1. Advanced Control Systems (ACS)
Lecture-6
State Space Modeling & Analysis
Dr. Imtiaz Hussain
URL :

2. Lecture Outline Basic Definitions State Equations State Diagram
State Controllability
State Observability
Output Controllability
Transfer Matrix
Solution of State Equation

3. Definitions
State of a system: We define the state of a system at time t0 as the amount of information that must be provided at time t0, which, together with the input signal u(t) for t  t0, uniquely determine the output of the system for all t  t0.
State Variable: The state variables of a dynamic system are the smallest set of variables that determine the state of the dynamic system.
State Vector: If n variables are needed to completely describe the behaviour of the dynamic system then n variables can be considered as n components of a vector x, such a vector is called state vector.
State Space: The state space is defined as the n-dimensional space in which the components of the state vector represents its coordinate axes.

4. Definitions
Let x1 and x2 are two states variables that define the state of the system completely .
State space of a Vehicle
Two Dimensional State space
State (t=t1)
Velocity
State (t=t1)
State
Vector
Position

5. 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.
The dynamic system must involve elements that memorize the values of the input for t> t1 .
Since integrators in a continuous-time control system serve as memory devices, the outputs of such integrators can be considered as the variables that define the internal state of the dynamic system.
Thus the outputs of integrators serve as state variables.
The number of state variables to completely define the dynamics of the system is equal to the number of integrators involved in the system.

6. State Space Equations
Assume that a multiple-input, multiple-output system involves 𝑛 integrators.
Assume also that there are 𝑟 inputs 𝑢 1 𝑡 , 𝑢 2 𝑡 ,⋯, 𝑢 𝑟 𝑡 and 𝑚 outputs 𝑦 1 𝑡 , 𝑦 2 𝑡 ,⋯, 𝑦 𝑚 𝑡 .
Define 𝑛 outputs of the integrators as state variables: 𝑥 1 𝑡 , 𝑥 2 𝑡 ,⋯, 𝑥 𝑛 𝑡 .
Then the system may be described by
𝑥 1 𝑡 = 𝑓 1 ( 𝑥 1 , 𝑥 2 ,⋯, 𝑥 𝑛 ; 𝑢 1 , 𝑢 2 ,⋯, 𝑢 𝑟 ;𝑡)
𝑥 2 𝑡 = 𝑓 2 ( 𝑥 1 , 𝑥 2 ,⋯, 𝑥 𝑛 ; 𝑢 1 , 𝑢 2 ,⋯, 𝑢 𝑟 ;𝑡)
𝑥 𝑛 𝑡 = 𝑓 𝑛 ( 𝑥 1 , 𝑥 2 ,⋯, 𝑥 𝑛 ; 𝑢 1 , 𝑢 2 ,⋯, 𝑢 𝑟 ;𝑡)

7. State Space Equations
The outputs 𝑦 1 𝑡 , 𝑦 2 𝑡 ,⋯, 𝑦 𝑚 𝑡 of the system may be given as.
If we define
𝑦 1 𝑡 = 𝑔 1 ( 𝑥 1 , 𝑥 2 ,⋯, 𝑥 𝑛 ; 𝑢 1 , 𝑢 2 ,⋯, 𝑢 𝑟 ;𝑡)
𝑦 2 𝑡 = 𝑔 2 ( 𝑥 1 , 𝑥 2 ,⋯, 𝑥 𝑛 ; 𝑢 1 , 𝑢 2 ,⋯, 𝑢 𝑟 ;𝑡)
𝑦 𝑚 𝑡 = 𝑔 𝑚 ( 𝑥 1 , 𝑥 2 ,⋯, 𝑥 𝑛 ; 𝑢 1 , 𝑢 2 ,⋯, 𝑢 𝑟 ;𝑡)
𝒇 𝒙,𝒖,𝑡 = 𝑓 1 ( 𝑥 1 , 𝑥 2 ,⋯, 𝑥 𝑛 ; 𝑢 1 , 𝑢 2 ,⋯, 𝑢 𝑟 ;𝑡) 𝑓 2 ( 𝑥 1 , 𝑥 2 ,⋯, 𝑥 𝑛 ; 𝑢 1 , 𝑢 2 ,⋯, 𝑢 𝑟 ;𝑡) ⋮ 𝑓 𝑛 ( 𝑥 1 , 𝑥 2 ,⋯, 𝑥 𝑛 ; 𝑢 1 , 𝑢 2 ,⋯, 𝑢 𝑟 ;𝑡)
𝒙 𝑡 = 𝑥 1 𝑥 2 ⋮ 𝑥 𝑛
𝒖 𝑡 = 𝑢 1 𝑢 2 ⋮ 𝑢 𝑟
𝒚 𝑡 = 𝑦 1 𝑦 2 ⋮ 𝑦 𝑚
𝒈 𝒙,𝒖,𝑡 = 𝑔 1 ( 𝑥 1 , 𝑥 2 ,⋯, 𝑥 𝑛 ; 𝑢 1 , 𝑢 2 ,⋯, 𝑢 𝑟 ;𝑡) 𝑔 2 ( 𝑥 1 , 𝑥 2 ,⋯, 𝑥 𝑛 ; 𝑢 1 , 𝑢 2 ,⋯, 𝑢 𝑟 ;𝑡) ⋮ 𝑔 𝑚 ( 𝑥 1 , 𝑥 2 ,⋯, 𝑥 𝑛 ; 𝑢 1 , 𝑢 2 ,⋯, 𝑢 𝑟 ;𝑡)

