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Finite Difference Output Feedback Control of a Class of Distributed Parameter Processes Wei Wu† Department of Chemical Engineering, National Yunlin University of Science and Technology, Touliu, Yunlin 640, Taiwan
Low-dimensional nonlinear feedback control schemes for distributed parameter processes are addressed. The main design procedure involves the use of a feasible numerical method to establish a finite difference-based differential equation system and to utilize the pure nonlinear feedback design for achieving uniform output regulation. When the almost uniform input distribution at steady-state conditions can be planed, the controller reduction can be involved in the proposed control design guidelines. Based on the nonlinear inverse design procedure, the low-dimensional and approximate feedback control law is developed such that the stable and bounded output tracking can be guaranteed. With the aid of internal model control strategy, the low-dimensional output feedback control scheme can ensure the asymptotic output regulation. Finally, those control methodologies are shown to be robust and effective in controlling a nonisothermal tubular reactor at possibly realistic situations. 1. Introduction The dynamic models of chemical processes are inherently nonlinear, and spatial gradients by transport involve variables in both time and space. Representative examples of processes with significant spatial variations have a plug-flow reactor (PFR) and a heat exchanger.12 The conventional approach to modeling and subsequent control design is based on various lumping techniques on the spatial discretization of the partial differential equation (PDE) system by orthogonal collocation or a finite difference scheme, and then the finite-dimensional controller as the practical control framework would be synthesized. Because the finite difference scheme is a simple and feasible technique for numerical approximation,13 many of the PDE systems should be reduced to a finite difference-based differential equation (FDDE) system. Although the established FDDE model is a typical finite-dimensional ODE model, the high-order approximation approach may lead to a complex controller framework, which still cannot be implemented in practice because of large computational requirements. Besides, uncertain model errors can strictly degrade the closed-loop stability as well as the tracking performance. Recently, significant research efforts have been focused on the development of geometric control for PDE systems that can directly account for their spatially distributed nature.2-4 The basic idea and concept is extended from the ODE to the PDE model. Christofides and Daoutidis2,4 developed the distributed feedback controllers that enforce output tracking and guarantee the overall stability of closed-loop systems. The method of characteristics is addressed on the basis of the PDE model itself and combination of the specified operators for controller reduction. Note that this specified input/ output operator can induce the finite-dimensional dominant structure. Moreover, Christofides and Daoutidis3 provided some comments and developments on the control of nonlinear distributed parameter systems. † Tel: 886-5-5342601 ext. 4620. Fax: 886-5-5312071. Email:
[email protected].
On the other hand, some recent works emphasized the methodologies of nonlinear control and transformation techniques. Hanczyc and Palazoglu9 used the method of characteristics such that the PDE system can be exactly transformed into a class of ODE models. Doyle et al.7 proposed a nonlinear controller design for a packed-bed reactor. Dochain et al.6 used the adaptive control scheme for a reduced nonlinear model by the orthogonal collocation method in place of an original fixed-bed bioreactor. Kurtz et al.11 proposed a nonlinear feedback control law which attenuates undesired oscillations on population balance models in the bioreactor system. In addition, Wu14 proposed a low-dimensional state feedback controller, which is implemented to suppress the effect of inlet disturbances of a nonisothermal PFR system. Gundepudi and Friedly8 considered that the manipulated input variable is denoted as the characteristic flow velocity and the controlled output is a function of time alone. Based on the feasible input/ output perspective, the distributed parameter system can be reduced to a nonlinear ODE model; thus, a discrete-time control can be implemented to compensate the process deadtime due to the fluid transportation. In light of developments, a general and practical framework for the analysis and control of nonlinear chemical processes associated with distributed nature is an important investigation. In this work, the novel control algorithm via the feasible controller reduction can deal with unknown disturbances. Based on the conventional numerical approximation, the nonlinear distributed parameter systems can be reduced to a nonlinear FDDE model, in which the spatial discretization method is assumed to capture the distributed nature of the PDE systems. Under the feedback linearization algorithm for the specified input/output relationship, the pure nonlinear feedback controller can induce the uniform state/output regulation. Moreover, if the almost uniform input distribution at steady-state conditions can be planed, the lowest possible number of actuators located at the selected positions can be determined. In virtue of the nonlinear inverse design procedure, the inverse-based feedback control law is
10.1021/ie9908783 CCC: $19.00 © 2000 American Chemical Society Published on Web 09/23/2000
Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000 4251
synthesized step-by-step. Within this “approximationthen-design” framework,12 the synthesis of state feedback and model-based output feedback control schemes can provide both stable output tracking and disturbance attenuation. Finally, simulation of a nonisothermal PFR system is presented to demonstrate the performance and robustness of proposed control methodologies.
