From d7733bd1b47530b5f6ddb9e3cfbad8922d5c9ae8 Mon Sep 17 00:00:00 2001 From: marinapotashenkova Date: Mon, 26 Feb 2018 12:27:39 +0300 Subject: [PATCH] issue #56: Changed the order of examples --- doc/chap_examples.rst | 437 +++++++++++++++++++++--------------------- 1 file changed, 218 insertions(+), 219 deletions(-) diff --git a/doc/chap_examples.rst b/doc/chap_examples.rst index 2d971b94..3f314a01 100644 --- a/doc/chap_examples.rst +++ b/doc/chap_examples.rst @@ -527,224 +527,7 @@ to the green region. Estimation with the Cell Transmission Model. In *Proceedings of the American Control Conference*, 3750–3755. Denver, Colorado, USA. - -LMI-based three-vehicle platoon -------------------------------- - -A platoon of vehicles includes typically a leader and a number -of followers. In a controlled platoon the controllers are -designed to maintain constant relative distances between -autonomous vehicles by tracking the trajectory of the -leader. The latter is manually driven and can be considered -as a reference input to the whole system. We are concerned -with the longitudinal control of a platoon of vehicles engaged -in following each other longitudinally by exchanging -information via a wireless local area network (WLAN) (see -Fig.1). - -.. _platoonfig1: - -.. figure:: /pic/chapter06_section08_platoon.png - :alt: platoon - :width: 50 % - - Platoon struture and notations. - -The spacing errors :math:`e_i` are defined as the difference -between the actual distance to the predecessor and a -(fixed) reference distance: -:math:`e_i(t) = d_i(t) - dref_i.` -Bounds for these reference distances will be stipulated by the result of -our safety verification. The effective acceleration ai of each -vehicle within the platoon is governed by the drivetrain -dynamics.According to Fig. 1 and with -the further approximation the resulting platoon model is -given by: - -.. math:: - :label: platoon1 - - \ddot{e}_i & = a_{i-1} - a_i. - -.. math:: - :label: platoon2 - - a_i & = -\frac{a_i}{\tau_i} + \frac{v_i}{\tau_i} - -where :math:`\tau_i` is the time constant of the drivetrain considered -here to be constant and vi the input signal. The -dynamics of the whole platoon with a state vector -:math:`x =[. . . e_i, \dot{e}_i, a_i . . .]^T` -can be expressed in state space form as -follows: - -.. math:: - :label: platoon3 - - \dot{x} & = A_sx+B_1a_L+B_2v. - -where the leading vehicle’s acceleration aL enters the dynamics -as a disturbance. The goal thereby is to stabilize -the platoon and realize a good disturbance rejection in -terms of small spacing errors at reasonable control effort. -These constraints comprise in particular maximum (absolute) -spacing errors to prevent collisions among platoon -members but also maximum amplification of velocity or -acceleration values to account for the existing saturation -effects that arise due to force limitation between road -and tire. This optimal control problem is applied to a -state feedback structure assuming that each vehicle has -information access to all other vehicles states. We obtain -as result an optimal matrix K verifying: - - -.. math:: - :label: platoon4 - - v & = K x - -The closed loop system is hence given by: - -.. math:: - :label: platoon5 - - \dot{x} & = Ax + Bu. - -where :math:`A = (A_s + B_2K), B = B_1` and :math:`u = a_L`. - -The main goal of this work, is to investigate the impact -of disturbances of the communication network on the performance -of the cooperative platoon. We are particularly -interested in worst cases, in which a loss of communication -between two/many or all vehicles occurs. The theory of -hybrid systems offers a convenient framework to model this -kind of systems. A hybrid automaton consists of states described -by continuous dynamics and discrete events which -trigger transitions between these states. Our application -can be modeled by a hybrid automaton. The controlled -platoon dynamics constitute thereby the continuous states -and the communication breakdowns trigger the discrete -switches from one continuous state to another. The interconnection -topology within