1. Analogue and digital techniques in closed loop regulation applications
Multivariable systems
State space analysis
Controllability and observability
An overview

2. Multivariable systems
• State variable technique
• State representation of a linear system
• Transition matrix

3. State variable technique
Linear
System
Output vector
Input vector
State vector

4. Multivariable linear system (MIMO)
Representation of x Bu + A +

5. State representation of a linear systemD A B x u C y
C= Output Matrix (r,n)
D= Direct Transmission Matrix (r,m)
y= Output Vector (r,1)
A= System Matrix(n,n)
B= Input Matrix (n,m)
x= State Vector (n,1)
u= Input Vector (m,1)

6. Transition matrix Assuming that the system is continuous and linear
that A and B are time-invariant and
Using Laplace transform
Taking the inverse Laplace transform of resolvent matrix
Transition matrix

7. Transition matrix
The state vector will take the following form (convolution)
Or more generally
The output vector will take the following form
Assuming that C and D are time-invariant

8. Transition matrix Properties of the transition matrix
The resolvent matrix

9. Transition matrix Inverse of sI-A
Determinant is a polynomial of degree n
The minors

10. Transition matrix

11. Application
Second order LRC circuit
• Monovariable
• Multivariable

12. Differential equation
LRC circuit: modelling
Presentation L R
u(t)
i(t)
v(t) C
Writing differential equations
Constant coefficient
Second order
Differential equation

13. LRC circuit: time domain analysis
Matrix form R C
u(t)
v(t) L
i(t)
Can be written

14. Time domain analysis For t=0 Step function with initial conditions
Matrix form
For t=0

15. Time domain analysis
Define M

16. Time domain analysis
General solution t
Example t
2t

17. Multivariable technique
LRC circuit
v(t)
i(t)
Equivalent circuit (Thevenin)
v(t)
i(t)

18. State variable technique
v(t)
i(t)
Basic circuit equation
Can be arranged
In matrix form

19. State variable technique
Solution: resolvent matrix

20. State variable technique

21. State-space for sampled systems
• Without zero-order-hold
• Representation

22. State-space for sampled systems
Without zero-order-hold
Linear
system
Ideal
sampler
u(t)
u’(t)
y(t)
x(t)
From
Evaluate

23. State-space for sampled systems
Some new definitions
With =1

24. State-space for sampled systems
With zero-order-hold
Linear
system
Ideal
sampler
u(t)
u’(t)
y(t)
x(t)
Zero-
order
Evaluate

25. State-space for sampled systems
Difference equation
Some new definitions
With =1
Notice

26. State variable technique Structural diagram D
I/z C F
A,B,C,D Time-invariant
Solution
F: System matrix or fundamental matrix
Difference equations

27. State variable technique
Evolution in between samplings
Without sample and hold circuit
With sample and hold circuit

28. State-space for sampled systems
Example
A linear system described by the following elements with
Zero-order hold
Transition matrix
Resolvent matrix

29. State-space for sampled systems
Evaluate F=(T)

30. State-space for sampled systems
Evaluate H=B

31. State-space for sampled systems
Evolution of state variables between sampling times
Can be transformed into
The matrices F() and H() determine the behaviour between
sampling times while matrices F(T) and H(T) determine
the values of the state variables at sampling times

32. State-space for sampled systems
y(k) D
x(k)
x(k+1) H
u(k)
H()
F()
x(k,)

33. Discrete system
D(z)
State equations

34. State variable technique
Digital (sampled or discrete) systems in series a b

35. • Controllability
• Observability

36. Controllability and observability
Definition:The process G is said to be controllable if every
state variable x of G can be affected or controlled in finite time
by some unconstrainted control signal u(t).
Observability
Definition:The process G is said to be observable if every
state variable x of G eventually affects some of the outputs
y of the process. Information on the state variables can be
obtained from the measurements of the outputs and the
inputs
The concept of observability is the dual of controllability

37. Controllability of a multivariable system
Consider the system described by:
Evaluation of x
Genera l case

38. Controllability of a multivariable system
Output
Depends on the
Initial conditions
Evolution due
To input
Depends directly
On input

39. Controllability of a monovariable system
The sum can be expressed as
The partitioned matrix
The input sequence
(k-vector)

40. Controllability of a monovariable system
The partitioned controllability matrix has n rows and k columns
Lets make k=n, assuming determinant different from zero
A monovariable linear system of order n can be driven from a
Initial state x[0] into a final state x[n] in exactly n sampling periods

41. Controllability of a multivariable system
The (n,mk) partitioned matrix
The input sequence
(mk-vector)

42. Controllability of a multivariable system
The matrix can be decomposed in
Square nXn
matrix
nXmk-n
matrix
The input vector is divided into
n-vector
(mk-n)-vector

43. Controllability of a multivariable system
The rank of this matrix
must be n
There are several possibilities to drive the system
from x[0] to x[n] via input sequence

44. Observability
Definition
A system is said observable if for any k0 any state x[k0] can
be determined from the knowledge of the output y[k] and
the input u[k]for k0<k<kN, where kN is some finite time
Monovariable system

45. Observability of a monovariable system
Define the two matrices

46. Observability of a multivariable system
Define the two matrices
Many possibilities

47. Thank you
The end