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Chapter 1
Introduction to Model Predictive Control

Please cite the book:

Maciej Ławryńczuk: Nonlinear Predictive Control Using Wiener Models:
Computationally Efficient Approaches for Polynomial and Neural Structures.
Studies in Systems, Decision and Control, vol. 389, Springer, Cham, 2022.

Abstract This Chapter is an introduction to the field of MPC. Its basic idea and the
rudimentary MPC optimisation problems are defined, at first for Single-Input Single-
Output (SISO) processes and next for Multiple-Input Multiple-Output (MIMO) ones.
A method to cope with infeasibility problems caused by constraints imposed on the
predicted controlled variables is presented. Next, parameterisation of the decision
variables using Laguerre functions in order to reduce the number of actually opti-
mised variables is described. Classification of MPC algorithms is given and com-
putational complexity issues are discussed. Finally, some example applications of
MPC algorithms in different fields are reported.

1.1 Formulation of the Basic MPC Problem

The objective of a good control algorithm is to calculate repeatedly on-line the value
of the manipulated variable (or the values of many manipulated variables) that leads
to good process behaviour [36]. Let us discuss the term good process behaviour
using two examples.
The first process example is a residential building equipped with an underfloor
radiant heating system based on electric heating foils [99]. From the point of view
of control engineering, the process is very simple since it has only one manipulated
variable (process input) which is the value of the current (or the voltage) applied
to the foils and only one controlled variable (process output) which is the average
temperature inside the building. There are two objectives of the controller:

3
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4 1 Introduction to Model Predictive Control

a) it must increase the temperature quickly when the user increases the temperature
set-point, i.e. the value of the required temperature,
b) it must stabilise the temperature when the outside temperature drops.
The first objective is set-point tracking, i.e. the process output must follow changes of
its set-point. The second objective is compensation of disturbances, i.e. the process
output must be (approximately) constant when the process is affected by external
disturbances, also called uncontrolled process inputs. In our simple example, it is
only possible to increase the temperature by increasing the current (or the voltage),
but it is impossible to reduce the temperature. It means that it works fine in the
two above situations, but when the user wants to reduce the set-point or the outside
temperature increases, the only possible action is to reduce heating, switch it off or
simply ventilate the building. Of course, in more advanced solutions, it is possible
to both heat and cool. Furthermore, it may be necessary to stabilise not only tem-
perature but also humidity. An important application of such a control system may
be found in greenhouses, where it is necessary to maintain constant temperature and
humidity values for the proper growth of plants. Different parts of the greenhouse
may be heated separately to obtain different local temperature conditions. In such
a case, there are many manipulated, controlled and disturbance variables. In addi-
tion to set-point tracking and compensation of disturbances, the calculated values of
the manipulated signals must satisfy some constraints. Typically, they have limited
values and rates of change caused by the physical limits of actuators. Moreover,
one may imagine that some constraints are imposed on the controlled variables, e.g.
temperature and humidity should be in some ranges.
The second process example is a car. Its control is significantly much more
complicated than the simple temperature control task discussed above. It is because
a driver must manipulate numerous variables, such as the accelerator, clutch and
brake pedals, the wheel and the gear lever. There are many controlled variables,
such as position on the road, speed, acceleration. The driver controls the car in
such a way that position, speed and acceleration set-point trajectories are followed.
Moreover, the influence of many external disturbances is compensated, e.g. variable
road slope, type of surface, side wind. Unlike the first process example, the driver
not only controls the process but also calculates the set-point trajectories on-line, i.e.
adjusts them to the current road conditions. Of course, there are numerous constraints
which must be taken into account during calculation of the values of the manipulated
variables and adjusting the trajectories. Both manipulated and controlled variables
must be constrained in this example.
The classical Proportional-Integral-Derivative (PID) controller in continuous-
time domain is described by the following rule

1 de(t)
 ∫ t 
u(t) = u0 + K e(t) + e(τ)dτ + Td (1.1)
Ti 0 dt

The control error is defined as the difference between the set-point and the current
measured value of the controlled variable, i.e. e(t) = y sp (t) − y(t). The value of
the manipulated variable u for the current time t is a linear function of three parts:
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1.1 Formulation of the Basic MPC Problem 5

the proportional part, which takes into account the current control error, e(t), the
integral part, which takes into account the past errors, and the derivative part, which
takes into account the rate of change of the error. The tuning parameters are: the
proportional gain K, the integration time-constant Ti and the derivative time-constant
Td . Using Euler’s backward differentiation and trapezoidal integration, in discrete-
time domain, the value of the manipulated variable for the current sampling instant
k is
u(k) = u(k − 1) + r0 e(k) + r1 e(k − 1) + r2 e(k − 2) (1.2)
where e(k), e(k − 1) and e(k − 2) denote the values of the control error at the
sampling instants k, k − 1 and k − 2, respectively, u(k − 1) is the value of the
manipulated variable at the sampling instant k − 1, r0 , r1 , r2 are parameters. They are
calculated for the settings K, Ti , Td and the chosen sampling time of the controller.
If properties of the process are (approximately) linear, the PID controller proves to
be very efficient in numerous applications. Nevertheless, the PID controller has the
following limitations:
1. The PID control law (1.1) or (1.2) is linear. In the case of nonlinear processes,
the possible quality of control may be not satisfactory, in particular when the
set-point changes are significant and fast or the external disturbances are strong.
2. The PID controller works fine when the process delay is not significant. Con-
versely, PID control of delayed dynamical systems is usually not good.
3. In its basic version, the PID controller does not include constraints. Although
simple limiters may easily enforce limits of the manipulated variable and con-
straints on its rate of change, there is no systematic way to enforce satisfaction of
constraints imposed on the controlled variable.
4. The PID controller is a natural choice when the controlled process has one ma-
nipulated variable and one controlled one. In the case of a dynamical process with
many inputs and many outputs, the basic problem is finding out which manipu-
lated variable has the strongest influence on each controlled one. Next, several
classical single-loop PID controllers are used. Such an approach works correctly
when the consecutive manipulated variables strongly impact the consecutive con-
trolled ones, but when one process input impacts two or more outputs, such a
control structure does not work. Moreover, the number of process inputs and
outputs must be equal.
5. It is interesting that the current value of the manipulated variable generated by
the PID controller depends on the current and past errors. It is clear when we
consider the discrete-time implementation (1.2). The derivative part tries to use
some information of the future control error but using only its current and previous
measurements.
6. The PID controller is tuned in practice using some simple rules, e.g. the famous
Ziegler and Nichols procedure, or simply by the trial and error approach. Although
interpretation of the continuous-time parameters K, Ti and Td is straightforward,
the parameters r0 , r1 and r2 of the discrete-time controller have no physical
interpretation.
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6 1 Introduction to Model Predictive Control

Having discussed the objectives of a good control algorithm and properties of

the PID structure, we will discuss the basic formulation of MPC. Let us recall the
problem of controlling a car by a driver. Humans do not use mathematical equations to
calculate values of the manipulated variables. Conversely, in our mind, we repeatedly
do the following:
1. We collect all possible information, i.e. we observe the road and monitor the car
dashboard.
2. Using a model of the car, i.e. knowing how the car reacts, we predict behaviour
of the car, i.e. its position, speed, acceleration, over some time horizon.
3. We optimise behaviour of the car, i.e. we find out how the car should be controlled
in order to satisfy all control objectives. We find not only the current values of
the manipulated variables, but we also assess their future values.
4. Prediction of the future car state, as well as optimisation of the current and future
control actions, are coupled, i.e. we have many possible control policies, we assess
how they are successful and we choose the best one.
5. We constantly repeat the above steps as we receive new information, we assess
the results of our actions and how the disturbances change. The traffic and road
conditions are never constant. The horizon is moved each time we start prediction
and optimisation.
Fig. 1.1 illustrates the above. Let us consider information collected by the driver A
(measurements) and the decisions taken for three different time instants denoted as
t1 , t2 and t3 , respectively. Initially, at time t1 , for the prediction horizon used, the
driver A is able to see the speed limit sign and his or her decision is to deduce
speed to 50 km/h. The prediction horizon is too short to notice the cars B and C.
Next, at time t2 , the prediction horizon makes it possible to notice the car B that
is approaching from the right-side road. Because the car B moves very slowly, the
driver A decides to continue driving with constant speed, he or she does not wait for
the car B to give way to it. For the prediction horizon used, the driver A does not
notice the car C. Finally, at time t3 , the driver A sees the car C. He or she is unable
to overtake it since the car D moves from the opposite direction, in the second lane.
Probably, provided that no other obstacles exist, overtaking will be possible shortly.
Let us point out that all decisions are made using predictions of future behaviour of
all drivers, possible drivers’ actions are predicted using some models, all existing
constraints are taken into account.
Now, it is time to formulate the basic MPC problem using mathematics. At first,
let us consider a SISO process. The input of the controlled process is denoted by u,
the output is denoted by y. At each consecutive sampling instant k, k = 1, 2, 3, . . .,
the vector of the future increments of the manipulated variable
 4u(k |k) 
.
 
