Compensation of Wave PDEs in Actuator Dynamics for Extremum Seeking Feedback

Available online at www.sciencedirect.com

ScienceDirect IFAC PapersOnLine 52-29 (2019) 134–139

Compensation of Wave PDEs in Actuator Compensation of Wave PDEs in Actuator Compensation of Wave PDEs in Actuator Dynamics for Extremum Seeking Feedback Compensation of Wave PDEs in Actuator Dynamics for Extremum Seeking Dynamics for Extremum Seeking Feedback Feedback Dynamics for Extremum Seeking Feedback Tiago Roux Oliveira ∗∗ Miroslav Krstic ∗∗ ∗∗

Tiago Roux Oliveira ∗∗ Miroslav Krstic ∗∗ Tiago Roux Oliveira ∗ Miroslav Krstic ∗∗ ∗∗ ∗ Tiago Roux Oliveira Miroslav Krstic of Electronics and Telecommunication Engineering, State ∗ ∗ Department Department of Electronics and Telecommunication Engineering, State ∗ University of Rio de Janeiro (UERJ), Rio de Janeiro – RJ, Brazil Department of Electronics and Telecommunication Engineering, State ∗ University of Rio de Janeiro (UERJ), Rio de Janeiro – RJ, Brazil Department of Rio Electronics and Telecommunication Engineering, State [email protected]) University of de(e-mail: Janeiro (UERJ), Rio de Janeiro – RJ, Brazil (e-mail: [email protected]) ∗∗ University of Rio de Janeiro (UERJ), Rio de Janeiro – RJ, Brazil Aerospace Engineering, University of (e-mail:and [email protected]) ∗∗ ∗∗ Department of Mechanical of Mechanical and Aerospace University of ∗∗ Department (e-mail: [email protected]) California - San Diego (UCSD), La Engineering, Jolla – CA, USA Department of Mechanical and Aerospace Engineering, University of ∗∗ California San Diego (UCSD), La Jolla – CA, USA Department of Mechanical and Aerospace University of (e-mail: [email protected]) California - San Diego (UCSD), La Engineering, Jolla – CA, USA (e-mail: [email protected]) California - San [email protected]) (UCSD), La Jolla – CA, USA (e-mail: (e-mail: [email protected]) Abstract: Abstract: Gradient Gradient extremum extremum seeking seeking for for compensating compensating wave wave actuator actuator dynamics dynamics in in cascade cascade with static scalar maps is addressed in the present paper. This class of Partial Differential Abstract: Gradient extremum seeking for compensating wave actuator dynamics in cascade with static scalar maps is addressed in the present paper. This class of Partial Differential Abstract: Gradient seeking wave actuator in cascade Equations forextremum extremum seeking has not been paper. studied yet. A dynamic feedback control with static(PDEs) scalar maps is addressed in for thecompensating present This class ofdynamics Partial Differential Equations (PDEs) for extremum seeking has not been studiedbackstepping yet. A dynamic feedback control with static scalar maps is addressed in the present paper. This class of Partial Differential law based on distributed parameters is proposed by employing transformation with Equations (PDEs) for extremum seeking has not been studied yet. A dynamic feedback control law based on distributed parameters is proposed employing backstepping transformation with Equations (PDEs) for extremum seeking has notby been studiedusing yet. A dynamic feedback control an appropriate target system and an adequate formulation Neumann interconnections. law based on distributed parameters is proposed by employing backstepping transformation with an appropriate target system and an adequate by formulation using Neumann interconnections. law based on distributed parameters is proposed employing backstepping transformation with Local stability and convergence to a an small neighborhood of theusing desired (but unknown) extremum an appropriate target system and adequate formulation Neumann interconnections. Local stability and convergence to a small neighborhood of the desired (but unknown) extremum an appropriate target system and an adequate formulation using Neumann interconnections. is proved by means of a Lyapunov and the theory averaging infinite dimensions. Local stability and convergence to afunctional small neighborhood of theofdesired (butinunknown) extremum is proved bysimulations means of aillustrate Lyapunov andresults. the theory averaging infinite dimensions. Local stability and convergence tothe afunctional small neighborhood of theof (butin extremum Numerical theoretical is proved by means of a Lyapunov functional and the theory ofdesired averaging inunknown) infinite dimensions. Numerical theoretical is proved bysimulations means of aillustrate Lyapunovthe functional andresults. the theory of averaging in infinite dimensions. Numerical simulations illustrate the theoretical results. © 2019, IFAC (International Federation Automatic results. Control) Hosting by Elsevier Ltd. AllAveraging rights reserved. Numerical simulations illustrate the oftheoretical Keywords: Keywords: Adaptive Adaptive Control, Control, Extremum Extremum Seeking, Seeking, Partial Partial Differential Differential Equation, Equation, Averaging Theory, Backstepping in Infinite Dimensions. Keywords: Adaptive Control, Extremum Seeking, Partial Differential Equation, Averaging Theory, Backstepping in Infinite Dimensions. Keywords: Adaptive Control, Extremum Seeking, Partial Differential Equation, Averaging Theory, Backstepping in Infinite Dimensions. Theory, Backstepping in Infinite Dimensions. 1. eigenvalues 1. INTRODUCTION INTRODUCTION eigenvalues are are on on the the imaginary imaginary axis, axis, and and due due to to the the fact fact that it has are a finite (limited) speed of and propagation (large 1. INTRODUCTION eigenvalues on the imaginary axis, due to the fact that it has a finite (limited) speed of propagation (large 1. INTRODUCTION eigenvalues onhelp) the imaginary axis, due to the fact control doesare (Krsti´cspeed , 2009). that it has anot finite (limited) of and propagation (large Extremum control doesanot help) (Krsti´cspeed , 2009).of propagation (large it has finite (limited) Extremum seeking seeking (ES) (ES) has has received received great great attention attention in in that control does not help) , 2009). the control seeking community, recognized one of the Our manuscript is the(Krsti´ first cccontribution of applying ES Extremum (ES) being has received greatasattention in control does not help) (Krsti´ , 2009). the control community, being