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Systems & Control Letters 50 (2003) 319 – 330 www.elsevier.com/locate/sysconle

Observer design for systems with multivariable monotone nonlinearities Xingzhe Fan∗ , Murat Arcak Department of Electrical, Computer and Systems Engineering, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180-3590, USA Received 17 October 2002; accepted 11 April 2003

Abstract Globally convergent observers are designed for a class of systems with multivariable nonlinearities. The approach is to represent the observer error system as the feedback interconnection of a linear system and a state-dependent multivariable nonlinearity. We 4rst extend an earlier design (Automatica 37 (12) (2001) 1923) to multivariable nonlinearities, satisfying an analog of the scalar nondecreasing property. Next, we exploit the structure of the nonlinearity to relax the positive real restriction on the linear part of the observer error system. This relaxed design renders the feasibility conditions less restrictive, and widens the applicability of the observer, as illustrated with examples. Finally, output nonlinearities are studied and the design is extended to be adaptive in the presence of unknown parameters. c 2003 Elsevier B.V. All rights reserved. Keywords: Nonlinear observer; Circle criterion; Multiplier

1. Introduction The prevalent approach in observer design is to dominate nonlinearities with high-gain linear terms, or to eliminate them via geometric transformations. High-gain observers have been developed by Khalil and other authors, as surveyed in [10]. To obtain linear observer error dynamics, Krener and Isidori [11] initiated a geometric design, which has been further studied by numerous authors, including Kazantzis and Kravaris [8], who proposed a less-conservative procedure. Other studies include open-loop observers, explored by Lohmiller and Slotine [12]; and an H∞ -like design for nonlinearities with linear growth bounds, introduced by Thau [16], and further developed by Hedrick and coworkers [7,15]. A recent design by Arcak and KokotoviAc [5] has eliminated a long-standing linear growth assumption from nonlinearities of the unmeasured states. This design represents the observer error system as the feedback interconnection of a linear system and a state-dependent sector nonlinearity. Convergence of the estimates to the true states is then achieved under two restrictions which allow the observer error system to satisfy the circle criterion: First, a linear matrix inequality (LMI) must be feasible, which guarantees a strict positive real ∗

This work was supported in part by NSF, under grant ECS-0238268. Corresponding author. Tel.: +1-518-276-8205; fax: +1-518-276-6261. E-mail addresses: [email protected] (Xingzhe Fan), [email protected] (M. Arcak).

c 2003 Elsevier B.V. All rights reserved. 0167-6911/03/$ - see front matter doi:10.1016/S0167-6911(03)00170-1

320

X. Fan, M. Arcak / Systems & Control Letters 50 (2003) 319 – 330

(SPR) property for the linear part of the observer error system. The second restriction is that the nonlinearities must be monotone nondecreasing functions of the unmeasured states. This implies a sector property for the state-dependent nonlinearity in the observer error system, and ensures convergence of the estimates from the circle criterion [9]. In this paper we extend the design in [5] in several directions: The 4rst result extends this observer to multivariable nonlinearities (·) : Rp → Rp which satisfy a multivariable analog of the monotonicity property: T 9 9 + ¿ 0 ∀v ∈ Rp : (1) 9v 9v With this property, the state-dependent nonlinearity that arises in the observer error system satis4es a multivariable sector condition. We then proceed with the observer design by modifying the LMI in [5] to satisfy the multivariable circle criterion. Property (1) has been employed in [1] to design observers for a class of Euler–Lagrange systems. In this paper, we present a systematic design procedure which is applicable to wider classes of systems. The second result of the paper is to exploit the structure of the multivariable nonlinearity by introducing a multiplier in the LMI. This multiplier broadens the applicability of the observer because it relaxes the SPR restriction on the linear part of the observer error system. This is illustrated on an example where, without a multiplier, the observer LMI is infeasible. However, by exploiting the decoupled structure of the nonlinearities and incorporating a multiplier, we show that the new LMI is feasible. The third result of the paper is to generalize the design to a class of systems with monotonic output nonlinearities. When these nonlinearities are perturbed by unknown parameters, we augment the design with an adaptive law, and recover observer convergence. The new design for multivariable nonlinearities is presented in Section 2, followed by the relaxed multiplier construction in Section 3. Output nonlinearities are discussed in Section 4 and a single-link Iexible joint robot example is studied to illustrate the new design tools. In Section 5 the adaptive design is presented. Conclusions are given in Section 6. This paper does not address the incorporation of the observer in feedback design, which is studied separately in our earlier work [5,6,14,2]. Likewise, analytical feasibility conditions for the observer LMIs can be derived following the methodology in [4]. 2. Circle criterion design for multivariable nonlinearities For our observer design, we consider the plant x˙ = Ax + G(Hx) + (y; u);

y = Cx;

