full-order observer design for non-linear dynamic control system

Journal of Al-Nahrain University

Vol.10(2), December, 2007, pp.154-161

Science

FULL-ORDER OBSERVER DESIGN FOR NON-LINEAR DYNAMIC CONTROL SYSTEM Shatha S. Sejad Department of Mathematics, College of Science, Al-Mustansiriyah University Baghdad, Iraq. Abstract The main theme of this paper is to design a full-order non-linear dynamic observer for estimating the state space estimator from its non-linear input-output dynamic control system. The quadratic Lyapunov function stabilization approach has been developed and the sufficient conditions for existence the dynamic observer for some class of non-linear input-output dynamic system have been presented and discussed. Illustrations are presented to demonstrate the validity of the of the presented procedure. Introduction Sometimes all state space variables not available for measurements or it is not practical to measure all of them, or it is too expensive to measure all state space variables.[5] The concept of observability allong with controllability first introduced by Kalman [2] plays an important role in both theoretical and practical concepts of modern control. Observability of a system as conceptualized by Kalman fundamentally answers the question whether it is possible to identify any state x(t ) at time t  in the state space by observing the output y(t , t f ) to a given input u(t , t f ) over a finite time interval t f  t (t f  t ) . Luenberger observer [5] is essentially a full-order observer, the state variables of which give a one-to-one estimation of the linear dynamical system state variables. In [1] the observer provides a smooth velocity estimate to be used by a trajectory tracking controller. The observer, controller and manipulator from a system where the observer error as well as the position and velocity tracking error tend to zero asymptotically. In [7], the authors have consider a class of uncertain non-linear control system, and its controllability and stablizabilty

Where x(t ) is n-state space, u (t ) is mcontrol signals which belong to a class of piecewise continuous function. f i ( x) : R n  R n to be continuous vector valued function,qi  Rris a vector of uncertain i  0,1,, r parameter, Ai  R nn , Bi  R nm ; are constant matrices. Consequently [7], the Lyapunove function that stabilize a class of a non-linear system can be obtained easily from the result of nominal linear dynamic system and under some assumption both Lyapunove function of linear and non-linear dynamic system are obtained to be identical. The work of this paper, focuses on the full order dynamic state observer for the uncertain non-linear control system  r  x (t )   qi Ai  x(t )   i 0 



y (t ) 

p

 j 1

r

r

 q B u (t )   f ( x) i

i

i

i

i 0

c j x j  g ( x) such that

i 1

p

c x j

j

 cT x

j 1

where x(t ) is n-state vector,y(t)is n –output vector u (t ) is m-control signals which belongs to a class of piecewise continuous function g ( x) , f i ( x) ; i  1,2,, r are non-linear vector functions, respectively of dimensions n and p and Ai  R nn , Bi  R nm cT  R1n ; i  0,1,, r are constant matrices. The following definitions and theorems are needed later on.

r r  r  x (t )    q i Ai  x(t )   q i Bi u i (t )   f i ( x) ...... (1) i 0 i 1  i 0 

154

Shatha S. Sejad

Definition The system

where aij is the absolute value of the matrix coefficient aij .

x  Ax y  Cx

where x is n-state vector, y is m-output vector, A  R nn , C  R mn is constant matrix, is said to be observable if every state x(t ) can be determined from the observation of y(t ) over a finite time interval t  t  t1 . Definition A (vector-valued) function f(x) is said to be Lipschitz function if there exists a constant L such that for all x1 , x 2  R n ,the following inequality holds f ( x1 )  f ( x2 )  L x1  x 2 In this case L is said to be the Lipschitz constant of f . Theorem The system is described by x  Ax  Bu y  Cx

