Global Finite-Time Stabilization by Output Feedback for Planar Systems Without Observable Linearization

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1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 6, JUNE Global Finite-Time Stabilization by Output Feedback for Planar Systems Without Observable Linearization Chunjiang Qian and Ji Li Abstract This note considers the problem of global finite-time stabilization by output feedback for a class of planar systems without controllable/observable linearization. A sufficient condition for the solvability of the problem is established. By developing a nonsmooth observer and modifying the adding a power integrator technique, we show that an output feedback controller can be eplicitly constructed to globally stabilize the systems in finite time. As a direct application of the main result, global output feedback finite-time stabilization is achieved for the double linear integrator systems perturbed by some nonlinear functions which are not necessarily homogeneous. Inde Terms Finite-time stabilization, nonlinear systems, output feedback, unobservable linearization. I. INTRODUCTION The problem of global stabilization by output feedback for a class of planar systems without controllable/observable linearization is an interesting and challenging problem in the nonlinear control field. The difficulty of the problem mainly stems from the lack of new observer for linearly unobservable systems to which the traditional Luenberger observer is inapplicable due to the unobservable linearization [6]. In attempting to solve the problem, several preliminary results have been achieved. In [3], a novel reduced-order observer was proposed to estimate the unmeasurable states. Combining this new observer with the smooth state feedback controller constructed by the adding a power integrator tool, a smooth output feedback controller was eplicitly constructed to render the planar system globally asymptotically stable under a high-order growth condition. However, this smooth result does not permit the presence of uncontrollable and unstable linearization, due to the violation of the well-known necessary condition for the eistence of a smooth controller (i.e., the linearized system has no uncontrollable modes whose eigenvalues are on the right-half plane [4]). Later, [4] removed this limitation by developing a nonsmooth design method. As a matter of fact, to overcome the intrinsic obstacle caused by the uncontrollable unstable linearization, nonsmooth adding a power integrator [], [] was used to construct the controller. In [3] and [4], the convergence of the closed-loop systems is asymptotic with infinite settling time. The issue of how to use output feedback to stabilize the planar system without observable linearization in finite time is still an open problem. The finite-time stabilization is of interest because systems with finite-time convergence demonstrate some nice features such as faster convergence and robustness to uncertainties [3]. However, the problem of finite-time stabilization by time-invariant controller is quite challenging even in the state feedback case. This problem has been studied by [], [6], [8], and [5], and seminal results have been achieved. Compared with the state feedback case, there are fewer results dealing with output feedback finite-time stabilization. An interesting result was given in [9], where the finite-time stabilization of the double integrator systems Manuscript received February 7, 004; revised October 8, 004. Recommended by Associate Editor W. Kang. This work was supported in part by the National Science Foundation under Grant ECS The authors are with the Department of Electrical Engineering, The University of Teas at San Antonio, San Antonio, TX 7849 USA ( chunjiang.qian@utsa.edu; jli@lonestar.utsa.edu). Digital Object Identifier 0.09/TAC was achieved by coupling a finite-time convergent observer with a