���BASIC CONTROL THEORY�LECTURE 5�� �
TOPICS
CONTROLLABILITY
The key issue in control is whether the input signal can influence all state variables independently.
The system is state-controllable if its state vector can be moved from the initial state to the arbitrarily specified end state by a control signal u over a finite time.
The individual components of the state variable must be independent of each other.
If this is true for the y output signal only, the system is output controllable.
Determination of controllability from the canonical form of the state equation:
The eigenvalues of matrix A are different.
None of the rows (elements) of matrix b (column vector) are zero.
BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI
2
2019
DETERMINING CONTROLLABILITY FROM THE GENERAL FORM OF THE STATE EQUATION (KALMAN CONDITION)
BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI
3
2019
Zero initial conditions are supposed.
Solution of the state equation:
As
(Cayley-Hamilton theorem)
The solution is:
The values of the integrals are constant.
The rank of the controllability hypermatrix should be n.
The rank of the output controllability matrix
should be equal to the number of the output signals.
OBSERVABILITY
Observability gives an answer to the question, whether the initial values of the state variables at the starting point of the measurements can be reconstructed by measuring the input and output signals during a certain time.
It is enough to perform the investigation only for u(t)=0, i.e. for the motion generated by the initial values of the state variables.
Determination of observability from the canonical form:
The eigenvalues of matrix A should be different.
None of the column vectors of matrix c is zero.
BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI
4
2019
DETERMINING OBSERVABILITY FROM THE GENERAL FORM OF THE STATE EQUATION (KALMAN CONDITION)�THE KALMAN DECOMPOSITION�
BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI
5
2019
The rank of the observability hypermatrix should be n.
THE KALMAN DECOMPOSITION
A system can be decomposed into four sub-systems, as:
controllable and observable
controllable and non-observable
non-controllable and observable
non-controllable and non-observable sub-systems.
The transfer function contains information only
on the controllable and observable sub-system.
EXAMPLES FOR DETERMINING CONTROLLABILITY AND OBSERVABILITY
BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI
6
2019
The canonical state equation of a system is:
Is the system state controllable? Is it output controllable? Is it observable? no yes no
Determine the transfer function!
EXAMPLES FOR DETERMINING CONTROLLABILITY AND OBSERVABILITY
BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI
7
2019
Given the block diagram of a control system
Equivalent scheme
The state equation with the assigned state variables:
Is the system state controllable, is it output controllable,
is it observable?
The rank is 1<2, not state controllable
The rank is 1, output controllable
The rank is 2, observable.
Explanation: pole cancellation
JORDAN FORM
BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI
8
2019
The transfer function of a system is:
Its block scheme is:
The state equation:
As the three poles are identical, the state variables
are not independent, so the system is not state controllable
and not observable.
Matrix A is not diagonal, this is the JORDAN form.
STATE FEEDBACK
The state equation:
The transfer function:
Generally d=0.
The poles are roots of the characteristic equation.
The control signal can be created by feeding back the state variables by constants.
or
The effect is acceleration.
BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI
9
2019
STATE FEEDBACK
BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI
10
2019
The characteristic equation of the system with state feedback:
The vector k is determined so that the poles are in the required position.
(Pole placement)
STATE FEEDBACK WITH POLE PLACEMENT
BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI
11
2019
ACKERMANN FORMULA:
Another principle: LQ controller, when the state feedback vector k
is calculated by minimising an integral criterion.
(It can be calculated easily from the observable canonical form.)
(command acker in matlab.)
EXAMPLE FOR STATE FEEDBACK
The parameter matrices of the state equation of a continuous system:
Solution
BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI
12
2019
The system is controlled with state feedback. Determine the state feedback vector ensuring that the poles of the closed loop system with state feedback be located at p1=-5, p2=-6.
or
Comparing the coefficients:
whence
és
EXAMPLE FOR STATE FEEDBACK
The transfer function of an unstable continuous system is:
Give the state equation in controllable canonical form.
Determine the stabilising state feedback vector. (Prescribing the poles of the closed loop system mirror the unstable pole to the imaginary axis, and let the stable pole at its location.)
BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI
13
2019
Solution
The state equation in controllable canonical form:
The prescribed poles of the feedback system: -6 and -3.
whence
és
Or:
An unstable system can be stabilised easily.
COMPENSATION OF THE STATIC ERROR
BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI
14
2019
The gain of the resulting transfer function is not 1.
The resulting transfer function of closed loop:
The callibration coefficient kr is:
Another solution compensating the static error is enhancing the control system
with an integrator.
be
STATE FEEDBACK BASED ON THE STATE EQUATION ENHANCED WITH AN INTEGRATOR
BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI
15
2019
The number of the state variables is increased by 1.
The state feedback vector has to be determined
considering the extended system.
STATE FEEDBACK BASED ON THE STATE EQUATION ENHANCED WITH AN INTEGRATOR
BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI
16
2019
Let us suppose d=0
The control system eliminates the static error
In case of step reference input.
STATE ESTIMATION
BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI
17
2019
If the state variables are not measurable, their values have to be estimated.
Another form of the state estimation circuit:
Its poles can be prescribed by the choice of L.
STATE FEEDBACK WITH STATE ESTIMATION
BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI
18
2019
The output signal with
feedback from the original
and from the estimated
state variables
The first state variable
and its estimation
REMARK ABOUT THE SPEED OF THE SYSTEM,� THE ESTIMATION AND THE CONTROL CIRCUIT
BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI
19
2019
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