the platoon is modeled with -a directed graph :math:`G = (V,E)`, defined by vertices V and -edges :math:`E`. The ith vertex represents the ith vehicle and -the edge (i, j) indicates that vehicle j receives information -from vehicle i. This graph is represented by the adjacent -matrix :math:`R = [r_{ij}]` referred to as the communication matrix -of the platoon. - -To take into account the communication failures in the -controller design, the loss of information is expressed by -forcing zeros in :math:`K`. Depending on the topology and the configuration -of the communication between vehicles given by -the matrix :math:`R`, many communication scenarios are possible. -Consequently, the hybrid automaton modeling this kind -of system will be complex. We focus our study on safetycritical -worst case scenarios.We consider the worst case in -Fig.2, in which the system switches from a full to a total -dropout of the communication between the vehicles within -the platoon. In general, our controlled hybrid automaton -has continuous states. - -.. _platoonfig2: - -.. figure:: /pic/chapter06_section08_automat.png - :alt: automat - :width: 50 % - - Hybrid automata modeling the worst case scenario. - -To each continuous state :math:`q` corresponds a new :math:`K_q` and consequently new -matrices :math:`A_q` and :math:`B_q` verifying the equation - -.. math:: - :label: platoon6 - - \dot{x}(t) = A_qx(t) + B_qu(t) - -where :math:`x(t) \in R^9` denotes the state vector, :math:`u(t) = a_L \in R` is - -the input vector and :math:`q \in {1, 2}` is the mode described -by :math:`(A_q,B_q) \in R^{9 \times 9} \times R^9.` - -Next, we give the brief description of the algorithm of the program and the program code itself. -First, we define the matrices and parameters of the linearized system (matrices, vectors, intervals). -Next, we solve the resulting system by methods of the Ellipsoidal Toolbox. -We build a tube of reachability, and simultaneously evolve the system where the situation requires. -Then we project the reach tube to the previously defined hyperplanes. -To test the system for a collision, we will follow the intersection of the tube and the hyperplane :math:`l_{1},\; l_{2},\; l_{3}`. -That is, if there is a crossing, then a collision is inevitable. - -.. literalinclude:: ../products/+elltool/+doc/+snip//s_chapter06_section08_snippet01.m - :language: matlab - :linenos: - -to demonstrate a qualitatively different situation in the further consideration of the collision problem, -let us set the other parameters of the system: - -.. literalinclude:: ../products/+elltool/+doc/+snip//s_chapter06_section08_snippet03.m - :language: matlab - :linenos: - -Construction of reachability tube: - -.. literalinclude:: ../products/+elltool/+doc/+snip//s_chapter06_section08_snippet02.m - :language: matlab - :linenos: - - -.. _tubefig1: - -.. figure:: /pic/chapter06_section08_tube1.png - :alt: tube1 - :width: 50 % - - Reach set for the first block of parameters. - - -.. _tubefig2: - -.. figure:: /pic/chapter06_section08_tube2.png - :alt: tube2 - :width: 50 % - - Reach set for the second block of parameters. - -Solving collision problem: - -.. literalinclude:: ../products/+elltool/+doc/+snip//s_chapter06_section08_snippet06.m - :language: matlab - :linenos: - - -.. literalinclude:: ../products/+elltool/+doc/+snip//s_chapter06_section08_snippet04.m - :language: matlab - :linenos: - -.. _colfig1: - -.. figure:: /pic/chapter06_section08_col1.png - :alt: col1 - :width: 50 % - - Result of solving collision problem. There are 3 intersections of planes of :math:`(e_1,e_2),\; (e_2,e_3),\; (e_3,e_1)` (red color) and our - reach set of system (blue color). According to geometrical interpretation of system this intersections means the conclisions. - - -.. raw:: html -

References

