MPC course materials

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Title: MPC course materials

1
Model Predictive Control Dr.Ir. Ton van den
Boom Prof.Dr.Ir. Ton Backx
November 29, 1999
2
Model Predictive Control

First lecture
Introduction on Model Predictive Control
Models and model characteristics
Prediction

3
Model Predictive Control

Model Predictive Control technology has its roots
in industry. The model predictive control concept
development started at the end of the sixties
Dynamic Matrix Control (Shell, Charlie Cutler et
al, 1980)
IDCOM (Jacques Richalet, 1978)
Quadratic Dynamic Matrix Control (Shell, Mike
Morshedi et al, 1984)
Shell Multivariable Optimizing Control (Shell,
1985)
IDCOM-M (Setpoint, 1986)
Setpoint Multivariable Control Architecture
(Setpoint, 1994)
Robust Multivariable Process Control Technology
(Honeywell, 1995)
Ipcos Novel Control Architecture (IPCOS, 1999)

Introduction on Model Predictive Control
Models and model characteristics
Prediction

4
Model Predictive Control

Model Predictive Control (MPC) technology is most
widely applied in Oil Refining and Petrochemical
industry applications today
The application objectives are
Maximization of throughput
Operation within permitted operating constraints
Pushing for best economic operating conditions
Interface between steady state, first principle
model based optimization and primary process
control

Introduction on Model Predictive Control
Models and model characteristics
Prediction

5
Model Predictive Control

Most of todays industrial applications of MPC
are primarily focussing on quasi steady state
behavior of processes
Compensation of low frequency components of
disturbances only due to low bandwidth
Low bandwidth controller tuning for robustness
reasons
Mostly a single, linear, non-parametric dynamic
model to describe process behavior for the
complete operating range

Introduction on Model Predictive Control
Models and model characteristics
Prediction

6
Model Predictive Control

Use of the MPC technology in most Chemical and
other Processing Industry applications requires
more attention for control system performance
6-sigma quality of specified product parameters
and critical process conditions
predictable and reproducible transition between
different operating conditions
market situation based process optimization and
transitions between product grades/product types
time critical operation as one step in a supply
chain

Introduction on Model Predictive Control
Models and model characteristics
Prediction

7
Model Predictive Control

Poor capital productivity -i.e. the money
generated with the invested capital- is a major
problem in most of the chemical processing, glass
manufacturing, steel production and several other
processing industries
Global competition resulting in strangling
pressure on prices and thus margins
World-wide saturation of markets leading to price
pressure and need for innovation
Tightening legislation on ecosphere load and
resource consumption resulting in increasing
complexity of processes and corresponding growing
difficulties with process operation

Introduction on Model Predictive Control
Models and model characteristics
Prediction

8
Model Predictive Control

Todays way of supply driven operation of
production is one of the causes of the poor
performance of the processing industries
Focus on increase of scale
Focus on reproducibility
Minimization of number of product types
Fixed grade slates, fixed recipe driven
changeovers
? Comparable situation to the Automotive and
Consumer Electronics Industries in the seventies
Part of the answer is demand driven production
This requires completely integrated high
performance technologies for process unit control
and plant optimization

Introduction on Model Predictive Control
Models and model characteristics
Prediction

9
Model Predictive Control

A constrained market situation asks for a demand
or market driven mode of process operation
Increase flexibility in processing of a broad
range of feedstock materials
Produce products that have market demand
Take price advantage of a scarce market
Minimize capital blocked in stored products and
intermediates
Increase capital turnaround by shortening
production-to-product delivery cycles
Each of the above effects directly contributes to
the required increase of the capital productivity

Introduction on Model Predictive Control
Models and model characteristics
Prediction

10
Model Predictive Control
Reduction of the order to delivery cycle
significantly increases capital productivity
mimp improved margin m margin per
cycle d margin improvement s speedup factor of
order to delivery cycle

Introduction on Model Predictive Control
Models and model characteristics
Prediction

