Nonlinear Bounded Control of a Nonisothermal CSTR - American

Ind. Eng. Chem. Res. 2000, 39, 3789-3798

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Nonlinear Bounded Control of a Nonisothermal CSTR Wei Wu Department of Chemical Engineering, National Yunlin University of Science and Technology, Touliu, Yunlin 640, Taiwan

Since many process control designs suffer degradation or failure from systems with input constraints and/or complex dynamic behavior, the control objective is to stabilize reactor systems which have constraints on both the magnitude and the rate of change of manipulated input. The developed constrained control schemes are composed of a nonlinear feedback for achieving the satisfactory tracking performance combined with a high-gain control to impair other possible large controller motions. On the basis of this two-input control scheme, the closed-loop stability can be guaranteed, and the bounded output regulation can also be achieved. Within the stable observation framework, the saturated output feedback control is developed. Finally, the proposed four designs are all evaluated through the simulations of the control of a nonisothermal CSTR with chaotic behavior. 1.Introduction Nonlinear chemical systems can show a variety of behavioral patterns, depending on operating conditions and the intrinsic features of the system. These dynamic behaviors may be strongly nonlinear or can perform complex oscillations or even chaotic motion.8 However, from a control point of view, these are processes where gains and time constants vary significantly over a state region of operation. Therefore, the use of conventional PI or PID control methods is inadequate, since they perform well only on linear or mildly nonlinear systems.5,6 In particular, when a system has a large number of embedded unstable orbits or unstable steady states, the servo and stable control problems become very challenging. During the past decade, a nonisothermal CSTR with continuous oscillation and even chaotic behavior was addressed.11 To stabilize this type of CSTR behavior, an adequate regulatory control using the proportional feedback technique at the desired orbit as the system performance was presented.4 Also the approach of proportional feedback control could directly stabilize the chaotic dynamics.7 Since proportional feedback does not need knowledge of the chaotic system, a high-gain feedback as well as large output offset is inevitable. However, to overcome these drawbacks, the feedback linearization control technique has been successfully applied for chemical processes with strong nonlinearities,15 improving tracking performance and reducing costs. The nonlinear internal model control (NIMC) strategy with derivative inputs is developed for controlling chaotic reactors.19 However, from a practical standpoint, the bounded controller motions must be taken into account, hence the use of linear or nonlinear designs must consider the saturation condition. The design of a saturating actuator for linear systems must include the limitations of the actuator, which have to be incorporated a priori into the control design. To this end, a low-and-high gain design technique to establish robust stabilization for stable systems was proposed,18 and a stabilizing controller through a Lyapunov-based approach for unstable linear systems or feedback linearizable systems was pre-

sented.12 Besides, the traditional nonlinear bounded control is circumvented by first designing a linearizing control for the no saturation case and then tuning its gains so that the bounds do not meet. Therefore, the successful implementation of nonlinear control relies on individual experience, as well as on extensive tuning and testing programs. However, the specified tuning algorithm on nonlinear controllers may be conservative and should be implemented case by case. While addressing a nonlinear bounded control problem for the case of exothermic reactor control, Calvet and Arkun6 reported performance degradation and stability problems due to input constraint. This problem was overcome in their work by means of on-line reconstruction and by eliminating cancellation the effect of saturation due to model mismatch. For a class of singleinput unstable chemical reactors, using global static nonlinear feedback, by partial or complete exact linearization, Alvarez et al.2 showed that the bounded control problem could be posed and solved at the structural level. Recently, the problem of designing a stabilizing feedback linearizable system with bounded inputs has also been addressed.3,13 Alonso and Banga3 proposed a general method for tuning global linearization control with a strategy that showed a low-gain static controller computed to preserve stability. Kendi and Doyle13 extended NIMC to systems with constrained inputs and measured disturbances, while the synthesis nonlinear controller must solve an instantaneous optimization to minimize the performance that is subject to actuator constraints. In addition, the nonlinear model predictive control (NMPC), which is composed of an input-output linearization strategy and optimization subject to constraints, has been discussed in recent papers.16,17,21 However, the MPC design may be infeasible due to the state inequality constraint22 and complex nonlinear dynamics. To avoid any computational algorithms, including optimization and/or prediction for systems with input constraints, Viel et al.22 showed a generic class of two-input exothermic chemical reactors could be globally stabilized by state feedback with bounded input at an unstable steady state.

