linear control of nonlinear processes

8/9/2019 Linear Control of Nonlinear Processes

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Linear Control of Nonlinear Processes:

Recent Developments and Future Directions

Michael Nikolaou and Pratik MisraChemical Engineering Dept.

University of HoustonHouston, TX 77204-4004

[email protected]://athens.chee.uh.edu

CEPAC 2001, Guaruj, Sao Paulo, Brazil, http://pqi.ep.usp.br/eventos/cepac2001.htm

Abstract

Virtually all chemical processes are nonlinear, but for several of them linear feedbackcontrol is adequate. Therefore, before nonlinear controller design is attempted, it is

natural to ask When is linear control adequate for a nonlinear process? to ensure

favorable cost/benefit ratio. Consequently, methods are needed that quantify thenonlinearity of a process within the context of assessing whether linear control is

adequate or nonlinear control is warranted. In this work we summarize efforts towards

this end and elaborate on our latest research in this area. Specifically, we present arigorous and general theoretical framework as well as an associated, heuristically

refined computational methodology that allow not only analysis but also synthesis of a

linear control system for a nonlinear process. Application to the multivariable case ispresented. Potential future developments within this framework are discussed.

1 Introduction

1.1 A basic question: Linear or nonlinear control?

Interest in nonlinear feedback control of chemical processes has been steadily increasing over the

last several years. This is due both to the pronounced nonlinear nature of several chemicalprocesses (whether in mature or emerging fields) and to the increased sensing and computationalcapabilities afforded by modern sensors, computers, algorithms, and software. Such capabilities

have been claimed and at times proven to offer benefits in better operation and control ofchemical processes. However, nonlinear control systems usually pose substantially higher data,

design, implementation, and maintenance demands than linear control systems. Therefore,before one develops and implements a nonlinear control system, one must carefully examine thepotential advantages of such a system in comparison to a linear one, by resolving the following

basic question, which will be the central theme of this proposal:Q1: Is linear control adequate or is nonlinear control necessary for a given process?

Rather than a mere Yes/No answer, insight is sought into how the answer to Q1 depends on


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various problem parameters, such as process model and control structure, experimental or routine

modeling information, process operating range, constraints, disturbances, etc., as discussed in thesequel.

1.2 All chemical processes are nonlinear, but not all of them require nonlinear control

Linear feedback control of chemical processes has a long history of research and diverseindustrial applications (Qin and Badgwell, 2000; Nikolaou, 1997; Kayihan, 1997; Edgar et al.,

1999). From single-input-single-output proportional-integral-derivative (SISO PID) controllersto plantwide model-predictive control (MPC) systems (Qin and Badgwell, 1997), there is an

abundance of feedback control systems which implicitly or explicitly assume that processdynamics are either inherently linear or almost linear owing to process operation close to asteady state. However, there are important instances for which the linearity assumption may be

violated. Such instances are not uncommon in chemicals, polymers, natural gas processing,pharmaceuticals, microelectronics, and pulp and paper plants (Qin and Badgwell, 2000), thus at

times necessitating nonlinear control algorithms (Sidebar 1).It can be argued that virtually all chemical processes are in principle nonlinear. However,

some are evidently more nonlinear than others. Moreover, feedback may drastically reducethe nonlinearity of a system, a fact recognized as early as during the invention of the feedbackamplifier by Black in the 1920s (Brittain, 1997). Therefore, methods are needed that quantify the

nonlinearity of a process within the context of assessing whether (and what) linear control is

adequate or whether (and what) nonlinear control is warranted(Harris and Seppala, 2001).

1.3 The interplay between nonlinearity and feedback can be quantified

Several and diverse sporadic attempts to quantify the nonlinearity of a dynamic process haveappeared over the last decade, as reviewed in section 3.1.

The PIs group (Eker and Nikolaou, 2000 and 2001; Nikolaou, 2001a; Misra andNikolaou, 2001) recently proposed a rigorous and general theoretical framework as well as an

associated, heuristically refined computational methodology that address the interplay betweennonlinearity and feedback. Within this framework, as explained more concretely in section 3.2,the need for nonlinear control is assessed in terms of a closed-loop nonlinearity measure.

Bounds on the closed-loop nonlinearity measure can be relatively simply evaluated in terms of(a)process nonlinearity and (b) a linear feedback controller. More importantly, this frameworkallows not only analysis but also synthesis of a linear control system for a nonlinear process.

This framework will provide the starting point of the proposed research.

1.4 Objectives of proposed research

The objective of the research work proposed in the sequel is to apply and expand the frameworkreferred to in section 1.3, in order to develop rigorous results and computational methodologies

that answer the following nontrivial questions of both theoretical and practical importance:Q2: Should constrained MPC employ a linear or nonlinear model of a controlled process?Q3: Can linear control stabilize an unstable nonlinear process? If so, over what operating

range?

The above questions Q2 and Q3 entail a number of specific research directions that will beelaborated on in section 5

Sidebar 1 The need for control of non linear processes

Over the latest few decades, studies on nonlinear control have generated a voluminous


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body of work, within both chemical engineering and other disciplines (Bequette, 1991;

Rawlings et al., 1994; Kumar and Daoutidis, 1999 and Christofides, 2001b emphasis on

geometric control of distributed parameter systems; Henson and Seborg, 1997 collection

of various contributions on nonlinear systems; Krstic et al., 1995 emphasis on

backstepping; Khalil, 1992; Slotine and Li, 1991; Nijmeijer and van der Schaft, 1990 and

Isidori, 1999 emphasis on geometric control; Vidyasagar, 1993; de Figueiredo and Chen,

1993; Brockett, 1996).It is interesting to note that an early academic publication that introduced what

today would be called nonlinear MPC, explicitly recognized and dealt with the issue of

nonlinearity for model-based control of a distillation column through on-line optimization

(Rafal and Stevens, 1968).

