e ) = 22 ( x21 - Semantic Scholar

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[9] D. Noh, N. Jo, and J. Seo, “Nonlinear observer design by dynamic observer error linearization,” ... servers for web machines,” Automatica, vol. 40, no. 9, pp.
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= 2121 ( 21)01 then 22 can be constructed as 22  ( ) =  (xe ). Therefore, we can pick any 21 without

If we define  22

x21 e

Partial-State Observers for Nonlinear Systems H. Trinh, T. Fernando, and S. Nahavandi

loss of generality.

REFERENCES [1] A. J. Krener and A. Isidori, “Linearization by output injection and nonlinear observers,” Syst. Control Lett., vol. 3, no. 1, pp. 47–52, Jun. 1983. [2] D. Bestle and M. Zeitz, “Canonical form observer design for non-linear time-variable systems,” Int. J. Control, vol. 38, no. 2, pp. 419–431, Aug. 1983. [3] A. J. Krener and W. Respondek, “Nonlinear observers with linearizable error dynamics,” SIAM J. Control Optim., vol. 23, no. 2, pp. 197–216, Mar. 1985. [4] X.-H. Xia and W.-B. Gao, “Non-linear observer design by observer canonical form,” Int. J. Control, vol. 47, no. 4, pp. 1081–1100, Apr. 1988. [5] X. Xia and W. Gao, “Nonlinear observer design by observer error linearization,” SIAM J. Control Optim., vol. 27, no. 1, pp. 199–216, Jan. 1989. [6] J. Rudolph and M. Zeitz, “Block triangular nonlinear observer normal form,” Syst. Control Lett., vol. 23, no. 1, pp. 1–8, Jul. 1994. [7] R. Marino and P. Tomei, “Adaptive observers with arbitrary exponential rate of convergence for nonlinear systems,” IEEE Trans. Autom. Control, vol. 40, no. 7, pp. 1300–1304, Jul. 1995. [8] A. F. Lynch and S. A. Bortoff, “Nonlinear observers with approximately linear error dynamics: The multivariable case,” IEEE Trans. Autom. Control, vol. 46, no. 6, pp. 927–932, Jun. 2001. [9] D. Noh, N. Jo, and J. Seo, “Nonlinear observer design by dynamic observer error linearization,” IEEE Trans. Autom. Control, vol. 49, no. 10, pp. 1746–1753, Oct. 2004. [10] W. Respondek, A. Pogromsky, and H. Nijmeijer, “Time scaling for observer design with linearizable error dynamics,” Automatica, vol. 40, no. 2, pp. 277–285, Feb. 2004. [11] N. Kazantzis and C. Kravaris, “Nonlinear observer design using Lyapunov’s Auxiliary Theorem,” Syst. Control Lett., vol. 34, no. 5, pp. 241–247, Jul. 1998. [12] J. P. Gauthier, H. Hammouri, and S. Othman, “A simple observer for nonlinear systems-applications to bioreactors,” IEEE Trans. Autom. Control, vol. 37, no. 6, pp. 875–880, Jun. 1992. [13] J. Schaffner and M. Zeitz, “Decentral nonlinear observer design using a block-triangular form,” Int. J. Syst. Sci., vol. 30, no. 10, pp. 1131–1142, Oct. 1999. [14] E. Delaleau and W. Respondek, “Lowering the orders of derivatives of controls in generalized state space systems,” J. Math. Syst. Estim. Control, vol. 5, no. 3, pp. 1–27, 1995. [15] A. F. Lynch, S. A. Bortoff, and K. Röbenack, “Nonlinear tension observers for web machines,” Automatica, vol. 40, no. 9, pp. 1517–1524, Sep. 2004. [16] R. Marino and P. Tomei, Nonlinear Control Design: Geometric, Adaptive, and Robust. Hertfordshire, U.K.: Prentice-Hall, 1995. [17] D. M. Dawson, J. Hu, and T. C. Burg, Nonlinear Control of Electric Machinery. New York: Marcel Dekker, 1998. [18] W. Rugh, Linear System Theory, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996.

