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An Educational Plant Based on The Quadruple-Tank Process 1
Model Predictive Control for Tracking Constrained Linear Systems Page 1
Model Predictive Control for Tracking Constrained Linear
Systems
D. Limon, I. Alvarado, A. Ferramosca,T. Alamo, E.F. Camacho
Dept. Ingenieria de Sistemas y Automatica Universidad de Sevilla
Model Predictive Control for Tracking Constrained Linear Systems Page 2
Outline
o Motivation
o Theoretical results
MPC for tracking
Optimal MPC for tracking
Robust tube-based MPC for tracking
Output Feedback Robust MPC for Tracking
o Real applications
Linear motor
ACUREX
The quadruple-tank process
o Conclusions and future works
An Educational Plant Based on The Quadruple-Tank Process 2
Model Predictive Control for Tracking Constrained Linear Systems Page 3
Motivation
Issues Constrained variables
Stability guarantee for every
operation point
Change in the dynamics
Traditional control scheme
Target optimizer
Plant
Adaptative strategy
Low-level Control Process
y
u
target
Advanced control
Processes with large changes in the operation point
Model Predictive Control for Tracking Constrained Linear Systems Page 4
Motivation
Proposed control scheme
Target optimizer
Plant
Predictive control
Low-level Control Process
y
u
target
Predictive Control Optimal performance
Constraint satisfaction
Stability guarantee
Robustness
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Model Predictive Control for Tracking Constrained Linear Systems Page 5
Motivation Standard MPC: (Muske and Rawlings., 1993 ;Mayne et al., 2000 )
RTO: For a given target, a fixed steady state xs is selected.
MPC:
where:
A stabilizing design requires that:
K and P such that:
is an admissible invariant set around xs
If the target changes, the feasibility may be lost and the MPC must be redesigned.
Model Predictive Control for Tracking Constrained Linear Systems Page 6
Motivation Loss of feasibility
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Model Predictive Control for Tracking Constrained Linear Systems Page 7
Outline o Motivation
o Theoretical results
MPC for tracking
• Problem Description
• Previous solutions
• Motivation example
• MPC for tracking
• Properties and an example
Optimal MPC for tracking
Robust tube-based MPC for tracking.
Output Feedback Robust MPC for Tracking
o Real applications
o Conclusions and future works
Model Predictive Control for Tracking Constrained Linear Systems Page 8
Problem Description: (Limon et al. Automatica 2008)
Consider the following discrete time LTI system
MPC for tracking
Objective: Given any target yt, design a control law such that:
y(k) tends to yt when k→
x(k) and u(k) are admissible for all k ≥ 0
Linear Process
u • x MPC for tracking
target yt
Target variable
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Model Predictive Control for Tracking Constrained Linear Systems Page 9
MPC for tracking
Some existing solutions to this problem:
• Translation to the new setpoint (Muske and Rawlings., 1993)
• MPC with an infinite horizon (Constrained LQR)
• Reference Governors (Gilbert et al., 1994; Bemporad et al., 1997)
• CSGPC adds an artificial reference as a decision variable and a contraction constraint to ensure the convergence (Rossiter et al., 1996)
• Dual mode strategy for tracking (Chisci and Zappa., 2003)
• The change of reference considered as a disturbance to be rejected (Pannocchia and Kerrigan., 2005)
10
XN
Projx()
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11
Invariant set for tracking
The set of states and targets such that the system starting from that state admissibly converges to the target
Model Predictive Control for Tracking Constrained Linear Systems Page 12
is an invariant set for tracking iff
Invariant set for tracking
The extended state xa is constrained to
Define the system:
Consider the stabilizing control law
Parametrization of the equilibrium point
Set of reachable targets
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Model Predictive Control for Tracking Constrained Linear Systems Page 13
MPC for tracking Standard MPC vs MPC for tracking:
Model Predictive Control for Tracking Constrained Linear Systems Page 14
Theorem:
Consider that is such that is stable
Q>0, R>0, and P such that:
is and admissible invariant set for tracking for the system subject to the following constraints
Let the feasibility region of the optimization problem
MPC for tracking
Then, for any feasible initial state i.e., x N the system is steered asymptotically to any reachable target yt Yt satisfying the constraints
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Model Predictive Control for Tracking Constrained Linear Systems Page 15
MPC for Tracking
Properties of the MPC for tracking with respect to the standard MPC:
1. Changing operation points. Since the constraints set doesn’t depend on the desired steady state and x(k) N for all k, the MPC is feasible for any admissible change of set point at any sample time
2. Larger domain of attraction. Since
3. Offset minimization. If the target is not reachable, then the system will converge to y*
s such that
7. Local optimality. For a parameter T big enough the cost function of the proposed MPC is arbitrarily close to the optimal one
8. Explicit solution. Due to only a QP have to be solved at each sample time, the explicit solution can be calculated
Model Predictive Control for Tracking Constrained Linear Systems Page 16
MPC for Tracking
Example: Consider the discrete time LTI system:
Subject to the following hard constraints:
Controller parameters:
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Model Predictive Control for Tracking Constrained Linear Systems Page 17
MPC for tracking
N=3
Model Predictive Control for Tracking Constrained Linear Systems Page 18
Outline o Motivation
o Theoretical results
MPC for tracking
Optimal MPC for tracking
• Drawback of the MPC for tracking
• Enhanced formulation
• Local optimality
• Example
Robust tube-based MPC for tracking.