11
Model Predictive Control

Realization of demand driven operation of
production processes requires new technologies
that enable
Flexible operation of plants over broad operating
ranges at minimum costs
dynamic optimization
Tight production at pre-specified Cpk values to
achieve quality constraints
high performance (model based) control systems
that enable significant reduction of variance of
critical process/product variables
Overall optimization of economic performance
integration of optimization of operation with
(model based) control
Extensive (re-)use of available a-priori
knowledge to minimize total application costs and
to enable economic feasibility

Introduction on Model Predictive Control
Models and model characteristics
Prediction

12
Model Predictive Control
Market driven objectives
Classical Steady-state model based
optimizationQuasi-steady state MPClocal
validity of MPC
NewTechnologyDynamic model based trajectory
optimizationHigh Performance MPCTrajectory
tracking MPC
Plant-Wide Model Based Optimizer
Plant-Wide Model
Optimal Process Conditions
Model Predictive Control
Model Predictive Control
Unit Model
Model Predictive Control
Unit Model
Unit Model
Optimal Reference Signals

Introduction on Model Predictive Control
Models and model characteristics
Prediction

DCS
DCS
DCS
Primary Control Signals
Process
Process
Process
13
Model Predictive Control
Model Predictive Control system

Introduction on Model Predictive Control
Models and model characteristics
Prediction

14
Model Predictive Control
Prioritised Optimisation
Fulfill operational requirements
Satisfy MV constraints Satisfy Priority 1 zones
and targets, if not possible balance between all
priority 1 requirements Satisfy priority i zones
and targets, if not possible balance between all
priority i requirements Bring CVs as close as
still possible to optimal values Bring MVs as
close as still possible to optimal values
Fulfill quality requirements

Introduction on Model Predictive Control
Models and model characteristics
Prediction

Fulfill economic performance conditions
15
Model Predictive Control

High performance operation of processes requires
a tight integration of optimization and control

Introduction on Model Predictive Control
Models and model characteristics
Prediction

16
Model Predictive Control

Model Predictive Control systems calculate the
future process manipulations by optimization of a
finite horizon objective function
Model Predictive Control
Generalized Predictive Control

Introduction on Model Predictive Control
Models and model characteristics
Prediction

17
Model Predictive Control

The model predictive control problem is mostly
formulated as a constrained optimization problem,
where constraints are imposed on
Inputs or manipulated variables
States and outputs or controlled variables

Introduction on Model Predictive Control
Models and model characteristics
Prediction

18
Model Predictive Control

Models are the heart of a model predictive
control system models are used for
Prediction of future process output behavior on
the basis of known past input signals, known past
disturbances and expected future disturbances
Calculation of the best future process
manipulations on the basis of a given criterion
function and specifications for the controlled
variables

Introduction on Model Predictive Control
Models and model characteristics
Prediction

Achievable performance and robustness of the
model predictive control system are governed by
model accuracy
19
Model Predictive Control

Essentially two type of modeling techniques are
applied today for modeling the dynamic behavior
of processes
First principles based modeling
modeling of known main process mechanisms
covering a broad operating envelope on the basis
of first principles (mass, energy and momentum
balances)
empirical estimates of physical properties,
reaction kinetics, reaction complex, ...
Empirical modeling
modeling of observed process behavior in response
to test signals and/or operator/disturbance
invoked process excitations
test signal content determines model validity and
coverage of operating envelope

Introduction on Model Predictive Control
Models and model characteristics
Prediction

20
Model Predictive Control
First Principle Modeling

Lack of detailed knowledge of all process
mechanisms and their parameter values that
contribute to relevant process input/output
behavior makes it difficult to develop purely
first principles based models for high
performance MPC
No methodology resulting in necessary accuracy of
the model for high performance control
Lack of structured model design methods to
analyze and assure inclusion of all relevant
mechanisms
Lack of methods that enable choosing required
model granularity and that enable appropriate
model reduction
Lack of methods for validation of first principle
models in a structured way

Introduction on Model Predictive Control
Models and model characteristics
Prediction

21
Model Predictive Control
Empirical Modeling

Empirical process models only reflect behavior
observed during testing. The models only have
limited validity and only cover a limited part of
the entire operating envelope of the process
Limitations in describing wide ranges in process
dynamics (slowest vs. fastest time constants)
Limitations in covering large gain ranges over
various directions in input and output spaces
Limited capability in accurately describing
non-linear process dynamics for a broad range of
input trajectories covering the full allowed
input space
Interpolation and especially extrapolation beyond
observed trajectories is risky and mostly results
in large modeling errors