10.1021/ie990186e CCC: $19.00 © 2000 American Chemical Society Published on Web 09/14/2000

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Ind. Eng. Chem. Res., Vol. 39, No. 10, 2000

On the basis of the illustrated reactor process with strong nonlinearity and parameter perturbation, the techniques for unreasonably large control motion on a P control, as well as conventional I/O linearization were investigated.10,23 If practical constraints on the magnitude and the rate of manipulated input are considered, undesirable control performance and even unstable responses may be inevitable. The present work addresses two-input control schemes for stabilizing the reactor temperature in the presence of dual input constraints. Through systematic design, the control structure is composed of a saturated nonlinear controller based on the I/O linearization approach, together with a high-gain controller that is bounded and varies depending on the value of the other input. The objective is to guarantee closed-loop stability as well as bounded output regulation. Moreover, within the stable observation framework, the saturated output feedback control is developed. Finally, the application and performance of the four derived control designs (including a modified static/dynamic state feedback controller, a simplified observer-based controller, and an extension of switching strategy as a high-gain action) are all demonstrated for a nonisothermal CSTR system that exhibits chaotic behavior and parameter perturbations. 2. A Nonisothermal Reactor

k1

irreversible first-order, and a series reaction, A 98 B k2

98 P, which one is endothermic and another is exothermic, takes place. In these reactions heat is removed through a coolant temperature Tc. Under the mass and energy balance, the reactor yields the following nonlinear model

dCA Fr ) (CAf - CA) - κ1 exp (-E1/RT)CA dt Vr dCB Fr ) (-CB) + κ1 exp (-E1/RT)CA dt Vr κ2 exp (-E2/RT)CB (1) (-∆H1) dT Fr ) (Tf - T) + κ1 exp (-E1/RT)CB dt Vr Fcp UA(T - Tc) (-∆H2) κ2 exp (-E2/RT)CB Fcp FcpVr by means of the following definitions:

CAf - CAf0 CA CB T - Tf x2 ) x3 ) φ v) CAf0 CAf0 Tf CAf0 φ)

Figure 2. Open-loop dynamics with the perturbed parameter (B ˆ) for the dimensionless reactor temperature (x3).

The system shown in (1) can be put into the dimensionless form

Consider the CSTR with volume Vr, where a volumetric flow rate Fr is processed. A nonisothermal,

x1 )

Figure 1. Nonisothermic CSTR with the two-input control scheme.

E1 Vr E2 τ) Da ) κ1τ exp (-φ) γ ) RTf Fr E1 κ2 (-∆H2) S ) exp (φ(1 - γ)) R ) κ1 (-∆H1) (-∆H1)CAf0 Tc - Tf UA B) φ β) u) φ FcpTf FcpFr Tf

x˘ 1 ) 1 - x1 - Dax1k1(x3) + v x˘ 2 ) -x2 + Dax1k1(x3) - DaSx2k2(x3)

(2)

x˘ 3 ) -x3 + BDax1k1(x3) - BDaRSx2k2(x3) β(x3 - u) where k1(x3) ) exp(x3/(1 + x3/φ)) and k2(x3) ) exp(γx3/(1 + x3/φ)). It is assumed that the dimensionless reactor temperature, x3, is a controlled output; the dimensionless coolant temperature u can be viewed as a manipulated input, and v is also considered as a bounded input that changes the value of the inlet concentration of the reactant A, CAf. For practical implementation, the amount of pure reactant A can be manipulated and add into the inlet flow to change CAf. The control scheme presented for this reactor system is shown in Figure 1 and the meanings of all of the above variables and parameters are defined in the Notation section. For discussion of the open-loop behavior dynamics of this CSTR, let us first assume that the process parameters are Da ) 0.26, φ ) 1000, β ) 7.995, γ ) 1, S ) 0.5, B ) 75.1, R ) 0.426, and zero inputs (u ) v ) 0). Thus, we can obtain one stable equilibrium point, (xS1 , xS2 , xS3 ) ) (0.03, 0.056, 4.85), as the initial operating point. Second, if we choose the “real” parameter B ˆ with an external disturbance, then B ˆ ) B(1 + δ) is considered where δ can be denoted as a percentage perturbation. Figure 2 demonstrates the effect of perturbation δ in the uncontrolled CSTR system, showing that for δ e -10%, continuous oscillation (limit cycles) and even chaotic behavior (δ ) -23%) result. This indicates that the CSTR dynamics are very complex and are combined with mixed limit cycles and chaos when the system parameters of eq 2 are perturbed in the described region.

