Sensitivity-Based Economic NMPC with a Path-Following Approach

Abstract

We present a sensitivity-based predictor-corrector path-following algorithm for fast nonlinear model predictive control (NMPC) and demonstrate it on a large case study with an economic cost function. The path-following method is applied within the advanced-step NMPC framework to obtain fast and accurate approximate solutions of the NMPC problem. In our approach, we solve a sequence of quadratic programs to trace the optimal NMPC solution along a parameter change. A distinguishing feature of the path-following algorithm in this paper is that the strongly-active inequality constraints are included as equality constraints in the quadratic programs, while the weakly-active constraints are left as inequalities. This leads to close tracking of the optimal solution. The approach is applied to an economic NMPC case study consisting of a process with a reactor, a distillation column and a recycler. We compare the path-following NMPC solution with an ideal NMPC solution, which is obtained by solving the full nonlinear programming problem. Our simulations show that the proposed algorithm effectively traces the exact solution.

Full text

Article
Sensitivity-Based Economic NMPC with
a Path-Following Approach
Eka Suwartadi 1, Vyacheslav Kungurtsev 2and Johannes Jäschke 1,*
1Department of Chemical Engineering, Norwegian University of Science and Technology (NTNU),
7491 Trondheim, Norway; eka.suwartadi@ntnu.no
2Department of Computer Science, Czech Technical University in Prague, 12000 Praha 2, Czech Republic;
vyacheslav.kungurtsev@fel.cvut.cz
*Correspondence: jaschke@ntnu.no; Tel.: +47-735-93691
Academic Editor: Dominique Bonvin
Received: 26 November 2016; Accepted: 13 February 2017; Published: 27 February 2017
Abstract:
We present a sensitivity-based predictor-corrector path-following algorithm for fast
nonlinear model predictive control (NMPC) and demonstrate it on a large case study with an economic
cost function. The path-following method is applied within the advanced-step NMPC framework to
obtain fast and accurate approximate solutions of the NMPC problem. In our approach, we solve
a sequence of quadratic programs to trace the optimal NMPC solution along a parameter change.
A distinguishing feature of the path-following algorithm in this paper is that the strongly-active
inequality constraints are included as equality constraints in the quadratic programs, while the
weakly-active constraints are left as inequalities. This leads to close tracking of the optimal
solution. The approach is applied to an economic NMPC case study consisting of a process with
a reactor, a distillation column and a recycler. We compare the path-following NMPC solution with
an ideal NMPC solution, which is obtained by solving the full nonlinear programming problem.
Our simulations show that the proposed algorithm effectively traces the exact solution.
Keywords:
fast economic NMPC; NLP sensitivity; path-following algorithm; nonlinear programming;
dynamic optimization
1. Introduction
The idea of economic model predictive control (MPC) is to integrate the economic optimization
layer and the control layer in the process control hierarchy into a single dynamic optimization layer.
While classic model predictive control approaches typically employ a quadratic objective to minimize
the error between the setpoints and selected measurements, economic MPC adjusts the inputs to
minimize the economic cost of operation directly. This makes it possible to optimize the cost during
transient operation of the plant. In recent years, this has become increasingly desirable, as stronger
competition, volatile energy prices and rapidly changing product specifications require agile plant
operations, where also transients are optimized to maximize profit.
The first industrial implementations of economic MPC were reported in [
1
,
2
] for oil refinery
applications. The development of theory and stability analysis for economic MPC arose almost
a decade afterwards; see, e.g., [
3
,
4
]. Recent progress on economic MPC is reviewed and surveyed
in [
5
,
6
]. Most of the current research activities focus on the stability analysis of economic MPC and do
not discuss its performance (an exception is [7]).
Because nonlinear process models are often used for economic optimization, a potential drawback
of economic MPC is that it requires solving a large-scale nonlinear optimization problem (NLP)
associated with the nonlinear model predictive control (NMPC) problem at every sample time.
Processes 2017,5, 8; doi:10.3390/pr5010008 www.mdpi.com/journal/processes

Processes 2017,5, 8 2 of 18
The solution of this NLP may take a significant amount of time [
8
], and this can lead to performance
degradation and even to instability of the closed-loop system [9].
To reduce the detrimental effect of computational delay in NMPC, several sensitivity-based
methods were proposed [
10
–
12
]. All of these fast sensitivity approaches exploit the fact that the NMPC
optimization problems are identical at each sample time, except for one varying parameter: the initial
state. Instead of solving the full nonlinear optimization problem when new measurements of the state
become available, these approaches use the sensitivity of the NLP solution at a previously-computed
iteration to obtain fast approximate solutions to the new NMPC problem. These approximate
solutions can be computed and implemented in the plant with minimal delay. A recent overview
of the developments in fast sensitivity-based nonlinear MPC is given in [
13
], and a comparison of
different approaches to obtain sensitivity updates for NMPC is compiled in the paper by Wolf and
Marquardt [14].
Diehl et al. [
15
] proposed the concept of real-time iteration (RTI), in which the full NLP is not
solved at all during the MPC iterations. Instead, at each NMPC sampling time, a single quadratic
programming (QP) related to the sequential quadratic programming (SQP) iteration for solving the
full NLP is solved. The real-time iteration scheme contains two phases: (1) the preparation phase and
(2) the feedback phase. In the preparation phase, the model derivatives are evaluated using a predicted
state measurement, and a QP is formulated based on data of this predicted state. In the feedback phase,
once the new initial state is available, the QP is updated to include the new initial state and solved
for the control input that is injected into the plant. The real-time iteration scheme has been applied
to economic NMPC in the context of wind turbine control [
16
,
17
]. Similar to the real-time iteration
scheme are the approaches by Ohtsuka [
18
] and the early paper by Li and Biegler [
19
], where one
single Newton-like iteration is performed per sampling time.
A different approach, the advanced-step NMPC (asNMPC), was proposed by Zavala and
Biegler [
10
]. The asNMPC approach involves solving the full NLP at every sample time. However,
the full NLP solution is computed in advance for a predicted initial state. Once the new state
measurement is available, the NLP solution is corrected using a fast sensitivity update to match
the measured or estimated initial state. A simple sensitivity update scheme is implemented in the
software package sIPOPT [
20
]. However, active set changes are handled rather heuristically; see [
21
] for
an overview. Kadam and Marquardt [
22
] proposed a similar approach, where nominal NLP solutions
are updated by solving QPs in a neighboring extremal scheme; see also [12,23].
The framework of asNMPC was also applied by Jäschke and Biegler [
24
], who use a multiple-step
predictor path-following algorithm to correct the NLP predictions. Their approach included measures
to handle active set changes rigorously, and their path-following advanced-step NMPC algorithm is
also the first one to handle non-unique Lagrange multipliers.
The contribution of this paper is to apply an improved path-following method for correcting the
NLP solution within the advanced-step NMPC framework. In particular, we replace the predictor
path-following method from [
24
] by a predictor-corrector method and demonstrate numerically that
the method works efficiently on a large-scale case study. We present how the asNMPC with the
predictor-corrector path-following algorithm performs in the presence of measurement noise and
compare it with a pure predictor path-following asNMPC approach and an ideal NMPC approach,
where the NLP is assumed to be solved instantly. We also give a brief discussion about how our
method differs from previously published approaches.
The structure of this paper is the following. We start by introducing the ideal NMPC and
advanced-step NMPC frameworks in Section 2and give a description of our path-following algorithm
together with some relevant background material and a brief discussion in Section 3. The proposed
algorithm is applied to a process with a reactor, distillation and recycling in Section 4, where we
consider the cases with and without measurement noise and discuss the results. The paper is closed
with our conclusions in Section 5.

