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Journal of Process Control journal homepage: www.elsevier.com/locate/jprocont

Bounded bilinear control of coupled ﬁrst-order hyperbolic PDE and inﬁnite dimensional ODE in the framework of PDEs with memory Sarah Mechhoud a,∗ , Taous-Meriem Laleg-Kirati b a

Department of Electrical Engineering, University of 20 August 1955 Skikda, El Hadaik, 21000 Skikda, Algeria Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia b

a r t i c l e

i n f o

Article history: Received 17 August 2018 Received in revised form 16 March 2019 Accepted 11 June 2019 Available online 8 August 2019 Keywords: Distributed parameter systems System of coupled PDE-inﬁnite ODE Energy-like bilinear control Bounded control Boundary observer design Bounded output feedback control

a b s t r a c t In this work, we consider the problem of bounded bilinear tracking control of a system of coupled ﬁrstorder hyperbolic partial differential equation (PDE) with an inﬁnite dimensional ordinary differential equation (ODE). This coupled PDE-inﬁnite ODE system can be viewed as a degenerate system of two coupled ﬁrst-order hyperbolic PDEs, the velocity of the ODE part vanishing. First, we convert this PDE-inﬁnite ODE system into a ﬁrst-order hyperbolic PDE with memory and investigate the bounded bilinear control problem in this framework. We consider as manipulated variable the constrained wave propagation velocity, which makes the control problem bounded and bilinear, and we take the measurements at the boundaries. To account for the actuator’s constraints, we develop conditions under which the bounded control law ensures stability and tracking performances. This leads to a speciﬁcation of the state-space region that enforces the desired system’s closed-loop behaviour. To overcome the lack of full-state measurements, we design an observer-based bounded output-feedback control law which guarantees the reference tracking and uniform asymptotic stability of the system in closed-loop. A strong motivation of our work is the control problem of the solar collector parabolic trough where the manipulated control variable (the pump volumetric ﬂow rate) is bilinear with respect to the PDE-inﬁnite ODE model, and the measurements are taken at the boundary (tube’s outlet). Simulation results illustrate the efﬁciency of the proposed control strategy. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Coupled PDE-ODE systems arise in many physical and industrial applications, to name a few, viscoelasticity, heat ﬂow in materials with memory, biochemical processes, heat exchangers and solar collectors. Two types of PDE-ODE coupling exist. The ﬁrst one, boundary PDE-ODE coupling, occurs in transport delay systems and when the boundary condition is function of an ODE. These constitute cascade PDE-ODE systems and their control has attracted some research work [1–3] (and references therein). In the other hand, there are in-domain coupled PDE-ODE systems where the coupling happens in the PDE domain. This arises either when the PDE coefﬁcients depend on the state of an ODE [4] or when the coupling is between a PDE and an inﬁnite dimensional ODE. This inﬁnite ODE

∗ Corresponding author. E-mail addresses: [email protected], [email protected] (S. Mechhoud), [email protected] (T.-M. Laleg-Kirati). https://doi.org/10.1016/j.jprocont.2019.06.006 0959-1524/© 2019 Elsevier Ltd. All rights reserved.

is a pure reaction PDE which can be viewed as a degenerate PDE with a vanishing velocity. In the literature, systems of coupled PDEs and inﬁnite dimensional ODEs are referred to as PDEs with memory term [5]. From a mathematical viewpoint, they are regarded as partial integrodifferential equations (PIDEs) or Volterra integro-differential equations. Many research papers have been dedicated to the control problem of coupled strict hyperbolic PDEs, e.g. [6], and to the problems of well-posedness, controllability and observability of PIDEs [7–9,5] (and references therein). However, to the authors’ best knowledge there is no work related to the control problem of our interest. In this paper, we consider the problem of constrained boundary reference tracking bilinear control of 1st-order hyperbolic PDE coupled with an inﬁnite dimensional ODE. The bounded manipulated variable (input) is the ﬂuid velocity and the objective is to drive the ﬂuid outlet temperature to track a target trajectory. This PDE-inﬁnite ODE system models heat transfer in a metallic pipe in which a ﬂuid ﬂows. We assume that the diffusion phenomenon in the pipe is negligible with respect to the advection