Nonlinearity, by definition referring to absence of linearity, can manifest itself in

various ways, such as nonlinear dynamics, constraints (saturation nonlinearity) and

changing modes of operation (see also Pearson (1999) for an interesting classification of

nonlinear behavior). While such nonlinearities are widespread, experience indicates that

they do not always require nonlinear control. On the other hand, important cases are also

known (e.g., polymer production changeover policies (Pavilion, 2001; Young, 2001)) for

which nonlinear control provides tangible advantages.

Discussed below are industrially important cases for which nonlinearity is usually

present and, consequently, knowing whether linear or nonlinear control is adequate wouldprovide obvious benefits.

- Biochemical production of chemicals. Enzyme kinetics (e.g., Michaelis-Menten) depend

nonlinearly on susbstrate concentration. With the rapidly increasing importance of

biotechnology, control of processes that produce and separate chemicals with the aid of

reactions in living cells will have obvious implications.

- Systems operating near constraints (e.g., petroleum refining, natural gas processing).

The saturation nonlinearity renders constrained MPC nonlinear, even if the model

employed is linear. While constrained MPC with linear model has been amply validated

in industrial practice, transition to constrained MPC with nonlinear model is all but trivial

and requires careful justification and execution.

- Non-routine operation situations (e.g., start-ups, shut-downs, change-overs, flares,

relief valve emissions). These are extremely important as the main contributors tosafety, quality, resource use, and environmental problems (Allen and Shonnard, 2001).

Because a process moves far from a steady state during non-routine operation,

nonlinear behavior is usually pronounced.

- Nonlinear distributed process systems (Kumar and Daoutidis, 1999; Christofides,

2001b) such as control of spatial profiles (CVD, etching, crystal growth, packed-bed

reactors) control of size distributions (aerosol production and particulate processes (Chiu

and Christofides, 1999), crystallization (Braatz and Hasebe, 2001), emulsion

polymerization, cell cultures (Daoutidis and Henson, 2001)), control of fluid flows

(mixing, wave suppression, drag reduction, separation delay, control of material

microstructure (thin-film growth, nano-structured coatings processing).

- Batch processes, particularly important for the production of fine chemicals andpharmaceuticals.

2 Motivating Examples

As already suggested, the main thesis of the proposed research is that lack of linearity in the

behavior of a process to be controlled does not necessarily make nonlinear control

indispensable. On the other hand, the nature of process nonlinearity may make linear control

completely inadequate and may require nonlinear control of a kind that may not even beimmediately obvious. The following two examples make the case (for more details and other


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interesting examples see Eker and Nikolaou, 2001).

2.1 Example 1 (The right) linear control may be adequate for a nonlinear process

Consider the exothermic reaction A B in a system of two jacket-cooled continuous stirred-tank reactors (CSTR) in series (Henson and Seborg, 1990). The concentration of the reactant at

the exit of the second CSTR, CA2, is the controlledvariable, and the coolant flow rate qc (common for bothreactors) is the manipulated variable.

Figure 1 Open-loop responses of CA2 (in

deviation form) to qc step change of +9.3, for

(a) the nonlinear system of two CSTRs (dashedline) and (b) its linearized model around the

nominal steady state -35.3 10 mol/L (solidline).

The step-response simulation in Figure 1 demonstrates

that the open-loop system is fairly nonlinear.

Figure 2 Closed-loop responses of CA2 to

pulse setpoint change 34.2 10+ mol/L for (a)the nonlinear process with linear IMC (filtertime constant 1 = ) (dashed line), and (b) theideal linear IMC loop (solid line).

However, when linear internal model control (IMC,

Morari and Zafiriou, 1989, p. 65) is used to steer CA2 tothe same value as in Figure 1, the resulting closed loop

is much less nonlinear or virtually linear (Figure 3).

Figure 3 Similar to Figure 2 except that the

IMC filter time constant 1 = .

But increasing the pulse setpoint change to 35.2 10+ mol/L generates an unstable closed-loop response farfrom the ideal linear response (Figure 4). Similar

behavior is eventually observed if is further reduced.

Figure 4 Similar to Figure 3, except that

pulse setpoint change is 35.2 10+ mol/L.

Note nonlinear loop oscillations that wouldhave been trivially missed if the pulse had

lasted less than ~70 time units.

Moreover, if measurement delay of 5 time units isadded, the ideal linear closed loop changes little in

comparison to absence of delay, while the nonlinearclosed loop shows drastic deterioration (Figure 5).

0 100 200time

300 400 500-1

0

1

2

3

4

5

CA2

-310

0 100 200time

300 400 500-1

0

1

2

3

5

CA2

-310

0 100 200time

300 400 500

2

4

6

8

CA2

10-3

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7-3

time

CA2


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0 1 2 3 4 5 6 7 8 9 10

3

4

5

7

8

10-3

CA2

This example suggests that

(a) Open- and closed-loop nonlinearities can be very different;(b)Process nonlinearity and controller design interact tightly and not monotonically;

(c) Process operating range profoundly affects nonlinearity and control;(d)Similarly to (if not more importantly than for) linear processes, modeling uncertainty must be

carefully taken into account when controllingnonlinear processes.

Figure 5 Closed-loop responses to setpoint

step change 34.5 10+ for (a) the nonlinearclosed loop with linear IMC, 0.1 = (dashedline), and (b) the ideal linear closed loop with

the same IMC (solid line).