Output Feedback Robust MPC for Tracking
o Real applications
o Conclusions and future works
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Model Predictive Control for Tracking Constrained Linear Systems Page 19
Drawback of the MPC for tracking • Potentially loss of local optimality.
• It does not consider as target operating points, states and inputs not consistent with the prediction model.
• Solution: enhanced formulation with a general convex function as offset cost function.
Model Predictive Control for Tracking Constrained Linear Systems Page 20
Enhanced formulation
Properties: • Offset minimization. If the target is not reachable, then the
system will converge to y*s such that
• Local optimality. The cost function of the proposed MPC equals the optimal one
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Model Predictive Control for Tracking Constrained Linear Systems Page 21
Local optimality
• The standard formulation of the MPC for regulation does not ensure the local optimality property, because the cost function is only minimized for a finite prediction horizon.
• Define:
• Then, 8 x 2 Υ(yt), the optimal value of the MPC for regulation equals the optimal one and the control laws are the same.
(Hu and Linnemann, 2002)
Model Predictive Control for Tracking Constrained Linear Systems Page 22
Local optimality (2)
• In the MPC for tracking, this property can be ensured thanks to the convex offset cost function.
• Property: There exists a ®*> 0 such that, for every ®1¸®*: – For all x 2 XNr(yt):
– If K is the one of the unconstrained LQR:
for all x 2 Υ(yt)
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Model Predictive Control for Tracking Constrained Linear Systems Page 23
Determination of ® *
• Consider the following regulation problem, deriving from the tracking problem, with the offset cost function posed as an equality constraint.
• ® * is the maximum of the Lagrange multipliers of the equality constraint of the previous problem, in the set of parameters
xp=(x,yt)2 XN £ Ys
Model Predictive Control for Tracking Constrained Linear Systems Page 24
Determination of ® * (2)
• Consider the standard formulation of a mp-QP problem:
• The Karush-Kuhn-Tacker optimality conditions for this problem are:
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Model Predictive Control for Tracking Constrained Linear Systems Page 25
Determination of ® * (3)
• ®* can be calculated by solving the following optimization problem:
• This problem is not easy to solve because the first constrained, the complementary slacknes, is not convex nor concave.
Model Predictive Control for Tracking Constrained Linear Systems Page 26
Optimal MPC for Tracking
Example: Consider the two cascade tanks system:
Subject to the following hard constraints:
Controller parameters:
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Model Predictive Control for Tracking Constrained Linear Systems Page 27
Optimal MPC for Tracking
Example: State and time evolution
Model Predictive Control for Tracking Constrained Linear Systems Page 28
Optimal MPC for Tracking
Example: Local optimality
The value of the Lagrange multiplier is 14.6611
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Model Predictive Control for Tracking Constrained Linear Systems Page 29
Outline o Motivation
o Theoretical results
MPC for tracking
Optimal MPC for tracking
Robust tube-based MPC for tracking
• Problem Description
• Preliminary Results
• Robust MPC for tracking
• Properties and an example
• Calculation of K
Output Feedback Robust MPC for Tracking
o Real applications
o Conclusions and future works
Model Predictive Control for Tracking Constrained Linear Systems Page 30
Robust MPC for Tracking
Consider the following discrete time LTI system with additive bounded uncertainties:
Objective: Given any admissible setpoint s, design a control law such that:
y(k) tends to the neighbourhood of yt when k→
x(k) and u(k) are admissible for all k ≥ 0 and all possible realizations of
Problem description
The system is constrained to: Linear
Process u x Robust MPC
for tracking
w
target yt
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Model Predictive Control for Tracking Constrained Linear Systems Page 31
Robust MPC for tracking
Existing solutions to the Robust tracking problem:
• Dual mode strategy (Rossiter et al., 1996; Chisci and Zappa, 2003)
• In the context of uncertain systems, the change of reference considered as a disturbance to be rejected. (Pannocchia 2004; Pannocchia et al., 2005; Magni et al. 2001)
• Reference Governors (Gilbert et al., 1999; Bemporad et al., 1997)
Existing solutions to the Robust MPC:
• Min-Max
• Tube-Based robust MPC