recognized as one of the Our manuscript is the first contribution of applying ES Extremum seeking (ES) has received great attention in powerful methodologies adaptive systems as to face to infinite-dimensional actuation dynamics governed ES by the control community,inbeing recognized one control of the Our manuscript is the first contribution of applying powerful methodologies in adaptive systems to face control to infinite-dimensional actuation dynamics governed by the control community, recognized one of the Our manuscript is the first contribution of applying ES problems where the plant is poorly modeledas itscontrol model wave PDEs. The problem studied here is more challenging powerful methodologies inbeing adaptive systems toor face to infinite-dimensional actuation dynamics governed by problems where the plant isuncertainties poorly modeled its model to wave PDEs. The problem studied here isal., more challenging powerful methodologies in adaptive toor face infinite-dimensional actuation dynamics governed by is by severe and unmodeled than the case (Feiling et 2018; Oliveira problems where the poorlysystems modeled itscontrol model wave The problem here more challenging is contaminated contaminated by plant severeis uncertainties andor unmodeled than PDEs. the diffusion diffusion case in instudied (Feiling et is al., 2018; Oliveira problems where the plant is poorly modeled or its model wave PDEs. The problem studied here is more challenging dynamics (Krsti´c by andsevere Wang,uncertainties 2000). et al.,the 2018) due tocase another difficulty the2018; PDEOliveira system is contaminated and unmodeled than diffusion in (Feiling et –al., dynamics (Krsti´c by andsevere Wang,uncertainties 2000). et al., 2018) due to time, another difficulty the PDE system is contaminated and unmodeled than diffusion inwhich (Feiling et ––al., is second order means that the Oliveira state is dynamics (Krsti´ c and Wang, 2000). employed et al.,the 2018) due intocase another difficulty the2018; PDE system Although ES has been successfully to many is second order in time, which means that the state is dynamics c and Wang, 2000). employed to many et al., 2018) due in to time, another difficulty – that the displacement PDE system Although (Krsti´ ES has been successfully “doubly infinite dimensional” (distributed is second order which means the state is engineeringESapplications, the authors employed in (Oliveira al., “doubly infinite dimensional” (distributed Although has been successfully to et many secondinfinite order velocity). in time, This which(distributed means much that displacement the problem state is and not of “doubly dimensional” engineering applications, the authors in (Oliveira et al., is Although ESapplications, employed many and distributed distributed velocity). This is is(distributed not so so muchdisplacement of a a problem 2017; Rusiti ethas al., been 2018, successfully 2019) pointed the to presence engineering the authors in out (Oliveira et al., “doubly infinite dimensional” displacement dimensionally, it is aThis problem in much constructing the distributed as velocity). is not so of a problem 2017; Rusiti et al., 2018, the 2019) pointed out the presence engineering applications, authors in out (Oliveira et al., and of as limiting in of dimensionally, as it for is compensating aThis problem inthe constructing the 2017; Rusiti et al., 2018, factor 2019) pointed the presence velocity). is not so much of adynamics problem of delay delay as one one limiting factor in the the application application of ES ES and statedistributed transformations PDE dimensionally, as it is a problem in constructing the 2017; Rusiti et al., 2018, 2019) pointed out the presence in situations. have state transformations for compensating the PDE dynamics of delay as one limitingThese factorpublications in the application of ES dimensionally, as it for is to a deal problem in constructing the in practical practical situations. These publications have provided provided (Krsti´ cc,, 2009). coupling of transformations compensating the PDE dynamics of delay as detailed one limiting factor in application ES state (Krsti´ 2009). One One has has to deal with with the the coupling of two two a full and analysis ofpublications thethe application of of delay in practical situations. These have provided state transformations for compensating the PDE dynamics infinite-dimensional states. (Krsti´ c , 2009). One has to deal with the coupling of two a full and detailed analysis of the application of delay in practical situations. have provided infinite-dimensional states. designsanalysis inThese ES for static and dynamic acompensation full and detailed ofpublications the application of maps. delay (Krsti´ c, 2009). One states. has to deal with the coupling of two compensation designsanalysis in ES for static and dynamic maps. a full and detailed of static the application of delay infinite-dimensional It provides a systematic and effective design for gradient The complete control design compensation designs in ES for and dynamic maps. infinite-dimensional states. It provides a systematic and effective design for gradient The complete control design employing employing aa compensator compensator for for compensation designsalgorithms. in ES for static and dynamic maps. and Newton seeking In particular, only firstthe wave actuation dynamics is developed via backstepIt provides a systematic and effective design for gradient The complete control design employing a compensator for and Newton seeking algorithms. In particular, only firstthe wave actuation dynamics is developed via backstepIt provides a seeking systematic andPartial effective design forEquations gradient complete controldynamics design employing compensator for order hyperbolic transport pingwave transformation by feeding thea estimates for the and Newton algorithms. InDifferential particular, only first- The the actuation isback developed via backsteporder hyperbolic transport Partial Differential Equations ping transformation by feeding back thederivatives) estimates for the and Newton seeking algorithms. In particular, only firstthe wave actuation dynamics is developed via backstep(PDEs) were originally assumed in (Oliveira et al., 2017) gradient