Processes 2017,5, 8 3 of 18
2. NMPC Problem Formulations
2.1. The NMPC Problem
We consider a nonlinear discrete-time dynamic system:
xk+1=f(xk,uk)(1)
where
xk∈Rnx
denotes the state variable,
uk∈Rnu
is the control input and
f:Rnx×Rnu→Rnx
is a
continuous model function, which calculates the next state
xk+1
from the previous state
xk
and control
input
uk
, where
k∈N
. This system is optimized by a nonlinear model predictive controller, which
solves the problem:
(Pnmpc): min
zl,vl
Ψ(zN) +
N−1
∑
l=0
ψ(zl,vl)(2)
s.t. zl+1=f(zl,vl)l=0, . . . , N−1,
z0=xk,
(zl,vl)∈ Z,l=0, . . . , N−1,
zN∈ Xf,
at each sample time. Here,
zl∈Rnx
is the predicted state variable;
vl∈Rnu
is the predicted control
input; and
zN∈ Xf
is the final predicted state variable restricted to the terminal region
Xf∈Rnx
.
The stage cost is denoted by
ψ:Rnx×Rnu→R
and the terminal cost by
Ψ:Xf→R
. Further,
Zdenotes the path constraints, i.e., Z={(z,v)|q(z,v)≤0}, where q:Rnx×Rnu→Rnq.
The solution of the optimization problem
Pnmpc
is denoted
z∗
0, . . . , z∗
N,v∗
0, . . . , v∗
N−1
. At sample
time
k
, an estimate or measurement of the state
xk
is obtained, and problem
Pnmpc
is solved. Then,
the first part of the optimal control sequence is assigned as plant input, such that
uk=v∗
0
. This first
part of the solution to Pnmpc defines an implicit feedback law uk=κ(xk), and the system will evolve
according to
xk+1=f(xk,κ(xk))
. At the next sample time
k+
1, when the measurement of the new
state
xk+1
is obtained, the procedure is repeated. The NMPC algorithm is summarized in Algorithm 1.
Algorithm 1: General NMPC algorithm.
1set k←0
2while MPC is running do
31. Measure or estimate xk.
42. Assign the initial state: set z0=xk.
53. Solve the optimization problem Pnm pc to find v∗
0.
64. Assign the plant input uk=v∗
0.
75. Inject ukto the plant (1).
86. Set k←k+1
2.2. Ideal NMPC and Advanced-Step NMPC Framework
For achieving optimal economic performance and good stability properties, problem
Pnmpc
needs
to be solved instantly, so that the optimal input can be injected without time delay as soon as the
values of the new states are available. We refer to this hypothetical case without computational delay
as ideal NMPC.
In practice, there will always be some time delay between obtaining the updated values of
the states and injecting the updated inputs into the plant. The main reason for this delay is the
time it requires to solve the optimization problem
Pnmpc
. As the process models become more

Processes 2017,5, 8 4 of 18
advanced, solving the optimization problems requires more time, and the computational delay cannot
be neglected any more. This has led to the development of fast sensitivity-based NMPC approaches.
One such approach that will be a adopted in this paper is the advanced-step NMPC (asNMPC)
approach [10]. It is based on the following steps:
1. Solve the NMPC problem at time kwith a predicted state value of time k+1,
2. When the measurement xk+1becomes available at time k+1, compute an approximation of the
NLP solution using fast sensitivity methods,
3. Update k←k+1, and repeat from Step 1.
Zavala and Biegler proposed a fast one-step sensitivity update that is based on solving a linear
system of equations [
10
]. Under some assumptions, this corresponds to a first-order Taylor
approximation of the optimal solution. In particular, this approach requires strict complementarity
of the NLP solution, which ensures no changes in the active set. A more general approach involves
allowing for changes in the active set and making several sensitivity updates. This was proposed
in [24] and will be developed further in this paper.
3. Sensitivity-Based Path-Following NMPC
In this section, we present some fundamental sensitivity results from the literature and then use
them in a path-following scheme for obtaining fast approximate solutions to the NLP.
3.1. Sensitivity Properties of NLP
The dynamic optimization Problem (2) can be cast as a general parametric NLP problem:
(PNL P): min
χF(χ,p)(3)
s.t. c(χ,p)=0
g(χ,p)≤0,
where
χ∈Rnχ
are the decision variables (which generally include the state variables and the control
input
nχ=nx+nu
) and
p∈Rnp
is the parameter, which is typically the initial state variable
xk
.
In addition,
F:Rnχ×Rnp→R
is the scalar objective function;
c:Rnχ×Rnp→Rnc
denotes the
equality constraints; and finally,
g:Rnχ×Rnp→Rng
denotes the inequality constraints. The instances
of Problem (3) that are solved at each sample time differ only in the parameter p.
The Lagrangian function of this problem is defined as:
L(χ,p,λ,µ)=F(χ,p)+λTc(χ,p)+µTg(χ,p), (4)
and the KKT (Karush–Kuhn–Tucker) conditions are:
c(x,p)=0, g(x,p)≤0(prim al f easibility)(5)
µ≥0, (dua l f easibility)
∇xL(x,p,λ,µ)=0, (stationary condition)
µTg(x,p)=0, (compl ementary slackness).
In order for the KKT conditions to be a necessary condition of optimality, we require a constraint
qualification (CQ) to hold. In this paper, we will assume that the linear independence constraint
qualification (LICQ) holds:
Definition 1
(LICQ)
.
Given a vector
p
and a point
χ
, the LICQ holds at
χ
if the set of vectors
n{∇χci(χ,p)}i∈{1,...,nc}∪ {∇χgi(χ,p)}i:gi(χ,p)=0ois linearly independent.