Ind. Eng. Chem. Res., Vol. 39, No. 10, 2000 3791

Figure 3. Reactor process with perturbation (δ ) -0.23) using P control (kc g 1) at time ) 4 for tracking set point (x3,set ) 4): (a) the output x3 response; (b) the feedback control u.

Figure 4. Reactor process with perturbation (δ ) -0.23) using I/O linearizing control (IOLC) at time ) 4 for tracking set point (x3,set ) 4): (a) the output x3 response; (b) the control u with c* ) 1.

3. Unconstrained Controls Since the illustrative CSTR system has particular dynamic behavior, how to stabilize it at a fixed point, as well as how to follow the desired output trajectory with the aid of control feedback will be investigated. In the past 10 years, Bandyopadhyay et al.4 and Chen et al.7 proposed a proportional feedback control (P control) to stabilize chaos, but an unfavorably large control effort resulted. Next, two control strategies, a traditional P control and a nonlinear feedback control by I/O linearization, are applied for output regulation of this reactor system with no input constraints. P Control. If x3 is a controlled output, the proportional feedback control with set point is of the form

u(t) ) kc(x3,set - x3)

(3)

where x3,set is the set point of the reactor temperature and kc > 0 is a tuning parameter. However, this control is a typical output feedback and does not require any knowledge of the system model. Consider the CSTR in eq 2 with constant inlet concentration (v ) 0) and the perturbed parameter B ˆ, the linear control input u(t) by eq 3 is equivalent to control heat and transfer direction such that the concentration of medium reactant B increases. If the control action for this CSTR is forced at time ) 4 and x3,set ) 4, Figure 3a shows that the CSTR with a perturbed parameter B ˆ (δ ) -23%) can be stabilized by the controller parameter kc g 1. In particular, a simple P control with high-gain feedback (kc ) 5) can almost suppress the oscillation such that this CSTR with chaotic dynamics can be stabilized. However, Figure 3b

shows that a controller with a large value of kc can produce a very fast and large response that can be unrealizable in practice. I/O Linearizing Control. On the basis of the application of the I/O linearization technique for many nonlinear processes,15 we choose a nonlinear state feedback control of the form

u(t) ) β-1[c*(x3,set - x3) - fd(x,B ˆ )]

(4)

where fd(x,B ˆ ) ) -x3 + B ˆ Da(x1k1(x3) - RSx2k2(x3)) - βx3 and c* > 0 is a tuning parameter. To demonstrate the control result, if the states of this CSTR can be completely measured and the perturbed parameter B ˆ is known, Figure 4a shows that asymptotic output tracking can be achieved by this I/O linearizing control (IOLC) forced at time ) 4, x3,set ) 4, and the controller parameter c* ) 1. The corresponding controller motion is shown in Figure 4b. It is noted that the effort for nonlinear control using state feedback is small compared to that for P control. Since the I/O linearizing control offers partial linearization, it is necessary to check the stability of the internal dynamics at the operating region. On the basis of this control scheme (4), the resulting closed-loop dynamics from eq 2 is

x˘ 1 ) 1 - x1 - Dax1k1(x3) x˘ 2 ) -x2 + Dax1k1(x3) - DaSx2k2(x3) x˘ 3 ) c*(x3,set - x3)

(5)

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If the input v is assumed to be a piecewise constant and have fast switching like the on-off control, then v is described by

v(u) ) 0 if |u| < U ) sgn(u) otherwise

(8)

where sgn is the sign function:

sgn(u) ) +1 if u > 0 ) -1 if u < 0 Under the measurable states and known perturbation, the saturated nonlinear control from eq 4 is shown as Figure 5. Phase plane of the closed-loop dynamics by using IOLC (c* ) 1).

