Nonlinear Control of the Viscous Burgers Equation: Trajectory Generation, Tracking, and Observer Design

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1 Miroslav Krstic Professor Department of Mechanical & Aerospace Engineering, University of California, San Diego, La Jolla, CA Lionel Magnis Ecole des Mines de Paris, 6 Boulevard Saint Michel, 756 Paris, France lionel.magnis@ensmp.fr Rafael Vazquez Assistant Professor Departmento de Ingeniería Aeroespacial, Universidad de Sevilla, Avenida, de los Descubrimientos s.n., 49 Sevilla, Spain rvazquez@ucsd.edu Nonlinear Control of the Viscous Burgers Equation: Trajectory Generation, Tracking, and Observer Design In a companion paper we have solved the basic problem of full-state stabilization of unstable shock-like equilibrium profiles of the viscous Burgers equation with actuation at the boundaries. In this paper we consider several advanced problems for this nonlinear partial differential equation (PDE) system. We start with the problems of trajectory generation and tracking. Our algorithm is applicable to a large class of functions of time as reference trajectories of the boundary output, though we focus in more detail on the special case of sinusoidal references. Since the Burgers equation is not globally controllable, the reference amplitudes cannot be arbitrarily large. We provide a sufficient condition that characterizes the allowable amplitudes and frequencies, under which the state trajectory is bounded and tracking is achieved. We then consider the problem of output feedback stabilization. We design a nonlinear observer for the Burgers equation that employs only boundary sensing. We employ its state estimates in an output feedback control law, which we prove to be locally stabilizing. The output feedback law is illustrated with numerical simulations of the closed-loop system. DOI:.5/.338 Introduction The viscous Burgers equation is considered a basic model of nonlinear convective-diffusive phenomena such as those that arise in Navier Stokes equations. We study several nonlinear control problems for the system u t = u u u with boundary conditions u,t = t, u,t = t where t and t are the controls and u,t is the state variable, for,. To save space, we drop the arguments,t whenever the contet allows. In a companion paper we studied the problem of stabilization of a family of stationary solutions called shock profiles or shocklike profiles, given by U = tanh / 3 which are parametrized by, unstable in open loop, i.e., with constant boundary conditions = = tanh / 4 and not stabilizable even locally by simple means, such as the radiation boundary conditions see Refs. 3 9 for the uses of nonlinear radiation boundary conditions for global asymptotic stabilization of neutrally stable equilibria of the Burgers equation and other PDEs. Let us denote the perturbation variable around the shock profile as ũ,t=u,t U, and t= t U, t= t U. The Burgers equation written in ũ is ũ t = ũ Uũ Uũ ũ ũ with boundary controls Corresponding author. Contributed by the Dynamic Systems, Measurement, and Control Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CON- TROL. Manuscript received January 4, 8; final manuscript received August, 8; published online February 9, 9. Assoc. Editor: Fen Wu. 5 ũ,t = t, ũ,t = t 6 In Ref. we presented a full-state boundary feedback law for stabilization of the shock-like unstable profiles of the Burgers equation and provided an estimate of the region of attraction for the closed-loop system. This estimate is finite, which is consistent with the fact that the Burgers system is not globally controllable. In this paper we first present results for trajectory generation and tracking for the Burgers equation. Our design is based on a nonlinear spatially-scaled Hopf Cole style transformation which is also used in the full-state design that transforms the system with the help of one of the two boundary controls into a linear reaction-diffusion PDE with nonlinear boundary conditions. For the resulting linear PDE, the general trajectory generation problem has been solved in Ref.. However, in this paper we introduce eplicit solutions to this problem for a particular class of reference functions of time. While we do not achieve a global result due to the lack of global