(2)

where x ∈ Rn is the state, y ∈ Rr is the measured output, u ∈ Rm is the control input, and the multivariable nonlinearity (·) : Rp → Rp satis4es (1). With this assumption, our observer has the same form as in [5] xˆ˙ = Axˆ + L(C xˆ − y) + G(H xˆ + K(C xˆ − y)) + (y; u):

(3)

Our task is to design the matrices K ∈ Rp×r and L ∈ Rn×r to make the observer error e = x − xˆ approach zero. From (2) and (3), the dynamics of the observer error e = x − xˆ are governed by e˙ = (A + LC)e + G[(v) − (!)];

(4)

v := Hx;

(5)

where ! := H xˆ + K(C xˆ − y):


321

To represent the observer error system (4) as the feedback interconnection of a linear system and multivariable sector nonlinearity, we view (v) − (!) as a function of v and z := v − ! = (H + KC)e; that is, a state-dependent multivariable nonlinearity in z: ’(v; z) := (v) − (!):

(6)

Substituting (6), we rewrite the observer error system (4) as e˙ = (A + LC)e + G’(v; z);

z = (H + KC)e:

(7)

To show that ’(v; z) satis4es a multivariable sector property, we make use of the Mean Value Theorem [13], and rewrite ’(v; z) as 1 1 9 9 ’(v; z) = (v) − (w) = (v − w) d = z d: (8) 9s 9s s=v−z 0 0 s=v+(w−v) Thus, from property (1) T 9 9 1 T 1 T + z ’(v; z) = z 2 9s 9s 0

d z ¿ 0:

(9)

s=v−z

Thanks to this sector property, asymptotic stability is guaranteed from the circle criterion if the linear system with input # = −’(v; z) and output z is SPR, that is, if a matrix P = P T ¿ 0, and a constant & ¿ 0 can be found such that (A + LC)T P + P(A + LC) + &I PG + (H + KC)T 6 0: (10) G T P + (H + KC) 0 Thus, the observer design for system (2) consists in solving (10), which is an LMI in P = P T ¿0; PL; K and & ¿ 0. Proposition 1. Consider system (2) and suppose x(t) exists for all t ¿ 0 and the multivariable function (·) satis9es (1). If a positive de9nite matrix P = P T and a constant & ¿ 0 can be found such that (10) holds, then the observer error converges to zero exponentially; that is, there exist constants ¿ 0 and ! ¿ 0 such that, for all t ¿ 0, |e(t)| 6 |e(0)|e−!t :

(11)

The proof follows from the Lyapunov function V = eT Pe, as detailed in Theorem 1. We illustrate the design procedure on the following example. Example 1. With the output y = x1 the system x˙2 = x2 − 13 x23 − x2 x32 ;

x˙1 = x2 ; is of form (2)  0   A=0 0 Because (·) =

with  1 0  1 0  ; 1 −1 1 3

x23 + x2 x32

x22 x3 + 13 x33

x˙3 = x2 − x3 − 13 x33 − x3 x22 

C = [1

0

0];

0

 G=  −1 0

0

(12)



 0   −1

and

H=

0

1

0

0

0

1

:

322


satis4es the monotonicity property T 2 x2 + x32 9 9 + = 9(x2 ; x3 ) 9(x2 ; x3 ) 2x2 x3

2x2 x3

x22 + x32

¿0

∀x2 ; x3 ∈ R;

our design is applicable. Indeed, LMI (10) is feasible, and a solution is     10 −2 −1 −3 −2       1 0  P= ;  −2  ; L =  −8  ; K = −1 −1 0 1 −4 which results in the observer xˆ˙1 = xˆ2 − 3(xˆ1 − x1 ); xˆ˙2 = xˆ2 − 8(xˆ1 − x1 ) − 13 (xˆ2 − 2(xˆ1 − x1 ))3 − (xˆ2 − 2(xˆ1 − x1 ))(xˆ3 − (xˆ1 − x1 ))2 ; xˆ˙3 = xˆ2 − xˆ3 − 4(xˆ1 − x1 ) − 13 (xˆ3 − (xˆ1 − x1 ))3 − (xˆ3 − (xˆ1 − x1 ))(xˆ2 − 2(xˆ1 − x1 ))2 :

(13)

It is important to note that in this example the LMI is infeasible when K is restricted to be zero. This means that the injection terms within the nonlinearities are crucial for feasibility.