Abstract—This note deals with the design of reduced-order observers for a class of nonlinear systems. The order reduction of the observer is achieved by only estimating a required partial set of the state vector. Necessary and sufficient conditions are derived for the existence of reduced-order observers. An observer design procedure based on linear matrix inequalities is given. A numerical example is given to illustrate the design method. Index Terms—Nonlinear duced-order observers.

systems,

partial-state

estimation,

re-

I. INTRODUCTION The state observers design problem for nonlinear systems has received widespread attention in the literature. The contributions made towards solving this problem can be broadly classified into two approaches. The first is known as the “output injection” approach. Here, the aim is to find a coordinate transformation so that the state estimation error dynamics are linear in the new coordinates and then linear techniques can be performed [1]–[5]. In [3], [4], necessary and sufficient conditions under which a nonlinear system can be transformed into an observer canonical form have been established. In [5], the class of nonlinear systems that can be transformed into a linear observable form has been identified. Overall, for this approach, the conditions for achieving the desired coordinate transformation are difficult to satisfy and the approach is applicable to a rather restrictive class of nonlinear systems. Significant research efforts have been directed towards developing transformation procedures that involve larger classes of nonlinear systems and canonical forms [5]–[7]. In the second approach, methods have been developed to design state observers for nonlinear systems without the need of the state transformation. State observers design using high-gain Luenberger-like observers for triangular nonlinear systems have been developed [8], [9]. For this method, the nonlinear system is brought into an observable form and a sufficiently large constant gain dominates the nonlinearity in the error dynamics equations. Dynamic output feedback stabilization using high-gain observers has also been studied for fully-linearizable systems [10] and for systems with “input-to-state stable” inverse dynamics [11]. The design of observers for nonlinear systems, where the estimation error decays irrespective of the input, has been reviewed and generalized in [12]. For the class of systems which are driven by nonlinear functions which are Lipschitz in nature, some fundamental insights into the design of observers and existence conditions of fullorder observers have been reported in [13]–[16]. Zhu and Han [17] showed that a (n 0 p) reduced-order state observer, where n is the system order and p is the number of outputs, can be designed under the same existence conditions as for a full-order state observer [16]. All the aforementioned methods involve the design of full-order or reduced-order observers to estimate the entire state vector of nonlinear Manuscript received December 20, 2004; revised June 28, 2005, February 9, 2006, and July 12, 2006. Recommended by Associate Editor M.-Q. Xiao. H. Trinh and S. Nahavandi are with the School of Engineering and Information Technology, Deakin University, Geelong 3217, Australia (e-mail: [email protected]; [email protected]). T. Fernando is with the School of Electrical, Electronic, and Computer Engineering, University of Western Australia, Crawley, WA 6009, Australia (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2006.884997

0018-9286/$20.00 © 2006 IEEE

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systems. There are applications where observers are designed to estimate only a linear combination of the states or a partial set of the states of nonlinear systems. Such observers are known as linear functional observers [18] and they can have a significantly lower order than that of full-order state observers. Following on from the work of [13]–[17], this note presents a method for designing reduced-order observers that can estimate a required partial set of states, r , of the state vector. Here, the order of the observer is the same as the number of partial states (r  (n 0 p)) to be estimated. Furthermore, if the order of the observer is (n 0 p), then the results in this note can handle a wider class of nonlinearities than those reported in [13]–[17]. A key feature of the design method in this note is the decomposition of the nonlinearities into two portions: one portion is assumed to be Lipschitz with respect to the partial states; the other portion comprises the remaining terms of the nonlinearities and is treated as unknown inputs. First, based on this decomposition, necessary and sufficient conditions are derived for the existence of reduced-order observers. Then, conditions are derived for the solvability of the design matrices of the proposed observer and for the stability of its dynamics. For the design computational efficiency, an asymptotic stability condition is developed using LMI formulation. The design procedure is illustrated by a numerical example.

