1. Basic search for design of control system and disturbance observer
Sang–Chul Lee
DCASL

2. contents POLE PLACEMENT STATE OBSERVER
EFFECTS OF THE ADDITION OF THE OBSERVER ON A CLOSED-LOOP SYSTEM
INTRODUCTION OF THE DISTURBANCE OBSERVER

3. closed-loop control system with u = -Kx
Pole placement
Open-loop system
closed-loop control system with u = -Kx
(Regulator system)
What is the difference between two figures ??
Second figure have a feedback loop
If this system is stable, output will be approaches u.
So what is the reference signal?
This figure can be changed like this.
We will consider this K matrix.

4. Pole placement
(1)
Where x = state vector (n-vector) u = control vector (sclar) A = n x n constant matrix B = n x 1 constant matrix C = 1 x n constant matrix
choose the Control signal ->
(2)
The row matrix x is multiplyed by –k matrix. And it will be the input signal.
So we can say this system have a controller.

5. Pole placement
Substituting Eq (2) into Eq (1) gives
If the eigenvalues of matrix (A-BK) are asymptotically stable, x(t) approaches 0 as t approaches infinity
Let’s see the solution of the system in time domain.
This is differential Eq.
Do you remember? Matrix exponential function. (행렬지수함수)

6. So, What is the pole-placement technique ??
The problem of placing the closed-loop poles at the desired location is called a pole-placement problem
So, What is the pole-placement technique ??
This is almost same to pole-assignment.
Last control lecture, we studied it. Right?? But this time, it is in state space.
So ~~
This technique is about designing K matrix.
Because A and B matrix is already given (it’s fixed).
So we have to design proper K matrix.
This is the core of pole-placement.

7. Pole placement - how to design the k matrix
Define T(transformation matrix) by M is the controllability matrix And
Refer to the notes please. Page. 1
This is how to design the K matrix.
We will use the transformation matrix.
Using the transformation matrix will give us some advantages.
It is convenient to transform the state Eq. into the controllable canonical form.
The are coefficients of the characteristic polynomial.

8. Pole placement - how to design the k matrix
Define a new state vector by Eq (1) can be modified to Where ,
(Controllable canonical form)
So we got the controllable canonical form.
If you cannot understand about controllable canonical form, refer to the notes please.
Refer to the notes please. Page.1

9. Pole placement - how to design the k matrix
Let us choose a set of the desired eigenvalues as
then the desired characteristic Eq. becomes
Let us write
when is used to control system
The characteristic Eq. is
(3)
(4)

10. Pole placement - how to design the k matrix
This characteristic Eq. is the same as the desired characteristic Eq. for the system, when ‘u = -Kx’ is used as the control signal.
We can rewrite the Eq. (4) as
Here is the prove that using the T matrix is reasonable.

11. Pole placement - how to design the k matrix
This is the characteristic Eq. for the system with state feedback. Therefore, it must be equal to Eq.(3)
Determinant will be like this, and we got the polynomial.

12. Pole placement - how to design the k matrix
So we get And then, we obtain
If we selected proper eigenvalue, we can get the proper K matrix,
Consequently total system will be stable.

13. State observer What is the STATE OBSERVER ??
- A state observer estimates the state variables
based on the measurements of the output
and control variables.
Why we use the STATE OBSERVER ??
- Not all state variables are available for feedback!
We assumed that all state variables are available for feedback in pole-placement technique.
That means the state observer can estimate the value that cannot be measured.

14. State observer (5) (6) Model of the state observer
Eq. (6) is the Eq. of state observer.
This State observer has two input (y, u) and estimated one output x’(capa).
In this fig, -K is from pole-placement, and ~part is observer.
Block diagram of the system and full-order state observer

15. State observer
Subtract Eq.(6) from Eq.(5). So we can obtain the observer error Eq.
(7)
Let’s Define
Then Eq.(7)becomes
The dynamic behavior of the error vector is determined by the eigenvalues of matrix
thus the problem here becomes the same as the pole-placement problem.

16. State observer It’s very similar to the pole-placement technique
Refer to the notes please.
It’s not easy . So you would cannot understand totally in this time.
But at least pleas remember the flow. And check the note please.

17. EFFECTS OF THE ADDITION OF THE OBSERVER ON A CLOSED-LOOP SYSTEM
Consider the system defined by
And, the observer Eq. is
The laplace transform of u is
The laplace transform of observer Eq. is
If x(0) = 0, then we can obtain
By substituting this Eq. into (8) , we get
(8)
This is the transfer function of observer and controller.

18. EFFECTS OF THE ADDITION OF THE OBSERVER ON A CLOSED-LOOP SYSTEM
Block diagram representation of system
Note that the transfer function acts as a controller for the system.
That block is sum of two parts. One is the controller and the other one is observer.

19. Disturbance observer
The application of disturbance observer to practice
J : inertia
Kt : torque coefficient of electric motor
Tl : load torque
When the disturbance ‘Tl’ is generated ??
How can we use the estimated value ‘Tdis’ ??
This upper part is the transfer function of the electric motor.
Input is current and output is angular position.
This lower part is the observer.
This observer is designed by Gopinath’s method .
So first let’s see about these two signals.
If I is increased then theta will be increased.
But when the end-effector is contacted with the surface.
Theta would cannot increase. How can it be??
Disturbance observer in motion control
(designed by Gopinath’s method)

20. Thank you!