• Stochastic approach
• LMI based solution
• It is based on nominal predictions
• All the computational load is made offline, suitable for fast systems
• Simple implementation, only requires the solution of a QP
Model Predictive Control for Tracking Constrained Linear Systems Page 32
Robust MPC for Tracking
Lemma (Langson 2004)
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Model Predictive Control for Tracking Constrained Linear Systems Page 33
• Considering the tighter set of constraints for the nominal system
The tube: (Langson 2004 ; Bertsekas 1972)
Robust MPC for Tracking
(Mayne et al., 2005)
Model Predictive Control for Tracking Constrained Linear Systems Page 34
Robust MPC for Tracking MPC for tracking vs Robust MPC for tracking:
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Model Predictive Control for Tracking Constrained Linear Systems Page 35
Robust MPC for Tracking
Then, for any feasible initial state i.e., x N and any reachable target, the uncertain system is steered asymptotically to the set for all possible realization of the disturbances, satisfying the constraints
Theorem: Consider that
is such that is stable
Q>0, R>0, and P such that:
at is an admissible invariant set for tracking for the nominal system
subject to the following constraints K is such that (A+BK) is stable and are not empty sets
Let be the feasibility region of the optimization problem
Model Predictive Control for Tracking Constrained Linear Systems Page 36
Consider the uncertain LTI discrete time system described by the matrices:
Subject to the following hard constraints:
The disturbance set is:
The controller parameters are:
Robust MPC for Tracking
Example:
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Model Predictive Control for Tracking Constrained Linear Systems Page 37 Model Predictive Control for Tracking Constrained Linear Systems Page 38
Cancellation of the tracking error: In the case that the disturbance w tends to a constant value
Robust MPC for Tracking
Linear Process
u x q Robust MPC for tracking
w
Disturbance Estimator
w ^ Hs
N yt ytc
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Model Predictive Control for Tracking Constrained Linear Systems Page 39
Robust MPC for Tracking
0 50 100 150 -10 -5
0 5
10 Output and Reference evolution
0 50 100 150 -0.5
0
0.5 Control action evolution
0 50 100 150 -1 -0.5
0 0.5
1 Disturbance evolution p.u.
Model Predictive Control for Tracking Constrained Linear Systems Page 40
Outline
o Motivation
o Theoretical results
MPC for tracking
Optimal MPC for tracking
Robust tube-based MPC for tracking
Output Feedback Robust MPC for Tracking
• Problem Description
• Preliminary Results
• Robust MPC for tracking
• Properties and an example
o Real applications
o Conclusions and future works
An Educational Plant Based on The Quadruple-Tank Process 21
Model Predictive Control for Tracking Constrained Linear Systems Page 41
Problem Description Consider the following uncertain discrete time LTI system
Objective: Given any admissible setpoint s, design a control law such that:
y(k) tends to neighbourhood of yt when k→
x(k) and u(k) are admissible for all k ≥ 0 and possible realization of and
Output feedback MPC for Tracking
The system is constrained to:
Linear Process
u y Robust MPC for tracking
Estimator x ^
w u target
yt
Model Predictive Control for Tracking Constrained Linear Systems Page 42
Output feedback robust MPC for tracking
Existing solutions to the Robust tracking problem:
• Standard stable estimator (that provides a measure of the state) plus a
stable robust controller (Magni et al. 2001)
• Using a set-membership state estimator plus a robust controller that takes
into account the set
1. MPC for tracking for constrained linear systems (Bemporad and
Garrulli, 1997). This MPC uses a switching strategy, because of that this is suboptimal solution
2. Reference governor proposed by (Angeli, Casavola and Mosca, 2001). It doesn’t takes into account the performance
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Model Predictive Control for Tracking Constrained Linear Systems Page 43
Output feedback MPC for Tracking
Let the observer system be
Let the state estimation error ee be defined by:
The state estimation error satisfies:
If AL is Hurtwitz and the disturbances are bounded, there exists an RPI
Thus if then
Preliminary results Estimation tube (Mayne et al., 2006)
Model Predictive Control for Tracking Constrained Linear Systems Page 44
Output feedback MPC for Tracking
Let the control error be
Then applying the control law:
The error satisfies:
If AK Hurwitz and the disturbances are bounded, there exist an RPI
Thus if then
Preliminary results