finite-time control law. When the double integrator system is perturbed by a nonhomogeneous function, local finite-time stabilization was achieved by homogeneous output feedback. In this note, we focus on the global finite-time stabilization by output feedback for a wider class of planar systems which are not necessarily linearly controllable/observable. The systems under consideration include the double integrator system as a special case and indeed cover a more general class of planar systems. We establish a sufficient condition under which the output feedback finite-time stabilization of the systems is achievable. The output feedback controller is comprised of a finite-time stabilizer, which is designed in the spirit of adding a power integrator [], and a new observer with nonsmooth structure that distinguishes it from those observers used in [3] and [4]. To overcome thedifficulties in thestability analysis caused by thenonsmoothness of the observer, a more rigorous design process has to be used to select an appropriate gain that guarantees the finite-time convergence of the observer. One notable feature of our design scheme is that it is not based on the commonly used separation principle. On the contrary, the observer itself will not converge unless a controller is implemented. The note is organized as follows. In Section II, we formulate the control objective of the note and give a sufficient condition for the eistence of finite-time stabilizers using output feedback. The main result is contained in Section III, where a design algorithm is presented for the eplicit construction of a reduced-order nonlinear observer as well as a nonsmooth state feedback control law. Eamples are also included in the section to demonstrate the significance of our output feedback stabilization results. In Section IV, we discuss a special case when the planar system is a generalized double integrator system. A global result is achieved for a class of planar systems with nonhomogeneous nonlinearities. II. PROBLEM STATEMENT AND PRELIMINARIES Throughout this note, we consider the following nonlinear system: _ = p + ( ) _ = u + ( ; ) y = (.) where i(), i =;, are C 0 functions with i(0) = 0, i =; and p is an odd integer. Theobjectiveof thenoteis to find a continuous output feedback controller of the form _z = (z; y); z IR u = u(z; y) (.) such that the closed-loop system (.) (.) is globally stable, moreover, all the trajectories will converge to the origin in finite time []. To be more precise, we adopt the following definitions and the Lyapunov-like Theorem [], [9] for the finite-time stabilization. Consider _y(t) =f (y(t)) (.3) where f :IR n! IR n is continuous and f (0) = 0. Definition. (Finite-Time Stability): Thezero solution of (.3) is finite-time convergent if there are an open neighborhood U of the origin and a function T : U nf0g! (0; ), such that 8 0 U, every solution (t; 0; 0 ) of system (.3) with 0 as theinitial condition is defined and (t; 0;0) U nf0g for t [0;T(0)), and lim t!t ( ) (t; 0; 0 ) = 0. Thezero solution of (.3) is finite-time /$ IEEE

2 886 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 6, JUNE 005 stable if it is Lyapunov stable and finite-time convergent. When U = IR n, then the zero solution is said to be globally finite-time stable. Theorem.: Supposethereeist a C positive definite and proper function V : IR n! IR and real numbers k > 0 and (0; ), such that V _ + kv is negative semi-definite, where _V () =(@V ()=@)f (). Then, the origin is a globally finite-time stableequilibrium of (.3). To solve the problem of finite-time stabilization via output feedback for (.), weassumethefollowing condition. Assumption.3: There is a ratio of odd integers h (0; ), such that for all ; IR j ( )jj j p=(p+0h) ( ) (.4) j ( ; )j(j j h=(p+0h) + j j h ) ( ) (.5) where ( ) 0 and ( ) 0 aresmooth functions. Remark.4: It should benoted that for any C function ( ) with (0) = 0, growth condition (.4) is automatically satisfied. As a matter of fact, by the C property of the function j ( )jj j' ( ) j j p=(p+0h) + ' ( ) where ' ( ) is a smooth function. Similarly, if () is only a C function of, condition (.5) also holds automatically. In the remainder of the section, we introduce three lemmas which will serve as bases for the development of an output feedback finitetime controller for (.). The first two lemmas are the key tools for adding a power integrator and their proofs can be found in []. Lemma.5: Let q be an odd integer or a ratio of odd integers. Then, the following inequality holds: ja 0 bj q q0 ja q 0 b q j 8 a IR; b IR: Lemma.6: Suppose n and m are two positive real numbers, and a 0, b 0 and 0 arecontinuous functions. Then, for any constant c>0 a n b m c a n+m + m n + m n c(n + m) n=m b n+m (n+m)=m : The net lemma plays a key role in constructing the observer for the planar systems. Lemma.7: Let the real number r (0; ) bea ratio of odd integers. Then, the following inequality holds for any real numbers 0 < l< and t t r +(0 t) r + l t +r ( r 0 )l 0r : (.6) Proof: When jtj, by thefact that thenumerator of +r is even, we have t r +(0 t) r + l t +r t r +(0 t) r : Calculating theminimum of t r +(0 t) r on thecompact set ftjjtj g yields t r +(0 t) r + l t +r ( r 0 ) l 0r ( r 0 ); jtj : (.7) In thecasewhen jtj >, using Young s Inequality, we have t r +(0t) r + l t +r = [t r +(0t) r =(+r) (+r)= ] + [l t +r =(0r) (0r)= ] (t r +(0 t) r )t 0r (+r)= l 0r : (.8) Let f (t) =(t r +(0 t) r )t 0r : Thetimederivativeof f (t) is f 0 (t) =+ r t 0 0 r t t 0 r0 ; jtj > : By using thefollowing inequality ( r 0 < r( 0 ) for > 0, 0 <r<, [7, p. 40]) 0 t it can beshown that 0r f 0 (t) =+ t 0 r0 0 < 0( 0 r) t ; jtj > < 0 t 0 r0 jtj > : t 0 0 r t 0 t 0r =0 This, in turn, implies that f (t) is decreasing on (0; 0) and (; ). Consequently, f (t) minff(0); lim t!+ f (t)g = minf r 0 ;rg: Applying theinequality r 0 <r( 0 ) [7] yields f (t) r 0. Therefore, when jtj >, (.8) can be estimated as t r +(0 t) r + l t +r ( r 0 ) (+r)= l 0r ( r 0 )l 0r : This, together with (.7), implies that for any t, t r +(0t) r +l t +r ( r 0 )l 0r. III. GLOBAL FINITE-TIME STABILIZATION BY C 0 OUTPUT FEEDBACK In this section, we develop a constructive design method for an output feedback controller, which stabilizes (.) in finite time. The fundamental design tools are a modified version of the adding a power integrator technique [], [] and a nonsmooth one-dimensional nonlinear observer that is inspired by but substantially different from the one used in [4]. Due to the implementation of this nonsmooth observer, a rigorous design process has to be used to select an appropriate gain that guarantees the finite-time convergence of the closed-loop system. The following theorem is the main result of the note. Theorem 3.: Under Assumption.3, there is a continuous output feedback controller of the form shown in [z + L( )] p=(p+0h) + ( ) u=u ; (z + L( )) =(p+0h) ; for a C L( ) (3.) such that the closed-loop system (.) (3.) is globally finite-time stable. Proof: The construction of output feedback finite-time stabilizer is accomplished by three steps. First, by modifying the adding a power integrator technique [], [], a finite-time stabilizer is eplicitly constructed using state feedback. Then, we reconstruct the one-dimensional observer proposed in [4] in such a way that the estimate of the unmeasurable state, with suitable gain and controller, will converge to the real value in finite time. Finally, we replace the unmeasurable state with the estimate recovered from the observer. An appropriate selection of the observer gain function will render the closed-loop system globally finite-time stable. A. Design of Finite-Time State Feedback Stabilizer Let V ( )=(=). Hence, by Assumption.3 _V p + +p=(p+0h) ( ):

