MODEL REDUCTION THROUGH IDENTIFICATION – APPLICATION TO SOME DIFFUSION-CONVECTION PROBLEMS IN HEAT TRANSFER, WITH AN EXTENSION TOWARDS CONTROL STRATEGIES Y. Rouizi1 , Y. Favennec2 , D. Petit1 , and Y. Jarny2 1
Institut P’ CNRS-ENSMA-Université de Poitiers, UPR 3346, Département Fluides, Thermique, Combustion. ENSMA Téléport 2. 1, avenue Clément Ader, BP 40109, F86961 FUTUROSCOPE CHASSENEUIL Cedex, France 2 LTN UMR CNRS 6607, Polytech’ Nantes, La chantrerie - 44306 NANTES Cedex 3, France
ABSTRACT
1.
In the framework of heat convection problems governed by Navier Stokes equations, coupled with the energy equation, the modeling (by finite elements for example) leads to very large systems, typically of order 105 to 109 . Of course, such systems cannot be used for on-line control algorithms that should deal with systems of order less than 100 (typically 10 actually). In order to take benefit of the knowledge of the studied system through such a fine modeling, model reduction techniques present a large interest to obtain a suitable low order model that can be then used in a control process. Our reduced models are obtained through the modal identification method. This method leans on the solution of an optimization problem of parameter estimation: one defines the structure of the reduced model formulation before estimating the related vectors and matrices through the minimization of a corresponding functional. When the optimization problem is well-posed and a priori convex, gradient-type iterative methods coupled with the adjoint-state method can be used. If the functional presents several local minima, zero-order (global) algorithms are used. After the reduced-order model is validated with input data different from those used within the identification process, a control algorithm can be then coupled with it. The feedback control laws can be based either on the reduced state or on the output. Next, a Kalman filter leads to evaluate the state through a limited number of measurements. The developed numerical application deals with a temperature field within a 2D stationary flow over a backward-facing step. The goal is to keep the outlet temperature as close as possible to a constant given temperature profile downstream the step whatever the pipe inlet temperature fluctuations. One thus searches some“optimal” heat fluxes upstream the step that counteract the inlet temperature variations.
Nowadays, the need to optimize system performance has become essential to reduce, for example, functions of production costs or production rates of pollution such as emissions of greenhouse gases. The industries are more and more confronted with this type of problem: the process control is becoming increasingly necessary. The optimal control of a system allows to reach a satisfying final goal while respecting certain constraints.
Key words: Optimization, Model Reduction, Modal Identification Method, Feedback Control, Backwardfacing step flow, Heat Transfer Convection.
INTRODUCTION
In the areas of heat transfer and fluid mechanics, modeling returns often to a discretization of the local equations (the energy equation and/or the Navier-Stokes equations) and then to a resolution of ordinary differential equations which result from this. This discretization, which is becoming increasingly fine to understand and to simulate some phenomena hitherto unknown, thus implies models with a large number of degrees of freedom. In spite of the development of the powers of the machines of calculations, simulations take more and more computing time and occupy more and more place memory. This makes impossible the use of such models for applications in inverse or optimal control problems. In both cases, the need to obtain reliable and fast answers is paramount. It is on this level that the methods of model reduction play a full role. The goal of the model reduction is thus to build a model of small size, allowing very short execution times and occupying a small place memory. They are particularly adapted to applications in real-time. This Reduced Order Model (ROM) should preserve the properties of the studied physical phenomena and must also capture the essential characteristics of the model of reference to have an acceptable precision. There is an extensive bibliography where one can find many model reduction techniques either for linear systems or not. The reader interested by the techniques and the comments of these methods can refer to the more detailed reviews and articles [1, 2, 3] (to cite a few). We focus here on the use of the Modal Identification Method (MIM) to build ROM.
