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control theory
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Collected & Sorted by Chang Hu
1 / 14
Control Theory Notable Points
!!!Only For Reference!!!
1. Matrix Multiplication:
Row Vector and Column Vector:
Square Matrix and Column Vector:
Collected & Sorted by Chang Hu
2 / 14
Square Matrices:
Rectangular Matrices:
2. Matrix Inversion:
Inversion of 22 matrices:
Collected & Sorted by Chang Hu
3 / 14
Inversion of 33 matrices:
3. Zeros (LU 4):
Invariant Zeros:
() = s0 are the invariant zeros
# All Invariant Zeros can be determined by solving Rosenbrock Matrix.
# If = () = 0 or, if () < ().
Transmission Zeros Zeros of the I/O - relation:
() < () s0 is a transmission zero.
# Transmission Zeros can be acquired from Transfer Function G(s).
# If = (0) = 0 or, if (0) < (0)
Decoupling Zeros:
Input Decoupling Zeros (IDZ):
[ ] < S0 is an input decoupling zero.
Output Decoupling Zeros (ODZ):
[
] < S0 is an output decoupling zero.
# Decoupling Zeros can be acquired from Transfer Function.
# Invariant Zeros are the summary of all zeros.
Collected & Sorted by Chang Hu
4 / 14
# Transmission Zeros and Decoupling Zeros are both Invariant Zeros with the following
differences: Transmission Zeros are not eigenvalues; Decoupling Zeros are the eigenvalues
that are not controllable (IDZ) / observable (ODZ).
# If the system is fully controllable and/or fully observable, there is no IDZ and/or ODZ, vice
versa.
4. Eigenvalues (LU 1-2):
( ) = are the eigenvalues of the system
# The Eigenvalues of the system matrix A denote a part of the inner characteristic of the
system.
# The locations of the eigenvalues yield the stability of the system.
# The eigenfrequencies and the damping constant can be derived from the eigenvalues.
5. Characteristic Polynomial of System Matrix A (LU 1-2):
( ) = +
+ + +
=
6. Poles (LU4):
Definition: A complex number is denoted as pole of the transfer function matrix (), if
a minimum of one element () of () has a pole at .
# Poles can be acquired from Transfer Function. Every pole of G(s) is an eigenvalue of the
system matrix A. However, not every eigenvalue of A is a pole of G(s).
7. Connections between Zeros, Eigenvalues and Poles (LU 4):
Invariant
Zeros Eigenvalues
Transmission
Zeros Poles
IDZ
ODZ
Collected & Sorted by Chang Hu
5 / 14
8. Rosenbrock System Matrix (LU 1-2/4):
() = [
]
9. Transfer Function (LU 1-2):
() =()
()= ( ) +
# Zeros are the roots of N(s) (the numerator of the transfer function) obtained by setting N(s)
= 0 and solving for s.
# Poles are the roots of D(s) (the denominator of the transfer function), obtained by setting
D(s) = 0 and solving for s.
# Because of our restriction above, that a transfer function must not have more zeros than
poles, we can state that the polynomial order of D(s) must be greater than or equal to the
polynomial order of N(s).
(Source: http://www.atp.ruhr-uni-bochum.de/rt1/syscontrol/node18.html)
: () =(+)(+)(+)
(+)(+)(+)
# Denominators are Eigenvalues sp1 = -1, sp2 = -2, sp3 = -4,
# Numerators are Zeros s01 = -1, s02 = -2, s03 = -3
# sp1 = -1, sp2 = -2 are eigenvalues but not poles, sp3 = -4 is eigenvalue and also pole.
# s01 = -1, s02 = -2 are invariant zeros and decoupling zeros, s03 = -3 is invariant zero and
transmission zero.
# The system has 3 eigenvalues, 2 decoupling zeros, 1 pole, 1 transmission zero.
If the system if fully controllable and observable (= no decoupling zeros), all eigenvalues
are poles.
Collected & Sorted by Chang Hu
6 / 14
10. Eigenvectors and Modal Matrix (LU 1-2):
Right Eigenvector (can be in some expressions):
= ( ) = =
[
]
Left Eigenvector (can be in some expressions):
=
( ) =
= [ ]
OR = ( ) = =
[
]
Modal Matrix V:
= [ ]
# The modal matrix V is the n n matrix formed with the eigenvectors of system
matrix A as the column in V.
=
# D is a n n diagonal matrix with the eigenvalues of A on the main diagonal of D and
zeros elsewhere.