3 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 6, JUNE Clearly, the C 0 virtual controller 3p = 0 p=(p+0h) ( ), ( ):==6 + ( ) renders _V 0 6 +p=(p+0h) + ( p 0 3p ) : Following [] and [], wethen construct thelyapunov function V ( ; )=V ( ) + s p+0h 0 3(p+0h) 0=(p+0h) which can beproved to bec, positivedefiniteand proper by using similar proofs in []. Moreover, it can be s p+0h 0 3(p+0h) c 0=(p+0h) p+0h 0 3(p+0h) with c =(0 =(p +0 h)) s p+0h 0 3(p+0h) = 0=(p+0h) where = p+0h 0 3p+0h = p+0h + ( ) (p+0h)=p : With these in mind, it is not difficult to show that _V 0 6 +p=(p+0h) + ( p 0 3p ) + 0=(p+0h) u + j j 0p=(p+0h) j j h=(p+0h) + j j h ( ) + N ( )j j j j p + ( )j j p=(p+0h) (3.) where N ( ) 3(p+0h) =@ is a smooth function. Net, we estimate the terms in (3.). First, by Lemmas.5 and.6, wehave j k p 0 3p jjj0p=(p+0h) j j p=(p+0h) ds 6 +p=(p+0h) + 0 +p=(p+0h) for a constant 0 > 0: (3.3) By the Young Inequality (Lemma.6), there is a smooth function ( ) 0 such that ( )j j 0(=+p0h) j j (h=+p0h) 6 +p=(p+0h) + ( ) +p=(p+0h) : (3.4) Note that by Lemma.5, it is easy to see that j jj 0 3 j + j 3 j 0=(p+0h) j j =(p+0h) + j j =(p+0h) =p ( ): Using this inequality and Lemma.6, one has j j 0=(p+0h) ( )j j h N ( )j j 0(=+p0h) j j h=(p+0h) + j j h=(p+0h) 6 +p=(p+0h) + ( ) +p=(p+0h) (3.5) for smooth functions N ( ) 0 and ( ) 0. Similarly N ( )j k j p N ( )j j j j p=(p+0h) + j j p=(p+0h) 6 +p=(p+0h) + 3( ) +p=(p+0h) (3.6) for smooth functions N ( ) 0 and 3( ) 0. Finally, thelast term in (3.) can be estimated using Lemma.6 N ( ) ( )j k j p=(p+0h) 6 +p=(p+0h) + 4( ) +p=(p+0h) for C 4 ( ) 0: (3.7) Substituting these estimates (3.3) (3.7) into (3.) yields _V ( ; ) 0 +p=(p+0h) + 0=(p+0h) u +( 0 + ( )+ ( )+ 3( ) + 4 ( )) +p=(+p0h) : Now, it is clear that the continuous controller yields _V ( ; )j u=u( ; ) u = 0 h=(p+0h) ( ) ( )=+ 0 + ( )+ ( ) + 3( )+ 4( ) (3.8) 0 +p=(p+0h) 0 +p=(p+0h) : (3.9) B. Nonlinear Observer Design Motivated by the reduced-order observer in [4], we now construct a one-dimensional nonsmooth compensator ) = 0 ([z + L( )] p=(p+0h) + ( )) where the nonlinear gain function L( ) )=@ > will be determined later. Let ^ p+0h = z + L( ) and e = p+0h 0 ^ p+0h. Then, a straightforward calculation yields _e =(p +0h) p0h (u + ( ; ) + ([z + L( )] p=(p+0h) + ) 0 ( + ()) =(p +0h) p0h (u + ( ; )) 0 (z + L( )) p=(p+0h) : ChoosetheLyapunov function V 3 (e) = (e =). Thetimederivative of V 3 is _V 3(e) =e(p +0 h) p0h (u + ( ; )) 0ek( ) p + e 0 +p0h where ((@L( ))=@ ):=k( ) >. p=(p+0h) (3.)

4 888 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 6, JUNE 005 Net, we use Lemma.7 to estimate the last term in (3.). When e 6= 0, substituting t = +p0h =e, r = p=( + p 0 h) and l = k 03=4 into (.6) yields ek p + e 0 +p0h p=(p+0h) + k 0= +p0h ( r 0 )k 0(3=4)((0h)=(+p0h)) e +p=(+p0h) : (3.) In addition, when e =0, (3.) holds automatically. Applying (3.) to (3.) gives _V 3(e) e(p +0h) p0h (u + ( ; )) + k 0= +p0h 0 ( r 0 ) k 0(3=4)((0h)=(+p0h)) e +p=(+p0h) : (3.3) By Assumption.3 and Lemma.6, it is easy to obtain e(p +0 h) p0h (u + ( ; )) jej j j p0h juj + j j p0h j ( ; )j (p +0 h) jej(juj p=h + j j p + j j p=(p+0h) )^ ( ) for a smooth function ^ ( ). Notethat j j p (j 0 3 j + j 3 j) p (j j p=(p+0h) + j j p=(p+0h) ) ~ ( ) (3.4) where ~ ( ) 0 is a smooth function. Thus e(p +0 h) p0h (u + ( ; )) jej juj p=h + j j p=(p+0h) + j j p=(p+0h) ( ) C ( ) 0: Applying Lemma.6 to the previous inequality yields e(p +0h) p0h (u + ( ; )) jekuj p=h + e +p=(p+0h) a ( ) + +p=(p+0h) + 6 +p=(p+0h) (3.5) for a smooth function a ( ) 0. On the other hand, it can be shown using a similar argument to (3.4) that k 0= +p0h k 0= c (j 0 3 j + j 3 j) +p0h c 3k 0= +p=(p+0h) + k 0= b( ) +p=(p+0h) (3.6) for a smooth function b( ) and nonnegative constants c, c 3. Substituting (3.5) (3.6) into (3.3) yields _V 3 (e) jekuj p=h + +p=(p+0h) + +p=(p+0h) + k0= b( ) 6 + c3k0= + e +p=(p+0h) a ( ) 0 ( r 0 ) k 0(3=4)(0h)=(+p0h) e +p=(+p0h) : (3.7) C. Determination of the Observer Gain L( ) Sincethestate is not measurable, the state feedback controller (3.8) is not directly