controllability mentioned above, for the case of tracking a sinusoid in time we give a bound that quantifies the trade-off between the maimum amplitudes and frequencies for which tracking is achieved. Then we turn our attention to an output-feedback problem. In Ref. output feedback for a marginally stable D Burgers equation with in-domain actuation was solved using nonlinear model reduction techniques. Our design of a nonlinear observer with gains computed using the backstepping observer design method 3 uses injection of the output estimation error from the boundaries into the interior of the PDE domain. We combine the observer with the full-state feedback design in Ref.. The resulting output feedback is fully collocated and decentralized, as in the case with radiation boundary conditions. However, while our feedback at one of the boundaries is static like the radiation feedback, at the other boundary it is dynamic and nonlinear.we prove the eponential stability of the linearized closed-loop system. The stabilization properties of the observer-based feedback laws are illustrated in simulations. Journal of Dynamic Systems, Measurement, and Control MARCH 9, Vol. 3 / - Copyright 9 by ASME

2 Full-State Stabilization of the Shock Profiles of the Burgers Equation This section summarizes the result in Ref.. In Ref. we designed the following full-state feedback law for achieving stabilization of the shocklike equilibria: = tanh/ũ + ũ / = ũ + + k, tanh/ũ k,y + tanh/k,y Gye y ũd ũydy 8 where cosh / G = 9 cosh/ and the kernel k,y in Eq. 8 is computed from k = k yy + tanh y / + tanh /k + ck k, = c tanh / + tanh/ 7 k y, = tanh/k, which is a linear hyperbolic PDE in the domain T=,y:y. The kernel k can be computed from Eqs. numerically or symbolically using procedures outlined in Ref. 4. The constant c in Eqs. and is a design parameter that is chosen positive if = and can be set to otherwise. 3 Trajectory Generation Given systems and, with the two inputs t and t, we consider a trajectory generation problem with u,t as the system s single output. Then, the problem of trajectory generation consists of finding open-loop control input functions r t, and r t to make u,t evolve eactly according to a given reference signal u r,t. We use the invertible transformation v,t = u,te / uy,tdy u,t = v,t vy,tdy which converts the Burgers system into the form v t,t = v,t + v,t t v,t v,t = t v,t v,t = vy,tdy t v,t vy,tdy From Eq. 3 we obtain v,t=u,t, hence we get that v r,t=u r,t. Thus the trajectory generation for the u-system can be approached as a trajectory generation problem for the v-system 5. Then we are looking for functions v r,t, r t, and r t that satisfy Eqs. 5 7 and v r,t=u r,t. We choose the control at = as r t = ur,t 8 which, substituted into Eq. 5, simplifies the nonlinear trajectory generation problem to the trajectory generation problem for the linear heat equation with nonlinear boundary conditions v r t,t = v r,t v r,t = v r,t = v r y,tdy r t v r,t v r y,tdy 9 A general infinite-series solution for v r,t in Eqs. 9 and eists for a very broad class of functions of time u r,t the Gevrey class, which has been developed in the framework of differential flatness. Furthermore, an eplicit solution can be derived for any function u r,t that can be written as an output of a linear eosystem. For eample, if u r,t = b + a sin t i.e., we want to track a sinusoid with a bias, then the eplicit solution for the reference state is v r,t = b + a Imcoshe jt 3 Once v r,t is found, the input reference r t is computed from Eq. as r t = vr,t + v r y,tdy v r,t 4 v y,tdy r Formula 4 requires that both the derivative and the integral of the state trajectory v r,t be known. In the case of the biased sinusoidal output reference they are easily obtainable as and v r,t = a Im sinh e jt v r y,tdy = a Im sinh e jt 5 + b 6 It can be shown that the following result holds for the particular case of the output given by Eq.. THEOREM. The following functions: b + a Imcoshe jt u r,t = b a r t = b + a sin t 8 je jt Imsinh 7 - / Vol. 3, MARCH 9 Transactions of the ASME