If we set x3,set ) 4, Figure 5 can be denoted as the phase plane plot of the closed-loop dynamics in eq 5 for c* ) 1. This indicates that this closed-loop dynamic has an asymptotically stable equilibrium point at (xS1 , xS2 ) ) (0.067, 0.117). Moreover, to set x3 ) x3,set in eq 5 the closed-loop zero dynamics is asymptotically stable and written as

x˘ 1 ) 1 - x1 - Dax1k1(x3,set) x˘ 2 ) -x2 + Dax1k1(x3,set) - DaSx2k2(x3,set)

(6)

These two control schemes can be successfully utilized for this chaotic CSTR, but the unconstrained I/O linearizing control can also provide satisfactory tracking performance, particularly for a system with large operating region. Moreover, it is worthwhile to investigate the controlling system with input constraints, such that the nonlinear feedback control is close to the practical design. 4. Constrained Nonlinear Controls It is well-known that many control designs with unconstrained or constrained schemes may be impaired due to process saturation. However, with the aid of the above control designs for the constrained and chaotic system will become inappropriate. If this CSTR system with the same perturbed parameter B ˆ is considered under new initial conditions (x1(0), x2(0), x3(0)) ) (0.067, 0.117, 4), for |u| e U and U ) 1, parts a and b of Figure 6 show the set point (x3,set ) 2) tracking and controller action by P control and I/O linearizing control, respectively. Note that these control schemes due to constrained action and a particular set point are no longer capable of stabilizing this nonisothermal CSTR. Hence, methods to avoid the undesirable closed-loop responses will be addressed in the following control designs. Design I. It is considered that the concentration of reactant A in the feed, CAf, can be manipulated, which should reduce the rate of nonisothermal reaction and can thus have a cooling effect on the system. Thus, the two-input reactor system with a perturbed parameter B ˆ is given by


(7)

x˘ 3 ) -x3 + B ˆ Da(x1k1(x3) - RSx2k2(x3)) - β(x3 - u)

u(t) ) sat {β-1[c*(x3,set - x3) - fd(x,B ˆ )]} |u|eU

(9)

where sat is the saturation function:

sat(u) ) u if |u| < U ) sgn(u)U otherwise The role played by the additional control v(u) is equivalent to the value of CAf from zero to double the base value of concentration of reactant A in the feed, 2CAf0, when the reactor temperature is too low or too high. Moreover, under the constrained nonlinear control of eq 9 and the additional input of eq 8, the closed-loop stability cannot be guaranteed due to the values of the controller parameter c*, the set point x3,set, and process constraints with the bounded input and perturbation. Therefore, we introduce the following assumption regarding the reactor system. Assumption 1: Under the input constraint |u| e U, for U > 0, and the known perturbation bound, |δ| e δ* for δ* > 0, the following inequality, for (x1,x2) ∈ [0,1], x3,set ∈ [x3-,x3+], and v ∈ {-1,1}, holds:

ˆ )|x3 ) x3,set| e U |β-1fd(x,B

(10)

Under the above inequality, the known conditions, U, δ* and process parameters in eq 10 can ensure that the set point x3,set is bounded and belongs to the specified reactor temperature interval. However, some set points or disturbances can result in the saturation problem, i.e., |u| g U. In this reactor control, the controller saturation can obviously degrade the close-loop stability due to the ineffective nonlinear control scheme. Consider the reactor system in eq 2 with two control inputs by eqs 8 and 9. If x3,set and δ are given, then the nonlinear control by the inequality of eq 10 is considered at saturation. Thus, the resulting perturbed dynamics from eq 7 are written as

x˘ 1 ) 1 - x1 - Dax1k1(x3,set) + v x˘ 2 ) -x2 + Dax1k1(x3,set) - DaSx2k2(x3,set) (11) Since v is a bounded control and the “nominal” zero dynamics in eq 6 is asymptotically stable in the large, the dynamics in eq 11 are asymptotically stable in the described bounded set for v ∈ {-1,1} and conditions in Assumption 1. Moreover, this implies that the constrained state feedback in eq 9 is a bounded input for stabilization, even though the control action of v is a fast discontinuous switching.


