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I n d . Eng. Chem. Res. 1995,34,557-566
Sliding Mode Control of Nonlinear Distributed Parameter Chemical Processes Eric M. Hanczyc and Ahmet Palazoglu' Department o f Chemical Engineering and Materials Science, University of California, Davis, California 95616
Sliding mode control theory is applied to distributed chemical processes modeled by nonlinear partial differential equations. The method of characteristics is used to transform exactly the distributed parameter system into a finite set of ordinary differential equations. Geometric methods are then used to extend sliding mode concepts for application to a class of partial differential equations. Through simulations, the method is shown to be effective in controlling typical chemical process units.
Introduction Many chemical processes are modeled as distributed parameter systems (DPS) in the presence of more than one independent variable. Under such circumstances, the governing equations of momentum, heat, and mass transport will contain temporally dependent terms as well as spatial gradients. Analytical solution of these (possibly nonlinear) partial differential equations (PDEs) is generally nontrivial and in many cases impossible. Typical distributed parameter chemical processes are described by first- and second-order PDEs that are commonly classified into three categories: hyperbolic, parabolic, and elliptic. All single first-order PDEs are considered hyperbolic. Higher order PDEs may be any one or a combination of the three. Models of plug flow reactors, steam heaters (Friedly, 1972),and crystallizers (Rawlings et al., 1992) are processes generally modeled by a single hyperbolic PDE. A double-pipe heat exchanger can be described by two hyperbolic PDEs (McCabe et al., 1985). The governing equations for a non-isothermal plug flow reactor are parabolic. The socalled heat equation (Romagnoli and Gani, 1983) and reactors with axial dispersion (Bonvin et al., 1983) are considered second-order parabolic systems. In process control literature, the modeling and control of distributed parameter systems have been addressed by various researchers (Ray, 1981; Foss et a1.,1980; Bonvin et al., 1983; Cinar, 1984; Budman et al., 1992). The most common approach to modeling and subsequent control design is based on various lumping techniques. The set of linear or linearized PDEs is transformed into a set of ordinary differential equations (ODEs) by orthogonal collocation, and typically the resulting lumped model is of very high order. Thus, further order reduction (Bonvin and Mellichamp, 1982)may be necessary to effectively use available control design methodologies. This naturally adds to the mismatch between the simple, low-order model and the original process. Despite its importance for feedback control, however, the issue of robustness is seldom addressed for DPS (Palazoglu and Owens, 1987; Hanczyc and Palazoglu, 1992; Budman et al., 1992). A recent approach to the control of DPS utilizes the method of characteristics and allows controller design for DPS described by nonlinear first-order partial differential equations (NFOPDEs). The DPS is transformed into a finite set of characteristic ODEs (Arnold, 19881, which, along with their Cauchy data (initial
* To whom all correspondence should be addressed. E-mail address for A. Palazoglu:
[email protected]. 0888-5885l95/2634-0557$09.QQlO
conditions), exactly describe the original DPS. Thus, control design may be subsequently performed on a set of nonlinear ODEs in place of the NFOPDEs without approximation. For example, one may design a sliding mode controller based on the characteristic equations (Sira-Ramirez, 1989, 1990). The method of characteristics is applicable to all hyperbolic systems, and in this paper, we shall elaborate on its use for control purposes. We shall essentially follow the work of Sira-Ramirez (1989,1990)but modify his control law by adding a tuning parameter. Practical implications of this control strategy as well as a spatial averaging of the control law will also be our contributions. In the next section, the method of characteristics is developed using a geometric approach to differential equations. The section following that discusses the application of distributed sliding mode control to nonlinear distributed parameter systems by exploiting the set of characteristic ODEs determined in the first part. Two examples from the chemical engineering literature illustrate the methodology. Finally, results are summarized and future work is discussed.