3 r t = b a j Imsinht e b + a Imcoshe jt + b a j Imsinht e a Im sinh e jt 9 ṽ,t = t u,t ṽ,t = vy,tdy t u,t v r,t First we choose the control t as the feedback law t = u,t which changes Eqs. 36 and 37 into ṽ t,t=ṽ,t, ṽ,t =, while Eq. 38 is unchanged. Net, we choose satisfy the nonlinear PDE u r t,t = u r,t u r,tu r,t u r,t = r t 3 3 u r,t = r t 3 and, in particular, u r,t=b+a sin t. Remark 3.. Functions 7 9 that solve the trajectory generation problem do not scale linearly with the amplitude a or the bias b. Furthermore, for values of a or b that are sufficiently large, the possibility eists of these functions taking infinite values for some,t pairs. 4 Trajectory Tracking While the open-loop system might be stable in rare cases around the trajectory found in the trajectory generation problem, usually this is not so. To solve the trajectory tracking problem, we need to find a feedback law that stabilizes the trajectory u r,t from any initial condition u,rather than just generating the desired motion from the special initial condition u r,. We introduce the state tracking error ṽ,t=v,t v r,t. Its linearization of Eqs. 5 7 around the reference trajectory v r,t is ṽ t,t = ṽ,t ur,tv r,tṽ,t ṽ,t = u r,tṽ,t ṽ,t = v r,t v r y,tdy ṽ,t ṽy,tdy v y,tdy vr,t r + v y,tdy r v,t r 35 This complicated linear time-varying PDE system in general will not be eponentially stable, thus we need to develop feedback control laws to stabilize the equilibrium ṽ. The nonlinear PDE governing the tracking error u,t ṽ,t is ṽ t,t = ṽ,t + v,t t 36 t = c u,t + u,t +e / uy,tdy v r,t + c v r,t 4 where c is a positive gain. Using Eqs. 3 and 4, it is found that this control law transforms Eq. 38 into ṽ,t= c ṽ,t. Hence the closed-loop ṽ-system is turned into a heat equation with one Neumann and one stabilizing Robin boundary condition ṽ t,t = ṽ,t ṽ,t = 4 4 ṽ,t = c ṽ,t 43 Using a Lyapunov estimate as in Ref., we find that the closedloop ṽ satisfies the following bound ṽt L ṽ L e c t 44 for some c whose eact value is not important. The solution to the plant state u,t is u,t = v r,t + ṽ,t v r y,tdy ṽy,tdy 45 where ṽ,t is the solution of 4 43 with initial condition ṽ =u,e / uy,dy v r,. Since lim t ṽ,t, we have that u,t converges to u r,t = v r,t v r y,tdy 46 However, the tracking result fails to be global i.e., to hold for all initial conditions because solution 45 is only valid if the condition v r y,tdy ṽy,tdy 47 + is verified for all and t. This condition holds when u r,t, namely, when v r,t which is a basic result of stabilizing u around the origin, which is global, however, it does not necessarily hold in the presence of a nonzero trajectory v r,t. The following theorem describes the behavior of the closedloop system with our tracking controller. THEOREM. Consider system () and () with control laws (39) and (4), where v r,t is a solution of the motion planning problem in Sec. 3. Let the reference trajectory v r,t be bounded for all,, t, and let v r,t be bounded for all t. If the following conditions hold Journal of Dynamic Systems, Measurement, and Control MARCH 9, Vol. 3 / -3

4 v r t L, t 48 4 u L h 4 49 where hr=re r, then the solution u,t is bounded for all, and t, the control inputs t and t are bounded for all t, and the function ũ,t=u,t u r,t converges to zero eponentially in t, for all,. Proof. We guarantee the boundedness of u,t by observing that ṽ,t is bounded as a solution to a heat equation, and by proving that where and v r t L + t L + t L 3 t =,t =,t = c,t = u e / u ydy t =,t =,t = c,t r = v 58 We first note that t L L, t, and t L L, t. We also have that hu L /4 implies that L /. Net, we observe that v r t L 4, t, implies that L /. As discussed in Ref., the solution of the error system ũ,t is defined in the space H,, which is the space of functions of and t whose derivatives from the zeroth up to t have bounded L norms in space-time. We now apply Theorem to our eample with a sinusoidal output Eq. with b= no bias. THEOREM 3. Consider the closed-loop Burgers systems () and () with the controls t = u,t t = c u,t + u,t 59 +e / uy,tdy a Im sinh + c coshe jt 6 If u L h /4 and a a ma = 8 cosh cos 6 where a ma is a positive, decreasing function with a ma =/8, then the state and the control inputs remain bounded and the state u,t converges to a Imcoshe jt u r,t = a Im jt sinhe 6 which means, in particular, that u,t converges to u r,t =a sin t. Proof. All we need to prove is that aim sinhe jt / 63 To do this, we use the fact that the absolute value of an imaginary