3. A relaxed construction for structured nonlinearities We now relax the feasibility of the LMI (10) by exploiting the decoupled structure of the multivariable nonlinearity (·). Suppose (v) has the form   [1] (v[1] )    [2] (v[2] )    ; (v) =  (14)   ..   .   [k] (v[k] ) where (v[1] ; v[2] ; : : : ; v[k] ), v[i] ∈ Rpi , is a partition of the vector v ∈ Rp that is p1 + p2 + · · · + pk = p; and each multivariable nonlinearity [i] : Rpi → Rpi satis4es (1). Then, with the multiplier   0 ··· 0 1 Ip1 ×p1     0 2 Ip2 ×p2 · · · 0    ; i ¿ 0; i = 1; 2; : : : ; k;  %= (15)  .. .. . . . .   . . . .   0 0 · · · k Ipk ×pk it is not diNcult to show that ’(v; z) := (v) − (!) satis4es (%z)T ’(v; z) ¿ 0:

(16)


323

Thus, the multivariable circle criterion holds if the linear block with input # and output %z is SPR; that is, if a matrix P = P T ¿ 0, a constant & ¿ 0, and a matrix % as in (15) exist such that (A + LC)T P + P(A + LC) + &I PG + (H + KC)T % 6 0: (17) G T P + %(H + KC) 0 The following theorem proves observer convergence with the multiplier %, and encompasses Proposition 1 in which % = I . Theorem 1. For nonlinear system (2), suppose x(t) exists for all t ¿ 0 and the multivariable function (·) has form (14) where each [i] satis9es (1). If a matrix % as in (15), a positive de9nite matrix P = P T , and a constant & ¿ 0 can be found such that (17) holds, then the observer error converges to zero exponentially; that is, there exist constants ¿ 0 and ! ¿ 0 such that (11) holds for some ¿ 0 and ! ¿ 0. Proof. The derivative of the Lyapunov function V = eT Pe is V˙ = eT (P(A + LC) + (A + LC)T P)e + 2eT PG’(v; z);

(18)

which, from the LMI (17), satis4es V˙ 6 − &eT e + 2eT (H + KC)T %’(v; z):

(19)

Substituting (H + KC)e = z and using (16), we obtain V˙ 6 − &eT e + 2(%z)T ’(v; z) 6 − &eT e;

(20)

from which (11) follows. Example 2. This example shows that the feasibility of the relaxed design (17) with the multiplier is less restrictive than that of the LMI (10). The system x˙1 = −x1 + x2 + tan−1 (x1 ); is of form (2) with −1 1 A= ; 0 −1

x˙2 = −x2 + e2x1 ;

C = [1

2];

G=

y = x1 + 2x2

1

0

0

1

;

−1

H=

1

0

2

0

v

and nondecreasing nonlinearities 1 (v) = tan (v) and 2 (v) = e . Without a multiplier, the LMI (10) is not feasible. To see this, note that the oO-diagonal components of (10) must be zero for feasibility; that is, G T P = −(H + KC):

(21)

Multiplying both sides of (21) from the right by G, and using positive-de4niteness of P, we conclude that −k1 − 1 −2k1 (22) − (H + KC)G = −k2 − 2 −2k2 must be symmetric and positive de4nite. However, with the symmetry constraint 2k1 = k2 + 2, the determinant of (22) is −4 regardless of the choice of k1 and k2 , which means that it cannot be positive de4nite. Because the nonlinearities are decoupled, the relaxed design is applicable. Indeed, a feasible observer is xˆ˙1 = −xˆ1 + xˆ2 − 4484(xˆ1 + 2xˆ2 − y) + tan−1 (xˆ1 − 5:04(xˆ1 + 2xˆ2 − y)); xˆ˙2 = −xˆ2 + 186(xˆ1 + 2xˆ2 − y) + e(2xˆ1 −2:83(xˆ1 +2xˆ2 −y)) ;

(23)