The many different facets of behavior of thissystem can be very well predicted by the theory

mentioned in section 1.3 and briefly discussed in

section 3.2.

2.2 Example 2 There are nonlinear processes that cannot be globally stabilized by any

continuous feedback law, linear or nonlinear

Meadows et al. (1995) showed that it is impossible to find any continuous static state feedback

law ( , )k k ku F x y= that can globally asymptotically stabilize the system3

1 1,k k k k k k x x u y y u+ += + = + (1)around (0,0), while the constrained nonlinear MPC

( )2

2 2 2

0

min i i ii

x y u+ + +=

+ + l l l (2)

subject to( , ) givenx yl l ; 1i i ix x u+ + + += +l l l ,

3

1i i iy y u+ + + += +l l l , 0,1,2i = ; 3 3 0x y+ += =l l (3)can globally asymptotically stabilize the above system around (0,0).

This example suggests that seemingly innocent (e.g., polynomial) nonlinearities maycreate great control challenges.

3 Related Background and Prior Results

Sidebar 2 Nonlinear operator analysis basics and notation

Basics of the input-output system framework can be found in Willems (1971), Desoer and

Vidyasagar (1975), and Zeidler (1990), among others. Within this framework the dynamic

behavior of a nonlinear system is described by an unbiased nonlinear operator (mapping)

: : : 0 (0) 0N U Y u y Nu N = =a a (4)which maps input signals u in the space U to output signals y in the space Y. Note thatthere is no unanimity in literature regarding uniqueness ofy given u. We will not assume

uniqueness in this work. The operator N is commonly realized through a set of ordinary orpartial differential equations and algebraic equations, such as

( ) ( ( ), ( )), ( ) ( ( ), ( ))dx t dt f x t u t y t h x t u t = = (5)Note that partial differential equations may be approximated by ordinary differential

equations such as (5) after spatial discretization using finite differences, finite elements,


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Karhunen-Loeve decomposition, spectral decomposition, etc. The norm (gain) and

incremental norm (incremental gain or local Lipschitz constant (Willems, 1971, p. 93)) of

:N U Y over the set V U are defined (Nikolaou and Manousiouthakis, 1989) as

, 0

supV

u V u

N Nu u

= (6)

and

1 2 1 2

1 2 1 2, ,

supV

u u V u u

N Nu Nu u u

= (7)

respectively, where the norm functions on the right-hand sides of eqns. (6) and (7) are

defined on the spaces Uand Y. The set V identifies these input signals that are physicallyimportant for the operator N, e.g., mole fractions in the interval [0,1] . An operator

:N U Y is bounded (stable) over the set Vwhen

VN

< . (8)

Note that the above definition of stability in eqn. (8) supercedes the standard stability

definitionV

N < , becauseV V

N N

. Note also that even for very simple nonlinear

operators it is possible to have1

VN

= and

2V

N

< (or1

VN = and

2V

N < ) for

two different sets V1 and V2 (Nikolaou and Manousiouthakis, 1989). The linearization of anoperator :N U Y around the input trajectory u0 is defined as the linear operator

0:uL U Y satisfying the equation 00 00lim ( ) 0uu N u u Nu L u u + = . For N defined via

eqn. (5),0u

L is a linear time-varying operator, defined by the equations

( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( ) ( )d x

t A t x t B t u t y t C t x t D t u t dt

= + = + (9)

where ( ) ( )0

0

( )( )

( ), ( ), ( ), ( ) , , ,f f h h

x tx u x uu t

A t B t C t D t = ; 0 ( )x t , 0( )u t satisfy eqn. (5); and (0) 0x = .

3.1 Efforts to quantify nonlinearity have spawned a plethora of creative approachesAs stated in section 1.2, nonlinearity quantification is needed in order to assess whether linear

control is adequate for a nonlinear process. Let us stress that nonlinearity quantification goesbeyond mere detection of lack of linearity, a topic that has been amply studied (e.g., see Haberand Unbehauen (1990), Pearson (1999) and references therein).

Efforts have appeared in literature to quantify the nonlinearity of a nonlinear operator N(eqn. (4)) by computing its distance from a suitably defined linear operator. The basis of this

idea has been that a fairly nonlinear operator would require nonlinear control.Desoer and Wang (1980) defined the nonlinearity measure of an operatorNas

infL

v N L

= (10)

where the above minimization in eqn. (10) must be performed over all linear operators L in theset , and the norm function can be any suitable norm. Computation of v in eqn. (10) can beextremely complicated, e.g., if the norm in eqn. (10) is an induced norm (eqn. (6)).

To address the computational issue, Nikolaou (1993) constructed an inner-product andcorresponding 2-norm theory for a class of nonlinear operators. Based on that theory, the

quantity v in eqn. (10), corresponding to the average discrepancy between outputs ofNandL forinputs within an explicitly specified set, can be trivially computed via Monte Carlo simulations.

In addition, explicit formulas for the optimal L in eqn. (10) can be derived. Using this theory,


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Nikolaou and Hanagandi (1998) quantified the nonlinearity of several open- and closed-loop

chemical process systems, and showed how different tunings of a linear IMC controller used tocontrol a nonlinear process may result in closed loops of significantly different nonlinearity

magnitudes. While making computations easy, the 2-norm is not an induced norm, therefore it

does not satisfy the submultiplicativity property ( 1 2 1 2N N N N ), thus limiting its use in

direct feedback controller synthesis.Allgwer (1996) tackled the computational problems of using an induced norm in eqn.

(10), by parametrizing the input signal u in eqn. (6) and the linear operator L in eqn. (10) though

finite-dimensional approximations, and by directly performing the optimization in eqn. (10), i.e.