part of a comple number is no greater than the modulus of the comple number, and that the modulus of a product of two comple numbers is no greater than twice the product of the moduli of the two comple numbers. Hence, Im sinhe jt sinhe j jt sinhe jt 4 sinh =4 cosh cos 64 Taking a derivative, it can be seen that the function in the numerator increases in. Hence, we get Im/ sinh e jt 4 cosh cos /, which proves Eq Simulation Eample of Trajectory Generation and Tracking We illustrate the solution to the trajectory tracking problem for the output reference u r,t=b+a sin t from Theorem. The eplicit state trajectory is u r,t =b + a e / sint + +e / sint b a e 4 / sint 4 + +e / sint 4 65 The presence of the b term in the denominator of Eq. 65 will increase the possibility of a blow-up of the trajectory when b, however b will help to keep the denominator away from. Figure shows the plot of the solution to the nonlinear trajectory generation problem. Figure shows the trajectory tracking starting from a zero initial condition for the PDE state, u,, which is different from the ideal reference initial condition obtained by setting t= in Eq. 65. The speed of response can be further improved by increasing the control gain c or by modifying the feedback law 4 to impart an arbitrarily fast decay, using the backstepping design tools in Ref / Vol. 3, MARCH 9 Transactions of the ASME

5 v = tanh/v + vy Gy dy ũ 7 6 Nonlinear Observer The control in Sec. requires us to know the state ũ,t for all,. We now design output feedback laws for t and t using the boundary measurement of ũ,t and ũ,t only, which is a two-input two-output TITO problem. We will design fully collocated decentralized feedback laws, i.e., using only the measurement of ũ,t for the control t, and only the measurement of ũ,t for t. Notice that the control law 7 is already an output feedback law requiring only the knowledge of ũ. We saw in Ref. that applying Eq. 7 and the mapping which has the inverse t v,t = Gũ,te / ũy,tdy v/g ũ = vy Gy dy transforms the plant into the linear system Fig. Solution to the nonlinear trajectory generation problem for a sinusoidal reference with b=, a=., =8 v t = v + cosh / v v = tanh/v.5 t Fig. Trajectory tracking: dashed line: output reference u r,t =. sin 8t ; and solid line: output response u,t with control gain c =5 Hence the problem reduces to the design of an observer-based feedback controller for Eqs using the measurement of ũ,t. We start with an observation that the boundary condition 7 contains a state of nonlinearity given by the integral term in vy. Our observer is designed as a copy of the nonlinear plant with the injection of the output error vˆ t = vˆ + cosh / vˆ + vˆy vˆ = tanh/vˆ vˆ = tanh/ + Gy dy ũ vˆ vˆy Gy dy ũ vˆ vˆy + Gy dy ũ + tanh/ũ where vˆ,t denotes the estimate of the state v,t. Notice that, using Eq. 67, vy Gy dy ũ = v/g = v 74 since G=. Hence the term vˆy Gy dy ũ 75 appearing in Eqs. 7 73, is an estimate of v and is used for output injection. The gains and are determined to ensure the convergence of vˆ to v. 7 Design of Output Injection Gains Using Backstepping To design and we use the backstepping method for observer design 3. First, we denote the observer error as e,t=v,t vˆ,t. Subtracting Eqs from Eqs we get e t = e + cosh / e vˆ + e ey Gy dy vˆy Gy dy ey Gy dy e e = tanh/e Journal of Dynamic Systems, Measurement, and Control MARCH 9, Vol. 3 / -5

6 e = tanh/ + e ey Gy dy k,y + tanh/k,y vˆydy vˆy Gy dy + k,vˆ + e + dy vˆy Gy dy ey Gy 78 = k, = tanh/ + c 9 Note that in Eq. 9 is the same function as the control gain in Eq. 8, by just changing the sign. Thus, the output injection gains can be computed from the control kernel. 