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In Discrete-time Control Engineering by Katsuhiko Ogata, it is stated that "A necessary and sufficient condition for complete state controllability is that no cancellation occurs in the pulse transfer function. If cancellation occurs, the system cannot be controlled in the direction of the cancelled mode."

The pulse transfer function relates a sequence of samples at the output of a system to the sequence producing it. In other words, it is the discrete-time equivalent of the transfer function in the continuous domain.

It is stated that, if we have a transfer function in which pole-zero cancellation occurs, converting the same to state space representation, the system can not be fully controllable and observable (at least one of them will be lost). It is also stated that the system cannot be controlled in the direction of the cancelled mode.

What does controlling in a direction mean?

Also, how is direction defined in state space?

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  • $\begingroup$ Please give more context to the question. Is a state space representation of the system involved ? If so, why convert to transfer function representation ? What is pulse transfer function ? How is the word mode defined ? Please add these details (and more). $\endgroup$
    – AJN
    Dec 12, 2022 at 13:21
  • $\begingroup$ Direction probably means direction in the state space. But your question does not mention state space. Question is too vague. $\endgroup$
    – AJN
    Dec 12, 2022 at 13:23
  • $\begingroup$ Hi @AJN, I am unsure how the mode is defined. Probably, it's referring to the cancelled pole(s) and zero(s). $\endgroup$ Dec 13, 2022 at 5:10
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    $\begingroup$ 1 are you familiar with state space representation ? 2 are you familiar with diagonalisation of matrices and change of co-ordinates with reference to state space representation of systems ? these concepts need to be known as pre requisites, I think. $\endgroup$
    – AJN
    Dec 13, 2022 at 12:13

1 Answer 1

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I'll try to explain using examples.

Basic setup

Discrete time system in state space.

$$ \begin{align} x_{n+1} &= A x_n + B u_n\\ x_1 &= A x_0 + B u_0\\ x_2 &= A^2 x_0 + AB u_0 + Bu_1\\ x_3 &= A^3 x_0 + A^2B u_0 + AB u_1 + Bu_2\\ x_n &= A^n x_0 + \sum_k^{n-1}{A^{n-k-1}B u_k} \end{align} $$[1]

Example 1

$$ x_{n+1} = \begin{bmatrix} 0.9 & 0\\ 0 & 0.8 \end{bmatrix} x_n + \begin{bmatrix} 1\\ 0 \end{bmatrix} u_n $$

For simplicity, assume zero initial condition. Then $$ \begin{align} x_1 &= \begin{bmatrix} u_1\\ 0 \end{bmatrix}\\ x_2 &= \begin{bmatrix} 0.9 & 0\\ 0 & 0.8 \end{bmatrix} \begin{bmatrix} u_1\\ 0 \end{bmatrix} + \begin{bmatrix} u_2\\ 0 \end{bmatrix} &&= \begin{bmatrix} 0.9 u_1 + u_2\\ 0 \end{bmatrix}\\ x_n &= \begin{bmatrix} \sum{0.9^{n-k-1}u_k}\\ 0 \end{bmatrix} &&= \left(\sum{0.9^{n-k-1}u_k}\right) \begin{bmatrix} 1\\ 0 \end{bmatrix} + 0 \begin{bmatrix} 0\\ 1 \end{bmatrix} \end{align} $$

It can be seen that, whatever we chose as the input sequence $u_n$, we can control the state only in the $[1,\quad 0]^T$ direction; i.e. we can influence only the first state variable and not the second one. The direction $[0,\quad 1]^T$ is not controllable.

Also note that, to reach this conclusion, we utilized the values of the column vectors $B,\ AB,\ A^2B, \dots$. This is why the controllability matrix is made from those column matrices.

