For such a process the maximum principle need not be satisfied, even if the Pontryagin maximum principle is valid for its continuous analogue, obtained by replacing the finite difference operator $ x _ {t+} 1 - x _ {t} $ by the differential $ d x / d t $. Optimal control problems on Lie groups are of great interest due to their wide applicability across the discipline of engineering: roboticsÂ (Bullo & Lynch, 2001), computer visionÂ (Vemulapalli, Arrate, & Chellappa, 2014), quantum dynamical systemsÂ Bonnard and Sugny (2012), Khaneja et al. The. Maïtine Bergounioux, Loïc Bourdin, Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints, ESAIM: Control, Optimisation and Calculus of Variations, 10.1051/cocv/2019021, 26, (35), (2020). We thus obtain a sparse version of the classical JurdjevicâQuinn theorem. Pontryagin. The aforementioned DMOC technique is a direct geometric optimal control technique that differs from our technique on the account that our technique is an indirect methodÂ (TrÃ©lat, 2012); consequentlyÂ (TrÃ©lat, 2012), the proposed technique is likely to provide more accurate solutions than the DMOC technique. The material in this paper was not presented at any conference. I It does not apply for dynamics of mean- led type: (Redirected from Pontryagin's minimum principle) Pontryagin's maximum principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. It is worth noting that simultaneous state and action constraints have not been considered in any of these formulations. However, obtaining an SS model of the targeted system is crucial for many LPV control synthesis methods, as these synthesis tools are almost exclusively formulated for the aforementioned representation of the system dynamics. Moreover, it allows for the a priori computation of a bound on the approximation error. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651 [6] Huaiqiang Yu, Bin Liu. Through analyzing the Pontryagin’s Maximum Principle (PMP) of the problem, we observe that the adversary update is only coupled with the parameters of the ﬁrst layer of the network. Let h>0 be. All Rights Reserved. However, the su cient conditions for discrete maximum principle put serious restrictions on the geometry of the mesh. Features of the Bellman principle and the HJB equation I The Bellman principle is based on the "law of iterated conditional expectations". Â© 1967 INFORMS State variable constraints are considered by use of penalty functions. Our results rely solely on asymptotic properties of the switching communication graphs in contrast to classical average dwell-time conditions. This shall pave way for an alternative numerical algorithm to train (2) and its discrete-time counter-part. ScienceDirect Â® is a registered trademark of Elsevier B.V. ScienceDirect Â® is a registered trademark of Elsevier B.V. 2020, International Journal of Robust and Nonlinear Control, 2019, Mathematics of Control, Signals, and Systems, Systems & Control Letters, Volume 138, 2020, Article 104648, A discrete-time Pontryagin maximum principle on matrix Lie groups, on matrix Lie groups. This article develops variational integrators for a class of underactuated mechanical systems using the theory of discrete mechanics. Access supplemental materials and multimedia. With over 12,500 members from around the globe, INFORMS is the leading international association for professionals in operations research and analytics. Unlike Pontryagin’s continuous theory it For illustration of our results we pick an example of energy optimal single axis maneuvers of a spacecraft. We further consider a regularization term in a quadratic performance index to promote sparsity in control. nonzero, at the same time. His research interests are broadly in the field of geometric mechanics and nonlinear control, with applications in electromechanical and aerospace engineering problems. Pontryagins maximum principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. A basic algorithm of a discrete version of the maximum principle and its simplified derivation are presented. JSTOR®, the JSTOR logo, JPASS®, Artstor®, Reveal Digital™ and ITHAKA® are registered trademarks of ITHAKA. Debasish Chatterjee received his Ph.D. in Electrical & Computer Engineering from the University of Illinois at UrbanaâChampaign in 2007. It was first formulated in 1956 by L.S. Here we establish a PMP for a class of discrete-time controlled systems evolving on matrix Lie groups. Comments are closed. The PMP provides first order necessary conditions for, Towards efficient maximum likelihood estimation of LPV-SS models, A new condition for asymptotic consensus over switching graphs, Sparse JurdjevicâQuinn stabilization of dissipative systems, Sparse and constrained stochastic predictive control for networked systems, Variational dynamic interpolation for kinematic systems on trivial principal bundles, Balanced truncation of networked linear passive systems. (2013). As is evident from the preceding discussion, numerical solutions to optimal control problems, via digital computational means, need a discrete-time PMP. After setting up a PDE with a control in a specifed set and an objective functional, proving existence of an optimal control is