Chapter 4 System Analysis
Xmath Control Design Module 4-4 ni.com
directly from the roots of the transfer function numerator. If Sys is in
state-space form, the definition of its zeros arises from the system matrix,
(4-3)
and its MIMO transfer function:
(4-4)
Defining n as the number of states in the system, p as the number of outputs,
and m as the number of inputs, the normal rank of S(λ) is n+ min(m,p).
If the rank of S(λ) equals the normal rank, the system is nondegenerate.
The values of λ, where R(λ) = 0 and S(λ) loses rank, are the invariant zeros
of the system. For degenerate cases in which the normal rank of S(λ) is less
than n+r, the zeros are defined analogously.
If a system is minimal (that is, no other system with lower order and the
same R(λ) exists), these invariant zeros are termed transmission zeros.
When the matrix in Equation 4-4 loses rank for some value λ=λ0, there
exists a vector [x0' u0']' of initial states and inputs such that:
Thus, there exists an initial state x0 such that the output y is zero for all
values of the input function defined over time t as u0eλt. Such zeros (λ0)
derive the name transmission zero, because their effect is to block
transmission of the system input to the output.
Note zeros = system zeros = {invariant zeros}∩{transmission zeros}.
For an example using zeros( ) with a state-space system, refer to
Example 4-2. For more details on this topic, refer to [Kai80] and [DeS74].
Example 4-2 Using zeros( ) with a State-Space System
Sys=system([-2.3,0.01,5.1;0,-0.35,-2;0,2,-.35],
[1,.25,.25]',[1.34,0,0],0);
[z,k] = zeros(Sys)
Sλ() λIA–B
C–D
=
Rλ() CλIA–()
1–BD+=
λ0IA–B
C–D
x0
u0
0=