4u(k) =  .. (1.3)
 

 4u(k + Nu − 1|k) 
 
 
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1.1 Formulation of the Basic MPC Problem 7

Fig. 1.1 Situations on the road and the driver’s A decisions for three example time instants t1 , t2 , t3
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8 1 Introduction to Model Predictive Control

is calculated on-line. The symbol 4u(k + p|k) denotes the increment of the manipu-
lated variable for the sampling instant k + p calculated at the current sampling instant
k, Nu is the control horizon which defines the number of decision variables (1.3).
The first increment is
4u(k |k) = u(k |k) − u(k − 1) (1.4)
and the following ones are

4u(k + p|k) = u(k + p|k) − u(k + p − 1|k) (1.5)

for p = 1, . . . , Nu − 1. The symbol u(k + p|k) denotes the value of the manipulated
variable for the sampling instant k + p calculated at the current sampling instant k,
u(k − 1) is the value of the manipulated variable used (applied to the process) at the
previous sampling instant. In the simplest case, the vector of decision variables (1.3)
is calculated on-line from an unconstrained optimisation problem

min {J(k)} (1.6)

4u(k)

Typically, the minimised objective function (the cost-function) consists of two parts
N Nu −1

(y sp (k + p|k) − ŷ(k + p|k))2 + λ (4u(k + p|k))2

Õ Õ
J(k) = (1.7)
p=1 p=0

The first part of the MPC cost-function measures the predicted quality of control
since the differences between the set-point trajectory and the predicted trajectory
of the output variable (i.e. the predicted control errors) over the prediction horizon
N ≥ Nu are taken into account. The set-point value for the sampling instant k + p
known at the current sampling instant k is denoted by y sp (k + p|k), the predicted
value of the output variable for the sampling instant k + p calculated at the current
instant is denoted by ŷ(k + p|k). The future values of the set-point are usually
not known, hence only the scalar set-point value for the current sampling instant,
denoted by y sp (k), is used, i.e. y sp (k + 1|k) = . . . = y sp (k + N |k) = y sp (k). Such
an approach is typically used in control of industrial processes in which changes
of the set-point are very rare, but the controller must compensate for changes of
the disturbances. However, in some applications, e.g. in autonomous vehicles and
robotics, the set-point trajectory may be not constant over the prediction horizon.
The second part of the MPC cost-function is a penalty term. It is used to reduce
excessive changes of the manipulated variable; λ > 0 is a weighting coefficient. The
greater its value, the lower the increments of the manipulated variable and, hence, the
slower control. Because in practice the control horizon is shorter than the prediction
one, it is assumed that u(k + p|k) = u(k + Nu − 1|k) for p = Nu, . . . , N, which means
that 4u(k + Nu |k) = . . . = 4u(k + N |k) = 0.
Although at each sampling instant as many as Nu future increments of the ma-
nipulated variable (1.3) are calculated, only the first element of this sequence is
actually applied to the process, i.e. the increment for the current sampling instant k.
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1.1 Formulation of the Basic MPC Problem 9

Fig. 1.2 The general structure of the MPC algorithm

Let the optimal vector calculated from the MPC optimisation problem be denoted
by 4u opt (k). The current optimal value of the manipulated variable is applied to the
process
u(k) = 4uopt (k |k) + u(k − 1) (1.8)
where 4uopt (k |k) is the first element of the vector 4u opt (k). In the next sampling
instant (k +1) the output value of the process is measured (the state variables may also
be measured or estimated), the prediction horizon is shifted one step forward and the
whole procedure described above is repeated. As a result, the MPC algorithm works
in the closed-loop, i.e. with feedback from the measured process output. Fig. 1.2
depicts the general structure of the MPC algorithm. It is assumed that the time
necessary to solve the MPC optimisation problem is much shorter than the sampling
time.
In practical applications, it is necessary to take into account existing constraints.
First of all, the magnitude of the manipulated variable may be constrained. Such
constraints result from the physical limits of the actuator

umin ≤ u(k + p|k) ≤ umax, p = 0, . . . , Nu − 1 (1.9)

where umin and umax are the minimal and maximal values of the manipulated variable,
respectively. It is interesting to notice the fact that all calculated values of the
manipulated variable over the whole control horizon are limited, not only the value
for the current sampling instant, i.e. u(k |k). Secondly, the rate of change of the
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10 1 Introduction to Model Predictive Control

manipulated variable may be constrained

4umin ≤ 4u(k + p|k) ≤ 4umax, p = 0, . . . , Nu − 1 (1.10)

where 4umin and 4umax are the maximal negative and maximal (positive) changes
of the manipulated variable, respectively (usually 4umin = −4umax ). All calculated
increments of the manipulated variable over the whole control horizon are limited,
not only the increment for the current sampling instant, i.e. 4u(k |k). Thirdly, the
predicted values of the process output variable may also be limited, which is usually
enforced by some technological reasons

y min ≤ ŷ(k + p|k) ≤ y max, p = 1, . . . , N (1.11)

where y min and y max are the minimal and maximal values of the predicted output
variable, respectively. All predictions over the prediction horizon N are constrained.
When the constraints are present, the vector of decision variables (1.3) is calculated
at each sampling instant from an optimisation problem in which the cost-function
(1.7) is minimised and all the constraints (1.9), (1.10) and (1.11) are taken into
account. Hence, the rudimentary MPC constrained optimisation problem is
 N Nu −1 
(y sp (k + p|k) − ŷ(k + p|k))2 + λ (4u(k + p|k))2
Õ Õ
min

 

J(k) =
4u(k)  
 p=1 p=0 
subject to (1.12)
umin ≤ u(k + p|k) ≤ umax, p = 0, . . . , Nu − 1
4umin ≤ 4u(k + p|k) ≤ 4umax, p = 0, . . . , Nu − 1
y min ≤ ŷ(k + p|k) ≤ y max, p = 1, . . . , N

The number of decision variables of the optimisation problem (1.12) is Nu , the

number of constraints is 4Nu + 2N.
All things considered, in the case of the SISO constrained MPC algorithm, at
each sampling instant k, the following steps are performed on-line:
1. The current value of the controlled variable, y(k), is measured; the state variables
may be measured or estimated when necessary.
2. The future sequence of increments of the manipulated variable is calculated from
the optimisation problem (1.12).
3. The first element of the determined sequence is applied to the process (Eq. (1.8)).
Having discussed the MPC formulation for the SISO case, we will concentrate on
a more general MIMO problem. Let us assume that the number of process inputs is
denoted by nu and the number of process outputs is denoted by ny . In this book we
use two notation methods: scalars and vectors. When possible, it is very convenient
to use vectors, but sometimes the consecutive scalar signals must be used. The vector
T
of manipulated variables is u = u1 . . . unu and the vector of controlled variables