Where ( A, C) is observable means the dynamic system is observability if and only if the observability matrix constant matrices is completely state observability if and only if the observability matrix C T C T AC T An1  has full rank equal to n ( T denoted to conjugate transpose matrix ). Theorem If all eignvalues of the system x  Ax , where A  Rnn is a constant matrix have negative real parts then the solution of this system is asymptotically stable ( exponentially stable ). Lemma Let A be a symmetric matrix and let min ( A) and max ( A) be the smallest and largest eignvalue of A, respectively, 2 2 then min ( A) x  xT Ax  max ( A) x , x  R n n

where x   xi , xi is the i-the component 2

2

i 1

of x. Remark The Euclidean norm of marix can be defined as : n

n

i 1

j 1

A  ( aij ) 2

1/ 2

155

Problem Formulation Consider the non-linear dynamical control system described as follows:   i 0 i 1  p p  T y (t )  c j x j  g ( x) such that cjxj  c x  j 1 j 1   x ( 0)  x 0    r  x (t )   q i Ai  x(t )   i 0 

r

r



 q B u (t )   f ( x)





i

i

i

i

.. (2)

Where x(t )  R n is unmeasurable state vector (see introduction [5]), u(t ) is the control input and y (t )  R n is the output vector. Suppose that the matrices A, B and C T are constant matrices g ( x(t )), f ( x(t )) , i  1,2, ,...r are non-linrear vector functions, respectively of dimensions n and p. Then a dynamical state observer of nonlinear dynamical control system (2) is constructed as follows : r r  r  xˆ (t )    qi Ai  xˆ (t )   qi Bi ui   f i ( xˆ )  k e y(t )  c T xˆ  g ( xˆ ) i 0 i 1  i 0 

............................... (3) Where the observed state is denoted by xˆ(t ) , and ke is the observer gain matrix. Define e(t )  x(t )  xˆ(t ) .............................. (4) e(t ) is the dynamical error between the actual state and state observer xˆ(t ) The dynamical error in state observer (4) of non-linear dynamical control system((2),(3)) has the following dynamic equation: r  r   e(t )    qi Ai   k e c T  e(t )    f i ( x)  f i ( xˆ)  k e g ( x)  g ( xˆ) i 1   i 0 

............................... (5) and as the following the linear part of equation (5)  r   e(t )    qi Ai   k e c T  e(t ) ................................ (6)   i 0 

e(0)  x(0)  xˆ(0) ................................... (7) If the dynamic behavior of dynamical error (5) is a symptotically stable, then the dynamical error (5) will tend to zero with an adequate speed as the time tend to infinite then

Journal of Al-Nahrain University

Vol.10(2), December, 2007, pp.154-161  x(t )  xˆ (t ) (exponentially

the state x(t) given in (2) will converge to the state observer xˆ(t ) .

stabilizable)

i.e x(t )  xˆ(t ) as t   .

Lemma Consider the linrear dynamical control system

Theorem Consider the linear dynamical system (8)  r  x (t )   qi Ai  x(t )   i 0 



   x (t )    qi Ai  x(t )   qi Bi u i (t ) i 0  i 0   ................. (8) T  y (t )  c x(t )  r

Science

r

y (t ) 

r

 q B u (t ) i

i

i

i 0

p

c x j

j

j 1

x(0)  x 0 r satisfied the following conditions   qi Ai , c T   i 0 

where x  R n , u  R m , y  R p , A  R nn , B  R nm , C  R pn and the state variables are not available for measurement . Consider the observer of linear dynamical control system (9)

is observable matrices and q  1 , qi  0 , i  1,2,, r are arbitrary, and let as the dynamical observer by r  r  xˆ    qi Ai  xˆ (t )   qi Bi u i (t )  k e y(t )  c T xˆ  ....... (9) i 0  i 0 

r  r  xˆ (t )    qi Ai  xˆ (t )   qi Bi u i (t )  k e y(t )  c T xˆ  i 0  i 0 

Then the state observer has the dynamical obsever state exponentially stabilization.

xˆ (0)  xˆ 0 satisfied the following conditions 

 r   e(t )  e exp   qi Ai   ke cT t   i 0 

  

 r  T   qi Ai   ke c  is Hurwitz stable matrix   i 0  

 q A   k c i

i

e

i 0

T

 t 

i

e

i 0

T

T

  r  T   P  P  q i Ai   k e c   Q    i 0 



the Lyapunove function V ( x  xˆ)  ( x  xˆ)T P( x  xˆ) Then the dynamical error is asymptotically stable via observer gain Ke

is 2

stable matrix norm  r  T   qi Ai   ke c  is   i 0 

Proof : Consider the linear dynamical system (8)