, 0

inf supL u V u

v Nu Lu u

= (11)

corresponding to the worst possible discrepancy between outputs ofNandL. For a number of

examples, he found that the value ofv is insensitive to the particular parametrization ofu.Helbig et al. (2000) defined a nonlinearity measure as

,0 ,0,0 ,0

,0 ,0 ,0,

inf sup inf [ , ] [ , ] [ , ] [0,1]L L

N N

N L N L x X u U x X

N u x L u x N u x

= (12)

corresponding to the worst discrepancy between outputs ofNandL as a function of both initialconditions and inputs. Exact computation of is practically infeasible. However, Helbig et al.(2000) have shown how to efficiently calculate good approximations or bounds of by finite-dimensional parametrization ofu and convex optimization.

To avoid having to directly optimize with respect toL in nonlinearity measures such as ineqns. (10) and (12), Sun and Kosanovich (1998) proposed to quantify nonlinearity as

max sup ,supupper lower u U u U

Nu L u Nu L u

(13)

whereLupperandLlower are linear operators such that they provide the smallest bounding envelop

on the output ofNas ( ) ( ) ( )( ) ( ) ( )lower upper L u t Nu t L u t for u U . This approach has many

similarities to identification for robust control (Helmicki et al., 1991, 1992).To better assess the need for nonlinear control, as opposed to just assessing the distance

of a nonlinear plant from a linear one, Stack and Doyle (1997) proposed to focus on thenonlinearity magnitude of an optimal nonlinear controller designed for a nonlinear process.

Quantification of the nonlinearity of that controller, using any method, was proposed as ameasure of the need for nonlinear control, the assumption being that a highly nonlinear controllerwould result in a highly nonlinear closed loop, hence rendering linear control inadequate and

necessitating nonlinear control. For static state feedback laws, these authors proposed to usecoherence analysis as the nonlinearity measure. (A related approach for static systems appearedin Yana et al. (1994) and for time series in Nielsen and Madsen (2001)). Stack and Doyle (1999)

also applied coherence analysis to assess the closed-loop nonlinearity of a nonlinear system

controlled by linear IMC. Emphasis was placed on observing the effect of different IMC tuningson closed-loop nonlinearity. An advantage of coherence analysis is that it entails trivialcomputational load, and the analysis may be conducted using experimental data, without detailedknowledge of a process model.

Departing from the notion of nonlinearity measures based on the distance (norm ofdifference) between a nonlinear operator from a suitable linear operator, a number of

investigators took different pathways towards quantifying nonlinearity.Guay et al. (1995) proposed to quantify the static nonlinearity of a system described by


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eqn. (5) in terms of the local geometry of the steady-state locus, i.e., by considering the first and

second derivatives of the steady-state map 0 ( , )s sf x u= with respect to us. Guay (1996)extended these results to quantification of dynamic nonlinearity.

Harris et al. (2000) proposed to quantify the nonlinearity of system as in eqn. (5) first by

approximating nonlinearities in the time domain by polynomials, and then by expressing the

solution as a functional infinite series in the Laplace-Borel domain. Their approach relies onrepeated application of the shuffle product (convolution). The nonlinearity of a system is large if

many and large higher-order terms are needed. Bounds of an accordingly definednonlinearity measure can be easily computed and used in controller design.

Hahn and Edgar (2001) compared the controllability and observability covariancematrices of a system that is linearized at its steady state operating point to covariance matricesthat are computed from data collected within an operating region of the nonlinear process. This

comparison results in two measures for the nonlinearity of the input-output behavior of aprocess: input-to-state (controllability) and state-to-output (observability). In addition to

identifying when a model is severely nonlinear, this approach can identify a model as beingWiener-like, Hammerstein-like, or both.

Choudhury (2001) used higher-order statistics on feedback error to detect closed-loopnonlinearities based on operating data. The approach is based on detecting nonlinearities intime-series, a topic with rich past (Kantz and Schreiber, 1997; Tong, 1990).

Rajput (2001) used nonlinearity measures for process monitoring and fault diagnosis.The premise of the majority of the above approaches is that if a nonlinear process (or the

optimal nonlinear controller) is close to a linear one, then a linear controller will be sufficient.

While that may frequently be true, proximity of a nonlinear process to a linear one is neithernecessary nor sufficient for good closed-loop performance. For example, Nikolaou and

Hanagandi (1998) have shown that a highly nonlinear process controlled by linear IMC mayresult in an almost linear closed loop, if IMC is suitably designed. Conversely, Schrama (1992)has shown that, even for a linear process, a controller design based on a linear model with close

proximity to a process may result even in closed-loop instability. The discussion in section 2corroborates the preceding point. Nevertheless, one would intuitively expect that there must be

process- and controller-dependent connections between open- and closed-loop nonlinearity.That intuition is indeed correct, as shown next.

3.2 Closed-loop nonlinearity depends on both the controlled process and the controller

Figure 6 Block diagram of IMC for a Nonlinear Process N.

Consider the IMC loop structure of Figure 6. The operator Ncorresponding to the stable-over-

set controlled plant is nonlinear, the (linear or nonlinear) operator L corresponds to the plant

Qsetpoint, r input, u

disturbance, doutput,y

noise, w

-

-

Controller

N

L


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model, and the (linear or nonlinear) operator Q is the Youla parameter of the controller. Note

that Q may have a closed form or may be defined implicitly, e.g., via on-line optimization, as isthe case for MPC (see section 5.1). Note also that this structure allows the development of

small-gain type of theorems (e.g., eqns. (17) and (18); Zheng and Zafiriou, 1999; Koung andMacGregor, 1992; Kothare and Morari, 1999) that are much less conservative than similar

theorems for classical feedback structures (Willems, 1971, p. 108; deFigueiredo and Chen, 1993,p. 96; van der Schaft, 2000, p. 11). Indeed, small-gain theorems on classical feedback structuresfail to capture important classes of stabilizing controllers, such as controllers with integral action.