T
is y = y1 . . . yny . The vector of decision variables of the MPC algorithm (1.3)

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1.1 Formulation of the Basic MPC Problem 11

is hence of length nu Nu . The minimised MPC cost-function for the MIMO case is
ny
N Õ
Õ
sp
2
J(k) = µ p,m ym (k + p|k) − ŷm (k + p|k)
p=1 m=1
Nu −1 Õ
nu
λ p,n (4un (k + p|k))2
Õ
+ (1.13)
p=0 n=1

In comparison with the SISO case (Eq. (1.7)), in the first part of the cost-function
(1.13), we consider the predicted control errors for all ny controlled variables over
the whole prediction horizon. Similarly, in the second part of the cost-function,
increments of all nu manipulated variables are taken into account over the whole
control horizon. The weighting coefficients µ p,m ≥ 0 make it possible to differentiate
the influence of the predicted control errors of the consecutive outputs within the
prediction horizon. The coefficients λ p,n > 0 are used not only to differentiate the
influence of the control increments of the consecutive inputs of the process within
the control horizon but to establish the necessary scale between both parts of the
cost-function.
The MPC cost-function and the resulting optimisation problems may be con-
veniently and compactly derived, formulated and implemented using vector-matrix
notation rather than scalars. The cost-function (1.13) may be expressed in the fol-
lowing form
N Nu −1
sp
p|k)k 2M p k4u(k + p|k)k 2Λ p
Õ Õ
J(k) = k y (k + p|k) − ŷ(k + + (1.14)
p=1 p=0

Now, the set-point vector for the sampling instant k + p known at the current sampling
instant k is denoted by y sp (k + p|k), the predicted vector of the output variables for
the sampling instant k + p calculated at the current sampling instant k is denoted by
ŷ(k + p|k), both vectors are of length ny . The matrix M p = diag(µ p,1, . . . , µ p,ny ) ≥
0 is of dimensionality ny × ny , the matrix Λ p = diag(λ p,1, . . . , λ p,nu ) > 0 is of
dimensionality nu × nu .
For the process with nu manipulated variables, the magnitude constraints are

umin max
n ≤ un (k + p|k) ≤ un , p = 0, . . . , Nu − 1, n = 1, . . . , nu (1.15)

where umin max

n and un are the minimal and maximal values of the manipulated variable
un , respectively. The constraints imposed on the rate of change of the manipulated
variables are

4umin max
n ≤ 4un (k + p|k) ≤ 4un , p = 0, . . . , Nu − 1, n = 1, . . . , nu (1.16)

where 4umin
n and 4un
max are the maximal negative and maximal (positive) changes of

the manipulated variable un , respectively. The constraints imposed on the predicted

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12 1 Introduction to Model Predictive Control

values of the process output variables are

min max
ym ≤ ŷm (k + p|k) ≤ ym , p = 1, . . . , N, m = 1, . . . , ny (1.17)

where ym min and y max are the minimal and maximal values of the predicted variable
m
ym , respectively. If we use the vector notation, the constraints are defined by the
following vectors of length nu
 umin   umax   4umin   4umax 
 1   1   1   1 
umin =  ...  , umax =  ...  , 4umin =  ...  , 4umax =  ...  (1.18)
       
 umin   umax   4umin   4umax 
       
 nu   nu   nu   nu 
and the following vectors of length ny

 y min   y max 
 1   1 
 ..  max  .. 
y min =  . , y = .  (1.19)
 min   max 
 yn   yn 
 y   y 
We may notice that the above 3 scalar constraints given by Eqs. (1.15), (1.16) and
(1.17) may be rewritten in the same way it is done for the SISO case, i.e. by Eqs.
(1.9), (1.10) and (1.11).
Now we may formulate the general MPC optimisation problem for MIMO pro-
cesses. Using the cost-function (1.14), the scalar constraints (1.15), (1.16), (1.17)
and the definitions (1.18)-(1.19), we have
N Nu −1
( )
sp 2 2
Õ Õ
min J(k) = k y (k + p|k) − ŷ(k + p|k)k M p + k4u(k + p|k)k Λ p
4u(k)
p=1 p=0

subject to (1.20)
min max
u ≤ u(k + p|k) ≤ u , p = 0, . . . , Nu − 1
min
4u ≤ 4u(k + p|k) ≤ 4umax, p = 0, . . . , Nu − 1
y min ≤ ŷ(k + p|k) ≤ y max, p = 1, . . . , N

where the norm is defined as k xk 2A = x T Ax (the matrix A is square). The above

optimisation problem corresponds with the task (1.12) for the SISO case. The number
of decision variables of the optimisation problem (1.20) is nu Nu , the number of
constraints is 4nu Nu + 2ny N.
Although at each sampling instant as many as nu Nu future increments of the
manipulated variables (1.3) are calculated, only the first nu elements of this sequence
are actually applied to the process, i.e. the increments for the current sampling instant
k. The current optimal values of the manipulated variables applied to the process
are calculated from Eq. (1.8), the same which is used in the SISO case, but now all
vectors, i.e. u(k), 4uopt (k |k) and u(k − 1), are of length nu .
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1.1 Formulation of the Basic MPC Problem 13

In the case of the MIMO constrained MPC algorithm, at each sampling instant k
the following steps are performed on-line:
1. The current values of the controlled variables, y1 (k), . . . , yny (k), are measured;
the state variables may be measured or estimated when necessary.
2. The future sequence of increments of the manipulated variables is calculated from
the optimisation problem (1.20).
3. The first nu elements of the determined sequence are applied to the process (Eq.
(1.8)).
Now, let us find a more compact representation of the rudimentary MIMO MPC
optimisation problem (1.20). Let us define the set-point trajectory vector
 y sp (k + 1|k) 
..
 
y sp (k) =  . (1.21)
 

 y sp (k + N |k) 
 
 
and the predicted output trajectory vector
 ŷ(k + 1|k) 
..
 
ŷ(k) =  . (1.22)
 

 
 ŷ(k + N |k) 
 
Both vectors are of length ny N. The MPC cost-function (1.14) may be rewritten in
the following compact form

J(k) = k y sp (k) − ŷ(k)k 2M + k4u(k)k 2Λ (1.23)

The matrices M = diag(M 1, . . . , M N ) ≥ 0 and Λ = diag(Λ0, . . . , Λ Nu −1 ) > 0 are of

dimensionality ny N × ny N and nu Nu × nu Nu , respectively.
It is necessary to find the relation between the future values of the manipulated
variables and their increments, which are calculated on-line in MPC. From the
definitions of increments (Eqs. (1.4) and (1.5)), we have

u(k |k) = 4u(k |k) + u(k − 1)

u(k + 1|k) = 4u(k |k) + 4u(k + 1|k) + u(k − 1)
..
.
u(k + Nu − 1|k) = 4u(k |k) + . . . + 4u(k + Nu − 1|k) + u(k − 1) (1.24)

which may be expressed as a general rule

p
Õ
u(k + p|k) = 4u(k + i|k) + u(k − 1) (1.25)
i=0
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14 1 Introduction to Model Predictive Control

for p = 0, . . . , Nu − 1. The above observation may be rewritten compactly

u(k) = J 4u(k) + u(k − 1) (1.26)

where
 u(k |k) 
.
 
u(k) =  .. (1.27)
 

 u(k + Nu − 1|k) 
 
 
is a vector of length nu Nu that corresponds to the vector of increments 4u(k). Using
Eq. (1.26), the scalar constraints (1.15) may be expressed compactly

u min ≤ J 4u(k) + u(k − 1) ≤ u max (1.28)

where the vectors

 umin   umax 
 .. 
=  ... 
   
min max
u =  . , u (1.29)
 
 umin   umax 
   
   
and
 u(k − 1) 
..
 
u(k − 1) =  . (1.30)
 

 u(k − 1) 
 
 
are of length nu Nu , the matrix

 I nu ×nu 0nu ×nu 0nu ×nu . . . 0nu ×nu 

. . . 0nu ×nu 

 I nu ×nu I nu ×nu 0nu ×nu
J= . .. .. .. .. 