Control tube (Mayne et al., 2006)
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Model Predictive Control for Tracking Constrained Linear Systems Page 45
Considering the tighter set of constraints for the
nominal system
The additional stabilizing constraint
If and then
applying
Output feedback MPC for Tracking Resulting tube
(Mayne et al., 2006)
Model Predictive Control for Tracking Constrained Linear Systems Page 46
Output feedback MPC for Tracking Robust MPC for tracking vs Output feedback Robust MPC for tracking:
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Model Predictive Control for Tracking Constrained Linear Systems Page 47
Theorem: Consider that
is such that is stable
Q>0, R>0, and P such that:
is an admissible invariant set for tracking for the nominal system subject to the following constraints T>0
K is such that (A+BK) is stable and are non-empty sets
The initial estimation error must be inside of the set ee
Let be the feasibility region of the optimization problem
Output feedback MPC for Tracking
Then, for any feasible initial estimated state i.e., and for any reachable target, the system is steered asymptotically to the set for all possible realizations of w and v satisfying the constraints
Model Predictive Control for Tracking Constrained Linear Systems Page 48
Output feedback MPC for Tracking Example Consider the LTI discrete time system:
Subject to the following hard constraints and the disturbance sets are:
The controller parameters:
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Model Predictive Control for Tracking Constrained Linear Systems Page 49
Output feedback MPC for Tracking
50
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Model Predictive Control for Tracking Constrained Linear Systems Page 51
Cancellation of the tracking error:
In the case that the disturbance w and v tends to a constant value
Output feedback MPC for Tracking
Linear Process
u y Robust MPC for tracking
Estimator
yt
x ^
w u
Disturbance Estimator
F
g ^
ytc
Model Predictive Control for Tracking Constrained Linear Systems Page 52
Output feedback MPC for Tracking
0 10 20 30 40 50 60 -50 -40 -30 -20 -10
0 10 Output and Reference evolution
0 10 20 30 40 50 60 -10
0
10 Control action evolution u u n
0 10 20 30 40 50 60 -1
0
1 Disturbance evolution p.u. w 1 w 2 v
yt
ys y
ytc
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Model Predictive Control for Tracking Constrained Linear Systems Page 53
Outline
o Motivation
o Theoretical results
o Real applications
Linear Motor
• Linear Positioning system
• The linear motor controlled by MPC for tracking
• The linear motor controlled by robust MPC for tracking
ACUREX
The Quadruple-Tank process
o Conclusions and future works
Model Predictive Control for Tracking Constrained Linear Systems Page 54
Linear Motor
The linear positioning system:
The controlled plant is a positioning system driven by a linear motor Linear
Positioning
sensor
Rails
Primary
Secondary
• Linear Motor: Siemens 1FN3 050-2W00-0AA
• Absolute position sensor: LC181 of Heidenhein. Precision of 5m • dSpace card in a PC: dSpace card DS1103 PPC, based on PowerPC
processor 604e that works at 400 MHz. This processor is programmed in Simulink using Real-Time Interface
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Model Predictive Control for Tracking Constrained Linear Systems Page 55
Linear Motor
The linear motor controlled by the MPC for tracking: From the response of the system to a PRBS, a linear discrete time model has been identified using least squares identification. The sample time is 10ms
The constraints:
The controller parameters are: Explicit solution:
The resulting MPC controller is defined by 514 regions. A binary search tree has been used with a depth of 13
Model Predictive Control for Tracking Constrained Linear Systems Page 56
Linear Motor
The linear motor controlled by the robust MPC for tracking: From the response of the system to a PRBS, a linear discrete time model has been identified using least squares identification. The sample time is 30ms
The constraints:
The controller parameters are: The disturbance set:
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Model Predictive Control for Tracking Constrained Linear Systems Page 57
Outline o Motivation
o Theoretical results
o Real applications
Linear Motor
ACUREX
• Acurex overview
• Identification
• Acurex controlled by RMPCT
• Acurex controlled by RMPCT with the output cancelation loop
The Quadruple-Tank process
o Conclusions and future works
Model Predictive Control for Tracking Constrained Linear Systems Page 58
ACUREX ACUREX is a solar plant that is located in the PSA of Almería in Taberna desert.
