Znd. Eng. Chem. Res. 1995,34, 4406-4412
4406
Nonlinear Control of a Distributed Parameter Process: The Case of Multiple Characteristics Eric M. Hanczyc and Ahmet Palazoglu* Department of Chemical Engineering and Materials Science, University o f California, Davis, California 9561 6
Sliding .mode control theory is applied to the special case of a distributed parameter chemical process modeled by two nonlinear partial differential equations. The method of characteristics is used to transform the hyperbolic system of equations into a set of first-order ordinary differential equations. Sliding mode control is used to control the resulting system.
Introduction A viable approach to the control of distributed parameter systems (DPS) is the use of the method of characteristics that allows design of a controller for DPS described by nonlinear first-order partial differential equations (NFOPDEs). One can transform the DPS into a finite set of characteristic ODEs (Arnold, 19881, which along with their initial conditions exactly describe the original DPS. The advantage of this approach is that the control design may be subsequently performed on this set of nonlinear ODEs in place of the original NFOPDEs without any approximation. Sira-Ramirez (1989, 1990), and Hanczyc and Palazoglu (1995) have demonstrated the use of the sliding mode control design based on this approach. In this paper, our goal is to present an application of the method t o the control of a double-pipe heat exchanger. The process chosen is modeled by a set of two hyperbolic PDEs, and such systems with multiple characteristics have not been previously studied in the literature. Sliding mode control is chosen as the design method as it is a nonlinear control strategy, and the manipulation of a desired operating surface (manifold) emerges naturally from the geometric interpretation of the characteristic direction fields.
We shall consider systems described by the following equations:
Here, u is the state of the system, t denotes time, x is the vector of n local spatial coordinate functions xi (i = 1,..., n), p is the n-dimensional vector with components pi denoting the spatial partial derivatives of the state, avl&i, and CP is a smooth function of its arguments. Further, q denotes the temporal partial derivative of the state, &/at, y is the system output defined by the smooth scalar function h, and u = u(u,x,t)is, in general, a distributed, smooth, time-varying feedback control law. It has been shown by Sira-Ramirez (1989) that a Lyapunov-like stability condition that drives the system trajectories to the sliding surface is a switching law, as the following feedback switching law determines the control action:
u+(u,x,t)fory u=(
>
O
or u-(u,x,t) for y < 0
(3)
Preliminaries In this section, we shall briefly review the sliding mode control concept within the framework of the method of characteristics. For details of the method, the reader is referred to Sira-Ramirez (19891, Hanczyc and Palazoglu (1995),and Hanczyc (1994). Utkin (1978) has originally introduced the sliding mode control as a method of discontinuous control for nonlinear systems, in which the key concept is t o define a surface on which the system has some attractive performance. Then, a Lyapunov-like stability condition is used t o guarantee that the distance t o the surface decreases along all system trajectories and constrains the trajectories to point toward the sliding surface. This is a robust control approach for nonlinear systems (Fernandez and Hedrick, 1987; Slotine and Li, 19911, and Sira-Ramirez (1992, 1994) has discussed the application of the technique to chemical process control systems.
* To whom all correspondence should be addressed. e-mail address:
[email protected]. FAX number: (916)7521031.
where u+(v,x,t)> u-(u,x,t). A distributed switch function v is defined as
v=[
1 fory>O or 0 fory < 0
(4)
that leads to the control law u = u-
+
V(U+
- u-)
(5)
with the objective of driving the output to the origin. The solution set defined as such is the sliding manifold or the sliding surface,
When the control law given by eq 5 is substituted into
0888-588519512634-4406$09.00/00 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34,No. 12, 1995 4407 writing eqs 10 and 11in new variables yields
where
This second-order system is classified as a set of hyperbolic PDEs. The method of characteristics leads to the following set of ODEs (Hanczyc, 1994):
F1 = q1 + pl(ul + 1 ) = -ho(Tl - 2'2) k, = u1
The following set of equations are obtained using the method of charateristics (Arnold, 1988; Hanczyc and Palazoglu, 1995):
+
+1
t, = 1
+
u = (Fp vGp)p q
(15) (16) (17)
P1= -P&o
(18)
+ vGp
S = Fp
t=l
(9) k, = -b(u2
0 = -(Fu+ Y G J P- (F, + YG,)
t, = 1
Noncharacteristic Cauchy data (initial conditions) completes the problem description. Equation 9 represents the controlled characteristic ODEs describing the original DPS.
