Which equations of a state space model influence the poles and zeroes?

Question

In regulation, if we represent a system with a state space model, we end with three matrices A, B and C (without taking into account feed-forward part) that define a specific part of the following equation:

$\dot{x}=Ax+Bu$

$y=Cx$

Is it also possible to define the transfer function $G=\frac{N}{D}=C(sI-A)^{-1}B$?

If we want to find the poles, we search when the denominator ($D$) is equal 0 and for the zeroes when the numerator ($N$) is 0.

My question is, if we want to define which part of the state space model influence whether the poles or the zeroes. I would say by looking at the previous equation that $A$ is the only one to influence the poles and $B,C$ act on the zeroes.

Unfortunately, the corrections made to my exam gave a different answer: $C$ influences the poles and all the matrices the zeroes.

I don't understand, why is that the case?

Answer 1

Half of your statement is correct: If we are considering the following system:

$$G : \begin{aligned} \dot{x} &= Ax+Bu \\ y &= Cx \end{aligned}$$

then $A$, $B$, and $C$ all influence the zeros of $G$, but $C$ certainly does not affect the poles. The poles $G$ are the eigenvalues of $A$.

This makes sense from a purely common-sense consideration: the matrix $C$ is the output matrix, it is only capable of scaling and adding the states to result in an output. It has no effect on the states themselves, therefore it won't affect things like stability of $G$ (which is related to the poles).

You can easily see this if you look at the transfer function equation you provided, with the inverse expressed as the determinant and the adjunct (or adjugate):

$G(s) = C(sI-A)^{-1}B = \frac{C \text{adj}(sI-A) B}{\text{det}(sI-A)}$

$C$, $B$, and the adjunct of $A$ all appear in the numerator, but only the determinant of $A$ appears in the denominator.

However...

Here's where your problem is likely coming from: If we consider $G$ to be a system plant being controlled through some feedback element with a transfer function $H(s)$ (like the following picture from Modern Control Engineering, 5th ed. by Ogata):

Then the overall transfer function for the entire system, from reference input $r$ to output $c$ is

$$\frac{C(s)}{R(s)} = \frac{G(s)}{1+G(s)H(s)}$$

Now, since $G(s)$ is in both the numerator and the denominator, you can see that $A$, $B$, and $C$ affect both the poles and zeros of the closed-loop transfer function.

In conclusion:

• Only $A$ affects the poles of the system plant $G$
• $A$, $B$, and $C$ affect the zeroes of the system plant $G$
• $A$, $B$, and $C$ affect both the poles and zeroes of the closed loop system

Reference: Modern Control Engineering, 5th ed. (2009) by Katsuhiko Ogata

Answer 2

The concept of poles and zeros in terms of the roots of the numerator and denominator is very crude and will not work for multivariable systems. Also computing them using the transfer function model is problematic because of pole-zero cancellations that can occur.

There are many notions of poles and zero and they are computed using both the transfer function and system matrix model. The latter is a generalization of the state-space model. I will provide a brief summary, and for more details you can refer to [1] or [2].

Transmission zeros: These are frequencies at which transmission through the system is blocked. They are computed from the Smith-McMillian form of the transfer function.

Transmissison poles: These are frequencies at which transmission through the system blows up. They are also computed from the Smith-McMillian form of the transfer function.

Input-decoupling zeros: These are the modes that are uncontrollable. They are computed from the system matrix model, and do not involve the output equations.

Output-decoupling zeros: These are the modes that are unobservable. They are computed from the system matrix model, and do not involve the inputs.

Input-output-decoupling zeros: These are the modes that are both uncontrollable and unobservable. They are computed from the system matrix model.

Invariant zeros: They are computed from the system matrix model. They involve the transmission zeros and the some of the input- and output-decoupling zeros.

System zeros: This is defined as {input-decoupling zeros, output-decoupling zeros, transmission zeros}-{input-output-decoupling zeros}

System poles: This is defined as {input-decoupling zeros, output-decoupling zeros, transmission poles}-{input-output-decoupling zeros}

The bottom-line is that it needs to be clear what type of pole or zero you are interested in, and the resulting computations may or may not include the various matrices of the state-space model.

Refs:

1. A. G. J. MacFarlane and N. Karcanias, Poles and zeros of linear multivariable systems: A survey of the algebraic, geometric and complex-variable theory, International Journal of Control 24(1):33-74, July 1976
2. H. H. Rosenbrock, State-space and multivariable theory.