Nonlinear Internal Model Control for Systems with ... - ACS Publications

Ind. Eng. Chem. Res. 1998, 37, 489-505

489

Nonlinear Internal Model Control for Systems with Measured Disturbances and Input Constraints Thomas A. Kendi† and Francis J. Doyle, III*,‡ School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907-1283

A nonlinear model-based control scheme is presented which extends nonlinear internal model control to systems with constrained inputs and measured disturbances. A method is proposed for a direct synthesis nonlinear controller which solves an instantaneous optimization to minimize the performance loss associated with enforcing the actuator constraints. Within this framework, the nonlinear controller can be represented as an optimal linear controller with an auxiliary feedback loop which cancels the effects of nonlinear dynamics and measured disturbances. The methods developed within the paper are applied to three simulation examples: a constrained linear example with measured disturbances, an isothermal reactor with Van de Vusse kinetics, and an industrial styrene polymerization reactor. 1. Introduction Within the internal model control (IMC) framework, synthesis results for several advanced control methodologies, such as feedforward control and constrained control (antiwindup control), have been proposed for linear systems. In spite of the successes of linear modelbased control in industry, it is recognized that chemical process systems tend to exhibit complex, nonlinear behavior which may require additional nonlinear compensation to achieve satisfactory closed-loop performance. According to Ogunnaike and Wright (1996), the following three trends dictate that chemical process systems are increasingly operated in regions where the nonlinear behavior is more prevalent: (i) increased throughput, (ii) tight quality specifications, and (iii) minimized (zero) emissions. In spite of the potential for closed-loop performance improvement, nonlinear control strategies should be applied with some caution due to the need for possibly complex nonlinear models and the commensurate increase in computational load. In addition, the nonlinear control scheme should be flexible enough to incorporate both fundamental nonlinear models, which may be difficult to obtain in practice, and empirical nonlinear models ( e.g., neural networks and Volterra models). The notion of incorporating constraints into nonlinear internal model control (NLIMC) has been explored by a number of investigators. Li and Biegler (1988) developed a single-step method which incorporated state and input constraints into the NLIMC framework. The control action was calculated using a specialized sequential quadratic programming problem and assumed that the model was perfect (plant equals model). Maner et al. (1996) developed a nonlinear model predictive control scheme for second-order Volterra models which was capable of enforcing input constraints by replacing the linear inverse in NLIMC with a linear model * To whom correspondence should be addressed. E-mail: [email protected]. Phone: (302) 831-0760. Fax: (302) 8311048. † Present address: International Paper Company, Mansfield, LA 71052. ‡ Present address: Department of Chemical Engineering, University of Delaware, Newark, DE 19716.

predictive (MPC) controller. In this approach, the control action is calculated using a horizon-based quadratic optimization problem. In both cases, the incorporation of constraints into NLIMC resulted in a controller which preserved the structure of the partitioned nonlinear inverse (a linear dynamic controller with an additive nonlinear correction); however, neither approach yielded a direct synthesis nonlinear controller. In addition, compensation for measured disturbances, standard for linear MPC approaches, was not considered. Wassick and Camp (1988) applied NLIMC to an industrial extruder and detailed several important considerations for the application of nonlinear control in industry. Advanced issues such as nonlinear feedforward control and manipulated variable constraints were addressed in an effective, but suboptimal, manner. State space methods provide an alternative framework for direct synthesis nonlinear control design for constrained systems. Alvarez et al. (1991) developed a nonlinear state feedback controller for a class of reactor systems (one input, two states) using input-output linearization and exact-state (Hunt-Su) linearization. Conditions for global asymptotic stability with bounded inputs were obtained. Kendi and Doyle (1997) developed an antiwindup scheme for input-output linearization which solved an instantaneous 1-norm minimization using the factorization scheme of Zheng, Kothare, and Morari (1994). A state-dependent constraint on the transformed input was obtained by inverting the input transformation, and nominal stability was evaluated using nonlinear µ-analysis. Spong et al. (1986) proposed a method for direct synthesis, constrained nonlinear control, related to input-output linearization, for a class of systems (minimum phase, single input-single output, and relative degree 1). The control action was calculated using a pointwise minimization (which reduced to a quadratic program). Soroush and Nikravesh (1996) propose a continuous time MPC algorithm (within the differential geometric framework) for nonlinear processes which employs a time series expansion of the reference trajectory over the “shortest possible, prediction horizon” to obtain a closed-form controller which minimizes a quadratic objective function for constrained systems. Additional classes of nonlinear problems, including systems with deadtime and incomplete state

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490 Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998

measurements, are addressed. Several practical issues concerning the application of state-space methods for chemical process systems which merit consideration are as follows: (i) full state information is often not available, necessitating the incorporation of state estimation or an observer into the control design; (ii) although several specialized approaches have been developed, input-output linearization is not applicable in general to nonminimum phase systems; (iii) the constraint mapping approach (Kendi and Doyle, 1997) required the evaluation of a complex state-dependent constraint operator and employed a potentially conservative conic sector stability analysis; (iv) the partitioned nonlinear inverse is better-suited for complex nonlinear inputoutput model structures. In the original NLIMC work, Economou et al. (1986) assumed that the nonlinear input-output model could be partitioned into a linear dynamic contribution with an additive nonlinear contribution. In the present work, measured disturbances are incorporated into the NLIMC design framework by the following: (i) including a third partition into the partitioned nonlinear model which describes the linear contribution of the measured disturbance and (ii) modifying the nonlinear partition to include nonlinear effects of the measured disturbances. For unconstrained systems, the proposed framework yields a nonlinear feedforward/feedback controller which, much like the conventional partitioned nonlinear controller, is composed of a conventional linear feedforward/ feedback controller with an additional loop to perform a nonlinear correction. It is important to note that the incorporation of feedforward compensation, while novel within the NLIMC/partitioned nonlinear inverse framework, has been applied to nonlinear systems using differential geometric methods. Calvet and Arkun (1988) developed a method for completely cancelling the effects of a class of measured disturbances (disturbances which enter the nonlinear state space model in the same manner as the input) and partially cancelling the effects of other measured disturbances for nonlinear synthesis using exact-state (Hunt-Su) linearization. Bartusiak et al. (1989) incorporated compensation for measured disturbances into reference system synthesis such that the effects of the nonlinear disturbances are exactly cancelled and the linear input-output behavior is preserved. Daoutidis and Kravaris (1989) classify measured disturbances (and the type of feedforward controller which is required) according to the disturbance relative degree, a parameter which is analogous to the inputoutput relative degree and corresponds to the number of times that the output must be differentiated until the disturbance appears explicitly. In this approach, the nonlinear feedforward terms exactly cancel the effects of the measured disturbances to preserve the linear relationship between the output and the transformed input. In this paper, a direct synthesis control scheme for constrained nonlinear systems is proposed. In the unconstrained case, the nonlinear controller perfectly “cancels” the effects of the nonlinear dynamics and measured disturbances. When the constraints are active, the nonlinear controller solves an instantaneous optimization to minimize the performance loss associated with enforcing input constraints. The nonlinear