Processes 2017,5, 8 5 of 18
The LICQ implies that the multipliers
(λ
,
µ)
satisfying the KKT conditions are unique.
If additionally, a suitable second-order condition holds, then the KKT conditions guarantee a unique
local minimum. A suitable second-order condition states that the Hessian matrix has to be positive
definite in a set of appropriate directions, defined in the following property:
Definition 2
(SSOSC)
.
The strong second-order sufficient condition (SSOSC) holds at
χ
with multipliers
λ
and
µ
if
dT∇2
χL(χ,p,λ,µ)d>
0for all
d6=
0, such that
∇χc(χ,p)Td=
0and
∇χgi(χ,p)Td=
0for
i
,
such that gi(χ,p)=0and µi>0.
For a given
p
, denote the solution to
(3)
by
χ∗(p)
,
λ∗(p)
,
µ∗(p)
, and if no confusion is possible,
we omit the argument and write simply
χ∗
,
λ∗
,
µ∗
. We are interested in knowing how the solution
changes with a perturbation in the parameter
p
. Before we state a first sensitivity result, we define
another important concept:
Definition 3
(SC)
.
Given a vector
p
and a solution
χ∗
with vectors of multipliers
λ∗
and
µ∗
, strict
complimentary (SC) holds if µ∗
i−gi(χ∗,p)>0for each i =1, . . . , ng.
Now, we are ready to state the result below given by Fiacco [25].
Theorem 1
(Implicit function theorem applied to optimality conditions)
.
Let
χ∗(p)
be a KKT point that
satisfies
(5)
, and assume that LICQ, SSOSC and SC hold at
χ∗
. Further, let the function
F
,
c
,
g
be at least
k+1-times differentiable in χand k-times differentiable in p. Then:
•χ∗is an isolated minimizer, and the associated multipliers λand µare unique.
•for pin a neighborhood of p0, the set of active constraints remains unchanged.
•
for
p
in a neighborhood of
p0
, there exists a
k
-times differentiable function
σ(p)=
hχ∗(p)Tλ∗(p)Tµ∗(p)Ti, that corresponds to a locally unique minimum for (3).
Proof. See Fiacco [25].
Using this result, the sensitivity of the optimal solution
χ∗
,
λ∗
,
µ∗
in a small neighborhood of
p0
can be computed by solving a system of linear equations that arises from applying the implicit
function theorem to the KKT conditions of (3):



∇2
χχL(χ∗,p0,λ∗,µ∗)∇χc(χ∗,p0)∇χgA(χ∗,p0)
∇χc(χ∗,p0)T0 0
∇χgA(χ∗,p0)T0 0





∇pχ
∇pλ
∇pµ


=−


∇2
pχL(χ∗,p0,λ∗,µ∗)
∇pc(χ∗,p0)
∇pgA(χ∗,p0)

.(6)
Here, the constraint gradients with subscript
gA
indicate that we only include the vectors and
components of the Jacobian corresponding to the active inequality constraints at
χ
, i.e.,
i∈A
if
gi(χ
,
p0) =
0. Denoting the solution of the equation above as
h∇pχ∇pλ∇pµiT
, for small
∆p
,
we obtain a good estimate:
χ(p0+4p)=χ∗+∇pχ4p, (7)
λ(p0+4p)=λ∗+∇pλ4p, (8)
µ(p0+4p)=µ∗+∇pµ4p, (9)
of the solution to the NLP Problem
(3)
at the parameter value
p0+∆p
. This approach was applied by
Zavala and Biegler [10].
If
∆p
becomes large, the approximate solution may no longer be accurate enough, because the SC
assumption implies that the active set cannot change. While that is usually true for small perturbations,
large changes in ∆pmay very well induce active set changes.

Processes 2017,5, 8 6 of 18
It can be seen that the sensitivity system corresponds to the stationarity conditions for a particular
QP. This is not coincidental. It can be shown that for
∆p
small enough, the set
{i:µ(¯
p)i>
0
}
is constant for
¯
p=p0+∆p
. Thus, we can form a QP wherein we are potentially moving off of
weakly-active constraints while staying on the strongly-active ones. The primal-dual solution of this
QP is in fact the directional derivative of the primal-dual solution path χ∗(p),λ∗(p),µ∗(p).
Theorem 2.
Let
F
,
c
,
g
be twice continuously differentiable in
p
and
χ
near
(χ∗,p0)
, and let the LICQ and
SSOSC hold at
(χ∗,p0)
. Then, the solution
(χ∗(p),λ∗(p),µ∗(p))
is Lipschitz continuous in a neighborhood
of (χ∗,λ∗,µ∗,p0), and the solution function (χ∗(p),λ∗(p),µ∗(p)) is directionally differentiable.
Moreover, the directional derivative uniquely solves the following quadratic problem:
min
4χ
1
24χT∇2
χχL(χ∗,p0,λ∗,µ∗)4χ+4χT∇2
pχL(χ∗,p0,λ∗,µ∗)4p(10)
s.t. ∇χci(χ∗,p0)T4χ+∇pci(χ∗,p0)T4p=0i=1, . . . nc
∇χgj(χ∗,p0)T4χ+∇pgj(χ∗,p0)T4p=0j∈K+
∇χgj(χ∗,p0)T4χ+∇pgj(χ∗,p0)T4p≤0j∈K0,
where
K+=j∈Z:µj>0
is the strongly-active set and
K0=j∈Z:µj=0 and gj(χ∗,p0)=0
denotes the weakly active set.
Proof. See [26] (Sections 5.1 and 5.2) and [27] (Proposition 3.4.1).
The theorem above gives the solution of the perturbed NLP
(3)
by solving a QP problem. Note
that regardless of the inertia of the Lagrangian Hessian, if the SSOSC holds, it is positive definite on
the null-space of the equality constraints, and thus, the QP defined is convex with an easily obtainable
finite global minimizer. In [
28
], it is noted that as the solution to this QP is the directional derivative
of the primal-dual solution of the NLP, it is a predictor step, a tangential first-order estimate of the
change in the solution subject to a change in the parameter. We refer to the QP
(10)
as a pure-predictor.
Note that obtaining the sensitivity via
(10)
instead of
(6)
has the advantage that changes in the active
set can be accounted for correctly, and strict complementarity (SC) is not required. On the other hand,
when SC does hold, (6) and (10) are equivalent.
3.2. Path-Following Based on Sensitivity Properties
Equation
(6)
and the QP
(10)
describes the change in the optimal solutions for small perturbations.
They cannot be guaranteed to reproduce the optimal solution accurately for larger perturbations,
because of curvature in the solution path and active set changes that happen further away from the
linearization point. One approach to handle such cases is to divide the overall perturbation into several
smaller intervals and to iteratively use the sensitivity to track the path of optimal solutions.
The general idea of a path-following method is to reach the solution of the problem at a final
parameter value
pf
by tracing a sequence of solutions
(χk
,
λk
,
µk)
for a series of parameter values
p(tk) = (1−tk)p0+tkpf
with 0
=t0<t1<
...
<tk<
...
<tN=
1. The new direction is found
by evaluating the sensitivity at the current point. This is similar to a Euler integration for ordinary
differential equations.
However, just as in the case of integrating differential equations with a Euler method,
a path-following algorithm that is only based on the sensitivity calculated by the pure predictor
QP may fail to track the solution accurately enough and may lead to poor solutions. To address this
problem, a common approach is to include elements that are similar to a Newton step, which force the
path-following algorithm towards the true solution. It has been found that such a corrector element can
be easily included into a QP that is very similar to the predictor QP
(10)
. Consider approximating
(3)