and Hessian (first and second of the order hyperbolic transport Partial Differential Equations ping transformation by feeding back the estimates for (PDEs) were originally assumed in (Oliveira et al.,the 2017) gradient and Hessian (first and second of the order hyperbolic transport Partial Differential ping transformation by feeding back thederivatives) estimatesstability for the to delays. This idea has destatic to be Our proofs for (PDEs) werepure originally (Oliveira et Equations al.,the 2017) and (first and of the to represent represent pure delays.assumed This key keyin idea has enabled enabled de- gradient static map map to Hessian be maximized. maximized. Oursecond proofsderivatives) for local local stability (PDEs) were originally assumed in (Oliveira et al., 2017) gradient and Hessian (first and second derivatives) of the velopment extensions other ofenabled PDEs, such as static of the map closed-loop system andOur theproofs convergence tostability a small to representofpure delays. to This key classes idea has the deto be maximized. for local velopment of extensions to other classes of PDEs, such as of the closed-loop system and the convergence to a small to represent delays. to This key classes idea has the deto be maximized. those describing diffusion phenomenon studied in (Feiling neighborhood of the extremum areproofs basedfor on local backstepping velopment ofpure extensions other ofenabled PDEs, such as static of the map closed-loop system andOur the convergence tostability a small those describing diffusion phenomenon studied in (Feiling neighborhood of the extremum are based on backstepping velopment extensions tophenomenon other of PDEs, such as of the closed-loop system and the convergence to a small et al., describing 2018;ofOliveira et al., 2018).classesstudied methodology for PDE control (Krsti´ c and Smyshlyaev, those diffusion in (Feiling neighborhood of the extremum are based on backstepping et al., describing 2018; Oliveira et al., 2018). methodology for PDE control (Krsti´ c and Smyshlyaev, those diffusion studied in (Feiling neighborhood of the extremum are based on backstepping 2008), of aa Lyapunov and et Oliveira et al.,phenomenon 2018). for PDE control (Krsti´ cfunctional and Smyshlyaev, 2008), the the construction construction of Lyapunov functional and the the In al., this2018; paper, we expand the class of PDEs for which methodology et al., 2018; Oliveira et al., 2018). methodology for PDE control (Krsti´ c and Smyshlyaev, use of averaging theoremof for infinite-dimensional systems 2008), the construction a Lyapunov functional and the In this paper, we expand the class of PDEs for which ES can be applied, by considering a wave dynamics in use of averaging theorem for infinite-dimensional systems In this paper, we expand the class of PDEs for which 2008), the Lunel, construction a the Lyapunov and the (Hale 1990). of To best of functional our knowledge, all of and averaging theorem for infinite-dimensional systems ES can with be applied, by considering a wave dynamics in use In this westatic expand themap classto PDEs for which cascade the optimized. The (Hale and Lunel,of 1990). To the best of oursystems knowledge, all ES can paper, be applied, by scalar considering aofbe wave dynamics in use of averaging theorem for infinite-dimensional systems cascade with the static scalar map to be optimized. The of these results distributed parameter for ES (Hale and Lunel, 1990). To the best of our knowledge, all ES can be applied, by considering a wave dynamics in problem with we tackle in the present be inspired of these of distributed parameter systems for ES cascade the static scalar mappaper to bemay optimized. The (Hale andresults Lunel, To the parameter best of oursystems knowledge, all with wave compensation are these of1990). distributed for ES problem we tackle in the present paper may be inspired cascade with the static scalar mappaper torelated bemay optimized. The of with waveresults compensation are novel. novel. by specific applications to problem we engineering tackle in the present be off-shore inspired of these results of distributed parameter systems for ES with wave compensation are novel. by specific engineering applications related to off-shore problem we engineering tackle in theapplications present paper may to be off-shore inspired drilling (Bekiaris-Liberis and Krsti´ c,related 2014) or even its by specific wave compensation are novel. drilling (Bekiaris-Liberis andKrstic, Krsti´c2019), ,related 2014) or off-shore evenrealits with by specific engineering applications to 1.1 Notation and Terminology optimal control (Aarsnes and where the drilling (Bekiaris-Liberis and Krsti´ c , 2014) or even its 1.1 Notation and Terminology optimal control (Aarsnes and Krstic, 2019), where the realdrilling (Bekiaris-Liberis and Krsti´ c2019), , affected 2014) orbythe even its 1.1 Notation and Terminology time optimization approach would be a wave optimal control (Aarsnes and Krstic, where realtime optimization approach would be affected by a wave andpartial Terminology optimal (Aarsnes and would Krstic,The 2019), where realWe Notation denote the derivatives of a function u(x, t) PDE optimization in control the actuation dynamics. wave equation is 1.1 time approach be affected bythe a wave We∂ denote the partial derivatives of=a ∂u(x, function u(x, t) PDE in thedue actuation dynamics. The wave equation is as time optimization approach would be affected by a wave challenging to the fact that all of its (infinitely many) u(x, t) = ∂u(x, t)/∂x, ∂ t)/∂t. The We∂xdenote the partial derivatives function u(x, t) PDE in thedue actuation dynamics. The wave equation is as challenging to the fact that all of its (infinitely many) u(x, t) = ∂u(x, t)/∂x, ∂ttt u(x, u(x, t) t)of =aa ∂u(x, t)/∂t. The x x We denote the partial derivatives of function u(x, t) PDE in the actuation dynamics. The wave equation is challenging due to the fact that all of its (infinitely many) as ∂x u(x, t) = ∂u(x, t)/∂x, ∂t u(x, t) = ∂u(x, t)/∂t. The as ∂x u(x, challenging due IFAC to the(International fact that all of its (infinitely many) t) = ∂u(x, t)/∂x, ∂t u(x, t) = ∂u(x, t)/∂t. The 2405-8963 © 2019, Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2019.12.634



