Properties of Rank of a Matrix (cntd.)

19. EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Eigenvalues of a Square Matrix

20. EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Eigenvalues of a Square Matrix
The n roots of the characteristic equation are called eigenvalues of A. They are also called the characteristic roots.
An n×n real matrix A does not necessarily possess real eigenvalues.
Since the characteristic equation is a polynomial with real coefficients, any compex eigenvalues must ocur in conjugate pairs.
If we assume the eigenvalues of A to be λi and those of to be μi then μi = (λi)-1

21. EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Eigenvectors of an n×n Matrix Similar Matrices

22. EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Diagonalization of Matrices

23. EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Jordan Canonical Form

24. EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Jordan Canonical Form

25. EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Jordan Canonical Form : only one eigenvector : two linearly independent eigenvectors : three linearly independent eignvectors

26. EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Jordan Canonical Form

27. EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Similarity Transformation When an n×n Matrix has Distinct Eigenvalues

28. EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Similarity Transformation When an n×n Matrix has Distinct Eigenvalues

29. EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Similarity Transformation When an n×n Matrix Has Multiple Eigenvalues

30. STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMS
Nonuniqueness of State Space Representations: For a given pulse transfer function syste the state space representation is not unique. The state equations, however, are related to each other by the similarity transformation. Let us define a new state vector by where P is a nonsingular matrix. By substituting to 1 2 2 1

31. STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMS
Nonuniqueness of State Space Representations: Let us define then Since matrix P can be any nonsingular nn matrix, there are infinetely many state space representations for a given system. If we choose P properly: (If diagonalization is not possible) ≡