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EI2mm.R Mathematical and Computer Modelling 38 (2003) 10X-1085 www.elsevier.com/locate/mcm

Burgers’ Equation: Galerkin Least-Squares Approximations and Feedback Control

B. B. KING AND D. A. KRUEGER Interdisciplinary Center for Applied Mathematics

Virginia Tech, Blacksbnrg, VA 24061-0531, U.S.A. bbking@icam.vt.edu dkruegeravt . edu

(Received and accepted February 2003)

Abstract-The standard Galerkin finite-element approximation of Burgers’ equation is known to be numerically unstable for small values of viscosity. One way to overcome this difficulty is to use a stabilized finite-element method, such as Galerkin least squares, and this is a standard approach taken in the simulation literature. In this paper, we investigate the effect of a stabilized finite-element approximation of Burgers’ equation in the context of feedback control design. The issue at hand is how the additional stabilizing terms affect the resulting controller. We show that the controller designed for the stabilized system varies little from the unstabilized controller. @ 2003 Elsevier Ltd. All rights reserved.

Keywords-Burgers’ equation, Galerkin least squares, Feedback control, Functional gains, St& bilized finite element method.

1. INTRODUCTION

The development of closed-loop active controllers for applications in fluid flow presents many challenges and has been the subject of much research throughout the past 15 years. The existence of controllers for certain applications can be proven using distributed parameter control theory. However, deriving such model-based controllers for implementation purposes requires applying computational fluid dynamics (CFD). It is well known that CFD is a challenging area within the simulation domain; using CFD codes for purposes of control design presents another layer of complexity which raises many research questions.

One challenge in using finite-element methods for the discretization of Navier-Stokes, and similarly Burgers’ equations, is that a “standard” basis will result in a spatial discretization that can be numerically unstable. One fix for this is using an approximating basis that satisfies the Babuska-Brezzi [l] condition. Another is using a stabilized finite-element code in which additional terms are incorporated into the model that stabilize the time integration and are negligible when one is close to the true solution. One type of stabilization method is Galerkin least squares (GLS) [2-41.

In [5], a GLS formulation of Burgers’ equation was used as a model for control design. However, the finite-element basis that was applied consisted of linear B-splines, and thus, stabilizing terms

This research was supported by NSF under Grant DMS-0072629.

0895-7177/03/i% - see front matter @ 2003 Elsevier Ltd. All rights reserved. doi: 10.1016/S0895-7177(03)00318-2

1076 I3 IS. KING AND D. A. KR~JEGER

involving higher-order derivatives were neglected. Here, we use cubic B-splines for the finite- element basis to more accurately study the effect of the stabilizing terms on the controller.

A fundamental issue in control design is that the controller should be robust to unmodeled dynamics, disturbances, and model variation. For controllers that are designed around physically- based models, one desires an accurate model of the dynamics to “minimize” the robustness demands that are placed on the controller. The use of a stabilized finite-element approximation for control design raises robustness questions regarding the use of a model that is different from the physics as a foundation for control design. Specifically, we consider how the control law is affected by using a GLS spatial discretization of Burgers’ equation as the underlying model for control design. To do this, we compare the controllers for a range of viscosity parameters.

The problem considered in this paper is the 1-D Burgers’ equation, sometimes referred to as the 1-D equivalent of the Navier-Stokes equations that model fluid flow. Burgers’ equation and distributed parameter controllers are presented in Section 2. The standard Galerkin finite-element approximation and stabilized GLS formulation are presented in Section 3. The numerical results are in Section 4, followed by the conclusions in Section 5.

2. BURGERS’ EQUATION AND CONTROL DESIGN

For t > 0 and 0 < 2 < L, Burgers’ equation with Dirichlet boundary conditions can be written as

wt(t, x) + w(t, 5)%(4 x) - wx(t, x) = b(z)u(t),

w(O, x) = we(z), (1) w(t, 0) = w(t, L) = 0.

The function u(t) is the control which will be determined by linear quadratic regulator (LQR) theory; the function b(z) acts to distribute the control throughout the spatial domain [0, L]. The parameter E can be thought of as viscosity and is analogous to the inverse of the Reynolds number in the Navier-Stokes equations.