Method of Characteristics We shall consider the following NFOPDE:
where 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, au/&i, and @ is a smooth function of its arguments. Further, q denotes the temporal partial derivative of the state, &/at. For the details of the method of characteristics applied t o general nonlinear PDEs, the reader is referred to Arnold (1988). Equation 1can be visualized as a 2n 2-dimensional hypersurface in the l-jet space, J', which is essentially the extension of the space x = (x,t ) in R(n+l)to the space with coordinates z = ( u , x, t, p, q ) in R(2n+3).Since p and q are the first derivatives of u , J1becomes the space which has as coordinate functions the state of the system, time, spatial coordinates, and all first derivatives of the state. This 2n 2 hypersurface is denoted by E and defined as
+
+
E = @-'(O)
{z
E
J1:@(z)= q + O(u,x,t,p)= 0) (2)
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558 Ind. Eng. Chem. Res., Vol. 34, No. 2, 1995
It has been shown by Arnold (1988) that, if E is nonchuructeristic a t all points z of the space J1,the set of ODEs which generate the jet characteristics for eq 1 can be computed in a straightforward manner. Typically, if the problem is well-posed, the hypersurface E is considered to be noncharacteristic, requiring that 4(z) is smooth and the initial conditions for the system are consistent. The jet characteristics of a PDE are the set of integral curves determined from the set of characteristic ODEs. These ODEs are found in the following manner (Arnold, 1988; Sira-Ramirez, 1990). Let us consider a non-zero vector 5,
a + dx a + dt a + dp a + dq a (3) 5 = dv -
ax
au
at
aP
aa
that belongs to the tangent plane of E. A subspace of the tangent space is a contact structure, a 2n 2-dimensional plane, in which the so-called l-form is satisfied,
+
a=du-pdx-qdt=O as discussed by Arnold (1988). Therefore, we can state that 5 lies in the contact structure, which is the field of contact planes, if it belongs to the null space of a:
a(5)= du
-p
d x - q dt = 0
(4)
Thus, 5 can be rephrased as
5 = (p dx +
a
d t k
a a + dp a + dq a + dx ax + dt at aP aa
x = @p = QP t = l P = - @U p - # x = - @UPq=-#q-#
(pS
a + dq a + 4)-aua + d x -axa + d t -ata + d p aP aa
= &z)
-9 d x - 4 dt + S d p + dq = O
u = p#p
+ q = pQp + q
(10)
+
Distributed Sliding Mode Control of NF'OPDE Sliding mode control is a method of discontinuous control for nonlinear systems (Utkin, 1978). The main idea is to define a surface on which the system has some desirable behavior. A Lyapunov-likestability condition guarantees that the distance to the surface decreases along all system trajectories and constrains the trajectories to point toward the sliding surface. Once on the surface, the trajectories remain there. The following extends this idea to distributed parameter systems described by a first-order PDE. We shall consider a system described by the following NFOPDE:
av at
-
+ @'(u,x,t,u,p)= 0 y = h(u,x,t)
(11) (12)
where u , t , x, p, q , and @ are defined as in the previous section, y is the system output defined by the smooth scalar function h, and u = u(u,x,t) is a distributed, smooth, time-varying feedback control law. A Lyapunov-like stability condition that drives the system trajectories to the sliding surface is a switching law (Sira-Ramirez, 1989). The following feedback switching law determines the control action: u+(u,x,t) for y > o or (13) u-(u,x,t) for y < 0 u-(u, x, t). Defining a distributed
u=[ with u+(u, x, t) switch function v:
1fory > 0
or Ofory- TSp(x)
(52)
The temperature set point profile, Tsp,is calculated from a desired concentration profile based on the analytic relationship derived from eqs 48 and 49:
c
+ T = co + To
(53)
where C O , and TOare the dimensionless inlet concentration and temperature, respectively. Discrete temperature measurements are made a t five locations along the reactor. Equations 48 and 49 are rephrased using the new variables
The following ODES are obtained using the method of characteristics:
(
+ 1)= R(c,T) 3i: = (u + 1)
T = q T +pT(u
t = l
(59)
(60)
(61) (62) (63)
n
This system is also quasilinear. Since this set of PDEs can be classified again as hyperbolic, eqs 56 and 61 are identical, which simplifies the calculation of the smooth control law. This means that the well-known characteristic curves of this set of quasilinear PDEs coincide. We have observed that for a step change in the set point (a 25% change based on the outlet temperature), the discontinuous control law causes a slight "chattering" effect as the temperature and concentration profiles progress through time. The control action, as expected, switches back and forth between u+ and u- causing the oscillatory behavior (Hanczyc, 1994). Figure 9 illustrates the effect of replacing the binary switching function with a continuous one. A smooth control is determined to be u = -1 -
-(R(c,T))
+
-T
(T - Ts*)
(66)
sp x=o
The reaction rate term (R(c,T))is calculated using the relationship in eq 53 substituted into eq 50. This allows an approximation for the reaction rate completely in terms of the temperature measured at points along the reactor. The tuning parameter A is set to 1. As expected, the temperature and concentration responses are smoother than those under switching control. The manipulated flow rate is also quite acceptable. Figures 10 and 11contain the simulation results for identical step changes in the set point profile for two alternate smooth control laws. The former contains an integral term, while the latter employs both an integral term and spatial weighting. Integral action increases the underdamped behavior as can be seen in Figure 10. The weighting makes the trajectories even more oscillatory (Figure 11). However, both attain the proper steady-state temperature values along the reactor. The sliding surfaces for these control laws are the same as eqs 39 and 46 from the previous example. The smooth integral controller is given by u = -1 -
For this case, A = 1 and
ti
= 0.1.
