Output Feedback Control of a Class of Nonlinear Systems: A Nonseparation Principle Paradigm

70 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 47, NO 0, OCTOBER 00 [7] F Martinelli, C Shu, and J R Perins, On the optimality of myopic production controls for single-server, continuous-flow manufacturing systems, IEEE Trans Automat Contr, vol 46, pp 69 73, Aug 00 [8] F Martinelli and P Valigi, The impact of finite buffers on the scheduling of a single machine two part-type manufacturing system: The optimal policy, Dipartimento di Ingegneria Elettronica e dell Informazione, Università di Perugia, Perugia, Italy, Tech Rep RDIEI-0-00, 00 Output Feedbac Control of a Class of Nonlinear Systems: A Nonseparation Principle Paradigm Chunjiang Qian and Wei Lin Abstract This note considers the problem of global stabilization by output feedbac, for a family of nonlinear systems that are dominated by a triangular system satisfying linear growth condition The problem has remained unsolved due to the violation of the commonly assumed conditions in the literature Using a feedbac domination design method which is not based on the separation principle, we explicitly construct a linear output compensator maing the closed-loop system globally exponentially stable Index Terms Global robust stabilization, linear growth condition, nonlinear systems, nonseparation principle design, output feedbac I INTRODUCTION AND DISCUSSION One of the important problems in the field of nonlinear control is global stabilization by output feedbac Unlie in the case of linear systems, global stabilizability by state feedbac plus observability do not imply global stabilizability by output feedbac, and therefore, the so-called separation principle usually does not hold for nonlinear systems Perhaps for this reason, the problem is exceptionally challenging and much more difficult than the global stabilization by state feedbac Over the years, several papers have investigated global stabilization of nonlinear systems using output feedbac and obtained some interesting results For example, for a class of detectable bilinear systems [4] or affine and nonaffine systems with stable-free dynamics [9], [0], global stabilization via output feedbac was proved to be solvable using the input saturation technique [9], [0] In [7], a necessary and sufficient condition was given for a nonlinear system to be equivalent to an observable linear system perturbed by a vector field that depends only on the output and input of the system As a consequence, global stabilization by output feedbac is achievable for a class of nonlinear systems that are diffeomorphic to a system in the nonlinear observer form [7], [8], and [5] In [], counterexamples were given indicating that global stabilization of minimum-phase nonlinear systems via output feedbac is Manuscript received August, 00; revised March, 00 Recommended by Associate Editor J M A Scherpen This wor was supported in part by the National Science Foundation under Grants ECS-987573, ECS-99068, DMS-997045, and DMS-003387 C Qian is with the Department of Electrical and Computer Engineering, The University of Texas at San Antonio, San Antonio, TX 7849 USA W Lin is with the Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, OH 4406 USA (e-mail: linwei@nonlinearcwruedu) Digital Object Identifier 009/TAC0080354 usually impossible, without introducing extra growth conditions on the unmeasurable states of the system Since then, much subsequent research wor has been focused on the output feedbac stabilization of nonlinear systems under various structural or growth conditions One of the common assumptions is that nonlinear systems should be in an output feedbac form [] or a triangular form with certain growth conditions [5], [3], [4], [], [3] The other