Theorem 1: Suppose that •Assumption 1 holds •the perturbed dynamics in eq 11 is stable •the controller parameter c* in eq 9, for x3,set ∈ [x3-,x3+] and (x1,x2) ∈ [0,1], satisfies

|β-1[c*(x3,set - x3) - fd(x,B ˆ )]| e U

(12)

Then, a constrained state feedback control in eq 9 combined with the bounded input in eq 8, the resulting closed-loop reactor system is obtained


(13)

x˘ 3 ) c*(x3,set - x3) Moreover, the output tracking is not offset, i.e., limx3 ) tf∞

x3,set. Proof: Since the changeable input v can reduce the rate of nonisothermal reaction as well as the variation of the coolant temperature, then |u| < U. Under assumption 1 and the condition of parameter c* in eq 12, the inequality becomes |β-1fd| < U. Note that the saturation of input u disappears and the closed-loop dynamics in eq 13 is obtained. Therefore, the asymptotic output regulation is obtained for an appropriate value of c*. To demonstrate this control scheme, we show in a simulation how the constrained control with the above two input control design, designated design I, can achieve the asymptotic output tracking. Under selected set points, Figure 7, parts a-c, shows that the desired state trajectories can be obtained. The saturated feedback control, u, with one parameter, c*, is shown in Figure 7d, and the feedback input, v, varies from the saturation of u as shown in Figure 7e. Since the action of input v compared to variations of states is minor, the corresponding states of closed-loop system can result in the almost smooth profiles. Note that the input response of u is very close to the saturation line due to the switching action of v. Design II. From a practical aspect, the constrained I/O linearizing control strategy still arrive at large and fast control action in transient response, as shown in Figures 4b and 7b. It is therefore apparent to stabilize the reactor system under dual input constraints, which involve both bounded control action, as has been considered in design I, and smooth controller condition. This means that the saturated control problem here comes from the assumption that the rate of control motion is limited. In other words, the derivative control input is likely to be bounded and written as

|u˘ | e Ud

(14)

for Ud > 0. Since the derivative input is limited to eq 14, the original static feedback in eq 4 necessarily leads to the dynamic state feedback. The well-established dynamic control input is through

ˆ Dak1(x3)]x˘ 1 - [B ˆ DaRSk2(x3)]x˘ 2 + x¨ 3 ) [B [B ˆ Da(x1k1′(x3) - RSx2k2′(x3)) - (1 + β)]x˘ 3 + βu˘ (15) where k1′(x3) ) exp (x3/(1 + x3/φ))(1 + x3/φ)-2 and

Figure 6. Reactor process under bounded input (U ) 1) using P control (kc ) 5) and IOLC (c* ) 5) for tracking set point (x3,set ) 2): (a) the output x3 responses; (b) the feedback control u.

k2′(x3) ) γ exp (γx3/(1 + x3/φ))(1 + x3/φ)-2. Consider that the derivative control input has the following form:

u˘ (t) ) β-1[(*)2c1(x3,set - x3) - *c2f3(x,u,B ˆ) fu(x,u,B ˆ )] (16) ≡ Φ(x,u,B ˆ) where

ˆ ) ) -x3 + B ˆ Da(x1k1(x3) - RSx2k2(x3) f3(x,u,B β(x3 - u) fu(x,u,B ˆ ) ) [B ˆ Dak1(x3)]γ˘ 3 - [B ˆ DaRSk3(x3)]x˘ 2 + [B ˆ Da(x1k1′(x3) - RSx2k2′(x3)) - (1 + β)]x˘ 3 and (*,c1,c2) ∈ R+ are tuning parameters. As shown in eq 16, this control is a dynamically I/O linearizing control (DIOLC). The resulting closed-loop dynamics by eq 16 are written as

x˘ 1 ) 1 - x1 - Dax1k1(x3) x˘ 2 ) -x2 + Dax1k1(x3) - DaSx2k2(x3)

(17)

∫0t[*c1(x3,set - x3) - c2f3] dτ

x˘ 3 ) *

Under the stable zero dynamics in eq 6 and the fixed parameters (c1,c2) being Hurwitz for * > 0, the above system in eq 17 is asymptotically stable at equilibrium points. Moreover, if the process with no constraints uses the above dynamic state feedback, the speed of the dynamic, x˘ 3, as well as the control performance, i.e., x3

3794


Figure 7. Reactor process under bounded input (U ) 1) using design I with respect to set points (x3,set ) 2, 5); (a) the state x1 responses; (b) the state x2 responses; (c) the state x3 responses; (d) the feedback control u; (e) the additional control (v).