If the model L and Youla parameter Q are linear, i.e. the classical feedback controller1( )C Q I LQ = is linear, then one can compare the resulting closed loop to the ideal linear

closed loop that would result if the controlled plant were linear and equal to the model L. Thedifference between the actual nonlinear and the ideal linear closed loop can be quantified by the

magnitude of the operator

( )1

N NQ I NQ LQ LQ

= + . (14)Eker and Nikolaou (2001) have proposed to use the incremental norm over a set (local Lipschitzconstant) to quantify W N

(where Wis a weighting filter) and have shown that if

( ) 1E

N L Q

= < (15)

then the closed-loop nonlinearity measureZ

W N

is bounded as

( )( ) ( )( )

1 ( ) ( )

( )( ) ( )( )

1 ( ) 1 ( )Closed-loop

nonlinearity

1

E E

E E

E E

Z

E E

W I LQ N L Q W I LQ N L Q

N L Q I N L Q

W I LQ N L Q N L QW N W I LQ

N L Q N L Q

+ +

1442443142431442443

(16)

where the setEcontains the signals (Figure 6) and the set [ ( ) ]( )Z I N L Q E = + .Note that the bounds of the closed-loop nonlinearity depend on both the process

nonlinearity, N L , and the controller elements Q andL. Note also that if Q can be designed ina way that makes 1


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1 1( )1Y

U

I RR

+

(18)

(b) Incremental norms can be computed using the approximation (Eker and Nikolaou, 2000)

( )0 00 0

0

'

constant

sup supu uVu V u V

u

M L L

= (19)

where the operator0

'

uL appearing in the right-hand side of eqn. (19) is the linearization of the

nonlinearoperatorM(eqn. (9)) around steady states (i.e. constant) u0 in the set V.

The importance of element (a) is that the existence and boundedness of the inverse of anonlinear operator are guaranteed over a set without explicit computation of that inverse. NoBanach space setting is needed either (Willems, 1971, p. 29). Note also the importance of the

sets Uand Vthat are associated with ranges of input signals.The importance of element (b) is that the associated computations, albeit approximate, do

not require a nonlinear dynamic process model but rather its linearization around a number ofsteady states, a fact that significantly eases modeling efforts.

The work proposed in the following section 5 will rely heavily on the above two elements.

4 The Multivariable Case

An important feature of the framework captured by eqns. (17), (18), and (19) is that it can be

easily applied to multivariable systems. Shown next are three examples that demonstrate that thepredictions of the theoretical framework for the multivariable case can range from conservativeto tight.

4.1 Example 3 Two nonisothermal CSTRs in series

Consider again the system of two

CSTRs studied in Example 1, with

coolant flow rate and feed flow rateto reactor 1 as process inputs, and

concentration and temperature ofthe second reactor as process

outputs.

Figure 7 Setpoint step-change responses of (a)

nonlinear closed loop with

linear IMC controller (solid

lines) and (b) ideal linearclosed loop (dashed lines).

Solid and dashed lines arevirtually indistinguishable.

Linear IMC, for certain setpoint

changes and for certain tuning, mayproduce a closed loop that is virtually linear, as Figure 7 indicates. However, for differentsetpoint changes, the same linear controller generates instabilities, as Figure 8 shows. Figure 7

0 10 20 30 40 50 60 70 80 90 100-6

-4

-2

0x 1 0

-5

concentration

time

0 10 20 30 40 50 60 70 80 90 100-5

0

5

10

15

20x 1 0

-3

temperature

time


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and Figure 8 correspond to setpoint

changes for which eqn. (15) is or isnot satisfied, respectively.

Figure 8 Similar toFigure 7 for different

setpoint changes. Noticethe discrepancies betweensolid and dashed lines.

Figure 9 shows the values of computed for various setpoint

changes in temperature andconcentration. The stabilitydomain indicated by Figure 9 is

conservative; setpoint changes ofmuch higher magnitude than

indicated by Figure 9 had to beeffected for instabilities to be noticeable. Note thatdeviation variables are used throughout.

Figure 9 Contour plot of the value of ,eqn. (15), as a function of temperature and

concentration setpoint changes.

It is also possible to work in the frequency domain

in order to gain qualitative insight into the behaviorof the closed loop, although the results will bequantitatively conservative for the multivariable

case. Nevertheless, the following analysis may behelpful in understanding the effect of controller design and operating range on closed-loop

stability and performance. Similar analysis may be performed for the subsequent Example 4 andExample 5, but is omitted for brevity.

Eqn. (15) is satisfied over an operating rangeEif

1 1( )

iL L L

F


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similar plots that have been developed in linear control

theory (Morari and Zafiriou, 1989) and provides a linkfor linear controller design for nonlinear systems.

Figure 10 1( )i

L L L (dashed lines for

different operating points) and 1F

(solid lines

for different values of the filter time constant,eqn. (21)) as a function of frequency.

4.2 Example 4 A fictitious problem

Consider the system described by the differential equations2

1 1 1 2 1

2

2 1 2 2 2

2 4

4 2

x x x x u

x x x x u

= +

= +

&

&(22)

where u1, u2 are the inputs andy1,y2 arethe outputs. Figure 11 shows closed-

loop responses for setpoint stepchanges around the equilibrium point

(0,0). Note that the closed loop is onlymildly nonlinear, if at all.