Case Study: A Double-Pipe Heat Exchanger This case study involves the energy balance model for a double-pipe counterflow heat exchanger (McCabe et al., 1985). Radial gradients and axial diffusion are assumed to be negligible. The flows through both sides of the heat exchanger are also assumed to be incompressible. Under these conditions, the system is given in deviation variables by,
with
T,(x=O) = 0, T2(x=l)= 1 where the states TIand T2 are the cold-side and hotside temperatures, respectively; u1 and u2 are the fluid flow rates, ho is the heat-transfer coefficient, t denotes time, and x is the spatial coordinate. Here, a and b are real constants that are functions of the process parameters. In this study, we have a = 2.35, b = 1.25, and ho = 0.70. The measured output for control is the hot-side temperature, y = 7"2(x,t)- T2,,(x)
+ 1)
(21) (22)
Since this set of PDEs is hyperbolic, calculation of the smooth control law is nontrivial. The hot-side temperature is the controlled variable, while the cold-side flow rate is the desired manipulated variable. This implies that the system's relative degree is 2; i.e., the output must be differentiated twice t o generate an explicit relationship between the output and the input (Slotine and Li, 1991). Previously, only systems with relative degree 1 have been considered (Sira-Ramirez, 1989). Thus, a new second-order sliding manifold needs to be developed. It should be noted that eqs 17 and 22 are identical equations, implying that tl and t 2 differ only by a constant which can be taken as zero without loss of generality. Before a smooth control law can be found, a description of the sliding regime must be given. A distributed sliding regime exists on an open set of the sliding manifold, if the total time derivative of the output behaves as (Slotine and Li, 1991): (25) and
(12)
Note that, for this problem, the inputs are not distributed, hence, somewhat simplifying the analysis. Re-
Here, AI and 22 are two tuning parameters. The detailed derivation of control laws for this system can be found
4408 Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995 lo1 Cold Sick
(bl
t 0.8 e
5
P
0
e
k 0.4 t u
r e
r
0
3
2
t
te 0'
0.2
-i
E
i i
3
tine
i i
+g-, u
r
-1
-1 I
1 2 3 tine
tine
l
4 5
1 tine 2 3
4
5
&!t Si&
IC1
t
e m 8.8
e
e
e
P
P
r
:i_
-1 -1 0
0
(cl ti& Si&
u
te"
0.2
e
t I
t
5 1
t
f 4
0.6
(bl
(a) Cald Sick
8.8
I_--_
0n6
;0,6
0.4
u
t
&F
1
0,4
P
-1
i
1
2 3 tine
4
5
Figure 1. (a) Set-point response of the cold-side temperature at and 1.0 under sliding mode control with the switching feedback control law. (b) Input response. (c) Set-point response of the hot-side temperature. Dashed lines represent set points.
x = 0.0, 0.2, 0.5, 0.8,
in the Appendix. The equivalent control function is given by
which includes both extreme characteristic vector fields. L,,, and L,,, are the Lie derivatives. Since the manipulated variable appears only in the cold-side energy balance (eq 101, only L,,, must be defined for both values of the binary switching function. For v = 0, the corresponding Lie derivative is Lv1-;for v = 1,it is Lvl+. The set of two hyperbolic PDEs (eqs 10 and 11)can be combined and expressed as a single second-order PDE in one variable. The leading second-order derivatives can be expressed by the two Lie derivatives:
The control law uses the two sets of extreme characteristic vector fields to define the equivalent sliding dynamics. In the following, the controller parameters are chosen to reflect the trade-off between performance and stability as determined by simulations. Parts a and c of Figure 1 show the cold-side and hot-side temperature and concentration profiles as a function of time for a step change in the set point (a 25% change based on the outlet hot-side temperature). The discontinuous control law causes a significant "chattering" effect in the cold-side temperature (Figure lb). The desirable input at steady state lies between 0.5 and -0.5. To eliminate
-1 I
1 2 3 4 5 tine Figure 2. (a) Set-point response of the cold-side temperature at x = 0.0, 0.2, 0.5, 0.8, and 1.0 under sliding mode control with the continuous feedback control law. (b) Input response. (c) Set-point response of the hot-side temperature.
chattering from the binary-valued switching law, we use the smooth control law,
Parts a and c of Figure 2 illustrate the smooth control for set-point tracking, with A1 = 1 2 = 2.5. As with the binary-valued compensator, the new set point profile is obtained in under 2 time units. The action of the coldside flow rate in Figure 2b is significantly better than that in Figure lb. We also wanted to see how the addition of integral action t o the sliding mode control would affect the performance of the controlled system. The control law with integral action is found to be
k2
+ -Ti b(1 + u,)T21EIt + Alb(l + u2)TzSpl~:A +
A -J(T2 la2
- T2 ) dt)i{ [A2 - u h , l T 2 1 ~-~A2TzSpl::~ ~ +
- ah,
Ti
'I
SP
}: I."
b ( l + u,)
(30)
Figure 3 illustrates this case with 11 = A2 = 2.5 and ti = 0.1. The trajectories exhibit a slight overshoot, and the system requires about 4 time units to attain the new set point. The responses are nonetheless acceptable.