Processes 2017,5, 8 7 of 18
by a QP, linearizing with respect to both
χ
and
p
, but again enforcing the equality of the strongly-active
constraints, as we expect them to remain strongly active at a perturbed NLP:
min
4χ,4p
1
24χT∇2
χχL(χ∗,p0,λ∗,µ∗)4χ+4χT∇2
pχL(χ∗,p0,λ∗,µ∗)4p+∇χFT4χ+∇pF4p+1
24pT∇2
ppL(χ∗,p0,λ∗,µ∗)4p(11)
s.t. ci(χ∗,p0)+∇χci(χ∗,p0)T4χ+∇pci(χ∗,p0)T4p=0i=0, . . . nc
gj(χ∗,p0)+∇χgj(χ∗,p0)T4χ+∇pgj(χ∗,p0)T4p=0j∈K+
gj(χ∗,p0)+∇χgj(χ∗,p0)T4χ+∇pgj(χ∗,p0)T4p≤0j∈1, ..., ng\K+.
In our NMPC problem
Pnmpc
, the parameter
p
corresponds to the current “initial” state,
xk
.
Moreover, the cost function is independent of
p
, and we have that
∇pF=
0. Since the parameter
enters the constraints linearly, we have that
∇pc
and
∇pg
are constants. With these facts, the above
QP simplifies to:
min
4χ
1
24χT∇2
χχL(χ∗,p0+4p,λ∗,µ∗)4χ+∇χFT4χ(12)
s.t. ci(χ∗,p0+4p)+∇χci(χ∗,p0+4p)T4χ=0i=1, . . . nc
gj(χ∗,p0+4p)+∇χgj(χ∗,p0+4p)T4χ=0j∈K+
gj(χ∗,p0+4p)+∇χgj(χ∗,p0+4p)T4χ≤0j∈1, ..., ng\K+.
We denote the QP formulation
(12)
as the predictor-corrector. We note that this QP is similar to
the QP proposed in the real-time iteration scheme [
15
]. However, it is not quite the same, as we enforce
the strongly-active constraints as equality constraints in the QP. As explained in [
28
], this particular QP
tries to estimate how the NLP solution changes as the parameter does in the predictor component and
refines the estimate, in more closely satisfying the KKT conditions at the new parameter, as a corrector.
The predictor-corrector QP
(12)
is well suited for use in a path-following algorithm, where the
optimal solution path is tracked from
p0
to a final value
pf
along a sequence of parameter points
p(tk) = (1−tk)p0+tkpf
with 0
=t0<t1<
...
<tk<
...
<tN=
1. At each point
p(tk)
, the QP is
solved and the primal-dual solutions updated as:
χ(tk+1) = χ(tk) + ∆χ(13)
λ(tk+1) = ∆λ(14)
µ(tk+1) = ∆µ, (15)
where
∆χ
is obtained from the primal solution of QP
(12)
and where
∆λ
and
∆µ
correspond to the
Lagrange multipliers of QP (12).
Changes in the active set along the path are detected by the QP as follows: If a constraint becomes
inactive at some point along the path, the corresponding multiplier
µj
will first become weakly active,
i.e., it will be added to the set
K0
. Since it is not included as an equality constraint, the next QP solution
can move away from the constraint. Similarly, if a new constraint
gj
becomes active along the path,
it will make the corresponding linearized inequality constraint in the QP active and be tracked further
along the path.
The resulting path-following algorithm is summarized with its main steps in Algorithm 2, and we
are now in the position to apply it in the advanced-step NMPC setting described in Section 2.2.
In particular, the path-following algorithm is used to find a fast approximation of the optimal NLP
solution corresponding to the new available state measurement, which is done by following the
optimal solution path from the predicted state to the measured state.

Processes 2017,5, 8 8 of 18
Algorithm 2: Path-following algorithm.
Input: initial variables from NLP χ∗(p0),λ∗(p0),µ∗(p0)
fix stepsize 4t, and set N=1
∆t
set initial parameter value p0,
set final parameter value pf,
set t=0,
set constant 0 <α1<1.
Output: primal variable χand dual variables λ,µalong the path
1for k←1to Ndo
2Compute step ∆p=pk−pk−1
3Solve QP problem ; /* to obtain ∆χ,∆λ,∆µ*/
4if QP is feasible then
5/* perform update */
6χ←χ+∆χ;/* update primal variables */
7Update dual variables appropriately; using Equations (8) and 9for the pure-predictor
method or (14) and (15) for the predictor-corrector method
8t←t+∆t;/* update stepsize */
9k←k+1
10 else
11 /* QP is infeasible, reduce QP stepsize */
12 4t←α14t
13 t←t−α14t
3.3. Discussion of the Path-Following asNMPC Approach
In this section, we discuss some characteristics of the path-following asNMPC approach presented
in this paper. We also present a small example to demonstrate the effect of including the strongly-active
constraints as equality constraints in the QP.
A reader who is familiar with the real-time iteration scheme [
15
] will have realized that the
QPs
(12)
that are solved in our path-following algorithm are similar to the ones proposed and solved in
the real-time iteration scheme. However, there are some fundamental differences between the standard
real-time iteration scheme as described in [15] and the asNMPC with a path-following approach.
This work is set in the advanced-step NMPC framework, i.e., at every time step, the full NLP is
solved for a predicted state. When the new measurement becomes available, the precomputed NLP
solution is updated by tracking the optimal solution curve from the predicted initial state to the new
measured or estimated state. Any numerical homotopy algorithm can be used to update the NLP
solution, and we have presented a suitable one in this paper. Note that the solution of the last QP
along the path corresponds to the updated NLP solution, and only the inputs computed in this last QP
will be injected into the plant.
The situation is quite different in the real time iteration (RTI) scheme described in [
15
]. Here,
the NLP is not solved at all during the MPC sampling times. Instead, at each sampling time, a single
QP is solved, and the computed input is applied to the plant. This will require very fast sampling
times, and if the QP fails to track the true solution due to very large disturbances, similar measures
as in the advanced-step NMPC procedure (i.e., solving the full NLP) must be performed to get the
controller “on track” again. Note that the inputs computed from every QP are applied to the plant and,
not as in our path-following asNMPC, only the input computed in the last QP along the homotopy.
Finally, in the QPs of the previously published real-time iteration schemes [
15
], all inequality
constraints are linearized and included as QP inequality constraints. Our approach in this paper,
however, distinguishes between strongly- and weakly-active inequality constraints. Strongly-active

Processes 2017,5, 8 9 of 18
inequalities are included as linearized equality constraints in the QP, while weakly-active constraints
are linearized and added as inequality constraints to the QP. This ensures that the true solution path is
tracked more accurately also when the full Hessian of the optimization problem becomes non-convex.
We illustrate this in the small example below.
Example 1. Consider the following parametric “NLP”
min
xx2
1−x2
2(16)
s.t. −2−x2+t≤0
−2+x2
1+x2≤0,
for which we have plotted the constraints at t =0in Figure 1a.
x1
x2
-2
2
2-2
-1
1
3-3 1-1
3
-3
g1:−2−x2+t≤0
g2:−2+x2
1+x2≤0
(1,-2)
(a)
x1
x2
-2
2
2-2
-1
1
3-3 1-1
3
-3
g1
g2
3−2∆x1−∆x2≥0
(1,-2)
(b)
Figure 1.
(
a
) Constraints of NLP
(16)
in Example 1 and (
b
) their linearization at
ˆ
x= (
1,
−
2
)
and
t=
0.
The feasible region lies in between the parabola and the horizontal line. Changing the parameter
t
from zero
to one moves the lower constraint up from x2=−2to x2=−1.
The objective gradient is
∇F(x)=(
2
x1
,
−
2
x2)
, and the Hessian of the objective is always indefinite
H= 2 0
0−2!
. The constraint gradients are
∇g(x) = 0−1
2x11!
. For
t∈[
0, 1
]
, a (local) primal solution
is given by
x∗(t)=(0, t−2)
. The first constraint is active, the second constraint is inactive, and the dual
solution is
λ∗(t)=(−2x2, 0)
. At
t=
0we thus have the optimal primal solution
x∗= (
0,
−
2
)
and the
optimal multiplier λ∗= (4, 0).
We consider starting from an approximate solution at the point
ˆ
x(t=
0
) = (
1,
−
2
)
with dual variables
ˆ
λ(t=
0
) = (
4, 0
)
, such that the first constraint is strongly active, while the second one remains inactive. The
linearized constraints for this point are shown in Figure 1b. Now, consider a change
∆t=
1, going from
t=
0
to t =1.
The pure predictor QP
(10)
has the form, recalling that we enforce the strongly active constraint as equality:
min
∆x
∆x2
1−∆x2
2
s.t. −∆x2+1=0.
(17)
This QP is convex with a unique solution ∆x= (0, 1)resulting in the subsequent point ˆ
x(t=1) = (1, −1).