Figure 11 Setpoint step-

change responses of (a)nonlinear closed loop with

linear IMC controller (solid

lines) and (b) ideal linear

closed loop (dashed lines).Solid and dashed lines are fairly

close.

However, for different setpointchanges, the instabilities of Figure 12emerge.

Figure 12 Similar to Figure

11 for different setpointchanges. Notice the

discrepancies between solid anddashed lines.

Such instabilities can be fairly well

predicted by the contour plot of Figure13, which depicts the value of log asa function of setpoint changes on x1 andx2. Note the difference between Figure

0 100 200 300 400 500 600 700 8000

0.1

0.2

0.3

0.4

x1

time

0 100 200 300 400 500 600 700 800

0

0.1

0.2

0.3

0.4

x2

time

0 100 200 300 400 500 600 700 8000

0.5

1

1.5

x1

time

0 100 200 300 400 500 600 700 800

0

0.5

1

1.5

x2

time

10-4

10-2

100

102

104

10-10

10-5

100

105

1010

=10=10

=5

=1


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13

13 and Figure 11 regarding the regions of setpoint

changes for which the closed-loop stabilitycondition, eqn. (15), is satisfied: In Figure 13 that

region appears to be open, while it is closed inFigure 11. Note also that the stability region

predicted by Figure 13 is much less conservativethan that predicted by Figure 11.

Figure 13 Contour plot of the value of

log , eqn. (15), as a function of x1 and x2setpoint changes.

4.3 Example 5 Nonisothermal CSTR

Consider the CSTR studied in Nikolaou and Hanagandi (1988), with feed flow rate and heatremoval rate as inputs and concentration and

temperature as process outputs. Figure 14 and

Figure 15 are the counterparts of Figure 11and Figure 12.

Figure 14 Setpoint step-change

responses of (a) nonlinear closed loop

with linear IMC controller (solid lines)and (b) ideal linear closed loop

(dashed lines). Solid and dashed lines

are virtually indistinguishable.

Figure 15 Similar to Figure 14 fordifferent setpoint changes (-200

mol/m3 , -30 K). Notice thediscrepancies between solid and

dashed lines.

The behavior of observed in Figure 14 and

Figure 15 can be explained by the contour plotof Figure 16. Note that predictions of thestability region are less conservative in this

Example than in Example 3.

0 50 100 150-150

-100

-50

0

concentration

time

0 50 100 150-5

-4

-3

-2

-1

0

temperature

time

0 1 0 20 30 40 50 60 70 80 9 0

-2000

0

2000

4000

6000

concentration

time

0 1 0 20 30 40 50 60 70 80 9 0

-200

0

200

400

600

temperature

time


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14

Figure 16 Contour plot of the value

of log , eqn. (15), as a function of x1and x2 setpoint changes.

5 Future Directions

The value of the approach discussed in thepreceding sections is that it is valid for generalclasses of nonlinear operators, and relies on

only two crucial elements, presented in section3.3. The results discussed above can be

further refined along a number of promisingdirections. We discuss below two such directions addressing two interrelated classes of systems,namely constrained MPCand control of unstable nonlinear processes. Issues particular to each

of these classes as well as issues common to both classes will be indicated.

5.1 Constrained MPC: Linear or nonlinear model?

5.1.1 Problem formulationMPC is currently enjoying enormous popularity in both academia and industry, as the workhorseof advanced process control (Nikolaou, 2001b). While static nonlinear elements (e.g., nonlinear

valves, inferential sensors, etc.) are included in several industrial MPC implementations, the vastmajority of them employ a lineardynamicmodel of the controlled process. Academic research

has produced a substantial amount of work on MPC with nonlinear model (Allgwer and Zheng,2000). Despite their obvious advantages in accuracy, nonlinear models have the disadvantagesof (a) necessitating on-line nonlinear optimization that is frequently non-convex, and (b)

seriously complicating model development. Would there be any advantages in using a (partly ortotally) nonlinear model in MPC for a given process or would a linear model be sufficient?

We propose to use the framework discussed in section 3 to address the above question.We demonstrate next that this framework can serve as basis for constrained nonlinear MPCanalysis and synthesis.

5.1.2 Preliminary analysisThe starting point of the research proposed in this section is the realization that an MPC system

can be cast in the form of Figure 6. To see that, consider the prototypical MPC on-lineoptimization

( )1

2 2

( | ),..., ( 1| ) 1 0

min ( | ) ( | )p m

u k k u k m k i i

r y k i k R u k i k

+ = =

+ + +

(23)

subject to

min max( | )y y k i k y + (24)

min max( | )u u k i k u + , min max( | )u u k i k u + (25)where a linear finite-impulse-response (FIR) model is used to predict future outputs as


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15

model

1 1

( )

( ) ( ) ( ) ( )

y

n n

j j

j j

d k k

y k i k h u k i j k y k h u k j= =

+ = + + 6447448

144424443

(26)

After substitution of eqn. (26) into eqns. (23) and (24), the above constrained MPC algorithmcorresponds to the IMC structure of Figure 6, with the operator :Q u a realized by

21

2

( | ),..., ( 1| )1 1 0

min ( ) ( ) ( | )p n m

ju k k u k m k

i j i

k h u k i j k R u k i k

+ = = =

+ + + +

(27)

subject to eqn. (25) and

min max

1

( ) ( )n

j

j

r y k h u k i j k r y=

+ + (28)where (Figure 17) model[ ( ) ( )] r y k y k = , and the operator model:L u ya realized by

model

1

( ) ( )n

j

j

y k h u k j=

=

(29)

Figure 17 Closed-loop MPC block diagram in IMC form, for a nonlinear process N.