Using distributed parameter theory, we can write Burgers’ equation in an abstract form that will be used as a foundation for control design. The details can be found in [6]. Specifically, (1) can be written as

G(t) = Aw(t) + N(w(t)) + Bu(t), w(O) = wo, (2)

where w(t) is the state in state space X = L2( [0, L]) and u(t) is in the control space U = R. The operator A = t& and is defined in domain D(A) = {w E X : w E H2(0, L), w(O) = 0 = w(L)}. The operator N contains the nonlinear part of the equation. The operator B = b(z). We restrict ourselves to the L&R problem for controller design, which uses a linearization of the system in (2).

LQR theory yields a full-state feedback control law of the form

u(t) = -Kw(t), (3)

where K comes from the solution to an algebraic Riccati equation. Specifically, there exists an optimal feedback control, u(t), that minimizes the cost function

J(u) = J oa [t&w(t), w(t)>x + tRu(t), 4t))al c-it over all u(.) E L2([0, 7’); W) subject to the dynamics given by the linearization of the constraint in (2) (see [6]). H ere, Q is a positive semidefinite state weighting operator, and R is a positive definite control weighting operator. This optimal control can be found by solving the algebraic Riccati equation

Al-I + IIA - TTBR-‘B*II + Q = 0.

Burgers’ Equation 1077

The feedback operator K is then given by K = -RWIB*lI. The controlled, nonlinear closed-loop system is then given by

15(t) = (A - BK)w(t) + N(w(t)), w(0) = WI). (4

For the distributed parameter control problems considered here the feedback operator K has an integral representation in terms of a feedback kernel /c(.). This kernel is called a feedback functional gain and provides a useful characterization of the control law. In particular, (3) can be written as

s

L u(t) = -Kw(t) = - k(x)w(t, x) dx. (5)

0

In subdomains over which Ic(.) is large, the state is amplified and is therefore important in the action of the controller. Conversely, in subdomains where Ic(.) is small, the state is neglected in the controller. In the section on numerical results, we will examine functional gains from stabilized and unstabilized systems to ascertain differences in the control laws.

3. GALERKIN LEAST-SQUARES APPROXIMATIONS

To compute the control law in (3), an approximation scheme is applied to the linearization of (2) which yields a finite-dimensional system. In this paper, we use a stabilized finite-element approximation and describe the derivation of the approximating equations in this section. As discussed in the introduction, such a code is used to stabilize the numerical integrations in simulations. We are interested in how the stabilizing terms, which are additional to the terms arising from the physics, affect the control design.

To develop a finite-element based spatial approximation of (2), we first consider the weak form of (l), given as

s

L

wt(t, x)4(x) dx 0

s

L

s

L

s

L (6)

=- ~(6 x)44 x>+(x) dx - E wz(t, +#G4 da: + f4xMWx) dx. 0 0 0

This form of Burgers’ equation holds for all w(t, .) E HA[O, L], $(.) E H,$([O, L]), and b(x) E

L2([0, LI) (see [Q Given a discretization of the domain [0, L] with N subintervals, the state w(t, x) is approxi-

mated by wN(t,x) = C~<iw~(t)h~(x), where {hr(x)}E!J’ is the set of N - 1 cubic splines satisfying the Dirichlkt boundary conditions. The use of cubic splines as opposed to the stan- dard linear splines will be made clear in the subsequent discussion. Let 4(x) = h?(x) for i=1,2,..., N - 1 in turn. The matrix representation of (6) can be written as

MtiN@) = -dcwN(t) - JiGI (wN(t)) + BOIL(t),

where

M= L 1

N-l

h~(x)hjN(x) dx , i,j=l

L 1 N-l

hy’(x)h,N’(x) dx )

i,j=l

N-l N-l

cc j=l kl

-, N-l

s L

WjN WJkN (G h:Y(x)h,N(x)h;'(x) dx ) 0

1 i,j=l

(7)

1078 B. B. KING AND D. A. KRUEGER

L a0 =

[i b(x)hY(x) dx

0 I

N-l

: and

i,j=l (7)jcont.)

wN(t) = [Wl(% wz(t), . , . , wN-l(t)lT The standard form of this approximation to (2) is then written as

tiN(t) = --~M-%w~(t) - M-‘JV-~ (wN(t)) + M-1130u(t)

= AWN(t) +N(wN(t)) + h(t),

wN(0) = wf. (8)

The linear part of this discretized form of Burgers’ equation would typically be used for LQR control design. However, for very small c (which is analogous to a very large Reynolds number in the Navier-Stokes equations), this approximation is numerically unstable in simulations. One method of stabilizing the numerical approximation is to use a GLS approach [2-41. The GLS is a combination of the standard finite-element approximation along with a weighted least-squares formulation on each element. The least-squares formulation is derived here. Explicit dependence on independent variables is omitted for notational brevity of this derivation.