• The purpose of this plant is to produce hot oil that can be used to produce high pressure steam for an electrical turbine or for a desalination plant
• The control goal is keeping the oil’s temperature close to the reference actuating on the flow despite the disturbances.
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Model Predictive Control for Tracking Constrained Linear Systems Page 59
ACUREX
The model that minimize the less square method error is a first order model without delay
Identification: Disturbances: • Solar radiation • Inlet oil temperature • Ambient temperature
Model Predictive Control for Tracking Constrained Linear Systems Page 60
ACUREX
Identification: To determine the set W the output of the model is compared with the real output as in the figure
Using the stored data of previous controllers W is obtained
The constraints sets are:
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Model Predictive Control for Tracking Constrained Linear Systems Page 61
ACUREX
ACUREX controlled by robust MPC for tracking:
Pyrometer sensor error disturbance:
This experiment demonstrates the robustness of the proposed controller to an additional and unmodelled disturbance Controller parameters:
Model Predictive Control for Tracking Constrained Linear Systems Page 62
ACUREX
11.2 11.4 11.6 11.8 12 12.2 12.4 12.6 12.8 13 220 230 240 250 260
Local time
MPC#1 T out T ref Tref art
11.2 11.4 11.6 11.8 12 12.2 12.4 12.6 12.8 13 220 240 260 280
Local time
Control Action
Tff
11.2 11.4 11.6 11.8 12 12.2 12.4 12.6 12.8 13 -50 0
50 100
Local time
Disturbances
Pyrometer sensor error
Rad/10 Rad cor /10 w est × 10 T in /10
Clouds
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Model Predictive Control for Tracking Constrained Linear Systems Page 63
ACUREX
ACUREX controlled by robust MPC for tracking with the output cancellation loop: Tin disturbance: A disturbance on the temperature at
the input of the collectors was introduced
The offset is removed due to the output cancellation loop
Controller parameters:
Model Predictive Control for Tracking Constrained Linear Systems Page 64
ACUREX
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Model Predictive Control for Tracking Constrained Linear Systems Page 65
Outline
o Motivation
o Theoretical results
o Real applications
Linear Motor
ACUREX
The Quadruple-Tank Process
• Description of the plant
• Identification
• The Quadruple-Tank Process controlled by robust MPC for tracking
o Conclusions and future works
Model Predictive Control for Tracking Constrained Linear Systems Page 66
The Quadruple-Tank process
Interesting features:
1. The linearized model has multivariable zeros which can be located in either the right or left half-plane by simply changing a couple of valves.
2. All the states are measurable.
3. The outputs are strongly coupled.
4. The system is nonlinear.
5. The states and inputs are constrained.
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Model Predictive Control for Tracking Constrained Linear Systems Page 67
The Quadruple-Tank process
Johansson, 2000
Model Predictive Control for Tracking Constrained Linear Systems Page 68
The Quadruple-Tank Process
Linearizing the model:
The system is open loop stable with 2 multivariable zeros. The nature of these zeros is determined by parameters γa and γb as follows:
• If 0≤γa+γb<1 The system has Right Half Plane transmission zeros (RHPZ)
• If 1<γa+γb≤2 The system has Left Half Plane transmission Zeros (LHPZ)
The sign of the real part of the zeros does not depend on the operating point.
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Model Predictive Control for Tracking Constrained Linear Systems Page 69
The Quadruple-Tank Process Identification:
The cross-section of the outlet holes can be adjusted, and the rest of the parameters are determined, so the only parameter to be identified is the set of the disturbances
Main source of disturbances:
• The linearization approximation error
• The discharge parameters are not constant
• The actuator dynamics
Model Predictive Control for Tracking Constrained Linear Systems Page 70
The Quadruple-Tank Process
Controller parameters:
The gain K and the minimal RPI have been calculated using the procedure aforementioned
Plant parameters:
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Model Predictive Control for Tracking Constrained Linear Systems Page 71
The Quadruple-Tank Process
Model Predictive Control for Tracking Constrained Linear Systems Page 72
The Quadruple-Tank Process
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Model Predictive Control for Tracking Constrained Linear Systems Page 73
The Quadruple-Tank Process Using the offset cancellation loop
Model Predictive Control for Tracking Constrained Linear Systems Page 74
Conclusions & Future works
Extension to other robust MPC techniques (min-max) Address the problem of tracking arbitrary references Generalization to more complex systems
Piece-wise affine systems Nonlinear systems
Novel MPC for tracking Feasibility under any change of the target Single QP Larger domain of attraction
Robust MPC for tracking based on tubes Nominal predictions Output feedback Offset-free control
Real applications