Ind. Eng. Chem. Res., Vol. 34,No. 12,1995 4409 (a) Cold
S a
(d Cold Si&
(b)
a-83 e
t
t
;0,6 e 7
0,4
u 0.2
r
e
- 1 0 1 2 3 4 5
1 2 3 tine
-1 0
tine
4
I
5
E - 1 0 1 2 3 4 5 tine
.
*l
-1 t l - r r n l -10 1 2 3 4 5 tine
t 1 e
m 008 P e
a
0.6
t
u 0.4
=0
.
-1
2
0
-
b
%nlll
,
882
1 2 3 4 5
- 1 0 1 2 3 4 5
tim Figure 3. (a)Set-point response of the cold-side temperature at x = 0.0, 0.2, 0.5, 0.8, and 1.0 under sliding mode control with the continuous feedback control law with integral action. (b) Input response. ( c ) Set-point response of the hot-side temperature.
tine Figure 4. (a) Set-point response of the cold-side temperature at x = 0.0, 0.2, 0.5, 0.8, and 1.0 under sliding mode control with the continuous, weighted, feedback control law with integral action. (b) Input response. ( c ) Set-point response of the hot-side temperature.
Next step is to consider the weighting of the measured output profile used in the smooth control law. This would help us penalize the error at a certain spatial location more than others. A weighted control law with integral action is given as
u1= -1 -
I[
I'
A, + 1, - ( a + l)h, + - ah,{W(T, -
(gT,)]+ ( 4 A 2 +
Ti
- T2sp))-
'&){W(T2 Ti
2 3 4 5 tine Figure 5. Open-loop response of the hot-side temperature at x = 0.0, 0.2, 0.5, 0.8, and 1.0 to an unmeasured disturbance in the hot-side fluid flow rate. -1
+
A2(wT2sp)I:I: + ah,(% T,) b(1
+ u,) x
The control parameters are 11 = 5.0, A2 = 4.0, and zi = 0.1, and the weighting function is given as
where w o= 10. The weighting emphasizes the hot-side outlet of the heat exchanger over the inlet by exp(l0). This is a rather large penalty; yet there is only a slight increase in the overshoot of the response. Parts a and c of Figure 4 illustrate weighted, integral control for the
0
1
set-point change. The temperatures take about 5 time units to achieve the new levels with a less damped response than the previous compensator. The system behavior is analogous t o linear time-delay systems under high gain compensation. The weighting function is responsible for the Utransformation''by penalizing the outlet error significantly heavier relative to the other points considered. The above control laws are also tested by an unmeasured change in operating conditions to illustrate the effect of disturbances. The open-loop effect of a disturbance in the hot-side fluid flow rate is seen in Figure 5.
4410 Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995
can be expressed in terms of the coordinate functions or the Lie derivatives:
0'51
,t 0.4
a
U
r
3
dt, = aT, Tl
2
+ Xx, ibc, + $=L V f
(A31
t j e
- 2 0 2 4 6 6 1 0 tine
0.4 e
0
and - 2 0 2 4 6 8 1 0 tine
A n example of one such function is the output function h. The equivalent control in terms of extreme characteristic vectors is given as
F-2
082
-2 0
2
4
6
B 10
tine
Figure 6. (a) Response of the cold-side temperature at x = 0.0, 0.2, 0.5, 0.8, and 1.0 to an unmeasured disturbance in the hotside fluid flow rate under sliding mode control with the continuous, weighted feedback control law with integral action. (b) Input response. (c) Response of the hot-side temperature.
The 10% increase in the flow rate causes an approximate 10% increase in the outlet hot-side temperature. The ability of each control law to effectively reject the unmeasured disturbance is determined from simulations and discussed by Hanczyc (1994) in detail. The basic smooth control law in eq 29 turns out to be totally ineffective, as the cold-side flow rate remains near the nominal value, and the hot-side temperature profile is indistinguishable' from the heat exchanger's open-loop response. The weighted sliding mode controller (eq 31) is implemented as the best alternative. Figure 6 shows the effectiveness of the weighting function. However, the response is sluggish and takes over 5 time units to reach steady state. Faster adjustment to disturbances can be made by altering the tuning parameters, but the change would result in a trade-off of poorer set-point tracking for better disturbance rejection.