f x3,set, only depends on tuning parameter * in eq 16. However, a large * may produce fast and large control action. Using the previous design and satisfying the smooth condition in eq 14, we present two-input control scheme with a constrained DIOLC and the same input v in eq 9 as follows:

u(t) )

∫0tΦ(x,u,Bˆ ) dτ}

sat { |u| e U

(18a)

where sat is defined as

sat(u˘ ) ) u˘ if |u˘ | < Ud

(19)

-1

) -β fu otherwise The roles played by the dual input constraints are that one is the bounded input (|u| e U) and the other is the limitation for the speed of controller motion (|u˘ | < Ud). Moreover, based on the control scheme in eqs 18 and 19, the stability of the resulting closed-loop system

x˘ 1 ) 1 - x1 - Dax1k1(x3) + v

with

u˘ (t) )

ˆ )} sat {Φ(x,u,B |u˘ | e Ud

x˘ 2 ) -x2 + Dax1k1(x3) - DaSx2k2(x3) (18b)

∫0t[*c1(x3,set - x3) - c2f3] dτ

x˘ 3 ) *

(20)


Figure 8. Reactor process under dual input constraints (U ) 1 and Ud ) 5) using design II and (c1,c2,*) ) (1,2,20) with respect to the set point (x3,set ) 2); (a) the output x3 responses; (b) the feedback control u; (c) the additional control v.

will be guaranteed under the variation of controller parameter *, set point x3,set, saturated conditions, and perturbation. If the derivative input in eq 18 is at saturation, this state feedback with * ) 0 in eq 18 is considered and yields y¨ ) 0 by eq 14 and |u| < U from eq 18. Although the switch strategy in eq 19 cannot ensure the stable closed-loop system due to * ) 0 and |u˘ | e Ud, this control scheme should be able to be utilized against large transient responses. Therefore, we introduce the following assumption regarding the reactor system. Assumption 2: Under dual input constraints |u| < U and |u˘ | e Ud, and the known perturbation bound |δ| e δ*, the appropriate controller parameters (*,c2) are such that the following inequalities, for (x1,x2) ∈ [0,1], x3,set ∈ [x3-,x3+], and v ∈ {-1,1}, hold:

|β-1[

∫0t(*c2f3 + fu) dτ]|x )x 3

eU 3,set

(21)

and -1

|β fu| < Ud

(22)

This assumption implies that the variation of *c2 in eq 21 can be decided and the inequality (22) can be held while * ) 0 in eq 18. The following theorem shows that this case of constrained reactor systems controlled by this two-input design can be asymptotically stable over a large operating region. Theorem 2: Suppose that •Assumption 2 holds •the perturbed dynamics in eq 11 is stable •the parameters (c1,c2) in eq 18 are Hurwitz

•the parameter *, for x3,set ∈ [x3-,x3+] and (x1,x2) ∈ [0,1], satisfies

∫0tΦ(x,u,Bˆ ) dτ| e U

|

(23)

Then, for the constrained DIOLC design in eqs 18 and 19 combined with the bounded input v ∈ {-1,1}, the resulting closed-loop reactor system in eq 20 is stable. Moreover, the output tracking is not offset, i.e., lim x3 tf∞

) x3,set. Proof: If (c1,c2) ) (1, 2) is selected, the controller parameters in eq 18 will be reduced to one parameter *, and the inequality (eq 23) can be reduced as

|β-1[

∫0t[(*)2(x3,set - x3) - 2*f3 - fu] dτ]| e U

(24)

Under |u| e U, |u˘ | e Ud, and * ) 0, by assumption 2 we obtain

|β-1fu| < Ud |β-1

∫0tfu dτ|x )x 3

3,set

(25) 0, the closed-loop system


the system is of relative order 1 and only the process output can be measured, we introduce a nonlinear observer

xˆ˙ 1 ) 1 - xˆ 1 - Daxˆ 1k1(xˆ 3) + v

(28)