 .. . . . . 

 I n ×n I n ×n I n ×n
 u u u u u u . . . I nu ×nu 

is of dimensionality nu Nu × nu Nu . The scalar constraints (1.16) may be expressed

compactly
4u min ≤ 4u(k) ≤ 4u max (1.31)
where the vectors
 4umin   4umax 
=  .  , 4u max =  ... 
 .. 
   
4u min (1.32)
 
 4umin   4umax 
   
   
are of length nu Nu . The scalar constraints (1.17) may be expressed compactly as

−y min ≤ ŷ(k) ≤ y max (1.33)

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1.1 Formulation of the Basic MPC Problem 15

where the vectors

 y min   y max 
y min =  ...  , y max =  ... 
   
(1.34)
   
 y min   y max 
   
   
are of length ny Ny . Taking into account the minimised cost-function (1.23) and the
constraints (1.28), (1.31), (1.33), the general MIMO MPC optimisation problem
(1.20) is rewritten in a very compact vector-matrix form

min J(k) = k y sp (k) − ŷ(k)k 2M + k4u(k)k 2Λ

n o
4u(k)

subject to (1.35)
min max
u ≤ J4u(k) + u(k − 1) ≤ u
min
4u ≤ 4u(k) ≤ 4u max
y min ≤ ŷ(k) ≤ y max

Since a mathematical model of the controlled process is used on-line for prediction
and optimisation of the control policy, the MPC algorithms have the following
advantages:
1. It is possible to control MIMO processes efficiently. When a series of classical
single-loop PID controllers are used for the MIMO process, the consecutive
controllers work independently; each of them has only one objective, i.e. control
of only one controlled variable. When cross-couplings in the process (interactions
of the consecutive manipulated variables with the consecutive controlled ones)
are strong, such single-loop PID controllers do not work properly. Conversely,
due to using a model for prediction, the MPC “knows” all interactions between
process variables and calculates the best possible control policy.
2. The MPC algorithms may be used when the number of process inputs is different
from the number of outputs. In such a case, it is practically impossible to use a
set of single-loop PID controllers.
3. It is possible to take into account constraints imposed on both manipulated and
predicted controlled variables in a simple way (MPC optimisation is simply
carried out subject to all necessary constraints).
4. It is possible to control “difficult” processes, i.e. with significant time-delays or
with the inverse step-response.
Additional advantages of MPC are:
1. Tuning of MPC algorithms is relatively easy. It is only necessary to select ap-
propriate horizons and some weighting coefficients. All these parameters have a
clear physical interpretation.
2. It is possible to take into account the measured disturbances of the process, i.e.
the uncontrolled inputs (the feed-forward action).
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16 1 Introduction to Model Predictive Control

3. Unlike the PID algorithm, future changes of the set-point trajectory over the
prediction horizon may be easily taken into account.
4. The core idea of MPC is straightforward, which is important when advanced
methods are introduced in industry [112, 177].
Let us emphasise the very significant role of the process model in MPC. The
model is used for prediction. Intuitively, the better the model, the better (potentially)
the resulting control accuracy. Moreover, without the model, it is impossible to use
MPC at all. Let us also mention some other advanced model-based computational
methods: fault diagnosis [81, 83, 145, 192] and fault-tolerant control [118, 145, 192].
An important question is how to assess the quality of control. In addition to
typically used indicators, such as the sum of squared errors, overshoot and setting
time, we can use more sophisticated indices, including fractal and entropy measures
[36]. Effectiveness of such methods is discussed in [38, 39, 41] (for MPC algorithms
based on linear models) and in [40, 42] (for nonlinear MPC algorithms). A review
of control performance assessment methods for MPC is given in [37].
We have presented above the classical formulation of MPC. In the next parts of
the book, we will detail computationally efficient nonlinear approaches. At this point
we have to mention a few important extensions of MPC. In numerous industrial
applications, when the objective is maximisation of production profits, set-point
optimisation that cooperates with MPC [50, 91, 89, 177, 181] and economic MPC
[49, 48, 107, 132] must be used. An excellent review of possible architectures
for distributed and hierarchical MPC is given in [163]. MPC algorithms may also
offer fault-tolerant control [118, 145, 167], which means that safe process operation
is guaranteed in the case of some faults, e.g. when a sensors’ or an actuators’
malfunction occurs. It is also possible to take into account in MPC not only control
accuracy and economic issues but also the remaining useful life of the system
considered (health-aware MPC) [150]. An important direction of theoretical research
is concerned with stable and robust versions of MPC algorithms [128, 129]. Different
versions of such approaches are presented in [58, 117, 146, 144, 145, 174, 159, 182].
In the last years MPC schemes for fractional-order systems have gained popularity
[43, 44, 45, 46, 135, 169]. The fractional-order approach makes it possible to control
processes for which classical differential (or difference) equations are insufficient as
models used for prediction in MPC.

1.2 How to Cope with Infeasibility Problem

In this work three different classes of constraints are taken into account in MPC
optimisation taks (1.12), (1.20) and (1.35). The constraints may be imposed on:
the values of the manipulated variables, the corresponding increments of those
variables and on the predicted values of the controlled variables. The first two
classes of constraints simply limit the feasible set of possible solutions of the MPC
optimisation task. The third type of constraints may cause some important problems.
Let us imagine that we require no overshoot. In order to achieve that, we use the
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1.2 How to Cope with Infeasibility Problem 17

constraints
ŷ(k + p|k) ≤ y sp (k), p = 1, . . . , N (1.36)
If the model used for prediction is precise and there are no external disturbances,
such constraints may work correctly provided that the constraints imposed on the
manipulated variables are not too restrictive. It is also possible that the constraints
(1.36) may be not satisfied because of the constraints imposed on the manipulated
variables, even in the case of a perfect model and no disturbances. When the model
is only a rough approximation of the process, which frequently happens, and/or the
process is affected by a strong disturbance, it is very likely that it is impossible
to calculate a decision variable vector which leads to satisfaction of the constraints
(1.36). When such problems occur, the feasible set of the MPC optimisation problem
is empty. In such a case, one may use for control at the current sampling instant the
signals applied to the process at the previous sampling instant, i.e. u(k − 1), or
the signals calculated at the previous sampling instant for the current sampling, i.e.
u(k |k − 1). A more mathematically sound approach is to use soft output constraints
[112, 177]. The original hard constraints (in the vector notation for a general MIMO
process)
y min ≤ ŷ(k + p|k) ≤ y max, p = 1, . . . , N (1.37)
are relaxed when they cannot be satisfied. It means that the predicted values of
the controlled variables may temporarily violate the hard constraints. As a result,
the feasible set is not empty. Using the soft constraints, the rudimentary MPC
optimisation problem (1.20) becomes
N
(
k y sp (k + p|k) − ŷ(k + p|k)k 2M p
Õ
min J(k) =
4u(k)
p=1
ε min (k), ε max (k)
N u −1

k4u(k + p|k)k 2Λ p
Õ
+
p=0 )
min min 2 max max 2
+ ρ ε (k) +ρ kε (k)k

subject to (1.38)
min max
u ≤ u(k + p|k) ≤ u , p = 0, . . . , Nu − 1
min
4u ≤ 4u(k + p|k) ≤ 4umax, p = 0, . . . , Nu − 1
y min − ε min (k) ≤ ŷ(k + p|k) ≤ y max + ε max (k), p = 1, . . . , N
ε min (k) ≥ 0ny ×1, ε max (k) ≥ 0ny ×1