Ind. Eng. Chem. Res., Vol. 34, No. 2, 1995 565 (b)
(a)
t
2
la8 P
1.6
;1.4 t 12 U
r e
-1
0
1
2 tine
3
4
-1
0
1
2
3
tine
4
1
0
.
-1
0
8
1
2
3
k 4
-1
0
1
2
3
4
tine
tine
IC)
1-21
- 1 0 1 2 3 4 tine
tine
Figure 9. Case study 2: (a) set point response of temperature at z = 0.0, 0.3, 0.6, and 1.0 under sliding mode control with the continuous feedback control law. Dashed lines represent set points. (b) Input response to a set point change with the continuous feedback control law. (c) Set point response of concentration at x = 0.0, 0.3, 0.6, and 1.0 under sliding mode control with the continuous feedback control law.
Figure 10. Case study 2: (a) set point response of temperature at x = 0.0, 0.3, 0.6, and 1.0 under sliding mode control with the continuous feedback control law with integral action. Dashed lines represent set points. (b) Input response to a set point change with the continuous feedback control law with integral action. (c) Set point response of concentration at x = 0.0,0.3, 0.6, and 1.0 under sliding mode control with the continuous feedback control law with integral action.
The weighted control law with integral action is
The weighting function is again given by eq 44. The parameters are A = 1, zi = 0.1, and wo = 0.05. The weighting constant implies that the error at the reactor outlet is 20 times more important than at the inlet. Although this very high penalty significantly increases the oscillatory behavior under feedback control, the resulting closed-loop response to the set point change is nonetheless acceptable. The advantage of this control law should be evident in cases in which parametric uncertainty causes steady-state offset. Next, we have studied the process for set point changes in the face of plant-model mismatch. A 10% discrepancy is assumed in the dimensionless activation energy, Eo. We have attempted to return the plant to its nominal steady state. The switching control law performs reasonably well in returning the profiles to their original steady-state values, and as in the previous case study, the average error is driven to zero (Hanczyc, 1994). Due to the parametric uncertainty, the exact set point profile is uncertain. Thus, the vanishing of the average error does not and cannot guarantee that the point error will also go to zero. Again, we have considered the smooth control since the switching law results in undesirable control actions, hence the flow rate response would be much more acceptable. However, for this control law, we have observed a significant steady-state offset (Hanczyc, 1994). Thus, an alternate control law became necessary.
(b)
- 1 0 1 2 3 4 tine
-1
0
1
2
3
- 1 0 1 2 3 4 tine
4
tine
Figure 11. Case study 2: (a) set point response of temperature at x = 0.0, 0.3, 0.6, and 1.0 under sliding mode control with the continuous, weighted, feedback control law with integral action. Dashed lines represent set points. (b) Input response to a set point change with the continuous, weighted, feedback control law with integral action. (a) Set point response of concentration at x = 0.0, 0.3, 0.6, and 1.0 under sliding mode control with the continuous, weighted, feedback control law with integral action.
The integral controller is expected to bring the temperature and concentration profiles as near to the set points as the switching law does, and the overall response becomes more underdamped than the smooth controller. Weighting is added to the integral control law to reduce the outlet error, since maintaining the exit concentration a t its set point is our control goal.
566 Ind. Eng. Chem. Res., Vol. 34,No.2, 1995 (0)
- 2 0 2 4 6 8 tine
(bl
- 2 0 2 4 6 a tine
(cl
- 2 0 2 4 6 8 tine
Figure 12. Case study 2: (a) set point response of temperature at z = 0.0, 0.3,0.6,and 1.0 with parameter uncertainty under sliding mode control with the continuous,weighted, feedback control law with integral action. Dashed lines represent set points. (b) Input response to a set point change with the switching feedback control law. (c) Set point response of concentration at x = 0.0, 0.3,0.6,and 1.0 with parameter uncertainty under sliding mode control with the continuous, weighted, feedback control law with integral action.
