MPC-based tracking with time-invariant constraints: an academic example

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The Shell Eco-Marathon around the world

The Shell Eco-Marathon also exists in America and Asia. The Shell Eco-Marathon Americas was launched in 2007 in the United States, involving teams from Canada to Brazil. The Shell Eco-Marathon Asia started in 2010, in Malaysia [3]. In Table 1.4, the Shell Eco-Marathon 2014 results for Electric Battery Prototype.

Electric vehicle dynamics

The dynamics of the vehicle can be described in terms of the force of trac-tion Ftraction[N], the external forces due to the aerodynamics Faerodynamics[N], the rolling (contact wheel-ground) resistance Frolling[N], and the slope resistance Fslope[N] due to the vehicle’s weight and the road slope θ[rad] (see Fig. 2.2) [60, 32]. All this forces are related by Newton’s laws of motion involving the mass m[kg] of the vehicle and its acceleration dx2(t) [m/s2] as follows dt m dx2(t) = Ftraction(t) − Faerodynamics(t) − Frolling(t) − Fg(t), (2.1).

Nonlinear grey-box identification

In System Identification, the mathematical model of a system can be estimated by using three kinds of information: the knowledge on the structure of the model (structural knowledge) or the physical knowledge obtained from first principles, the data taken from intentional experiments performed on the system, and the assumptions made over the validity of the model [34].
If the model of the system is found using only structural knowledge or physical knowledge, then the model is a white-box model [34]. The white-box models are usually described by differential equations. If the models are estimated to fit the experimental data regardless of the structure of the model, then they are black-box models [34]. In between are the grey-box models, in which the structure of the model is known a priori and the data are used to estimate the values of the unknown parameters of the model [34, 48].
The System Identification ToolboxTM of Matlab R allows to perform grey-box identification. The continuous-time nonlinear differential equations of the model having been properly defined, the unknown parameters can be identified using iterative Prediction-Error Minimization techniques (PEM) for continuous-time linear and nonlinear models [48].

Parameter estimation

To perform the estimation of the unknown parameters η, Cr and CdAf , several accelerating and decelerating tests were performed to collect the data required for the identification process. In Fig. 2.5, the initial experimental data for a flat path (θ = 0) used to perform the nonlinear grey-box identification are depicted in black.
The known parameters are kt = 0.0604Nm/A, gr = 8.5, m = 90kg, rw = 0.24m, ρ = 1.225kg/m3 and g = 9.81m/s2, and the known continuous-time non-linear structure of the model is the one given by (2.6). Then, the parameters η, Cr and the product CdAf have been identified with the PEM method. The best fit to the experimental data has given η = 0.97, CdAf = 0.1031m2 and Cr = 8.1549 × 10−4. (2.8).

Low consumption driving strategy

The dynamics of the vehicle being fully identified, the problem of the energy-management can now be addressed. The problem amounts to defining how using the available energy sources so that the energy efficiency can be maximized. The previous question can be rephrased as how the vehicle must be driven so that the minimum quantity of energy is used during the driving task (a driving task can be single or multiple repetitions of a prescribed circuit, or a common route [61]). The answer to this questions is precisely the driving strategy [60].
Since the battery provides all the traction power (see Fig. 2.3), the problem of the low consumption strategy becomes an electrical resource management prob- lem, where the energy level of the battery (only the discharging of battery being taken into account) is the critical variable in the formulation of the optimization problem that leads to the driving strategy solution [41, 61].

Constraints of the Optimality problem

The search for the driving strategy is an optimization problem which, given the model of the vehicle, the road profile (slope, curves, straight lines, etc.) and the constraints in terms of maximum velocity allowed at each curve, maximum time of the race, and total number of kilometres, must provide (x∗1(t), x∗2(t), Ibatt∗(t)) at time t. The optimization problem must consider the following constraints:
• The maximum time tfmax [s] allowed to complete the race.
• The total distance x1total [m] to run.
• The maximum battery current Ibattmax [A].
• The maximum speed x2max [m/s2] for the vehicle.
• The limits of the centrifugal force Fc[N].
In a curve, the centrifugal force Fc(N ) over the car is Fc = m(x2curve )2/rcurve, with rcurve[m] the radius of the curve (see Fig. 2.6) and x2curve [m/s2] the velocity of the vehicle in the curve. This force must not be larger than the total frictions forces wheel-road Ft[N] to prevent the car slipping over, i.e. x2curve ≤ Ftrcurve/m [60].
Notice that the constraint over the velocity x2curve in the curves is concerned exclusively to the position x1 where there is precisely a curve, otherwise the constraint is only x2 ≤ x2max. As a result, one has x2 (t) ≤ x2 max, if the vehicle is in a straight line, (2.14).
The optimality problem that includes the aforementioned constraints and that leads to the driving strategy for driving through a known path in a finite time tf , with the minimum electrical consumption, is stated in the following.