(4) A simple technique for approximating the PDE system is based on the finite difference scheme; e.g., the second-order approximation with the five-point formula is used and shown as
2. Problem Formulation We consider a quasi-linear hyperbolic PDE system described by a state-space description of the form
∂xj ∂xj + vl ) ˜f(xj,u j) ∂t ∂z y0(t) ) h(xj)|z)L xj(0,t) ) x0(t) xj(z,0) ) xs(z)
(1)
where xj(z,t) ∈ H[(0,L),Rn] represents state variables and the symbol H[‚] is an infinite-dimensional Hilbert space of states defined on the interval (0, L) and z ∈ (0, L) ⊂ R; vl is the system parameter (e.g., it can be described as the flow velocity of PFR systems); x0 ∈ Rn is the boundary input of systems; xs ∈ Rn is the vector of initial state profiles; ˜f(‚) ∈ Rn is a smooth vector field; y0 ∈ R represents the process output located at the terminal position (z ) L); and u j ∈ Rm is the vector of internal inputs. In fact, the compositions of feed in many chemical processes are often perturbed by external disturbance. So, the perturbed boundary input can be denoted as x0 ) σ(ω), where ω ∈ Rd represents an exogenous input and σ(0) ∈ Rn is the nominal boundary input. To resolve the distributed term of eq 1, some numerical methods can provide the accuracy of approximation, e.g., the finite difference scheme and orthogonal collocation technique, such that the PDE system can be approximated to a finite-dimensional nonlinear model. Under the specified finite difference scheme, the socalled FDDE model is obtained
x˘ k ) -vl
|
∂x + f(xk) + g(xk) uk, k ) 1, 2, ..., p ∂z k
3. Input/Output Linearization Approach
y0 ) h(xp) xk(0) ) xsk
Above the finite difference scheme is composed of the forward finite difference terms in eq 3a, center finite difference terms in eq 3b, and backward finite difference terms in eq 3c. Particularly, the center finite difference of eq 3b possesses the symmetrical combination. Remark 1. Without loss of generality, the large number p and the high-order finite difference scheme can directly minimize numerical errors. Naturally, we can assume that the FDDE system (2) can be substituted for the original PDE system as the numerical system. Obviously, this approximate approach still leads to a very high-dimensional dynamical system. To address some practical issues, the low-dimensional controller and observer syntheses with the aid of the effective model reduction have been addressed.14,15 Remark 2. According to the recent results including the finite-dimensional robust control5 and adaptive control1 for a class of distributed parameter systems, these developments can ensure the robust stability of closed-loop systems on the basis of the finite-dimensional control framework. Therefore, we will seek that the developed control scheme can be robust against numerical errors as well as unknown perturbations at the inlet streams.
(2)
where p denotes the number of spatial points and xk ∈ Rn represents the estimated state variables at the node point k. The term (∂x/∂z)k ∈ Rn represents the resolved discretization term. u ∈ Rp represents the virtual control input at spatial points. Moreover, this approximate model with the following main assumptions should be made: (1) If the selected approach for the approximation is feasible and the number of node points is sufficiently enough, then xj(k∆z,t) ) xk can hold, where ∆z is the fixed interval in the z direction. (2) The finite-dimensional model of eq 2 is affine in u. (3) Part of the inputs can be considered as the manipulated inputs. This means that only ul, for l ∈ {i|1 e i e p}, can represent the implemented control input at the selected position (z ) l∆z).