Tiago Roux Oliveira et al. / IFAC PapersOnLine 52-29 (2019) 134–139

2-norm of a finite-dimensional (ODE) state vector ϑ(t) is denoted by single bars, |ϑ(t)|. In contrast, norms of functions (of x) are denoted by double bars. We denote the spatial L2 [0, D] norm of the PDE state u(x, t) as D u(t)2L2 ([0,D]) := 0 u2 (x, t)dx, where we drop the index L2 ([0, D]) in the following, hence  ·  =  · L2 ([0,D]) , if not otherwise specified. As defined in (Khalil, 2002), a vector function f (t, ) ∈ Rn is said to be of order O() over an interval [t1 , t2 ], if ∃k, ¯ : |f (t, )| ≤ k, ∀ ∈ [0, ¯] and ∀t ∈ [t1 , t2 ]. In most cases we give no estimation of the constants k and ¯, then O() can be interpreted as an order of magnitude relation for sufficiently small . 2. PROBLEM FORMULATION 2.1 Basic extremum seeking for static maps In the simplest case of ES for static maps, the goal is to find and maintain the optimum of an unknown nonlinear static map Q(·) with optimal unknown output y ∗ , unknown optimizer θ∗ , measurable output y and input θ. Without loss of generality, we consider maximization problems. The method of sinusoidal perturbation (Krsti´c and Wang, 2000) varies the input parameter θ of the static map to obtain an estimate of the gradient G for the static map. Hence, the additive dither S(t) = a sin(ωt), (1) with amplitude a and frequency ω, is added to the estiˆ The multiplicative mation of the optimizer θ ∗ , given by θ. dither signal to estimate the gradient of the static map is chosen as 2 (2) M (t) = sin(ωt). a The idea of choosing the dither signals as (1) and (2) as ˆ˙ well as the adaptation law θ(t) = kG(t) is to achieve the averaged signal of the gradient estimate G given by Gav (t) = H θ˜av = H(θˆav − θ∗ ), where H is the unknown negative Hessian of the static map and θ˜ = θˆ − θ∗ the estimation error. This yields to the averaged error ˙ dynamics θ˜av = KH θ˜av , with adaption gain K > 0. The average system is exponentially stable and by the averaging theorem in (Khalil, 2002), the original error dynamics is exponential stable with respect to a small residual set. 2.2 Wave actuation dynamics and output signal We consider actuation dynamics which are described by a wave process, i.e., a wave PDE with the actuator θ(t) and the propagated actuator Θ(t) given by (3) Θ(t) = ∂x α(0, t) (4) ∂tt α(x, t) = ∂xx α(x, t), x ∈ [0, D] α(0, t) = 0 (5) ∂x α(D, t) = θ(t), (6) where the domain length D is known. The measurement is defined by the unknown static map with input (3), such that y(t) = Q(Θ(t)). (7) For the sake of simplicity, we assume the following.

θ(t) = αx (D, t)

αtt (x, t) = αxx (x, t)

135

Θ(t) = αx (0, t)

y(t)

Q( · )

M (t) G(t) +

ˆ θ(t)

1 s

U (t)

Wave PDE Compensator

ˆ H(t)

× ×

S(t) N (t)

Fig. 1. Gradient ES with actuation dynamics governed by a wave PDE with compensating controller (40), additive and multiplicative perturbation signals (2), (24) and (26), respectively. Assumption 1. The unknown nonlinear static map is locally quadratic, i.e., H (8) Q(Θ) = y ∗ + (Θ − Θ∗ )2 , 2 ∗ in the neighborhood of the extremum, where besides Θ ∈ R and y ∗ ∈ R being unknown, the scalar H < 0 is the unknown Hessian of the static map. Hence, the output of the static map is given by H (9) y(t) = y ∗ + (Θ(t) − Θ∗ )2 . 2 Combining the above actuation dynamics and the basic ES scheme, further adapting the proposed scheme in (Oliveira et al., 2017), the closed-loop ES with actuation dynamics governed by a wave PDE system under unknown wave compensation controller is shown in Fig. 1. 2.3 System and signals As in the basic ES scheme, we define the unknown optimal input θ∗ of θ(t) with respect to the static map and the wave process, with the relation Θ∗ = θ∗ . Since our goal is to find the unknown optimal input θ∗ , we define the estimation errors ˜ := θ(t) ˆ − θ∗ , (10) θ(t) ∗ ˆ where θ(t) is the estimation of θ . To make (10) consistent with the optimizer of the static map Θ∗ , we introduce the ∗ ˆ through the propagated estimation error ϑ(t) := Θ(t)−Θ wave PDE domain ϑ(t) := ∂x α ¯ (0, t) (11) ¯ (x, t) = ∂xx α ¯ (x, t), x ∈ [0, D] (12) ∂tt α α ¯ (0, t) = 0 (13) ˜ ¯ (D, t) = θ(t). (14) ∂x α From the control loop in Fig. 1 we get ˆ˙ = U (t). θ(t) (15) Taking the time derivative of (11)-(14) and with the help of (10) and (15), the propagated error dynamics is written as the following cascade of a wave PDE and ODE (integrator) with Neumann interconnection (Susto and Krstic, 2010): ˙ = ∂x u(0, t), ϑ(t) (16) ∂tt u(x, t) = ∂xx u(x, t), u(0, t) = 0, ∂x u(D, t) = U (t),

x ∈ [0, D]