Introduction on Model Predictive Control
Models and model characteristics
Prediction

22
Model Predictive Control

In industrial practice processes are showing
three types of dynamic characteristics with much
potential for performance improvement, if they
can be exploited
Broad frequency range covered by various process
mechanisms and accessible for process operation
and disturbance compensation
Large differences in gain for various
input/output directions in multivariable
processes
Non-linear process behavior both for steady state
as well as for transfer dynamics

Introduction on Model Predictive Control
Models and model characteristics
Prediction

23
Model Predictive Control
Directionality of process transfer

Introduction on Model Predictive Control
Models and model characteristics
Prediction

24
Model Predictive Control
Controller Robustness
Controller performance

Introduction on Model Predictive Control
Models and model characteristics
Prediction

Model inaccuracy
25
Model Predictive Control

Model accuracy, controller performance and
controller robustness need to be selected in line
with the imposed performance specifications over
the entire relevant operating range
steady state gain errors of the model generally
result in slow responses towards targeted steady
state
high frequency errors of the model may result in
instabilities of the controller, if performance
is pushed
tuning of the control system to cope with model
inaccuracies results in stable but sluggish
control with poor disturbance rejection

Introduction on Model Predictive Control
Models and model characteristics
Prediction

26
Model Predictive Control

This course considers model predictive control
systems using linear, causal, time-invariant,
discrete time, finite dimensional models only
with G0(q) - the plant model
F0(q) - the disturbance model
H0(q) - the noise model
y(k) - the output signal
u(k) - the input signal
d0(k) - the known or measured disturbance
signal
e0(k) - the (zero-mean, white) noise signal

The Input Output (IO) model

Introduction on Model Predictive Control
Models and model characteristics
Prediction

27
Model Predictive Control

If we assume G0(q) to be strictly proper, a state
space representation for this system is given by

The corresponding transfer functions are

Introduction on Model Predictive Control
Models and model characteristics
Prediction

28
Model Predictive Control
State Space representation of the IO-model

Introduction on Model Predictive Control
Models and model characteristics
Prediction

29
Model Predictive Control
Working with input signal increments in stead of
input signals has some attractive advantages in
MPC as will become clear later
with Gi(q) - the increment plant
model Fi(q) - the increment disturbance
model Hi(q) - the increment noise
model y(k) - the output signal ?u(k) u(k) -
u(k-1) di(k) d0(k) - d0(k-1) ei(k) e0(k)
- e0(k-1)
The Increment Input Output (IIO) model

Introduction on Model Predictive Control
Models and model characteristics
Prediction

30
Model Predictive Control

The state space representation of the IIO model
is given by

with

Introduction on Model Predictive Control
Models and model characteristics
Prediction

31
Model Predictive Control

The IIO-model has a very positive influence on
the steady state behavior of the controller The
controller will have steady state error 0 due to
integrating behavior.

IO Model

Introduction on Model Predictive Control
Models and model characteristics
Prediction

32
Model Predictive Control

For the IIO model the input increment ?u becomes
zero after the control horizon, due to the fact
that the input u is stabilized at uss

IIO Model

Introduction on Model Predictive Control
Models and model characteristics
Prediction

33
Model Predictive Control

Most industrially applied model predictive
control systems make use of a Finite Impulse
Response (FIR) or a Finite Step Response (FSR)
model
FIR model
with g(j) - the j-th sample of the Impulse
Response
FSR model
with s(j) - the j-th sample of the Step
Response

Introduction on Model Predictive Control
Models and model characteristics
Prediction

34
Model Predictive Control

The relationship between the FIR model and the
FSR model is given by

Introduction on Model Predictive Control
Models and model characteristics
Prediction

35
Model Predictive Control

The complete truncated impulse response model is
given by
The complete truncated step response model is
given by

Introduction on Model Predictive Control
Models and model characteristics
Prediction

Observation This is an IIO model!
36
Model Predictive Control

In order to find the transformation from FIR
model to a corresponding State-Space model the
FIR model is written as

