A note on the control of processes exhibiting input multiplicity Robert J. Lovelett1 Yorgos M. Psarellis2 Ioannis G. Kevrekidis12

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A note on the control

of processes exhibiting input multiplicity

Robert J. Lovelett1, Yorgos M. Psarellis2, Ioannis G. Kevrekidis1,2,

and Manfred Morari3,*

1Department of Chemical and Biological Engineering, Princeton

University

2Department of Chemical and Biomolecular Engineering, Johns

Hopkins University

3Department of Electrical and Systems Engineering, University of

Pennsylvania

*Correspondence: morari@seas.upenn.edu

October 6, 2022

Abstract

Steady state multiplicity can occur in nonlinear systems, and this

presents challenges to feedback control. Input multiplicity arises when

the same steady state output values can be reached with system inputs at

diﬀerent values. Dynamic systems with input multiplicities equipped with

controllers with integral action have multiple stationary points, which may

be locally stable or not. This is undesirable for operation. For a 2x2 ex-

ample system with three stationary points we demonstrate how to design

a set of two single loop controllers such that only one of the stationary

points is locally stable, thus eﬀectively eliminating the “input multiplic-

ity problem” for control. We also show that when MPC is used for the

example system, all three closed-loop stationary points are stable. De-

pending on the initial value of the input variables, the closed loop system

under MPC may converge to diﬀerent steady state input instances (but

the same output steady state). Therefore we computationally explore the

basin boundaries of this closed loop system. It is not clear how MPC or

other modern nonlinear controllers could be designed so that only speciﬁc

equilibrium points are stable.

Dedication: RJL completed his PhD under Tunde Ogunnaike’s supervision

at the University of Delaware. He is grateful that despite Tunde’s many roles—

professor, dean, author, journal editor, and more—Tunde reserved time and

attention to support his students’ lives and careers. He will miss Tunde’s kind

mentorship and thoughtful guidance, and the opportunity to discuss and share

arXiv:2210.01965v1 [math.OC] 5 Oct 2022

new research ﬁndings and life updates. IGK (who will miss Tunde’s trade-

mark Greek greeting “παλιοφιλε,” old friend) kept in contact with him from

the DuPont days to University of Delaware. His last invitation to Tunde for a

seminar at Hopkins was initially delayed by Covid, never to materialize. MM

was on the faculty at the University of Wisconsin when Tunde was there for his

graduate studies. In his typical kind and inquisitive manner Tunde contributed

to MM’s research about the Relative Gain Array (RGA) on which the ideas in

this note are based It was the starting point for many stimulating exchanges in

the following 40 years.

1 Introduction

In this note, we are studying the control of nonlinear time invariant square

systems with manipulated input u∈Rnand controlled output y∈Rn. The

system is assumed to be stable in the sense that any asymptotically constant

input uwill result in an asymptotically constant output y. Thus the system

deﬁnes a static map Grelating the constant input vector uto the eventually

constant output vector y:

y=G(u).(1)

We are interested in systems where the gain matrix, i.e., the Jacobian

G0(u) = ∂G

∂u

uis singular for some us, i.e. det ∂G

∂u

us= 0. A simple SISO

example is shown in Fig. 1. The map G(u) has either an extremum or a sad-

dle point at u=us. In either case it is impossible to control the system at

a reference r=ysbecause the controller gain vanishes. In the MIMO case,

the condition det ∂G

∂u

u= 0 implies that the gain vanishes in a certain input

direction, again indicating that control is not possible.

The output variable ymay be associated with process economics so that its

maximization is sought. In this case extremum seeking control schemes would

be used, which have been studied extensively [1] and are not the subject of this

note. Here we study only regulatory control, i.e. control of the system at a

speciﬁed setpoint y=r.

It should be emphasized that this lack of (practical) controllability at r=ys

is a property of the system and cannot be ﬁxed by any clever controller design.

Instead, it is necessary to change the system itself, for example, by choosing

a diﬀerent manipulated variable or by changing the design parameters of the

system such that the singular point is pushed out of the region of interest.

The system may be diﬃcult to control even at setpoints r6=yswhere the

gain does not vanish. In Fig. 1 we observe “input multiplicity”, i.e. that a

particular desired output y=G(u) = rcan be reached for multiple inputs

u. In practical operation it may not be easily predictable which input will be

ultimately “chosen” by the controller. This could be undesirable because it

would make it very diﬃcult to monitor the process for “correct” operation.

It appears that the control issues surrounding input multiplicity were ﬁrst

discussed by Koppel [7]. In this pioneering paper and several follow-ups [8, 4]

Figure 1: A simple SISO system that exhibits input multiplicity. For setpoint

r, there are two values of u(u1and u2) for which y=G(u) = r. For smooth

G, a necessary condition for this phenomenon is that G0(u) = 0 for some value

of u.

he presents various examples, which suggest that input multiplicity can occur

commonly in process control. The fact that it has not been discussed in the

literature more broadly in the past several decades may simply mean that the

erratic behavior associated with input multiplicity has been observed in practical

operation, but not diagnosed correctly. Such a focused analysis would have

required an accurate process model which may not have been available.

In our note we will analyze two diﬀerent approaches to design controllers for

systems with input multiplicities.

1. Multi-loop integral control. Our control objective is slow control which

eliminates oﬀ-set. We will achieve this by a set of pure integral controllers,

which guarantee local asymptotic stability for a speciﬁc input instance.

2. Model Predictive Control (MPC). We determine the control action by

minimizing a quadratic objective over a moving ﬁnite horizon. If the

closed loop system is locally stable, then there will be no oﬀ-set for our

particular choice of objective.

2 Multi-loop integral control

Based on his experience, Koppel argues convincingly that in practical process

control the open loop speed of response is usually satisfactory, we rarely wish

to make it faster. Even if we wanted to, any speed-up would require a more

accurate dynamic process model, which is typically not available. We do want

oﬀ-set free tracking, however, which we can simply achieve with integral control

action.

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AnoteonthecontrolofprocessesexhibitinginputmultiplicityRobertJ.Lovelett1,YorgosM.Psarellis2,IoannisG.Kevrekidis1,2,andManfredMorari3,*1DepartmentofChemicalandBiologicalEngineering,PrincetonUniversity2DepartmentofChemicalandBiomolecularEngineering,JohnsHopkinsUniversity3DepartmentofElectricalandSyste...

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A note on the control of processes exhibiting input multiplicity Robert J. Lovelett1 Yorgos M. Psarellis2 Ioannis G. Kevrekidis12

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