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process in the ﬂuid. A practical example of such a model is the solar collector parabolic trough (SC) [10], [11,chapter 2]. It consists of a parabolic surface which focuses the solar irradiance onto a receiver tube placed in the focal line of the parabola. In this tube a ﬂuid absorbs this concentrated heat while it is transported. It can be used then for many purposes, generally it is pumped to a power generating system, a desalination plant or to an oil-cooling system. The solar irradiance plays the role of the source term and it is measured by a heliometer [12]. The SC is modelled by a 1st-order hyperbolic PDE coupled with an inﬁnite dimensional ODE. The PDE describes the ﬂuid’s temperature while the inﬁnite ODE deﬁnes the variations of the metal tube’s temperature. The manipulated variable of the SC is the pump volumetric ﬂow rate which is, from a modelling perspective, the transportation speed. From a practical viewpoint, the ﬂow rate is upper bounded due to physical limitations and lower bounded to ensure good operation conditions. Also the ﬂow rate has to be positive to ensure the well-posedness of the PDE’s solution. In addition the temperature measurements are available only at the boundary, i.e. at the tube’s outlet. As a result this leads to a bounded bilinear control problem with boundary measurements. Most of control methods devoted to the SC have been based on a simpliﬁed model by neglecting the heat losses. The resulted model was a 1st-order hyperbolic PDE and the control was addressed in the early lumping approach [10,11]. To handle the input constraints in [13] and [14] the MPC strategy was applied to the discretized PDE model. In [15] and [16] the unconstrained bilinear control problem of the 1st-order hyperbolic PDE modelling the SC was considered in the late lumping framework. However, to the best of the author’s knowledge, the constrained bilinear control problem of the full model, i.e., PDE-inﬁnite ODE, was not discussed yet. This motivates us to investigate this problem in its original framework (inﬁnite dimensional). We start ﬁrst by discussing the well-posedness of the PDEinﬁnite ODE model. Then using the mild solution of the inﬁnite ODE component, we convert the coupled PDE-inﬁnite ODE system into a 1st-order hyperbolic PIDE. After that we formulate the control problem in the late lumping framework, and analyse its performance using Lyapunov stability theory. The control of bilinear systems is generally dealt with using geometric or optimal approaches, e.g. [17,18]. However, in the geometric control, the stability of the zero dynamics cannot be proved analytically and is generally checked after discretization in simulations [19]. For the design of optimal control strategies, it is well known in the literature that there is no guarantee that an optimal solution exists, and at best, we end up with a sub-optimal controller [18]. A novel approach based on the relationship between delay equations and 1st-order hyperbolic PDEs was proposed in [20] where the bilinear stabilization problem of age-structured chemostat models was considered and the manipulated variable was the dilution rate (reaction coefﬁcient). This control method is very technical and from the authors’ point of view its adaptation to coupled PDE-ODE systems is not straightforward. In [15], to control the bilinear 1storder hyperbolic PDE, the average tracking error is selected instead of the outlet (boundary) temperature as the controlled variable. However, the model was a hyperbolic PDE and the convergence was proved in the steady-state. This new controlled variable proposed in [15] has inspired the work in [16] and [21] where it was improved by including a condition which guarantees the convergence of the distributed and outlet tracking errors. However, the added condition restricts the set of reachable states and the input constraints were not analytically imposed, i.e. only in simulations. In this work, we consider the full PIDE model and rather then using the average tracking error, we employ the L1 -norm of the tracking error as the controlled variable. Moreover, the problem of bounded control will be handled explicitly.

It is well known that due to input constraints the system’s behaviour cannot be freely modiﬁed and undesired dynamics may result such as oscillations, overshoot or generally the “windup” effect [22]. To avoid such phenomena we synthesize sufﬁcient conditions that characterize the state-space for which the closedloop stability and target reference tracking are satisﬁed under input constraints. However, since the measurements are located at the boundaries, we design a Luenberger-like observer which guarantees the uniform asymptotic stability of the state estimation error. This enables us to design a bounded output-feedback controller which insures the uniform asymptotic stability of the state estimation and tracking errors. The main contributions of our work are (i) the use of the L1 -norm of the tracking error as the controlled variable, (ii) the elaboration of a control law which explicitly accounts of the actuator’s limitations, (iii) the design of an observer for the PIDE model and (iv) the construction of a bounded output-feedback controller which guarantees reference tracking and uniform asymptotic stability for our 1st-order hyperbolic PIDE. After the presentation of the model’s well-posedness and the control problem in Section 2, the bilinear control design in the case of full-state measurements is considered in Section 3. First, we address the problem of unbounded control and then present how to comply with the input constraints. Section 4 treats the observer design and Section 5 studies the bounded output-feedback control properties. Simulation results are depicted in Section 6. Notations and deﬁnitions: Throughout this paper, we will omit the functions’ time and space dependencies whenever it is necessary to alleviate the notation. Let R+ = [0, ∞[, R+ ∗ =]0, ∞[, I ⊆ R be an interval and =]0, 1]. L2 () denotes the space of square integrable functions, endowed with its standard norm ||.||. The norm

1

1/2

||.|| is a weighted L2 -norm deﬁned by f = ( 0 e−x f (x)2 dx) , > 0 (||.|| is equivalent to ||.||). Hs () is the usual Sobolev space of functions on whose derivatives up to order s lie in L2 (). We use the standard notation of space-time spaces, e.g. (I, Hs ()) refers to the set of functions f : (I) → Hs (). If space and time spaces are identical then we use the following notation (0, ∞, ). To denote a space-time function f ∈ (I, Hs ()) which takes values in R+ ∗ , we ). use the notation f ∈ (I, H s (); R+ ∗ 2. Problem formulation We consider the following 1st-order hyperbolic PDE and inﬁnite ODE system which represents, as in the SC, heat transfer between a metallic tube at temperature Tm and a ﬂuid, at temperature T, circulating inside this tube [12]

⎧ ∂t Tm = −ˇ(Tm (x, t) − T (x, t)) + f (t), ⎪ ⎪ ⎪ ⎪ ⎨ ∂t T = −u(t)∂x T + ˛(Tm (x, t) − T (x, t)), 0, ⎪ Tm (x, 0) = Tm ⎪ ⎪ ⎪ ⎩

T (x, 0) = T0 (x),

x ∈ ]0, 1]

(1)

T (0, t) = g(t),

where (x, t) ∈ Qxt := ×]0, ∞[is the normalized space-time domain and u(t) is the control input (manipulated variable). The metallic tube is heated by the source term f(t). Positive constants (˛, ˇ) are respectively the metal and ﬂuid heat transfer coefﬁcients. 0 is We assume that in the metallic tube, the initial temperature Tm 0 is space independent (a conuniformly distributed, therefore Tm stant), and that the diffusion process is negligible with respect to the reaction one. The initial and boundary conditions in (1) satisfy the zero-order compatibility condition g(t = 0) = T0 (x = 0).