Considering that the PDE system can be approximated to a finite-dimensional model like eq 2, the new control methodology on the basis of FDDE models can be constructed step-by-step. First, a simple linearization approach for FDDE systems is introduced, and the variable for tracking errors is defined as
e1 ) y1 - y2 l ek ) yk - yk+1 l ep-1 ) yp-1 - y0 ep ) y0 - ysp
(4)
where ysp is a constant setpoint; yk ≡ h(xk) can be supposed to virtual multioutputs. Moreover, the time derivative of tracking errors from eqs 2 and 4 can be expressed as
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is considered, the complicated and nonsystematical controller structure is inevitable. Note that the simple linearization procedures in eqs 4-9 are treated as the beginning preparation. Besides, this finite-dimensional control scheme is perfect, so it cannot be implemented in practice. Therefore, the proper control design can be developed as a result of uniform state/output regulation, and thus the low-dimensional control technique would be systematically established as follows. 4. Low-Dimensional Control Design
and a compact form can be written as
e˘ ) A(x) + B(x) u
(6)
where e ∈ Rp, A(x) ∈ Rp, and B(x) ∈ Rp×p. If the matrix B(x) is nonsingular, the nonlinear state feedback can be directly expressed as
u ) B(x)-1[-c1e - A(x)]
(7)
where c1 ∈ R is a constant controller parameter. Moreover, the closed-loop subsystems by eqs 2, 6, and 7 are shown as
e˘ ) -c1e η˘ k ) q(xk)
(8)
∂ηk g(xk) ) 0 ∂xk
(9)
Generally, the approximate control design may cause the numerical errors for the implemented systems, and the closed-loop system probably tends to be unstable. Therefore, the feasible low-dimensional design procedures for FDDE systems are essential to determine the appropriate numbers and positions of the actuators. In light of recent developments of the low-dimensional nonlinear control approach,4,8,14,15 we seek a controller reduction technique that can provide some degree of tracking performance and robustness. If the controlled outputs in eq 4 are denoted as functions of one state, then the uniform output regulation of FDDE systems in eq 10 can result in a uniform state at steady state, i.e.
lim Hx1 ) Hx2 ) ... ) Hxp
where H ∈ R1×n is a constant unit vector and Hxi ) h-1(yi), for 1 e i e p. If the uniform steady state in eq 12 is satisfied, the center difference terms in eq 3b would be close to zero
with satisfying
where ηk ∈ Rn-1 and q(xk) ) (∂η/∂xk) [-vl (∂x/∂z)|k + f(xk)]. If the parameter c1 > 0 and the internal dynamics, η˘ k, is stable, then the uniform and asymptotic error tracking at the spatial points can be achieved, i.e.
