Page : 1 EE406 Control Systems Lecture 17 : State Space Analysis II
UCSI University Faculty of Engineering
Kuala Lumpur, Malaysia Department of Mechatronics
Lecture 17
State Space Analysis II
Mohd Sulhi bin Azman Lecturer
Department of Mechatronics UCSI University [email protected]
1 August 2011
Contents
• Review : Rank of a matrix
• Controllability & Observability
• Controller design
• Observer design
Page : 3 EE406 Control Systems Lecture 17 : State Space Analysis II
Review : Rank of a Matrix
• A rank of matrix can be found by finding the
highest order square sub-matrix that is
non-singular.
• In another word, the rank of a (n x m) matrix A,
is the order of the largest square sub-matrix
whose determinant is non-zero.
Introduction
• In control systems, two basic questions are
usually asked:
1. Can we transfer the system from any initial state (say t=0) to any other desired state in finite time by application of a suitable control force?
2. If we know the output vector for a finite length of time, can we then determine the initial state of the system?
Page : 5 EE406 Control Systems Lecture 17 : State Space Analysis II
Answer to Question 1
• Question 1:
– Can we transfer the system from any initial state (say t=0) to any other desired state in finite time by application of a suitable control force?
• Answer :
– A system is said to be completely controllable if it is possible to transfer the system state from any initial state, say x(t0) to any desired state x(t) in a
specified finite time by one control vector, u(t).
• This is the concept of controllability.
Alternative Definition : Controllability
• Per Nise (2007)
– If an input to a system can be found that takes every state variable from a desired initial state to a
desired final state, then the system is said to be controllable. Otherwise, the system in uncontrollable.
• Per Dorf & Bishop (2008)
– A system is said to be controllable if there exists an unconstrained control u(t) that can transfer any initial state x(t0) to any other desired location x(t) in
Page : 7 EE406 Control Systems Lecture 17 : State Space Analysis II
Answer to Question 2
• Question 1:
– If we know the output vector for a finite length of time, can we then determine the initial state of the system?
• Answer :
– A system is said to be completely observable if every state x(t0) can be completely identified by
measurements of the outputs y(t) over a finite time interval.
• This is the concept of observability.
Alternative Definition : Observability
• Per Nise (2007)
– Observability is the ability to deduce the state variables from a knowledge of the input u(t) and the output of y(t).
• Per Dorf & Bishop (2008)
Page : 9 EE406 Control Systems Lecture 17 : State Space Analysis II
Observers
• The heart of the observer is an accurate model of the system and around this model, a feedback loop is added.
• The use of an observer is to estimate the system by subjecting the model to the same input as the plant itself and make the model’s output to follow accurately the measured output of the plant.
• The observer is sometime called estimator. The concept of an observed can be extended to a stochastic system where there exists uncertainty about true state
variable values because of noise (unwanted signals) entering both the plant and the measurements.
Analogy
• Let the following state space matrix describes
the overall behaviour of a person, say Daniel:
• And let:
– “A” be the matrix describing Daniel;
– “B” be the matrix describing the input to Daniel’s behaviour (i.e. parents, friends, hobbies etc) – “C” be the output i.e. Daniel’s current behaviour.
Ax
Bu
y
Cx
=
+
Page : 11 EE406 Control Systems Lecture 17 : State Space Analysis II
Analogy
• Let us depict Daniel’s behaviour in a block
diagram:
• Now comes the question of controllability and
observability.
Daniel’s Behaviour
(A)
Input (Daniel’s Parents)
(B)
Output (Daniel’s Behaviour)
Analogy : Controllability
• The concept of controllability relates to the concept of suitable input(s) that should be given to a system. These suitable inputs would then “control” the system
effectively.
• In another word, what input should be given in order to control a system in yielding the desired outcome?
Daniel
Input
•What is the suitable input in order to control/change Daniel’s behaviour?
Output
Page : 13 EE406 Control Systems Lecture 17 : State Space Analysis II
Analogy : Observability
• In our analogy, suppose that we know Daniel’s current behaviour. Now, what was his initial behaviour? Was he a bad guy before? Or was he a damn good guy that he is now?
• This is now the concept of observability. We are “observing” his current behaviour, but we wanted to “estimate” his initial behaviour.
Daniel’s initial behaviour
Daniel’s current behaviour
present 10 years ago
More Questions
• Q : How to tell if a system is controllable or
observable?
Page : 15 EE406 Control Systems Lecture 17 : State Space Analysis II
Controllability Matrix
• Consider a system defined by the following state equations:
• We can determine the controllability of this system by examining the algebraic conditions:
• It is noted that A and B are of the order (nxn) and (nx1),
respectively. For a SISO system, the controllability matrix is given as:
• A system is controllable if det(CM)≠0.
x
ɺ
=
Ax
+
Bu
2
rank
B
AB
A B
⋯
A B
n
=
n
2 n
M
C
=
B
AB
A B
⋯
A B
Determinant Test for Controllability
• We can test the controllability of the system by using the determinant test.
• Theorem:
– A system of “n” states and single input is controllable if the determinant of the controllability matrix CMis nonzero.
• Note: for a single input systems, CMis a square matrix (n
x n). If the determinant of a matrix is non-zero, the matrix is invertible or non-singular and hence attains a full rank. Hence, there exists the inverse of the matrix.
( )
Page : 17 EE406 Control Systems Lecture 17 : State Space Analysis II
Example 1
• Test the controllability of the following system:
[
0]
1 20
1
0
0
0
0
1
0
1
1 0
0
u
a
a
a
y
=
+
−
−
−
=
x
x
x
ɺ
Solution to Example 1
• Obtain the controllability matrix, C
M(the
workings are left for you as an exercise):
• And since det(C
M)=-1, then the system is
controllable.
2 2 2 2 1
0
0
1
0
1
1
M
C
a
a
a
a
=
−
−
−
Page : 19 EE406 Control Systems Lecture 17 : State Space Analysis II
Alternative Test
• Another alternative test for controllability of a
system is by using the rank test.
• Theorem:
– A system of “n” states and “r” inputs is controllable if the controllability matrix CMhas a rank of “n”.
• The rank test is valid for both rectangular and
square matrices.
( )
rank
C
M=
n
Observability Matrix
• Recall that the observability is a concept that is
used to estimate the state variable. In another
word, we are looking at the detectable of the
system.
• We can tell the observability of the system
simply by computing the observability matrix:
2 n T
M
Page : 21 EE406 Control Systems Lecture 17 : State Space Analysis II
Example 2
• We re-consider the system as defined in
Example 1. Test the observability of the system.
[
]
0 1 2
0
1
0
0
0
0
1
0
1
1 0
0
u
a
a
a
y
=
+
−
−
−
=
x
x
x
ɺ
Solution to Example 2
• Obtain the obervability matrix, O
M(the
workings are left for you as an exercise):
• And since det(C
M)=1, then the system is
completely observable.
1
0
0
0
1
0
0
0
1
M
O
=
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Duality Property
• We take note of the following property:
– The pair AB in the concept of controllability implies that the pair ATBTis observable.
– The pair AC in the concept of observability implies that the pair ATCTis controllable.
• Thus, the concept of controllability and
observability are dual concepts.
Classification of State Variables
Controllable and observable
Controllable but not observable
Observable but not controllable
Neither controllable nor observable Control
Inputs
Measurable Outputs
u(t) y(t)
Page : 25 EE406 Control Systems Lecture 17 : State Space Analysis II
Next Step
• Textbook reference : Chapter 12.
• Homework 16 has been posted on the course
website. Attempt them. You do not have to
submit Homework 16 as it will not be graded.