(17) (18) (19)

Tiago Roux Oliveira et al. / IFAC PapersOnLine 52-29 (2019) 134–139

136

˜˙ ˆ˙ where θ(t) = θ(t), since θ∗ is constant. As in extremum seeking without actuation through a wave PDE domain, the perturbation signal S(t) should add a sin(ωt) to Θ(t), thus compensates the wave process. Hence, a sin(ωt) with perturbation amplitude a and frequency ω is applied as follows: (20) S(t) := ∂x β(D, t) (21) ∂tt β(x, t) = ∂xx β(x, t), x ∈ [0, D] β(0, t) = 0 (22) ∂x β(0, t) = a sin(ωt). (23) Equations (20)-(23) describe a trajectory generation problem as in (Krsti´c and Smyshlyaev, 2008, Chapter 12). The explicit solution of (20) is given by a (24) S(t) = sin(ωD) sin(ωt). ω The relation among the propagated estimation error ϑ(t), the propagated input Θ(t), and the optimizer of the static map Θ∗ is given by (25) ϑ(t) + a sin(ωt) = Θ(t) − Θ∗ , ˆ + S(t) which can be easily proven, considering θ(t) = θ(t) along with the solutions of (3)-(6), (11)-(14) and (20)-(23). It remains to define the dither signal N (t) which is used to estimate the Hessian of the static map by multiplying it with the output y(t) of the static map. In (Ghaffari et al., 2012), the Hessian estimate is derived as 8 ˆ H(t) = N (t)y(t) with N (t) = − 2 cos(2ωt). (26) a Note that the dither signal M (t), to estimate the gradient, is the same as in the basic ES (see (2)), such that G(t) = M (t)y(t). (27) 3. CONTROLLER DESIGN AND CLOSED-LOOP SYSTEM In this section, we present the proposed filtered boundary control with perturbation-based (averaging-based) estimates of the gradient and Hessian used for wave compensation in the closed-loop extremum seeking feedback of Fig. 1. 3.1 Wave compensation via Hessian’s estimation

obtained by evaluating the backstepping transformation (28) at x = D as   ¯ U (t) = c¯ Ku(D, t) − ∂t u(D, t) + ρ(D)ϑ(t)  D + ρ(D − σ)∂t u(σ, t)dσ , (35) 0

¯ where ρ(s) = K[0 I]eAs [0 I]T . However, the proposed control law in (35) is not applicable directly, because we have no measurement on ϑ(t). Thus, we introduce an important result of (Ghaffari et al., 2012): the averaged version of the gradient (27) and Hessian estimate (26) are ˆ av (t) = H, Gav (t) = Hϑav (t), H (36) if a quadratic map as in (8) is considered. For the proof of (36), see (Ghaffari et al., 2012). Regarding (36), we average ¯ = KH with K > 0, such that (35) and choose K Uav (t) = c¯ [KHuav (D, t) − ∂t uav (D, t)] + ρ¯(D)KHϑav (t)  D + KH ρ¯(D − σ)∂t uav (σ, t)dσ , (37) 0

with

ρ¯(s) = [0 I]eAs [0 I]T ,

A=



0 0 I 0



.

(38)

By plugging the averaged estimate (36) into (37), we obtain Uav (t) = c¯ [KHuav (D, t) − ∂t uav (D, t)] + ρ¯(D)KGav (t)  D + KH ρ¯(D − σ)∂t uav (σ, t)dσ , (39) 0

Due to technical reasons in the application of the averaging theorem for infinite-dimensional systems (Hale and Lunel, 1990) in the following stability proof, we introduce a lowpass filter to the controller. Finally, we get the averagebased infinite-dimensional control law to compensate the wave process:    c ˆ U (t) = c¯ K H(t)u(D, t)−∂t u(D, t) + ρ¯(D)KG(t) s+c   D ˆ +K H(t) ρ¯(D − σ)∂t u(σ, t)dσ , (40) 0

We consider the PDE-ODE cascade (16)-(19). As in (Susto and Krstic, 2010), we use the backstepping transformation  x l(x, σ)ut (σ, t)dσ − γ(x)ϑ(t), (28) w(x, t) = u(x, t) − 0

with the gain kernels l(x, σ) = γ(x − σ) ,

¯ γ(x) = K[0 I]eAx [I 0]T ,

A=



0 0 I 0



,

where c > 0 is chosen later. For notational convenience, we mix the time and frequency domain in (40), where the low-pass filter acts as an operator on the term between braces.

(29)

3.2 Closed-loop system

(30)

Substituting (40) into (19), we can write the closed-loop system (16)–(19) as

which transforms (16)-(19) into the target system ˙ = Kϑ(t) ¯ < 0, ¯ (31) ϑ(t) + wx (0, t), K ∂tt w(x, t) = ∂xx w(x, t), x ∈ [0, D], (32) w(0, t) = 0, (33) wx (D, t) = −¯ cwt (D, t), c¯ > 0. (34) Since the target system (31)-(34) is exponentially stable, the controller which compensates the wave process can be

˙ ϑ(t) = ∂x u(0, t) , ∂tt u(x, t) = ∂xx u(x, t) , u(0, t) = 0 , c ∂x u(D, t) = s+c ˆ +K H(t)



D 0





(41) (42) (43)

x ∈ [0, D] ,



ˆ c¯ K H(t)u(D, t)−∂t u(D, t) + ρ¯(D)KG(t)

ρ¯(D − σ)∂t u(σ, t)dσ



.