control design decomposes into a two-step design procedure. In the first step, an optimal unconstrained controller is obtained using nonlinear internal model control (NLIMC). The unconstrained nonlinear controller is composed of the partitioned nonlinear inverse, calculated using nonlinear operator theory, and a linear IMC filter as proposed by Economou et al. (1986). In the second step, the unconstrained optimal nonlinear controller is factored into two subcontrollers for antiwindup compensation. Two attractive qualitative characteristics of the resultant nonlinear antiwindup controller are as follows: (i) the use of the partitioned nonlinear inverse requires explicit calculation of the linear inverse only, and (ii) the nonlinear antiwindup controller can be decomposed into the conventional linear antiwindup controller with an additive nonlinear perturbation. It is not difficult to envision scenarios in which the decomposition property of the nonlinear antiwindup controller can be exploited. One such instance would be in the event that the process is operating outside of the confidence limits of the nonlinear model. Under these circumstances, the operator can disable the nonlinear correction and operate with the optimal linear controller. The paper is organized as follows. In section 2, several results from NLIMC and the partitioned nonlinear inverse are reviewed to motivate the use of a novel partitioned nonlinear inverse for nonlinear feedforward/feedback synthesis. A direct synthesis nonlinear antiwindup scheme, based upon the IMC-based antiwindup approach for linear systems (Zheng et al., 1994), is proposed. In section 3, the methods are applied to three case studies: (i) a linear numerical example which demonstrates the effectiveness of the linear antiwindup controller for systems with measured disturbances; (ii) a nonminimum phase reactor example to demonstrate the application of nonlinear antiwindup with a locally stable pseudoinverse; (iii) a realistic process example, a multivariable polymerization reactor with input constraints and a measured disturbance. Conclusions and a summary follow in section 4. 1.1. Nomenclature. In this work, nonlinear state space systems of the following form are considered: p

x3 ) f(x) + g(x)u +

wi(x)di ∑ i)1

x ∈ Rn, u ∈ Rm, y ∈ Rm, d ∈ Rp

(1)

y ) h(x) where

e ui e umax umin i i

i ) 1, ..., m

(2)

Equation 1 represents a control-affine state space model, and eq 2 indicates that the manipulated inputs have upper and lower bounds.

Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 491

The saturation operator for a vector signal is defined as follows:

{

sat(u1) sat(u) ) l sat(um)

Figure 1. Block diagram realization of the partitioned nonlinear inverse.

{

umax if ui > umax i i e ui e umax if umin where sat(ui) ) ui i i min min ui if ui < ui The Lie Derivative of the scalar field h along a vector field f is defined as

[]

f1(x) ∂h ∂h ∂h ∂h f2(x) , f ) Lf h(x) ) , , ..., ∂x ∂x1 ∂x2 ∂xn l fn(x)

〈

〉 [

]

(3)

2. Results It has been shown by Doyle et al. (1995) that a nonlinear model-based controller which is designed using second-order Volterra models can be decomposed into two elements: (i) an optimal linear controller and (ii) a loop with a second-order Volterra term to compensate for the nonlinear dynamics. This decomposition arises due to the use of the partitioned nonlinear inverse within the controller design and has been shown to be applicable for input-constrained (as well as unconstrained) systems. The results in this section are organized as follows. In section 2.1, a LIMC-based antiwindup scheme which computes the control action from the implicit solution of an instantaneous minimization is presented for linear systems with measured disturbances. In section 2.2, a NLIMC controller for unconstrained nonlinear systems with measured disturbances is discussed in detail. In section 2.3, an antiwindup controller for nonlinear systems with measured disturbances is proposed. The NLIMC-based antiwindup scheme is built upon the results from sections 2.1 and 2.2. Essentially, the partitioned nonlinear inverse is used to incorporate compensation for the nonlinear dynamics and measured disturbance effects into the LIMC-based antiwindup scheme. To avoid confusion and to precisely differentiate between linear and nonlinear IMC, observe that the following convention has been adopted and applied throughout the paper (LIMC is linear IMC and NLIMC is nonlinear IMC). 2.1. Constrained Linear Synthesis. For linear systems with measured disturbances, the following objective function is solved to achieve constrained nominal performance:

min|(FA‚y)(t) - (FA‚y′)(t)|1 u ˜

zation problem posed in eq 4 is solved instantaneously at each time t, in contrast to many of the traditional performance metrics, such as integral squared error (ISE), which are calculated over a horizon. It is important to note that the solution of an instantaneous optimization in (4) does not guarantee optimal performance over a horizon. The output of the constrained system can be written as

(4)

where y is the output of the unconstrained system, y′ is the output of the constrained system, and FA is a linear filter. This objective function was originally proposed by Zheng et al. (1994) for constrained linear systems. In the present work, compensation for measured disturbances is incorporated into the design of antiwindup controllers for linear systems. The optimi-

y′ ) (PL‚u ˜ )(t) + (PD‚d)(t)

(5)

where PL and PD are linear operators describing the response of the process output to a change in the system input and the measured disturbance, respectively. If one assumes that the effects of the measured disturbances can be perfectly cancelled, i.e., PL is minimum phase and that P-1 L PD is causal, then the output of the unconstrained system with feedforward control can be written as

h ‚e)(t) - (PLP ˜ -1 ˜ D‚d)(t) + (PD‚d)(t) (6) y ) (PLQ L P where Q h is the unconstrained LIMC (feedback) control˜ D is the disturbance ler, P ˜ L is the plant model, and P model. Equation 6 can be constructed using the linear portion of the block diagram shown in Figure 4. For ˜ L, PD ) P ˜ D), the objective the nominal case (PL ) P function of the instantaneous optimization can be rewritten as

h ‚e)(t) - (FAPL‚u ˜ )(t) - (FAPD‚d)(t)|1 min|(FAPLQ u ˜

(7)

The new objective function can be interpreted as follows. In the nominal unconstrained case, the effects of measured disturbances are cancelled exactly by the feedforward/feedback controller. Therefore, the unconstrained output can be written as a strict function of the error. In the constrained case, the effect of measured disturbances cannot be exactly cancelled, necessitating two terms in the constrained output prediction equation, a term which accounts for the constrained input and a term which predicts the effects of measured disturbances. In the following derivations, we will alternate usage of the Q and P notation for both time-domain operators in the objective function and the corresponding Laplace domain operator for control derivation. The latter will be distinguished by the (s) argument on process variables. The controller which solves the instantaneous minimization is shown in Figure 2. Observe that the linear antiwindup controller is composed of three linear subh 2, and Q h 2*. The proposed antiwindup controllers, Q h 1, Q methodology is a modification of the unconstrained optimal linear feedforward/feedback controller. Therefore, when the constraints are not active, the antiwindup controller is reduced to the optimal linear feedforward/feedback controller and the following is true


Figure 2. Block diagram of LIMC-based antiwindup with feedforward compensation.

(provided that Q h is biproper): ~ Qe(s) – PL–1PDd(s)

=

feedforward/feedback controller

h , it follows that For Q h 1 ) FAPLQ

(I + Q2)–1 Q1e(s) + (I + Q2)–1 Q*2d(s)

(8)

unconstrained antiwindup controller

Equation 8 can be solved to obtain the following expressions for Q h 2 and Q h *2:

Q h2 ) Q h 1Q h -1 - I

(9)

h 1Q h -1P ˜ -1 Q h 2* ) -Q L PD

(10)

From the block diagram in Figure 2, u can be written as

h 2u ˜ (s) - Q h 2*d(s) u(s) ) Q h 1e(s) - Q

(11)

The difference u(s) - u ˜ (s) can be written as

h 2 + I)u ˜ (s) - Q h 2*d(s) u(s) - u ˜ (s) ) Q h 1e(s) - (Q h 1Q h -1u ˜ (s) )Q h 1e(s) - Q h -1P ˜ -1 ˜ Dd(s) (12) Q h 1Q L P In the following lemma, conditions for parameterizing Q h 1, such that the instantaneous optimization stated in (4) is implicitly solved by the antiwindup controller, are stated. Lemma 1. Assume that the following conditions hold: (A1) the plant is stable and minimum phase; (A2) the LIMC controller Q h is biproper (Q h and Q h -1 are proper); (A3) the feedforward compensator P-1 L PD is ˜ L and PD ) P ˜ D. Then the anticausal; (A4) PL ) P windup architecture in Figure 2 solves the minimization problem (shown in eq 4) instantaneously if the following are true: h. 1. Q h 1 ) FAPLQ 2. FAPL|s)∞ is a diagonal nonsingular matrix with finite elements. Proof 1. Consider the time domain version of eq 12:

h 1Q h -1‚u ˜ )(t) u(t) - u ˜ (t) ) (Q h 1‚e)(t) - (Q h -1P ˜ -1 ˜ D‚d)(t) (Q h 1Q L P

u(t) - u ˜ (t) ) (FAP ˜ LQ h ‚e)(t) - (FAP ˜ L‚u ˜ )(t) (FAP ˜ D‚d)(t) The dependence of the above expression on the measured disturbance arises from the use of a feedforward compensator. From eq 7, it follows that