324


which is obtained from the LMI (20) with the multiplier 1:26 0 %= : 0 15:29 This example showed that the feasibility of the relaxed design (17) with the multiplier is less restrictive than that of the LMI (10). Even if (10) is feasible, the use of a multiplier widens the set of admissible observer parameters K and L, thus oOering further design Iexibility. 4. Design for output nonlinearities We now consider a more general class of systems in which nonlinearities also exist in the outputs: x˙ = Ax + G1 (Hx) + (y; u);

y1 = C1 x;

y2 = 2 (C2 x):

(24)

The output consists of y1 ∈ Rr1 and y2 ∈ Rr2 , and the multivariable nonlinearity 1 (·) : Rp1 → Rp1 satis4es (1), and 2 (·) : Rp2 → Rp2 satis4es T 92 92 + ¿ 'I ∀v ∈ Rp (25) 9v 9v for some constant ' ¿ 0. To design an observer for this system, one possible approach would be to compute the inverse −1 2 (·) of the output nonlinearity, which exists because, from (25), the Implicit Function Theorem [9] holds. However, in practice it may be diNcult or impossible to obtain an analytical expression for −1 2 (·). The approach taken in this paper is to avoid cancelling the nonlinearity 2 (·), and to directly incorporate it in the observer: xˆ˙ = Axˆ + L1 (C1 xˆ − y1 ) + L2 (2 (C2 x) ˆ − y2 ) + G1 (H xˆ + K1 (C1 xˆ − y1 )) + (y; u):

(26)

The task now is to determine the observer matrices K1 ∈ Rp×r , L1 ∈ Rn×r1 and L2 ∈ Rn×r2 to make the observer error e = x − xˆ converge to zero. From (24) and (26), the dynamics of the observer error e = x − xˆ are governed by e˙ = (A + L1 C1 )e + G[1 (Hx) − 1 (H xˆ + K1 (C1 xˆ − y1 ))] + L2 [2 (C2 x) − 2 (C2 x)] ˆ 1 (Hx) − 1 (H xˆ + K1 (C1 xˆ − y1 )) 1 = (A + L1 C1 + 2 ' L2 C2 )e + [G L2 ] 2 (C2 x) − 12 ' C2 x − (2 (C2 x) ˆ − 12 ' (C2 x)) ˆ

1 2

1 (Hx) − 1 (H xˆ + K1 (C1 xˆ − y1 ))

= (A + L1 C1 + ' L2 C2 )e + [G

L2 ]

= (A + L1 C1 + 12 ' L2 C2 )e + [G

L2 ][(v) − (!)];

where

v=

Hx C2 x

and

(v) =

;

1 (v1 ) ˜2 (v2 )

!= :

H xˆ + K1 (C1 xˆ − y1 ) C2 xˆ

˜2 (C2 x) − ˜2 (C2 x) ˆ

;

˜2 (s) = 2 (s) − 2' s

(27)


325

Viewing (v) − (!) as a function of v and H + K 1 C1 z := v − w = e; C2 we have ˜ e˙ = (A + L1 C1 )e + G’(v; z); where

H˜ =

H + K 1 C1 C2

Because 92 ' − I+ 9v 2

z = H˜ e;

(28)

and

92 ' − I 9v 2

’(v; z) := (v) − (!):

T ¿0

∀v ∈ Rp ;

(29)

’(v; z) still satis4es the multivariable sector property. Theorem 2. For the nonlinear system (35), suppose x(t) exists for all t ¿ 0 and the multivariable functions 1 (·) and 2 (·) satisfy (1) and (25), respectively. If a positive de9nite matrix P = P T and constants & ¿ 0, 1 ¿ 0 and 2 ¿ 0 can be found such that  T  (H + K C ) 1 1 1  (A + L1 C1 + 1 ' L2 C2 )T P + P(A + L1 C1 + 1 ' L2 C2 ) + &I P[G L2 ] +  2 2   2 C2     6 0;   1 (H + K1 C1 )   T 0 [G L2 ] P + 2 C2 (30) then the observer error converges to zero exponentially; that is, (11) holds for some Proof. Because the multivariable nonlinearity 1 (v1 ) (v) = 2 (v2 ) is decoupled, we recall from Section 3 that, with the multiplier 1 Ip1 ×p1 0 %= ; i ¿ 0; i = 1; 2; 0 2 Ip2 ×p2

¿ 0 and ! ¿ 0.