8. State Space Modeling 𝒙 𝑡 =𝒇(𝒙,𝒖,𝑡) 𝒚 𝑡 =𝒈(𝒙,𝒖,𝑡)
State space equations can then be written as
If vector functions f and/or g involve time t explicitly, then the system is called a time varying system.
𝒙 𝑡 =𝒇(𝒙,𝒖,𝑡)
State Equation
𝒚 𝑡 =𝒈(𝒙,𝒖,𝑡)
Output Equation

9. State Space Modeling
If above equations are linearised about the operating state, then we have the following linearised state equation and output equation:

10. State Space Modeling
If vector functions f and g do not involve time t explicitly then the system is called a time-invariant system.
In this case, state and output equations can be simplified to

11. Example-1 𝑚 𝑦 (𝑡)+𝑏 𝑦 (𝑡)+𝑘𝑦(𝑡)=𝑢(𝑡) 𝑥 1 𝑡 =𝑦(𝑡) 𝑥 2 𝑡 = 𝑦 (𝑡)
Consider the mechanical system shown in figure. We assume that the system is linear. The external force u(t) is the input to the system, and the displacement y(t) of the mass is the output. The displacement y(t) is measured from the equilibrium position in the absence of the external force. This system is a single-input, single-output system.
From the diagram, the system equation is
𝑚 𝑦 (𝑡)+𝑏 𝑦 (𝑡)+𝑘𝑦(𝑡)=𝑢(𝑡)
This system is of second order. This means that the system involves two integrators. Let us define state variables 𝑥 1 (𝑡) and 𝑥 2 (𝑡) as
𝑥 1 𝑡 =𝑦(𝑡)
𝑥 2 𝑡 = 𝑦 (𝑡)

12. Example-1 𝑚 𝑦 (𝑡)+𝑏 𝑦 (𝑡)+𝑘𝑦(𝑡)=𝑢(𝑡) 𝑥 1 𝑡 =𝑦(𝑡) 𝑥 2 𝑡 = 𝑦 (𝑡)
𝑥 1 𝑡 =𝑦(𝑡)
𝑥 2 𝑡 = 𝑦 (𝑡)
𝑚 𝑦 (𝑡)+𝑏 𝑦 (𝑡)+𝑘𝑦(𝑡)=𝑢(𝑡)
Then we obtain Or
The output equation is
𝑥 1 𝑡 = 𝑥 2 (𝑡)
𝑥 2 𝑡 =− 𝑏 𝑚 𝑦 𝑡 − 𝑘 𝑚 𝑦 𝑡 + 1 𝑚 𝑢 (𝑡)
𝑥 1 𝑡 = 𝑥 2 (𝑡)
𝑥 2 𝑡 =− 𝑏 𝑚 𝑥 2 𝑡 − 𝑘 𝑚 𝑥 1 𝑡 + 1 𝑚 𝑢 (𝑡)
𝑦 𝑡 = 𝑥 1 𝑡

13. Example-1 𝑥 2 𝑡 =− 𝑏 𝑚 𝑥 2 𝑡 − 𝑘 𝑚 𝑥 1 𝑡 + 1 𝑚 𝑢 (𝑡) 𝑥 1 𝑡 = 𝑥 2 (𝑡)
𝑥 2 𝑡 =− 𝑏 𝑚 𝑥 2 𝑡 − 𝑘 𝑚 𝑥 1 𝑡 + 1 𝑚 𝑢 (𝑡)
𝑥 1 𝑡 = 𝑥 2 (𝑡)
𝑦 𝑡 = 𝑥 1 𝑡
In a vector-matrix form,

14. Example-1 𝑢(𝑡) 1/s 1/s 𝑦(𝑡) State diagram of the system is
𝑥 1 𝑡 = 𝑥 2 (𝑡)
𝑥 2 𝑡 =− 𝑏 𝑚 𝑥 2 𝑡 − 𝑘 𝑚 𝑥 1 𝑡 + 1 𝑚 𝑢 (𝑡)
𝑦 𝑡 = 𝑥 1 𝑡
-k/m
-b/m
1/m
𝑥 2
𝑢(𝑡)
𝑥 1
1/s
1/s
𝑦(𝑡)
𝑥 2 = 𝑥 1