condition is that the system can nonlinearly depend on the output of the system but is linear in the unmeasurable states [] The latter was relaxed recently in [3] by only imposing the global Lipschitz-lie condition on the unmeasurable states In this note, we consider a class of single-input single-output (SISO) time-varying systems _x = x + (t; x; u) _x = x 3 + (t; x; u) _xn = u + n(t; x; u) y = x () where x =(x ; ;xn) T IR n, u IR and y IR are the system state, input, and output, respectively The mappings i: IRIR n IR! IR, i =; ;n, are continuous and satisfy the following condition Assumption : For i =; ;n, there is a constant c 0 such that j i (t; x; u)j c(jx j + + jx i j): () Under this hypothesis, it has been shown in [4] that global exponential stabilization of nonlinear systems () is possible using linear state feedbac The objective of this note is to prove that the same growth condition, namely Assumption, guarantees the existence of a linear output dynamic compensator _ = M + Ny; M IR nn ; N IR n u = ; IR n (3) such that the closed-loop system () (3) is globally exponentially stable (GES) at the equilibrium (x; ) =(0; 0) It must be pointed out that systems () satisfying Assumption represent an important class of nonlinear systems that cannot be dealt with by existing output feedbac control schemes such as those reported in [4], [], [], and [3] To mae this point clearer, in what follows we examine three seemingly simple but nontrivial examples The first example is a planar system of the form _x = x + ln( + u x x ) +u x _x = u + x ( 0 cos(x u)) y = x (4) which obviously satisfies Assumption However, it is not in an output feedbac form (see, eg, []) nor satisfies the structural or growth conditions in [4], [], and [3] Therefore, global stabilization of the planar system (4) by output feedbac appears to be open and unsolvable by existing design methods Notably, due to a nontriangular structure, uniformly observability of nontriangular systems lie (4) cannot be guaranteed in general In particular, the lac of a lower-triangular structure maes the conventional observer-controller based output feedbac design not feasible 008-986/0$700 00 IEEE

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 47, NO 0, OCTOBER 00 7 The next example illustrates that a nonlinear system () with Assumption, eg, _x = x _x = u + x sin x y = x (5) may fail to satisfy the global Lipschitz-lie condition given in [3] Consequently, global stabilization of (5) via output feedbac cannot be solved by the approach of [3] In fact, it is easy to verify that x sin x is not global Lipschitz with respect to the unmeasurable state x, although Assumption holds For this type of nonlinear systems, most of the existing results on output feedbac stabilization are not applicable and a Luenberger-type observer, which consists of a copy of (5) plus an error correction term, does not seem to wor because convergence of the error dynamics is hard to prove Finally, in the case when the system under consideration involves parametric uncertainty, the problem of output feedbac stabilization becomes even more challenging Few results are available in the literature dealing with nonlinear systems with uncertainty that is associated with the unmeasurable states For instance, consider the uncertain system _x = x + d (t)x _x = u + d (t) ln( + x 4 ) sin x y = x (6) which satisfies Assumption, where jd i(t)j, i =;, are unnown continuous functions with nown bounds (equal to one in the present case) When d (t) 0, global stabilization of the uncertain system (6) can be easily solved using output feedbac However, when d (t) 6= 0, all the existing methods cannot be used because the presence of d (t) maes the design of a nonlinear observer extremely difficult The examples discussed thus far have