Example 2

$$ x_{n+1} = \begin{bmatrix} 0.9 & 0\\ 0 & 0.9 \end{bmatrix} x_n + \begin{bmatrix} 1\\ 1 \end{bmatrix} u_n $$ Doing the same procedure as above, $$ \begin{align} x_1 &= \begin{bmatrix} u_1\\ u_1 \end{bmatrix}\\ x_2 &= \begin{bmatrix} 0.9 & 0\\ 0 & 0.9 \end{bmatrix} \begin{bmatrix} u_1\\ u_1 \end{bmatrix} + \begin{bmatrix} u_2\\ u_2 \end{bmatrix} &&= \left(0.9 u_1 + u_2\right) \begin{bmatrix} 1\\ 1 \end{bmatrix}\\ x_n &= \begin{bmatrix} \sum{0.9^{n-k-1}u_k}\\ \sum{0.9^{n-k-1}u_k} \end{bmatrix} &&= \left(\sum{0.9^{n-k-1}u_k}\right) \begin{bmatrix} 1\\ 1 \end{bmatrix} + 0 \begin{bmatrix} 1\\ -1 \end{bmatrix} \end{align} $$ It can be seen that, whatever we chose as the input sequence $u_n$, we can control the state only in the $[1,\quad 1]^T$ direction; i.e. we can influence the first state variable and second state variable by the same amount. We cannot increase first state variable while decreasing the second state variable; i.e. The direction $[1,\quad -1]^T$ is not controllable.

Example 3

$$ x_{n+1} = \begin{bmatrix} 0.9 & 0\\ 1 & 0.8 \end{bmatrix} x_n + \begin{bmatrix} 1\\ 0 \end{bmatrix} u_n $$ Doing the same procedure as above, $$ \begin{align} x_1 &= \begin{bmatrix} u_1\\ 0 \end{bmatrix}\\ x_2 &= \begin{bmatrix} 0.9 & 0\\ 1 & 0.8 \end{bmatrix} \begin{bmatrix} u_1\\ 0 \end{bmatrix} + \begin{bmatrix} u_2\\ 0 \end{bmatrix} &&= u_1 \begin{bmatrix} 0.9\\ 1 \end{bmatrix} + u_2 \begin{bmatrix} 1\\ 0 \end{bmatrix}\\ &= u_1 \begin{bmatrix} 0\\ 1 \end{bmatrix} + \left(u_2 + 0.9 u_1\right) \begin{bmatrix} 1\\ 0 \end{bmatrix}\\ \end{align} $$

Here we can see that by suitably choosing $u_1$ and $u_2$, we can change the state of the system in the direction $[1,\quad 0]^T$ as well as $[0,\quad 1]^T$ and in any direction which is a linear combination of these. Since this is a two dimensional state space, and we have two independent directions, their linear combination can actually generate any and all directions in the state space. So this system is fully controllable.

One more thing to note here is that, by looking at the original system equation, we see that the input directly influences only the first state variable due to $B$ matrix being of the form $[1,\quad 0]^T$. But by looking at the second time step, we realise that input can influence the second state variable also (but indirectly). This is why this system is fully controllable; the input signal can directly and indirectly influence all the state variables.

How to identify the directions ?

Left as an exercise to the reader. Look at the eigen vectors of the $A$ matrices in the examples above.

What if the system being analysed is different from the examples?

The examples above were chosen to be easy to explain. But, we can transform all systems to the format of one of the above examples (diagonal or nearly diagonal) by using eigen decomposition of the $A$ matrix. This transformation is also called change of variable or change of coordinates.

Transfer functions, cancellation of modes and controllability

Continuing from the example 1, and assuming that its $C$ matrix is $[1,\quad 1]$, $$ \begin{align} C(zI-A)^{-1}B &= \begin{pmatrix} 1 & 1 \end{pmatrix} \frac{1}{(z-0.8)(z-0.9)} \begin{pmatrix} z-0.8 & 0\\ 0 & z-0.9 \end{pmatrix} \begin{pmatrix} 1\\ 0 \end{pmatrix}\\ &= \frac{(z-0.8)}{(z-0.8)(z-0.9)}\\ &= \frac{1}{(z-0.9)}\\ \end{align} $$

We notice that the modes correspond to the poles of the system and that one of the poles (the one at $z=0.8$) cancelled out. This is the mode that turns out to be uncontrollable. But, just by looking at the transfer function, we can't tell the direction in which the mode $z=0.9$ acts along, and the direction in which mode $z=0.8$ acts along.

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    $\begingroup$ Thanks, @AJN, this clarifies everything. One thing, in example 3, you have specified the directions. The 2nd direction should be $$[0, 1]^T$$ $\endgroup$ Dec 23, 2022 at 9:36
  • $\begingroup$ Corrected to $[0,\ 1]^T$. $\endgroup$
    – AJN
    Dec 23, 2022 at 12:58

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