(2)

The output-equation is given by y(t) = T (x = 1, t),

(3)

S. Mechhoud, T.-M. Laleg-Kirati / Journal of Process Control 81 (2019) 223–231

which states that the measurements are taken at the tube’s outlet (x = 1). By writing the mild solution of the inﬁnite dimensional ODE (ﬁrst equation) and replacing its expression in the second equation of system (1) we obtain the following PIDE equation

⎧ ∂t T (x, t) = −u(t)∂x T (x, t) − ˛T (x, t) ⎪ ⎪ ⎪ t ⎪ ⎨

e−ˇ(t−s) T (x, s) ds + S(t),

+˛ˇ

⎪ ⎪ ⎪ ⎪ ⎩ T (0, t)

0

= g(t),

x ∈ [0, 1].

e−ˇ(s−) T (x, )2 d ds =

0 t

t

0 t

2

T d + ˛ˇ

e

0

0

−ˇ(s−) 2

T (x, ) ds d

(10) dx

where we have proceeded to a change of the integration’s order in the last term. After some straightforward computations we ﬁnd that

t

(5)

s

T (x, ) d + ˛ˇ

1 0

0

0

t

t

s

e−ˇ(s−) T (x, )2 d ds =

T (x, )2 d + ˛ˇ 0

t

0

(11)

e−ˇ(t−) T (x, ) d ≤ 0.

−˛ 0

Therefore inequality (9) becomes

2.1. Well-posedness

The well-posedness of system (1) and thus Eq. (4) is derived from the work of [8] and [23]. It can be summarized in the following proposition. Proposition 1. If f ∈ L2 (0, ∞) ∩ L∞ (0, ∞), g ∈ Hs (0, ∞) ∩ L∞ (0, ∞), T0 ∈ Hs (0, 1) ∩ L∞ (0, 1), u ∈ L∞ (0, ∞, R+ ∗ ), ˛ > 0, ˇ > 0 and the rcompatibility conditions (see [23] for more details) are satisﬁed then Eq. (4) (and thus system of Eqs. (1)) admits a unique solution (Tm , T) ∈ Cr+1 (0, ∞, Hs−r (0, 1)) × Cr (0, ∞, Hs−r (0, 1)), 0 ≤ r ≤ s. Proof. To prove the well-posedness of PIDE (4), we exploit the results presented in the work of [23]. To this end, ﬁrst we need to prove that PIDE (4) satisﬁes the energy estimate inequalities (1.2) and (1.3) presented in [23].

T (x, t)

2

≤

+˛ˇ

e

T (x, 0) + umax

t

t

0

T 2 (0, ) d (12)

S 2 () 0

where we have applied the differential form of Gronwall’s inequality. Note that inequality (12) represents the same energy estimates of a standard transport equation. Moreover this inequality asserts that the solution of PIDE (4) satisﬁes both inequalities (1.2) and (1.3) of [23]. Furthermore, using the weighted L2 -norm ||.|| and following the same procedure which yielded inequality (12), we can prove that when S(t) = g(t) = 0 then the solution of PIDE (4) is exponentially stable in the L2 -sense. Namely

T (x, )T (x, t) d

t

e−ˇ(t−) T (x, )2 d

+˛ˇ

(13)

0

(6)

t −ˇ(t−)

2

+

u − ∂x T 2 − ˛T 2 + S(t)T (x, t) 2

et

∂t T 2 ≤ −umin T 2 − ue− T 2 (1, t) − ˛T 2

Multiplying Eq. (4) by T gives 1 ∂t T 2 = 2

−˛

−˛ 0 e−ˇ(t−s) f (s) ds + ˛ e−ˇt Tm .

S(t) = ˛

t

t 2

where

0

t ∈ ]0, ∞],

= T0 (x),

T (x, 0)

By analysing the last two terms of the right-hand side of inequality (9), we ﬁnd that

−˛ (4)

225

which leads to

0

By applying Young’s inequality on the two last terms we obtain the following inequality 1 ∂t T 2 2

u ˛ ˛ˇ ≤ − ∂x T 2 − T 2 + 2 2 2

t

e−ˇ(t−) T 2 (x, ) d 0

(7)

1 1 + S 2 (t) + T 2 (x, t) 2 2 Integrating (7) over gives the following inequality

∂t T 2

≤ −u(T 2 (1, t) − T 2 (0, t)) − ˛T 2

t

e−ˇ(t−) T (x, )2 d + S 2 (t) + T 2

+˛ˇ

+

t

0

S 2 () d +

t

T 2 (0, ) d 0

t

T (x, )2 d 0

t

t

−˛

T (x, ) d + ˛ˇ 0

(9)

s

e−ˇ(s−) T (x, )2 d ds

2

0

0

Therefore, using Theorem 3.1 and Proposition 5.1 of [23] we conclude that if the r-compatibility conditions are satisﬁed, f ∈ L2 (0, ∞) ∩ L∞ (0, ∞), g ∈ Hs (0, ∞) ∩ L∞ (0, ∞) and T0 ∈ Hs (0, 1) then T is in Cr (0, ∞, Hs−r (0, 1)). From Tm ’s mild solution, Tm is then in Cr+1 (0, ∞, Hs−r (0, 1)). 䊐 In all what will follow, we consider Eq. (4) since it is an equivalent representation of our original model (1). 2.2. Formulation of the control problem

Let us integrate (8) in time, while taking into consideration that u ∈ L∞ (0, ∞, R+ ∗ ), then

(14)

(8)

0

T (x, t)2 − T (x, 0)2 ≤ umax

T (x, t)2 ≤ T (x, 0)2 e−(umin )t

Let T* denotes the target temperature proﬁle, such that T* satisﬁes Eq. (4) while replacing the parameters (u, g, T0 ) by their unknown correspondent (u∗ , g ∗ , T0∗ ). The output equation is given 0 ) are positive, in (3), and we assume that the parameters (˛, ˇ, Tm known constants and that f(t) is measured. The objective of this work is to design a bounded positive control input u(t) ∈ L∞ (0, ∞, R+ ∗ ) such that the closed-loop system is stable and lim (T ∗ (1, t) − T (1, t)) = 0 while providing, at the same t→∞

time, a characterization of the initial state region for which the closed-loop and tracking performances are guaranteed. Note that the selection of u as the manipulated variable renders the control problem constrained and bilinear.