˜ LQ h ‚e)(t) - (FAP ˜ L‚u ˜ )(t) u(t) - u ˜ (t) ) (FAP ˜ D‚d)(t) (FAP ) (FA‚y)(t) - (FA‚y′)(t) ) ˆ yf(t) - y′f(t)

(13)

If eq 13 is rewritten in vector form as

ui(t) - u˜ i(t) ) yfi(t) - y′fi(t)

i ) 1, 2, ..., n

The consequence of choosing FA such that FAPL|s)∞ is diagonal, nonsingular, and finite is that y′fi is instantaneously affected by only u˜ i for any value of the measured disturbance. Since |yfi(tk) - y′fi(tk)| is affected linearly by u˜ i(tk) (and not instantaneously by u˜ j(tk), j * i), the objective function is a convex function of u˜ i(tk). The solution to a convex optimization occurs at the or ui(tk) < umin ˜ i(tk) boundary. Thus for ui(tk) > umax i i , u ) sat(ui)(tk) minimizes |yfi(tk) - y′fi(tk)|. The same argument can be replied recursively for all n channels to show that the controller implicitly solves the instantaneous optimization. Remark 1. The lemma demonstrates a method for incorporating feedforward compensation into LIMCbased antiwindup. Nominal performance for setpoint changes and measured disturbances is guaranteed through the solution of an instantaneous optimization (whose objective function was originally proposed in Zheng et al. (1994)). Although an additional control element is introduced for systems with measured disturbances, the lemma and proof follow directly from the corresponding result in Zheng et al. (1994). Remark 2. The antiwindup filter FA is parameterized such that the following three conditions are upheld: 1. FA is diagonal (to prevent introducing a change in the output direction). ˜ L|s)∞ is a diagonal nonsingular matrix with 2. FAP finite elements. ˜ L - I are 3. The diagonal terms in the matrix FAP relative degree 1. If condition 2 cannot be met for a diagonal FA, an approximate plant model, P ˜ †L which is arbitrarily close


to P ˜ L, is chosen such that condition 2 is upheld. In this instance,

Q h 1 ) FAP ˜ †LQ h ˜ †L - I Q h 2 ) FAP Figure 3. Block diagram realization of the partitioned nonlinear inverse.

˜ †LP ˜ -1 ˜D Q h *2 ) FAP L P Remark 3. If, for a given channel, the disturbance relative degree is less than the manipulated variable relative degree, the feedforward compensator Q h *2 will not be proper (or equivalently, Q h 2* is not causal). An approximate disturbance model, P ˜ †D which is arbitrarily close to P ˜ D, can be chosen and incorporated into the feedforward compensator design such that Q h *2 is implementable. Observe that the objective function in eq 4 consists of two terms, the (filtered) unconstrained output and the (filtered) constrained output. It can be shown that the controller parameterization yields an implicit solution to the minimization for an arbitrary P ˜ †D. However, the response of the filtered unconstrained output, ˜ †D since noted as FAy(t) in eq 4, changes on the basis of P the measured disturbances are no longer exactly canceled. Since the reference trajectory for the optimiza˜ †D, the tion, FAy(t), is parameterized by the choice of P solution to the minimization changes accordingly. Thus for a P ˜ †D which detunes the response of the feedforward compensator for measured disturbances, the unconstrained output exhibits a more sluggish response to measured disturbances which leads to a more sluggish response in the constrained system. 2.2. Unconstrained Nonlinear Synthesis. In this section, the design of a NLIMC controller for unconstrained systems with measured disturbances is considered. It is assumed that the nonlinear system can be represented as the following general dynamic inputoutput model:

y ) P[u,d]

(14)

where d is the measured disturbances. For systems without measured disturbances, Economou et al. (1986) have shown that the NLIMC controller is composed of two elements: (i) an inverse of the nonlinear system and (ii) a linear filter for setpoint tracking. In the next theorem, a partitioned nonlinear inverse is obtained for the nonlinear system with measured disturbances. Theorem 1. Assume that the nonlinear operator can be partitioned as follows:

P[u,d] ) PL[u] + PD[d] + PNL(d)[u]

y ) PL[u] + PD[d] + PNL(d)[u] ) ξ Subtracting the linear contribution of the disturbance isolates the operators on the manipulated input:

PL[u] + PNL(d)[u] ) ξ - PD[d] This transforms the system into an equation that resembles the traditional partitioned nonlinear system and the inverse is calculated as follows:

PL(I + P-1 L PNL(d))[u] ) ξ - PD[d] -1 (I + P-1 L PNL(d))[u] ) PL [ξ - PD[d]] -1 -1 u ) (I + P-1 L PNL(d)) PL [ξ - PD[d]] ) H[ξ,d]

Remark 4. When the decomposition shown in eq 15 is performed, no assumptions are made about the structure of the operators PL, PD, and PNL. Therefore, this partition is a general result and not a severe limitation for the proposed framework. A NLIMC controller which cancels the effects of measured disturbances can be obtained by adding a filter to the error signal, ξ, within the partitioned nonlinear inverse. A schematic of the NLIMC controller with feedforward compensation is shown in Figure 4. Observe that the controller is composed of three subcontrollers: the LIMC (feedback controller) Q h , the linear feedforward compensator Q h D, and a nonlinear compensator Q*. For stable minimum phase systems,

Q h ) P-1 L F Q h * ) F-1PD Q* ) F-1PNL

(15)

where PL and PD are linear operators and PNL is a nonlinear operator which operates on u and is updated using the measured disturbance parameter, d. If the inverse of the linear operator PL exists, the left inverse of P, denoted as H|ξ,d], is equal to the following:

H[ξ,d] ) (I + PNL(d))-1P-1 L [ξ - PD[d]]

Proof 2. It is necessary to find an input which drives the output, y, to an arbitrary value, ξ, which is demonstrated mathematically below:

(16)

which is shown in block diagram form in Figure 3.

Remark 5. Figure 4 illustrates that the NLIMC controller can be decomposed into the following: 1. An optimal linear feedforward/feedback controller Q h F. 2. A loop with a nonlinear correction term Q* which compensates for the process nonlinearity. Doyle et al. (1995) have observed a similar decomposition for NLIMC controllers which use Volterra models. Remark 6. The NLIMC controller with feedforward compensation is consistent with previous results in the limiting cases.


Figure 4. Block diagram of the NLIMC with feedforward architecture.