When the original hard constraints (1.37) cannot be satisfied, they are temporarily
violated. It is done by relaxing the minimal and maximal predicted values of the
controlled variables by ε min (k) and ε max (k), respectively. The MPC algorithm cal-
culates not only the future control increments 4u(k) but also the vectors ε min (k) and
ε max (k) of length ny . Because it is natural that the original hard output constraints
should be relaxed only when necessary, the degree of violations of the hard con-
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18 1 Introduction to Model Predictive Control

straints is minimised in the cost-function by additional penalty terms; ρmin, ρmax > 0
are penalty coefficients. Additionally, the last two constraints require that the de-
gree of constraints’ violation is non-negative. The number of decision variables
of the optimisation problem (1.38) is nu Nu + 2ny , the number of constraints is
4nu Nu + 2ny N + 2ny .
Using the vector-matrix notation, the rudimentary MPC optimisation problem
with soft output constraints (1.38) may be easily transformed to the following task
in a compact vector-matrix notation, similar to the task (1.35)

J(k) = k y sp (k) − ŷ(k)k 2M + k4u(k)k 2Λ

n
min
4u(k)
2
ε min (k), ε max (k)
+ ρmin ε min (k) + ρmax kε max (k)k 2
o

subject to (1.39)
min max
u ≤ J 4u(k) + u(k − 1) ≤ u
min
4u ≤ 4u(k) ≤ 4u max
y min − ε min (k) ≤ ŷ(k) ≤ y max + ε max (k)
ε min (k) ≥ 0ny ×1, ε max (k) ≥ 0ny ×1

where the vectors of length ny N are

 ε min (k)   ε max (k) 

ε min (k) =  ...  , ε max (k) = ..
   
. (1.40)
   
 
 ε min (k)   ε max (k) 
   
   
In the soft output approach it is possible to allow that the degree of relaxation of
the same controlled variable may change over the prediction horizon. In such a case,
in the optimisation problem (1.38), the soft constraints are

y min − ε min (k + p) ≤ ŷ(k + p|k) ≤ y max + ε max (k + p), p = 1, . . . , N (1.41)

The vectors of additional decision variables of the MPC optimisation task are now
 ε min (k + 1|k)   ε max (k + 1|k) 
.. ..
   
ε min (k) = 
 max
.  , ε (k) = . (1.42)
  
 
 ε min (k + N |k)   ε max (k + N |k) 
   
   
Unfortunately, the number of decision variables increases to nu Nu +2ny N, the number
of constraints is 4nu Nu + 4ny N. In practical applications of MPC, the assumption
that the output constraints are relaxed by the same degree for the whole prediction
horizon (for the consecutive controlled variables) and only 2ny additional variables
are used gives very good results, very close to those possible when as many as 2ny N
additional variables are necessary [96].
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1.3 Parameterisation of Decision Variables 19

1.3 Parameterisation of Decision Variables

Laguerre, Kautz and other orthonormal functions may be successfully used for mod-
elling of dynamical systems in linear [137] and nonlinear [138] cases, respectively.
Application of orthonormal Laguerre functions to parameterise the calculated future
sequence of the manipulated variables may be used in MPC algorithms based on
linear state-space models: in continuous-time [186] and discrete-time [187] versions,
respectively, as well as in the DMC algorithm, in which a step-response model is
used for prediction [178]. A systematic tuning methodology to find parameters of
Laguerre functions in parameterised MPC is discussed in [61, 75]. MPC algorithms
with Laguerre parameterisation have been developed for different technological pro-
cesses. Example applications include: buildings [19], wave energy converters [69],
magnetically actuated satellites [76], wind turbines [84], hexacopters [104] and
power systems [202]. All cited MPC algorithms use linear models for prediction. In
this book, the Laguerre functions are used to parameterise the decision vector of all
discussed nonlinear MPC algorithms, i.e. to reduce the number of decision variables
that are actually optimised on-line.
At first, let us consider the SISO case. Let l1 (k),. . . ,lnL (k) denote nL Laguerre
functions. The transfer function of the Laguerre function of the order n is [185]

1 − (aL )2 1 − aL z n−1
p  
G n (z) = (1.43)
z − aL z − aL

where aL is a scaling factor, often named a Laguerre pole. For stability, the condition
0 ≤ aL < 1 must be satisfied. The transfer functions G n (z) satisfy the following
orthonormality conditions

1
∫ π
G n (e jω )G n (e jω )∗ dω = 1 (1.44)
2π −π
1
∫ π
G m (e jω )G n (e jω )∗ dω = 0 for m , n (1.45)
2π −π

where G n (e jω )∗ denotes complex conjugate of the transfer function G n (e jω ). The

Laguerre functions are defined as inverse Z-transforms of the transfer functions
G n (z)
ln (k) = Z −1 (G n (z)) (1.46)
Taking into account the structure of the obtained Laguerre functions, it may be found
that [187]
L(k + 1) = ΩL(k) (1.47)
where the vector of length nL is
T
L(k) = l1 (k) . . . lnL (k) (1.48)

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20 1 Introduction to Model Predictive Control

and the matrix of dimensionality nL × nL is

 aL 0 0 ... 0 
β 0 ... 0
 
 L a L 
 −aL βL β ... 0
 
L a L 
Ω=  aL2 βL −aL βL βL ... 0 (1.49)
 

.. .. .. .. ..
 
. . . . .
 
 
 (−aL )nL −2 β (−aL )nL −3 β . . . βL
 
 aL 

The initial condition is


 1 

 −a L 
aL2
 
q  
L(0) = 1 − aL2  −aL3 (1.50)
 

..
 
.
 
 
 (−aL )nL −1
 

 

and βL = 1−aL2 . The orthonormality conditions (1.44)-(1.45) may also be formulated

for the discrete-time description
∞
Õ
li (k)l j (k) = 0 for i , j (1.51)
k=0
Õ∞
li (k)l j (k) = 1 for i = j (1.52)
k=0

The idea of parameterisation is to eliminate the necessity of calculating at each

sampling instant as many as Nu future increments 4u(k |k), . . . , 4u(k + Nu − 1|k), i.e.
the whole vector 4u(k) (Eq. (1.3)). The future control increments are parameterised
using the Laguerre functions in the following way [187]
nL
Õ
4u(k + p|k) = li (p)ci (k) (1.53)
i=1

Using the vector notation, we have

4u(k + p|k) = L T (p)c(k) (1.54)

where the vector of coefficients is

T
c(k) = c1 (k) . . . cnL (k) (1.55)

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1.3 Parameterisation of Decision Variables 21

For the whole vector of future increments of the manipulated variable over the control
horizon, we have
4u(k) = Lc(k) (1.56)
where the matrix of dimensionality Nu × nL is
 l1 (0) l2 (0) ... lnL (0) 
...
 
 l1 (1) l2 (1) lnL (1) 
L= .. .. .. .. (1.57)
 
. . . .

 
 l1 (Nu − 1) l2 (Nu − 1) . . . lnL (Nu − 1) 
 

In parameterised MPC the vector of decision variables is c(k), not 4u(k). Since
nL < Nu , the number of decision variables used in the MPC optimisation problem
solved on-line is reduced. Having calculated the optimal vector copt (k) from the MPC
optimisation problem, using Eq. (1.56) and taking into account the structure of the
matrix L given by Eq. (1.57), the current optimal value of the manipulated variable
is calculated from

u(k) = l1 (0) l2 (0) . . . lnL (0) copt (k) + u(k − 1) (1.58)

 

and applied to the process.