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Assumption 1. (i) (ii) (iii) (iv)

Let us consider the following assumptions

ing error ı (and thus the distributed tracking error) exponentially towards zero. Let u(t) be given by

⎧

1 ⎪ −1 ⎪ u(t) = (t) sgn(ıT (x, t)) S(t) − ∂t T ∗ − ˛T (x, t) ⎪ ⎪ ⎪ ⎪ t0 ⎨

˛ and ˇ are positive constants. g ∈ Qg := {g ∈ H1 (0, ∞) :0 < g ≤ gmax }. T0 ∈ QT0 := {T ∈ H1 (0, 1) : 0 < T ≤ Tmax }. f ∈ QS = {f ∈ L2 (0, ∞) :0 < f < fmax }.

+˛ˇ

From Proposition 1 these assumptions ensure that T ∈ QT := C(0, ∞, H1 ()) and thus T is also in C(0, ∞, ) (from the embededness of H1 () in C()). Instead of using the outlet temperature error (T* (1, t) − T(1, t)) as the controlled variable, we use the following variable ı(t) deﬁned by

1

|T ∗ (x, t) − T (x, t)| dx.

ı(t) =

(15)

e−ˇ(t−s) T (x, s) ds

dx − kp ı(t) ,

⎪ ⎪ 1 0 ⎪ ⎪ ⎪ ⎪ ⎩ (t) = sgn(ıT (x, t))∂x T (x, t) dx.

(16)

0

where ıT(x, t) := T* (x, t) − T(x, t) is the distributed tracking error, sgn is the signum function and kp is a strictly positive gain. Theorem 1. Consider PIDE (4) under assumptions (i)–(iv). The control input u(t) given by Eq. (16) ensures that ı(t) converges to zero exponentially for t ≥ t* where t ∗ = u 1 is the maximum control time min (the maximum transportation time from x = 0 to x = 1).

0

which is the L1 -norm of the tracking error. Note that the convergence in the L1 -norm does not lead to the point-wise convergence. Nonetheless under assumptions (i)–(iv) then T ∈ QT := C(0, ∞, H1 ()) and therefore we can ensure that ∗

lim ı(t) = 0 ⇒ lim (T (x, t) − T (x, t)) = 0,

t→∞

t→∞

∀x ∈ ]0, 1]

Proof. tional

Let us consider the following candidate Lyapunov func-

V1 (t) =

1 ı(t)2 2

(17)

The derivative of ı(t) in (15) is given by

˙ = ı(t)

1

sgn(ıT )

Remark 2. The proposed control problem is bilinear. In the pioneer work of [24] it was pointed out that exact controllability of bilinear PDEs cannot be achieved, i.e. we cannot expect to control PIDE (4) to a full neighbourhood of the target state T* (x, t). Therefore, only approximate controllability can be expected (more details can be found in [24] and [25]).

−S) dx + u(t)

0

1

t

e−ˇ(t−s) T (s) ds

∂t T + ˛T − ˛ˇ

0

Remark 1. The objective of the present work is to design a bilinear baseline controller (with no model uncertainties) for systems described by coupled hyperbolic PDE and inﬁnite ODE. Therefore, as already stated, we assume that the metallic tube’s initial tem0 is known. Although this assumption appears to be perature Tm 0 is restrictive, from the expression of S(t) in Eq. (5) the effect of Tm transient and exponentially vanishing. As a result, if the designed controller stabilizes the nominal model then it will be robust to any 0 . Note also that it is possible bounded uncertainties on this term Tm to consider the source term S(t) as an unknown parameter and handle the control problem using robust adaptive control techniques. However, this is beyond the scope of the present work and will be investigated in future.

∗

(18)

sgn(ıT )∂x T dx 0

where we have omitted the (x, t) notation for brevity. By replacing expression (16) in Eq. (18), the derivative of V1 (t) writes as V˙ 1 (t) = −kp ı2 (t) ≤ −2kp V1 (t), and thus V1 is a Lyapunov function and V1 (t) ≤ V1 (0) e−2 kp t and ı(t) converges exponentially to zero. Therefore ıT is also exponentially converging to zero. 䊐 Remark 3. Note that the control law in Eq. (16) can be described as the combination of a feed-forward plus a proportional feedback. This latter can be augmented to the form of a PID feedback. 3.2. Bounded control It is natural, from a physical point of view to consider that the control input u has to be positive, upper and lower bounded, i.e. 0 < umin ≤ u(t) ≤ umax

3. Bilinear controller design As stated in Section 2.2, the objective is to design a bounded control input 0 < umin ≤ u(t) ≤ umax which assigns a desired closedloop dynamic for the system described by PIDE (4). To achieve this, we consider that assumptions (i)–(iv) are satisﬁed, and we select ı deﬁned in (15) as the controlled variable. First we begin by presenting the unbounded nonlinear control strategy and prove its performances in terms of closed-loop stability and tracking error convergence. Later, we formulate sufﬁcient conditions under which the aforementioned nonlinear controller subject to saturation ensures the same performances.

3.1. Unbounded control In this section we show that under no constraints there exists a nonlinear control law which can steer the L1 -norm of the track-

where umin and umax are the positive constraints on u. A straightforward approach to handle input constraints is to use a saturation projection, i.e.

u(t) =

⎧ u if (t) ≤ umin ⎪ ⎨ min ⎪ ⎩

(t)

if umin ≤ (t) ≤ umax

umax

if (t) ≥ umax

(19)

where (t) is the control input given in Eq. (16) which ensures the global exponential stability of the tracking error ıT without control saturation. It is known that input saturation may deteriorate the nominal stability and tracking properties, and stimulate undesired closedloop behaviour such as critical points and/or limit cycles [26]. Therefore, in this subsection, we aim to construct sufﬁcient conditions that guarantee nominal stability and tracking performances while avoiding the unsuitable phenomena due to input saturation.