Processes 2017,5, 8 10 of 18
The predictor-corrector QP
(12)
, which includes a linear term in the objective that acts as a corrector, is
given for this case as
min
∆x
∆x2
1−∆x2
2+2∆x1+4∆x2
s.t. −∆x2+1=0
−3+2∆x1+∆x2≤0.
(18)
Again this QP is convex with a unique primal solution
∆x= (−
1, 1
)
. The step computed by this predictor
corrector QP moves the update to the true optimal solution ˆ
x(t=1) = (0, −1) = x∗(t=1).
Now, consider a third QP, which is the predictor-corrector QP
(12)
, but without enforcing the strongly
active constraints as equalities. That is, all constraints are included in the QP as they were in the original
NLP (16),
min
∆x
∆x2
1−∆x2
2+2∆x1+4∆x2
s.t. −∆x2+1≤0
−3+2∆x1+∆x2≤0.
(19)
This QP is non-convex and unbounded; we can decrease the objective arbitrarily by setting
∆x= (
1.5
−
0.5
r
,
r)
and letting a scalar
r≥
1go to infinity. Although there is a local minimizer at
∆x= (−
1, 1
)
, a QP
solver that behaves “optimally” should find the unbounded “solution”.
This last approach cannot be expected to work reliably if the full Hessian of the optimization problem
may become non-convex, which easily can be the case when optimizing economic objective functions. We note,
however, that if the Hessian
∇xxL
is positive definite, QP it is not necessary to enforce the strongly active
constraints as equality constraints in the predictor-corrector QP (12).
4. Numerical Case Study
4.1. Process Description
We demonstrate the path-following NMPC (pf-NMPC) on an isothermal reactor and separator
process depicted in Figure 2. The continuously-stirred tank reactor (CSTR) is fed with a stream
F0
containing 100% component
A
and a recycler
R
from the distillation column. A first-order reaction
A → B
takes place in the CSTR where
B
is the desired product and the product with flow rate
F
is fed
to the column. In the distillation column, the unreacted raw material is separated from the product and
recycled into the reactor. The desired product
B
leaves the distillation column as the bottom product,
which is required to have a certain purity. Reaction kinetic parameters for the reactor are described
in Table 1. The distillation column model is taken from [
29
]. Table 2summarizes the parameters
used in the distillation. In total, the model has 84 state variables of which 82 are from the distillation
(concentration and holdup for each stage) and two from the CSTR (one concentration and one holdup).
VersionFebruary 3, 2017 submitted to Processes 11 of 19
F0
MB
MD
XF
VB
LT
D
B
F
Figure 2. Diagram of CSTR and Distillation Column
Table 2. Distillation column A parameters
Parameter Value
αAB 1.5
number of stages 41
feed stage location 21
The stage cost of the economic objective function to optimize under operation is
J=pFF0+pVVB−pBB, (20)
where pFis the feed cost, pVis steam cost, and pBis the product price. The price setting is pF=
1 $/kmo l,pV=0.02 $/k mol,pB=2 $/kmo l. The operational constraints are the concentration of
the bottom product (xB≤0.1), as well as the liquid holdup at the bottom and top of the distillation
column and in the CSTR (0.3 ≤M{B,D,C STR}≤0.7) kmol. The control inputs are reflux flow (LT),
boilup flow (VB), feeding rate to the distillation (F), distillate (top) and bottom product flow rates (D
and B). These control inputs have bound constraints as follows







0.1
0.1
0.1
0.1
0.1







≤






LT
VB
F
D
B







≤






10
4.008
10
1.0
1.0







[kmol /min].
First, we run a steady-state optimization with the following feed rate F0=0.3 [kmol/min].
This gives us the optimal values for control inputs and state variables. The optimal steady state
input values are us=h1.18 1.92 1.03 0.74 0.29 iT. The optimal state and control inputs are
used to construct regularization term added to the objective function (20). Now, the regularized stage
becomes
Jm=pFF0+pVVB−pBB−pDD+(z−xs)TQ1(z−xs)+(v−us)TQ2(v−us). (21)
The weights Q1and Q2are selected to make the rotated stage cost of the steady state problem strongly
convex, for details see [22]. This is done to obtain an economic NMPC controller that is stable.
Figure 2. Diagram of continuously-stirred tank reactor (CSTR) and distillation column.

Processes 2017,5, 8 11 of 18
Table 1. Reaction kinetics parameters.
Reaction Reaction Rate Constant (min−1) Activation Energy (in J/mol)
A → B 1×1086×104
Table 2. Distillation Column A parameters.
Parameter Value
αAB 1.5
number of stages 41
feed stage location 21
The stage cost of the economic objective function to optimize under operation is:
J=pFF0+pVVB−pBB, (20)
where
pF
is the feed cost,
pV
is the steam cost and
pB
is the product price. The price setting is
pF=1 $/kmol,pV=0.02 $/kmol,pB=2 $/kmol
. The operational constraints are the concentration
of the bottom product (
xB≤
0.1), as well as the liquid holdup at the bottom and top of the distillation
column and in the CSTR (0.3
≤M{B,D,CST R}≤
0.7 kmol). The control inputs are reflux flow (
LT
),
boil-up flow (
VB
), feeding rate to the distillation (
F
), distillate (top) and bottom product flow rates
(Dand B). These control inputs have bound constraints as follows:







0.1
0.1
0.1
0.1
0.1







≤






LT
VB
F
D
B







≤






10
4.008
10
1.0
1.0







(kmol/min).
First, we run a steady-state optimization with the following feed rate F0=0.3 (kmol/min). This
gives us the optimal values for control inputs and state variables. The optimal steady state input
values are
us=h1.18 1.92 1.03 0.74 0.29 iT
. The optimal state and control inputs are used to
construct regularization term added to the objective function
(20)
. Now, the regularized stage becomes:
Jm=pFF0+pVVB−pBB−pDD+(z−xs)TQ1(z−xs)+(v−us)TQ2(v−us). (21)
The weights
Q1
and
Q2
are selected to make the rotated stage cost of the steady state problem
strongly convex; for details, see [
24
]. This is done to obtain an economic NMPC controller that is stable.
Secondly, we set up the NLP for calculating the predicted state variables
z
and predicted control
inputs
v
. We employ a direct collocation approach on finite elements using Lagrange collocation to
discretize the dynamics, where we use three collocation points in each finite element. By using the
direct collocation approach, the state variables and control inputs become optimization variables.
The economic NMPC case study is initialized with the steady state values for a production rate
F0=
0.29 kmol/min, such that the economic NMPC controller is effectively controlling a throughput
change from
F0=
0.29 kmol/min to
F0=
0.3 kmol/min. We simulate 150 MPC iterations, with
a sample time of 1 min. The prediction horizon of the NMPC controller is set to 30 min. This setting
results in an NLP with 10,314 optimization variables. We use CasADi [
30
] (Version 3.1.0-rc1) with
IPOPT [31] as the NLP solver. For the QPs, we use MINOS QP [32] from TOMLAB.