1. In the absence of measured disturbances, the proposed design method yields a nonlinear controller which is consistent with Economou et al. (1986). 2. In the absence of nonlinear dynamics, the proposed design method yields the a LIMC controller with feedforward compensation which is consistent with Morari and Zafiriou (1989). 2.3. Constrained Nonlinear Synthesis. In this section, the synthesis of nonlinear controllers for constrained systems is considered. The proposed scheme has the following characteristics: 1. When the constraints are not active, one recovers the unconstrained nonlinear controller discussed in section 2.2. 2. When the constraints are active, nominal performance is guaranteed through the solution of the instantaneous minimization in eq 4 from section 2.1. The output of the constrained system can be written as

y′ ) P(u ˜ ,d) ˜ )(t) + (PD‚d)(t) + PNL(d)[u ˜ ](t) ) (PL‚u

(17)

For unconstrained systems, note that the nonlinear feedforward/feedback controller from section 2.2 is recovered. Thus,

u ) Q[e,d] The output of the unconstrained system (with feedforward control) can be written as

y ) P[Q[e,d],d](t) ˜ L, PD ) P ˜ D, PNL ) P ˜ NL), For the nominal case (PL ) P

y ) P[Q[e,d],d](t) h ‚e(t) ) PLQ

(18)

In parallel with the development of the linear antiwindup control design in section 2.1, it is assumed in

eq 18 that the unconstrained nonlinear controller perfectly cancels the effects of measured disturbances. For nonlinear systems, this assumption requires that the feedforward part of the controller is causal and that the system is minimum phase. For many nonlinear systems, it is sufficient to show that PL is minimum phase (nonlinear system is locally minimum phase) and that P-1 L PD is realizable. The linear causality test is sufficient to establish nonlinear causality in the absence of singularities which cause the linear and nonlinear relative degrees to differ. The objective function of the instantaneous optimization, eq 4, can be rewritten as

min|(FAPLQ h ‚e)(t) - (FAPL‚u ˜ )(t) - (FAPD‚d)(t) u ˜

(FAPNL[u ˜ ,d])(t)|1 (19) The controller which solves the instantaneous minimization is shown in Figure 5. The antiwindup controller is composed of three linear subcontrollers, Q h 1, Q h 2, and Q h 2* (which are exactly equivalent to the the ones in section 2.1), and Q*2, which includes nonlinear disturbance and dynamic effects. Lemma 2. Assume that the following conditions hold: (A1) the plant is stable and minimum phase; (A2) the LIMC controller Q h is biproper; (A3) PL ) P ˜ L and PD ) P ˜ D; (A4) the feedforward compensator P-1 L PD is causal; and (A5) the plant is control affine and has characteristic matrix with the following structure:

[

C(x) ) Lg1Lrf 1-1h1(x) 0 0 · · · 0

Lg2Lrf 2-1h2(x) · · · 0

· · · 0 · · · 0 · · · · · · r -1 · · · LgmLf m hm(x)

]

Then the antiwindup architecture in Figure 5 is the solution of the instantaneous minimization problem in eq 4 if the following are true:


Figure 5. Block diagram for NLIMC-based antiwindup with feedforward compensation.

[

1. The controller is parameterized as follows:

FA1 · · · FA2 0 FA ) · · · · · · 0 · · ·

Q h 1 ) FAP ˜ LQ h Q h 2 ) FAP ˜L - I ˜D Q h *2 ) FAP

· · · 0 · · · 0 · 0 · · · · · FAn

Application of this operator to the nonlinear system results in the following:

Q* ˜ NL 2 ) FAP ˜ L|s)∞ is a diagonal nonsingular matrix with 2. FAP finite elements. Proof 3. In the absence of constraints, it is not difficult to show via loop algebra that the NLIMC-based antiwindup controller is equivalent to the optimal NLIMC controller with feedforward compensation (from section 2.2). It must be shown that the proposed factorization yields an implicit solution to the instantaneous optimization. From Figure 5, observe that

u(t) ) Q h 1‚e(t) - Q h 2‚u ˜ (t) - Q h 2*‚d(t) - Q* ˜ ,d](t) 2[u which, given the proposed factorization, is equivalent to

˜ LQ h ‚e(t) - FAP ˜ L‚u ˜ (t) + u ˜ (t) - FAP ˜ D‚d(t) u(t) ) FAP FAP ˜ NL(d)[u ˜ ](t) Subtracting u ˜ (t) from both sides yields the following:

˜ LQ h ‚e(t) - FAP ˜ L‚u ˜ (t) - FAP ˜ D‚d(t) u(t) - u ˜ (t) ) FAP ˜ NL(d)[u ˜ ](t) FAP ) yf(t) - y′f(t)

]

The antiwindup filter can be represented as follows:

(20)

Much like the proof for the linear synthesis results, it has been shown that the proposed parameterization yields an equivalence between minimizing the input u and the filtered output. The final element of the proof is showing that the quantity |yfi(t) - y′fi(t)| is instantaneously a convex function of u˜ i.

[ ]

FA1y1 · FAy ) · ·F y

An n

If r1 is the input-output relative degree of y1, then it can be shown that a minimal order realization of the antiwindup filter requires that FA1 is relative degree -r1. ˜ L|s)∞ is a (This is due to the requirement that FAP diagonal nonsingular matrix with finite elements.) Under the assumption that the system is control affine and possesses a diagonal, nonsingular characteristic matrix the following are true: 1. yfi is instantaneously a function of ui. 2. yfi is a linear function of ui (follows directly from the definition of relative degree for control affine systems). Therefore, |yfi(tj) - y′fi(tj)| is a convex function of u˜ i(tj). As was the case for linear systems, the optimization can be solved channel by channel. Using the same arguments that were used for linear systems, it can be shown that u˜ i(tj) ) sat(ui(tj)) is the solution to the optimization and that the proposed control structure implicitly solves the stated minimization problem. Remark 7. The NLIMC-based antiwindup controller differs from the corresponding LIMC-based antiwindup controller by the Q*2 term (which compensates for the nonlinear dynamics and measured disturbance effects). However, the antiwindup filter FA is parameterized entirely from PL using the three conditions outlined in section 2.1. Remark 8. Doyle et al. (1995) proved that receding horizon performance criteria and penalties on the manipulated variables can be incorporated into NLIMC by replacing the exact linear inverse with an appropriate linear pseudoinverse in the control design. In


Figure 6. An equivalent representation of the NLIMC-based antiwindup controller.

addition, it was demonstrated that an optimal nonlinear controller can be obtained by proving the optimality of the linear pseudoinverse. Figure 6 shows an equivalent representation of the NLIMC-based antiwindup controller. The alternate representation illustrates that constraints are incorporated into NLIMC by replacing the unconstrained LIMC controller with the LIMC-based antiwindup controller. Within the context of the results by Doyle et al. (1995), the LIMC-based antiwindup controller functions as an optimal linear pseudoinverse. Although an instantaneous optimization is posed for NLIMC-based antiwindup, the resultant controller architecture is consistent with other results which employ more conventional optimization methods. The assumption regarding the structure of the characteristic matrix (diagonal and nonsingular) is somewhat restrictive; however, for linear systems it is straightforward to solve the optimization for an arbitrarily close approximate plant which possesses the desired structure. An approximate linear model, P ˜ †L, is † ˜ L|s)∞ is a diagonal nonsingular chosen such that FAP

matrix with finite elements. For nonlinear systems, the following approximation is recommended:

P ˜† ) P ˜ †LP ˜ -1 ˜ L P ˜ †LP ˜ -1 ˜ †LP ˜ -1 )P ˜ †L[u] + P L PD[d] + P L PNL(d)[u]

(21)

The characteristic matrix for the approximate plant shown in eq 21 is guaranteed to be diagonal through order 1. Therefore, a suboptimal (first-order) solution of the instantaneous optimization is obtained. A more rigorous solution can be obtained by finding an approximate nonlinear model y ) P ˜ †[u,d] such that the characteristic matrix is diagonal and nonsingular. Two potential difficulties arise from this approach: 1. Determination of an approximate nonlinear partition for P ˜ †[u,d] is not as straightforward as in the linear case. 2. A more complicated control structure is obtained since the subcontroller Q1 is no longer guaranteed to be linear.