asymptotically

 r  x (t )   q i Ai  x(t )   i 0 



stable thus there exist positive numbers  ,  such that

y (t ) 

e(t ) 1   e 1 expt 

 e(t ) 1   expt 

i

5. assuming 2

1

since



r

 q A   k c

as a unique positive definite solution P for arbitrary positive definite matrix Q

is Eucledian norm and 

where 

is asymptotically stable

matrix 4. The Riccati equation

Due to the observability condtions of system (8) the matrix

r



 r  T    qi Ai   k e c    i 0 

solution

 exp  



(8) is completely state observer 2. The uncertain condition q  1 , qi  0 ; i  1,2,, r are arbitrary 3. The observer gain k e selected such that

   e(t )    qi Ai   k e c T  e(t ) . ........................ (10)   i 0  with initial condition e(0)  e has the

1

r

 i 0

r

e(t ) 1  e



1.   qi Ai , C T  of linear dynamical system

Proof : Let e(t )  x(t )  xˆ(t ) from equations (8),(9) we have

since  ,  are positive numbers e(t ) 1  0 as t   Then and e(t ) 1  0 (exponentially)

 q B u (t ) i

i

j

j

i

i 0

p

c x j 1

where     e

r

suchthat

CT x 

p

c x j

j

j 1

x ( 0)  x 0 1

the state observer of linear dynamical control system (8) is given as follows (9)

hence

r  r  xˆ (t )    qi Ai  xˆ (t )   qi Bi u i (t )  k e y(t )  c T xˆ  i 0  i 0 

156

Shatha S. Sejad

xˆ (0)  xˆ 0 Let e(t )  x(t )  xˆ(t ) We have dynamical error in state observer (9) of linear dynamical control system (8) is obtained to subtract (8) from (9) as follows     

 r   e(t )    qi Ai   k e c T  e(t )   i 0  e(0)  e0

................. (10)

To examine the stability of e(t), we consider the following quadratic Lyapunove function V (e(t ))  e T (t ) Pe(t ) thus   r  V (e)     q i Ai   k e c T    i  0

T

  r    e Pe  e T P    q i Ai   k e c T i  0       

  e  

  r   r       V (e)     qi Ai   k e c T  eT  Pe  eT P    qi Ai   k e c T e     i 0   i 0     T   r      r V (e)  eT   qi Ai   k e c T  P  P   qi Ai   k e c T   e    i 0   i 0   T

............................... (6) r since   qi Ai , c T  is observable matrices

 i 0





r









and hence   qi Ai   ke cT  stabilizable (see  i 0

Remark (2) [2] ) hence the following equation has a unique positive definite solution P such that T

 r  r   T  T   qi Ai   k e c  P  P  qi Ai   k e c   Q .......... (7)    i 0   i 0 





Thus from equation (6) is positive V (e)  e T Qe ; Q  solution and then V (e)  0 ,  e , t since

definite

and of V (e)  e T Pe  0 as e  0 when as e  0

 The error dynamic system (10) is asymptotically stable via observer gain k e Thus x(t )  xˆ (t ) as t   .

Theorem Consider the linear dynamical system (2) r r  r  x (t )    qi Ai  x(t )   qi Bi u i (t )   f i ( x) i 0 i 1  i 0 

y (t )  C x  g ( x)

xˆ (0)  xˆ 0 satisfied all the conditions (1,2,3,4 and 5) in theorem (2) and two conditions as the following The non-linearity function f i ( x) : R n  R n ; i  1,2,, r and

g ( x) : R n  R n are assumed to be Lipschitz condition with Lipschitz constant Li ; i  1,2, , r and respectively r

i.e ,

r



f i ( x)  f i ( xˆ )   Li x  xˆ

i 1

, Li  0

i 1

and g ( x)  g ( xˆ )  L x  xˆ , L  0 , xˆ  R n  r   (Q) If   Li   L  min 



i 1



2 max ( P)