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Table of contents :

1 General Introduction 
1.1 The European Shell Eco-Marathon
1.1.1 Categories of participation
1.1.2 The Shell Eco-Marathon around the world
1.2 The EcoMotionTeam
1.3 Motivation of the work
1.4 Outline
2 The Low Consumption Vir’volt Electric Vehicle 
2.1 Introduction
2.2 The Vir’volt prototype
2.2.1 Electric vehicle dynamics
2.2.2 Parameter identification
2.2.2.1 Nonlinear grey-box identification
2.2.2.2 Parameter estimation
2.3 Low consumption driving strategy
2.3.1 Energetic considerations
2.3.2 Optimization problem
2.3.2.1 Constraints of the Optimality problem
2.3.2.2 The Multi-phase Optimality problem
2.4 Nonlinear discrete-time model
2.5 Real-time tracking of the optimal driving strategy
2.5.1 Linearised model
2.5.2 Linear Parametric Varying model
2.6 Benchmark
2.7 Conclusions
3 Tracking Model Predictive Control 
3.1 Introduction
3.2 Preliminaries
3.2.1 Polytopic constraints
3.2.2 Problem formulation
3.3 Design of the invariant terminal set
3.4 MPC-based tracking with time-invariant constraints: an academic example
3.4.1 Computation of the terminal invariant set
3.4.2 Closed loop response
3.5 MPC-based tracking with time-invariant constraints: application to the Vir’volt vehicle
3.6 Conclusions
4 Tracking under time-varying polytopic constraints 
4.1 Introduction
4.2 Preliminaries
4.3 Homothetic transformation of the invariant set
4.3.1 Principle of the homothetic transformation
4.3.2 Computation of the homothetic factor
4.4 MPC with homothetic transformation of the invariant set
4.5 MPC-based tracking with time-varying constraints: an academic example
4.5.1 Problem statement
4.5.2 Results
4.5.3 Computational resources
4.6 MPC-based tracking with time-varying constraints: application to the Vir’volt vehicle
4.6.1 Results
4.7 Conclusions
5 Real-time Robust Model Predictive Control for LPV systems 
5.1 Introduction
5.2 Problem formulation
5.3 Robust constrained MPC for LPV systems
5.3.1 Explicit MPC for LPV systems with off-line computation of LMIs
5.3.1.1 Asymptotically stable invariant ellipsoids
5.3.1.2 The explicit MPC algorithm
5.4 Robust MPC for LPV systems: application to the Vir’volt vehicle
5.4.1 Robust MPC for LPV systems with on-line computation of LMIs
5.4.2 Explicit MPC for LPV systems with off-line computation of LMIs
5.4.3 Comparison of the fully on-line MPC and explicit MPC .
5.5 MPC for LPV systems with Parameter dependent Lyapunov function
5.6 Explicit MPC using the PDLF
5.6.1 The asymptotically stable invariant set
5.6.2 Algorithm of the explicit MPC based on PDLF
5.7 Robust MPC for LPV systems with PDLF: application to the Vir’volt vehicle
5.7.1 Robust MPC with PDLF with on-line computation of LMIs
5.7.2 Explicit MPC for LPV using PDLF
5.7.3 Comparison of fully on-line MPC based on PDLF and explicit MPC based on PDLF
5.8 Explicit MPC for LPV using PDLF: application to the benchmark
5.9 Conclusions
6 Robust adaptive real-time control based on an on-off driving srategy 
6.1 Introduction
6.2 Parameter identification
6.2.1 Problem formulation
6.2.2 Identification of parameter a
6.2.2.1 Algorithm of the off-line identification
6.2.3 Identification of parameters b and c
6.3 Low consumption driving strategy
6.3.1 Problem formulation
6.3.2 Former results
6.3.3 Periodic low consumption driving strategy
6.4 Robust adaptive real-time control based on an on-off driving strategy: application to the Vir’volt vehicle
6.4.1 Off-line identification of the parameter a
6.4.2 On-line adaptative real-time control
6.5 Conclusions
7 General conclusions and Perspectives 
A Rotterdam’s Ahoy circuit 
B Proofs of Chapter 5 
B.1 Schur complement
B.2 Proof of equation (5.27)
B.3 Proof of equation (5.28)
B.4 Proof of equation (5.31)
B.5 Proof of equation (5.32)
C Proofs of Chapter 6 179
C.1 Proof of equation (6.13)
References 

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