Observer design for descriptor linear systems

OBSERVER DESIGN FOR DESCRIPTOR SYSTEMS
PD Observer Design for Linear Descriptor Systems
Mahendra Kumar Gupta
Research Scholar
Department of Mathematics
Indian Institute of Technology Patna, India
Coauthors: Nutan Kumar Tomar1 & Shovan Bhaumik2
1Department of Mathematics, Indian Institute of Technology Patna, India
2Department of Electrical Engineering, Indian Institute of Technology Patna, India
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Introduction
Linear Descriptor System
Ex_ (t) = Ax(t) + Bu(t);
y = Cx (1)
System Description
State x(t) 2 Rn
Control u(t) 2 Rm
Output y(t) 2 Rp
E; A 2 Rnn, B 2 Rnm
and C 2 Rpn
Dual Normalizable
The linear square descriptor system (1) is called Dual Normalizable if, 9
matrix K of compatible dimension such that matrix (E + KC) is non-singular.
Detectable
The linear square descriptor system (1) is said to be Detectable if, 9 matrix L
of compatible dimension such that matrix pair (E; A + LC) is stable.
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Descriptor system
Ex_ (t) = Ax(t) + Bu(t); y = Cx (1)
Checking Criteria:
For descriptor system (1), some useful terms are defined by the following
conditions:

(a) rank
EC

= n,
(b) rank

E A
C

= n 8 2 C+.
Where C+ = fsjs 2 C; Re(s) 0g is the closed right half complex plane.
(1) System (1) is called dual normalizable if condition (a) holds.
(2) System (1) is said to be detectable if condition (b) holds.
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Observer Design: Need and definition
Need of Observer
Knowledge of the states of the system is important for feedback. But it is not
always possible or necessary to measure all the state variables. In such
cases, the states can be estimated from the output of another dynamical
system, which is called an observer for the given system.
Definition of Observer
An observer is a mathematical realization which uses the input and output
information of a given system and its output asymptotically approaches to the
truth state values of the given system.
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Descriptor linear system
Ex_ = Ax + Bu;
y = Cx; (2)
where E; A 2 Rnn, B 2 Rnm
C 2 Rpn, x 2 Rn and u 2 Rm
Assumptions:

(H1) rank
EC

= n,
(H2) rank

E A
C

= n 8 2 C+.
Lemma
Let any matrix pair (E;C), where E 2 Rnn and C 2 Rpn satisfies following
condition:
EC

= n
Then there exists a nonsingular matrix R such that
rank

I RE
C

= rank(C) (3)
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Algorithm to find the matrix R
1. Determine
p := rank of matrix C
n :=order of matrix E.
2. Check
(i) If rank

I E
C

= p. Take R = In and stop.

(ii) If rank
EC

= n, then go to steps 3-8.
3. Carry out the singular value decomposition (SVD) of matrix
C = U1

D1 0

VT
1 .
4. Calculate P = V1

D1
1 UT
1 0
0 Inp

.
5. Calculate ~E = EP

0
Inp

.
6. Carry out the SVD of matrix ~E = U2

D2
0

V2.
7. Calculate R0 =

0 Ip
VT
2 D1
2 0

UT
2 .
8. Calculate R = PR0.
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Lemma 2
Under the assumption of Lemma 1, the following statements are equivalent.
(1) Descriptor system (1) is detectable.
(2) Matrix pair (RA;C), is detectable.
Proof.
It is obvious that equation (3) implies the existence of M 2 Rnp such that
RE = I MC: (4)
Thus for any 2 C+, we have
rank

E A
C

= rank

RE RA
C

= rank

I MC RA
C

= rank

I RA
C

:
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Detectability and Observer Design for Linear Descriptor Systems
Restricted System Equivalent
Ex_ = Ax + Bu;
y = Cx; (5)
Observer
_^
x = Nx^ + RBu + Ly + My_ (6)
Theorem
If assumptions (H1) and (H2) hold for the system (2). Then there exists
matrices N, L, and M of compatible dimensions such that the system (6) is
observer for the system (5).
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Proof
From systems (6) and (5) the error
e = x ^x (7)
gives the dynamics:
_ e = _ x _^
x
= x_ (Nx^ + RBu + Ly + MCx_ )
= REx_ (Nx^ + RBu + LCx)
= R(Ax + Bu) (N^x + RBu + LCx)
= Ne + (RA LC N)x = Ne: (8)
here, we have assumed that
RE = I MC (9)
N = RA LC (10)
where N is stable.
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Example
E =
2
4
0 0 0
0 1 0
0 0 1
3
5 A =
2
4
3
5 B =
1 0 0
0 2 0
0 0 3

1 1 1
T , C =

1 0 0
0 1 0

:
Results
This
system is not completely observable but completely detectable and
rank
I E
C

= 2.
Hence R = I3 and M =
2
4
1 0
0 0
0 0
3
5.
L =
2
4
3
5. Thus N =
1:5 0
0 1:5
0 0
2
4
:5 0 0
0 :5 0
0 0 3
3
5
Taking x0 =

0 1 0
T , z0 =

10 11 12
T , and u = t2.
10 / 12

Detectability and Observer Design for Linear Descriptor Systems
Truth and estimated value of first state
20
0
-20
First State
Turth x(1)
Estimated x(1)
-40
-60
-80
-100
0 1 2 3 4 5 6 7 8 9 10
Time (Sec)
Truth and estimated value of second state
50
40
30
Second State
Turth x(2)
Estimated x(2)
20
10
0
-10
0 1 2 3 4 5 6 7 8 9 10
Time (Sec)
Truth and estimated value of third state
40
30
20
Turth x(3)
Estimated x(3)
10
0
-10
0 1 2 3 4 5 6 7 8 9 10
Third State
Time (Sec)
11 / 12

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Thank you.
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Observer design for descriptor linear systems

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