Then the dynamical error (5) r  r   e(t )    qi Ai   k e c T  e(t )    f i ( x)  f i ( xˆ)  k e g ( x)  g ( xˆ) i 1   i 0 

is asymptotically stable via observer gain parameter k e . Proof: The linear dynamical system (2)  r  x (t )   qi Ai  x(t )  i  0   y (t )  C T x  g ( x)



r

r

 q B u (t )   f ( x) i

i

i

i

i 0

i 1

The state observer of non-linear dynamical control system (2) is given as follows (3) r r  r  xˆ ( t )    q i A i  x ( t )   q i Bi u i ( t )   f i ( x )  i 0 i 1  i 0  T k e y( t )  ( x )  C xˆ  g( xˆ )





x (0)  x 0 The dynamical error instate observer (3) of non-linear linear dynamical control system is obtained to subtract (2) from (3) as follows:  r   e ( t )    q i A i   k e c T  e    i 0 

p

suchthat

r  r  xˆ (t )    qi Ai  xˆ (t )   qi Bi u i (t )  k e y(t )  c T xˆ  g ( xˆ ) i 0  i 0 

x(0)  x0

V (e)  e T Pe  0 , P  P T  0 and V (e)  0

T

where x  R n , u  R m , y  R p , A  R nn , B  R nm , C  R pn , f : Rn  Rn, g : Rn  Rn Consider the observer of linear dynamical control system (2) is given in equation (3)

C x  cj xj T

j 1

r

 f ( x )  f ( xˆ )  k g( x )  g( xˆ )

x(0)  x0

i 1

157

i

i

e

Journal of Al-Nahrain University

Vol.10(2), December, 2007, pp.154-161

Let e(t )  x(t )  xˆ(t ) Now, we consider the following quadratic Lyapunove function For examining the stability of e(t ) Let V (e)  eT Pe V (0)  0 , V (e)  0 And V (e)  eT Pe  eT Pe

r 2 2  e  Li   2 L P e  i 1   r  2 2    min (Q) e  2 Li   L  max ( P) e  i 1   r     2     min (Q)  2 Li   L  max ( P)  e  i 1    





on using Lemma(1) and condition (2) , we

T

Since P is unique positive definite solution and it is clear that V (e)  0 , V (0)  0 and by  V (e)  0 we have conclude that error dynamic system (5) is asymptotically stable via observer gain parameter t   as x(t )  xˆ (t ) Thus

r  r     e P   qi Ai   k e c T  e(t )    f i ( x)  f i ( xˆ )  k e g ( x)  g ( xˆ )  i 0  i 1  

  e T P  

 r

i 0

 i 0

T

  q i Ai   k e c T  e T (t )   

  q i Ai   k e c T  e(t )   

 r

i 1



  f ( x)  f ( xˆ)  k g ( x)  g ( xˆ) Pe r

T

i

T

T

i

e



i 1

  f i ( x)  f i ( xˆ)  k e g ( x)  g ( xˆ) 

Algorithem Step(1). Consider the linear and non-linear dynamical system (8)-(2) with respectively. Step(2). Check the pair   q A , C  is

T  r  r     T   T   V (e)  e  qi Ai   k e c  P  P  qi Ai   k e c T       i 0   i 0  



 V (e)  0

get

T

r



r

T

 r T T  T   f i ( x)  f i ( xˆ )  k e g ( x)  g ( xˆ )  Pe   i 1 



 i 0

observable Step(3). Check conditions

 r  e T P   f i ( x)  f i ( xˆ )  k e g ( x)  g ( xˆ ) i  1  



r



by using theorem (2) we have  V (e)  e Qe    T

T   f i ( x)  f i ( xˆ)  ke T g ( x)  g ( xˆ)T Pe   i 1 

i 1

r



T





but





e



 f i ( x)  f i ( xˆ) e

i

i

i 1

2 P 2 P



 e  

 e   2



 L  x  xˆ r

r

f i ( x)  f i ( xˆ )   Li x  xˆ i 1

t R ,

for

 r  T    qi Ai   k e c  asymptotically   i 0 

stable

using pole placement (see[3]) Step(5). Let the dynamic e(t )  x(t )  xˆ(t ) and e(0)  e0

by error

e(0)  e0

i

i 1

 r

i 1

 Li  

Step(6). Set V (t )  V (e(t ))  e T (t ) Pe(t ) k e where P is the unique positive definite solution of