indicated that nonlinear systems () with Assumption cover a class of nonlinear systems whose global stabilization by output feedbac does not seem to be solvable by any existing design method, and therefore is worth of investigation The main contribution of the note is the development of a feedbac domination design approach that enables one to explicitly construct a linear dynamic output compensator (3), globally exponentially stabilizing the entire family of nonlinear systems () under the growth condition () It must be pointed out that our output feedbac control scheme is not based on the separation principle That is, instead of constructing the observer and controller separately, we couple the high-gain linear observer design together with the controller construction An obvious advantage of our design method is that the precise nowledge of the nonlinearities or uncertainties of the systems needs not to be nown What really needed is the information of the bounding function of the uncertainties, ie, the constant c in () This feature maes it possible to stabilize a family of nonlinear systems using a single output feedbac compensator In other words, the proposed output feedbac controller has a universal property In the case of cascade systems, our design method can deal with an entire family of finite-dimensional minimum-phase nonlinear systems whose dimensions of the zero-dynamics are unnown II OUTPUT FEEDBAC DESIGN In [4], it was proved that a class of nonlinear systems satisfying Assumption is globally exponentially stabilizable by linear state feedbac When dealing with the problem of global stabilization via output feedbac, stronger conditions such as lower-triangular structure, differentiability of the vector field (t; x; u) =( (); ; ()) T and the global Lipschitz condition were assumed [4] In this section, we prove that Assumption suffices to guarantee the existence of a globally stabilizing output feedbac controller This is done by using a feedbac domination design which explicitly constructs a linear output feedbac control law In contrast to the nonlinear output feedbac controller obtained in [4], the dynamic output compensator we propose is linear with a simple structure (3) Theorem : Under Assumption, there exists a linear output feedbac controller (3) maing the uncertain nonlinear system () globally exponentially stable Proof: The proof consists of two parts First of all, we design a linear high-gain observer motivated by [], [5], without using the information of the system nonlinearities, ie, i(t; x; u); i =; ;n: This results in an error dynamics containing some extra terms that prevent convergence of the high-gain observer We then construct an output controller based on a feedbac domination design to tae care of the extra terms arising from the observer design This is accomplished by choosing, step-by-step, the gain parameters of the observer and the virtual controllers in a delicate manner At the last step, a linear output dynamic compensator can be obtained, maing the closed-loop system globally exponentially stable Part I Design of a Linear High-Gain Observer We begin with by designing the following linear observer _^x =^x + La (x 0 ^x ) _^x n0 =^x n + L n0 a n0(x 0 ^x ) _^x n = u + L n a n (x 0 ^x ) () where L is a gain parameter to be determined later, and a j > 0 and j =; ; n; are coefficients of the Hurwitz polynomial p(s) = s n + a s n0 + + a n0s + a n Define " i =(x i 0 ^x i )=L i0, i =; ;n A simple calculation gives where " = _" = LA" + " " " n A = (t; x; u) L (t; x; u) L n0 n(t; x; u) 0a 0 0a n0 0 0a n 0 0 : () Clearly, A is a Hurwitz matrix Therefore, there is a positive definite matrix P = P T > 0 such that A T P + PA = 0I:

7 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 47, NO 0, OCTOBER 00 Consider the Lyapunov function V 0(") = (n +)" T P" By Assumption, there is a real constant c > 0, which is independent of L, such that _V 0(") =0(n +)L" +(n +)" T P 0(n +)L" (t; x; u) (t; x; u) Ḷ n(t; x; u) L n0 + c " jx j + L jx j + + L n0 jx nj : Recall that x i = ^x i + L i0 " i Hence L i0 x i L i0 ^x i + j" i j; i =; ;n: With this in mind, it is not difficult to deduce that _V 0 (") 0 (n +)L 0 c p n " + c " j^x j + L j^x j + + L n0 j^x nj p n 0 (n +)L0 c n 0 c " + c ^x + L ^x + + L n0 ^x n : (3) Part II Construction of an Output Feedbac Controller Initial Step: Construct the Lyapunov function V ("; ^x )=V 0 (")+ (^x =) A direct calculation gives _V 0 (n +)L 0 p n + n c " + c ^x + L ^x + + +^x (^x + La " ) 0 nl 0 p n + n c " + c L ^x + + L n0 ^x n L n0 ^x n +^x ^x + 4 La ^x + c ^x : Let L c and define =^x 0 ^x 3 with ^x 3 being a virtual control Observe that c ^x L ^x ; c L ^x c L + c L ^x3 : With this in mind, we have _V ("; ^x ) 0 nl 0 p n + n c " + c L 4 ^x 3 + + L n0 ^x n + c L Choosing the virtual controller results in + c L ^x3 +^x +^x ^x 3 + L 4 a + ^x 3 = 0Lb ^x ; b := n + 4 a + > 0 _V 0 nl 0 p n + n c " 0 nl 0 c b ^x ^x : (4) +c L 4 ^x 3 + + L n0 ^x n + c L +^x : (5) Inductive Step: Suppose at step, there exist a smooth Lyapunov function V ("; ; ; ) which is positive definite and proper, and a set of virtual controllers ^x 3 ; ; ^x 3 +, defined by x 3 = 0, = ^x 0 x 3 and ^x 3 i = 0Lb i0 i0 i =^x i 0 ^x 3 i ; i =; ;+ with b i > 0 being independent of the gain constant L, such that _V 0 (n +0 )L 0 p n + n c " 0 j= L j0 (n +0 )L 0 c b j j + c L + ^x + + + L n0 ^x n + c L + + L (0) + : (6) Now, consider the Lyapunov function V + ("; ; ; + ) = V ("; ; ; )+ L +; Observe that =^x + Lb 0^x 0 + L b 0 b 0^x 0 Then, it is straightforward to show that d dt L + = L + ^x + + L + a + " + Lb i= + =^x + 0 ^x 3 +: + + L 0 b 0 b 0 b ^x : @ @ ^x i ^x i+ + L i a i " = L + ^x + + L + a + " + i= i+ 0 Lb i i + L i a i " L 0i+ b b i = L + ^x + + L + d 0" + L + d + L d + + Ld + + (7) where d 0; ;d + ; are suitable real numbers that are independent of the gain constant L, and d + > 0 Putting (6) and (7) together, we have _V + 0 (n +0 )L 0 p n + n c " 0 j= L j0 (n +0 )L 0 cb j j + c L +4 ^x +3 + + + c L + + + c ^x3 L+ L n0 ^x n + + L + + + L + ^x 3 + + + d 0 L 0 " + d L 0 + d L 0 + + d 0 L 03 0 + d + L 0 + d + + c L L 0 +

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 47, NO 0, OCTOBER 00 73 0 (n 0 )L 0 p n + n c " 0 (n 0 )L 0 c b 00 (n 0 )L 0 c b + c L +4 ^x +3 + + + c L + + + c ^x3 L+ + L + ^x 3 + + + 0 L 0 L n0 ^x n + + L + + L 0 d0 4 + d 4 + + d + (d +) + d + + : 4 4 (8) From the previous inequality, it follows that the linear controller with ^x 3 + = 0Lb + + b + = n 0 + d 0 4 + d 4 + + d 0 + (d +) + d + + > 0 4 4 being independent of L, renders _V + 0 (n 0 )L 0 p n + n c " 0 (n 0 )L 0 c b 00 (n 0 )L 0 c b + + + c L +4 ^x +3 + + L L n0 ^x n + c L + + + L + + : (9) This completes the inductive argument Using the inductive argument step by step, at the nth step one can design the linear controller u = 0 Lb n n = 0 Lb n (^x n + Lb n0 (^x n0 + + Lb (^x + Lb ^x ) ) (0) with b i > 0, i =; ;nbeing real numbers independent of the gain parameter L, such that _V n 0 L 0 p n + n c " 0 L 0 c b 00 L n04 L 0 cb n0 n0 0 L n0 L n () where V n is a positive definite and proper function defined by V n ("; ; ; n )=V 0 (")+ n i= L (i0) i : If we choose the gain constant L > L 3 := maxf; ( p n + (n=))c, c b ; ;c b n0g, the right-hand side of () becomes negative definite Therefore, the closed-loop system is globally exponentially stable Remar : The novelty of Theorem is two-fold: on one hand, in contrast to the common observer design that usually uses a copy of (), we design only a linear observer () for the uncertain nonlinear system () Such a construction, under