Figure 12 displays the simulations for this control law. The response is somewhat more oscillatory than under other control laws, however, the system does settle within approximately the same time frame.
Conclusions The combination of the method of characteristics and sliding mode control was shown to be an effective methodology to design controllers for a class of nonlinear systems described by first-order partial differential equations. The examples chosen displayed a single characteristic direction, however, we have shown that systems with multiple characteristics can also be treated with this methodology (Hanczyc, 1994). Research is also underway to extend these concepts to more general PDE classes using the symmetry groups (Olver, 1986).
Bonvin, D.; Rinker, R. G.; Mellichamp, D. A. On Controlling an Autothermal Fixed-bed Reactor at an Unstable Steady State. Chem. Eng. Sci. 1983,38,233-244. Bonvin, D.; Mellichamp, D. A. A Unified Derivation and Critical Review of Modal Approaches to Model Reduction. Znt. J. Control 1982,35,829-839. Budman, H. M.;Webb, C.; Holcomb, T. R.; Morari, M. Robust Inferential Control for a Packed-Bed Reactor. Ind. Eng. Chem. Res. 1992,31,1665-1679. Cinar, A. Controller Design for a Tubular Catalytic Reactor. Can. J . Chem. Eng. 1984,62,746-754. Crider, J. E.; Foss, A. S. An Analytic Solution for the Dynamics of a Packed Adiabatic Chemical Reactor. AIChE J. 1968,14,7784. Foss, A. S.;Edmunds, J. M.; Kouvaritakis, B. Multivariable Control System for Two-bed Reactors by the Characteristic Loci Method. Ind. Eng. Chem. Fundam. 1980,19,109-117. Friedly, J. C. Dynamic Behavior of Processes; Prentice-Hall: Englewood Cliffs, NJ, 1972. Gruhle, W.-D.; Isermann, R. Modelling and Control of a Refrigerant Evaporator. Proc. ACC 1985,287-292. Hanczyc, E. M. Modeling and Control of Chemical Process Systems Described by Partial Differential Equations. Ph.D. Dissertation. University of California, Davis, 1994. Hanczyc, E. M.;Palazoglu, A. Eigenvalue Inclusion for Model Approximations to Distributed Parameter Systems. Znd. Eng. Chem. Res. 1992,31,2538-2546. McCabe, W. L.; Smith, J. C.; Harriott, P. Unit Operations of Chemical Engineering; McGraw-Hill: New York, NY,1985. Olver, P. J. Applications of Lie Groups to Differential Equations; Springer Verlag: New York, NY,1986. Palazoglu, A.; Owens, S. E. Robustness Analysis of a Fixed-bed Tubular Reactor: Impact of Modeling Decisions. Chem. Eng. Commun. 1987,59,213-227. Ray, W. H. Advanced Process Control; McGraw-Hill: New York, NY,1981. Rawlings, J. B.; Witkowski, W. R.; Eaton, J. W. Modelling and Control of Crystallizers. Powder Technol. 1992,69,3-9. Romagnoli, J. A,; Gani, R. Studies of Distributed Parameter Systems: Decoupling the State-Parameter Estimation Problem. Chem. Eng. Sci. 1983,38,1831-1843. Sira-Ramirez, H. Distributed Sliding Mode Control in Systems Described by Quasilinear Partial Differential Equations. Syst. Control Lett. 1989,13,177-181. Sira-Ramirez, H. Distributed Sliding Regimes in Systems Described by Nonlinear First Order Partial Differential Equations. Presented at the Fourth Latin American Congress of Automatic Control: Puebla, Mexico, 1990. Slotine, J.-J.E.; Li, W. Applied Nonlinear Control; Prentice Hall: Englewood Cliffs, NJ, 1991. Utkin, V. I. Sliding Modes and Their Application in Variable Structure Systems; MIR Publishers: Moscow, Russia, 1978.
Received for review January 14, 1994 Revised manuscript received September 2 , 1994 Accepted October 27, 1994@
IE940024C
Literature Cited Arnold, V. I. Geometric Methods in the Theory of Ordinary Differential Equations; Springer Verlag: New York, NY,1988.
Abstract published in Advance ACS Abstracts, January 15, 1995. @