MIM uses concepts stemming from the automatics community and has been developed in heat transfer to identify ROM in linear thermal diffusion processes from experimental data [4], and then has been extended to non-linear heat transfer [5, 6]. An extension of the method to fluid flows has been proposed [7, 8], the identification method being reformulated using the adjoint state method to compute the gradient of the cost function to be minimized [9].
where ε = Y − Yc is the tracking error that integrates the difference between the output model Y and the desired output Yc . I denotes a time range. Q is a weighting matrix and `2p is a diagonal matrix `2 Ip (with Ip the identity matrix of dimension p), and ` is a parameter that penalizes excessive control inputs magnitudes. In order to obtain the optimal command (i.e. the minimum of cost function J ) we introduce the Hamiltonien [10] :
The Modal Identification Method relies on two main steps:
H = εT Q ε + uT `2p u + λT (AX + Bu)
• The first stage is to define a structure of equations for the ROM. Starting from the local partial differential equations describing the physics, a state space representation which arises from space discretization is considered as a general detailed model structure. Eventually, the ROM equations are similar to those of the state space representation but under the modal form. • Once the structure is chosen, the next step is to identify the parameters that define the ROM through minimization of a cost function that measures the difference between the outputs of the ROM and the outputs of the system (detailed model or real process). The identification process therefore leans on the solution of an inverse problem of parameter estimation, where the data to be fitted are coming either from simulations performed with the detailed model or from in-situ measurements.
2.
Consider a linear time-invariant state space model ˙ X(t) = AX(t) + Bu(t) Y (t) = CX(t) p
(1) (2)
q
where X ∈ R ,u ∈ R , and Y ∈ R are the state, the control input, and the controlled output, respectively, and A, B and C are the corresponding matrices. The optimal feedback control consists in acting on the control input u so that the outputs of the system Y present the wished characteristics. The use of optimal feedback control is particularly important when the dynamics of the system is known in spite of the disturbances, and for Multi-Inputs / Multi-outputs (MIMO) systems. Optimal control deals with the problem of finding a control law for a given system such that a certain optimality criterion is satisfied. Here we will focus on the feedback control with a quadratic cost function LQ (Linear Quadratic). In the present work, two optimal feedback control problems are considered: the regulator and the tracking problems. Firstly, we consider the tracking problem where the aim is to find the optimal command that makes the system follow a prescribed trajectory. This is performed minimizing the following cost function Z T ε Q ε + uT `2p u dt (3) J = I
where λ is the co-state variable. The differentiation of H leads to the equations of optimality [10] −λ˙ = 0
=
X˙
=
∂H = AT λ + C T QCX − C T QYc (5) ∂X ∂H = `2p u + B T λ (6) ∂u ∂H = AX + Bu (7) ∂λ
In order to perform the optimal feedback control, we define a feedback relationship between the state X and its related co-state λ λ = PX + L
(8)
Inserting this relationship into the adjoint equation (5) and integrating the optimality condition (6), one obtains the expression of the optimal control law −1
u = −(`2p )
−1
B T P X − (`2p )
BT L
(9)
where P and L are the so-called Riccati matrices, solution of the differential Riccati equations
OPTIMAL CONTROL
n
(4)
−1 P˙ + P A + AT P − P B(`2p ) B T P + C T QC = 0 −1 L˙ + AT L − P B(`2p ) B T L − C T QYc = 0 (10)
Let us consider now the regulation problem. As in [11, 12], the regulator problem consists in finding the control law that minimizes the deviation δX between the ¯ (relative to actual state X and the steady optimal state X the target Yc ). This can be mathematically rephrased as minimizing Z T 1 J = δX Q δX + δuT `2p δu dt (11) 2 I
¯ and δu = u − u where δX = X − X ¯ are the deviations from the optimal target. In short, the problem of regulation tries through its cost function to minimize the deviation compared to the optimal trajectory The control u solution of the optimal control (11) is given by [12] −1 T δu(t) = − `2p + B T ΠB B ΠA δx(t) (12)