1809

Define M =

C L

and using SVD, then  (t) can be written as

 (t) = M x(t) = [ S

0 ]V x(t) = [ S

where q (t) = V x(t) 2 n , matrix S and V 2 n2n is an unitary matrix.

2

0 ]q (t)

(p+r)2(p+r)

(4)

is nonsingular

q1 (t)

, q1 (t) 2 Let us now partition q (t) according to q (t) = q2 (t) (p+r) and q (t) 2 n0(p+r) , then  (t) and f (x; u) can be expressed 2

as

 (t) = Sq1 (t)

(5)

and f (x; u) = f

V

T q1 (t) ; u :

(6)

q2 (t)

Let the right-hand side of (6) be decomposed as follows: f

V

T

q1 (t) ;u q2 (t)

= f1 (; u) + f~

q1 (t) ;u q2 (t)

(7)

where II. PROBLEM STATEMENT f1 (; u) = f

Consider the following class of nonlinear systems described by: x_ (t) = Ax(t) + f (x; u) + Bu(t)

(1a)

y (t) = Cx(t)

(1b)

z (t) = Lx(t)

(1c)

where x(t) 2 n , u(t) 2 m and y (t) 2 p are the state, input and the output vectors, respectively. f (x; u) 2 n is a real nonlinear vector function. z (t) 2 r is the vector to be estimated. Matrices A, B , C and L are real constant and of appropriate dimensions. We assume that the pair (A; B ) is controllable and also without loss of generality, it is C assumed that rank(C ) = p, rank(L) = r and rank L n.

= (p + r )



The problem to be addressed in this note is the design of an r th-order observer to estimate the partial state vector z (t). Let us consider the following reduced-order observer for the system (1): !_ (t) = N ! (t) + Jy (t)+ Hf1

y (t) ; u + HBu(t) z^(t)

z^(t) = ! (t) + Ey (t)

where ! (t) f1

2

y (t) ;u z^(t)

(2a) (2b)

III. MAIN RESULTS 1) Decomposition of f (x; u): Let the nonlinear vector function f (x; u) be decomposed as follows: f (x; u) = f1 (; u) + Df2 (x; u)

where  (t) =

y (t) z (t)

=

C L

x(t)

2

(p+r )

(3) , D

2

n2d ,

rank(D) = d and 0  d  n. The functions f1 (; u) and f2 (x; u) can be obtained by following the decomposition procedure as outlined as follows.

T

S 01  (t) 0

;0

+ f (0; u)

(8)

and f~(x; u) = f V

T q1 (t) ; u q2 (t)

0 f1 (; u) = f (x; u) 0 f1 (Mx; u):

(9)

Furthermore, f~(x; u) can always be expressed as follows: f~(x; u) = Df2 (x; u)

(10)

where f2 (x; u) 2 d is regarded as an unknown input (or disturbance) vector and 0  d  n is the number of independent unknown inputs. Matrix D 2 n2d is a full-column rank matrix. There is no loss of generality in assuming that matrix D is of full-column rank. Otherwise, the following rank decomposition can be applied to the matrix D : Df2 (x; u) = D1 D2 f2 (x; u), where D1 is now a full-column rank matrix and f2 (x; u) = D2 f2 (x; u) can now be considered as a new unknown input vector. The aforementioned development completes the decomposition of the nonlinear function f (x; u) into the form (3). 2) Convergence of Observer (2): Let H 2 r2n be a full-row rank matrix and let us define the error vectors "(t) 2 r and e(t) 2 r as

r , matrices N , J , H , E , and nonlinear function 2 n are to be determined such that z^(t) converges

asymptotically to z (t).