e¨ ) -2*e˘ - (*)2e is also stable for the polynominal, e + 2*e˘ + (*)2e, being Hurwitz, and e ≡ x3 - x3,set. To illustrate the result of Theorem 2, it is shown by a simulation how this design, designated design II, can be achieved. Figure 8a shows that the desired output trajectory can be asymptotically achieved, considering the case of either no constraints or constraints by the bounded input (U ) 1) together with the bounded derivative input (Ud ) 5). The corresponding saturated nonlinear state feedback design by eqs 18 and 19 with * ) 20 is shown in Figure 8b. Note that under dual input constraints, the manipulated coolant temperature (u) is smoother than in the previous design. The additional control for the reactor process is depicted in Figure 8c. Design III. Inspired by the above saturated designs, design III still has certain drawbacks, including state feedback and measurable disturbances, etc. To achieve practical implementation, the developed saturated output feedback control scheme is introduced. Since this CSTR has complex oscillations and unstable open-loop dynamics, the state-space internal model control design is inappropriate.9 Therefore, based on the stable observation and control of an unstable CSTR within the I/O linearization framework,1,24 when

xˆ˙ 2 ) -xˆ 2 + Daxˆ 1k1(xˆ 3) - DaSxˆ 2k2(xˆ 3)

(29)

xˆ˙ 3 ) -xˆ 3 + BDa(xˆ 1k1(xˆ 3) - RSxˆ 2k2(xˆ 3)) - β(xˆ 3 - u) + Q(Lo,xˆ )(y - h(xˆ )) where (xˆ 1,xˆ 2,xˆ 3) represent the state estimation. The function h(xˆ ) represents the observer output. Q(k1,xˆ ) ) Lo(∂h(xˆ )/∂xˆ )-1 is denoted as a nonlinear observer gain, and Lo is a positive parameter. Obviously, the nonlinear output function and the nominal parameter (B) are considered here. Considering that the system relative order is 1, the previous design, design I, with state estimation becomes

u(t) )

-1 sat {(β∂h(xˆ )/∂xˆ ) [c*(yset - y) |u| e U (∂h(xˆ )/∂xˆ )fd(xˆ ,B)]} (30)

where yset can be treated as the desired output or set point. Moreover, to check the stability of the reactor within the controller-plus-observer framework of eqs 29 and 30, we employ the following theorem. Theorem 3: Suppose that •Assumption 1 holds •the perturbed dynamics in eq 11 is stable •the controller parameter c* in eq 12 holds •the term, Q(Lo,xˆ ) ) Lo(∂h(xˆ )/∂xˆ )-1, exists for Lo > 0


•the inequality, |(∂h(xˆ )/∂xˆ )f3(xˆ ,u,B) - (∂h(x)/∂x)f3(x,u,B ˆ )| e Bd, holds, where Bd g 0. Then, for the constrained observer-based controller in eqs 29 and 30 combined with the bounded input v ∈ {-1,1}, the resulting closed-loop reactor system is stable and the output regulation is bounded. Proof: Since this reactor is of relative order 1, let z ) h(x) and zˆ ) h(xˆ ), yielding

z˘ ) (∂h(x)/∂x)f3(x,u,B ˆ)

(31)

zˆ˙ ) (∂h(xˆ )/∂xˆ )f3(xˆ ,u,B) + Lo(z - zˆ )

(32)

and

To define the estimation error, ξ ) zˆ - z, by eqs 31 and 32, we obtain

ˆ )] ξ˙ ) -Loξ + [(∂h(xˆ )/∂xˆ )f3(xˆ ,u,B) - (∂h(x)/∂x)f3(x,u,B (33) Since the magnitude of |(∂h(xˆ )/∂xˆ )f3(xˆ ,u,B) - (∂h(x)/∂x)ˆ )| is bounded, the error ξ in eq 33 is also bounded f3(x,u,B for Lo > 0. Thus, the observer-based controller using eqs 29 and 30 can stabilize the perturbed process. According to the previous description of this reactor, ∂h(xˆ )/∂xˆ ) ∂h(x)/∂x ) constant, the resulting output tracking form can be expressed as

ˆ ) - fd(xˆ ,B) e˘ o ) -c*eo + fd(x,B

(34)