In summary, the proposed approximate solution strategy trades off rigor in the solution of the optimization problem to obtain a controller which is straightforward to design and implement. The simulation results indicate that the presence of nonlinear terms on the off-diagonal elements of the characteristic matrix do not significantly affect closed-loop performance. 2.4. Nonminimum Phase Synthesis. Nonminimum phase systems constitute a challenging class of systems for control design since the exact system inverse is unstable. For linear systems, the LIMC design procedure employs an allpass factorization to partition the process model into a minimum phase part and a nonminimum phase part. It has been shown that a controller which utilizes the inverse of the minimum phase part is ISE optimal for setpoint changes. Doyle et al. (1995) have shown that a controller which employs the allpass factorization for calculation of the linear inverse within the partitioned nonlinear inverse yields superior closed-loop performance over the corresponding linear controller for nonminimum phase systems. Similar efforts to obtain a pseudoinverse with input-output linearization (Doyle, Allgo¨wer, and Morari, 1996; Kravaris and Daoutidis, 1990) have yielded results for specific classes (maximally nonminimum phase and second-order systems, respectively); however, a general result has not yet been obtained. In this section, we present results which are applicable to a large class of nonlinear systems, including some systems for which input-output linearization is not applicable. A given nonminimum phase system can be represented as a partitioned nonlinear plant using eq 15. Furthermore, PL(s) can be factored into an allpass and a minimum phase contribution as outlined in Morari and Zafiriou (1989):

where d(s) is a measured disturbance. First, convert to the notation from section 2, as follows:

PL ) PAL PM L

Q* ) P ˜ NL ) 0

(22)

where PAL denotes the allpass and PM L denotes the minimum phase portion. For stable linear systems, an H2 optimal pseudoinverse can be obtained by inverting the minimum phase portion of the plant. Application of the allpass factorization within the nonlinear antiwindup design equations from the previous section yields the following set of subcontrollers:

Q h 1 ) FAF

(23)

Q h 2 ) FA‚PM L - I

(24)

Q h *2 ) FA‚PD

(25)

Q*2 ) FA‚PNL

(26)

for application within the architecture illustrated in Figure 5. Remark 9. In extending NLIMC-based antiwindup to nonminimum phase systems, an approximate plant is found which is locally minimum phase and upholds the conditions of Lemma 2. For this approximate process description, the instantaneous optimization is exactly solved to guarantee nominal performance. To be precise, the factorization for nonminimum phase systems shown in eqs 23-26 implicitly solves a related instantaneous minimization exactly. This philosophy is inspired by the all-pass factorization for IMC controller synthesis for nonminimum phase systems from

Morari and Zafiriou (1989) and extended to unconstrained nonlinear systems in Doyle, Ogunnaike, and Pearson (1995). 3. Case Studies 3.1. Numerical Example. In this section, we illustrate an application of the LIMC results derived in section 2.1. Consider the following linear process description (a modified example from Zheng et al. (1994)):

y(s) )

2 5 u(s) + d(s) 100s + 1 10s + 1 |u| e 1

P ˜ L(s) )

(27) (28)

2 100s + 1

P ˜ D(s) )

5 10s + 1

In the unconstrained case, an optimal linear feedforward/feedback controller can be obtained and implemented in the architecture shown in Figure 4. Recall that for unconstrained systems,

Q h )P ˜ -1 L F )

0.5(100s + 1) λs + 1

˜D ) Q h * ) F-1P

5(λs + 1) 10s + 1

For the subsequent simulations, the LIMC tuning constant is λ ) 20. For the constrained case, an optimal linear antiwindup controller for systems with measured disturbances is synthesized and implemented as illustrated in Figure 5. Recall that,

˜ LQ h ) Q h 1 ) FAP

50(s + 1) 20s + 1

˜L - I ) Q h 2 ) FA‚P ˜D ) Q h *2 ) FA‚P

99 100s + 1

250(s + 1) 10s + 1

The antiwindup filter is designed such that the instantaneous optimization is well-posed for setpoint changes ˜ L|s)∞ is finite and FAP ˜ L - 1 is relative degree 1). (FAP The results of several closed-loop simulations for a step disturbance of magnitude 0.1 are shown in Figure 7. The following three controllers were applied to the system: (i) LIMC-based antiwindup, (ii) LIMC-based antiwindup with feedforward compensation, and (iii) standard LIMC with feedforward. The addition of feedforward compensation to a LIMC-based antiwindup controller results in a significant improvement in closedloop performance. The LIMC-based antiwindup controller and the LIMC controller with feedforward compensation have unacceptably large settling times. When

498 Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 Table 1. Parameters for the Isothermal CSTR k1 k2 k3 CAf V

0.05 min-1 0.10 min-1 0.01 L mol-1 min-1 10 mol L-1 1000 L

mass balances on components A and B as follows:

F C˙ A ) -k1CA - k3CA2 + (CAf - CA) V F C˙ B ) k1CA - k2CB - CB V y ) CB The control objective of this problem is to regulate the concentration of component B by manipulating the inlet flow rate. This example exhibits the well-known phenomena that the gain of the system changes sign at the point of maximum conversion (this can be observed from a plot of the gain locus). In addition, the dynamics of the system also change from nonminimum phase to minimum phase at the point of maximum conversion. The specific values of the parameters for the CSTR corresponding to an operating point in the nonminimum phase regime are shown in Table 1. The values of the states and input at the operating point are

CA0 ) 3.0 CB0 ) 1.117 F0 ) 0.034 286 V Figure 7. Closed-loop comparison of linear antiwindup with feedforward compensation to other linear controllers for a step disturbance of magnitude 0.1. Solid line, unconstrained reference; dotted line, LIMC with feedforward (clipped); dash-dot line, LIMC-based antiwindup; dashed line, LIMC-based antiwindup with feedforward.

the closed-loop response of the controllers with feedforward compensation are compared, it is clear that the LIMC controller duplicates the large initial response of the LIMC-based antiwindup controller to the disturbance; however, the LIMC-based antiwindup controller produces an input which stays on the lower constraint for nearly 3 times longer than the LIMC controller. The improved closed-loop response of the LIMC-based antiwindup controller with feedforward compensation can be attributed to the combination of the feedforward compensation (which is responsible for the aggressive initial response) and the antiwindup compensation (which determines the optimal time for the controller to saturate). 3.2. CSTR with Van de Vusse Kinetics. The second example is an isothermal CSTR with the following well-known sequence of reactions (where component B is the desired product):

AfBfC 2A f D A model for the process can be obtained by performing

Substituting the values for the constants and converting the variables to deviation form results in the following state-space process description:

x˘ 1 ) -0.05x1 - 0.1x12 + u(0.01 - 0.001x1) x˘ 2 ) 0.05x1 - 0.1x2 + u(-0.001x2) y ) x2 -0.03 e u e 0.03

(29)

where x1 is the deviation variable for the concentration of component A, x2 is the deviation variable for the concentration of component B, and u is the deviation variable for inlet flow. Unconstrained Synthesis. In the initial stage of the nonlinear control design, the effects of the input constraints are not considered. A linear approximation of the nonlinear model is obtained from a first-order Taylor series expansion:

x˘ 1 ) -0.05x1 + 0.01u x˘ 2 ) 0.05x1 - 0.1x2 yL ) x2

(30)

The linear model, shown above, is a state-space version of the P ˜ L operator with an equivalent Laplace domain


representation shown below:

P ˜ L(s) )

-s + 0.17 0.9s2 + 0.25s + 0.017

(31)

A state-space representation of the nonlinear partition, PNL, is as follows:

x˘ 1 ) -0.05x1 - 0.1x12 + u(0.01 - 0.001x1) x˘ 2 ) 0.05x1 - 0.1x2 + u(-0.001x2)

x˘ 3 ) -0.05x3 - 0.1x32 + 0.01u x˘ 4 ) 0.05x3 - 0.1x4 yNL ) x2 - x4

(32)

Observe that the first two state equations are the fundamental modeling equations and the second two state equations are the linearized fundamental modeling equations. The nonlinear contribution is isolated by the difference operation which is performed on the output map, effectively P ˜ NL ) P ˜ -P ˜ L. To obtain the unconstrained nonlinear controller, one applies the allpass factorization to P ˜ L:

P ˜ L(s) ) PAL PM L PAL )

PM L )

s - 0.17 -s - 0.17

-s - 0.17 -(0.9s2 + 0.25s + 0.017)

Figure 8. Closed-loop simulation of IMC controllers for a step setpoint change of -0.50 mol L-1. Solid line, unconstrained reference; dotted line, LIMC; dashed line, NLIMC; dash-dot line, LIMC (clipped); solid-circle line, NLIMC (clipped). M such that FAPM L |s)∞ is finite and FAPL - 1 is relative degree 1.