lim ek ) 0 tf∞
(10)
Remark 3. Because the p-dimensional FDDE model can be described as the multi-input (uk) and multioutput (yk) nonlinear system, based on the “multivariable” nonlinear feedback (7) and transformations in eqs 4 and 9, the closed-loop system in eq 8 can induce a linear input/output response. However, the condition of relative degree 1, (∂h/∂x) g(x) * 0, should be satisfied a priori. If its condition cannot hold, we shall choose other transformations, ζ1 ) e and ζ2 ) A, such that the FDDE system can be transformed to
ζ˙ 1 ) ζ2 ζ˙ 2 ) C(x) + D(x) u
(11)
If the matrix D(x) is nonsingular, the condition of relative degree 2 can hold. However, the nonlinear feedback derived from eq 11 must be more complicated than before. If the higher relative degree formulation
(12)
tf∞
lim H tf∞
|
∂x ≈ 0, for 3 e j e p ∂z j
(13)
Similarly, the term limtf∞ [∂h(xj)/∂xj] (∂x/∂z)|j approach to zero. However, the boundary point x0 in eq 3a,b can induce nonzero difference terms at steady state
lim tf∞
|
|
∂x ∂x * lim *0 ∂z 1 tf∞ ∂z 2
(14)
Moreover, we can show that the following property is evident from the above algorithm. Proposition 1. Considering that the uniform steady state of the closed-loop FDDE system can be achieved, the steady-state input can be expressed as
[ |
usk ) lim g(xk)-1 vl tf∞
∂x - f(xk) ∂z k
]
(15)
If limtf∞ (∂x/∂z)|1 * limtf∞ (∂x/∂z)|2 * 0 and limtf∞ f(xk) ≈ xss can hold, where xss ∈ R represents the constant value, the steady-state input distribution can be written as
us1 * us2 * us3
and
us3 ≈ us4 ... ≈ usp
(16)
Moreover, the control input at steady state in eq 15 can
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be reduced to
[ | [ |
] ]
us1 ) lim g(x1)-1 vl
∂x - f(x1) ∂z 1
us2 ) lim g(x2)-1 vl
∂x - f(x2) ∂z 2
tf∞
tf∞
usj ) lim g(xj)-1[-f(xj)]
(17)
tf∞
Remark 4. Obviously, the result of eq 15 can be directly derived from eq 2. Based on the almost zero values of difference terms in eq 13 and two nonzero difference terms in eq 14, the steady-state input at us1, us2, and us3 would be obviously different. Using the conditions of limtf∞ f(xk) ≈ xss, the remainding steadystate inputs usj can be written as
usj ) lim g(xj)-1[-f(xj)] ≈ x˜ ss, tf∞
for 3 e j e p and x˜ ss ∈ R (18)
Hence, the almost uniform steady-state input in eq 16 can be obtained. Remark 5. Based on the previous nonlinear feedback control approach, the state trajectory and dynamic behavior of the closed-loop FDDE system can be investigated. If the state variable by inverse of the output function can be uniformly achieved at steady state, then the condition of limtf∞ f(xk) ≈ xss can be well-founded. Moreover, the steady-state input distribution with almost equivalent values like eq 16 can be established. In fact, the variety of concentrations for many reaction processes is often slighter than the change of the reactor temperature. Moreover, the steady-state inputs at spatial discretization of the FDDE systems would be almost uniform like eq 16. Inspired by proposition 1, we think that p-dimensional control inputs at spatial operating points can be further reduced. Based on the established steady-state input distribution in eq 16, the three-dimensional control input can be treated as the approximate state feedback
ui ) Ri(x,ui+1), i ) 1-3 )
(
)[ -1
( |
) )]
∂h(xi) ∂h(xi) ∂x g(xi) -c1ei + vl - f(xi) ∂xi ∂xi ∂z i ∂h(xi+1) ∂x vl - f(xi+1) - g(xi+1) ui+1 (19) ∂xi+1 ∂z i+1
( |
Because usj in eq 18 has an almost equivalent relationship, the third control input (u3) is assumed to replace the (p - 2)-dimensional control inputs. However, the input u3 in eq 19 cannot be directly implemented because of unknown input u4 involved. Using the result of proposition 1 and a reduced-dimensional feedback controller structure, we provide an inverse-based lowdimensional feedback control
where yi, with i ) 1-3, represents the addition of controlled outputs. This proposed low-dimensional control technique for the reduction of the FDDE system is clearly depicted in Figure 1. Remark 6. For above controls combination, the first actuator, u1, is an individual input and its location is determined by the finite difference scheme of the FDDE systems. The other two inputs are derived from the inverse procedure by eq 19. Besides, the (p - 2)dimensional control uj can be reduced to one-dimensional control u3, so the proposed feedback control law is low-dimensional and as similar as an approximate feedback. Moreover, the resulting closed-loop system by eqs 2 and 20 can be written as
y˜˘ 1 ) -c1y˜ 1 y˜˘ 2 ) -c1y˜ 2 y˜˘ 3 ) -c1y˜ 3 x˘ l ) -vl
|
∂x + f(xl) + g(xl) u3 ∂z l η˘ k ) q(xk)
(21)
where 3 < l e p, y˜ 1 ≡ y1 - ysp, y˜ 2 ≡ y2 - y1, and y˜ 3 ≡ y3 - y2. Furthermore, the above design procedure can be summarized as follows. Proposition 2. Consider that the uniform state trajectory of the closed-loop FDDE system at the start time can be planed
Hx1(0) ) Hx2(0) ) ... ) Hxp(0)
(22)
and the steady-state input distribution in eq 16 is used; thus, an inverse-based low-dimensional feedback control in eq 20 can yield a partially input/output linearized closed-loop system in eq 21. Moreover, if the nonlinearized dynamics of eq 21 is stable, then the output tracking can be asymptotically achieved, i.e.