Introduction on Model Predictive Control
Models and model characteristics
Prediction

37
Model Predictive Control

The relation between FIR and state space model is
given by

With

Introduction on Model Predictive Control
Models and model characteristics
Prediction

38
Model Predictive Control

Writing this model as an IIO model gives

With

Introduction on Model Predictive Control
Models and model characteristics
Prediction

39
Model Predictive Control

The IIO state space model representation block
diagram of the FIR model is given by

Introduction on Model Predictive Control
Models and model characteristics
Prediction

40
Model Predictive Control

Transformation of FSR model to state-space model

With

Introduction on Model Predictive Control
Models and model characteristics
Prediction

41
Model Predictive Control

The block diagram of the state space
representation of the FSR model is given by

?u
s(n)
s(n-1)
s(2)
s(1)
x3
x2
x1
Xn
y
q-1
q-1
q-1
ei
t(n)
t(n-1)
t(2)
t(1)

Introduction on Model Predictive Control
Models and model characteristics
Prediction

di
42
Model Predictive Control

Polynomial description of a SISO system

Relates with

Introduction on Model Predictive Control
Models and model characteristics
Prediction

43
Model Predictive Control
The corresponding IIO model is
With

Introduction on Model Predictive Control
Models and model characteristics
Prediction

44
Model Predictive Control

The first step in each control cycle of the Model
Predictive Control system is the Prediction step.
In the prediction step the expected future output
responses of the system are calculated.

Introduction on Model Predictive Control
Models and model characteristics
Prediction

With x(k) - state vector e(k) - zero-mean
white noise v(k) - control vector (u(k) or
?u(k) w(k) - measured external
disturbances z(k) - prediction signal vector
45
Model Predictive Control

At each sample instant k the performance index J
is evaluated over the prediction horizon N by
making a set of j-step ahead predictions

With - the so called free response - the
response on future input signals

Introduction on Model Predictive Control
Models and model characteristics
Prediction

46
Model Predictive Control

The signals that determine the future output
response of the process are
Applied past input signals
Measured past disturbances
Response to noise signal
Future disturbances
Future input signals

Introduction on Model Predictive Control
Models and model characteristics
Prediction

47
Model Predictive Control

Using successive substitution the prediction can
be found to be for the noiseless case

Introduction on Model Predictive Control
Models and model characteristics
Prediction

48
Model Predictive Control

Substitution of the predicted state vector into
prediction vector z(k) gives

Introduction on Model Predictive Control
Models and model characteristics
Prediction

49
Model Predictive Control

Defining the following matrices

Introduction on Model Predictive Control
Models and model characteristics
Prediction

The prediction vector can be written as
50
Model Predictive Control

In case the noise e(k) is no longer assumed to be
equal to zero, but is assumed to be zero mean
white noise, the prediction becomes

Introduction on Model Predictive Control
Models and model characteristics
Prediction

51
Model Predictive Control

Defining in addition

The prediction vector now becomes

Introduction on Model Predictive Control
Models and model characteristics
Prediction

52
Model Predictive Control

The prediction mechanism of the MPC is one of the
three mechanisms that are fundamental for the
performance and for the success of this
technology
Predictions on the basis of non-parametric models
(FIR, FSR, ) as discussed ? Bandwidth
limitations due to limited complexity of the
models
Predictions on the basis of parametric models
(Polynomial, State-Space, ) enable inclusion of
all relevant system dynamics ?
Large bandwidth feasible with limited model
complexity
Connection with (parametric) first principles
based models
Predictions of the free response can be
done using a detailed simulation model that
includes all relevant behavior

Introduction on Model Predictive Control
Models and model characteristics
Prediction

53
Model Predictive Control

To enable predictions using polynomial models we
make use of the following property that holds for
any rational polynomial function

For any rational polynomial
polynomials can be found that satisfy
the following equation

Introduction on Model Predictive Control
Models and model characteristics
Prediction

This is the so called Diophantine equation
54
Model Predictive Control

Using the Diophantine equation predictions of
future outputs of a CARIMA (Controlled
Auto-Regressive, Integrating Moving Average)
models can be calculated

Solving