This document discusses state space analysis and related concepts. It defines state as a group of variables that summarize a system's history to predict future outputs. The minimum number of state variables required is equal to the number of storage elements in the system. These state variables form a state vector. The document also covers state space representation, diagonalization, solving state equations, the state transition matrix, and concepts of controllability and observability.

1. Unit-5
State Space Analysis

2. • Concepts of state
• state variables and state model
• state space representation of transfer function
• Diagonalization
• solving the time invariant state equations
• State Transition Matrix and it’s Properties
• concepts of controllability and observability.

4. State :
It is a group of variables, which summarizes the history of the system
in order to predict the future values (outputs).
State Variable:
The number of the state variables required is equal to the number of
the storage elements present in the system.
State Vector:
Terms used in State Space Model
It is a vector, which contains the state variables as elements.
 The state space model can be obtained from any one of these two
mathematical models
 Those are the differential equation model and the transfer function model .

5. State Variable: The state of a dynamic system is the smallest set of
variables called state variables such that the knowledge of these variables at
time t=to (Initial condition), together with the knowledge of input for ≥𝑡0 ,
completely determines the behavior of the system for any time 𝑡≥𝑡0 .
State vector: If n state variables are needed to completely describe the
behaviour of a given system, then these n state variables can be considered the n
components of a vector X. Such a vector is called a state vector.
State space: The n-dimensional space whose co-ordinate axes consists of the
x1 axis, x2 axis,.... xn axis, where x1 , x2 ,..... xn are state variables: is called a
state space

6. Advantages of SSM
 It can analyze multi input and multi output systems easily
 It can gives the information about controllability of the system
 It can describes all dynamic Systems
 The analysis of the system can be possible by considering the
initial conditions of the system
 Accurate than Transfer Function
Dis- Advantages:
 More computations are required
 Techniques are more complex compared to T.F,,D.E

7. STATE SPACE MODEL
Lets consider a multi input & multi output system is
having
rinputs 𝑢1 (t) ,𝑢2 (t) ,…….𝑢𝑟(𝑡)
m no of outputs 𝑦1 (𝑡) ,𝑦2 (𝑡) ,…….𝑦𝑚(𝑡)
n no of state variables 𝑥1 (t) ,𝑥2 (t) ,…….𝑥𝑛(𝑡)

8. STATE SPACE MODEL

9. STATE Equation:
State Equation In Matrix Form

10. Output Equation:
Output Equations In Matrix Form

11. STATE SPACE MODEL

12. STATE Equation:
State Equation In Matrix Form

13. Output Equation:
Output Equations In Matrix Form

14. STATE MODEL

15. The behavior of x(t) TO y(t):
1. Homogeneous solution of x(t)
2. Non-homogeneous solution of x(t)

16. Homogeneous solution
• We are having the solution for state variables
without considering the input variables
U(t)=0

19. Properties of
State Transition Matrix

20. Non-homogeneous solution

22. Homogeneous solution:
U(t)=0
Non-homogeneous solution:

23. Problem:
Solution:

24. ZIR:

25. Given

26. ZSR:

28. Problem:
Solution:
Solution of Homogeneous equation
is

31. Controllability and Observability
Controllability: A control system is said to be controllable if the initial
states of the control system are transferred (changed) to some other desired
states by a controlled input in finite duration of time.
We can check the controllability of a control system by using Kalman’s test.
Write the matrix Qc in the following form.
 Find the determinant of matrix Qc and if it is not equal to zero, then the
control system is controllable.
i.e., │ Qc │ ≠ 0 Then it is Controllable
and │ Qc │= 0 Then it is un-Controllable

32. Observability: A control system is said to be observable if it is able to
determine the initial states of the control system by observing the outputs in
finite duration of time.
We can check the observability of a control system by using Kalman’s test.
• Write the matrix Qo in following form.
Find the determinant of matrix QoQo and if it is not equal to zero, then
the control system is observable
i.e., │ Qo │ ≠ 0 Then it is observable
│ Qo │= 0 Then it is non-observable

33. Problem on
controllability and observability
verify the controllability and observability of a control system
which is represented in the state space model as,
Solution:

35. OBSERVABILITY:

37. Problem on
controllability and observability
verify the controllability and observability of a control system
which is represented in the state space model as,
Solution:

38. CONTROLLABILITY:

39. OBSERVABILITY:

41. Problem:2
Solution:

48. DIAGONALISATION
Diagonalisation of the matrix is also called as
the canonical form representation of state model

51. Procedure for
Diagonalizing a Matrix:
Step 1 Find n linearly independent eigenvectors of A, say M1,
M2, ..., Mn.
Step 2 Form the matrix M having M1, M2, ..., Mn as its column
vectors.
Step 3 The matrix M−1AM will then be diagonal with λ1, λ2, ...,
λn as its diagonal entries, where λi is the eigenvalue
corresponding to Mi , for i = 1, 2, ..., n.

52. Example Problem on
Diagonalisation of Matrix

53. Solution:

60. Canonical Representation of State
model is

61. Transfer Function from SSM
We know the state space model of a Linear Time-
Invariant (LTI) system is
Apply Laplace Transform on both sides of
the state equation.
Apply Laplace Transform on both sides of the
output equation

62. Substitute, X(s) value in the above equation

63. Calculate the transfer function of the system represented in the
state space model as,
PROBLEM:
SOLUTION:

66. Transfer Function from SSM
We know the state space model of a Linear Time-
Invariant (LTI) system is
Apply Laplace Transform on both sides of
the state equation.
Apply Laplace Transform on both sides of the
output equation

67. Substitute, X(s) value in the above equation

68. Calculate the transfer function of the system represented in the
state space model as,
PROBLEM:
SOLUTION:

71. SSM from Transfer Function
Find the state space model for the system having transfer function.
Problem:
SOLUTION:

74. MODEL:2

78. Problem:

81. Electrical Circuit to SSM
Derive the State Space Model for the given
Electrical Network

82. From the Circuit

84. PROBLEM:2

85. APPLY KVL FOR 1ST LOOP

86. APPLY KVL FOR 2ND LOOP

88. STATE DIAGRAM
• State Diagram can be represent by using three
basic components
Scalar
Adder
Integral
1. SCALAR
2. ADDER:
3. INTEGRAL:

90. • Concepts of state
• state variables and state model
• state space representation of transfer function
• Diagonalization
• solving the time invariant state equations
• State Transition Matrix and it’s Properties
• concepts of controllability and observability.