The corresponding closed-loop process output is1

( ) ( )LinearModel L L Ly d NQ I NQ LQ r d = + + (30)

Now, if MPC employs a nonlinear model to predict future process outputs as

( )model

( )

( ) ( 1 ),..., ( ) ( ) ( ( 1),..., ( ))

y

d k k

y k i k f u k i k u k i n k y k f u k u k n+ = + + + 64444744448

1444442444443(31)

then a structure analogous to that of Figure 17 is trivially obtained, and the closed-loop process

output, assuming no process/model mismatch, is

( )NonlinearModel N y d NQ r d = + (32)

Therefore, eqns. (32) and (30) yield[ ] 1[ ( ) ] [ ( ) ] ( )

( )

LinearModel NonlinearModel L N L L

MPC

y y NQ NQ I N L Q I N L Q r d

N r d

= + + =

= (33)

To compare MPC with linear model and MPC with nonlinear model, we can use anapproach similar to that in section 3.2 and the two crucial elements of section 3.3 to show and

perform computations on the counterpart of eqn. (16), i.e.

r ud

y

-

-

Controller

NQL

L

Eqns. (27), (28), (25)

Eqn. (29)ymodel


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16

[ ( ) ] [ ( ) ]

( ) 1 ( )

L N L L N LE EMPC Z

L LE E

NQ NQ I N L Q NQ NQ I N L QN

I N L Q N L Q

+ +

+ (34)

provided that ( ) 1L EN L Q < . If a satisfactory controller QL can be designed that makes thebounds in eqn. (34) small enough, then MPC with nonlinear model is not necessary.

5.1.3 MPC stability revisitedFor the analysis presented in the preceding section 5.1.2 to be valid, closed-loop stability must be

ensured. For that, it is necessary and sufficient that both :LQ u a and :NQ u a be input-outputstable over corresponding sets, i.e. there exists a constant 0K> such that

u K . (35)Constrained MPC stability has been amply studied in literature (Nikolaou, 2001b) after

the seminal paper of Rawlings and Muske (1993). Virtually all analyses take a Lyapunovstability approach. The main result of this approach can be summarized as feasibility of the on-line optimization guarantees closed-loop stability (Mayne et al., 2000). While the Lyapunov

stability approach is an extremely important advancement in our understanding of MPCbehavior, an input-output stability approach (Willems, 1971, p. 101) could also provide

fundamental insight, by determining sets of signals (e.g., disturbances, setpoint changes, etc.)over which stability is maintained.

5.1.4 Issues to address and potential paths of development- Establishing input/output stability over set for QL and QN, is not trivial, because no closed-

form expressions exist for either QL or QN. Any computational approach attempting explicit

enumeration of all controllers resulting from corresponding sets of constraints being activewould be hopeless for problems of industrial size. A more promising approach could be touse the Karush-Kuhn-Tucker optimality conditions and attempt to find (preferably the

smallest) K that satisfies eqn. (35) using parametric programming ideas (a conceptsuccessfully used by Bemporad et al. (2000) for small problems). The input-to-state stability

(ISS) concept (Sontag, 2000) may also be used to blend input/output and Lyapunov stabilityover sets of inputs .

- Computation of incremental norms through eqn. (19) must account for the fact that QL andQN, realized through on-line optimization, are almost everywhere linearizable, i.e. pointscorresponding to changes in the set of active constraints have one-sided derivatives.

- Given that MPC is a multivariable control scheme, it is natural to ask what parts of the modelemployed by MPC need to be nonlinear and what parts can be handled as linear. Blending of

physical process understanding (e.g., sparsity, monotonicity) with algorithms might provepromising in addressing the computational complexity of the problem.

- Simulation tests on multivariable test problems should reveal the limits of the approach.

5.2 Unstable nonlinear processes: Over what range can they be stabilized by linear control?

5.2.1 Problem formulationQuantification of nonlinearity and determination of whether linear control is adequate for anunstable system is complicated by the fact that unstable systems produce unbounded signals and,consequently, distances between unstable operators cannot be measured in terms of standard


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17

norms, but require different metrics. For example, eqn. (10) yields an unbounded quantity. The

gap metric, introduced by Zames and El-Sakkary (1980) is a suitable metric that relies onfractional representation of operators (Vidyasager, 1985) and quantifies distances of operators in

terms of distances of fraction components. We demonstrate next that the fractionalrepresentation approach fits naturally within the framework of section 3, and can be used to

assess when linear control is adequate for an unstablenonlinearprocess.

5.2.2 Preliminary analysisTo stabilize a nonlinear process by a linear controller C, one can start with the standard Youla

parametrization of all linear stabilizing controllers for a linear plant L and examine whether thiscontroller stabilizes the actual nonlinear plant. In addition, the operating domain over whichstabilization is possible should also be examined.

To see how the basic results of section 3 can be used in this analysis, parametrize all

linearcontrollers Cthat stabilize L as 1 1( ) ( ) ( )( )r l r l l r l r C Y QA X QB X B Q Y A Q = + = + . The

Youla parameter Q must be a proper and stable operator and the proper stable linear operators Ar,Al, Br, Bl, Xr, Al, Yr, Yl, for which construction formulas are well known (Vidyasagar, 1985) ,

constitute a doubly coprime factorization ofL as 1 1r r l lL A B B A = = .

Figure 18 Block diagram of linear

control for a nonlinear process N.Because N is unstable, input disturbances

v must be explicitly considered to ensure

closed-loop stability.

When the above controller C is used with the plant Nas in Figure 18, the closed-loop operatorsfrom the external signals rand v to the signals u andy (needed to assess closed-loop stability andperformance) can be explicitly expressed as

1

( ) ( )u I CN v Cr

= + + ,1

( ) ( )y N I CN v Cr

= + + (36)Note that becauseNis nonlinear, the effects of rand v on u andy in the right-hand side of eqn.(36) cannot be separated.