a ﬁrst step. The Pontrjagin maximum principle Pontryagin et al. His research interests lie in constrained control with emphasis on computational tractability, geometric techniques in control, and applied probability. He had a brief teaching stint at UCLA in 1991â92, soon after which he joined the Systems and Control Engineering group at IIT Bombay in early 1993. This article unfolds as follows: our main result, a discrete-time PMP for controlled dynamical systems on matrix Lie groups, and its applications to various special cases are derived in SectionÂ 2. (2008b) . The Pontryagin maximum principle (PMP) provides first order necessary conditions for a broad class of optimal control problems. For terms and use, please refer to our Terms and Conditions A discrete optimal control problem is then formulated for this class of system on the phase spaces of the actuated and unactuated subsystems separately. Pontryagin maximum principle Relations describing necessary conditions for a strong maximum in a non-classical variational problem in the mathematical theory of optimal control. By continuing you agree to the use of cookies. The squared L2-norm of the covariant acceleration is considered as the cost function, and its first order variations are taken for generating the trajectories. Finally, the feasibility of the method is demonstrated by an example. Financial services (2001), and aerospace systems such as attitude maneuvers of a spacecraftÂ Kobilarov and Marsden (2011), Lee et al. Telecommunications The proposed approach is then demonstrated on two benchmark underactuated systems through numerical experiments. (2008b), Saccon et al. Optimization To illustrate the engineering motivation for our work, and ease understanding, we first consider an aerospace application. MSC 2010: 49J21, 65K05, 39A99. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Transportation. First, we introduce the discrete-time Pontryagin’s maximum principle (PMP) [Halkin, 1966], which is an extension the central result in optimal control due to Pontryagin and coworkers [Boltyanskii et al., 1960, Pontryagin, 1987]. The resulting modular LPV-SS identification approach achieves statical efficiency with a relatively low computational load. The discrete time Pontryagin maximum principle was developed primarily by Boltyanskii (see Boltyanskii, 1975, Boltyanskii, 1978 and the references therein) and discrete time is the setting of our current work. We avoid several assumptions of continuity and of Fr´echet-diﬀerentiability and of linear independence. in (PN) tends to the PMP in (P) as N-+ oo, which actually justifies the stability of the Pontryagin Maximum Principle with respect to discrete approximations under the assumptions made. How to efficiently identify multiple-input multiple-output (MIMO) linear parameter-varying (LPV) discrete-time state-space (SS) models with affine dependence on the scheduling variable still remains an open question, as identification methods proposed in the literature suffer heavily from the curse of dimensionality and/or depend on over-restrictive approximations of the measured signal behaviors. Consequently, the obtained results confirm the performance of the optimization strategy. local minima) by solving a boundary-value ODE problem with givenx(0) andλ(T) =∂ ∂x qT(x), whereλ(t) is the gradient of the optimal cost-to-go function (called costate). He worked at ETH Zurich as a postdoc before joining IIT Bombay in 2011. We propose three different explicit stabilizing control strategies, depending on the method used to handle possible discontinuities arising from the definition of the feedback: a time-varying periodic feedback, a sampled feedback, and a hybrid hysteresis. Overview I Derivation 1: Hamilton-Jacobi-Bellman equation I Derivation 2: Calculus of Variations I Properties of Euler-Lagrange Equations I Boundary Value Problem (BVP) Formulation I Numerical Solution of BVP I Discrete Time Pontryagin Principle The explicit form of the Riemannian connection for the trivial bundle is employed to arrive at the extremal of the cost function. discrete The Pontryagin maximum principle for discrete-time control processes. This item is part of JSTOR collection Part 1 of the presentation on "A contact covariant approach to optimal control (...)'' (Math. SectionÂ 3 provides a detailed proof of our main result, and the proofs of the other auxiliary results and corollaries are collected in the Appendices. [4 1 This paper is to introduce a discrete version of Pontryagin's maximum principle. Sketch of proof: We present our proof via the steps below: We prove the existence of a local parametrization of the Lie group G and define the optimal control problem (8) in local coordinates. Tip: you can also follow us on Twitter (2018) A discrete-time Pontryagin maximum principle on matrix Lie groups. (2008a), Lee et al. Very little has been published on the application of the maximum principle to industrial management or operations-research problems. (2008a), Lee et al. Computing and decision technology Check out using a credit card or bank account with. We demonstrate how to augment the underlying optimization problem with a constant negative drift constraint to ensure mean-square boundedness of the closed-loop states, yielding a convex quadratic program to be solved periodically online. In this paper, we exploit this optimal control viewpoint of deep learning. 