where Π is the solution of the differential Riccati equation −1 T Π = AT ΠA − AT ΠB `2p + B T ΠB B ΠA + C T C (13)
The optimal control laws defined by (9) and (12) requires the knowledge of the state at each time. In practice, we have not a complete knowledge of the state of the system. In this case it is necessary to introduce an estimator of state, which will allow to give an approximation of the state X starting from some measurements. Here the Kalman filter is used to provide an optimal estimate of the state from stochastic state propagation and from noisy observations. Let us consider the following stochastic state model X˙ (t) = AX (t) + Bu (t) + Gw (t) (14) Y (t) = CX (t) + v (t) where w is the state noise and v is the observation noise. Both noises are assumed to be white zero mean Gaussian noises and to be independent: E[w (t) w (t + τ )] = W δ(τ ) E[v (t) v (t + τ )] = V δ(τ ) E[w (t) v (t + τ )] = 0
(15) (16) (17)
where E[·] is the esperance operator, and δ(`) = 1 only if the integer ` = 0, and δ(`) = 0 otherwise, and matrices W and V are the covariance matrices associated with noises w and v. The Kalman filter enables us under some assumptions to have on line estimating, or tracking, of unobservable sigˆ of nals. The filtering problem consists in estimating X the state X taking into account of outputs Y available until this moment. We consider the estimation error ex (t) defined by ˆ ex (t) = X(t) − X(t)
(18)
The Kalman filter allows to establish the structure of the ˆ such as the average of the estilinear system providing X mation error ex (t) tends towards 0 when t tends towards the infinity. It consists of two differential equations, one for the state estimate and one for the covariance matrix [10] h i h i ˆ (t) ˆ˙ (t) = AX ˆ (t) + Bu (t) + Kf X (t) − C X X (19) where Kf (t) = ΣC T V −1 is the Kalman gain matrix and Σ is solution of the differential Riccati equation Σ˙ = ΣAT + AΣ − ΣC T V −1 CΣ + GW GT
(20)
Now that the the control law by state feedback u is defined as well as the Kalman filter equations giving the ˆ we will present an example of estimation of the state X, application using ROM to illustrate the method.
3.
THE STUDIED SYSTEM
This section describes the 2D incompressible laminar flow over a backward facing step with an expansion ratio of 2.0, along with some numerical results of simulations. A schematic diagram of the considered geometry
is shown in Figure 1. The step height is h =1 cm, the upstream height is also h =1 cm, hence the downstream height is 2h. The chosen fluid, assumed to be Newtonian and incompressible is air, with the following constant properties: dynamic viscosity µ = 1.81 × 10−5 kg/(m s), density ρ = 1.205 kg/m3 , heat capacity Cp = 1005 J/(kg K), and thermal conductivity λ = 0.0262 W/(m K). The coordinate system is defined as shown schematically in this → → figure, where the e1 and e2 coordinate directions denote respectively the streamwise and the transverse directions. The flow geometry was considered by Armaly et al. [13] who chose to define the Reynolds number Re, as follows: Re =
U∞ 2h ν
(21)
where U∞ is the inlet mean velocity and ν = µρ is the kinematic viscosity of the fluid. The use of 2h as the characteristic length scale is attributed to the fact that the Reynolds number is based upon the hydraulic diameter of the inlet channel. The flow entering the channel is assumed to be fully-developed and is described using a u1 -velocity parabola for laminar flow, for x1 = −4h, x2 ∈ [h : 2h]: (
Re ν u ¯1 (−4h, x2 ) = 3 3 (x2 − h)(2h − x2 ) h u ¯2 (−4h, x2 ) = 0.
(22)
where (u1 , u the components of the velocity vec2→) are → tor v in the e1 , e2 coordinate directions. In the present work, the outflow boundary condition that assumes that the flow is fully developed at the outlet section is used. This can be done when the outlet is located far enough 1 downstream, assuming that ∂u ∂x1 = 0. On all other boundaries a null velocity is prescribed. In the present study, the flow is driven at a fixed Reynolds number equaling 500. Hence, only the energy problem is considered. At the entrance of the pipe, the fluid is at temperature T∞ and the ∂T = 0. Next, two outflow boundary condition leads to ∂x 1 heaters (in terms of fluxes) can heat the fluid. Heaters respectively located at x1 ∈ [2h : 4h] bottom and top (see Fig. 1) heat the flow at reason of flux density equaling respectively ϕ1 and ϕ2 . All other boundaries but the inflow and outflow are considered as insulated. The thermal initial condition is T (t = 0) = 300 K.