- -.. [1] Ibtissem Ben Makhlouf, Hilal Diab, Stefan Kowalewski. Safety Verification of a Controlled - Cooperative Platoon Under Loss of - Communication Using Zonotopes. Proceedings of the 4th IFAC Conference on - Analysis and Design of Hybrid Systems (ADHS 12) - June 6-8, 2012 - -.. [2] Jan P. Maschuw, Günter C. Keßler, D. Abel. LMI-based control of vehicle platoons for - robust longitudinal guidance. Proceedings of the 17th World Congress - The International Federation of Automatic Control - Seoul, Korea, July 6-11, 2008 - + Two-link planar manipulator ------------- @@ -1024,4 +807,220 @@ and taking its projection onto :math:`(z_1, z_2)` subspace. :language: matlab :linenos: - \ No newline at end of file + +LMI-based three-vehicle platoon +------------------------------- + +A platoon of vehicles includes typically a leader and a number +of followers. In a controlled platoon the controllers are +designed to maintain constant relative distances between +autonomous vehicles by tracking the trajectory of the +leader. The latter is manually driven and can be considered +as a reference input to the whole system. We are concerned +with the longitudinal control of a platoon of vehicles engaged +in following each other longitudinally by exchanging +information via a wireless local area network (WLAN) (see +Fig.1). + +.. _platoonfig1: + +.. figure:: /pic/chapter06_section08_platoon.png + :alt: platoon + :width: 50 % + + Platoon struture and notations. + +The spacing errors :math:`e_i` are defined as the difference +between the actual distance to the predecessor and a +(fixed) reference distance: +:math:`e_i(t) = d_i(t) - dref_i.` +Bounds for these reference distances will be stipulated by the result of +our safety verification. The effective acceleration ai of each +vehicle within the platoon is governed by the drivetrain +dynamics.According to Fig. 1 and with +the further approximation the resulting platoon model is +given by: + +.. math:: + :label: platoon1 + + \ddot{e}_i & = a_{i-1} - a_i. + +.. math:: + :label: platoon2 + + a_i & = -\frac{a_i}{\tau_i} + \frac{v_i}{\tau_i} + +where :math:`\tau_i` is the time constant of the drivetrain considered +here to be constant and vi the input signal. The +dynamics of the whole platoon with a state vector +:math:`x =[. . . e_i, \dot{e}_i, a_i . . .]^T` +can be expressed in state space form as +follows: + +.. math:: + :label: platoon3 + + \dot{x} & = A_sx+B_1a_L+B_2v. + +where the leading vehicle’s acceleration aL enters the dynamics +as a disturbance. The goal thereby is to stabilize +the platoon and realize a good disturbance rejection in +terms of small spacing errors at reasonable control effort. +These constraints comprise in particular maximum (absolute) +spacing errors to prevent collisions among platoon +members but also maximum amplification of velocity or +acceleration values to account for the existing saturation +effects that arise due to force limitation between road +and tire. This optimal control problem is applied to a +state feedback structure assuming that each vehicle has +information access to all other vehicles states. We obtain +as result an optimal matrix K verifying: + + +.. math:: + :label: platoon4 + + v & = K x + +The closed loop system is hence given by: + +.. math:: + :label: platoon5 + + \dot{x} & = Ax + Bu. + +where :math:`A = (A_s + B_2K), B = B_1` and :math:`u = a_L`. + +The main goal of this work, is to investigate the impact +of disturbances of the communication network on the performance +of the cooperative platoon. We are particularly +interested in worst cases, in which a loss of communication +between two/many or all vehicles occurs. The theory of +hybrid systems offers a convenient framework to model this +kind of systems. A hybrid automaton consists of states described +by continuous dynamics and discrete events which +trigger transitions between these states. Our application +can be modeled by a hybrid automaton. The controlled +platoon dynamics constitute thereby the continuous states +and the communication breakdowns trigger the discrete +switches from one continuous state to another. The interconnection +topology within the platoon is modeled with +a directed graph :math:`G = (V,E)`, defined by vertices V and +edges :math:`E`. The ith vertex represents the ith vehicle and +the edge (i, j) indicates that vehicle j receives information +from vehicle