1. Pole Placement for State Space Model

If one matrix is controllable canonical form

If we choose

For any state space model, according to Ackermanns Formula, we have

For a uncontrollable system, if we use state feedback control, some poles cannot be changed.

If uncontrollable poles are stable, we can still control some poles.

2. Observer

Common used observer

Deadbeat Observer

From Ackermans Formula

3. Disturbance Rejection

Augmented state vector

If we choose ,

Then

Therefore, we only need

If the above cannot be satisfied, we can use the following method,

if the disturbance is constant

If we use state feedback controller, then we will have,

If the disturbance is not constant, then frequency analysis should be used.

Example

We can get the augmented state space model

Step1. Disturbance Rejection

Then we have

If not satisfied and the disturbance is constant, we will have

Step2. Observer

Then we have

Step3. Pole Placement

4. Feedforward Controller

We can check the transfer function from feedforward input to Output y

Therefore, we can choose

We can achieve perfect tracking if and only if B is stable. Then the input of the feedforward controller is bounded.

5. Example

Where, disturbance v is constant. Design controllers such that influence of v can be eliminated in s.s. in the following cases.

a) The state and v can be measured.

The augmented state space model is

We can choose the controller,

Step 1. Disturbance Rejection.

We cannot get a to solve this equation.

Step 2. Feedback Controller

Then we have

Then

We can get

b) The state can be measured.

We set k=0.

Because states can be measured, we can solve v

Then the controller is the same as a).

c) The input can be measured.

Now we need to design an Deadbeat observer.

Then we have

Now our controller should be

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