8 Output Feedback Law Using control law 7 and the estimate vˆ which is transformed into an estimate of ũ using Eq. 67 in control law 8, we obtain our nonlinear output feedback control laws, which are defined as follows t = tanh/ũ,t + ũ,t 93 We now linearize Eqs around the origin, obtaining e t = e + cosh / e e 79 e = tanh/e 8 e = tanh/ + e 8 We need to design the gains and so that the system 79 8 is eponentially stable. The plant Eqs is a linear D reaction-diffusion PDE with Robin boundary conditions, so the backstepping observer design method in Ref. 3 can be applied. We map the state e,t into a new variable,t using the transformation e = p,yydy 8 with verifying the observer error target system t = tanh / c 83 = tanh/ 84 = tanh/ 85 The system was shown in Ref. to be eponentially stable in the L norm. Following the method in Ref. 3, we find the kernel p,y appearing in Eq. 8 from p p yy = tanh / + tanh y /p cp 86 p, = tanh / + tanh/ + c 87 p,y = tanh/p,y 88 Once p,y is computed, it is used to compute the output injection gains as follows: = p y, + tanh/p, 89 = p, 9 Comparing Eqs with Eqs., we deduce that p,y=ky,. Hence it is not necessary to solve Eqs and we can use the solution for k in Eqs. 89 and 9, obtaining = k, + tanh/k, 9 t = ũ,t + tanh/ + k,ũ,t k,y + tanh/k,y + vˆy,tdy vˆy,t Gy dy 94 Thus we have obtained a diagonal TITO compensator ũ,t ũ,t t t 95 with one static channel and one dynamic channel. If we linearize the compensator, we obtain the following. For Channel, = tanh/ũ 96 For Channel, = tanh/ + ũ yvˆydy vˆ t = vˆ + cosh / vˆ vˆ + ũ vˆ = tanh/vˆ vˆ = tanh/ + vˆ yvˆydy The dynamic channel Channel is of relative degree zero note the throughput term in Eq. 97 and one can think of it as an infinite dimensional transfer function whose Bode plot is shown in Fig. 3 for =5. 9 Stability of the Linearized Output-Feedback System THEOREM 4. Consider the linearized ũ,vˆ system given by ũ t =ũ + tanh /ũ + cosh /ũ and Eqs. (6) and (96) (). The equilibrium ũvˆ is eponentially stable in the L norm, i.e., there eists C,c such that ũt L +vˆt L C e c t ũ L +vˆ L, for all t. Proof. We just outline the proof. It is lengthy but it follows the standard constructions for linear reaction-diffusion systems stabilized by output feedback using the backstepping method see -6 / Vol. 3, MARCH 9 Transactions of the ASME

7 u(,t) Frequency (rad/s) 5.5. t.4 Fig. 3 =5 The Bode plot of the compensator Eqs. 97 for Fig. 5 Convergence of the closed-loop system under the linear output-feedback controller Eqs. 96 for =5 Refs. 3,5,6 for details. It starts by considering the linearized e,vˆ system given by Eqs and vˆ t=vˆ + /cosh / vˆ +e, with Eqs. 99 and. The key element in the proof is to use the invertible controller and observer backstepping transformations, ŵ,t =vˆ,t k,yvˆy,tdy and Eq. 8, to study the ŵ, system instead of the vˆ,e system. The system is an eponentially stable autonomous system, whereas the ŵ system not given here; its autonomous part is identical to Eqs is eponentially stable and driven by. Using the Lyapunov function ŵ +M, for sufficiently large M, we obtain eponential stability for the ŵ, system. Using the direct and inverse backstepping transformations we get eponential stability estimates for vˆ,e. Using the fact that v=vˆ +e, we get estimates for vˆ,v. Finally applying the origin-preserving transformation ũ,t =v,t/g, we get the estimates for vˆ,ũ stated in Theorem 4. Simulations With Output Feedback All simulations in this section have been produced using a mied Cranck Nicholson/Runge Kutta three scheme in time and a second order finite difference method in space. The open-loop system,, and 4 is unstable, as shown in Ref.. A numerical study of the linearized system around the shock profile shows the presence of one positive though possibly small eigenvalue for any. In Fig. 4 one can see an apparent finite time u(,t) Fig. 4 Simulation of the open-loop system with constant inputs, Eq. 4 for =3, and u =U ++ U U 4 appears to ehibit a finite-time blow-up t blow-up of the open-loop system for =3. For the same initial conditions, we show in Fig. 5 the numerical evolution of the system with the linear observer-based backstepping controller in Sec. 6. This result is achieved for c=; with c the hump in the transient can be eliminated which might be preferable mathematically, though not as interesting visually. Conclusions We have solved the problems of motion planning, trajectory generation, observer design, and output feedback stabilization for the fully nonlinear viscous Burgers equation. The problem of fullstate stabilization was solved in a companion paper. Our results are based on a nonlinear feedback linearizing transformation, which allows us to use trajectory generation for the heat equation, the linear backstepping control design method 4, and observer design method 3. Due to the eplicit nature of the transformation