Processes 2017,5, 8 12 of 18
4.2. Comparison of the Open-Loop Optimization Results
In this section, we compare the solutions obtained from the path-following algorithm with
the “true” solution of the optimization problem
Pnmpc
obtained by solving the full NLP. To do this,
we consider the second MPC iteration, where the path-following asNMPC is used for the first time to
correct the one-sample ahead-prediction (in the first MPC iteration, to start up the asNMPC procedure,
the full NLP is solved twice). We focus on the interesting case where the predicted state is corrupted by
noise, such that the path-following algorithm is required to update the solution. In Figure 3, we have
plotted the difference between a selection of predicted states, obtained by applying the path-following
NMPC approaches, and the ideal NMPC approach.
Version February 3, 2017 submitted to Processes 12 of 19
0 20 40 60 80 100 120
collocation points
-4
-3
-2
-1
0
1
Difference [-]
10-4 Distillation: Top composition
pf-NMPC one-step pure-predictor 1% noise
pf-NMPC one-step predictor-corrector 1% noise
pf-NMPC four-step pure-predictor 1% noise
pf-NMPC four-step predictor-corrector 1% noise
0 20 40 60 80 100 120
collocation points
-1
0
1
2
3
4
5
6
Difference [-]
10-4 Distillation: Bottom composition
0 20 40 60 80 100 120
collocation points
-6
-4
-2
0
2
4
Difference [-]
10-4 CSTR: Concentration
0 20 40 60 80 100 120
collocation points
-2
-1
0
1
2
3
Difference [kmol]
10-3 CSTR: Holdup
Figure 3. The difference in predicted states variables between iNMPC and pf-NMPC from second
iteration.
Secondly, we set up the NLP for calculating the predicted state variables zand predicted control
inputs v. We employ a direct collocation approach on finite elements using Lagrange collocation to
discretize the dynamics, where we use three collocation points in each finite element. By using the
direct collocation approach, the state variables and control inputs become optimization variables.
The economic NMPC case study is initialized with the steady state values for a production rate
F0=0.29 kmol/min, such that the Economic NMPC controller is effectively controlling a throughput
change from F0=0.29 kmol/min to F0=0.3 kmol/min. We simulate 150 MPC iterations, with a
sample time of 1 minute. The prediction horizon of the NMPC controller is set to 30 minutes. This
setting results in an NLP with 10314 optimization variables. We use CasADi [38] (version 3.1.0-rc1)
with IPOPT [23] as NLP solver. For the QPs, we use MINOS QP [39] from TOMLAB.
4.2. Comparison of Open-loop Optimization Results
In this section we compare the solutions obtained from the pathfollowing algorithm with the
“true” solution of the optimization problem Pnmpc obtained by solving the full NLP. To do this,
we consider the second MPC iteration, where the pathfollowing asNMPC is used for the first time
to correct the one-sample ahead-prediction (in the first MPC iteration, to start up the asNMPC
procedure, the full NLP is solved twice). We focus on the interesting case where the predicted state
is corrupted by noise, such that the pathfollowing algorithm is required to update the solution.
In Figure 3we have plotted the difference between a selection of predicted states, obtained by
applying the pathfollowing NMPC approaches, and the ideal NMPC approach. We observe that
the one-step pure-predictor tracks the ideal NMPC solution worst, and the four-step pathfollowing
with predictor-corrector tracks best. This happens because the predictor-corrector pathfollowing QP
has an additional linear term in the objective function and constraint for the purpose of moving closer
to the solution of the NLP (the "corrector" component), as well as tracing the first order estimate of
the change in the solution (the "predictor"). The four-step pathfollowing performs better because a
smaller step size gives finer approximation of the parametric NLP solution.
Figure 3.
The difference in predicted state variables between ideal NMPC (iNMPC) and path-following NMPC
(pf-NMPC) from the second iteration.
We observe that the one-step pure-predictor tracks the ideal NMPC solution worst and
the four-step path-following with predictor-corrector tracks best. This happens because the
predictor-corrector path-following QP has an additional linear term in the objective function and
constraint for the purpose of moving closer to the solution of the NLP (the “corrector” component),
as well as tracing the first-order estimate of the change in the solution (the “predictor”). The four-step
path-following performs better because a smaller step size gives finer approximation of the parametric
NLP solution.
This is also reflected in the average approximation errors given in Table 3. The average approximation
error has been calculated by averaging the error one-norm
χpath−f ol lowin g −χideal NMPC 1
over all
MPC iterations.
We observe that in this case study, the accuracy of a single predictor-corrector step is almost as
good as performing four predictor-corrector steps along the path. That is, a single predictor-corrector
QP update may be sufficient for this application. However, in general, in the presence of larger noise
magnitudes and longer sampling intervals, which cause poorer predictions, a single-step update may
no longer lead to good approximations. We note the large error in the pure-predictor path-following
method for the solution accuracy of several orders of magnitude.