Having discussed the SISO case, we will consider parameterisation using La-
guerre functions for MIMO processes. In order to obtain a flexible solution, we
assume that for the consecutive manipulated variables separate Laguerre poles
aL1 , . . . , aLnu are used. Furthermore, we also assume that for the consecutive variables
different numbers of Laguerre functions are possible, i.e. nL1 , . . . , nLnu . Similarly to
Eq. (1.53) used in the SISO case, the future control increments are parameterised in
the following way
1
nL
Õ
4u1 (k + p|k) = l1,i (p)c1,i (k) (1.59)
i=1
..
.
nu
nL
Õ
4unu (k + p|k) = lnu,i (p)cnu,i (k) (1.60)
i=1

In place of Eq. (1.54), we have

4u1 (k + p|k) = L1T (p)c1 (k) (1.61)

..
.
4unu (k + p|k) = LnTu (p)cnu (k) (1.62)
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22 1 Introduction to Model Predictive Control

where the vectors of coefficients, of length nL1 , . . . , nLnu , respectively, are

 c1,1 (k)   cnu,1 (k) 

.. ..
   
c1 (k) =  .  , . . . , cnu (k) = . (1.63)
   
 
 c1,n1 (k) 
   
 cn ,n nu (k) 
 L   u L 
For all manipulated variables and the whole vector of future increments over
the control horizon, for the MIMO process we also obtain Eq. (1.56), the same as
in the SISO case, but now the vector 4u(k) is of length nu Nu and the matrix of
dimensionality nu Nu × (nL1 + . . . + nLnu ) has the general structure

 L 1 0 N ×n2 0 N ×n3
u u
. . . 0 Nu ×n nu 
 L L L 
0
 Nu ×nL1 L 2 0 Nu ×n3 . . . 0 Nu ×n nu 
L L 
L =  0 Nu ×nL1 0 Nu ×nL2 L3 . . . 0 Nu ×n nu (1.64)
 
L

.. .. .. .. ..

. . . . .
 
 
. . . L nu
 
 0 N ×n1 0 N ×n2 0 N ×n3 
 u L u Lu L 
where the consecutive submatrices of dimensionality Nu × nLn are

 ln,1 (0) ln,2 (0) ... ln,nLn (0) 

 ln,1 (1) ln,2 (1) ... ln,nLn (1)
 

Ln =  . .. .. .. (1.65)
 
. .

 . . . 
 ln,1 (Nu − 1) ln,2 (Nu − 1) . . . ln,nLn (Nu − 1)
 

 

for n = 1, . . . , nu . The vector of optimised decision variables is of length nL1 +. . .+nLnu

and has the structure
 c1 (k) 
c(k) =  ... 
 
(1.66)
 
 
 cn (k) 
 u 
where the subvectors are defined by Eqs. (1.63).
Having calculated the optimal vector copt (k) from the MPC optimisation problem,
using Eq. (1.56) and taking into account the matrices L and L n , given by Eqs. (1.64)
and (1.65), respectively, the current optimal values of the manipulated variables are
calculated from
opt
h i
u1 (k) = l1,1 (0) l1,2 (0) . . . l1,nL1 (0) c1 (k) + u1 (k − 1) (1.67)
..
.
opt
h i
unu (k) = lnu,1 (0) lnu,2 (0) . . . lnu,nLnu (0) cnu (k) + unu (k − 1) (1.68)

and applied to the process.

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1.4 Computational Complexity of MPC Algorithms 23

1.4 Computational Complexity of MPC Algorithms

In the simplest case, a linear model is used in MPC for prediction and no constraints
are taken into account. A few different such MPC methods have been developed, with
different structures of linear models. To name the most important MPC approaches
based on linear models, we have to mention the following ones:
1. The Predictive Functional Control (PFC) algorithm (also known under the name
Model Heuristic Predictive Control (MHPC)) [156, 157] in which the impulse-
response process representations are used.
2. The Dynamic Matrix Control (DMC) algorithm [29] in which the step-response
models are used.
3. The Generalized Predictive Control (GPC) algorithm [27] in which the discrete-
time transfer functions are used.
4. The MPC algorithm with state-space models (MPCS) [112, 177] in which the
classical linear state-space models are used.
The use of a linear model implies that the predicted trajectory of the manipulated
variables (Eq. (1.22)) is a linear function of the decision variable vector (1.3).
Remembering that the typical minimised MPC cost-function is of the quadratic type
(Eq. (1.13)), we obtain an unconstrained quadratic optimisation problem. It may
be solved analytically, without on-line optimisation. The future increments of the
manipulated variables are linear functions of the following: the model parameters,
some values of the manipulated variables computed at the previous sampling instants
and the values of the process controlled variables measured at the previous sampling
instants. Hence, such unconstrained MPC methods are named unconstrained linear
explicit MPC algorithms.
If a linear model is used for prediction, but the constraints must be taken into
account, at each sampling instant, it is necessary to solve on-line a quadratic op-
timisation task (a quadratic minimised cost-function and linear constraints). Such
methods are named constrained linear MPC algorithms or, better, constrained MPC
algorithms based on linear models since in the constrained case, the explicit linear
solution does not exist, the optimal solution is obtained as a result of on-line opti-
misation. Depending on the model used, we obtain constrained MHPC, DMC, GPC
and MPCS algorithms. For linear models, provided that µ p,m ≥ 0 and λ p,n > 0, the
optimisation task has only one solution, which is the global one. Different approaches
may be used to find the solution of the quadratic optimisation MPC problem [171]:
the active-set methods, the interior-point ones and the first-order ones. It is necessary
to point out that many very computationally efficient quadratic optimisation solvers
are available, e.g. qpOASES [52], CVXGEN [126] and OSQP [171]. To speed up
calculations, advanced quadratic optimisation algorithms may be specially tailored
for MPC, i.e. the special form of the MPC optimisation task may be exploited. They
may be used not only for industrial control applications [13] but also in embedded
systems [16, 78, 158], for which sampling times are very short, of the order of
hundreds, teens or even single milliseconds.
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24 1 Introduction to Model Predictive Control

As described in Section 1.3, some basis functions, e.g. Laguerre orthonormal

functions, may be used to reduce the number of decision variables of the MPC
optimisation problem. The sequence of future manipulated variables is parameterised
using a set of basis functions. The optimisation routine does not directly calculate the
future manipulated variables or the corresponding increments but the coefficients of
the basis functions. In the literature, a few variants of MPC algorithms which use that
concept are described [178, 186, 187]. The parameterisation approach may be used
in unconstrained linear explicit MPC algorithms and constrained MPC algorithms
based on linear models. A similar approach is used in the PFC algorithm, which also
uses linear models for prediction [156]. Finally, parameterisation may be used in the
nonlinear MPC algorithms [98] which are discussed in the following chapters of this
book.
Although the classical quadratic optimisation MPC problem is quite simple, in
some applications, it would be best to eliminate the necessity of on-line optimisation
at all. It can be proven that for a linear model and the typical quadratic cost-function,
the optimal solution of the constrained quadratic optimisation MPC problem is a
function of the state [15, 179]. That observation leads to constrained linear explicit
MPC algorithms. The whole state domain is divided into a number of sets. For
each set, the explicit control law is derived off-line. During on-line control, it is
only necessary to determine to which set the current state of the process belongs
and to use the corresponding precalculated control law; no on-line optimisation is
necessary. Although the idea seems to be generally simple and intuitive, it may turn
out that many (dozens or even hundreds) of sets and local control laws are required
for typical processes.
When a general nonlinear model is used for prediction, the predicted trajectory
(1.22) is a nonlinear function of the decision variable vector (1.3). Thus, the min-
imised cost-function (Eq. (1.13)) is not quadratic but nonlinear. The constraints
imposed on the magnitude and on the rate of change of the manipulated variables are
linear, but the constraints put on the predicted values of the controlled variables are
nonlinear. The general class of the discussed approach is known as fully-fledged con-
strained nonlinear MPC algorithms or constrained MPC algorithms with nonlinear
optimisation. A constrained nonlinear MPC optimisation problem must be solved
on-line at each sampling instant. There are two difficulties of that approach. Firstly,
nonlinear optimisation algorithms must be used. They are much more complicated
than the classical quadratic optimisation ones. Solution of a constrained nonlinear
optimisation task may need a lot of time. It is particularly important in the case of fast
dynamical systems, for which very short sampling times are required. Secondly, it is
possible that not only one global but several local minima exist. When a suboptimal
solution is used for control, the resulting control quality may be lower than expected.
Typically, the Newton-like nonlinear optimisation algorithms are used. The Se-
quential Quadratic Programming (SQP) [151] and Interior Point (IP) [20] methods
are the most frequently used ones in nonlinear MPC. Efficient implementation meth-
ods for SQP and IP algorithms have been developed which exploit the particular
structure of the MPC optimisation task [53, 153]. Specialised nonlinear optimisation
methods, developed with the aim of being used to solve MPC optimisation problems,
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1.4 Computational Complexity of MPC Algorithms 25