The solution to Burgers’ equation minimizes

s

L J(w) = (wt + ww, - ewzz)‘dx,

0

for all w(t) E D(A). If 4 E D(A) is an admissible variation, then

; J(w + 64) = 0, 6=0

which implies

2 oL(wI -t wwz - EW,, s

- bu)(w+’ + q5w, - q5”) dx = 0.

After multiplying, regrouping, and integrating by parts, we obtain the following least-squares formulation of Burgers’ equation:

2 J

L (wtwqb’ + W&W, - W@p” + w2w,# + w(w,)24 - WW,qb” 0

- EWING’ - ~w,,qiw, + E~w,,$” + EbqS’u + ~bwqSu) dx = 0.

Combining the weak form of Burgers’ equation in (6) and the weighted least-squares formulation above and linearizing about the equilibrium w(t, x) = 0, we have the GLS formulation

s oL(wt$ + cwz$’ - b$u) dx + IL T(W) (qiSwtz + ~~w,~q5” -I- cbq%) dx = 0. (9)

0

An appropriate weighting parameter 7 must be chosen. The -r applied here is the same version used in [5], that which was developed for the incompressible, time-dependent Navier-Stokes equa- tion in [2-41 for problems involving advection and diffusion. The weighting parameter r depends on E, w, and the finite-element approximation grid size. Define (for fixed t)

T(W(X,Q) = 2E IwCx7 9’ L a-l’

h=$

1 m= -,

3 where L is the length of the domain and N is the number of subintervals.

Burgers’ Equation 1079

We now approximate the additional terms in the GLS formulation with the finite-element method described previously. The weak form of GLS in (9) contains second spatial derivatives that are one order higher than the first-order derivatives that appear in the weak form of Burgers’ equation in (6). It is these terms that motivate the cubic spline basis used in the approximation scheme. In [5], the standard linear B-spline basis was used in the approximation of (9), and hence, the second derivative terms were ignored. The implications of these stabilizing terms will be clear below. Using the approximations of zu(t, .) as before and letting 4 range over the basis functions, the matrix representation of the GLS formulation is

Mg+) + &2irN(t) = -&UN(t) - E2SwN(t) + &u(t) - 4’Ll(t)7

where

L K, =

[/ T(w)h:(x)h;(x) dx

0 1 N-l

,

i,j=l

L N-l

a, = [i T(W)b(X)h~(X) dx ) 0 ,I i,j=l

r “1, -, N-l

s= - +)h~(x)h;(x) dx , i,j=l

and the other matrices are as in (7). The standard form of this matrix equation is

(M + &&i~~(t) = (--EK - ~~S)tu~(t) + (B,, + &,)u(t)

zbN(t) = (M + K,)-‘(--elc - e2S)wN(t) + (M + &J1(B,, + &,)u(t)

= .4tawN(t) + &tabu(t)

wN(0) = wt.

This is the stabilized finite-element approximation on which we will build our control design. As we can see, the use of linear splines in [5] implies that B, and 5’ are identically zero. Thus, the only difference in their stabilized matrix system lies in the stabilized mass matrix. In the case of cubic splines, B, and 5’ are not’zero and can impact the controller design.

4. NUMERICAL RESULTS FOR CONTROL OF BURGERS’ EQUATION

In this section, we present numerical results for a series of viscosity parameters, E. We compare the eigenvalues of the closed-loop system and functional, gains for stabilized and unstabilized codes and include simulations of open- and closed-loop systems. Again, the fundamental issue for control design is that the stabilized system contains terms that are derived from other con- siderations than the physics of Burgers’ equation. In all numerical results, b(x) = eZ. Unless otherwise specified, a discretization of the interval [0, L] was chosen with iV = 50 subintervals.

In Figure 1, we show eigenvalues and functional gains for stabilized and unstabilized systems for L k 1 and E = 1. The eigenvalues for the stabilized and unstabilized codes are virtually indistinguishable; the functional gains show only a slight difference between the two systems.