Q h 1 ) FAF ) The unconstrained linear LIMC controller, Q h , is obtained by inverting the minimum phase portion of the plant and adding a first-order filter the make to controller realizable.

Q h )

0.9s2 + 0.25s + 0.017 (-s - 0.17)(λs + 1)

For the subsequent closed-loop simulations, the LIMC filter constant is λ) 12 min. The nonlinear control design is completed by adding the nonlinear feedback loop, where Q* ) F-1PNL, around the optimal linear controller as depicted in Figure 4. Antiwindup Synthesis. Following the procedure outlined in section 2.3, the nonlinear controller can be factored for application to constrained nonlinear systems. The following antiwindup filter is selected:

FA ) 0.9 (s + 0.1)

Q h 2 ) FAPM L - I )

0.9(s + 0.1) 12s + 1 -0.001(7s + 1.7)

0.9s2 + 0.25s + 0.017

Q h 2* ) 0 Q* ˜ NL 2 ) FAP Simulation Results. In Figure 8, the closed-loop response of the LIMC and NLIMC controllers are shown for a -0.50 mol/L step setpoint change. An important observation is that, in the unconstrained case, the NLIMC controller does not perfectly track the reference. This can be attributed to the use of a pseudoinverse (as opposed to an exact inverse) in the controller synthesis due to the nonminimum phase behavior of the process. However, in the unconstrained case, the NLIMC controller shows improved closed-loop performance over the corresponding linear LIMC controller (as evidenced by the reduction in overshoot and oscillation). Both of the


discussion), it is not surprising that the incorporation of antiwindup compensation, as shown in Figure 9, did not significantly affect the closed-loop performance. The antiwindup filter was parameterized to recover the more aggressive response of the unconstrained controller; however, a modified parameterization could be employed to achieve the closed-loop response of the clipped system. The closed-loop performance of the nonlinear controller showed similar performance improvement over that of the linear controller observed previously (closer correspondence to the reference signal and less overshoot/oscillation). The second example illustrates several important points: 1. The nonlinear model-based control scheme can handle nonminimum phase systems. In each of the three control schemes (unconstrained, clipped, and antiwindup) applied to the CSTR, the incorporation of nonlinear compensation resulted in a closed-loop performance improvement. 2. The nonminimum phase behavior may impose a fundamental limitation on an achievable closed-loop performance which cannot be overcome with antiwindup compensation. 3.3. Free-Radical Polymerization Reactor. The final example is a jacketed CSTR for the free-radical solution polymerization of styrene. The process model, from Maner et al. (1996), is shown below:

Figure 9. Closed-loop simulation of antiwindup controllers for a step setpoint change of -0.50 mol L-1. Solid line, unconstrained reference; dotted line, LIMC-based antiwindup; dashed line, NLIMC-based antiwindup.

unconstrained LIMC controllers violate the proposed lower limit on the inlet flow rate. One method for incorporating input constraints into LIMC is to “clip” the output of the controller. If the constraint is applied to the input of the plant and the plant model, the closed-loop stability is guaranteed for stable plants; however, in many instances the closedloop performance is suboptimal. Figure 8 illustrates the result of applying clipping to LIMC and NLIMC controllers. It is clear that the nonlinear model-based controller shows a closer correspondence to the reference system, while the linear LIMC controller exhibits a closed-loop response with increased oscillation and overshoot. Neither of the model-based controllers exhibit any of the adverse characteristics associated with enforcing input constraints (increased oscillation, sluggishness). In addition, the closed-loop response of both the LIMC controller and the NLIMC controller with constraints tracks the reference signal more closely than those of the corresponding unconstrained controller. This somewhat unexpected result is a confirmation that the fundamental performance limitation for this reactor system is the nonminimum phase behavior rather than the input constraint. On the basis of the closed-loop response of the constrained LIMC/NLIMC controllers (and the ensuing

d[I] (Qi[If] - Qt[I]) ) - kd[I] dt V

(33)

d[M] (Qm[Mf] - Qt[M]) ) - kp[M][P] dt V

(34)

dT Qt(Tf - T) (-∆Hr) hA ) + (T - Tc) k [M][P] dt V FCp p FCpV (35) dTc Qc(Tcf - Tc) hA + (T - Tc) ) dt Vc FcCpcVc

(36)

dD0 QtD0 ) 0.5kt[P]2 dt V

(37)

QtD1 dD1 ) Mmkp[M][P] dt V

(38)

D1 D0

(39)

y2 ) T

(40)

y1 )

where

[P] )

[ ] 2fkd[I] kt

ki ) Aiexp(-Ei/T),

0.5

i ) d, p, t

Qt ) Qi + Qs + Qm Qs ) 1.5Qm - Qi

Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 501 Table 2. Parameters for the Polymerization Reactor Model f Ad Ed At Et Ap Ep -∆Hr hA FCp FCpc Qs Qm V Vc [If] [Mf] Tf Tcf Mm

0.6 5.95 × 1013 1/s 14 897 K 1.25 × 109 L/(mol s) 843 K 1.06 × 107 L/(mol s) 3557 K 16 700 cal/mol 70 cal/(K s) 360 cal/(K L) 966.3 cal/(K L) 0.1275 L/s 0.105 L/s 3000 L 3312.4 L 0.5888 mol/L 8.6981 mol/L 330 K 295 K 104.14 g/mol

Table 3. Steady-State Operating Conditions for the Polymerization Reactor 6.6832 × 10-2 mol/L 3.3245 mol/L 323.56 K 305.17 K 2.7547 × 10-4 mol/L 16.110 g/L 0.03 L/s 0.131 L/s 58 481 g/mol 323.56 K

x1 ) [I] x2 ) [M] x3 ) T x4 ) Tc x5 ) D0 x6 ) D1 u1 ) Qi u2 ) Qc y1 y2

Figure 10. Schematic of control scheme for the styrene polymerization reactor.

y1 is the number average molecular weight (NAMW), y2 is the reactor temperature, u1 is the initiator flow rate Qi, and u2 is the coolant flow rate Qc. The feed temperature, Tf, acts as a measured disturbance. The parameters and initial conditions for the polymerization reactor model are found in Tables 2 and 3, respectively. For a complete list of modeling assumptions, the interested reader is referred to the original reference. The control objectives for the polymerization reactor are as follows: (i) to manufacture a uniform polymer of a target number average molecular weight (NAMW) and (ii) to regulate the temperature of the reactor for both safety and economic considerations. The control policy is carried out by manipulating the flow of coolant and the flow of initiator into the reactor. A schematic of the control strategy for reactor is shown in Figure 10. Unconstrained Synthesis. Recall that for systems with measured disturbances, the nonlinear operator P[u,d] is partitioned as follows:

y ) P[u,d]