15. Example-1
State diagram in signal flow and block diagram format
1/s
𝑢(𝑡)
𝑦(𝑡)
-k/m
-b/m
𝑥 2
1/m
𝑥 2 = 𝑥 1
𝑥 1

16. Example-2 State space representation of armature Controlled D.C Motor.
ea is armature voltage (i.e. input) and  is output. ea ia T Ra La J  B eb
Vf=constant 

17. Example-2 Choosing 𝜃, 𝜃 𝑎𝑛𝑑 𝑖 𝑎 as state variables
Since 𝜃 is output of the system therefore output equation is given as
𝑑 𝑑𝑡 𝜃 𝜃 𝑖 𝑎 = − 𝐵 𝐽 𝐾 𝑡 𝐽 0 − 𝐾 𝑏 𝐿 𝑎 − 𝑅 𝑎 𝐿 𝑎 𝜃 𝜃 𝑖 𝑎 𝐿 𝑎 𝑒 𝑎
𝑦 𝑡 = 𝜃 𝜃 𝑖 𝑎

18. State Controllability
A system is completely controllable if there exists an unconstrained control u(t) that can transfer any initial state x(to) to any other desired location x(t) in a finite time, to ≤ t ≤ T.
uncontrollable
controllable

19. State Controllability
Controllability Matrix CM
System is said to be state controllable if

20. State Controllability (Example)
Consider the system given below
State diagram of the system is

21. State Controllability (Example)
Controllability matrix CM is obtained as
Thus
Since 𝑟𝑎𝑛𝑘(𝐶𝑀)≠𝑛 therefore system is not completely state controllable.

22. State Observability
A system is completely observable if and only if there exists a finite time T such that the initial state x(0) can be determined from the observation history y(t) given the control u(t), 0≤ t ≤ T.
observable
unobservable

23. State Observability Observable Matrix (OM)
The system is said to be completely state observable if

24. State Observability (Example)
Consider the system given below
OM is obtained as
Where

25. State Observability (Example)
Therefore OM is given as
Since 𝑟𝑎𝑛𝑘(𝑂𝑀)≠𝑛therefore system is not completely state observable.

27. Output Controllability
Output controllability describes the ability of an external input to move the output from any initial condition to any final condition in a finite time interval.
Output controllability matrix (OCM) is given as

28. Home Work
Check the state controllability, state observability and output controllability of the following system

29. Transfer Matrix (State Space to T.F)
Now Let us convert a space model to a transfer function model.
Taking Laplace transform of equation (1) and (2) considering initial conditions to zero.
From equation (3)
(1)
(2)
(3)
(4)
(5)

30. Transfer Matrix (State Space to T.F)
Substituting equation (5) into equation (4) yields

31. Example#3
Convert the following State Space Model to Transfer Function Model if K=3, B=1 and M=10;

32. Example#3
Substitute the given values and obtain A, B, C and D matrices.

33. Example#3

34. Example#3

35. Example#3

36. Example#3

37. Example#3

38. Home Work
Obtain the transfer function T(s) from following state space representation.
Answer

39. Forced and Unforced Response
Forced Response, with u(t) as forcing function
Unforced Response (response due to initial conditions)

40. Solution of State Equations
Consider the state equation given below
Taking Laplace transform of the equation (1)
(1)

41. Solution of State Equations
Taking inverse Laplace
State Transition Matrix

42. Example-4
Consider RLC Circuit obtain the state transition matrix ɸ(t). Vc + - Vo iL

43. Example-4 (cont...) State transition matrix can be obtained as
Which is further simplified as

44. Example-4 (cont...)
Taking the inverse Laplace transform of each element

45. Home Work
Compute the state transition matrix if
Solution

46. State Space Trajectories
The unforced response of a system released from any initial point x(to) traces a curve or trajectory in state space, with time t as an implicit function along the trajectory.
Unforced system’s response depend upon initial conditions.
Response due to initial conditions can be obtained as

47. State Transition
Any point P in state space represents the state of the system at a specific time t.
State transitions provide complete picture of the system
P(x1, x2) t0 t1 t2 t3 t4 t5 t6

48. Example-5
For the RLC circuit of example-4 draw the state space trajectory with following initial conditions.
Solution

49. Example-5 (cont...)
Following trajectory is obtained

50. Example-5 (cont...)

51. Equilibrium Point
The equilibrium or stationary state of the system is when

52. Solution of State Equations
Consider the state equation with u(t) as forcing function
Taking Laplace transform of the equation (1)
(1)

53. Solution of State Equations
Taking the inverse Laplace transform of above equation.
Natural Response
Forced Response

54. Example#6 Obtain the time response of the following system:
Where u(t) is unit step function occurring at t=0. consider x(0)=0.
Solution
Calculate the state transition matrix

55. Example#6
Obtain the state transition equation of the system

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