implementable. To get an implementable controller, wereplace in thecontroller (3.8) by (z + L( )) h=(p+0h) : This leads to u = 0 ^ p+0h + ( ) (p+0h=p) h=(p+0h) ( ) = 0 ( 0 e) h=(p+0h) ( ): (3.8) Note that under this controller (3.8), jekuj p=h in (3.7) can beestimated as follows: jekuj p=h cjej(j j p=(p+0h) + jej p=(p+0h) ) p=h ( ) a ( )e +p=(+p0h) + 6 +p=(+p0h) (3.9) for a smooth function a ( ) 0. Substituting (3.9) into (3.7) yields _V 3 0 ( r 0 )k 0(3=4)(0h=+p0h) 0 e +p=(p+0h) k0= c 3 +p=(p+0h) + + k0= b( ) i= a i ( ) +p=(p+0h) : (3.0) On the other hand, under the new controller (3.8), inequality (3.9) is no longer true. Instead, we have _V j u=u( ;^ ) 0 +p=(p+0h) 0 +p=(p+0h) + 0=(p+0h) h=(p+0h) 0 ( 0 e) h=(p+0h) ( ): By Lemmas.5 and.6 _V j u=u( ;^ ) 0 +p=(p+0h) 0 +p=(p+0h) + 0h=(p+0h) j j 0=(p+0h) jej h=(p+0h) ( ) 0 +p=(p+0h) 0 +p=(p+0h) + 6 +p=(p+0h) for a smooth function a 3 ( ) 0. Putting (3.0) and (3.) together gives _V j u=u( ;^ ) + _ V 3 (e) + a 3 ( )e +p=(p+0h) (3.) 0 ( r 0 )k 0(3=4)(0h=+p0h) 0 3 i= a i ( ) e +(p=p+0h) 0 0 k0= b( ) +p=(p+0h) 0 0 k0(=) c 3 +p=(p+0h) : (3.) Obviously, thechoiceof k( ) such that and k( ) > maf4b( ); 4c 3; g ( r 0 )k 0(3=4)(0h=+p0h) a ( ) + a ( )+a 3( )+ 4

5 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 6, JUNE renders _V j u=u( ;^ ) + _ V 3 (e) 0 4 +p=(p+0h) Notethat by Lemma.5, wehave Hence s p+0h 0 3(p+0h) + +p=(p+0h) + e +p=(p+0h) : (3.3) 0=(p+0h) ds (p0h)=(p+0h) j j : +p=(p+0h) + +p=(p+0h) + e +p=(p+0h) + c (+p=(p+0h))= +(e ) (+p=(p+0h))= V (+p=(p+0h))= + V (+p=(p+0h)= 3 c (V + V 3) (+p=(p+0h))= for a positiveconstant c >0. With this in mind, it can be deduced from (3.3) that _V + _ V 3 + c 4 (V + V 3 ) (+p=(p+0h))= 0: Therefore, by Theorem. (let V = V + V 3 and = ( + p=(p + 0 h))=, the closed-loop system (.) (3.0) (3.8) is globally finite-time stable. Moreover, for a given initial condition 0 thesettling timet can be estimated as T V ( 0 ) 0 =((c=4)( 0 )). Remark 3.: The output feedback control scheme proposed in this note differs from the one in [4] in two aspects: ) Although it is inevitable that both state feedback controllers are nonsmooth due to the intrinsic obstacle caused by uncontrollable unstable linearization, the structure of the controller (3.8) is very different from the asymptotic state feedback stabilizer in [4]; and ) in [4], the observer is still smooth although state feedback controller is nonsmooth. It is known that C system has unique solution which conversely implies impossibility of finite-time convergence []. Therefore, to guarantee the finite-time convergence of the error dynamic, we introduce and implement a novel one-dimensional observer (3.0) that has a nonsmooth structure. Remark 3.3: As pointed in the preceding remark, the observer has to be nonsmooth to guarantee the finite-time convergence of the observer. However, the nonsmoothness of the observer will make the stability analysis of the design process very difficult. To overcome the difficulty, a new tool, namely Lemma.7, is introduced to help us rigorously select a gain that guarantees the finite-time convergence of the observer. In what follows, we show that Theorem 3. can be used to design an output feedback controller globally stabilizing the following system with uncontrollable unstable linearization in finite time. Eample 3.4: Consider the well-known benchmark system [0] _ = 3 + _ = u: (3.4) Due to the presence of the uncontrollable unstable linearization, the stabilization of (3.4) is very challenging. The state feedback stabilization has been solved using homogeneous controller (see [5] and [0]). If weassumey =, the problem of output feedback stabilization of (3.4) is more difficult because of the unobservable linearization of the system. Recently, in [4] a nonsmooth output feedback controller was eplicitly constructed to globally stabilize the system. However, the controller in [4] is only an asymptotic stabilizer instead of a finite-time