make it possible to carry out parallel calculations [31, 199]. When the model used
for prediction is comprised of a set of differential-algebraic equations, specialised
optimisation methods must be used [33]. An excellent review of possible approaches
to nonlinear optimisation in MPC is given in [34]. Very infrequently, for nonlinear
optimisation other algorithms may be used, e.g. the golden section method [114, 193]
or the branch-and-bound approach [195].
When the process dynamics is slow, which makes it possible to use relatively
long sampling periods, we may use heuristic global optimisation algorithms. For
example, applications of genetic algorithms to solve the constrained nonlinear MPC
optimisation task may be found in [103, 149]. Specialised genetic operators (mutation
and crossover) are used, tailored for the nature of MPC. An alternative is to use the
particle swarm optimisation algorithm [25, 191]. Another option is to use simulated
annealing for nonlinear optimisation [1]. It must be stressed that application of
heuristic optimisation methods is limited.
There are, however, some deterministic global optimisation methods [164] that
may be used in MPC [47]. The cited method is based on a convex relaxation of the
MPC cost-function. It is reported to significantly reduce dimensionality of the MPC
optimisation task, which lower the overall computational burden. To further reduce
computational complexity, a neural multi-model is used rather than one dynamical
model applied recurrently.
In practice, fuzzy MPC is a very important alternative. To control a nonlinear
process, a set of simple local MPC controllers is used. The local controllers are
switched on-line, taking into account the current operating point of the process
and/or the set-point. Both the unconstrained linear explicit MPC methods and the
constrained MPC algorithms based on linear models may be used as local controllers.
It is important that the local controllers are developed off-line. During on-line control,
it is only necessary to combine the values of the manipulated variables computed
by the local controllers in a fuzzy way. Fuzzy DMC algorithms [30, 119, 125]
and fuzzy GPC methods are given as examples of the described approach [177].
Advanced methods utilised for prediction generation in the fuzzy DMC algorithm
are discussed in [123, 124]. A similar idea is to use multi-linear models for prediction
in MPC [200]. A specialised procedure is used to determine the multi-linear process
representation from nonlinear Hammerstein or Wiener models.
There are numerous attempts to simplify the general nonlinear MPC optimisation
task that must be solved at each sampling instant on-line. The following methods are
reported in the literature:
1. The first nu elements of the future control policy are computed from a nonlinear
optimisation task, whereas the remaining ones are found from an explicit control
law [201]. As a result, the optimisation problem is still nonlinear, but the number
of decision variables is equal to nu , not to nu Nu as in the rudimentary approach.
2. The technique named move blocking may be used [21]. The degree of freedom
is reduced by fixing the manipulated variables or their derivatives to be constant
over several time-steps. Some of such methods guarantee stability and satisfaction
of constraints.
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26 1 Introduction to Model Predictive Control

3. Compression of the constraint set is possible [102]. It simplifies the MPC opti-
misation task. Such an approach may be used together with the move blocking
technique.
4. The domain of the calculated manipulated variable may be discretised [115] (in the
cited approach, the control horizon is equal to 1). A simple procedure determines
its best value and on-line optimisation is not necessary. A more advanced graph
search method for finding the control policy is used in [155].
5. In the case of the cascade models, the inverse of the static part of the model
may be used to make an attempt to cancel the effect of nonlinearity. It makes
it possible to formulate the classical quadratic optimisation MPC problem. For
the Hammerstein structure, such an approach is discussed in [54], for the Wiener
structure in [4, 23, 70, 133, 134, 168]. The same method may also be used for
cascade models with 3 blocks, e.g. the Hammerstein-Wiener ones as described
in [35, 63, 147]. As pointed out in Section 3.1, the discussed approach has
important structural disadvantages and limitations. Moreover, as demonstrated
in simulations discussed in this book, it is very sensitive to model errors and
disturbances.
6. In the fast MPC algorithm [190] the MPC optimisation task is not solved precisely
but in an approximate way. Although it may have a negative effect on the resulting
control quality, the time of calculations necessary at each sampling instant is likely
to be significantly reduced. As proved in [165], for stability, it is sufficient to use
a feasible control strategy, i.e. the one that satisfies all the existing constraints,
not the optimal one.
7. The numerical optimisation procedure used in the MPC algorithm may be re-
placed by a specially designed neural network which acts as a neural optimiser.
There are a few neural structures which solve the quadratic optimisation problem
[109, 188]. The network described in [109] is used for optimisation in an MPC
algorithm based on a linear model [141] and in an MPC algorithm with on-line
model linearisation [140].
8. The MPC algorithm may be replaced by a specially designed neural network
which acts as a neural approximator that attempts to mimic the whole MPC
algorithm [2, 142]. At first, the classical nonlinear MPC algorithm is developed
and run on-line (or off-line in simulations) for different operating conditions and
set-points. A data set is collected and next used to train a neural approximator. For
a given operating point of the process, determined by measurements of the process
input and output variables, as well as the set-point, the approximator finds the
current values of the manipulated variables. An approximator may also be used
to find the initial solution of the MPC optimisation problem [180]. Finding the
initial solution is likely to significantly shorten the calculation time in embedded,
microprocessor-based systems [77].
9. The prediction and control horizons may be equal to 1 and the current value of
the manipulated variable may be computed by a simple binary search algorithm
[160].
10. The Experience-driven Predictive Control (EPC) algorithm constructs a database
of feedback controllers that are parameterised by the system dynamics [32].
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1.4 Computational Complexity of MPC Algorithms 27

When, for given conditions, the control law does not exist, it is calculated by
a conventional MPC algorithm based on a linear model. In order to obtain a
quadratic optimisation task, for prediction Locally-Weighted Projection Regres-
sion (LWPR) models are used, which allow for easy on-line model adaptation.
11. The nonlinear optimisation MPC problem is relaxed into a Mixed Integer Linear
Programming (MILP) one. Next, the solution of the MILP problem is taken as a
starting point of the nonlinear one [189].
12. Constrained explicit nonlinear MPC algorithms are possible [57, 71]. Unfortu-
nately, a huge number of local control laws may be necessary.
13. A specialised model may be used in which the output values for the consecutive
sampling instants within the prediction horizon are linear functions of the calcu-
lated future manipulated variables, but they are nonlinear functions of the past (the
quasi-linear model) [106]. Such an approach results in a quadratic optimisation
MPC task. Neural networks are used for modelling.
14. When Linear Parameter Varying (LPV) models are used for prediction, the general
nonlinear optimisation problem is replaced by a convex Linear Matrix Inequal-
ities (LMIs) optimisation task [203, 205, 204]. Neural networks may calculate
coefficients of the LPV models.
15. Model convexity may be achieved when Input Convex Neural Networks (ICNNs)
are used [8]. ICNNs are obtained by explicitly constraining the model outputs to
be convex functions of the inputs during model development. As a result, convex
MPC optimisation problems are obtained: unconstrained [26] or constrained ones
[196].
16. A class of linear predictors may be used to describe a nonlinear system [127]. The
key step in obtaining such accurate predictions is to lift (or embed) the nonlinear
dynamics into a higher dimensional space in which its evolution of this lifted state
is (approximately) linear. The idea corresponds to the Koopman operator [79, 80].
When such a model is used in MPC, we obtain a quadratic MPC optimisation
task [82, 127]. An alternative method, named polyflows, is discussed in [72].
Finally, on-line linearisation must be discussed as the method which makes it
possible to significantly reduce computational burden of nonlinear MPC. Details of
numerous such MPC methods are presented in Chapters 3 and 7 for input-output
and state-space Wiener process descriptions, respectively. Let us now only give a
short literature review. In general, two categories of computationally efficient MPC
algorithms may be distinguished: with on-line model linearisation and with on-line
trajectory linearisation. In both cases, we obtain computationally simple quadratic
optimisation problems, the necessity of on-line nonlinear optimisation is eliminated.
In the simplest approach, a linear approximation of the nonlinear model is com-
puted on-line for the current operating point of the process. Typically, model lineari-
sation is performed at each sampling instant but, for some “less nonlinear” processes
or when changes of the set-point are slow and infrequent, model linearisation may
be repeated less frequently. Next, the obtained linearised model is used to calculate
the predicted trajectory of the controlled variables. Thanks to linearisation, the pre-
dicted trajectory is a linear function of the vector of decision variables (1.3), which
is a characteristic feature of the classical MPC algorithms based on linear models.
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28 1 Introduction to Model Predictive Control