ˆ ) - fd(xˆ ,B)| is where eo ≡ y - yset. Obviously, if |fd(x,B finite, the bounded output regulation can be achieved. When the concentrations of the reactor and the perturbation δ are unknown but bounded, parts a-c of Figure 9 show that the system uses the saturated output feedback design, designated design III, with controller parameters c* ) 5 and Lo ) 1000. If the constraint U ) 1 and the unmeasured perturbation δ are considered, Figure 9a shows that stable output regulation can be achieved. The corresponding saturated output feedback controller is shown in Figure 9b and the discontinuous control is depicted in Figure 9c. It is noted that the proposed design cannot entirely eliminate the output offset due to B ˆ * B in eq 33. Also the discontinuously switching design for avoiding the process saturation is still difficult to implement. Moreover, we can provide a simple extension for design III by selecting the continuous input v ∈ [-1,1], shown as

v(u) ) u/U if |u| < U

(35)

and the saturated output feedback by eq 30 with integral action is written as -1 sat {β [c*(Tset - x3) + |u| e U

∫0t(Tset - x3) dτ - fd(xˆ )]}

τ*

ously, if the resulting closed-loop system


) sgn(u) otherwise

u(t) )

Figure 10. Reactor process under dual input constraints (U ) 1 and Ud ) 5) for unmeasured states and perturbation using design IV and (c*,Lo,τ*) ) (5,1000,5) with respect to the set point (x3,set ) 2); (a) the output x3 responses; (b) the feedback control u; (c) the additional control v.

(36)

where τ* > 0 is an adding controller parameter. Obvi-

(37)

x˘ 3 ) fd(x,B ˆ ) - fd(xˆ ,B) + c*(x3,set - x3) +

∫0t(x3,set - x3) dτ

τ*

for appropriate parameters (c*,τ*), is stable, then the derivative of x3 is bounded. Moreover, the integral term ∫0t (x3,set - x3) dτ exists and is finite, thus by Barbalat’s lemma14 x3 f x3,set as t f ∞. Finally, design III can be extended by eqs 35 and 36 and designated as design IV. Figure 10a shows that

3798


asymptotic output regulation can be achieved due to the smooth control design (eq 35) and the integral action (τ* ) 5) under the same constraints and perturbations. The corresponding manipulated coolant temperature is shown in Figure 10b, and the smooth control is depicted in Figure 10c. 5. Conclusion The control of chaotic CSTR systems is challenging due to these systems’ embedded unstable orbits and their tendency to lead to thermal runways. In this study, the static and dynamic state feedback developed for the constrained and chaotic CSTR can achieve the asymptotic stabilization if the manipulated coolant temperature required boundedness as well as sufficiently smooth trajectory. The systematic design of bounded inputs with two controllers for the three-state reactor process is such that the main results of theorems 1 and 2 can be evaluated and explained in detail. During implementation, the saturated and dynamic feedback control developed here for systems with dual input constraints can provide low controller action and regulate the process states. Moreover, as the result of theorem 3, the implemented output feedback control scheme is bounded with two control inputs, such that the observer-based closed-loop system can be stabilized and robust against unknown disturbances. It is noteworthy that the presented design IV is a practical design since it provides the output feedback design and smooth control inputs, and it is without offset performance. Although the illustrated example is a typical series reaction in a CSTR, we think that the proposed methodologies could be extended to a generic class of chemical reactors with multiple reactions. Furthermore, the compact saturated control design for more general systems will be addressed in future works. Acknowledgment The author acknowledges Hwa Hsia College of Technology and Commerce for its partial financial support. This work is supported by the National Science Council of Republic of China under Grant Number NSC-872214-E-146-002. Notation CA ) concentration of reactant A CB ) concentration of reactant B CAf ) inlet concentration of reactant A CAf0 ) base value of inlet concentration of reactant A cp ) heat capacity of reacting mixture E1,E2 ) activation energy of reaction A f B and B f P, respectively Fr ) volumetric feed rate P ) desired product R ) ideal gas constant T ) reactor temperature Tc ) coolant temperature Tf ) inlet stream temperature Tset ) set point U ) heat-transfer coefficient Vr ) reactor volume

F ) density of process liquid τ ) reactor residence time

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Greek Letters (-∆H1), (-∆H2) ) heat of reaction A f B and B f P, respectively κ1,κ2 ) frequency factor for reaction A f B and B f P, respectively

Received for review March 15, 1999 Revised manuscript received March 13, 2000 Accepted July 22, 2000 IE990186E

Nonlinear Bounded Control of a Nonisothermal CSTR - American

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