Systems Theory
Poles and Zeros of MIMO Systems
M.Varatharajan
Asst.Professor
Department of Electrical & Electronics Engineering
Thiagarajar College of Engineering
Madurai
December 12, 2012
Transfer function
Consider the system
Ẋ = AX + BU
Y = CX + DU
where, x ∈ R n , u ∈ R m and y ∈ R p
Transfer function is given by H(s) = C [SI − A]−1 B + D
I In SISO system m = p = 1
I Relative degree = Degree of denominator - Degree of
numerator
I H(s) is
I Strictly proper if Relative degree ≥1
I Proper if Relative degree ≥0
MIMO Systems
N(s)
I Here H(s) = C [SI − A]−1 B + D = ∆(s) , where N(s) is a
matrix and ∆(s) = [SI − A] is a polynomial
−1
I Roots of |[SI − A]−1 | = 0 are the poles of the system
I Different kinds of system zeros Given in terms of Rosenbrock
system matrix
1. Transmission zeros
2. Input decoupling zeros
3. Output decoupling zeros
Zeros of Multivariable Systems
Transmission Zeros:
I Zeros of |N(s)| = 0
I Depends only on the transfer function
I Available for square matrix. i.e. p=m
I If input u(t) is applied at the frequency of the transmission
zero, output will not contain that frequency component
I Also called as blocking zeros
Rosenbrock System Model
Zeros are given in terms of Rosenbrock’s system matrix.
sI − A B
P(s) =
−C D
The Rosenbrock system matrix is important in that it captures the
structure of a dynamical system and provides a unifying point of
view for analysis and design.
0
sI − A B −X (s)
=
−C D U(s) Y (s)
Zeros of Multivariable Systems
Input Decoupling Zeros:
The input-decoupling zeros are those values of s for which the n ×
(n + m) input- coupling matrix PI (s) = [sI −AB],loses rank, i.e.
has rank less than n.
Note that this matrix can lose rank only where (sI − A) loses rank,
so the input-decoupling zeros must be a subset of the poles.
Output Decoupling Zeros:
The output-decoupling zeros are those values of s for which the (n
sI − A
+ p ) × n output-coupling matrix Po (s) = loses rank,
−C
i.e. has rank less than n.
Note that this matrix can lose rank only where (sI − A) loses rank,
so the output-decoupling zeros must be a subset of the poles.
Zeros of Multivariable Systems
Input Decoupling Zeros:
The input-decoupling zeros are those values of s for which the n ×
(n + m) input- coupling matrix PI (s) = [sI −AB],loses rank, i.e.
has rank less than n.
Note that this matrix can lose rank only where (sI − A) loses rank,
so the input-decoupling zeros must be a subset of the poles.
Output Decoupling Zeros:
The output-decoupling zeros are those values of s for which the (n
sI − A
+ p ) × n output-coupling matrix Po (s) = loses rank,
−C
i.e. has rank less than n.
Note that this matrix can lose rank only where (sI − A) loses rank,
so the output-decoupling zeros must be a subset of the poles.
System Zeros & Poles
System zeros = transmission zeros( zeros of transfer function) +
input-decoupling zeros + output-decoupling zeros – input/output
decoupling zeros.
These coupling zeros are dependent on the state model. They
cancel out while finding the transfer function
System Zeros & Poles
System zeros = transmission zeros( zeros of transfer function) +
input-decoupling zeros + output-decoupling zeros – input/output
decoupling zeros.
These coupling zeros are dependent on the state model. They
cancel out while finding the transfer function
System poles = Poles of transfer function + input-decoupling zeros
+ output-decoupling zeros – input/output decoupling zeros.
Meaning of Input Decoupling Zeros
If rank of PI (s) < n, it means that there exist a frequency so and a
vector w such that
w T [s0 I −A B] = 0
or,
w T [s0 I −A] = 0
It means that s0 is an eigven value of sI − A and w is the eigen
vector.
Input & Ouput Decoupling Zeros
Note that (sI −A)−1 B is the right-hand or input portion of the
transfer function. We shall see that the input decoupling zeros
mean a loss of control effectiveness at that frequency, and we
cannot fully control the system with the given inputs. We should
design systems with no input-decoupling zeros, i.e. with a fully
effective set of inputs.