one. With the help of Theorem 3., now we are able to stabilize the system in finite-time using output feedback since ( )= satisfies Assumption.3 as illustrated in Remark.4. The net eample illustrates that our design method is applicable to nonhomogeneous system. Eample 3.5: Consider the following planar system: _ = 3 + e ; _ = u +ln + ; y = : (3.5) Apparently, (3.5) is not a homogeneous system. However, it is easy to verify that Assumption.3 is satisfied. Specifically, by mean-value theorem (with z = 7=9 ) In addition ( ; ) = ln( + z 8=7 ) 8 7 jzj j j h 7 ; h = 7 9 : =7 + 8=7 [0;jzj] j ( )j = j e jj j 7=9 ( ) ( )= + e : Therefore, by Theorem 3., (3.5) is globally stabilizable in finite time by output feedback. Remark 3.6: For thesakeof simplicity of theproof, weassumethat h in Assumption.3 is a rational number with odd numerator and odd denominator. This requirement can be removed by replacing function s a with sign(s)jsj a for any real number a>0. For instance, the new observer for any real number 0 <h< will be _z sign(z + L( ))jz + L( )j p=(p+0h) + ( ) : Using this new function in the controller and observer, similar finitetime stabilization result can be achieved under Assumption.3 for any real number h (0; ). IV. SPECIAL CASE FEEDBACK LINEARIZABLE SYSTEMS IN THE PLANE When p =, the planar system (.) reduces to the following feedback linearizable system: _ = + ( ); _ = u + ( ; ); y = (4.) which is comprised of a double integrator perturbed by a lower triangular nonlinear vector field. In this case, Theorem 3. has the following [z + L( )] =(0h) + ( ) u=u ; (z + L( )) =(0h) ; for a C function L( ) (4.)

6 890 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 6, JUNE 005 interesting consequence which leads to a global solution to the problem of output feedback finite-time stabilization of (4.). Corollary 4.: Under Assumption.3 with p =, there is a continuous output feedback controller of the form shown in (4.) at the bottom of the previous page, rendering (4.) globally finite-time stable. Proof: Apparently, (4.) is a special case of (.) when p = and, therefore, Corollary 4. is a direct consequence of Theorem 3.. This result is an etension of the previous work [9] where an output feedback finite-time stabilizer was eplicitly constructed for the double integrator system (i.e., i() =0). Reference [9] also showed that local output feedback finite-time stabilization of (4.) was achievable. As an etension, Corollary 4. provides a sufficient condition under which the nonlinear planar system (4.) can be globally stabilized in finite time by output feedback. Moreover, the design method in this note is quite different from the homogeneous design in [9] and can be used to deal with nonhomogeneous systems. In what follows, weusean eampleto illustratethat Corollary 4. can be used to deal with nonhomogeneous nonlinear systems. Eample 4.: Consider the following uncertain planar system: _ = _ = u + 7=9 sin(u); y = : (4.3) Due to the presence of sin(u) in ( ;u), (4.3) is not homogeneous. On the other hand, it is easy to verify that Assumption.3 holds since ( )=0, ( )=, and h =(7=9). By Corollary 4., (4.3) is globally finite-time stabilizable by output feedback. In fact, the output feedback controller can be constructed as follows: _z = 0 c (z + c ) 9= u = 0 7:87 z + c = [3], Finite-time stability of continuous autonomous systems, SIAM J. Control Optim., vol. 38, no. 3, pp , 000. [4] R. W. Brockett, Asymptotic stability and feedback stabilization, in Differential Geometric Control Theory, R. W. Brockett, R. S. Millman, and H. J. Sussmann, Eds. Boston, MA: Birkäuser, 983, pp [5] W. P. Dayawansa, C. F. Martin, and G. Knowles, Asymptotic stabilization of a class of smooth two dimensional systems, SIAM J. Control Optim., vol. 8, no. 6, pp , 990. [6] V. T. Haimo, Finite-time controllers, SIAM J. Control Optim., vol. 4, no. 4, pp , 986. [7] G. Hardy, J. E. Littlewood, and G. Pólya, INEQUALITIES, nd