3.1. The reference model The computational fluid dynamics (CFD) software R 6.3.26) [14] has been used to carry out com(Fluent putations. Grid independence tests have been performed using several grid densities, and the reattachment location on the stepped wall has been used as the criterion. In the present study, 144, 247 nodes have been used for the computations. The experimental study provided by Armaly et al. [13] and numerical studies provided by Erturk
[15] and Zang et al. [16] have been used to validate our model. It can be observed that several benchmark solutions are available in the literature when the heat flux is applied in the stepped wall. However, to our knowledge, there is no benchmark solutions available when the heat flux is applied in the stepped wall just before the sudden expansion region. Once the fluid simulation methodology is validated, the thermal simulation in forced convection problem with the CFD code should be accurate. Note that the aim of the present paper is not to give a complete study of the test problem but to give an example to illustrate a optimal control with a ROM. The objective of the present study is to control the temperature profile on a vertical line located at x1 = 16h through the two heaters when the inlet temperature (at x1 = 0) varies in time. More precisely we would like the temperature on the considered line to be as close as possible to the temperature T˜ whatever the inlet temperature T∞ (t) ≤ T˜.
have a very different meaning since T in (23) represents the space-continuous temperature variable while T in (24) represents the semi-discretized temperature variable (e.g. on the nodes of a Finite Element mesh). Eventually, u in (24) is the vector of control variables, i.e. T u(t) = [ϕ1 (t), ϕ2 (t)] . The solution of (24) gives the temperature response T (t) from given inlet temperature T∞ (t) and input flux u(t), and with a given initial condition T (0) = T0 . The number N of differential equations present in (24) depends on the discretization type and fineness. This set of ordinary differential equations (24) is coupled with an output equation when space selection is required: Y = CT
(25)
where Y ∈ Rq , q ≤ N . The output Y actually represents the quantity of interest for control applications detailed later on. Note that whatever the discretization type and fineness, the number N of differential equations in (24) is too large most for any real on-line control application. A way to reduce the number of differential equations in (24) (or in any other equivalent system) is to consider model reduction approaches.
3.2. The governing equation The energy problem for a given velocity distribution v is given by the following convection-diffusion equation ∂T ρCp + v · ∇T − ∇ · (λ∇T ) = 0 (23) ∂t where T = T (x, t) is the temperature distribution for t ≥ 0 and x ∈ Ω. Within the scope of this paper, Ω represents the backward-facing step pipe presented schematically in Fig. 1. The boundary conditions are set as follows. The time-dependent inlet temperature is assumed to be x2 independent, hence T (x, t) = T∞ (t) for x ∈ ∂Ωin . The heaters on boundaries ∂Ωϕk deliver a known prescribed flux density ϕk , hence λ∇T · n = −ϕk for x ∈ ∂Ωϕk , k = 1, 2 with n the outward unit normal vector. A zero diffusion flux condition is applied at the outflow boundary as the flow is assumed to be fully-developed. Next, we consider a null flux on all other boundaries, then ∇T · n = 0 for x ∈ ∂Ωwall \ (∪k ∂Ωφk ∪ ∂Ωin ). An initial condition T (x, 0) = T0 (x) is also considered. The space discretization of the partial differential equation (23) with above-considered boundary conditions leads to the set of ordinary differential equations T˙ = AT + Bu + GT∞
(24)
where A gathers both the convection operator (v · ∇) and the diffusion operator (∇ · λ∇), B is the flux input operator, and G is the operator that applies the inlet Dirichlet boundary condition onto the system. Note that the temperature variables T in (24) and T in (23)
4.