lim y˜ i ) 0, for i ) 1-3 tf∞
(23)
Remark 7. In fact, the above inverse-based controls may reveal the problem of feasibility. Note that the approximate feedback control may cause the closed-loop instability because of the effects of numerical errors or unknown perturbations. Moreover, the lowest numbers of actuators at the selected position refer to the steady-
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Figure 1. Inverse-based low-dimensional feedback control scheme.
Figure 2. Low-dimensional output feedback control scheme.
state input distribution and the internal stability of the closed-loop system in eq 21. To analyze these point controls for the stable output regulation, the nominal system should be open-loop stable. Thus, the asymptotic output regulation in eq 23 can induce the three steadystate inputs, us1, us2, and us3, and the subsystem of eq 21
x˘ l ) vl
|
∂x + f(xl) + g(xl) u3 ∂z l
(24)
resembles steady-state values in xl. Furthermore, the process output regulation is bounded, i.e.
lim (y0 - ysp) e tf∞
(25)
where g 0 is a small constant. 5. Output Feedback Control Scheme Although the previous low-dimensional control technique possesses some practical elements, the estimation of the state variable at the specified points is inevitable. To release the previous constraint of accurate state information, a low-dimensional output feedback control scheme for the FDDE systems would be developed here. Inspired by the significant result of the nonlinear internal model control strategy,10 a finite-dimensional FDDE model can be considered as
|
∂xˆ + f(xˆ k) + g(xˆ k) uˆ k xˆ˘ k ) -vl ∂z k xˆ k(0) ) xsk
(26)
Figure 3. Using the nonlinear feedback control scheme (c1 ) 2, ysp ) Tsr(1)) at 100 spatial points: (a) response of the reactor temperature at spatial points; (b) corresponding concentration of the reactant; (c) steady-state input distribution.
time derivative of eq 27 can be written as Note that this model can be denoted as the nominal system of eq 2, i.e., difference terms of eq 26 with x0 ) σ(0). Moreover, the new variable for setpoint tracking is defined as
(28)
If the nonlinear feedback like eq 7 is considered,
uˆ ) B(xˆ )-1[-c1eˆ - A(xˆ )]
eˆ 1 ) h(xˆ 1) - h(xˆ 2)
≡ Rˆ (xˆ ,ym)
l eˆ p-1 ) h(xˆ p-1) - h(xˆ p) eˆ p ) ym - ysp
eˆ˘ ) A(xˆ ) + B(xˆ ) uˆ
(29)
a linearized FDDE model can be expressed as
(27)
where ym ) h(xˆ p) is denoted as the observer output. The
|
∂xˆ + f(xˆ k) + g(xˆ k) Rˆ k(xˆ ,yˆ m) xˆ˘ k ) -vl ∂z k
(30)
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Remark 8. Obviously, the result of proposition 3 is the extension of the nonlinear internal model control scheme,10 so the linearized FDDE model in eq 31 can be denoted as an open-loop state observer. Under the results and conditions of propositions 1 and 2, the lowdimensional output feedback controller can be expressed as
uj ) Rˆ (xˆ ,uj+1) )
(
∂h(xˆ j) g(xˆ j) ∂xˆ j
)[ -1
-c1eˆ j +
( |
∂h(xˆ j+1) ∂xˆ vl ∂xˆ j+1 ∂z
j+1
( |
)
∂h(xˆ j) ∂xˆ vl - f(xˆ j) ∂xˆ j ∂z j
)]
- f(xˆ j+1) - g(xˆ j+1) uj+1
(32)
Figure 4. Under the setpoint, ysp ) 310 K, and using a lowdimensional state feedback controller: (a) response of the reactor temperature in the process outlet; (b) corresponding concentration of the reactant in the process outlet; (c) corresponding manipulated inputs.