f i ( x)  f i ( xˆ ) is satisfied Lipschitze

i 1

condition and k e g ( x)  g ( xˆ)T Pe  eT Pk e g ( x)  g ( xˆ) 2

Lipschitz

r  r   e(t )    q i Ai   k e c T  e    f i ( x)  f i ( xˆ )  k e g ( x)  g ( xˆ ) i 1   i  0 

 f ( x)  f ( xˆ) r

r



i 1

2 P

since

following

i

xR And design the observer dynamic by (9)-(3) ke Step(4). Select that makes

 r   r   V (e)   e T Qe    f i ( x)  f i ( xˆ ) Pe  eT P   f i ( x)  f i ( xˆ )   i 1   i 1  T T ˆ ˆ ke g ( x)  g ( x) Pe  ke g ( x)  g ( x)e P

r

i

n



T

the

g ( x)  g ( xˆ )  L x  xˆ

r  eT P  f i ( x)  f i ( xˆ )  k e g ( x)  g ( xˆ )  i 1 

 r   r  T   f i ( x)  f i ( xˆ ) Pe  e P   f i ( x)  f i ( xˆ )  2 P  i 1   i 1 



 V (e)  e T Q e  2 P

r  r    V (e)    qi Ai   k e c T  e(t )    f i ( x)  f i ( xˆ )  k e g ( x)  g ( xˆ ) Pe  i 0  i 1  

   V (e)   

Science

 r  r   T  T   qi Ai   k e c  P  P  qi Ai   k e c   Q    i 0   i 0 



P e k e g ( x)  g ( xˆ )  2 P e L x  xˆ 2L P e



for arbitrary positive definite matrix Q

2

158

Shatha S. Sejad

Step(7).

 r  min (Q)  ,   Li   k e L  2 max ( P)  i 1  

Check r

where

L ,

L and k e are found in step(3),

i

i 0

step(4) with respectively and min (Q) denotes the smallest eigenvalue of Q, max ( P) denotes the largest eigenvalue of P. Illustration (1) Consider the linear dynamical control observation error description  2   e(t )    qi Ai   ke cT  e(t )   i 1 

 1  k e1  2   1   2  2 0    T     k e       qi Ai   k e c    1  ke 2  0   1   i 0   0  3  2  k e 2  3 1  k e1  0



 2      qi Ai   ke cT  is diagonal matrix   i 0   2      qi Ai   ke cT  is stable matrix   i 0 

 2 0  we have , e ( t )    e( t ) 0  3   Now, find the positive defined solution P P   11  P21

P12  P22 

where q  1 , q1 , q2  0 ; for n  2 gives the matrices

equation

 3 0  0 1 A   , A1   ,    2  1 22  1  3 22

 2 0   0  3  

1 0  A2    2 1 22

where q  1 , q1  1 , q2  2 ; where

Let us examine state observability of the matrix  2    qi Ai , c T  , notice that  i 0  1 0  T T rank c T   AT c T   rank  2 1  1  the cT T , AT cT T vectors





2

 i 0

i

i

T

  is observable . 

Then, the system is completely state observable  r      qi Ai   ke cT  is asymptotically stable   i 0 

(see problem formulation) Now, by using the formula  2  T   qi Ai   ke c    E22 such that   i 0  22

E 22 is

arbitrary diagonal matrix.   1 1   k e1    2 0     1 1     k 1  2    e2   0 3 

 1  ke1   1  ke 2

algebraic

 P11 P12   P11 P12   2 0  1 0 P P   P P      Q ; Q  0 1 0  3     21 22   21 22  

1 1 , P12  0 , P21  0 , P22  4 6 set the Lyapunove function V (e)  eT Pe , P11 

1  where P   4 0  V (e)  e1

are independent and the rank of the matrix cT T  AT cT T is two

   q A , c

following

and thus we have

 1 1  qi Ai  q A  q1 A1  q2 A2  A  A1  2 A2     i 0  1  2 2



the

 2P11  2P12    2P11  3P12   1 0        3P21  3P22   2P21  3P22   0  1

cT  1 112



T

of

 0 1  6 1   4 0   e1  e2  1  e2  0  6 

1   e1  1 V (e)   e1 e2   4 6  e2   1 2 1 2 2 2 V (e)  e1 e2  V (e)  0.25 e1  0.1666666 e2 4 6  V (e) positive defined function and have

the system (10) is stabilized by using Lyapunove V (e) and observer gain ke In example (1) we apply the proceeding main theorem (2) and we find Linear Dynamical control observation error (5) is asymptotically stabilizable . Now, we illustrate the proceeding main theorem (3), we consider the non-linear dynamical control observation error. Illustration (2) where g ( x) , f i ( x) ; i  1,2 are non linear vector functions and