The control law uses the two sets of extreme characteristic vector fields to define the equivalent sliding dynamics. Since the model is a set of quasilinear hyperbolic PDEs, eq A5 can be rewritten as
As the hot-side temperature is the output variable, the derivation continues with the Lie derivative term corresponding to the hot-side energy balance:
where the following characteristic equations have been employed:
Appendix Equation 29 is the simplest of the smooth feedback control laws in this study. The integral and weighted sliding mode controllers are derived similarly. We exploit the quasilinear relationship between the input and the hyperbolic set of PDEs. For a second-order hyperbolic system, there exist two extreme characteristic vector fields:
ql=X - a+ - + aT libc, at
.
a
la^,
t, = 1
(A101
The following characteristic equations
TI
= 41tl + Plkl
(All)
and
q 2 = k 2 -a + - +aT ibc,
at
a a ~ ,
-
The total time derivative of functions in these spaces
t, = 1
(A131
are used in taking the Lie derivative in eq A7 corre-
Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995 4411 sponding to the cold-side energy balance:
=
the energy balance on the hot side (eq 11). Thus, eq A17 becomes
{[?++
-ah
(1 u
O
l a x
at +(l+u)-b(1
a2h + u2)(l + u,)- ax,axlc, +A
u,)]
1
- - b ( l + u2) at aah x2
"1
+ (1+ u ) + A1&h } ax1
+
(A14
Since h is a smoothly defined with respect to both characteristic surfaces, the convention of distinguishing between the spatial coordinate functions is omitted. Note that the set of hyperbolic PDEs can be rephrased as a single equation without distinguishing between the characteristic coordinate functions; i.e.
a2T2
aT21
+b(l+
+ (1+ ul)ax + [Alb(l +
a2h atax,
["
1
-+(1+u)-+hO(Tl-T2) aT2 [aT2 at
~ 2 -) A 2 ( 1
2
+
= 0 (AM)
Like terms are collected and terms containing the input are placed on the left-hand side
a2T2
aT2
ax2
ax
+
aT21 +
[ b ( l + u2)3- A2 2 (1,- aho)- (1
ax
Using eq 11 to eliminate the partial derivatives with respect to time gives
a2T2
+ ((1+ ul)- b(1 + u 2 ) }- b(1 + u2)(l + at2 at ax ax2
aT2 + (1+ ul)ax + h(T, - T2)]
Notice that the second-order terms in eq A15 can be expressed by the Lie derivatives with relpect to the two characteristic fields. Also these terms appear explicitly in the right-hand side of eq A14. The output function is expressed as
h = T2 - T2SP
(A16)
Integration over the spatial domain averages eq A19. Recall that the fluid is incompressible so that the input is a function of time only. Rearranging eq A20, we obtain for the smooth control law
Substituting h into eq A14 and using the invariance condition (eq A6) yields
Literature Cited
The first three terms are the leading derivatives in the single equation expression (eq A17) for the model. The seventh term contains the first-order derivatives from
Arnold, V. I. Geometric Methods in the Theory of Ordinary Differential Equations; Springer-Verlag: New York, 1988; pp 59-88. Fernandez, B.; Hedrick, J. K. Control of Multivariable Non-linear Systems by the Sliding Mode Method. Int. J . Control 1987,46, 1019-1040. Hanczyc, E.M.Modeling and Control of Chemical Process Systems Described by Partial Differential Equations. Ph.D. Dissertation, University of California, Davis, CA, 1994. Hanczyc, E. M.;Palazoglu, A. Sliding Mode Control of Nonlinear Distributed Parameter Chemical Processes. Ind. Eng. Chem. Res. 1996, 34, 557-566.
4412 Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995 McCabe, W. L.; Smith, J. C.; Harriott, P. Unit Operations of Chemical Engineering, 4th ed.; McGraw-Hill: New York, 1985; p 311. Sira-Ramirez, H. Distributed Sliding Mode Control in Systems Described by Quasilinear Partial Differential Equations. Sys. Control Lett. 1989, 13, 177-181. Sira-Ramirez, H. Distributed Sliding Regimes in Systems Described by Nonlinear First Order Partial Differential Equations. Fourth Latin American Congress of Automatic Control, Puebla, Mexico, 1990. Sira-Ramirez, H. On the Dynamical Sliding Mode Control Strategies in the Regulation of Nonlinear Chemical Processes. Int. J . Control 1992, 56, 1-21. Sira-Ramirez, H. Dynamical Discontinuous Feedback Strategies in the Regulation of Nonlinear Chemical Processes. IEEE Trans. Control Syst. Technol. 1994, 2, 11-21.
Slotine, J.-J.E.; Li, W. Applied Nonlinear Control; Prentice Hall: Englewood Cliffs, NJ, 1991; p 218. Utkin, V. I. Sliding Modes and Their Application in Variable Structure Systems; MIR Publishers: Moscow, 1978.
Received for review February 22, 1995 Revised manuscript received July 10, 1995 Accepted July 25, 1995@ IE950127A
Abstract published in Advance ACS Abstracts, October 15, 1995. @