V

0 Hx(t) e(t) = z^(t) 0 z (t): "(t) = ! (t)

(11a) (11b)

Theorem 1: z^(t) in (2) is an asymptotic estimate of z (t) for the decomposition of the nonlinearity as in (3) for all x(0), z^(0), u(t) and all possible set of nonlinear functions f (x; u) 2 n if and only if the following conditions hold. 3) Condition 1: The error "(t) determined by the observer error system "_(t) = N "(t) + H f1

y (t) ;u "(t) + z (t)

converges asymptotically to zero. 4) Condition 2: HD = 0. 5) Condition 3:

0

N H + JC HA = 0 H L + EC = 0:

0

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0 f1

y (t) ;u z (t)

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Proof: Sufficiency: From (11a) and (1)–(3), the following error dynamics equation is obtained: "_(t) = !_ (t) =

0 Hx t

_( )

0

0

N "(t) + (N H + JC HA)x(t) HDf2 (x; u) y (t) y (t) + H f1 ;u f1 ;u : (12) z^(t) z (t)

0

8) Condition C: There exist matrices P = P T > 0, G and positive scalars 1 and 2 such that the following linear matrix inequality is satisfied 1 P H1 GH2 H1T P 0 1 In 0 < 0 (16) 0 0 2 In H2T GT where

0

From (11b), the error vector e(t) can be expressed as e(t) = "(t) + (H

0L

+

P N1 + N1T P

EC )x(t):

(13)

From (12) and (13), e(t) ! 0 as t ! 1 if Conditions 1–3 of Theorem 1 are satisfied. Necessity: If Condition 1 is not satisfied then even for u(t) = 0, 0 and also e(t) 0 as x(0) = 0 and f (x; u) = 0 we have "(t) t ! 1. Now that the necessity of Condition 1 is established, we have to establish the necessity of Condition 2 and Condition 3. If Condition 2 is not satisfied, then we can find a f (x; u) in n and, therefore, a f2 (x; u) in d to make "(t) 0 and also e(t) 0 as t ! 1. If Condition 3 is not satisfied and the pair (A; B ) is controllable, even 0 for f (x; u) = 0 we can find a u(t) to generate a x(t) to make e(t) as t ! 1. This completes the proof of Theorem 1. Remark 1: Note that since matrix D is a full-column rank matrix, the structure of matrix D is fixed (i.e., the number of rows and columns of matrix D is fixed), however the choice of elements in matrix D to satisfy (10) is not unique. It should be noted that irrespective of the choice of elements in D , the conditions for asymptotic convergence in Theorem 1 are unaltered because all different choices for matrix D also alter f2 (x; u) in such a way that the product Df2 (x; u) still remains the same and is the nonlinearity f~(x; u) as in (10). From (12), it is clear that the error dynamics depends on the product Df2 (x; u). The following Theorem provides a procedure for determining matrices N , H , J and E so that the Conditions 1–3 of Theorem 1 are satisfied. Theorem 2: The estimation error e(t) = (z^(t) 0 z (t)) of observer (2) converges asymptotically to zero if the following conditions are satisfied. 6) Condition A: See (14a)–(14b)

0 GN2 0 N2T GT

2 ( 1 + 2 )Ir (17) + + + + CAL + CAL N1 = LAL 9

; N2 = (I2p

) CL+ CL+ (18) C + C + H1 = L 9

; H2 = (I2p

) (19) 1 =

0

=

0 0

i) rank

CD 0

LD

=

rank

0

CA C L

+

CA(In L L) C (In L+ L)

CD

(14a)

0

ii) rank

sL LA CA C

0LD

CD

=

rank

0

8s 2 ; < s  ( )

CA C L

where K

N

=

H

=

E

=

J

=

=

0 P 01 GN2 H1 0 P 01 GH2 +

9

K +

9

+

+

(14b) 0

7) Condition B: The nonlinear function f1 (; u) 2 n is Lipschitz in its first argument with a Lipschitz constant , i.e.,

kf1 ; u 0 f1 ; u k  k 0 k; 8u (

)

y (t) z (t) denotes the norm symbol.

where  (t)

=

2

(^

( p+ r )