S. Mechhoud, T.-M. Laleg-Kirati / Journal of Process Control 81 (2019) 223–231

To do this we combine the strategies presented in [26], [27] and [22] to deal with input control constraints and extend them to the case of distributed parameter systems. First, let us consider the following assumptions. The target temperature proﬁle T* is selected such

Assumption 2. that

(20)

2

T ∗ ∈ [T1 , T2 ] and

(21)

Assumption 3 can be viewed as a feasibility condition. It states that the control input law in (16) is guaranteed to respect its upper and lower bounds at least for a neighbourhood around the target state T* . This can be regarded as a characterization of the space of solutions for which closed-loop stability and tracking are guaranteed under bounded control. Proposition 2. Consider PIDE (4). If Assumption 1 is satisﬁed, T* and T2 are selected as in Assumptions 2 and 3 then the interval]0, T2 ] is positively invariant. Proof. From assumption (iv) we have ∂t T(T = 0) > 0 and since ∀t ≥ 0 : g(t) > 0, S(t) > 0 and ∀x ∈ ]0, 1] : T0 (x) > 0, then T(x, t) > 0, ∀ t ≥ 0, ∀ x ∈ [0, 1]. When T = T2 then ∂t T (T = T2 ) = −u(t)∂x T (x, t) − ˛T (x, t) + t ˛ˇ 0 e−ˇ(t−s) T (x, s) ds + S(t) and u(t) = min(umax , (t)) ≥ umin . From Assumption 2, we know that

+˛ˇ

e 0

0

t −ˇ(t−s)

1

T2 ds

S − ∂t T ∗ − ˛T2

dx + kp ı(t)

e−ˇ(t−s) T2 ds 0

1

∂x T2 dx −

1

0

T2 (1, t) − T2 (0, t) t

S − ∂t T ∗ − ˛T2

0

e−ˇ(t−s) T2 ds

(23)

dx + kp ı(t)

0

1

∂t T ∗ dx − kp ı(t)

≤

1

∂t (T (T = T2 ) − ∂t T ∗ ) dx ≤ −kp 0

1

(T2 − T ∗ ) dx

(24)

0

From this last inequality we conclude that when the control input u = umax and the temperature proﬁle T reaches T2 , T converges exponentially towards T* . This implies that ∂t T(T = T2 ) ≤ 0 (since T2 > T* and thanks to Assumption 1). On the other hand, if u(t) = (t) then from Eq. (18) we have ˙ ı(t) = −kp ı(t) and then using the same previous arguments we conclude that ∂t T(T = T2 ) ≤ 0. Hence the interval]0, T2 ] is positively invariant. 䊐

0

0

and thus we end up by having

⎧ ∀T ∈ [T1 , T2 ] : ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ umin ≤ (t)−1 sgn(ıT (x, t)) S(t) − ∂t T ∗ − ˛T (x, t) ⎪ ⎨ 0 t −ˇ(t−s) ⎪ e T (x, s) ds dx − kp ı(t) ≤ umax , +˛ˇ ⎪ ⎪ ⎪ ⎪ 0 1 ⎪ ⎪ ⎪ ⎪ ⎩ (t) = sgn(ıT (x, t))∂x T (x, t) dx.

1 T2 (1, t) − T2 (0, t)

t

∂t T (T = T2 ) dx ≤ S − ˛T2 + ˛ˇ

Assumption 3. ∃ (T1 , T2 ) ∈ C(0, ∞, H 1 ()) :

1

0

Eq. (20) states that at least when T = T* the control input satisﬁes its constraints. By a continuity argument we can assert that since T* satisﬁes Assumption 2, then there exist T1 ∈ C(0, ∞, H1 ()) and T2 ∈ C(0, ∞, H1 ()) such that T* ∈ [T1 , T2 ] and the following assumption holds for all T ∈ [T1 , T2 ]

≥

+˛ˇ

T ∗ (t) = T ∗ (1, t) − T ∗ (0, t).

umax

Therefore, if u = umax then

1 ⎧ 1 ⎪ ⎪ u ≤ S(t) − ∂t T ∗ − ˛T ∗ (x, t) min ⎪ ∗ ⎪ T (t) ⎪ 0 ⎨ t −ˇ(t−s) ∗ ⎪ +˛ˇ e T (x, s) ds dx ≤ umax ⎪ ⎪ ⎪ 0 ⎪ ⎩

227

(22)

Proposition 3. Consider PIDE (4). If Assumptions 1–3 are satisﬁed then the control input in Eq. (19) guarantees the exponential stability of the tracking error (relatively to the domain]0, T2 ]) provided that kp is sufﬁciently large. Proof. The proof follows from the work of [27]. From Assumption 2 and a continuity argument there exist T¯ 1 ∈ C(0, ∞, H 1 (); R+ ) and T¯ 2 ∈ C(0, ∞, H 1 (); R+ ) such that T1 < T¯ 1 < T ∗ < T¯ 2 < T2 and Eq. (21) is satisﬁed ∀T ∈ [T¯ 1 , T¯ 2 ]. As in [27], the idea is to show that the interval [T¯ 1 , T¯ 2 ] is positively invariant. Let us ﬁrst examine the case when 0 < T0 (x) < T* . If T ∈ ]0, T1 [then u(t) = max(umin , (t)) ≤ umax provided that kp is selected large enough. Whereas, if T ∈ [T1 , T¯ 1 [ then for all kp > 0 we have u(t) = max(umin , (t)) ≤ umax . Now, if u = umin , from Assumption 3 we have umin

1 ≤ T (1, t) − T (0, t)

+˛ˇ

0

t

e

−ˇ(t−s)

1

T ds

S − ∂t T ∗ − ˛T

(25)

dx − kp ı(t)