and the design methods we use, we were able to derive formulas for the reference trajectories, feedback laws, and observer gains. Since our nonlinear transformation is not globally invertible, our results are not global, which is consistent with the lack of global controllability shown in Ref.. Our result does not apply to the inviscid Burgers equation, which is a completely different problem, with well posedness considerations that may, as time and state evolve, allow one, two, or no boundary conditions/controls. In our stabilization efforts, such as in Secs. 6 9, we focus on the equilibrium 3. Actually many other monotonically increasing profiles, which are the solutions of the equilibrium problem u u u=, are allowed. We consider only the symmetric problem for notational simplicity, conceptual clarity, and due to the fact that this is actually the most difficult equilibrium shape from the stabilization point of view the shock, which is the cause of instability, is the furthest away from both boundary actuators. Dirichlet actuation is crucial for applications in flow control. Unfortunately, the present result does not etend from Neumann to Dirichlet actuation. The actuation at = can be changed to Dirichlet, but not the actuation at =, which has to be of Neumann type. Every single one of our previous designs for backstepping boundary control could be interchangeably implemented through either Dirichlet or Neumann, but not this one. Interesting problems for future research include a design using only one control input,, etensions to more general parabolic PDEs with convective nonlinearities, to convective nonlinearities of a more general form, to Burgers equations in higher dimensions, and to other PDEs with convective nonlinearities such as Kuramoto Sivashinsky and Navier Stokes. Journal of Dynamic Systems, Measurement, and Control MARCH 9, Vol. 3 / -7

8 Acknowledgment The work was supported by NSF. References Krstic, M., Magnis, L., and Vazquez, R., 8, Nonlinear Stabilization of Shock-Like Unstable Equilibria in the Viscous Burgers PDE, IEEE Trans. Autom. Control, 53, pp Engelberg, S., 998, The Stability of the Shock Profiles of the Burgers Equation, Appl. Math. Lett., 5, pp Balogh, A., and Krstic, M.,, Burgers Equation With Nonlinear Boundary Feedback: H Stability, Well Posedness, and Simulation, Math. Probl. Eng., 6, pp Balogh, A., and Krstic, M.,, Boundary Control of the Korteweg-de Vries-Burgers Equation: Further Results on Stabilization and Numerical Demonstration, IEEE Trans. Autom. Control, 45, pp Krstic, M., 999, On Global Stabilization of Burgers Equation by Boundary Control, Syst. Control Lett., 37, pp Liu, W.-J., and Krstic, M.,, Backstepping Boundary Control of Burgers Equation With Actuator Dynamics, Syst. Control Lett., 4, pp Liu, W.-J., and Krstic, M.,, Stability Enhancement by Boundary Control in the Kuramoto Sivashinsky Equation, Nonlinear Anal.: Real World Appl., 43, pp Liu, W.-J., and Krstic, M.,, Adaptive Control of Burgers Equation With Unknown Viscosity, Int. J. Adapt. Control Signal Process., 5, pp Liu, W.-J., and Krstic, M.,, Global Boundary Stabilization of the Korteweg de Vries Burgers Equation, Comput. Appl. Math.,, pp Fernandez-Cara, E., and Guerrero, S., 7, Null Controllability of the Burgers Equation With Distributed Controls, Syst. Control Lett., 56, pp Laroche, B., Martin, P., and Rouchon, P.,, Motion Planning for the Heat Equation, Int. J. Robust Nonlinear Control,, pp Baker, J., Armaou, A., and Christofides, P. D.,, Nonlinear Control of Incompressible Fluid Flow: Application to Burgers Equation and D Channel Flow, J. Math. Anal. Appl., 5, pp Smyshlyaev, A., and Krstic, M., 5, Backstepping Observers for Parabolic PDEs, Syst. Control Lett., 54, pp Smyshlyaev, A., and Krstic, M., 4, Closed Form Boundary State Feedbacks for a Class of Partial Integro-Differential Equations, IEEE Trans. Autom. Control, 49, pp Krstic, M., Guo, B.-Z., Balogh, A., and Smyshlyaev, A., 8, Output- Feedback Stabilization of an Unstable Wave Equation, Automatica, 44, pp Vazquez, R., and Krstic, M., 7, Eplicit Output Feedback Stabilization of a Thermal Convection Loop by Continuous Backstepping and Singular Perturbations, Proceedings of the 7 American Control Conference, pp / Vol. 3, MARCH 9 Transactions of the ASME