1. Multivariable Control Systems
Ali Karimpour
Assistant Professor
Ferdowsi University of Mashhad
<<<1.1>>>
###Control System Design###
{{{Control, Design}}}

2. Topics to be covered include:
Chapter 2
Introduction to Multivariable Control
Topics to be covered include:
Multivariable Connections
Multivariable Poles
Multivariable Zeros
Directions of Poles and Zeros
Smith Form of a Polynomial Matrix
Smith-McMillan Forms
Matrix Fraction Description (MFD)
Scaling
Performance Specification
Trade-offs in Frequency Domain
Bandwidth and Crossover Frequency

3. Multivariable Connections
Cascade (series) interconnection of transfer matrices
Generally
Parallel interconnection of transfer matrices

4. Multivariable Connections
Feedback interconnection of transfer matrices
A useful relation in multivariable is push-through rule
Exercise 1: Proof the push-through rule

5. Multivariable Connections
MIMO rule: To derive the output of a system, start from the
output and write down the blocks as you meet them when
moving backward (against the signal flow) towards the input.
If you exit from a feedback loop then include a term
or according to the feedback sign where L is the
transfer function around that loop (evaluated against the
signal flow starting at the point of exit from the loop).
Parallel branches should be treated independently and their
contributions added together.

6. Multivariable Connections
Example 2-1: Derive the transfer function of the system shown in figure

7. Multivariable Poles ( State Space Description )
Definition 2-1: The poles of a system with state-space description
are eigenvalues of the matrix A.
The pole polynomial or characteristic polynomial is defined as
Thus the system’s poles are the roots of the characteristic polynomial
Note that if A does not correspond to a minimal realization then the poles by this definition will include the poles (eigenvalues) corresponding to uncontrollable and/or unobservable states.

8. Multivariable Poles ( Transfer Function Description )
Theorem 2-1
The pole polynomial corresponding to a minimal realization of
a system with transfer function G(s) is the least common denominator
of all non-identically-zero minors of all orders of G(s).
A minor of a matrix is the determinant of the square matrix obtained by
deleting certain rows and/or columns of the matrix.

9. Multivariable Poles ( Transfer Function Description )
Example 2-4
The non-identically-zero minors of order 1 are
The non-identically-zero minor of order 2 are
By considering all minors we find their least common denominator

10. Multivariable Poles Exercise 2: Consider the state space realization
a- Find the poles of the system directly through state space form.
b- Find the transfer function of system ( note that the numerator and the denominator
of each element must be prime ).
c- Find the poles of the system through its transfer function.

11. Multivariable Zeros ( State Space Description )
Laplace transform
The zeros are then the values of s=z for which the polynomial system
matrix P(s) loses rank resulting in zero output for some nonzero input.
Let
Then the zeros are found as non trivial solutions of
This is solved as a generalized eigenvalue problem.

12. Multivariable Zeros ( State Space Description )
For square systems with m=p inputs and outputs and n states, limits
on the number of transmission zeros are:

13. Multivariable Zeros ( State Space Description )
Example 2-5: Consider the state space realization

14. Multivariable Zeros ( Transfer Function Description )
Definition 2-2
is a zero of G(s) if the rank of is less than the normal rank
of G(s). The zero polynomial is defined as
Where is the number of finite zeros of G(s).

15. Multivariable Zeros ( Transfer Function Description )
Theorem 2-2
The zero polynomial z(s) corresponding to a minimal realization of the
system is the greatest common divisor of all the numerators of all
order-r minors of G(s) where r is the normal rank of provided that
these minors have been adjusted in such a way as to have the pole
polynomial as their denominators.

16. Multivariable Zeros ( Transfer Function Description )
Example 2-9
according to example 2-4 the pole polynomial is:
The minors of order 2 with as their denominators are
The greatest common divisor of all the numerators of all order-2 minors is

17. Multivariable Poles Exercise 3: Consider the state space realization
a- Find the zeros of the system directly through state space form.
b- Find the transfer function of system ( note that the numerator and the denominator
of each element must be prime ).
c- Find the zeros of the system through its transfer function.

18. Directions of Poles and Zeros
Zero directions:
Let G(s) have a zero at s = z, Then G(s) losses rank at s = z and
there will exist nonzero vectors and such that
is input zero direction and is output zero direction
We usually normalize the direction vectors to have unit length

19. Directions of Poles and Zeros
Pole directions:
Let G(s) have a pole at s = p. Then G(p) is infinite and we may
somewhat crudely write
is input pole direction and is output pole direction

20. Directions of Poles and Zeros
Example:
It has a zero at z = 4 and a pole at p = -2 .
Output zero direction
Input zero direction

21. Directions of Poles and Zeros
Example:
It has a zero at z = 4 and a pole at p = -2 .
??!!
Output pole direction
Input pole direction

22. Smith Form of a Polynomial Matrix
Suppose that is a polynomial matrix.
Smith form of is denoted by , and it is a pseudo diagonal
in the following form
is a factor of and

23. Smith Form of a Polynomial Matrix
is a factor of and
gcd {all monic minors of degree 1}
gcd {all monic minors of degree 2}
………………………………………………………..
………………………………………………………..
gcd {all monic minors of degree r}

24. Smith Form of a Polynomial Matrix
The three elementary operations for a polynomial matrix are used
to find Smith form.
Multiplying a row or column by a constant;
Interchanging two rows or two columns; and
Adding a polynomial multiple of a row or column to another
row or column.