MODEL REDUCTION
Among the large number of model reduction methods, identification methods have proved to be efficient for diffusion-convection problems. For instance, the Modal Identification Method consists in identifying the components of a matrix system equivalent to (24) but written in its modal base. Only a small number of modes are considered for the identification. Usually, the MIM is coupled with iterative methods using gradient-type methods and the adjoint-state method for the computation of the objective function gradient. We describe hereafter the main features related to the MIM. The reader could refer to [17, 8, 18, 6, 9, 5] for more details on recently developed numerical methods as well as related applications. Let Aˆ be the diagonal matrix of A involved in (24) and Σ the matrix of eigenvectors of A such that Aˆ = Σ−1 AΣ, and let perform the change of variable T = ΣX, X ∈ RN where N is the number of degrees of freedom involved in (24). Doing so, (24) becomes: ˆ + Bu ˆ + GT ˆ ∞ X˙ = AX
(26)
where Bˆ = Σ−1 B and Gˆ = Σ−1 G. The output relationship is also expressed to as: ˆ Y = CX
(27)
where one used Cˆ = CΣ. The modal formulation (26)(27) is still of order N . In many reduction methods, the
reduced models are obtained by selecting the most dominant modes, say n N modes. Instead of computing these modes and selecting them afterwards, the identification method consists in identifying the components of a matrix system of the form (26)-(27) but of smaller dimension, say with the reduced order variable x ∈ Rn : x˙ = ax + bu + gT∞ y = cx
(28) (29)
Note that the dimension of y in (29) and Y in (27) are the same. The identification algorithm is to identify n components for the diagonal matrix a, n × p components for the input matrix b, n components for the vector g, and q × n components for the selection matrix c. Note that for the particular application presented afterwards, the number p equals 2. An iterative gradient-type algorithm searches the components of vectors and matrices involved in (26)(27) comparing the related output data with those from a given model, say in this case a detailed model issued from a commercial software. The used gradient-type algorithm is the well-known BFGS algorithm [19]. At each identification iteration step, the gradient of a cost function is computed along with the descent direction. Only the components of a, b and g are taken into account for the cost gradient. A line search is performed. At this stage, say at the minimum of the cost function along the descent direction, the matrix c is actualized by using the least squares so that both reduced and detailed models fit again better. The cost function is written as q K 2 1 XX J (θ) = yj (θ, tk ) − Yj∗ (tk ) 2 j=1
ϕ2
x2
ϕ1 x1
Figure 1. The backward facing-step scheme. The upstream tube is 4h long ; the downstream tube is 30h long. the dashed line represents the place where the sensors are and where the optimization is performed on.
Algorithm 1 The ROM identification algorithm. The outer loop increases the reduced model order while the inner loop iterates searching the system (28)-(29) components 1. Form the output Y ∗ from one detailed model simulation 2. Let n = 1 \\ n = dim x
(30)
k=1
where the parameters vector θ = {aii , bij , gi , cki }, i = 1, . . . , n, j = 1, . . . , p, k = 1, . . . , q gathers all unknown components of (28)-(29). The identification of the ROM is schematically presented in Algorithm 1. The number K used in the step (3b) of Algorithm 1 is the number of time steps used for the time discretization of the total time range of simulation. In order to get representative data of the dynamic of the system needed for the reduction process, some input fluxes are applied on boundaries ∂Ωϕ1 and ∂Ωϕ2 . When, as in this considered case, the physics of the problem is linear, the method of superposition can be used. It consists in identifying submodels (28). One sub-model (28) is required for each input. Thus for our considered case, one should build three sub-models, one related to ϕ1 , one related to ϕ2 , and one related to T∞ . Then, all sub-models are gathered within the output equation (29). Though this method is efficient in linear model reduction with easy implementation, the resulting formulation seems inappropriate when coupled with feedback optimal control problems where the control laws are likely to depend on the state rather than on the output. Then, since the matrix a in (28) is diagonal, there is no coupling between states x, and only the matrix d in (29) plays the coupling between distinct inputs,
3. ` = 0, initialize a = a0n , b = b0n , and g = gn0 ; (a) Compute the reduced model (28) to get x; (b) Identify the matrix c through the minimization of the quadratic norm of cχ − Υ where χ is the collection of x, and Υ is the collection of Y ∗ for k = 1, . . . , K, i.e. through −1 ct = (χ · χt ) χΥt ; (c) Compute the cost function J comparing y solution of (29) and Y ∗ from step (1) (d) Compute the cost function gradient (e) Compute the new parameters from a gradienttype algorithm (BFGS quasi-Newton algorithm); (f) If satisfactory result : end, else ` ← ` + 1 and return to step (3a). 4. If satisfactory result : end, else n ← n+1 and return to step (3).
(defined by (30)) as a function of the increasing reduced model order and, for each order, the decreasing cost function value with respect to the inner iterations (see Algorithm 1). This figure shows that for a given order the cost function is still decreasing and that the cost function value is generally decreasing with respect to the reduced model order at the end of the optimization iterations. 2009 iterations in total were necessary for global convergence until order 10.
12 Armally Zang Erturk Present 10
8
25 X2/h Erturk X2/h Present X3/h Erturk X3/h Present
6 20 15 4
108
10
2 100
200
300
400
500
500
600
600
700
800
Figure 2. Size of the main recirculation region length X1 /h as a function of the Reynolds number Re. The locations of the detachment point X2 /h and the reattachment point X3 /h as a function of the Reynolds number Re are shown in the inset.