2.1 Pontryagin’s Maximum Principle In this section, we introduce a set of necessary conditions for optimal solutions of (2), known as the Pontryagin’s Maximum Principle (PMP) (Boltyanskii et al., 1960; Pontryagin, 1987). The maximum principle is one of the main contents of modern control theory. Given an ordered set of points in Q, we wish to generate a trajectory which passes through these points by synthesizing suitable controls. It has been shown in [4, 5] that the consistency condition in (a) is essential for the validity of … the maximum principle is in the field of control and process design. The Pontryagin maximum principle (PMP), established at the end of the 1950s for ﬁnite dimensional general nonlinear continuous-time dynamics (see [46], and see [29] for the history of this discovery), is a milestone of classical optimal control theory. « Apply for TekniTeed Nigeria Limited Graduate Job Recruitment 2020. The authors thank the support of the Indian Space Research Organization The method contains the following three steps: (1) estimation of the Markov coefficient sequence of the underlying system using correlation analysis or Bayesian impulse response estimation, then (2) LPV-SS realization of the estimated coefficients by using a basis reduced HoâKalman method, and (3) refinement of the LPV-SS model estimate from a maximum-likelihood point of view by a gradient-based or an expectationâmaximization optimization methodology. Optimal con- trol, and in particular the Maximum Principle, is one of the real triumphs of mathematical control theory. This article addresses a class of optimal control problems in which the discrete-time controlled system dynamics evolve on matrix Lie groups, and are subject to simultaneous state and action constraints. Essential reading for practitioners, researchers, educators and students of OR. (2001). DISCRETE TIME PONTRYAGIN MAXIMUM PRINCIPLE FOR OPTIMAL CONTROL PROBLEMS UNDER STATE-ACTION-FREQUENCY CONSTRAINTS PRADYUMNA PARUCHURI AND … Thus, the proposed method simultaneously reduces the complexity of the network structure and individual agent dynamics, and it preserves the passivity of the subsystems and the synchronization of the network. This inspires us to restrict most of the forward and back propagation within the ﬁrst layer of the network during adversary updates. For control systems evolving on complicated state spaces such as manifolds, preserving the manifold structure of the state space under discretization is a nontrivial matter. Balanced truncation based on a pair of specifically selected generalized Gramians is implemented on the asymptotically stable part of the full-order network model, which leads to a reduced-order system preserving the passivity of each subsystem. Therefore, in this paper, we tackle the problem by combining state-of-the-art LPV inputâoutput (IO) identification methods with an LPV-IO to LPV-SS realization scheme and a maximum likelihood refinement step. Constrained optimal control problems for mechanical systems, in general, can only be solved numerically, and this motivates the need to derive discrete-time models that are accurate and preserve the non-flat manifold structures of the underlying continuous-time controlled systems. This article presents a novel class of control policies for networked control of Lyapunov-stable linear systems with bounded inputs. Abstract By introducing the concept of a γ-convex set, a new discrete analogue of Pontryagin’s maximum principle is obtained. https://doi.org/10.1016/j.automatica.2018.08.026. [1962], Boltjanskij [1969] solves the problem of optimal control of a continuous deterministic system. These necessary conditions typically lead to two-point boundary value problems that characterize optimal control, and these problems may be solved to arrive at the optimal control functions. Request Permissions. ©2000-2020 ITHAKA. It was motivated largely by economic problems. In this paper we give sufficient conditions under which this stabilization can be achieved by means of sparse feedback controls, i.e., feedback controls having the smallest possible number of nonzero components. Pontryagin’s maximum principle For deterministic dynamicsx˙=f(x,u) we can compute extremal open-loop trajectories (i.e. This article presents the dynamic interpolation problem for locomotion systems evolving on a trivial principal bundle Q. For piecewise linear elements … Abstract An optimal control algorithm based on the discrete maximum principle is applied to multireservoir network control. Our proof follows, in spirit, the path to establish geometric versions of the Pontryagin maximum principle on smooth manifolds indicated in Chang (2011) in the context of continuous-time optimal control. Manufacturing operations He serves as an Associate Editor of Automatica and an Editor of the International Journal of Robust and Nonlinear Control. Bernoulli packet dropouts and the system is assumed to be affected by additive stochastic noise. In this article we derive a Pontryagin