In Figure 2, we show functional gains for systems in which viscosity is decreased to E = .Ol and E = .OOl, respectively. In both cases, the eigenvalues are again nearly identical and are therefore not shown. For these values of E, the functional gains for both systems are also nearly identical.

In Figure 3, we show functional gains for the system in which the viscosity is chosen to be E = .OOOOl, and E = .OOOOOOl. Surprisingly, there is no significant difference in the eigenvalues,

1080 l3 13. KING AND D. A. KRUEGER

1

0.8 -

I I I

x Closed Loop Eigmvalues 0 Closed Loop Stabilized Eigenvalues

0.6

t

. -0.4

t -0.6

t 1 -11 -2.5 -2 -1.5 -1 -0.5 0

(a) Eigenvalues.

0.8

0.7

0.6

3 0.5

0.4

0.3

0.2

0.1

C I 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

X

(b) Functional gains.

Figure 1. t = 1, L = 1.

Burgers’ Equation

-0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 X

1

(a) E = .Ol, L = 1.

4.5 I I I I I I I I I - Unstabilized Functional Gains - - Stabilized Functional Gains

4- I

I I I I I I I I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1

-1081

X

(b) E = .OOl, L = 1.

Figure 2. Functional gains.

1082 B. B. KING AND D. A. KRUEGER

I I I I I I I - Unsta ibilized Functional Gains 1 1

2.5 -

5? P

-- Stabilized Functional Gains 1 /

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1 X

(a) E = .OOOOl, L = 1.

I I I I I I I 1 I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1

X

(b) e = .OOOOOOl, L = 1.

Figure 3. Functional gains.

Burgers’ Equation 1083

X

(a) Uncontrolled dynamics.

. . . , .

. . “ . . ‘,

t 0 0 X

(b) Closed loop dynamics.

Figure 4. e = .Ol, L = 1.

1084 B. B. KING AND D. A. KRUEGER

t

,.5,,.. .“I.

1,; .,.‘i

: : o.5 . ‘...

jz 0, d s

4.5, .’ ” ;

,. ” -, .,, ,... ..,: ‘.

.

X

(a) Uncontrolled dynamics.

0.6

. . .

‘_

. : .

. :

: . . . ‘. ‘.

. . . : .

t 0 0 X

(b) Closed loop dynamics.

Figure 5. E = ,001, L = 1.

Burgers’ Equation 1085

which are not shown, or in the functional gains, yet these are viscosity values for which the unstabilized code is numerically unstable in simulations. We do see a roughness in the unstabilized gains near the boundaries that is smoothed out in the gains from the stabilized code.

To show the behavior of the system when the controller is applied, we show two simulations. The closed-loop systems are simulated using the stabilized control law in the unstabilized code. The first is for the parameter choice c = .Ol and L = 1. The simulations of the open-loop and closed-loop systems are shown in Figure 4. The plots show the state of the system, w, as a function of time, t, and position, CC. As expected, the results show that the solution is decaying to zero faster when the controller is applied.

The effect the controller can have on the numerical stability of the system can be seen in Figure 5. The numerically un&able open-loop system is shown on the top while the closed system stabilized, (numerically) by inclusion of the controller, is shown on the bottom. Such results were previously documented in [5].

5. CONCLUSIONS The sampling of numerical results presented above have provided some insight into the effect

of a GLS finite-element approximation as a foundation for control design. As opposed to pre- senting a “robustness” challenge in designing the controller around a model that is different from the physics-based model, the stabilized code appears to provide gain convergence for the small viscosity parameter values for which gain computation is difficult. The controller can numerically stabilize the system, as was documented in [5]. This paper is intended to raise questions about the use of stabilized models for the purpose of controller design. These preliminary results clearly indicate that, in addition to offering alternative methods for simulation, stabilized finite-element methods offer the potential for producing convergent controller designs even for large Reynolds number fluid flows. This will be the subject of future work.

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diffusive model, Computer Methods in Applied Mechanics and Engineer&g 95, 253-276, (1992). 5. J.A. Atwell and B.B. King, Stabilized finite element methods and feedback control for Burgers’ equation,

Proceedings of the American Control Conference, 2745-2749, (June 2000). 6. J.A. Burns and S. Kang, A control problem for Burgers’ equation with bounded input/output, Nonlinear

Dynamics 2, 235-262, (1991).