(31)

(32)

the multivariable circle criterion is satis4ed if the feedforward block with input # and output %z is SPR; that is, if a positive de4nite matrix P = P T and constants & ¿ 0, 1 ¿ 0 and 2 ¿ 0 exist such that (30) is satis4ed. Using this property, the result follows from the Lyapunov function V = eT Pe, as in Theorem 1. In Theorem 2, we employed multiplier (32) because 1 (v1 ) and 2 (v2 ) in (31) are decoupled. By further exploiting the structure in 1 (v1 ) and 2 (v2 ), we can apply the results of Section 3, and make the multiplier (32) even less restrictive. Example 3 (Single-link Iexible joint robot): The design for output nonlinearities is now illustrated on a Iexible joint robot link example studied in [15]. As depicted in Fig. 1 below, the system nonlinearities are due to joint Iexibility, modelled as a stiOening torsional spring, and the gravitational force.

326


Torsional Spring

DC Motor

l

m Jl

Jm

Fig. 1. The Iexible joint robot link. Joint Iexibility is modelled as a stiOening torsional spring.

Denoting by (m , !m , (l and !l , the motor and link position and velocities respectively, the dynamic equations are given by (˙m = !m ;

!˙ m =

1 B K* *− !m + u; Jm Jm Jm

(˙l = !l ;

!˙ l = −

1 mgh *− sin((l ); Jl Jl

(33)

where Jm is the inertia of the dc motor, Jl is the inertia of the link, 2h and m represent the length and mass of the link, B is the viscous friction, and K* is the ampli4er gain. The torque due to the stiOening spring is *=

1 ((l

− (m ) +

2 ((l

− ( m )3 ;

where 1 and 2 are positive constants. We suppose that the position and velocity of the motor are measured, but due to the position sensor nonlinearity, instead of (m , 2 ((m ) is available, where 2 (·) is a known function with slope greater than one. To design an observer for [(l !l ] from the output y = [2 ((m ) !m ], we note that the system nonlinearities ((l −(m )3 and (l −sin((l ) and the output nonlinearity 2 ((m ) are nondecreasing and rewrite (33) as in (24) with     0 1 0 0 0   B  K   1  * − 1 − 0 C1 0 1 0 0     Jm J J m m     J = ; A= ; B =  m ; 0 0 1 1 0 0 0 C2  0   0       1 mgh 1 0 0 − 0 Jl Jl Jl   0 0    1    0 −1 0 1 0  Jm  ; H= ; G=   0  0 0 1 0  0    1 mgh  − Jl Jl and

1 (z1 ; z2 ) =

3 2 z1

z2 − sin(z2 )

:

(34)


327

Fig. 2. Angular rotation of the link (l (dashed) and its observer estimate (ˆl (solid).

Fig. 3. Angular velocity of the link !l (dashed) and its observer state !ˆ l (solid).

Because the entries 1 (z1 ; z2 ) are decoupled, we modify the multiplier in (32) as % = diag{11 ; 12 ; 2 }. To further relax the feasibility conditions, we exploit the upper bound on the slope of 2 (z2 ) and modify the LMI (30) as in [6]. With the physical values Jm = 3:7 × 10−3 kg m2 , J1 = 9:3 × 10−3 kg m2 , h = 1:5 × 10−1 m, m = 0:21 kg, B = 4:6 × 10−2 m and K* = 8 × 10−2 N m V−1 , the new LMI is feasible and a solution is     −2:6295 −0:4107      −932:8074   −2:6337  −0:3149     K1 = ; L1 =   ; L2 =  :     0:4242 −222:0269 −1:2400     −774:3624

−2:1259

From Theorem 2, observer (26) ensures exponential convergence of the state estimates (ˆl and !ˆ l , as illustrated with simulations in Figs. 2 and 3.

328


5. An adaptive extension As in every observer design, model uncertainty poses a challenge for achieving convergence of state estimates. We now address the situation in which system (24) is perturbed by unknown parameters ( ∈ Rp2 : x˙ = Ax + G1 (Hx) + (y; u) + Q(;

y = 2 (Cx + ()

(35)

and develop an adaptive variant of our observer. The structure of system (35) is in part motivated by a fuel cell model in [3], where the unknown hydrogen level ( at the inlet of the anode aOects both the dynamics of the level x at the outlet, and the voltage output y, which depends nonlinearly on x and (. In this section, we generalize the design in [3] to the class of systems de4ned by (35). Our adaptive design for this class is ˆ xˆ˙ = Axˆ + G (H x) ˆ + (y; u) + Q(; (ˆ˙ = −k/(y; y); ˆ (36) 1

where yˆ = C xˆ + (ˆ and the adaptive law has the form /(y; y) ˆ = 01 (y; 02 (y) ˆ − 0(y)); with the functions 01 (·; ·) and 02 (·) to be designed such that 902 (y) 01 (y; x) ¿ 1 ∀x =

