(Computer Control)3. Analys. of Discrete Time Sys.
1. Stability
Jurys Stability Test
For the system
Using Jurys Stability Test
If the system is stable, we have
2. Controllability
Controllable: The system can be driven from any initial state to any other destination in finite time.
For the system
We have the initial state $x(0)$
Then
If the system can be driven to any state in finite steps, we just need to ensure
is NONSINGULAR.
If we cannot get to one state with n-1 steps, we cannot to get to one state with n steps.
Proof
Then we have
Similarly,
We know is linear combination of .
Therefore, we still cannot get to arbitrary desired state.
Controllable Canonical Form
3. Observability
Observer
Observability Matrix
Observable Canonical Form
4. How to Create State Space From Difference Equation
Example
First, we choose
Then we have
Then we choose
Therefore, we have
Therefore,
5. Some Useful Conclusions
1. The minimal realization has the same order as that of the transfer function. If higher, it is called non-minimal realization.
2. The minimal realization can be always cast into the form of observable canonical form. Therefore, the minimal realization is always observable.
3. If there are common zeros and poles, uncontrollable, otherwise, controllable.
6. Example
Example 1
a) What is the Transfer Function?
b) is it possible to realize the system such that it is observable but not controllable.
It is not controllable, therefore, it should have common zeros and poles.
The system is observable, therefore, we realize the system in observable canonical form.
Therefore, the system is observable but not controllable.
However, we need to check if this space state is that transfer function. It is easy to find that it is exactly the G(z) instead of the transfer function in the question. Therefore, it is **impossible** to realize this system in observable but not controllable form.
c) is it possible to realize the system such that it is controllable but not observable.
Same as the above, it is possible to realize.
d) is it possible to realize the system such that it is both controllable and observable.
Obviously, it is possible.
Example 2.
The questions are same as the above.
a)
b) it is possible observable but not controllable
c) it is not possible controllable but not observable
d) it is not possible controllable and observable
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