To proceed with the analysis, an obvious first step is to cons ider the cases 0v = and0r= separately, and establish closed-loop stability and performance on the basis of non-singular

perturbation arguments. Thus, on the basis of linearity of the controller C, it can be shown aftera series of manipulations that,

10 ( )

N

r r lv y A D X QB r = = + and 1( )Nr r lu B D X QB r

= + (37)10 ( )Nr r lr y A D Y QA v

= = and 1( )Nr r lu B D Y QA v= (38)

where ( ) ( )1 1

N N N N

r r l lN A B B A

= = is a doubly coprime factorization of the nonlinear operator N

(van der Schaft, 2000, note 9, p. 161; Ball and Verma, 1994; Baramov and Kimura, 1997) and

( ) ( )N N

r l r r l r D Y QA B X QB A= + + (39)Eqns. (37) and (38) can be used in closed-loop stability and performance analysis, as follows.

Stability. The stability of the closed loop depends on the existence and stability of 1D

over a corresponding set. According to element (a) in section 3.3, if

1E

D I

= < (40)

Cr e u

vy

- Nq


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18

then 1D exists, is stable, and 11

1ZD

. Element (b) in section 3.3 can be used in all

computations.

Performance. If the controlled plant were equal to L, then a series of manipulations canshow that it would be

1LinearPlant LinearPlant D D I = = (41)

and that the resulting ideal linear closed loop would be

( ) ( )LinearPlant r r l r r ly A Y QA v A X QB r = + + , ( ) ( )LinearPlant r r l r r lu B Y QA v B X QB r = + + (42)Comparison of eqns. (42) with eqns. (37) and (38) provides a measure of closed-loop

nonlinearity for unstable nonlinear plants. For example, when 0v = , we have1( ) ( )

N

LinearPlant r r r ly y A A D D X RB r Mr = + = (43)

and if eqn. (40) holds, then we can show that

1( )

1

N

r r r lZ RW M W A A D X QB

+

(44)

which is the counterpart of eqn. (16) for unstable nonlinear systems. Again, as in section 5.1.2,if a linear controller Q can be designed that makes the bound in eqn. (44) small enough, thennonlinear control is not necessary.

5.2.3 Issues to address and potential paths of development- Eqns. (40) and (44) indicate that linear control will have to render ( )Nr r R

W A A D

and

small. Design limitations stemming from this fact should be investigated both in terms ofgeneral rigorous results development and simulation tests.

- Computing a nonlinearcoprime factorization ( ) ( )1 1N N N N

r r l lN A B B A

= = may be feasible

but cumbersome (van der Schaft, 2000). However, use of element (b) (p. 10) can greatly

simplify computations by requiring only a series of linear coprime factorization of N (cf.element (b), p. 10).

- The relationship between the sets R andZ in eqn. (44) will have to be better quantified, inorder to determine sets over which stability and performance are ensured (cf. Kapoor and

Daoutidis, 1997; Scibile and Kouvaritakis, 2000; Cantoni, 1999; Zheng and Morari, 1995).- Simulations should test and compare the approach on unstable systems that have been

studied in literature (e.g., Downs and Vogel, 1993).

5.3 Issues common to both directions of research

- How conservative is the approximation in eqn. (19) for the computation of the incremental

norm of a nonlinear operator?

- Nikolaou and Manousiouthakis (1989) have shown how the incremental norm can becomputed through the exact equality in eqn. (19), using nonsmooth optimal control. Their

approach is rigorous but cumbersome for large problems. Could it be made more efficient?- How conservative are the inequalities (17) and (18) in the various instances of element (a) (p.

10) such as eqns. (15) and (16), (34), (40) and (44)?- How can the proposed approach be generalized when disturbances are not additive?- Is it more efficient (in terms of experimental data needed) to linearize a nonlinear model or to

develop a series of empirical linear models in eqn. (19)?


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19

- Could steady-state information obtainable from commercial process simulators be combined

with monotonicity arguments to make the proposed approach easier to apply (see Vinson andGeorgakis (2000) for related work)?

6 Conclusions

In this paper we addressed the issue of linear controller design for nonlinear plants. We gave abrief exposition of a general theory that quantifies the interplay between nonlinearity and

feedback control. The main ingredients of this theory, summarized in section 3.3, are the indices

, eqn. (15), and 1

=

, eqn. (16). The theory presented is important for controller analysis

and, more importantly, efficient controller synthesis. A number of potential extensions and

refinements were suggested. We believe that the proposed framework can be widely applicable,especially if it is improved according to theoretical possibilities and industrial needs. Asdiscussed in Sidebar 1 (p. 2), the need for control of nonlinear systems exists both in mature

high-volume industries (for which the tendency towards wider integration of operationsinevitably leads to encompassing nonlinearities) as well as in emerging industries (for which

control may be essential for feasible rather than efficient process operation). However, controlsystems for nonlinear processes do not have to be overly complex. Moreover, if such systemshave to be internally complex, the complexity of the design (e.g. translation of qualitative

engineering requirements to design parameter specifications), operation, and maintenance ofsuch systems by process engineers and operators should be low, to ensure successful

implementation (Birchfield, 1997; Downs, 2001). Our vision is to develop related tools that arebased on rigorous concepts but are not more complicated than necessary, and delegatecumbersome computations to the computer while allowing the designer to concentrate on

important design concepts.

Acknowledgement Partial financial support from the National Science Foundation (Grant CTS

9896231) is gratefully acknowledged.

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