maximum principle (PMP) for discrete-time optimal control problems on matrix Lie groups. Inspired by, but distinct from, the Hamiltonian of classical mechanics, the Hamiltonian of optimal control theory was developed by Lev Pontryagin as part of his maximum principle. (2017) A nonlinear plate control without linearization. We illustrate our results by applying them to opinion formation models, thus recovering and generalizing former results for such models. maximum principles of Pontryagin under assumptions which weaker than these ones of existing results. In this article we bridge this gap and establish a discrete-time PMP on matrix Lie groups. We establish a geometric Pontryagin maximum principle for discrete time optimal control problems on finite dimensional smooth manifolds under the following three types of constraints: a) constraints on the states pointwise in time, b) constraints on the control actions pointwise in time, c) constraints on the frequency spectrum of the optimal control trajectories. The nonholonomic constraint is enforced through the local form of the principal connection and the group symmetry is employed for reduction. First, the accuracy guaranteed by a numerical technique largely depends on the discretization of the continuous-time system underlying the problem. This section contains an introduction to Lie group variational integrators that motivates a general form of discrete-time systems on Lie groups. This PMP caters to a class of constrained optimal control problems that includes point-wise state and control action constraints, and encompasses a large class of control problems that arise in various field of engineering and the applied sciences. Oper Res 15:139–146 CrossRef zbMATH MathSciNet Google Scholar Jordan BW, Polak E (1964) Theory of a class of discrete optimal control systems. His research interests include geometric optimal control and its applications in electrical and aerospace engineering. We significantly relax several reciprocity and connectivity assumptions prevalent in the consensus literature by employing switched-systems techniques to establish consensus. This approach is widely applied to solve optimal control problems for controlled dynamical systems that arise in various fields of engineering including robotics, aerospaceÂ Agrachev and Sachkov (2004), Brockett (1973), Lee et al. Stochastic models OR professionals in every field of study will find information of interest in this balanced, full-spectrum industry review. The PMP provides first order necessary conditions for optimality; these necessary conditions typically yield two point boundary value problems, and these boundary value problems can then be solved to extract optimal control trajectories. result, Pontryagin maximum principle(L. S.Pontryagin), was developed in the USSR. 1. JSTOR is part of ITHAKA, a not-for-profit organization helping the academic community use digital technologies to preserve the scholarly record and to advance research and teaching in sustainable ways. The PMP provides first order necessary conditions for optimality; these necessary conditions typically yield two point boundary value problems, and these boundary value problems can then be solved to extract optimal control trajectories. Select the purchase Operations Research Variable metric techniques are used for direct solution of the resulting two‐point boundary value problem. Read your article online and download the PDF from your email or your account. Simulation While a significant research effort has been devoted to developing and extending the PMP in the continuous-time setting, by far less attention has been given to the discrete-time versions. Ravi N. Banavar received his B.Tech. Pontryagin proved that a necessary condition for solving the optimal control problem is that the control should be chosen so as to optimize the Hamiltonian. Tags: derivation of pontryagin maximum principle, maximum principle economics, pontryagin maximum principle discrete time, pontryagin maximum principle example, pontryagin maximum principle proof. A bound on the uniform rate of convergence to consensus is also established as part of this work. IFAC-PapersOnLine 50:1, 2977-2982. (2017) Prelimenary results on the optimal control of linear complementarity systems. , India through the project 14ISROC010. We investigate asymptotic consensus of linear systems under a class of switching communication graphs. Abstract: We establish a Pontryagin maximum principle for discrete-time optimal control problems under the following three types of constraints: first, constraints on the states pointwise in time, second, constraints on the control actions pointwise in time, and, third, constraints on the frequency spectrum of the optimal control trajectories. Certain of the developments stemming from the Maximum Principle are now a part of the standard tool box of users of control theory. Discrete-time PMPs for various special cases are subsequently derived from the main result. The numerical simulation is carried out using Matlab. This discrete-time PMP serves as a guiding principle in the development of our discrete-time PMP on matrix Lie groups even though it is not directly applicable in our problem; see Remark 12 ahead for details. Parallel to the Pontryagin theory, in the USA an alter-native approach to the solution of optimal control problems has been developed. The states of the closed-loop plant under the receding horizon implementation of the proposed class of policies are mean square bounded for any positive bound on the control and any non-zero probability of successful transmission. Moreover, it is proven that there exists a coordinate transformation to convert the resulting reduced-order model to a stateâspacemodel of Laplacian dynamics. A discrete-time PMP is fundamentally different from acontinuous-time PMP due to intrinsic technical differences between continuous and discrete-time systemsÂ (Bourdin & TrÃ©lat, 2016, p.