Processes 2017,5, 8 13 of 18
On the other hand, given that the optimization vector
χ
has dimension 10,164 for our case
study, the average one-norm approximation error of ca. 4.5 does result in very small errors on the
individual variables.
Table 3.
Approximation error using path-following algorithms. asNMPC, advanced-step NMPC
(asNMPC); QP, quadratic programming.
Average Approximation Error between ideal NMPC and Path-Following (PF) asNMPC
PF with predictor QP, 1 step 4.516
PF with predictor QP, 4 steps 4.517
PF with predictor-corrector QP, 1 step 1.333 ×10−2
PF with predictor-corrector QP, 4 steps 1.282 ×10−2
4.3. Closed-Loop Results: No Measurement Noise
In this section, we compare the results for closed loop process operation. We consider first the case
without measurement noise, and we compare the results for ideal NMPC with the results obtained
by the path-following algorithm with the pure-predictor QP
(10)
and the predictor-corrector QP
(12)
.
Figure 4shows the trajectories of the top and bottom composition in the distillation column and the
reactor concentration and holdup. Note that around 120 min, the bottom composition constraint in the
distillation column becomes active, while the CSTR holdup is kept at its upper bound all of the time
(any reduction in the holdup will result in economic and product loss).
Version February 3, 2017 submitted to Processes 14 of 19
0 50 100 150
Time [minutes]
0.8
0.85
0.9
0.95
Concentration [-]
Distillation: Top composition
iNMPC
pf-NMPC one-step pure-predictor
pf-NMPC one-step predictor-corrector
pf-NMPC four-step pure-predictor
pf-NMPC four-step predictor-corrector
0 50 100 150
Time [minutes]
0
0.05
0.1
0.15
Concentration [-]
Distillation: Bottom composition
0 50 100 150
Time [minutes]
0.65
0.66
0.67
0.68
0.69
0.7
Concentration [-]
CSTR: Concentration
0 50 100 150
Time [minutes]
0.65
0.7
0.75
Holdup [kmol]
CSTR: Holdup
Figure 4. Recycle composition, bottom composition, reactor concentration, and reactor holdup.
0 50 100 150
Time [minutes]
1.1
1.2
1.3
1.4
LT [m3/minute]
reflux flow (LT)
iNMPC
pf-NMPC one-step pure-predictor
pf-NMPC one-step predictor-corrector
pf-NMPC four-step pure-predictor
pf-NMPC four-step predictor-corrector
0 50 100 150
Time [minutes]
0.7
0.75
0.8
0.85
0.9
D [m3/minute]
recycle flowrate (D)
0 50 100 150
Time [minutes]
0.2
0.25
0.3
0.35
B [m3/minute]
bottom flowrate (B)
0 50 100 150
Time [minutes]
1.9
2
2.1
2.2
VB [m3/minute]
boilup flow (VB)
0 50 100 150
Time [minutes]
1
1.05
1.1
1.15
1.2
F [m3/minute]
feeding rate (F)
Figure 5. Optimized control inputs.
Figure 4. Recycle composition, bottom composition, reactor concentration and reactor holdup.
In this case (without noise), the prediction and the true solution only differ due to numerical
noise. There is no need to update the prediction, and all approaches give exactly the same closed-loop
behavior. This is also reflected in the accumulated stage cost, which is shown in Table 4.

Processes 2017,5, 8 14 of 18
Table 4. Comparison of economic NMPC controllers (no noise). Accumulated stage cost in $.
Economic NMPC Controller Accumulated Stage Cost
iNMPC −296.42
pure-predictor QP:
pf-NMPC one step −296.42
pf-NMPC four steps −296.42
predictor-corrector QP:
pf-NMPC one step −296.42
pf-NMPC four steps −296.42
The closed-loop control inputs are given in Figure 5. Note here that the feed rate into the
distillation column is adjusted such that the reactor holdup is at its constraint all of the time.
Version February 3, 2017 submitted to Processes 14 of 19
0 50 100 150
Time [minutes]
0.8
0.85
0.9
0.95
Concentration [-]
Distillation: Top composition
iNMPC
pf-NMPC one-step pure-predictor
pf-NMPC one-step predictor-corrector
pf-NMPC four-step pure-predictor
pf-NMPC four-step predictor-corrector
0 50 100 150
Time [minutes]
0
0.05
0.1
0.15
Concentration [-]
Distillation: Bottom composition
0 50 100 150
Time [minutes]
0.65
0.66
0.67
0.68
0.69
0.7
Concentration [-]
CSTR: Concentration
0 50 100 150
Time [minutes]
0.65
0.7
0.75
Holdup [kmol]
CSTR: Holdup
Figure 4. Recycle composition, bottom composition, reactor concentration, and reactor holdup.
0 50 100 150
Time [minutes]
1.1
1.2
1.3
1.4
LT [m3/minute]
reflux flow (LT)
iNMPC
pf-NMPC one-step pure-predictor
pf-NMPC one-step predictor-corrector
pf-NMPC four-step pure-predictor
pf-NMPC four-step predictor-corrector
0 50 100 150
Time [minutes]
0.7
0.75
0.8
0.85
0.9
D [m3/minute]
recycle flowrate (D)
0 50 100 150
Time [minutes]
0.2
0.25
0.3
0.35
B [m3/minute]
bottom flowrate (B)
0 50 100 150
Time [minutes]
1.9
2
2.1
2.2
VB [m3/minute]
boilup flow (VB)
0 50 100 150
Time [minutes]
1
1.05
1.1
1.15
1.2
F [m3/minute]
feeding rate (F)
Figure 5. Optimized control inputs.
Figure 5. Optimized control inputs.
4.4. Closed-Loop Results: With Measurement Noise
Next, we run simulations with measurement noise on all of the holdups in the system. The noise
is taken to have a normal distribution with zero mean and a variance of one percent of the
steady state values. This will result in corrupted predictions that have to be corrected for by the
path-following algorithms. Again, we perform simulations with one and four steps of pure-predictor
and predictor-corrector QPs.
Figure 6shows the top and bottom compositions of the distillation column, together with the
concentration and holdup in the CSTR. The states are obtained under closed-loop operation with the
ideal and path-following NMPC algorithms. Due to noise, it is not possible to avoid the violation of
the active constraints in the holdup of the CSTR and the bottom composition in the distillation column.
This is the case for both the ideal NMPC and the path-following approaches.