0

as a result, following the same computation’s steps as in the previous proof 3 we get ı(t) ≤ − kp ı(t). While if u(t) = (t) then ı(t) = −kp ı(t). Therefore ı(t) is exponentially converging. Moreover, since T < T* and both are smooth (Assumption 1) then we conclude that ∂t T ≥ 0 and T converges exponentially to T* . Second, if T* < T0 (x) < T2 then u(t) = min(umax , (t)) ≥ umin . When u = umax then ı(t) ≤ −kp ı(t) (see the previous proof) and when u(t) = (t) then ı(t) = −kp ı(t). Thus, since T > T* and both in C(0, ∞, H 1 (); R+ ) (Assumption 1) then we conclude that ∂t T ≤ 0 and T converge exponentially to T* . In summary, we have proved that ∀T ∈ ]0, T2 ] the temperature proﬁle is trapped within an arbitrary small interval around T* and moreover that the L1 -norm of the tracking error converges exponentially towards zero. From Assumption 1 we conclude that the distributed tracking error is also exponentially converging to zero. 䊐

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S. Mechhoud, T.-M. Laleg-Kirati / Journal of Process Control 81 (2019) 223–231

Remark 4. Assumptions 2 and 3 provide sufﬁcient conditions under which the desired performances are guaranteed.

−ˇ(t−s) ˆ

T (x, s) ds + S(t),

e

The integration of inequality (32) gives

T (x, s) ds

(27)

0

= k T (1, t).

= Tˆ0 (x) − T0 (x).

Theorem 2. Consider the system described by Eqs. (4) under assumptions (i)–(iv). Provided that u ∈ L∞ (R+ ∗ ) and if the tuning parameter k is such that 0 < k < 1 then the state observer given in Eq. (26) ensures the global uniform asymptotic stability of the L2 -norm of the state estimation error T(x, t). Let us consider the following candidate Lyapunov func-

1

e−x

0

t

e−ˇ(t−s) T 2 (x, s) ds

T 2 (x, t) + ˛

dx

(28)

0

which is inspired from the combination of the work of [28], [29] on ﬁrst-order hyperbolic PDEs and the work of [8] on the stability of integro-differential ODEs. Using Eq. (27), the derivative of V2 is given by

V˙ 2 (t)

1

=

e

−x

−2u(t) T (x, t)∂x T (x, t) − 2˛ T 2 (x, t)

0

e−ˇ(t−s) T (x, s) T (x, t) ds + ˛ T 2 (x, t)

0 t

≤

e

0

+˛ˇ

t

1

e−ˇ(t−s) T 2 (x, s) ds dx e−x [−u(t)∂x T 2 (x, t) − ˛e−ˇt T 2 (x, t)] dx

≤ 0

5. Bounded output-feedback control In this section we state the main result. First, let us deﬁne the following variables ˆ (x, t) = T ∗ (x, t) − Tˆ (x, t), ıT

1

(34)

1

ˆ (x, t)| dx. |ıT

ˆ = ı(t) 0

Theorem 3. Consider the PIDE (4), the observer given in Eq. (26), Assumptions 1–3 and the following control law

⎧ u if (t) ≤ umin ⎪ ⎨ min ⎪ ⎩

(t) ˆ

if umin ≤ (t) ≤ umax

umax

if (t) ≥ umax

(35)

e−ˇ(t−s) Tˆ (x, s) ds

+˛ˇ

0

ˆ dx − kp ı(t)

,

1

ˆ (x, t))∂x Tˆ (x, t) dx, sgn(ıT

=

(30)

ˆ and ı(t) is deﬁned as in (34). If the controller’s gain kp is chosen sufﬁciently large as in Proposition 3 then 1. The interval]0, T2 ] is positively invariant. 2. The tracking error ı(t) and the L2 -norm of the state estimation error T converge uniformly asymptotically to zero relatively to]0, T2 ].

e−x ∂x T 2 dx

≤ −u(t) 0

(36)

0

e−ˇ(t−s) ( T 2 (x, s) + T 2 (x, t)) ds

0

(33)

Remark 5. Since T and Tˆ are both in C(0, ∞, (0, 1)), from the convergence of the L2 -norm of the state estimation error T, we can conclude the pointwise uniform asymptotic stability of T(x, t), ∀x ∈ ]0, 1].

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ˆ ⎩ (t)

−u(t)∂x T 2 (x, t) − ˛ T 2 (x, t)

−˛ˇ

u()e−x T 2 dx d 0

where

−x

0 t

1

⎧ 1 ⎪ −1 (t) ˆ (x, t)) S(t) − ∂t T ∗ − ˛Tˆ (x, t) ˆ ⎪ (t) ˆ = sgn(ıT ⎪ ⎪ ⎪ ⎪ 0t ⎨

Using Young’s inequality, V˙ 2 becomes V˙ 2 (t)

and the second member of inequality (33) deﬁnes a function of class-K (with respect to T) (more details on class-K functions can be found in [30]). Therefore from [31] we conclude that the L2 norm of the state estimation error T converges globally uniformly asymptotically to zero. 䊐

(29)

0

1

t+tf

t

u(t) =

e−ˇ(t−s) T 2 (x, s) ds dx

−˛ˇ

V2 (tf + t) − V2 (t) ≤ −

t

+2˛ˇ

e−x T 2 (x, t) dx 0

⎧ ∂t T (x, t) = −u(t)∂x T (x, t) − ˛ T (x, t) ⎪ ⎪ ⎪ t ⎪ ⎨ −ˇ(t−s) +˛ˇ

(32)

1

≤ −u(t)

= k T (1, t) + g(t), = Tˆ0 (x),

V2 (t) =

e−x T 2 (x, t) dx

(26)

where T (x, t) := Tˆ (x, t) − T (x, t) is the state estimation error, k is a tuning parameter, Tˆ0 , g and S satisfy assumptions (i)–(iv) while the control input u is assumed to be positive. These ensure that Tˆ ∈ C(0, ∞, (0, 1)) ∩ C(0, ∞, L2 (0, 1)). The derivative of T is written as

Proof. tional

1

≤ −u(t)

− u(t) T 2 (1, t)(e− − k) e

T (x, 0)

(31)

− T 2 (0, t)).