(44)



Tiago Roux Oliveira et al. / IFAC PapersOnLine 52-29 (2019) 134–139

4. STABILITY ANALYSIS The availability of Lyapunov functionals via backstepping transformation (Krsti´c and Smyshlyaev, 2008) permits the stability analysis of the complete feedback system with a cascade representation of ODE-PDE equations and the infinite-dimensional control law. Theorem 1. Consider the system in Fig. 1 with the dynamic system being represented by the nonlinear quadratic map in (9) in cascade with the actuation dynamics governed by the wave PDE in (3)–(6). For a sufficiently large c > 0, there exists some ω ¯ (c) > 0, such that ∀ω > ω ¯ , the closed-loop system (41)-(44) with states ϑ(t), u(x, t), has an unique locally exponentially stable periodic solution in t of period Π := 2π/ω, denoted by ϑΠ (t), uΠ (x, t), satisfying ∀t ≥ 0:       ϑΠ (t)2 + ∂x uΠ (t)2 + ∂t uΠ (t)2  2 1/2 + ∂x uΠ (D, t) ≤ O (1/ω) . (45) Furthermore, lim sup |θ(t) − θ∗ | = O (a/ω + 1/ω) ,

(46)

t→∞

lim sup |Θ(t) − Θ∗ | = O (a + 1/ω) , t→∞   lim sup |y(t) − y ∗ | = O a + 1/ω 2 .

(47) (48)

t→∞

Proof : The proof is structured into Step 1 to 6, analogously to what has been done in (Feiling et al., 2018). First, in Step 1-4, we show the exponential stability of the average closed-loop system of (41)-(44) via a backstepping transformation. Then, we invoke the averaging theorem for infinite-dimensional systems (Hale and Lunel, 1990, Section 2) in Step 5 to show the exponential stability of the original closed-loop system (41)-(44). Finally, Step 6 shows the convergence of (θ(t), Θ(t), y(t)) to a small neighborhood of the extremum (θ∗ , Θ∗ , y ∗ ). Step 1: Average closed-loop system The average version of the system (41)-(44) for ω large is ϑ˙ av (t) = ∂x uav (0, t), (49) ∂tt uav (x, t) = ∂xx uav (x, t), x ∈ [0, D] (50) (51) uav (0, t) = 0 d ∂x uav (D, t) = −c∂x uav (D, t) dt  − c c¯ [KHuav (D, t) − ∂t uav (D, t)] + ρ¯(D)KHϑav (t) +KH



D 0



ρ¯(D − σ)∂t uav (σ, t)dσ ,

(52) where the low-pass filter is represented in state-space form. Step 2: Backstepping transformation into target system With some abuse of notation we also use w(x, t) to denote the average transformed state. From (28), the backstepping transformation

137

w(x, t) = uav (x, t)  x − γ(x − σ)∂t uav (σ, t)dσ − γ(x)ϑav (t)

(53)

0

maps the average closed-loop system (49)-(52) into the exponentially stable target system (shown in Step 3) (54) ϑ˙ av (t) = KHϑav (t) + wx (0, t), (55) ∂tt w(x, t) = ∂xx w(x, t), x ∈ [0, D], w(0, t) = 0, (56) 1 wt (D, t) = − ∂x w(D, t), c¯ > 0, (57) c¯ 1 (58) ∂x w(D, t) = − ∂t ∂x uav (D, t). c The target system (54)-(58) can be derived by plugging the inverse backstepping transformation (Susto and Krstic, 2010): uav (x, t) = w(x, t) + KHn(x)ϑav (t)  x + KH n(x − σ)wt (σ, t)dσ,

(59)

0

with



 0 (KH)2 , (60) n(x) = [0 I]e [I 0] , I 0 into the average closed-loop system (49)-(52). Additionally taking the time derivative of the backstepping transformation (53) along with (52) and its inverse (59), we arrive at (58) remiding that U˙ av (t) = ∂t ∂x uav (D, t), or equivalently, ¯ Ax

T

A¯ =

∂t wx (D, t) = −cwx (D, t) + KHw(D, t) +(KH)2 n(D)ϑav (t)+(KH)2



D

n(D−σ)wt (σ, t)dσ .

(61)

0

Step 3: Exponential stability of the target system We start by introducing the system norms Ω(t)=∂x uav (t)2+∂t uav (t)2+|ϑav (t)|2+|∂x uav (D, t)|2 ,

(62)

Ξ(t)=wx (t)2+wt (t)2+|ϑav (t)|2+|wx (D, t)|2 .

(63)

To prove the stability of the closed-loop system, we consider the Lyapunov-Krasovskii functional b ϑ2 (t) + aE(t) + wx2 (D, t), V (t) = av (64) 2 2 where the parameters a, b > 0 are to be chosen later and the functional E(t) is defined by (Susto and Krstic, 2010):  1 E(t) = ||wx (t)||2 + ||wt (t)||2 + 2  D +δ (1 + y) wx (y, t)wt (y, t)dy , (65) 0

where δ > 0 is also a parameter to be chosen later. We observe that (66) θ1 Ξ ≤ V ≤ θ2 Ξ , where   b 1 a , [1 − δ (1 + D)] , , (67) θ1 = min 2 2 2   b 1 a θ2 = min , [1 + δ (1 + D)] , . (68) 2 2 2 We choose 1 0<δ< (69) 1+D