A state-space realization of PL[u] + PD[d] is obtained by performing a Jacobian linearization on the fundamental model of the polymerization reactor (after introducing the scaled variables). The model for nonlinear partition PNL(d)[u] is obtained using the state-space method demonstrated for the previous example (the isothermal CSTR with Van de Vusse kinetics). The Laplace transforms of PL[u] and PD[d] are as follows:

yˆ (s) ) P ˜ Lu ˆ (s) + P ˜ Dd(s) 1 n11(s) n12(s) P ˜L ) B(s) n21(s) n22(s)

[

P ˜D )

]

[

1 w11(s) B(s) w21(s)

]

B(s) ) 100000s6 + 182000s5 + 136000s4 + 53200s3 + 11400s2 + 1280s + 56.9 n11(s) ) -5000s4 - 6460s3 - 3010s2 - 594s - 40.9 n12(s) ) 438s3 + 436s2 + 144s + 15.8 n21(s) ) 111s4 + 129s3 + 56.1s2 + 10.8s + 0.76

) PL[u] + PD[d] + PNL(d)[u]

(41)

To minimize the likelihood of encountering singularities in the computation of the control action, the inputs and outputs were scaled by the nominal values to yield the following scaled variables:

y1 yˆ 1 ) y10

y2 yˆ 2 ) y20

u1 uˆ 1 ) u10

u2 uˆ 2 ) u20

n22(s) ) -104s4 - 133s3 - 63.8s2 - 13.6s - 1.08 w11(s) ) -135000s4 - 164000s3 - 7400s2 14700s - 1080 w21(s) ) 32100s5 + 48100s4 + 28700s3 + 8520s2 + 1260s + 74 The relative degree information for the polymerization reactor is summarized in Table 4. Observe that the disturbance relative degree for the second output is less than the input-output relative degree. The following

502 Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 Table 4. Summary of Relative Degree (Input-Output and Disturbance) for the Polymerization Reactor output

vector relative degree

disturbance relative degree

2 2

2 1

y1 y2

approximate disturbance model is chosen for the system:

P ˜ †D )

[ ]

w11(s) 1 w (s) B(s) 21 γs+1

The parameter γ can be used as an additional tuning parameter to adjust the aggressiveness of the feedforward controller. For the subsequent calculations, γ is fixed to a value of 0.5 hr. The unconstrained linear LIMC controller, Q h , is obtained by calculating the inverse of the plant transfer function matrix, P ˜ -1 L (s) and adding a filter:

Q h )P ˜ -1 L F(s) where F(s) is a diagonal transfer function matrix of the following form:

F(s) )

[

1 0 (λ1s + 1)2 1 (λ2s + 1)2

0

]

Since both of the outputs are vector relative degree 2, second-order filters are required to make the controller realizable. For this example, λ1 ) λ2 ) 2 h. The linear feedforward controller is obtained by multiplying the disturbance model with the inverse of the LIMC filter:

Q h * ) F-1P ˜ †D

FA )

[

0 -20.02s2 - 500.0s - 500.0 0 957.86s2 - 30000s - 30000

]

upholds the first and third conditions (FA is diagonal and ˜ L - I are relative the diagonal terms in the matrix FAP degree 1) of the filter selection criteria. However, an approximate plant P ˜ †L of the following form

1 ) B(s)

[

n11(s) n12(s) n21(s) n22(s) 0.01s + 1

˜ †L|s)∞ is diagonal. The must be chosen such that FAP characteristic matrix has the proper structure for the first output; however, the second output has a relative degree of 2 for both of the input channels and must be modified to obtain an approximate plant model. The nonlinear antiwindup scheme is constructed using the architecture shown in Figure 5, where

Q h 1 ) FAP ˜ †LQ h (s) ˜ †L - I Q h 2 ) FAP ˜ †LP ˜ -1 ˜ †D Q h *2 ) FAP L P Q* ˜ †LP ˜ -1 2 ) FAP L PNL[d]

Observe that when γ is equal to zero, the feedforward controller Q h * is not proper. Thus, γ * 0 is required to make the feedforward controller realizable. The linear feedforward/feedback controller Q h * is equal to the sum of the linear feedforward controller and the linear feedback controller Q h )Q h e(s) - Q h *d(s). With the addition of an additional feedback loop, the unconstrained nonlinear feedforward/feedback controller is obtained and implemented as shown in Figure 4 where ˜ NL. Q* ) F-1P Antiwindup Synthesis. Following the procedure outlined in section 2, the nonlinear controller can be factored for application to constrained nonlinear systems. The following antiwindup filter

P ˜ †L

Figure 11. Closed-loop response of constrained/unconstrained NLIMC controllers for a step setpoint change of +25000 g/mol in NAMW and +5 °C in temperature setpoints. Solid line, unconstrained reference; dotted line, LIMC; dashed line, NLIMC; dashdot line, LIMC (clipped); solid-circle line, NLIMC (clipped).

]

Simulation Results. For a simultaneous change in the NAMW setpoint and the temperature setpoint, shown in Figure 11, the unconstrained NLIMC controller perfectly tracks the reference for both outputs. The setpoint response of the unconstrained LIMC controller exhibits significant overshoot in the NAMW channel and a minor overshoot in the temperature channel. The lower constraint for the initiator flow and the coolant flow is violated by both the NLIMC controller and the LIMC controller. From a practical perspective, it is desired to make the NAMW setpoint change as quickly as possible while simultaneously minimizing the overshoot. In the unconstrained case, the NLIMC controller better meets the practical grade change criteria. If the overshoot of the LIMC controller is deemed unacceptable, the controller would need to be detuned which is also undesirable. The performance of the unconstrained NLIMC controller and LIMC controller was compared for a -5 °C step change in the feed temperature. The closed-loop simulation results are shown in Figure 12. The unconstrained NLIMC controller perfectly cancels the effects of the disturbance in the NAMW channel and the temperature channel, resulting in negligible overshoot in both channels. The unconstrained LIMC controller rejects the disturbance with an overshoot of approximately 4% of the desired value in the NAMW channel


Figure 12. Closed-loop response of constrained/unconstrained NLIMC controllers for a step of -5 °C in feed temperature. Solid line, unconstrained reference; dotted line, LIMC; dashed line, NLIMC; dash-dot line, LIMC (clipped); solid-circle line, NLIMC (clipped).

and a small overshoot in the temperature channel. The lower constraint for the coolant flow is violated by both the NLIMC controller and the LIMC controller. The initial response of the coolant flow to feed temperatures can be modified on the basis of upon the choice of the tuning parameter γ. In the unconstrained case, the NLIMC controller perfectly tracks the reference signal and cancels the effects of measured disturbances. This result is not unexpected since it is assumed that the plant and model are identical. However, when the output of the NLIMC controller is clipped such that the constraints on the coolant flow and initiator flow are upheld, the closedloop response of the NLIMC controller exhibits overshoots in the NAMW on the same order as the LIMC controller, as indicated in Figure 11. In addition, the constrained NLIMC controller reaches the setpoint in nearly twice the time as the constrained LIMC controller. The constrained NLIMC controller is noticeably more sluggish than the unconstrained NLIMC controller and takes nearly twice the time to reach the setpoint as the constrained LIMC controller. Both controllers, but the NLIMC controller to a greater degree, exhibit the effects of windup as a result of the suboptimal implementation of the controller with constraints. As shown in Figure 12, enforcing the input constraints has a similar effect on the performance of the NLIMC and LIMC controllers for measured disturbance rejection. The observation that both the LIMC controller and the NLIMC controller exhibit the effects of windup for setpoint changes applies for measured disturbances also. As was also the case for setpoint changes, the constrained NLIMC controller exhibits a more noticeable performance loss than the LIMC controller for disturbances. Figure 13 shows the closed-loop response for the linear and nonlinear antiwindup controllers for a simultaneous change in the NAMW setpoint and the temperature setpoint. The closed-loop response of the system with nonlinear antiwindup shows a noticeable improvement over the response of constrained NLIMC (Figure 11) for setpoint changes of identical magnitudes. In addition, the closed-loop response of the linear antiwindup controller, while improved over the response

Figure 13. Closed-loop response of constrained/unconstrained NLIMC-based antiwindup controllers for a step setpoint change of +25 000 g/mol in NAMW and +5 °C in temperature setpoints. Solid line, unconstrained reference; dotted line, LIMC-based antiwindup; dashed line, NLIMC-based antiwindup.