= [
]
11. Stability (LU 3):
Re { i} < 0 for all i of A The system is asymptotically stable.
Re { i} 0 for all i of A The system is boundary stable.
# If a system is asymptotically stable, it is also BIBO stable. If a system is BIBO stable, then
the output will be bounded for every input to the system that is bounded.
# System that start out near an equilibrium point stays near forever, then is
Lyapunov stable, if is Lyapunov stable and all solutions that start out near converge to
, then is asymptotically stable;
Collected & Sorted by Chang Hu
7 / 14
# The system is I/O stable, if all poles of the system are stable. Only need to check the
poles but not the eigenvalues (pi 0 < 0
Hurwitz Matrix H & Hurwitz Determinants Hi:
=
[
]
> 0
= >
= |
| > 0
= |
| > 0
# The system can be determined asymptotically stable, if and are both satisfied.
# If is not true, the system can be viewed non stable. It is not necessary to continue
the procedure.
# Hurwitz Criteria can be applied to check if the system is asymptotic stability but not
to boundary stability (i = 0).
13. Lyapunov Approach/Equation (Base for Optimal Control Method) (LU 9-10):
+ =
# If the equation can be solved (= is positive definite) System is asymptotical
stable.
# If exists, system is asymptotical stable Weighting Matrix is positive definite.
# If exists, system is stable Weighting Matrix is positive semidefinite.
Collected & Sorted by Chang Hu
8 / 14
14. Controllability and Observability (LU 5):
The system is fully controllable, if it is possible to shift the system state in definite time
from an arbitrary initial state 0 to the arbitrary final state [0,] using a suitable
input function [0,].
The system is controllable then the eigenvalues of the system can be placed arbitrarily
via the input .
The system is fully observable, if it is possible beginning with the initial state x0 during
the definite time interval [0, ] using the known input time behavior [0,] and the
output time behavior [0,] to reconstruct the system state during the time interval
[0,].
The system is observable, if the state vector can be reconstructed by the measured
output .
15. Connections between Stabilizability & Controllability (IDZs), Detectability &
Observability (ODZs):
The system is stabilizable, if all unstable eigenvalues are controllable or the system is
already fully controllable.
If all unstable eigenvalues of the system are controllable, the system is stabilizable.
If the system has no IDZ, the system is fully controllable; if the eigenvalues of the
system is unstable, they can be stabilizable because the system is controllable.
If the system has IDZs, the system is not fully controllable; if all/part unstable
eigenvalues are IDZs, the system is not stabilizable because the unstable
eigenvalues are not controllable.
The system is detectable, if all unstable eigenvalues are observable or the system is
already fully observable.
If all unstable eigenvalues of the system are observable, the system is detectable.
Similar to description of Stabilizability & Controllability above.
16. Kalman Criterion (LU 5):
Controllability:
= [ ]
Observability:
=
[
]
# If = System is fully observable.
# If < System is not fully observable.
# If = System is fully controllable.
# If < System is not fully controllable.
Collected & Sorted by Chang Hu
9 / 14
17. Gilbert Criterion (Canonical Coordinates) (LU 5):
Controllability:
=
Observability:
=
18. Hautus Criterion (LU 5):
Controllability:
[ ] = for all of A System is fully controllable
# If [ ] < , the i-th mode is not controllable the related
eigenvalue is an input decoupling zero.
Observability:
[
] = for all of A System if fully observable
# If [
] < , the i-th mode is not observable the related eigenvalue
is an output decoupling zero.
19. Original Hautus Criterion (LU 5):
Controllability (Left Eigenvectors):
for all of A System is fully controllable.
# wiT B 0 the i-th mode is controllable.
# wiT A = 0 the i-th mode is not controllable the related eigenvalue is an
input decoupling zero.
Observability (Right Eigenvectors):
for all of A System is fully controllable.
# C vi 0 the i-th mode is observable.
# C vi = 0 the i-th mode is not observable the related eigenvalue is an
output decoupling zero.
# If has no zero row and the corresponding rows of complex conjugate
eigenvalues are linear independent. System is fully controllable.
# If has no zero column and the corresponding column of complex
conjugate eigenvalues are linear independent. System is fully observable.