106 105 104 103 102
say ϕ1 , ϕ2 and T∞ on one hand, and the outputs y on the other hand. When compared to the superposition method, the direct method used for nonlinear systems leads to consider smaller models with no lost of accuracy [20]. As a consequence, the different inputs are applied successively within the same test, and a global ROM is identified, the single constraint being the distinction of the input-output relationships. The initial temperature is taken constant at 300 K with inlet temperature also equal to 300 K and null fluxes ϕ1 and ϕ2 . At time zero the flux ϕ1 jumps from 0 to 300 W/m2 then stays constant. At time t = 40 s the flux ϕ2 jumps from 0 to 300 W/m2 then stays constant. Then, at time t = 80 s, the inlet temperature jumps to 325 K then stays constant. This procedure is presented in Fig. 3. 400
330
350
325
300 ϕ1 (t), ϕ2 (t)
200
315
150
310
100
Tin (t)
320
250
0
500
1000 1500 iterations
2000
2500
Figure 4. Evolution of the cost function value J with respect to the optimization iterations for reduced model orders increasing until n=10. Note that roughly 200 iterations are needed for convergence for each given reduced model order. Table 1 gives the evolution of the cost function and the mean quadratic error σ (such that σ 2 = 2(K × q)−1 J ) with respect to the reduced model order after reached inner-loop convergence. order 1 2 3 4 5 6 7 8 9 10
J 0.214×106 0.221×105 0.783×104 0.234×104 0.872×103 0.517×103 0.334×103 0.296×103 0.128×103 0.666×102
σ 0.141×101 0.452×100 0.269×100 0.147×100 0.899×10−1 0.692×10−1 0.556×10−1 0.523×10−1 0.344×10−1 0.248×10−1
Table 1. Evolution of the cost function value J and of the mean quadratic errors σ with respect to the reduced model order n.
305
50 ϕ1 (l.h.s.) ϕ2 (l.h.s.) Tin (r.h.s.)
0 -50 -40
n=1 n=2 n=3 n=4 n=5 n=6 n=7 n=8 n=9 n=10
107
700
J
5 400
0
40
80
120
300 295 160
time (s)
Figure 3. Time evolution (in s) of ϕ1 , ϕ2 and T∞ as known input. Both flux densities ϕ1 and ϕ2 are expressed in W/m2 and are projected on the left hand side of the graph. The inlet temperature T∞ is expressed in K and is to be projected on the right hand side of the graph. Fig. 4 presents the evolution of the cost function value J
Next, the aim is to validate the ROM finding out if the identified ROM is able to reproduce with accuracy the output Y of the original detailed model when other input evolutions are prescribed. According to Table 1, it can be seen that the best minimization is obtained for the highest order. The increase of the order model for 6 to 10 does not improve significantly the cost function. For our study order 6 gives enough satisfying results. The input fluxes used for validation are the ones presented in figure 5. The resulting temperature evolutions in some outputs for both the detailed model and the reduced model are also presented on figure 5. One can see that the temperature evolutions are very close, the maximum temperature differ-
1
ence between both models being negligible when compared to orders of magnitude of the temperature steps. Further, the obtained reduced model of order 6 will be the one used for solving optimal control problems, this model being assumed to approximate sufficiently well the physical behavior of the heated flow over the concerned backward-facing step. It is emphasized that the required computation time for solving the full original flow problem along with the related energy problem was about a couple of hours. In the other hand, the solution of the ROM requires less than 0.1 s of computation, including all file readings and writings.
Position x2/h
0.75
0.5
0.25
0 318 318.5 319 319.5 320 320.5 321 321.5 322 Temperature (K)
Figure 6. Temperature profile obtained u¯ and T∞ = 300 K. vector y is as close as possible to a constant vector determined by the user (e.g yc = 320 K). When considering the regulation problem, the control law is based on the deviation δx between the actual state x and the optimal steady-state x¯ rather than on output y. In order to find u ¯ and x ¯ implicitly involved in (11), the following overdetermined system (with no perturbation) is solved ST S u ¯ = S T yc (31)
where the least squares solution is given by Figure 5. Top: inlet temperature evolution (K); middle: input flux evolution (W/m2 – solid line : ϕ1 , dashed line: ϕ2 ); bottom left: temperature evolution (K) for both the detailed model (dashed line, l.h.s.) and the six-order reduced model (solid line, l.h.s.) for location x1 = 16h and x2 = 0, and the absolute difference between both models (dotted, r.h.s.); bottom right: temperature profile (K) at t=4 s and x1 = 16h for both the detailed model (dashed line) and the six-order reduced model (solid line).