1  ke1    2 0     2  ke 2   0  3

159

Journal of Al-Nahrain University

A1  R 2 and

Vol.10(2), December, 2007, pp.154-161

, c T  R 2 ; i  0,1,2

T

 2 0   2 0  1 0  0  3 P  P  0  3  Q , Q  0 1      

 2  2 0   T   qi Ai   ke c     is stable matrix   i 0   0  3 0.03 x sin x   0.2 cos x  f 1 ( x)    , f 2 ( x)   0.02 cos x 0 . 05 x     x and g ( x)  are defined on the region 5 D  t , x  R: t  1 , x  0.01

we have max ( P)  0.25 and min (Q)  1 since L1  0.0803 , L2  0.22 , L  0.4  2   min (Q) 2   Li   L   0.0997  2 max ( P)  i 1    r   (Q) the condition   Li   L  min is satisfied 

0.05 x  0.05 xˆ  0.03 x sin x  x

 0.03 x cos x  0.03 sin x



x  xˆ 

L1  0.0803

[1] Erlic, M and Lu, W.S “Reduced-order Observer for Underwater Manipulators Using Potentiometers for Position Measurement”, Proceedind of the conference on Oceans, Victoria, BC,Can 1993. [2] Kalman, R.E, “On the General Theory of Control Systems” Proc. IFAC, vol. 1, pp. 481-492, Butter worth, London, 1961. [3] Katsuhiko Ogata, “Modern Control Engineering”, fourth edition, Prentice-Hall of India, 1996. [4] Katsuhiko Ogata, “State Space Analysis of Control Systems”, Prentice-Hall of Canada, Ltd. Upper Saddle River. [5] Luenberger, D.G., “Observers for Multivariable system”, IEEE Trans, 1966, AC-II. [6] Rao M. Roma Mohana, “Ordinary Differential Equations”, printed and bound in Great Britain at the Pitman Press, Bath, 1980. [7] Radih A.Zaboon and Shatha S.Sejad, “Lyapunov Function Approach for Stablizability a Class of non-linear Dynamical Control System”, to appear. [8] Tsong-Chi and Chen, “Linear System Theory and Design”, HRW Series in Electrical and Computer Engineering.

 0.02 cos x  0.02 cos xˆ x  xˆ  0.02 x  xˆ

 0.2 x  xˆ  0.02 x  xˆ  0.22 x  xˆ

 Lipschitz condition and g ( x)

x 5

,

L2  0.223

  1  k e     k e g ( x)    1 

x

 k e g ( x)  g ( xˆ )  

5



xˆ 5



x   5  x  5 

x 5



xˆ 5

2 x  xˆ 5 2  x  xˆ 5 2 L   0.4 5 



finally, we can discussed the condition  2   min (Q) as the following   Li   L   i 1   2 max ( P) 0 0.25  since the matrix P   0.1666666  0

2

References

f 2 ( x )  f 2 ( xˆ )   0.2 cos x  0.2 cos xˆ   0.2 cos x  x

2 max ( P)

Conclusion  Sufficient conditions were given for the design observers for a class of non-linear system  This system is characterized by non-linear functions which are Lipschitz in nature.

 0.0803 x  xˆ





2

0.05 x  xˆ

 Lipschitz constant



 V ( x  xˆ )  0.25 e1  0.1666666e2

x  xˆ 

0.05 x  xˆ

i 1

the system (4) is asymptotically stable in the large using the Lyapunove function 2 2 V (e)  0.25 e1  0.1666666e2

f 1 ( x )  f 1 ( xˆ )  0.03 x sin x  0.03 xˆ sin xˆ 



Science

been

found by solution 160

Shatha S. Sejad

(input-output)

161