Figure 14. Closed-loop response of constrained/unconstrained NLIMC-based antiwindup controllers for a step of -5 °C in feed temperature. Solid line, unconstrained reference; dotted line, LIMC-based antiwindup; dashed line, NLIMC-based antiwindup.

of the constrained LIMC controller (Figure 11), exhibits a noticeable overshoot and some oscillations in the NAMW channel and the temperature channel. Observe that the NAMW and the temperature responses for the nonlinear antiwindup controller rise at the maximum rate allowed by the constraints until intersecting the unconstrained reference. For a -5 °C step change in the feed temperature, the nonlinear antiwindup controller perfectly cancels the effect of the disturbance in the NAMW channel, as illustrated in Figure 14. The response of the constrained NLIMC controller (Figure 12) exhibited a significant overshoot and a long settling time for an identical measured disturbance. The linear antiwindup controller is unable to perfectly cancel the effect of the disturbance in the NAMW channel and exhibits an overshoot of approximately 4% in magnitude. The closed-loop response of both the linear antiwindup controller and the nonlinear antiwindup controller has an initial deviation from the desired value of the

504 Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 Table 5. ISE Error Analysis for the Polymerization Reactor setpoint tracking

disturbance rejection

controller type

ISE NAMW

ISE temperature

ISE NAMW

ISE temperature

unconstrained linear unconstrained nonlinear linear feedforward nonlinear feedforward linear antiwindup nonlinear antiwindup

1.01 × 10-1 4.68 × 10-7 2.15 × 10-1 3.50 × 10-1 6.44 × 10-2 8.00 × 10-3

1.75 × 10-4 4.68 × 10-7 9.09 × 10-4 1.50 × 10-3 7.60 × 10-4 5.83 × 10-4

1.02 × 10-1 5.11 × 10-7 6.50 × 10-1 6.37 × 10-1 8.12 × 10-2 3.05 × 10-2

7.62 × 10-4 4.77 × 10-9 3.40 × 10-3 3.40 × 10-3 2.54 × 10-3 8.77 × 10-5

temperature; however, the closed-loop response of the linear antiwindup controller exhibits an additional overshoot in the positive direction. The proposed nonlinear model-based control scheme was tested on a realistic process example with input constraints and measured disturbances. On the basis of the desired operating criteria for the polymerization reactor during grade changes (quick transitions and minimized overshoots), the nonlinear antiwindup scheme outperforms the linear antiwindup controller as well as the constrained NLIMC controllers. In addition, the utility of performing compensation for the nonlinear dynamics in addition to antiwindup compensation was demonstrated. Although the primary objective of the derived controllers is instantaneous minimization, this is very difficult to evaluate for a particular example. Instead, the integral squared error (ISE), tabulated in Table 5, is used to confirm that the nonlinear antiwindup controller exhibits the best closed-loop response for measured disturbance rejection and setpoint tracking. 4. Conclusions In this paper we have presented a novel approach for direct synthesis nonlinear model-based control. The proposed scheme, which incorporates features from nonlinear internal model control and LIMC-based (linear) antiwindup, is applicable to a broad class of nonlinear systems, including multivariable and nonminimum phase systems. The foundation of the proposed scheme is an optimal linear antiwindup controller with nonlinear compensation provided through an additional feedback loop. An additional capability of the proposed nonlinear synthesis scheme is optimal nonlinear compensation for measured disturbances. Compensation for measured disturbances entailed extending the notion of a partitioned nonlinear inverse to systems with measured disturbances and incorporating feedforward compensation into the LIMC-based (linear) antiwindup scheme. The performance of the nonlinear model-based controller was evaluated using three examples. 1. A simple linear example demonstrated the effectiveness of a new method for performing constrained linear feedforward/feedback control. Since the foundation of the proposed nonlinear synthesis scheme is an optimal linear controller, it was essential to demonstrate the effectiveness of the constrained linear feedforward/ feedback controller. 2. For a well-known reactor problem, the isothermal CSTR with Van de Vusse kinetics, it was shown that nonlinear control yielded significantly better closed-loop performance over linear control. In addition, it was demonstrated that the proposed approach can be extended to nonminimum phase systems and new insight was gained into the fundamental performance limitations of this class of nonlinear systems.

3. For an industrial process example, a polymerization reactor, it was shown that the addition of the nonlinear feedback loop provided a significant performance improvement for measured disturbance rejection and setpoint changes. Also, it was shown that incorporation of antiwindup compensation into the nonlinear control design was critical for input-constrained systems. Since the nonlinear corrective action is implemented by feedback, the additional difficulty in implementing a NLIMC-based antiwindup controller lies entirely in obtaining an accurate nonlinear model, whereas other popular approaches to nonlinear model-based control require the development of additional compensators, such as coordinate transformations for differential geometric methods or efficient algorithms for solving nonlinear optimization problems online. Although obtaining fundamental nonlinear models for complex industrial processes may prove tedious, it would be straightforward to develop NLIMC-based antiwindup controllers with empirical model structures, such as Volterra models or neural networks, as successful applications of unconstrained NLIMC have been reported in the literature for empirical models. Two issues which merit additional consideration are the application of NLIMC-based antiwindup to unstable systems (and the underlying stability analysis problem) and the robustness of the proposed approach (which to date has been demonstrated by simulation). Acknowledgment Financial support for this work has been provided from an NSF NYI award (CTS-9257059) and from the E. I. DuPont deNemours & Co., Inc. Notation CSTR ) continuous stirred tank reactor IMC ) internal model control ISE ) integral squared error LFT ) linear fractional transformation LIMC ) linear internal model control MPC ) model predictive control NAMW ) number average molecular weight NLIMC ) nonlinear internal model control SISO ) single input-single output d ) measured disturbance vector (p × 1) e ) error (p × 1) FA ) antiwindup filter PD ) linear disturbance partition (plant) PL ) linear partition (plant) PNL ) nonlinear partition (plant) P ˜ D,L,NL ) partitions of model P ˜ †D,L,NL ) approximate partition of model Q h ) unconstrained LIMC (feedback) controller Q h * ) unconstrained feedforward correction Q* ) unconstrained nonlinear correction Q h 1, Q h 2 ) LIMC-based antiwindup subcontrollers

Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 505 Q h 2* ) LIMC-based antiwindup subcontroller (feedforward term) Q h 2* ) NLIMC-based antiwindup subcontroller (nonlinear term) rm ) relative degree of mth output (ym) Rm ) m-dimensional Euclidean space u ) process input vector (m × 1) u ˜ ) constrained process input vector (n × 1) x ) state vector (n × 1) y ) output vector (m × 1) y′ ) constrained output (m × 1) yf ) filtered output (m × 1) Greek Letters γ ) disturbance filter constant λ ) IMC filter constant σ j ) maximum singular value ω ) frequency Superscripts max ) upper limit min ) lower limit

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Received for review May 19, 1997 Revised manuscript received October 23, 1997 Accepted October 27, 1997 IE970357K

Nonlinear Internal Model Control for Systems with ... - ACS Publications

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