)

^

(15)

, is a positive real scalar and k:k

(21) (22)

01 G(I2p 0

+ ) + P

Ip

(24)

P 01 G(I2p

Proof: Let us substitute H of Theorem 1 to give N L = LA K

=

(23)

p2p

0

NE

J

0

L

=

0

[

+



0

E

p2p

0

. Ip EC into Conditions 2 and 3 )

K]

0 NE

CA C

(25) (26)

ECD = LD:

(27)

Post-multiply both sides of (25) by a full-row rank matrix L+ (In 0 L+ L) ] gives

[

E

K]

0 0

CA(In L+ L) C (In L+ L)

=

LAL+

0

LA(In

0L

[

E

K]

CAL+ CL+

(28) =

+

L):

(29)

Now, (29) and (27) can be written in an augmented matrix equation as [

0

LD ]

N1

CD 0

0

; 9 = [ LA(In L+ L)

0

(L+ denotes the generalized matrix inverse of L) and is the Lipschitz constant defined in (15). Furthermore, matrices N , H , E and J of the observer (2) are then determined as

[

0

0

(20)

CD 0

0

0

0

N CA C LA L

+

E

K ] = 9

(30)

where and 9 are as defined in (20). It is clear from the previous equations that the knowledge of [E K ] is necessary and sufficient for the determination of matrices N , H and J . From (30), a solution for [ E K ] exists if and only if the following condition holds [19]: rank

9

=

rank( ); i:e:;

0 0 0

CA(In L+ L) CD 0 C (In L+ L) LA(In L+ L) LD CA(In L+ L) CD = rank : 0 C (In L+ L)

rank

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0 0

(31)

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1811

Post-multiply both sides of (14a) by a full row-rank matrix L+ (In 0 L+ L) 0 , it is easy to show that (14a) is equivalent 0 0 Id to (31). Accordingly, (30) has the following solution [19]: [

E K ] = 9 + + Z (I2p 0

+ )

(32)

where Z 2 r22p is an arbitrary matrix. From (32), matrices N and H can be expressed as

N = N1 0 ZN2 H = H1 0 ZH2

(33) (34)

where N1 , N2 , H1 , and H2 are as defined in (18) and (19). Incorporating (33) and (34) into the observer error system given in Condition 1 of Theorem 1 gives

"_(t) = (N1 0 ZN2 )"(t) + (H1 0 ZH2 )f~1 where for simplicity of notation,

f1

y(t) ; u z(t)

is referred to as f

f1

y(t) "(t) + z(t) ; u

(35)

0

Fig. 1. Responses of z (t) and z^(t).

~1 .

To ensure that "(t) in (35) converges asymptotically to zero, it is first necessary that matrix N = (N1 0 ZN2 ) is Hurwitz. Accordingly, N is Hurwitz if and only if the pair (N2 ; N1 ) is detectable, i.e., rank

sIr 0 N1 N2

=

r; 8s 2 ; 0. Taking its time derivative gives

V_ ("; t) = "T (t) P (N1 0 ZN2 ) + (N1 0 ZN2 )T P "(t) T T T +" (t)P (H1 0 ZH2 )f~1 + f~1 (H1 0 ZH2 ) P "(t):

It is easy to show that the right-hand side of (14b) is equivalent to

CA CD C 0 L 0

(39)

(38)

Therefore, (36) is satisfied if (14b) holds and hence matrix N is Hurwitz. Thus, in the absence of the nonlinear function f1 (; u) (i.e., f~1 = 0), condition (14b) is the necessary and sufficient condition for the determination of a matrix Z such that "(t) in (35) converges asymptotically to zero. When f~1 6= 0, then (14b) is not sufficient to ensure asymptotic convergence of (35). The stability of the nonlinear differential equation of the type (35) has been extensively studied in the literature and

P (N1 0 ZN2 ) + (N1 0 ZN2 )T P + 1 P H1 H1T P +

1

HT ZT P

2 P ZH2 2

1

+

2 ( 1 + 2 )Ir :