ed. Cambridge, U.K.: Cambidge Univ. Press, 99. [8] Y. Hong, Finite-time stabilization and stabilizability of a class of controllablesystems, Syst. Control Lett., vol. 46, no. 4, pp. 3 36, 00. [9] Y. Hong, J. Huang, and Y. Xu, On an output feedback finite-time stabilization problem, IEEE Trans. Autom. Control, vol. 46, no., pp , Feb. 00. [0] M. Kawski, Homogeneous stabilizing feedback laws, Control Theory Adv. Technol., vol. 6, no. 4, pp , 990. [] C. Qian and W. Lin, A continuous feedback approach to global strong stabilization of nonlinear systems, IEEE Trans. Autom. Control, vol. 46, no. 7, pp , Jul. 00. [], Non-Lipschitz continuous stabilizers for nonlinear systems with uncontrollableunstablelinearizatiion, Syst. Control Lett., vol. 4, no. 3, pp , 00. [3], Smooth output feedback stabilization of planar systems without controllable/observable linearization, IEEE Trans. Autom. Control, vol. 47, no., pp , Dec. 00. [4], Nonsmooth output feedback stabilization of a class of genuinely nonlinear systems in the plane, IEEE Trans. Autom. Control, vol. 48, no. 0, pp , Oct [5] E. R. Rang, Isochrone families for second order systems, IEEE Trans. Autom. Control, vol. 8, no. AC-, pp , Jan [6] X. Xia and W. Gao, On eponential observers for nonlinear systems, Syst. Control Lett., vol., no. 4, pp , 988. for a largeenough constant c>0. V. CONCLUSION This note has addressed the problem of global finite-time stabilization via output feedback for a class of planar nonlinear systems. A sufficient condition has been established for the solvability of the problem. Moreover, a nonsmooth design methodology is presented for the eplicit constructions of a reduced-order observer and a state feedback controller, which render the closed-loop systems globally finite-time stable. It is worthwhile pointing out that the design scheme proposed in this note, namely feedback domination, is quite different from the homogeneous methods used in most of the eisting results. As a special case, a global result on the output feedback finite-time stabilization of the double integrator perturbed by nonlinear functions is also presented in this note. Due to the use of the feedback domination design, we are able to achieve global result for a wider class of planar systems, which is not necessarily homogeneous. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for many useful suggestions. REFERENCES [] S. P. Bhat and D. S. Bernstein, Finite-time stability of homogeneous systems, in Proc. Amer. Control Conf., 997, pp [], Continuous finite-time stabilization of the translational and rotational doubleintegrators, IEEE Trans. Autom. Control, vol. 43, no. 5, pp , May 998. Solving a Dynamic Resource Allocation Problem Through Continuous Optimization Yingdong Lu Abstract A class of dynamic resource allocation problems with infinite planning horizon are studied. We observe special structures in the dynamic programming formulation of the problem, which enable us to convert it to continuous optimization problems that can be more easily solved. Structural properties of the problems are discussed, and eplicit solutions are given for some special cases. Inde Terms Dynamic programs, knapsack problems, optimization. I. INTRODUCTION Suppose that we have a fied number of units W of certain resource availablefor demands that arriveat timeepochs t = ; ;... The composition of each demand arrival can be described by a random pair (P; Q), which represents the price and quantity. Random variable (P; Q) follows joint probability distribution f j (p), where j refers to thedemand sizeq j and p the price. Also let F j and F c j denote the cdf and ccdf of thepricegiven that thedemand sizeis q j. Manuscript received June, 003; revised July, 004 and February 4, 005. Recommended by Associate Editor D. Li. The author is with the IBM T. J. Watson Research Center, Yorktown Heights, NY 0598 USA ( yingdong@us.ibm.com). Digital Object Identifier 0.09/TAC /$ IEEE