Similarly, the above design algorithms are also derived from the input/output linearization approach. Furthermore, some properties of the observer-based controller can be summarized as follows. Proposition 3. When a p-dimensional FDDE system as a nominal model is considered, using the transformation in eq 27 and p-dimensional nonlinear feedback in eq 29, a linearized FDDE model like eq 30 is given by
|
∂xˆ xˆ˘ k ) -vl + f(xˆ k) + g(xˆ k) Rˆ k(xˆ ,y0) ∂z k
(31)
Then the process output y0 can asymptotically track to the setpoint; i.e., limtf∞ y0 ) ysp.
Thus, the so-called observer-based controller is composed of a linearized DDE model in eq 31 and lowdimensional nonlinear feedback in eq 32. However, the nominal FDDE system must be open-loop stable, but only the process output, y0, is available. If unknown disturbances appearing in the inlet direction are introduced, the near start point of the actuators by eq 32 should achieve the asymptotic output regulation. Particularly, this model-based controller only relies on one tuning parameter for the improvement of the control performance. The proposed control framework is depicted in Figure 2. Remark 9. The configuration of the proposed output feedback control is a typically model-based control framework. The introduced internal model aims to urge the asymptotic output regulation of the FDDE systems, but it cannot guarantee the convergence of state estimation. Therefore, its dynamics of the observer must be open-loop stable a priori as the constraint of this output feedback scheme. Under the same FDDE model and the low-dimensional design procedure, the low-dimensional output feedback controller can be constructed. However, this reduced-dimensional controller needs to involve some errors from controller reduction and plant/model mismatch. Generally, the control performance would become worse than the previous state feedback design. To rectify the degradation, the additional numbers of actuators could be naturally useful. Consequently, the proposed steady-state input distribution in proposition 1 and low-dimensional control framework given in proposition 2 are not unique results, but they provide the novel controller reduction algorithm for nonlinear FDDE systems. Via numerical implementation and the illustrated example, we conceive that the synthesis of the controller should be workable and useful. 6. Plug-Flow Reactor System Consider a plug-flow reactor with a volume Vr, where k1
an exothermic, irreversible first-order reaction, A 98 B, takes place. The inlet stream consists of a pure reactant A, CA0, and temperature TA0. The reaction rate is assumed to be of the form:
rA ) -k0e-E1/RTrCA where k0 and E1 represent the frequency factor and the activation energy of the reaction, respectively. Under assumptions of perfect radial mixing, no diffusion in the
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reactor, and insulate on the reactor outer, via the mass and energy balances, the following reactor model can be described:
∂CA ∂CA ) -vl - k0e-E1/RTrCA ∂t ∂z U0 ∂Tr ∂Tr (-∆H1) -E1/RTr ) -vl + k0 e CA + (T - Tr) ∂t ∂z Fcp FcpVr s (33) subject to the boundary conditions
CA(0,t) ) CA0, Tr(0,t) ) TA0(1 + δ)
(34)
and the initial condition
CA(z,0) ) CsA(z), Tr(z,0) ) Tsr(z)
(35)
where CA denotes the concentration of reactant A, Tr is the temperature of the reactor, δ denotes the inlet disturbance of the reactor temperature, and Ts is the surface temperature of the naked reactor. Besides, the meanings of the above variables and parameters have been defined in the notation section. Based on the model reduction algorithm by virtue of the numerical approximation, a nonlinear FDDE system can be expressed as
x˘ 1,k ) -vl x˘ 2,k ) -vl
|
|
∂x1 + f1,k(x1,k,x2,k) ∂z k
∂x2 + f2,k(x1,k,x2,k) + g2,k(x1,k,x2,k) Ts ∂z k x1,k(0) ) CsA(k∆z) x2,k(0) ) Tsr(k∆z)
Table 1. Nominal Parameter Values for the PFR Model
(36)
where ∆z ≡ L/p is the fixed interval in the z direction, x1,k ) CA(k∆z,t), and x2,k ) Tr(k∆z,t). The nonlinear functions are shown as
f1,k ) -k0 exp(-E1/Rx2,k)x1,k f2,k )
Figure 5. Dynamic profiles of the PFR system for a 5% increase in the inlet flow-rate temperature: (a) open-loop response of the reactor temperature; (b) corresponding concentration of reactant A.