5.
CONTROL APLLICATION
The goal here is to show the feasibility of the approach combining the reduction of model and the optimal control theory. There exists many applications combining these two approaches, among which one can quote [21, 22, 23].
u ¯ = ST S
(32)
Figure 6, shows the optimal temperature profile along x1 = 16h obtained with u¯ and without perturbation (i.e. T∞ = 300 K). We note that without disturbance, the command u¯ allows a close temperature profile to the set 320 K. However, in presence of disturbances, i.e. when the temperature at the entrance of channel T∞ varies , the command u ¯ is not valid any more. The objective now is, when the inlet temperature T∞ is disturbed with the signal of the figure 8, to seek on-line for the command allowing to correct this influence and to make the obtained temperature close to 320 K. ϕ2 X3
X2 h
One formulates the optimal control problem: whatever the supposed unknown disturbance T∞ , one seeks the optimal inputs u = [ϕ1 (t) , ϕ2 (t)]T so that the output
S T yc
and S is the sensibility matrix S = −ca−1 b. Then x¯ is obtained by x ¯ = −a−1 b¯ u.
For our part, we propose to use a ROM identified by the MIM to determine a control law in order to control a temperature profile. We are interested in the reduced model of order n = 6 identified and developed in the previous section. Recall that this model is based on three inputs, and has an observable vector of dimension 135 located on the line 16h. The flow is heated by both flux densities placed just upstream of the step and there is a temperature fluctuation around a nominal value (that must be counteracted) at the channel entrance.
−1
T∞
Point C
Re = cte
Point B
h
ϕ1 4h
0
Point A
e2 e1 X1
30h
x1/h = 16
h
Figure 7. Studied system. Kalman filter is used here to estimate the state x from 3 temperature measurements placed at x1 = 16h and for vertical positions x2 = h2 , x2 = h and x2 = 3h 2 . The obtained optimal input u = [ϕ1 (t), ϕ2 (t)]T is reported in figure 9.
4500
310
`22 `2 `2 `2 `2 `
Flux density (W/m2)
Temperature T∞ (K)
4000 305
300
295
3500 3000
= 10+0 = 10−1 = 10−2 = 10−3 = 10−4 = 10−5
2500 2000 1500 1000 500
290
0
50
100
150 200 Time (s)
250
0
300
10
15 Time (s)
20
25
30
ϕ1 ϕ2
3500 3000
2800
`22 `2 `2 `2 `2 `
2500
Flux density (W/m2)
Flux density (W/m2 )
5
Figure 11. Contribution of the `2p parameter on the evolution of the input ϕ1 .
Figure 8. Perturbation T∞ (t). 4000
0
2000 1500 1000 500 0
5
10
15 Time (s)
20
25
30
3h 2
is plotted en figure The temperature evolution at x2 = 10. We can note that the no-controlled temperature is not close to 320 K. The obtained optimal law allows to reduce drastically this discrepancy. These results are relative to a regulation problem with a cost value `2p = 10−4 . In this part one analyzes the sensitivity of the results to this parameter. Let us recall that in the criterion to be minimized, the parameter `2p comes to balance the effect of the commande u : to have a weak δu it is necessary to give a raised value to `2p . This avoids the use of excessive controls. For different values of `2p , one presents in figures 11 and 12 the evolutions of the optimal commands and in figure 13 the evolution of the corresponding space-mean output temperature. As it was planned, the variations of u decrease as `2p increase (10−5 to 1). The value `2p = 1 crushes the varia-
2200 2000
1600
0
5
10
15 Time (s)
20
25
30
Figure 12. Contribution of the `2p parameter on the evolution of the input ϕ2 .
tion of the obtained command u. For a low value of `2p (for example `2p = 10−5 ) the optimal commands present strong fluctuations. In return, this value allows to obtain the best result with respect to the set point c.f. figure 13). According to all these curves, we thus see that the adjustment of the value of `2p corresponds to a compromise between acceptable fluctuations in inputs and deviations from the target. In what precedes, we presented the results of optimal command concerning a problem of regulation. We propose now in figure 14 the optimal command solution of the tracking problem. The evolution of the mean temperature obtained by this command is given and compared with the results of the regulation problem in figure 15.