138

Tiago Roux Oliveira et al. / IFAC PapersOnLine 52-29 (2019) 134–139

in order to ensure that θ1 and θ2 are non-negative and so the Lyapunov function V in (64) is positive definite. Next, we compute the time derivative of E(t)  δ ˙ E(t) = − ||wx (t)||2 + ||wt (t)||2 + wx (0, t)2 + 2   δ + (1 + D) wt (D, t)2 + wx (D, t)2 + 2 (70) + wx (D, t)wt (D, t) . From (57), we substitute the boundary condition wx (D, t) = −¯ cwt (D, t) and get  δ ˙ E(t) = − ||wx (t)||2 + ||wt (t)||2 + wx (0, t)2 + 2    1+D  1 + c¯2 wt (D, t)2 . − c¯ − δ (71) 2 Choosing now 2¯ c δ< (72) (1 + D)(1 + c¯2 ) we have that the constant between brackets in the second term of (71) is positive. Now, computing the complete derivative of V (t), associated with the solution of the target system (54)-(58), we have: V˙ (t) = KHϑ2av (t) + ϑav (t)wx (0, t) ˙ (73) + aE(t) + bwx (D, t)∂t wx (D, t) . By applying Young’s inequality to the second term in (73), we can write   δ 1 KH 2 ϑav (t) + − a wx (0, t)2 + V˙ (t) ≤ 2 2|KH| 2  δ − a ||wx (t)||2 + ||wt (t)||2 + bwx (D, t)∂t wx (D, t) . 2 (74) By choosing 1 a≥ , (75) δ|KH| we now obtain  δ KH 2 ϑav (t) − a ||wx (t)||2 + ||wt (t)||2 V˙ (t) ≤ 2 2 (76) + bwx (D, t)∂t wx (D, t) . Finally, substituting (61) into (76), the last term in the right-hand side can be treated analogously to what have been done in references (Feiling et al., 2018) and (Oliveira et al., 2018) for diffusion processes or even as carried out in (Oliveira et al., 2017) with pure delays. From After lengthy calculations, applying Young’s, Poincare’s, Agmon’s and Cauchy-Schwarz’s inequalities (more than one) and the the help of integration by parts, we conclude that there exist c∗ > 0 (depending on KH and D) and η > 0 such that, for c > c∗ sufficiently large in (40), one has V˙ (t) ≤ −ηV (t) and the target system (54)-(58) is exponentially stable in the norm 1/2  , i.e., in |ϑav (t)|2 + wx (t)2 + wt (t)2 + |wx (D, t)|2 the transformed variables (ϑav , w). Step 4: Exponential stability estimate (H1 ) of the average closed-loop system In the last step, we arrive at the estimate V (t) ≤ e−ηt V (0) .

(77)

To prove stability of the closed-loop system in its original variables (ϑav , uav ) from (77) we provide inequalities relating the variables u(x, t) and w(x, t). From the inverse transformation (59) we obtain that  x φ (x − y)w(y, t)dy+ ∂x uav (x, t) = wx (x, t) + 0  x  + n (x − y)wt (y, t)dy + ψ(x) ϑav (t) , 0



(78)

x

∂t uav (x, t) = wt (x, t) + φ(x − y)wt (y, t)dy+ 0  x + n (x − y)w(y, t)dy + ψ(x)KHϑav (t) . 0

(79) Applying Poincare’s, Young’s and the Cauchy-Schwartz inequalities, we get ||∂x uav (t)||2 ≤ α1 ||wx (t)||2 + α2 ||wt (t)||2 + α3 |ϑav (t)|2 , ||∂t uav (t)||2 ≤ β1 ||wx (t)||2 + β2 ||wt (t)||2 + β3 |ϑav (t)|2 , (80) where (81) α1 = 4(1 + 4D3 ||φ ||2 ) ,  2 α2 = 4D||n || , (82)  2 (83) α3 = 4||ψ || ,  2 (84) β1 = 4||n || (85) β2 = 4(1 + 4D3 ||φ||2 ) , 2 β3 = 4||ψKH|| . (86) Applying (80), we obtain (87) Ω(t) ≤ θ4 Ξ(t) , where (88) θ4 = max {α1 + β1 , α2 + β2 , α3 + β3 } . With the help of time and space derivatives of (28) – see (Susto and Krstic, 2010) for more details – and applying again Poincare’s, Young’s and the Cauchy-Schwartz inequalities, we obtain the following inequalities: ||wx (t)||2 ≤ a1 ||∂x uav (t)||2 + a2 ||∂t uav (t)||2 + a3 |ϑav (t)|2 , 2

2

2

2

||wt (t)|| ≤ b1 ||∂x uav (t)|| + b2 ||∂t uav (t)|| + b3 |ϑav (t)| ,

(89) (90)

by means of them, we obtain (91) θ3 Ξ ≤ Ω(t) , where 1 . (92) θ3 = max {a1 + b1 , a2 + b2 , a3 + b3 + 1} With the help of (66), (77), (87) and (91), we get θ1 θ 3 Ω(t) ≤ Ω(0)e−ηt , (93) θ2 θ4 which completes the proof of exponential stability of the average closed-loop system in sense of the norm (62) in the variables (ϑav , uav ). Steps 5 and 6: Due to space limitations, we are not able to go into details in Step 5 (Averaging Theorem) and Step 6 (convergence to extremum), but the demonstration is similar to those presented in (Oliveira et al., 2017) and (Feiling et al., 2018). According to (93), the origin of the average closed-loop system with wave PDE is exponentially stable. Applying the averaging theorem for infinite-dimensional systems developed in (Hale and Lunel,



Tiago Roux Oliveira et al. / IFAC PapersOnLine 52-29 (2019) 134–139

1990, Section 2), for ω sufficiently large, (49)-(52) has a unique exponentially stable periodic solution around its equilibrium satisfying (45). Along with (45), it is not difficult to show ˜ = O (1/ω) . (94) lim sup |θ(t)| t→∞