(42)

From (41) and (42), the LMI (16) is obtained by using the Schur decomposition and by letting G = P Z . This completes the proof of Theorem 2. Remark 2: For the case when matrix M is square, i.e., when the number of states to be estimated reaches r = (n 0 p), then the decomposition (3) gives f1 (; u) = f (x; u) = f (M 01 ; u), f2 (x; u) = 0 and D = 0. If the nonlinearity f (x; u) is Lipschitz as assumed in [13]–[17], then f1 (; u) is also Lipschitz and vice-versa. The procedure

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in this note can be used to design an (n 0 p) order observer. Furthermore, if the nonlinearity f (x; u) is non-Lipschitz then the procedure in this note allows the design of an (n 0 p) order observer if the decomposition (3) produces a Lipschitz function f1 (; u). IV. A NUMERICAL EXAMPLE 1) Example 1: Consider an example of [18] with the following matrices:

A=

C=

01

0

0

2

0

1

0

3

0

0

0

0

01

1

0

0

0

1

0

0

03

1

1

0

0

1

1

0

0

0

0

0

1

1

0

0

0

0

0

1

0

0

1

1

0

0

1

01

0

0

0

01

0

02

01 0

0

1

0

0

0

0

0

1

0

0

0

0

and L = [ 0 0 0 1 1 0 0 ] :

For the case where there are no nonlinearities to enter the system, we can use the design procedure presented in [18] to design a first-order observer to estimate the state z (t) = Lx(t) = x4 (t) + x5 (t). Note that the method of [18] cannot cope with added nonlinearity. Now, to illustrate the design method in this note, let us also design a first-order observer and with an added nonlinear function f (x; u), where

f (x; u)

= [ 4x4 x7 0

2 0:1x4 x7 0:45 sin (x4 + x5 )

0

x4 x7 ]T :

0

The nonlinear function f (x; u) can now be decomposed according to (3) so that

f1 (; u) = [ 0 D = [4

0

0

0

0:1

2 0:45 sin (4 ) 0

0

0

0

T

0

T

0]

1]

f2 (x; u) = x4 x7 , where f1 (; u) is Lipschitz with a Lipschitz constant

= 0:45 and the nonlinearity function f2 (x; u) = x4 x7 is treated as unknown input. Now, the conditions of Theorem 2 are used to design a first-order functional observer. It is easy to verify that (14a)–(14b) of Condition A of Theorem 2 are satisfied. With the Lipschitz constant = 0:45, the LMI problem (16) is feasible with the following parameters:

1 = 2 = 1; P = 0:2796 and G = [ 6:0444 03:6266 1:2089 1:6622 2 103 :

2:2666

This note has presented a method for the design of reduced-order observers to estimate a required partial set of states of the state vector of a class of nonlinear systems. The order of the observer is the same as the number of partial states to be estimated. A key feature of the design method in this note is the decomposition of the nonlinearities into two portions: one portion is assumed to be Lipschitz with respect to the partial states; the other portion comprises the remaining terms of the nonlinearities and is treated as unknown inputs. Necessary and sufficient conditions have been derived for the existence of reduced-order observers. An observer design procedure based on LMIs has been given. The proposed observer design procedure has been illustrated through a numerical example.

REFERENCES

0

0

V. CONCLUSION

0:1511 ]

Accordingly, using (21)–(24), a first-order observer (2) for z (t) =

x4 (t) + x5 (t) is obtained where

N = 0 1:4875; J = [ 00:7252 05:2127 02:9750 ] H = [ 0:0250 01:4875 01 1 1 0 0 ] E = [ 00:5125 1:4875 1 ] y(t) and f1 = ; u = [0 0 0 0:45 sin2 (^z ) 0 0 0]T : z^(t)

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Fig. 1 shows the simulated responses of z (t) and z^(t), which shows that z^(t) ! z (t).

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