25. Smith Form of a Polynomial Matrix
Example 2-11
Exercise 4: Derive R(s) and L(s) that convert to

26. Smith-McMillan Forms Theorem 2-3 (Smith-McMillan form)
Let be an m × p matrix transfer function, where
are rational scalar transfer functions, G(s) can be represented by:
Where Π(s) is an m× p polynomial matrix of rank r and DG(s) is the least
common multiple of the denominators of all elements of G(s) .
Then, is Smith McMillan form of G(s) and can be derived directly by

27. Smith-McMillan Forms
Theorem 2-3 (Smith-McMillan form)

28. Smith-McMillan Forms
Example 2-13

29. Matrix Fraction Description (MFD)
polynomial
polynomial
This is a Right Matrix Fraction Description (RMFD)
polynomial
polynomial
This is a Left Matrix Fraction Description (LMFD)

30. Matrix Fraction Description (MFD)
A model structure that is related to the Smith-McMillan form is matrix fraction description (MFD).
Right matrix fraction description (RMFD)
Left matrix fraction description (LMFD)
Let is a matrix and its the Smith McMillan is
Let define:

31. Matrix Fraction Description (MFD)
We know that
This is known as a right matrix fraction description (RMFD) where

32. Matrix Fraction Description (MFD)
We know that
This is known as a left matrix fraction description (LMFD) where

33. Matrix Fraction Description (MFD)
We can show that the MFD is not unique, because, for any nonsingular
matrix we can write G(s) as:
Where is said to be a right common factor. When the only right
common factors of and is unimodular matrix, then, we
Say that the RMFD is irreducible.
It is easy to see that when a RMFD is irreducible, then

34. Matrix Fraction Description (MFD)
Example 2-16
RMFD:

35. Matrix Fraction Description (MFD)
Example 2-16
LMFD:

36. Scaling

37. Performance Specification
Nominal stability NS: The system is stable with no model uncertainty.
Nominal Performance NP: The system satisfies the performance
specifications with no model uncertainty.
Robust stability RS: The system is stable for all perturbed plants
about the nominal model up to the worst case model uncertainty.
Robust performance RP: The system satisfies the performance
specifications for all perturbed plants about the nominal model up
to the worst case model uncertainty.

38. Performance Specification
Time Domain Performance
Frequency Domain Performance

39. Time Domain Performance
Although closed loop stability is an important issue,
the real objective of control is to improve performance, that is,
to make the output y(t) behave in a more desirable manner.
Actually, the possibility of inducing instability is one of the
disadvantages of feedback control which has to be traded off
against performance improvement.
The objective of this section is to discuss ways of evaluating
closed loop performance.

40. Time Domain Performance
Step response of a system
Rise time, tr , the time it takes for the output to first reach 90% of
its final value, which is usually required to be small.
Settling time, ts , the time after which the output remains within
5% ( or 2%) of its final value, which is usually required to be small.
Overshoot, P.O, the peak value divided by the final value, which
should typically be less than 20% or less.

41. Time Domain Performance
Step response of a system
Decay ratio, the ratio of the second and first peaks, which should
typically be 0.3 or less.
Steady state offset, ess, the difference between the final value and the desired final value, which is usually required to be small.

42. Time Domain Performance
Step response of a system
Excess variation, the total variation (TV) divided by the overall
change at steady state, which should be as close to 1 as possible.
The total variation is the total movement of the output as shown.

43. Time Domain Performance
ISE : Integral squared error
IAE : Integral absolute error
ITSE : Integral time weighted squared error
ITAE : Integral time weighted absolute error

44. Frequency Domain Performance
Let L(s) denote the loop transfer function of a system which is
closed-loop stable under negative feedback.
Bode plot of

45. Frequency Domain Performance
Let L(s) denote the loop transfer function of a system which is
closed-loop stable under negative feedback.
Nyquist plot of

46. Frequency Domain Performance
Stability margins are measures of how close a stable closed-loop system
is to instability.
From the above arguments we see that the GM and PM provide stability
margins for gain and delay uncertainty.
More generally, to maintain closed-loop stability, the Nyquist stability
condition tells us that the number of encirclements of the critical point
-1 by must not change.
Thus the actual closest distance to -1 is a measure of stability

47. Complementary sensitivity
Frequency Domain Performance
One degree-of-freedom
configuration
Complementary sensitivity
function
Sensitivity
function
The maximum peaks of the sensitivity and complementary sensitivity
functions are defined as

48. Frequency Domain Performance
Thus one may also view Ms as a robustness measure.

49. Frequency Domain Performance
There is a close relationship between MS and MT and the GM and PM.
For example, with MS = 2 we are guaranteed GM > 2 and PM >29˚.
For example, with MT = 2 we are guaranteed GM > 1.5 and PM >29˚.

50. Complementary sensitivity
Trade-offs in Frequency Domain
One degree-of-freedom
configuration
Complementary sensitivity
function
Sensitivity
function

51. Trade-offs in Frequency Domain
Performance, good disturbance rejection
Performance, good command following
Mitigation of measurement noise on output
Small magnitude of input signals
Physical controller must be strictly proper
Nominal stability (stable plant)
Stabilization of unstable plant

52. Bandwidth and Crossover Frequency
Definition 2-3
Definition 2-4
Definition 2-5

53. Bandwidth and Crossover Frequency
Specifically, for systems with PM < 90˚ (most practical systems) we have
In conclusion ωB ( which is defined in terms of S ) and also ωc
( in terms of L ) are good indicators of closed-loop performance, while ωBT ( in terms of T ) may be misleading in some cases.

54. Bandwidth and Crossover Frequency
Example 2-17
: Comparison of and as indicators of performance.

55. Bandwidth and Crossover Frequency
Example 2-17
: Comparison of and as indicators of performance.