Collected & Sorted by Chang Hu
10 / 14
=
[
]
=
[
]
= [ ]
20. Comparison of the Noted Criteria (LU 5):
Kalman Criterion:
+ Ideal for small systems, easy to understand and use, good for analytical
considerations;
- Not usable for numerical purposes (large systems);
Gilbert Criterion:
+ Good for numerical considerations and therefore for large systems;
Hautus Criterion (Original Hautus):
+ Good for analytical considerations, optimal to analyze individual eigenvalue related
modes;
21. System Matrix A in Canonical Form:
Normal Canonical Form:
: () =
+ + +
+ + + + +
( ) = +
+ + +
Control Normal Form:
# Coefficients have different signs ( > 0 < 0) System is unstable.
Observer Normal Form:
=
=
=
= [
]
=
[
]
= [
]
= [ ]
Collected & Sorted by Chang Hu
11 / 14
22. State Space Representation (LU 1/P.E. OBT):
Equations:
() = () + ()
() = () + ()
System Matrix : n n;
Input Matrix : n r
Output Matrix : m n
Direct-Transmission Matrix : m r (in most cases equals to 0)
23. Sate Feedback Control (Feedback from Input) (LU 7/P.E. OBT):
In the state feedback control the state vector is multiplied with the feedback gain matrix
and is fed back to the input of the system.
With the state feedback
= +
The equation for the controlled system is given by
= ( ) +
The first part of the state feedback represents the real control. In addition a new
reference signal is multiplied with the amplification matrix and is inserted. If the
system is controllable, the eigenvalues of the controlled system can theoretically be placed
at any position via appropriate parameters of , so the system dynamics can be
manipulated. Because of the limited input energy the design of the system dynamics is
practically bounded.
The feedback gain matrix can be designed using one of the following two method.
B C
A
u(t) (t) x(t) y(t)
-
+ B C
A
(t) x(t) y(t) V
K
(t)
Block Diagramm:
Block Diagramm:
Collected & Sorted by Chang Hu
12 / 14
24. Pole Placement Method (LU7/P.E. OBT):
Controllability Check: The system has to be either fully controllable or stabilizable to
perform feedback control via pole placement method.
Characteristic Polynomial with the desired eigenvalues:
( ) = +
+ + +
Characteristic Polynomial of the controlled system with feedback gain matrix K:
( ( )) = +
+ + +
Comparison of Coefficients:
( = ), , ,
, , ,
= [ ]
25. Optimal Control Method (Based on Lyapunov Approach) (LU 9-10/P.E. OBT):
Minimum of the quality function:
= [ + ]
=
Choose the Weighting Matrices and :
# and are the weighting matrices which must be symmetric and positive definite.
# = , = (Dimension: )
Calculation of the Matrix P by applying Riccati-Equation:
+ + =
# P is a symmetric matrix = [
11 1212 22
1 2
1 2
]
Calculation of the Feedback Gain Matrix K:
=
Collected & Sorted by Chang Hu
13 / 14
26. Comparison Between Pole Placement & Optimal Control (P.E. OBT):
Using pole placement method the system dynamics can be arbitrary designed, if the system
is fully controllable of stabilizable. Possibly the control does not work energy optimal, the
energy to displace the eigenvalues could be very high.
# The system is adjusted energy optimal with respect to desired goals by the optimal
feedback considering the quadratic integral criterions. Less energy cost than Pole
Placement Method.
27. State Feedback Control with Observed States (Feedback from Input + Output)
(LU 8/P.E. OBT):
Using state feedback control, the whole state vector is fed back by the state feedback
control, which sets up the availability of the state vector as measurements. If the system
is observable, the whole state vector can be reconstructed by the output signals. Then
the control can be implemented with the reconstructed state vector.
The state observer is represented by the equation
= + ( ) +
=
The feedback gain matrix L can be designed by either Pole Placement or Optimal Control
Method. The vector differential equation for the observed state is obtained
= ( ) + +
The control is given by the equation
= + .
- +
B C
A
(t) (t) (t)
V K (t)
B C
A
L
(t)
(t)
(t)
Block Diagramm:
+
-
Plant
Control
Observer
Collected & Sorted by Chang Hu
14 / 14
The state Model for the whole system is
[] = [
] [] + [
]
The error of the observer is defined by
=
The error equation and the controlled system reset to
[] = [
] [] + [
]
The system matrix of the Equation has an upper triangular form. It can be seen that the
eigenvalues of the controlled system can be designed apart from the eigenvalues of the
observer
=
=
This characteristic is called Separation in control theory. The design requires that the
observer is faster than the control.
# Separation Principle: The eigenvalues should definitely lie left to the
eigenvalues of the close-loop system on the left half space of the complex plane,
Where is the leftmost eigenvalue of the system.
(, ) = +
{} {} or |{.}| = (~) |{}|
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