˜ + S(t), From (10) and Fig. 1, we can write θ(t) − θ∗ = θ(t) and recalling S(t) in (24) is of order O (a/ω), we finally get with (46). The convergence of the propagated actuator Θ(t) to the optimizer Θ∗ is easier to prove. Using (25) and taking the absolute value, one has (95) |Θ(t) − Θ∗ | = |ϑ(t) + a sin(ω(t))| . As in the convergence proof of the parameter θ(t) to the optimal input θ∗ above, we write (95) in terms of the periodic solution ϑΠ (t) and follow the same steps by applying Young’s inequality and reminding that ϑ(t) − ϑΠ (t) → 0 exponentially according to the averaging theorem (Hale and Lunel, 1990, Section 2). Hence, along with (45), we finally get (47). To show the convergence of the output y(t) of the static map to the optimal value y ∗ , we replace Θ(t) − Θ∗ in (9) by (25) and take the absolute value:   H 2 (96) |y(t) − y ∗ | =  [ϑ(t) + a sin(ω(t))]  . 2 Expanding the quadratic term in (96) and applying Young’s to the resulted equation, one has |y(t)−  inequality   y ∗ | = H ϑ(t)2 + a2 sin2 (ωt) . As before, we add and subtract the periodic solution ϑΠ (t) and use the convergence of ϑ(t)−ϑΠ (t) → 0 via averaging theorem (Hale and Lunel, 1990, Section 2). Hence, again with (45), we get (48).  5. NUMERICAL SIMULATIONS Consider a quadratic static map as in (8), with Hessian H = −0.2, optimizer Θ∗ = 2, and optimal value y ∗ = 5. The domain length of the wave PDE is set to D = 1. The parameters of the proposed ES are chosen as ω = 10, a = 0.2, c = 10, c¯ = 0.5, and K = 0.4. The simulation results of the closed-loop are illustrated in Fig. 2. We observe that the variable Θ converges to a neighborhood of the optimum Θ∗ , as expected. 3

2

1

0

-1

-2 0

5

10

15

20

25

30

(a) Parameter Θ(t) converging to Θ∗ .

Fig. 2. ES with a wave PDE in the actuation dynamics. 6. CONCLUSIONS This work has provided a full and detailed analysis of the application of wave compensation designs in extremum seeking. We have presented Gradient extremum seeking

139

method for scalar static maps with actuation dynamics governed by wave PDEs. The actuation dynamics must be known, but no knowledge is assumed for the map parameters. The average control law to compensate the wave actuation dynamics was constructed by combining stateof-the-art analysis techniques in the area of boundary control of PDEs via backstepping methodology and employed perturbation-based estimates of the gradient and Hessian of the static map. The additive perturbation-dither signal has also taken the wave dynamics into account and compensated it. At last, local exponential stability and convergence to a small neighborhood of the desired extremum were guaranteed. The proposed approach has also broad applicability in practice since the presence of PDE models is often listed as a major limiting factor in the application of extremum seeking in some practical situations. ACKNOWLEDGEMENTS This work was supported by CNPq, CAPES, and FAPERJ. REFERENCES Aarsnes, U.J. and Krstic, M. (2019). Extremum seeking for real-time optimal drilling control. In American Control Conference, 2926–2931. Philadelphia. Bekiaris-Liberis, N. and Krsti´c, M. (2014). Compensation of wave actuator dynamics for nonlinear systems. IEEE Transactions on Automatic Control, 59(6), 1555–1570. Feiling, J., Koga, S., Krsti´c, M., and Oliveira, T.R. (2018). Gradient extremum seeking for static maps with actuation dynamics governed by diffusion PDEs. Automatica, 95(9), 197–206. Ghaffari, A., Krsti´c, M., and Neˇsi´c, D. (2012). Multivariable Newton-based extremum seeking. Automatica, 48(8), 1759–1767. Hale, J.K. and Lunel, S.M.V. (1990). Averaging in infinite dimensions. J. Integral Equations Appl, 2(4), 463–494. Khalil, H.K. (2002). Nonlinear Systems. Prentice-Hall. Krsti´c, M. (2009). Compensating a string PDE in the actuation or sensing path of an unstable ODE. IEEE Transactions on Automatic Control, 54(6), 1362–1368. Krsti´c, M. and Smyshlyaev, A. (2008). Boundary control of PDEs: A course on backstepping designs. SIAM. Krsti´c, M. and Wang, H.H. (2000). Stability of extremum seeking feedback for general nonlinear dynamic systems. Automatica, 36(4), 595–601. Oliveira, T.R., Feiling, J., Koga, S., and Krsti´c, M. (2018). Scalar Newton-based extremum seeking for a class of diffusion PDEs. In IEEE Conference on Decision and Control, 2926–2931. Miami. Oliveira, T.R., Krstic, M., and Tsubakino, D. (2017). Extremum seeking for static maps with delays. IEEE Transactions on Automatic Control, 62(4), 1911–1926. Rusiti, D., Evangelisti, G., Oliveira, T.R., Gerdts, M., and Krstic, M. (2019). Stochastic extremum seeking for dynamic maps with delays. IEEE Control Systems Letters, 3(1), 61–66. Rusiti, D., Oliveira, T.R., Mills, G., and Krstic, M. (2018). Newton-based extremum seeking for higher derivatives of unknown maps with delays. European Journal of Control, 41(5), 72–83. Susto, G.A. and Krstic, M. (2010). Control of PDE–ODE cascades with Neumann interconnections. Journal of the Franklin Institute, 347(1), 284–314.