Hence, a quadratic optimisation problem is formulated when the constraints must be

taken into account or even the explicit unconstrained solution is possible.
The MPC algorithms with on-line model linearisation may be divided into two
categories [91, 177]. In the first one, the time-varying linear approximation of the
rudimentary nonlinear model is used to calculate future predictions and the influence
of the past, i.e. the free trajectory. In the second approach to MPC with successive
linearisation, the linearised model is only used to calculate the future predictions,
whereas the nonlinear model is used to find the nonlinear free trajectory. The first
approach is used to control a spark-ignition engine in [28] and an aircraft gas turbine
engine in [130]. Applications to a polymerisation reactor and a distillation column are
presented in [85]. When necessary, the nonlinear model may be retrained on-line as
shown in [3], applications of the algorithm to a fluidised bed furnace reactor and the
autopilot of the F-16 aircraft are described. An application to a boiler-turbine unit in a
power plant described by a state-space process model is detailed in [96], two variants
of soft constraints are considered. Although the algorithm may be implemented for
practically any differentiable model, a straightforward calculation is possible for
Wiener structures since the linearised model is found in a simplified way, as a
multiplication of the linear dynamic part and the time-varying gain of the nonlinear
static block [5]. A similar calculation method is possible for the Hammerstein model.
The second approach, i.e. with the nonlinear free trajectory, is used to control a solar
power plant in [9, 17], a spark-ignition engine [162], a yeast fermentation reactor
[91], a polymerisation reactor and a distillation column [85]. Also in the second
approach simple calculations are possible when Hammerstein [91, 121] or Wiener
[87, 91, 120, 122] models are used.
In more advanced MPC algorithms with on-line trajectory linearisation, not the
model itself is linearised, but a linear approximation of the predicted trajectory of the
controlled variables over the whole prediction horizon is directly calculated. Unlike
the simple MPC algorithms with model linearisation, linearisation is not performed
for the current operating point of the process, defined by past measurements of the
process input and output signals, but carried out along some future trajectory of
the manipulated variables defined for the whole control horizon. Similarly to the
simple algorithm with on-line model linearisation, a quadratic optimisation problem
is next formulated. The explicit unconstrained solution is also possible. In practice,
the classical MPC algorithm with model linearisation may be used when the process
is close to the desired set-point. If it is not true, the calculated solution defines the
future trajectory of the manipulated variables along which a linear approximation
of the predicted trajectory of the controlled variables is calculated. Such a hybrid
MPC structure is presented in [88, 91], an application to a high-pressure distillation
column is discussed. An application of the algorithm to a solid oxide fuel cell is
presented in [97], the method of coping with infeasibility caused by linearisation
of nonlinear technological constraints (fuel utilisation) are discussed. The MPC
algorithm with trajectory linearisation is also of course possible when the process
is described by cascade models, including: Hammerstein [91] (for a polymerisation
reactor benchmark), Wiener [94] (for a neutralisation reactor) and [100] (for a
proton exchange membrane fuel cell), Hammerstein-Wiener [93] as well as Wiener-
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1.5 Example Applications of MPC Algorithms 29

Hammerstein [95] (for a heat exchanger) structures. Although all cited works are
concerned with the input-output process representation, the MPC algorithm with
trajectory linearisation is, of course, possible for the state-space representation [101]
(implementation details for the Wiener model are given).
Finally, let us mention computationally efficient MPC algorithms with on-line
linearisation and approximation. The approximator is used in order to eliminate
some calculations that must be repeated at each sampling instant. They are neces-
sary in the classical MPC algorithms with on-line linearisation. Successive model
linearisation and prediction calculation may be simplified using an approximator
which directly estimates, at each sampling instant, the time-varying matrix of step
response coefficients of the linearised model [91]. An application of that approach to
a simulated distillation column is detailed in [90]. The same approximation method
may be used in the nonlinear DMC algorithm [86, 91]. A significant reduction of
computational complexity in comparison with the classical MPC algorithms with
on-line linearisation may be obtained when explicit unconstrained versions of the
discussed algorithms are considered. It may be proved [91, 92] that in such a case,
the optimal vector of the decision variable vector (1.3) is a linear function of the
set-point, model parameters and some past measurements. The time-varying vector
of coefficients of the control law is determined on-line by a neural approximator
for the current operating point. As a result, on-line model linearisation and some
other calculations are not necessary, which significantly speeds up calculations. A
simulation study concerned with a high-pressure distillation process is presented in
[91, 92]. In all mentioned cases, neural networks are used as approximators, although
other structures are also possible.

1.5 Example Applications of MPC Algorithms

MPC is regarded as the only one among the advanced control techniques, defined
as more advanced than the classical PID controller, which is successfully used in
numerous industrial applications [152]. Let us cite a number of typical applications.
Traditionally, MPC algorithms may be successfully used for controlling the following
industrial processes:
– chemical reactors [64, 166, 175, 198],
– distillation columns [11, 65, 111, 74, 116, 148, 184],
– combustion in pulverized-coal-fired boilers (in power plants) [62],
– greenhouses [60],
– hydraulic systems [12],
– solar power stations [9, 55],
– waste water treatment plants [131],
– electromagnetic mills [136],
– cement kilns [170].
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30 1 Introduction to Model Predictive Control

Typically, the sampling period of industrial MPC algorithms used in process con-
trol is quite long, of the order of seconds, a dozens of seconds or even minutes.
Programmable Logic Controllers (PLCs) are used for implementation of MPC al-
gorithms in industrial process control. In addition to that, thanks to availability of
fast microcontrollers, it is possible to develop MPC algorithms for fast dynamical
systems (in embedded systems). In contrast to the mentioned industrial applications,
they require short sampling times, shorter than one second, typically of millisecond
order. Example applications of fast MPC include:
– fuel cells [59],
– active vibration attenuation [176],
– combustion engines [28, 73, 154],
– robots [183, 22, 139],
– servomotors [24],
– quadrotors [7],
– stratospheric airships [108],
– power converters [194],
– electrical inverters [110],
– induction machines [51].
Many research works are concerned with automotive applications. A few exam-
ples are: autonomous driving [105, 173], autonomous racing [6], traction control
[68], vehicle roll-over [67].
There are some applications of MPC in medicine, e.g. muscle relaxant anaesthesia
[114] and artificial pancreas [66].
In addition to industrial and embedded applications of MPC, it is interesting to
mention a few original and less frequent applications in which MPC algorithms also
turn out to be very efficient:
– drinking water transport networks [143],
– supermarket refrigeration systems [161],
– traffic on highways [14],
– high energy physics accelerators [18],
– inventory management in hospitals [113].
Important applications of MPC are concerned with building control. Typically,
only temperature control (stabilisation despite changes of the outside temperature,
which is a disturbance) is considered [56, 172]. In more advanced solutions, thermal
comfort is controlled [197], i.e. temperature, humidity and other factors. MPC may
cooperate with on-line energy optimisation which determines optimal set-points for
MPC [10].
It is important to emphasise that all cited works in Chapter 1.5 discuss real
applications only. In addition to that, hundreds or even thousands of works annually
discuss simulation results.
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