i. This graph is represented by the adjacent +matrix :math:`R = [r_{ij}]` referred to as the communication matrix +of the platoon. + +To take into account the communication failures in the +controller design, the loss of information is expressed by +forcing zeros in :math:`K`. Depending on the topology and the configuration +of the communication between vehicles given by +the matrix :math:`R`, many communication scenarios are possible. +Consequently, the hybrid automaton modeling this kind +of system will be complex. We focus our study on safetycritical +worst case scenarios.We consider the worst case in +Fig.2, in which the system switches from a full to a total +dropout of the communication between the vehicles within +the platoon. In general, our controlled hybrid automaton +has continuous states. + +.. _platoonfig2: + +.. figure:: /pic/chapter06_section08_automat.png + :alt: automat + :width: 50 % + + Hybrid automata modeling the worst case scenario. + +To each continuous state :math:`q` corresponds a new :math:`K_q` and consequently new +matrices :math:`A_q` and :math:`B_q` verifying the equation + +.. math:: + :label: platoon6 + + \dot{x}(t) = A_qx(t) + B_qu(t) + +where :math:`x(t) \in R^9` denotes the state vector, :math:`u(t) = a_L \in R` is + +the input vector and :math:`q \in {1, 2}` is the mode described +by :math:`(A_q,B_q) \in R^{9 \times 9} \times R^9.` + +Next, we give the brief description of the algorithm of the program and the program code itself. +First, we define the matrices and parameters of the linearized system (matrices, vectors, intervals). +Next, we solve the resulting system by methods of the Ellipsoidal Toolbox. +We build a tube of reachability, and simultaneously evolve the system where the situation requires. +Then we project the reach tube to the previously defined hyperplanes. +To test the system for a collision, we will follow the intersection of the tube and the hyperplane :math:`l_{1},\; l_{2},\; l_{3}`. +That is, if there is a crossing, then a collision is inevitable. + +.. literalinclude:: ../products/+elltool/+doc/+snip//s_chapter06_section08_snippet01.m + :language: matlab + :linenos: + +to demonstrate a qualitatively different situation in the further consideration of the collision problem, +let us set the other parameters of the system: + +.. literalinclude:: ../products/+elltool/+doc/+snip//s_chapter06_section08_snippet03.m + :language: matlab + :linenos: + +Construction of reachability tube: + +.. literalinclude:: ../products/+elltool/+doc/+snip//s_chapter06_section08_snippet02.m + :language: matlab + :linenos: + + +.. _tubefig1: + +.. figure:: /pic/chapter06_section08_tube1.png + :alt: tube1 + :width: 50 % + + Reach set for the first block of parameters. + + +.. _tubefig2: + +.. figure:: /pic/chapter06_section08_tube2.png + :alt: tube2 + :width: 50 % + + Reach set for the second block of parameters. + +Solving collision problem: + +.. literalinclude:: ../products/+elltool/+doc/+snip//s_chapter06_section08_snippet06.m + :language: matlab + :linenos: + + +.. literalinclude:: ../products/+elltool/+doc/+snip//s_chapter06_section08_snippet04.m + :language: matlab + :linenos: + +.. _colfig1: + +.. figure:: /pic/chapter06_section08_col1.png + :alt: col1 + :width: 50 % + + Result of solving collision problem. There are 3 intersections of planes of :math:`(e_1,e_2),\; (e_2,e_3),\; (e_3,e_1)` (red color) and our + reach set of system (blue color). According to geometrical interpretation of system this intersections means the conclisions. + + +.. raw:: html +

References

+ +.. [1] Ibtissem Ben Makhlouf, Hilal Diab, Stefan Kowalewski. Safety Verification of a Controlled + Cooperative Platoon Under Loss of + Communication Using Zonotopes. Proceedings of the 4th IFAC Conference on + Analysis and Design of Hybrid Systems (ADHS 12) + June 6-8, 2012 + +.. [2] Jan P. Maschuw, Günter C. Keßler, D. Abel. LMI-based control of vehicle platoons for + robust longitudinal guidance. Proceedings of the 17th World Congress + The International Federation of Automatic Control + Seoul, Korea, July 6-11, 2008