With control Without control
326
2400
1800
Figure 9. Optimal command u = [ϕ1 (t), ϕ2 (t)]T .
328
2600
= 10+0 = 10−1 = 10−2 = 10−3 = 10−4 = 10−5
Température (K)
324 322 320 318 316 314 312 0
5
10
15 Temps (s)
20
25
30
Figure 10. Temperature evolution at x2 =
3h 2 .
One notes a light discrepancy between the solutions of both problems (tracking and regulation), while keeping the same dynamics. The introduction of an integrator would allow to correct this static discrepancy. We recall that the regulation problem requires the solution of a Riccati equation. Whereas the tracking problem requires the solution of two equations of Riccati, however the tracking problem allows to follow a target which evolves in the time.
`22 `2 `2 `2 `2 `
Mean temperature (K)
326 324 322
328
= 10+0 = 10−1 = 10−2 = 10−3 = 10−4 = 10−5
320 318 316 314 312 310
5
10
15 Time (s)
20
25
30
322 320 318 316 314
3500 3000 2500 2000
0
5
10
15 20 Times (s)
25
30
Figure 15. Mean temperature evolution for all cases.
3. A.C. Antoulas. An overview of approximation methods for large-scale dynamical systems. Annual Reviews in Control, 29(2):181 – 190, 2005.
1500 1000 500 0
310
2. D.J. Lucia, P.S. Beran, and Silva W.A. Reducedorder modeling: new approaches for computational physics. Progress in Aerospace Sciences, 40(1-2):51 – 117, 2004.
ϕ1 ϕ2
4000 Flux density (W/m2)
324
312 0
Figure 13. Contribution of the `2p parameter on the evolution of the mean temperature.
0
5
10
15 Time (s)
20
25
30
Figure 14. Evolution of inputs ϕ1 (t) and ϕ2 (t) as a function of time for the tracking problem.
6.
Tracking problem Regulation problem Without control
326 Mean temperature (K)
328
CONCLUSION
It has been presented in this study tow feedback optimal control resolutions, using a Reduced Order Model, applied on a diffusion-convection problem. One concerned the purely regulation problem while the other one dealt with the tracking problem. A Kalman filter has been designed to evaluate the state from a few measurement points. If this system had been treated by a classic modelling, the corresponding model would not have allowed the control. That is the reason why a reduction technique has been used. The Modal Identification Method, which leans on an inverse problem of parameter estimation, is well suited for convection diffusion processes. This study also showed that the Modal Identification Method seems to be efficient when being coupled with feedback controllers, integrating unknow perturbations and a filter to reconstruct the consequently unknown state. Next steps will include the coupling with some fluid mechanics reduced models in order to integrate some input velocity within the command.
4. R. Pasquetti and D. Petit. Analyse modale d’un processus de diffusion thermique : identification par thermographie infrarouge. International Journal of Heat and Mass Transfer, 31(3):487–496, 1988. 5. M. Girault and D. Petit. Identification methods in nonlinear heat conduction. part i: Model reduction. International Journal of Heat and Mass Transfer, 48:105–118, January 2005. 6. O. Balima, Y. Favennec, and D. Petit. Model reduction for heat conduction with radiative boundary conditions using the modal identification method. Numerical Heat Transfer, Part B, 52:107–130, 2007. 7. O. Balima, Y. Rouizi, Y. Favennec, and D. Petit. Reduced modelling through identification on 2D incompressible laminar flows. Inverse Problems in Science and Engineering, 17(3):303–319, 2009. 8. Y. Rouizi, Y. Favennec, J. Ventura, and D. Petit. Numerical model reduction of 2D steady incompressible laminar flows: Application on the flow over a backward-facing step. Journal of Computational Physics, 228(6):2239–2255, 2009. 9. Y. Favennec, M. Girault, and D. Petit. The adjoint method coupled with the modal identification method for nonlinear model reduction. Inverse Problems in Science and Engineering, 14:153–170, 2006.
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