0

0

⎪ ⎪ ⎪ T (0, t) ⎪ ⎩

t

+˛ˇ

T (x, 0)

e−x T 2 (x, t) dx

If Tˆ (0, t) is chosen as in Eq. (26), u(t) ∈ L∞ (R+ ∗ ), while the tuning parameter 0 < k < 1 then we can always ﬁnd a > 0 such that (e− − k) is positive. Eq. (32) becomes V˙ 2 (t)

= −u(t)∂x Tˆ (x, t) − ˛Tˆ (x, t)

⎪ ⎪ ⎪ Tˆ (0, t) ⎪ ⎪ ⎩ˆ

1

−u(t)

0 −u(t)( T 2 (1, t)e−

In order to apply the control law proposed in Eq. (16), the distributed state proﬁle T(x, t) is required, however only the inlet and outlet states are available. In this section an observer is designed in order to estimate the full-state proﬁle T(x, t). Let us consider the following Luenberger-like observer

V˙ 2 (t) ≤

4. Observer design

⎧ ∂t Tˆ (x, t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

Using integration by parts, we obtain the following inequality

The proof of theorem 3 relies on the following lemma.

S. Mechhoud, T.-M. Laleg-Kirati / Journal of Process Control 81 (2019) 223–231

229

Fig. 2. Outlet and distributed target temperature proﬁles.

Fig. 1. Source term f(t).

Lemma 1. If Assumption 1 is satisﬁed and if u(t) = (t) ˆ > 0 then the tracking error ı(t) and the L2 -norm of the state estimation error T converge uniformly asymptotically to zero. Proof.

The proof can be found in Appendix A.

䊐

Fig. 3. Outlet state estimation and its relative error.

At this stage we can present the proof of Theorem 3. Proof. We start by proving that the interval]0, T2 ] is positively invariant. From the analysis presented in proof of Proposition 2 we know that T is positive. When T = T2 then u(t) = min(umax , (t)) ˆ ≥ umin > 0. If u = umax then following the same reasoning presented in Proposition 2’s proof we get ∂t T(T = T2 ) ≤ 0. Also if u(t) = (t) ˆ then from Lemma 1 we obtain the uniform asymptotic convergence of ı(t). Therefore, since T is in C(0, ∞, (0, 1)) then this implies that when T = T2 it converges then to T* . As a result we conclude that ∂t T(T = T2 ) ≤ 0 and the domain]0, T2 ] is positively invariant. To prove part 2 of Theorem 3 we employ the same steps as in Proposition 2’s proof. The key is to demonstrate that the interval [T¯ 1 , T¯ 2 ] is positively invariant where T¯ 1 and T¯ 2 are deﬁned as in Proposition 2’s proof. The only difference compared to this latter proof is when u(t) = (t). ˆ However we have already proved in Lemma 1 that in this case, i.e. u(t) = (t) ˆ then ı and || T|| are both uniformly asymptotically converging to zero. 䊐

6. Simulation results To check the performance of the proposed control design, we propose the following example which represents the model of a one-loop solar collector parabolic through [10]

⎧ ∂t Tm = −ˇ(Tm (x, t) − T (x, t)) + f (t), (x, t) ∈ Qxt ⎪ ⎪ ⎪ ⎪ ⎨ ∂t T = −u(t)∂x T + ˛(Tm (x, t) − T (x, t)), (x, t) ∈ Qxt 0 , T (x, 0) = T (x), x ∈ ]0, 1] ⎪ Tm (x, 0) = Tm 0 ⎪ ⎪ ⎪ ⎩

The target temperature proﬁle T* (x, t) and the outlet target T* (1, t) proﬁles are given in Fig. 2. To generate the desired proﬁle T* (x, t) ﬁrst we solved the steady-state solution of Eq. (20) given by x

Sss Sss ≤ T ∗ (x) − T ∗ (0) ≤ x . umax umin

then this solution is multiplied by a desired temporal proﬁle such that Assumption 2 is satisﬁed. We select T* (0) = 20◦ . In this regard, 0 and the boundary condition T(0, t) and the initial conditions Tm 0 = T(x, 0) are ﬁxed at the ambient temperature, i.e. T (0, t) = Tm T (x, 0) = 20◦ to ensure that T(0, t) ≤ T* and therefore that T(0, t) ≤ T2 . The proﬁles of f(t) and T* (x, t) are inspired from practical applications of the solar collector plant such as [10] and [32]. Implementation of Eqs. (37), (26) and (36) is carried out in Matlab/Simulink® using the quadratic B-splines Galerkin method [33,34]. We choose the observer tuning gain equals to k = 0.8 and the controller gain kp = 10. Note that since the control law design was not based on Galerkin truncation, there is no risk of spillover instabilities. To implement the observer given in PDE (26), which is a PIDE, we transform PIDE (26) into the following system

⎧ ∂t z = −ˇz(x, t) + ˛ˇTˆ (x, t) + ˛f (t), (x, t) ∈ Qxt ⎪ ⎪ ⎪ ⎪ ⎨ ∂t Tˆ = −u(t)∂x Tˆ − ˛Tˆ + z(x, t), (x, t) ∈ Qxt 0, T ⎪ ˆ (x, 0) = Tˆ 0 (x), x ∈ ]0, 1] z(x, 0) = ˛Tˆm ⎪ ⎪ ⎪ ⎩

(38)

Tˆ (0, t) = k T (1, t) − g(t).

where (37)

T (0, t) = g(t), t ∈ [0, ∞[

PDE (37) is similar to PDE (1), where the variables (x, t) are the normalized space and time variables, the source term f(t) is selected as depicted in Fig. 1 and the control input u(t) is the pump volumetric ﬂow rate. The maximum ﬂow rate is umax = 12 L s−1 where as the minimum tolerated is umin = 2 L s−1 (see [10] for more details).