Assumption, avoids dealing with difficult issues caused by the uncertainties or nonlinearities of the system On the other hand, the gain parameter L of the observer () is designed in such a way that it depends on the parameters At the last step, the design of the controller u is slightly different from that of inductive argument, because all the jun terms (eg, ^x, j n) in (6) have already been canceled at Step n 0 of the controller [ie, b i;i=; ;n, in (0)] In fact, the observer and controller designs in Theorem are heavily coupled with each other This is substantially different from most of the existing wor where the designed observer itself can asymptotically recover the state of the controlled plant, regardless of the design of the controller, ie, the controller design is independent of the observer design nown as the separation principle Theorem has an interesting consequence on output feedbac stabilization of a family of time-varying nonlinear systems in the form a ; (t; y) 0 0 _x = x + u a n0; (t; y) a n0; (t; y) 0 a n; (t; y) a n; (t; y) a n; n(t; y) y = x () where a i; j (t; y), = ; ;i, i = ; ;n, are unnown continuous functions uniformly bounded by a nown constant Obviously, Assumption holds for () Thus, we have the following result Corollary 3: For the uncertain time-varying nonlinear system (), there is a linear dynamic output compensator (3), such that the closed-loop system () and (3) is globally exponentially stable Note that this corollary has recovered the output feedbac stabilization theorem in [], for the time-invariant triangular system _x = x + a ; (y)y _x = x 3 + _x n = u + = n = a ; (y)x a n; (y)x y = x (3) with globally bounded a i; j (y) s (see []) In the rest of this section, we use examples to illustrate applications of Theorem Example 4: Consider a continuous but nonsmooth planar system of the form _x = x + x sin x _x = u + x =3 x =3 y = x : (4) Due to the presence of (t; x ;x )=x sin x, system (4) is not in a lower-triangular form Moreover, (t; x ;x ) = x =3 x =3 is only a non-lipschitz continuous function Neither the differentiability nor the global Lipschitz condition [4], [3] is fulfilled As a result, global output feedbac stabilization of (4) cannot be solved by the methods proposed in [4] and [3] On the other hand, it is easy to verify that j (x ;x )j jx j j (x ;x )j jx j + jx j: That is, Assumption holds By Theorem, the output feedbac controller _^x =^x + L(y 0 ^x ) _^x = u + L (y 0 ^x ) u = 0Lb (^x + Lb ^x ) (5) with a suitable choice of the parameters L, b and b (eg, b ==4, b = 0 and L 00), globally exponentially stabilizes system (4) The simulation shown in Fig demonstrates the GES property of the closed-loop system (4) and (5)

74 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 47, NO 0, OCTOBER 00 to transform (7) into Fig Transient responses of (4) and (5) with (x (0); x (0), ^x (0), ^x (0)) = (5; 50; ; 000) Example 5: Consider the three-dimensional system with uncertainty x x 3 _x = x + +u + x 3 _x = x 3 + x sin x _x 3 = u + d(t) ln( + jx x 3 j) y = x (6) where the unnown function d(t) is continuous, belonging to a nown compact set (eg, =[0; ]) Since (t; x ;x ;u) (x )=x sin x, j (x )j jx j Note that, however, there is no smooth gain function c(x ) 0 satisfying j (x ) 0 (^x )jc(x )jx 0 ^x j; 8 x IR; ^x IR ie, the global Lipschitz condition required in [3] does not hold and, therefore, the existing output feedbac control schemes such as [], [4], and [3] cannot be applied to (6) On the other hand, (4) is globally exponentially stabilizable by the linear output feedbac controller () (0) as Assumption is obviously satisfied Example 6: Consider a single-lin robot arm system introduced, for instance, in [6] The state-space model is described by _z = z _z = J N z 3 0 F (t) z 0 z 0 mgd cos z J J J _z 3 = z 4 _z 4 = u + J J N z 0 J N z 3 0 F (t) z 4 J y = z (7) where J ;J ;;N;m;g;dare nown parameters, and F (t) and F (t) are viscous friction coefficients that may vary continuously with time Suppose F (t) and F (t) are unnown but bounded by nown constants The control objective is to globally stabilize the equilibrium (z ;z ;z 3 ;z 4 )=(0; 0; mgdn=;0) by measurement feedbac In the current case, z the lin displacement of the system is measurable and, therefore, can be used for feedbac design To solve the problem, we introduce a change of coordinates x = z ; x = z ; x 3 = mgd z3 0 ; x 4 = J N J J N z4 and a prefeedbac v = J N u 0 mgd J J _x = x _x = x 3 0 F (t) x 0 x 0 mgd (cos x 0 ) J J J _x 3 = x 4 _x 4 = v + J J N x 0 F(t) x3 0 x 4 J N J y = x : (8) Since F (t) and F (t) are unnown, most of the existing results are not applicable to the output feedbac stabilization problem of (8) Observe that Assumption holds for (8) because j cos x 0 j jx j F (t) J x c jx j F (t) J x 4 c jx 4j; for constants c ;c : Using Theorem, it is easy to construct a linear dynamic output compensator of the form () (0), solving the global stabilization problem for (8) We end this section by pointing out when Assumption fails to be satisfied, global stabilization of () by output feedbac may be impossible For instance, the nonlinear system _x =x _x =x 3 _x 3 =u + x 3 y =x (9) is not globally stabilizable by any continuous dynamic output compensator This fact can be proved using an argument similar to the one in [] III UNIVERSAL OUTPUT FEEDBAC STABILIZATION From the design procedure of Theorem, it is clear that there is a single linear output feedbac controller (3) maing the entire family of nonlinear systems () simultaneously exponentially stable, as long as they satisfy Assumption with the same bound c This is a nice feature of our output feedbac control scheme, due to the use of the feedbac domination design For example, it is easy to see that the output feedbac controller (5) designed for the planar system (4) also globally exponentially stabilizes the following system: _x = x + (x + x sin(ux)) 4 _x = u + x sin(ux ) y = x (3) which was proved to be globally stabilizable by linear state feedbac [4] However, the problem of output feedbac stabilization was not solved in [4] because (3) violates the growth conditions (B) (B) of [4] The universal stabilization idea above can be extended to a family of C 0 minimum-phase nonlinear systems Specifically, consider m cascade systems with the same relative degree r _z = F (z )+G (t; z ;x ;u) _x = x + (t; z ;x ;u) _x i = x i+ + i (t; z ;x ;u) _x r = u + n(t; z ;x ;u); y = x ; =; ;m (3)

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 47, NO 0, OCTOBER 00 75 where z IR s and s is an unnown nonnegative integer Theorem 3: Suppose for each individual system (3), _z = F (z ) is GES at z =0and jg (t; z ;x ;u)j c jx j j i (t; z ;x ;u)j c (z + jx j + + jx i j); =; ;m: Then, there is a universal linear output feedbac controller _ = M + Ny; M IR rr ; N IR r u = ; IR r (33) that simultaneously exponentially stabilizes the m cascade systems (3) Proof: Since _z = F (z ) is globally exponentially stable, by the converse theorem there is a positive definite and proper function V (z ) such that This, in turn, implies @V (z ) @z @V (z ) @z F (z ) 0z ; @V (z ) @z ^c z with ^c > 0: F (z )+G (t; z ;x ;u) 0 3 4 z + c ^c jx j ; =; ;m: (34) Now, one can construct a single r-dimensional observer () with the gain parameter L to be determined later and a Lyapunov function V0(" )=(r + )(" ) T P" with " i = x i 0 ^x i L i0 ;i =; r: Similar to the proof of Theorem, there is a real constant ~c > 0 satisfying _V (z )+ _ V0(" ) 0 3 4 z + c ^c jx j 0 (r +)L" +~c " z + jx j + jx j L By the completion of squares, it is easy to see that + + jx r j L r0 : (35) ~c " z (~c ) " + 4 z (36) ~c " jx j j L j0 ~c " j^x j j L j0 +~c " j" j j ~c " +~c (^x j ) L (j0) +~c " j" j j: (37) Substituting (36) and (37) into (35) yields _V (z )+ V0(" _ ) 0 z 0 (r +)L" + (~c ) +~c p r + r ~c " +(c ^c ) ^x +~c ^x + L ^x + + L r0 ^x r 0 z 0 (r +)L 0 (c) 0 c p r 0 r c " r + c j= L j0 ^x j (38) where c is a uniform constant defined as c := max (c ^c ) +~c ; =; ;m : Observe that (38) is almost identical to inequality (3) in the proof of Theorem Using a recursive design procedure similar to the one in Theorem, we can obtain at the rth step the following globally exponentially stabilizing controller: u = 0Lb r (^x r + Lb r0(^x r0 + + Lb(^x + Lb^x) ) (39) where b i > 0, i = ; ;r, are real numbers independent of the gain parameter L, and L>L 3 := maxf; c +( p r +(r=))c; cb; ;cbr0g Because (38) holds uniformly for the m systems with a common constant c > 0, it is not difficult to prove that the feedbac control law (39) together with the single r-dimensional observer () stabilizes the m systems (3) simultaneously IV CONCLUSION We have presented a new output feedbac control scheme for a class of nonlinear systems whose global stabilization problem via output feedbac cannot be handled by existing methods The proposed output dynamic compensator is linear and can stabilize simultaneously a family of nonlinear systems which are dominated by a chain of integrators perturbed by a triangular vector field with linear growth condition Moreover, the result can also be applied to a finite number of globally exponentially minimum-phase systems (say, for instance, m controlled plants), in which the dimensions of the zero dynamics can be different and unnown REFERENCES [] G Besancon, State affine systems and observer-based control, in Proc NOLCOS 98, vol, July 998, pp 399 404 [] H halil and F Esfandiari, Semiglobal stabilization of a class of nonlinear systems using output feedbac, IEEE Trans Automat Contr, vol 38, pp 4 45, Sept 993 [3] H halil and A Saberi, Adaptive stabilization of a class of nonlinear systems using high-gain feedbac, IEEE Trans Automat Contr, vol 3, pp 03 035, Nov 987 [4] J P Gauthier and I upa, A separation principle for bilinear systems with dissipative drift, IEEE Trans Automat Contr, vol 37, pp 970 974, Dec 99 [5] J P Gauthier, H Hammouri, and S Othman, A simple observer for nonlinear systems, applications to bioreactocrs, IEEE Trans Automat Contr, vol 37, pp 875 880, June 99 [6] A Isidori, Nonlinear Control Systems, 3rd ed New Yor: Springer- Verlag, 995 [7] A J rener and A Isidori, Linearization by output injection and nonlinear observer, Syst Control Lett, vol 3, pp 47 5, 983 [8] A J rener and W Responde, Nonlinear observers with linearizable error dynamics, SIAM J Control Optim, vol 3, pp 97 6, 985 [9] W Lin, Input saturation and global stabilization of nonlinear systems via state and output feedbac, IEEE Trans Automat Contr, vol 40, pp 776 78, Apr 995 [0], Bounded smooth state feedbac and a global separation principle for nonaffine nonlinear systems, Syst Control Lett, vol 6, pp 4 53, 995 [] R Marino and P Tomei, Dynamic output feedbac linearization and global stabilization, Syst Control Lett, vol 7, pp 5, 99 [] F Mazenc, L Praly, and W D Dayawansa, Global stabilization by output feedbac: Examples and counterexamples, Syst Control Lett, vol 3, pp 9 5, 994 [3] L Praly, Asymptotic stabilization via output feedbac for lower triangular systems with output dependent incremental rate, in Proc 40th IEEE Conf Decision and Control, Orlando, FL, 00, pp 3808 383 [4] J Tsinias, A theorem on global stabilization of nonlinear systems by linear feedbac, Syst Control Lett, vol 7, pp 357 36, 99 [5] X H Xia and W B Gao, Nonlinear observer design by observer error linearization, SIAM J Control Optim, vol 7, pp 99 6, 989