(-∆H1) U0x2,k k0 exp(-E1/Rx2,k)x1,k Fcp FcpVr U0 g2,k ) FcpVr
Under the known parameters as shown in Table 1 and sufficient discretized points (p ) 100, ∆z ) 0.01), this FDDE system in place of the PFR system can be assumed to a numerical model. Using the suitable steady-state profiles as initials, the high reactor temperature can facilitate the high conversion of reactant. Moreover, it is supposed that the reactor temperature can be directly controlled, yk ) Tr(0.01k,t), while the surface temperature Ts at the surface position of the reactor can be varied or manipulated, ul ≡ Ts(l∆z,t), for l ∈ {i|1 e i e p}. Steady-State Input Distribution. Based on the input/output linearization approach, the pure nonlinear feedback by eq 7 can be employed. Figure 3a shows that the uniform reactor temperature in the entire reactor
CA0 ) 3 TA0 ) 300 vl ) 1.0 L ) 1.0 E1 ) 2.0 × 104 k0 ) 5.0 × 1012
kmol/m3 K m/min m kcal/kmol min-1
-∆H1 ) 10 000 F ) 900 Vr ) 2 U0 ) 2000 cp ) 0.2 R ) 1.987
kcal/kmol kg/m3 m3 kcal/min‚K kcal/kg‚K kcal/kmol‚K
can be asymptotically achieved, i.e.
lim Tr(0.01k,t) ) Tsr(z)|z)1 rf∞
Obviously, the uniform reactor temperature can induce the highest conversion of reactant depicted in Figure 3b. Under the specified finite difference and satisfying condition limtf∞ |x1,k - x1,k+1| , limtf∞ |x2,k - x2,k+1|, the steady-state input distribution is shown in Figure 3c. According to the proposed controller reduction algorithm, the inverse-based low-dimensional feedback control by eq 20 can be introduced. The transformed internal dynamics of eq 33 is shown as
∂CA ) -µ(Tr)CA ∂θ
(37)
with θ ≡ (t + z/vl)/2 and the function µ > 0. Obviously, this dynamic can be stabilized by exponential decay. State Feedback Implementation. To demonstrate the proposed low-dimensional state feedback, Figure 4a shows that the reactor temperature at the process outlet can asymptotically track to the new setpoint, ysp ) 310
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Figure 6. Attenuation of inlet disturbances, δ ) (10%, using a low-dimensional state feedback controller: (a) response of the reactor temperature in the process outlet; (b) corresponding concentration of the reactant in the process outlet.