0 z(x, t) = ˛e−ˇt Tˆm +˛

1

e−ˇ(t−s) (ˇTˆ (x, s) + f (s)) ds 0

0 = 40◦ . The same approach is used for the implementation of and Tˆm the control law in Eq. (36) while the sign function is approximated by the hyperbolic tangent function to avoid the chattering effect. 0 = 0 to check the validity of Remark 1. Note that Tˆm / Tm The outlet estimation error T(1, t) is depicted in Fig. 3 where it shows that the observer in PIDE (26) performs well. Fig. 4, left

230

S. Mechhoud, T.-M. Laleg-Kirati / Journal of Process Control 81 (2019) 223–231

Conﬂict of interest The authors declare that they have no known competing ﬁnancial interests or personal relationships that could have appeared to inﬂuence the work reported in this paper. Acknowledgement Research reported in this publication has been supported by the King Abdullah University of Science and Technology (KAUST). Fig. 4. Outlet temperature tracking and its relative error.

Appendix A. Proof of Lemma 1 We have

ı(t)

=

1

|ıT (x, t)| dx

0 1

|T ∗ (x, t) − Tˆ (x, t) + Tˆ (x, t) − T (x, t)| dx

=

0

1

ˆ + ≤ ı(t)

1

1

ˆ +ω ≤ ı(t)

t

e−ˇ(t−s) T 2 ds dx

T 2 dx + ω˛

Fig. 5. Proﬁles of the control input u(t) and the average tracking error ı(t).

(A.1)

| T | dx (Using triangular inequality)

0

0

0

0

(RH)

panel, shows that using the proposed control law in Eqs. (35)–(36), the outlet temperature T(1, t) tracks the target T* (1, t) efﬁciently. Note that the control problem is bilinear and the convergence of the tracking error is just uniformly asymptotic, thus the convergence to zero needs a longer time window. Nevertheless, the simulation results demonstrate the tracking error’s tendency to converge towards zero. At steady-state the tracking error in Fig. 4, right panel, is less than 1% whereas at the ramp segment it does not exceed 5% which is admissible for our application. Also, the controlled variable ı(t) in Fig. 5 (right panel) converges rapidly and in ﬁnite time. The control input u(t) shown in Fig. 5 (left panel) is admissible and physically coherent, i.e. it respects it upper and lower bounds. It states that to decrease the outlet temperature, the ﬂow rate should increase in order to discharge rapidly the excess of heat. Conversely to increase the temperature, the ﬂow rate should decrease in order to increase the period of time in which the ﬂuid is exposed to the solar irradiance.

7. Conclusion In this paper we studied the control problem of a system of a 1st-order hyperbolic PDE coupled with an inﬁnite dimensional ODE. In practice this system represents the model of a one-loop solar collector parabolic trough. The control problem is constrained and bilinear, and the measurements are located at the boundaries. By re-writing the system in the form of a PIDE, we designed an observer-based bounded controller which guarantees the uniform asymptotic stability of the system in closed-loop. Simulation results show the efﬁciency of the proposed method and claim that the convergence occurs even in ﬁnite time. In the future, we will investigate the problem of in-domain and boundary disturbance rejection. Note that since the manipulated variable is the ﬂow velocity, applying already available results on disturbance rejection, e.g. [35], is not straightforward and remains an open problem. We will also consider the case where the diffusion term in the inﬁnite ODE is not negligible which will lead to a system of coupled hyperbolic and parabolic PDEs.

where we can always ﬁnd 1 ≤ ω <∞ such that | T| ≤ ω T2 and the second term added in the last inequality will be justiﬁed later in inequality (A.4). In the stability analysis below, we employ this inequality (A.1) instead of ı(t). ˆ written as The time derivative of ı(t) ˆ˙ ı(t) =

1

ˆ ) sgn(ıT

t

∂t T ∗ + ˛Tˆ − ˛ˇ

0

e−ˇ(t−s) Tˆ (s) ds 0

1

(A.2)

ˆ )∂x Tˆ dx sgn(ıT

−S(t)) dx + u(t) 0

Now if u(t) is given by Eq. (36) then Eq. (A.2) becomes ˆ˙ ˆ ı(t) = −kp ı(t).

(A.3)

ˆ towards zero. Eq. (A.3) states the exponential convergence of ı(t) Since T and Tˆ are in C(0, ∞, (0, 1)) then we can infer the exponential ˆ towards zero. pointwise convergence of ıT Now, the derivative of the right-hand side of inequality (A.1) is given by ˙ (RH)

ˆ − u(t)ω(1 − k) T 2 (1, t) = −kp ı(t)

−˛(1 − e−ˇt )ω

+ω

e−ˇ(t−s) T 2 ds dx

1

0

˛ 0

T 2 dx 0

t

+ω˛ˇ

1

ˆ ≤ −kp ı(t)

t

e−ˇ(t−s) T 2 ds

T 2 − ˇ

(A.4)

dx

0

where the second term in the ﬁrst inequality is obtained using the observer boundary condition in system (26). From inequality (A.4) we can already conclude on the boundedness of ı(t) i.e.

∃ > 0 : ı(t) < < ∞, t ∈ [0, ∞].

S. Mechhoud, T.-M. Laleg-Kirati / Journal of Process Control 81 (2019) 223–231

Now, let us deﬁne the following Lyapunov equation V (t) =

1 2

ˆ +ω ı(t)

1

1

t

e−ˇ(t−s) T 2 ds dx

T dx + ω˛ 0

2

0

0

+V2 (t), (A.5) where 1 ≤ ω <∞ and V2 (t) is the Lyapunov function deﬁned in Eq. (28). Using inequality (A.4), the time derivative of V(t) can be bounded as follows.

V˙ (t)

1

e−x T 2 dx

≤ −kp ıˆ 2 (t) − u(t)

(A.6)

0

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