0; ∀y and ¿1 01 (y; 0) = 0; x 9y

(37)

∀y:

(38)

The general form of injection term /(y; y) ˆ uses the “weighting functions” 01 (·; ·) and 02 (·) for additional design Iexibility. This form includes the choice /(y; y)= ˆ yˆ −y, in which these functions are identity. De4ning e = xˆ − x and (˜ = (ˆ − (, we obtain the observer error system e˙ = Ae + Q(˜ + G( (H x) ˆ − (Hx)); (˜˙ = −k/(y; y); ˆ (39) 1

1

which is semiglobally asymptotically stable; that is, the region of attraction of the origin can be enlarged arbitrarily by increasing the observer gain k. Theorem 3. Consider system (35), suppose x(t) is bounded for all t ¿ 0, and the multivariable function (·) satis9es (1). If a positive de9nite matrix P = P T and constants & ¿ 0 and ¿ 0 can be found such that (A − QC)T P + P(A − QC) + &I PG + H T 6 0; (40) GT P + H 0 ˜ that includes the origin, we can 9nd a constant k ∗ such that, with then for any compact set 1 of (e; () ∗ ˜ = 0 with a k ¿ k , the adaptive observer (36) guarantees uniform exponential stability of the origin (e; () region of attraction that includes 1. Proof. With the new variable ˜ 2 = Ce + (;

(41)

we let V = eT Pe + 2T 2 and note that its derivative along the trajectory of (35) is V˙ = eT (PA + AT P)e + 2(Q(˜ + G(1 (H x) ˆ − 1 (H x))) ˆ T Pe ˆ 6 eT (P(A − QC) + (A − QC)T P)e − 2k2T /(2 (Cx + (); 2 (C xˆ + ()) + 22T Q2 + 22T ((CA − CQC + QT P)e + CG(1 (H x) ˆ − 1 (Hx))):

(42)


We now show that, for any compact set I , we can 4nd constant c ¿ 0 such that ˆ ¿ c2T 2: 2 ∈ I ⇒ 2T /(2 (Cx + (); 2 (C xˆ + ()) To prove this, we denote := Cx + ( and recall from (35) and (1) that 9( ) 9y = ¿ 0: 9 9 Next we note from (37), (38) and the chain rule that 902 (y) 902 (y) 9y 9y = ¿ : 9 9y 9 9 Finally, using ˆ − = 2, we obtain from the Mean Value Theorem, 1 902 ((x)) ˆ 02 (y) ˆ − 02 (y) = 02 (( )) − 02 (( )) = 2 d: x 0 x= +2

329

(43) (44)

(45)

(46)

From (44) and (45), the partial derivative in (46) admits a lower bound cV ¿ 0, because and 2 belong to compact intervals. This means that, for all 2 = 0, 02 (( ˆ )) − 02 (( )) ¿ c: V (47) 2 Then, using the sector property of the function 01 (·; ·) in (38), we obtain 01 (y; 02 (y) 01 (y; 02 (y) ˆ − 0(y)) ˆ − 0(y)) 02 (y) ˆ − 0(y) /(y; y) ˆ = ¿ c; V (48) = 2 02 (y) ˆ − 0(y) 2 2 from which (43) follows with c = c. V To complete the proof, we let Br be the ball of radius r around the origin, where r ¿ 0 is to be selected, and let I be a bounded interval such that ˜ ∈ Br ⇒ 2 = (ce + () ˜ ∈ I: (e; () And also because x(t) is bounded, we can 4nd a constant K ¿ 0 such that 22T ((CA − CQC + QT P)e + CG(1 (Hx) − 1 (H x))) ˆ 6 K 2 e in Br . This is because 1 (H x) ˆ − 1 (Hx) =

0

1

91 (&) He d & &=Hx+He

(49) (50)

and the partial derivative in (50) admits an upper bound since e and x are bounded by assumption. Then, substituting (43) and (49), we note that V˙ 6 eT (P(A − QC) + (A − QC)T P)e − 2T (2kcI − 2Q)2 + K 2 e ; which, from the LMI (40), satis4es V˙ 6 − &eT e − 2T (2kcI − 2Q)2 + K 2 e :