K. Note that the large value of parameter, c1 ) 2, can provide the satisfactory tracking performance. The outlet concentration of the reactant as shown in Figure 4b can also asymptotically achieve a low and new steady-state value. Moreover, the corresponding individual inputs are depicted in Figure 4c. When the weak disturbance effect for this FDDE system, an inlet perturbed temperature (δ ) 5%), is considered, Figure 5a shows that the temperature profile distributed in the reactor is changed. The corresponding reactant concentrations in the selected position will tend to unbounded oscillations depicted in Figure 5b. Note that the dynamic behavior of the numerical PFR system is very sensitive for a small inlet perturbation at the boundary point. With the aid of the proposed control scheme, Figure 6a shows that the bounded output regulation can be achieved while large inlet perturbations of temperature, δ ) (10%, are considered, respectively. The corresponding concentration of the reactant in the process outlet is depicted in Figure 6b. Note that both temperature and concentration variations at the process outlet can maintain smooth trajectories such as no disturbance effects. This is the reason that the controller location is close to the start point of reactor, and the disturbance appearing in the inlet space can be immediately eliminated as soon as possible. Output Feedback Implementation. From a practical viewpoint and using the result of proposition 3, the output feedback control scheme by eqs 31 and 32 can be considered. Under the same operating conditions and
Figure 7. Under the setpoint, ysp ) 310 K, and using a lowdimensional output feedback controller: (a) response of the reactor temperature in the process outlet; (b) corresponding concentration of reactant A in the process outlet; (c) corresponding manipulated inputs.
tuning parameters, Figure 7a shows that the reactor temperature at the process outlet can asymptotically track to the new setpoint, ysp ) 310 K, and the large value of the parameter, c1 ) 2, can provide the satisfactory tracking performance. The satisfactory outlet concentration of reactant is shown in Figure 7b. Moreover, the corresponding individual inputs are depicted in Figure 7c. When the perturbations of temperature are considered, Figure 8a shows that the asymptotic output regulation can be achieved. Based on a reduced modelbased controller and internal model control scheme, the no offset tracking as well as disturbance attenuation can be guaranteed. However, Figure 8b shows that the
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output feedback controllers can guarantee the stable and bounded output regulation. More importantly, the presented control strategy can effectively eliminate the inlet disturbances, so the nonlinear control design technique has strong robustness. With the aid of successive simulations for comparison, we are convinced that the developed controller design can perform well and fit in with the practical implementation. Acknowledgment This work was supported by the National Science Council of Republic of China under Grant NSC-89-2214E-146-001. Notation CA ) concentration of reactant A CA0 ) inlet concentration of reactant A cp ) heat capacity of the reacting mixture c1, c2 ) selected parameters of the controller E1 ) activation energy of the reaction R ) ideal gas constant Tr ) reactor temperature Ts ) surface temperature of the naked reactor TA0 ) inlet stream temperature U0 ) heat-transfer coefficient Vr ) reactor volume vl ) velocity of the inlet stream to the reactor Figure 8. Attenuation of inlet disturbances, δ ) (10%, using a low-dimensional output feedback controller: (a) response of the reactor temperature in the process outlet; (b) corresponding concentration of the reactant in the process outlet.
concentration of the reactant in the process outlet almost becomes unstable; obviously the disturbance in the inlet space cannot be suppressed by the developed model-based control scheme. Consequently, a low-dimensional and observer-based controller synthesis is a practical design. Particularly, the state feedback control strategy can provide both good tracking performance and robustness, and the observer-based feedback controller can guarantee the asymptotic output regulation. Although the methodology must be based on the numerical approximation and the steady-state profiles of closed-loop systems, via the above demonstration we think that the proposed control schemes for a class of PFR systems is proper and reliable.
7. Conclusion Under the sufficient numbers of spatial points and the appropriate finite difference scheme for the numerical approximation, the established nonlinear FDDE system could directly capture the characteristic of PDE systems. When the uniform state regulation of the closed-loop system can be achieved, the steady-state input distribution of numerical processes can provide a criterion for the controller reduction. Moreover, the lowdimensional design procedures can be treated as the cascaded nonlinear control framework, and the state/
Greek Letters (-∆H1) ) heat of the reaction κ0 ) frequency factor for the reaction δ ) inlet temperature perturbation F ) density of the process liquid
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Received for review December 6, 1999 Revised manuscript received June 2, 2000 Accepted August 2, 2000 IE9908783