Processes 2017,5, 8 15 of 18
Version February 3, 2017 submitted to Processes 15 of 19
0 50 100 150
Time [minute]
0.8
0.85
0.9
0.95
Concentration [-]
Distillation: Top composition
iNMPC 1% noise
pf-NMPC one-step pure-predictor 1% noise
pf-NMPC one-step predictor-corrector 1% noise
pf-NMPC four-step pure-predictor 1% noise
pf-NMPC four-step predictor-corrector 1% noise
0 50 100 150
Time [minutes]
0
0.05
0.1
0.15
Concentration [-]
Distillation: Bottom composition
0 50 100 150
Time [minutes]
0.65
0.66
0.67
0.68
0.69
0.7
Concentration [-]
CSTR: Concentration
0 50 100 150
Time [minutes]
0.5
0.55
0.6
0.65
0.7
0.75
Holdup [kmol]
CSTR: Holdup
Figure 6. Recycle composition, bottom composition, reactor concentration, and reactor holdup.
Table 5. Comparison of economic NMPC controllers. Accumulated stage cost is in $.
economic NMPC controller Acc. stage cost
iNMPC -296.82
pure-predictor QP:
pf-NMPC one step -297.54
pf-NMPC four steps -297.62
predictor-corrector QP:
pf-NMPC one step -296.82
pf-NMPC four steps -296.82
4.4. Closed-loop Results – With Measurement Noise
Next, we run simulations with measurement noise on all the holdups in the system. The
noise is taken to have a normal distribution with zero mean and a variance of one percent of the
steady state values. This will result in corrupted predictions that have to be corrected for by the
pathfollowing algorithms. Again, we perform simulations with one and four steps of pure-predictor
and predictor-corrector QPs.
Figure 6shows the top and bottom compositions of the distillation column, together with the
concentration and holdup in the CSTR. The states are obtained under closed-loop operation with the
ideal and pathfollowing NMPC algorithms. Due to noise it is not possible to avoid violation of the
active constraints in the holdup of the CSTR and the bottom composition in the distillation column.
This is the case for both the ideal NMPC and the pathfollowing approaches.
The input variables shown in Figure 7are also reflecting the measurement noise, and again we
see that the fast sensitivity NMPC approaches are very close to the ideal NMPC inputs.
Finally, we compare the accumulated economic stage cost in Table 5. Here we observe that our
proposed predictor-corrector pathfollowing algorithm performs identically to the ideal NMPC. This
is as expected, since the predictor-corrector pathfollowing algorithm is trying to reproduce the true
NLP solution. Interestingly, in this case, the larger error in the pure predictor pathfollowing NMPC
Figure 6. Recycle composition, bottom composition, reactor concentration and reactor holdup.
The input variables shown in Figure 7are also reflecting the measurement noise, and again, we
see that the fast sensitivity NMPC approaches are very close to the ideal NMPC inputs.
Version February 3, 2017 submitted to Processes 16 of 19
0 50 100 150
Time [minutes]
1.1
1.2
1.3
1.4
LT [m3/minute]
reflux flow (LT)
iNMPC
pf-NMPC one-step pure-predictor
pf-NMPC one-step predictor-corrector
pf-NMPC four-step pure-predictor
pf-NMPC four-step predictor-corrector
0 50 100 150
Time [minutes]
0.7
0.8
0.9
D [m3/minute]
recycle flowrate (D)
0 50 100 150
Time [minutes]
0.2
0.25
0.3
0.35
B [m3/minute]
bottom flowrate (B)
0 50 100 150
Time [minutes]
1.9
2
2.1
2.2
VB [m3/minute]
boilup flow (VB)
0 50 100 150
Time [minutes]
1
1.1
1.2
F [m3/minute]
feeding rate (F)
Figure 7. Optimized control inputs.
leads to a better economic performance of the closed loop system. This behavior is due to the fact
that the random measurement noise can have positive and negative effect on the operation, which is
not taken into account by the ideal NMPC (and also the predictor-corrector NMPC). In this case, the
inaccuracy of the pure predictor pathfollowing NMPC led to better economic performance in closed
loop. But it could also have been the opposite.
5. Discussion and Conclusion
We applied the pathfollowing ideas developed in Jäschke et al. [22] and Kungurtsev and
Diehl [30] to a large-scale process containing a reactor, a distillation column and a recycle stream.
Compared with single-step updates based on solving a linear system of equations as proposed by [14]
our pathfollowing approach requires somewhat more computational effort. However, the advantage
of the pathfollowing approach is that active set changes are handled rigorously. Moreover, solving a
sequence of a few QPs can be expected to be much faster than solving the full NLP, especially since
they can be initialized very well, such that the computational delay between obtaining the new state
and injecting the updated input into the plant is still sufficiently small. In our computations, we have
considered a fixed step-size for the pathfollowing, such that the number of QPs to be solved is known
in advance.
The case without noise does not require the pathfollowing algorithm to correct the solution,
because the prediction and the true measurement are identical, except for numerical noise. However,
when measurement noise is added to the holdups, the situation becomes different. In this case, the
prediction and the measurements differ, such that an update is required. All four approaches track the
ideal NMPC solution to some degree, however, in terms of accuracy the predictor-corrector performs
consistently better. Given that the pure sensitivity QP and the predictor-corrector QP are very similar
in structure, it is recommended to use the latter in the pathfollowing algorithm, especially for highly
nonlinear processes and cases with significant measurement noise.
We have presented basic algorithms for pathfollowing, and they seem to work well for the
cases we have studied, such that the pathfollowing algorithms do not diverge from the true
solution. In principle, however, the path-following algorithms may get lost, and more sophisticated
implementations need to include checks and safeguards. We note, however, that the application of
Figure 7. Optimized control inputs.
Finally, we compare the accumulated economic stage cost in Table 5.

Processes 2017,5, 8 16 of 18
Table 5. Comparison of economic NMPC controllers (with noise). Accumulated stage cost in $.
Economic NMPC Controller Accumulated Stage Cost
iNMPC −296.82
pure-predictor QP:
pf-NMPC one step −297.54
pf-NMPC four steps −297.62
predictor-corrector QP:
pf-NMPC one step −296.82
pf-NMPC four steps −296.82
Here, we observe that our proposed predictor-corrector path-following algorithm performs
identically to the ideal NMPC. This is as expected, since the predictor-corrector path-following
algorithm is trying to reproduce the true NLP solution. Interestingly, in this case, the larger error in
the pure predictor path-following NMPC leads to a better economic performance of the closed loop
system. This behavior is due to the fact that the random measurement noise can have a positive and
a negative effect on the operation, which is not taken into account by the ideal NMPC (and also the
predictor-corrector NMPC). In this case, the inaccuracy of the pure-predictor path-following NMPC
led to better economic performance in the closed loop. However, it could also have been the opposite.
5. Discussion and Conclusions
We applied the path-following ideas developed in Jäschke et al. [
24
] and Kungurtsev and Diehl [
28
]
to a large-scale process containing a reactor, a distillation column and a recycle stream. Compared
with single-step updates based on solving a linear system of equations as proposed by [
10
], our
path-following approach requires somewhat more computational effort. However, the advantage
of the path-following approach is that active set changes are handled rigorously. Moreover, solving
a sequence of a few QPs can be expected to be much faster than solving the full NLP, especially since
they can be initialized very well, such that the computational delay between obtaining the new state
and injecting the updated input into the plant is still sufficiently small. In our computations, we have
considered a fixed step-size for the path-following, such that the number of QPs to be solved is known
in advance.
The case without noise does not require the path-following algorithm to correct the solution,
because the prediction and the true measurement are identical, except for numerical noise. However,
when measurement noise is added to the holdups, the situation becomes different. In this case, the
prediction and the measurements differ, such that an update is required. All four approaches track the
ideal NMPC solution to some degree; however, in terms of accuracy, the predictor-corrector performs
consistently better. Given that the pure sensitivity QP and the predictor-corrector QP are very similar
in structure, it is recommended to use the latter in the path-following algorithm, especially for highly
nonlinear processes and cases with significant measurement noise.
We have presented basic algorithms for path-following, and they seem to work well for the
cases we have studied, such that the path-following algorithms do not diverge from the true
solution. In principle, however, the path-following algorithms may get lost, and more sophisticated
implementations need to include checks and safeguards. We note, however, that the application of the
path-following algorithm in the advanced-step NMPC framework has the desirable property that the
solution of the full NLP acts as a corrector, such that if the path-following algorithm diverges from the
true solution, this will be most likely for only one sample time, until the next full NLP is solved.
The path-following algorithm in this paper (and the corresponding QPs) still relies on the
assumption of linearly-independent constraint gradients. If there are path-constraints present in
the discretized NLP, care must be taken to formulate them in such a way that LICQ is not violated.

Processes 2017,5, 8 17 of 18
In future work, we will consider extending the path-following NMPC approaches to handle more
general situations with linearly-dependent inequality constraints.
Acknowledgments:
Vyacheslav Kungurtsev was supported by the Czech Science Foundation Project 17-26999S.
Eka Suwartadi and Johannes Jäschke are supported by the Research Council of Norway Young Research Talent
Grant 239809.
Author Contributions:
V.K. and J.J. contributed the algorithmic ideas for the paper; E.S. implemented the
algorithm and simulated the case study; E.S. primarily wrote the paper, with periodic assistance from V.K.; J.J.
supervised the work, analyzed the simulation results and contributed to writing and correcting the paper.
Conflicts of Interest: The authors declare no conflict of interest.
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