(51)

Substituting the Young Inequality & K T K 2 e 6 − eT e − (52) 2 2; & ¿ 0 2 2& into (51) we obtain & K V˙ 6 − eT e − 2T 2kcI − 2Q − I 2; 2 2& which is negative de4nite in Br if we choose k large enough such that 2kcI − 2Q − KI=(2&) ¿ 0. Finally, selecting r such that Br includes a level set of V in (42) that encompass 1, we conclude that 1 ∈ Br is in ˜ = 0. the region of attraction of the equilibrium (e; ()

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6. Conclusion We have extended the observer design in [5] to multivariable nonlinearities both in the system equations and in the output. Global convergence of the nonlinear observer is achieved under two restrictions which allow the observer error system to satisfy the multivariable circle criterion: First, a linear matrix inequality must render the linear part of the observer error system SPR and, next the nonlinearity must satisfy a multivariable monotonicity property. The SPR restriction on the linear part is then relaxed by employing a multiplier in the LMI, which preserves the sector property of the multivariable nonlinearity. Finally, we extend these designs to output nonlinearities and make it adaptive in the presence of unknown parameters. It would be of interest to extend this design with dynamic multipliers, which would further relax the feasibility conditions. As pointed out in the Introduction, we only addressed the observer design problem, and assumed that the state variable x(t) exists for all t ¿ 0. Observer-based feedback designs that guarantee this existence property are being developed separately, with preliminary results reported in [5,6,14,2]. References [1] O.M. Aamo, M. Arcak, T.I. Fossen, P.V. KokotoviAc, Global output tracking control of a class of Euler–Lagrange systems with monotonic nonlinearities in the velocities, Internat. J. Control 74 (7) (2001) 649–658. [2] M. Arcak, A global separation theorem for a new class of nonlinear observers, Proceedings of the 41st Conference on Decision and Control, Las Vegas, Nevada, December 2002, pp. 676–681. [3] M. Arcak, H. Gorgun, L.M. Pedersen, S. Varigonda, An adaptive observer design for fuel cell hydrogen estimation, Proceedings of the American Control Conference, Denver, Colorado, June 2003, to appear. [4] M. Arcak, P. KokotoviAc, Feasibility conditions for circle criterion design, Systems Control Lett. 42 (2001) 405–412. [5] M. Arcak, P. KokotoviAc, Nonlinear observers: a circle criterion design and robustness analysis, Automatica 37 (12) (2001) 1923– 1930. [6] M. Arcak, P. KokotoviAc, Observer-based control of systems with slope-restricted nonlinearities, IEEE Trans. Automat. Control 4 (7) (2001) 1146–1151. [7] A. Howell, J.K. Hedrick, Nonlinear observer design via convex optimization, in: Proceedings of the American Control Conference, Anchorage, Alaska, 2002, pp. 2088–2093. [8] N. Kazantzis, C. Kravaris, Nonlinear observer design using Lyapunov’s auxiliary theorem, Systems Control Lett. 34 (1998) 241– 247. [9] H.K. Khalil, Nonlinear Systems, 2nd Edition, Prentice-Hall, Englewood CliOs, NJ, 1996. [10] H.K. Khalil, High-gain observers in nonlinear feedback control, in: H. Nijmeijer, T.I. Fossen (Eds.), New Directions in Nonlinear Observer Design, Springer, Berlin, 1999, pp. 249–268. [11] A.J. Krener, A. Isidori, Linearization by output injection and nonlinear observers, Systems Control Lett. 3 (1983) 47–52. [12] J.-J. Lohmiller, W. Slotine, On contraction analysis for nonlinear systems, Automatica 34 (1998) 683–696. [13] J.M. Ortega, W.C. Rheinboldt, Interative Solution of Nonlinear Equations in Several Variables, SIAM, Philadelphia, 2000. [14] L. Praly, M. Arcak, On certainty-equivalence design of nonlinear observer-based controllers, Proceedings of the 41st Conference on Decision and Control, Las Vegas, Nevada, December 2002, pp. 1485–1490. [15] S. Raghavan, J.K. Hedrick, Observer design for a class of nonlinear systems, Internat. J. Control 59 (1994) 515–528. [16] F.E. Thau, Observing the state of non-linear dynamic systems, Internat. J. Control 17 (1973) 471–479.