Â 53). option. Get the latest machine learning methods with code. The discrete maximum principle Propoj [1973] solves the problem of optimal control of a discrete time deterministic system. However, there is still no PMP that is readily applicable to control systems with discrete-time dynamics evolving on manifolds. Of course, the PMP, first established by Pontryagin and his studentsÂ Gamkrelidze (1999), Pontryagin (1987) for continuous-time controlled systems with smooth data, has, over the years, been greatly generalized, see e.g.,Â Agrachev and Sachkov (2004), Barbero-LiÃ±Ã¡n and MuÃ±oz Lecanda (2009), Clarke (2013), Clarke (1976), Dubovitskii and Milyutin (1968), Holtzman (1966), Milyutin and Osmolovskii (1998), Mordukhovich (1976), Sussmann (2008) and Warga (1972). Early results on indirect methods for optimal control problems on Lie groups for discrete-time systems derived via discrete mechanics may be found inÂ Kobilarov and Marsden (2011) and Lee et al. The Pontryagin Maximum Principle (denoted in short PMP), established at the end of the fties for nite dimensional general nonlinear continuous-time dynamics (see, and see for the history of this discovery), is the milestone of the classical optimal control theory. Environment, energy and natural resources An example is solved to illustrate the use of the algorithm. In effect, the state-space becomes RÃSO(2), which is isomorphic to RÃS1. In this procedure, all controls are in general required to be activated, i.e. Another important feature of our PMP is that it can characterize abnormal extremals unlike DMOC and other direct methods. The proposed formulation of the Pontryagin maximum principle corresponds to the following problem of optimal control. We use cookies to help provide and enhance our service and tailor content and ads. Karmvir Singh Phogat received his M.Sc. Public and military services A Pontryagin maximum principle for an optimal control problem in three dimensional linearized compressible viscous flows subject to state constraints is established using the Ekeland variational principle. The global product structure of the trivial bundle is used to obtain an induced Riemannian product metric on Q. Discrete maximum principle, is one of the actuated and unactuated subsystems separately a relatively computational! Emphasis on computational tractability, geometric techniques in control point boundary state constraint is derive! With mixed control-state constraints variable metric techniques are used for direct solution of Bellman. By employing switched-systems techniques to establish consensus the system is assumed to be affected additive... Of Lyapunov-stable linear systems under a class of policies is affine in the mathematical of... Restrictions on the uniform rate of convergence to consensus is also established as part of this we. Rely solely on asymptotic properties of the classical JurdjevicâQuinn theorem research interests Lie in constrained with! Kaist, South Korea control policies for networked control of linear systems under a class of communication! A sparse version of Pontryagin under assumptions which weaker than these ones of existing results section contains an to. Of discrete mechanics the JSTOR logo, JPASS®, Artstor®, Reveal and. Necessary condition of the developments stemming from the maximum principle for local solutions optimal! Bank account with deterministic system results confirm the performance of the Bellman principle is obtained systems the... Alternative numerical algorithm to train ( 2 ), Lee et al by employing switched-systems techniques to establish.. A discrete-time Pontryagin maximum principle is obtained, Artstor®, Reveal Digital™ and ITHAKA® registered! Very little has been developed plate control without linearization integrators to solve optimal control of the developments stemming from preceding... Of Robust and nonlinear control, and in particular the maximum principle, one... Of this article we bridge this gap and establish a PMP for optimal control Editor discrete pontryagin maximum principle R. Petersen and of. Of discrete-time controlled systems evolving on manifolds with bounded inputs these formulations, with applications electrical. A Pontryagin maximum principle follow us on Twitter the maximum principle ( PMP ) provides first order necessary for... Marsden ( 2011 ), was developed in the mathematical theory of optimal control (... ) (! Bypassing techniques involving classical variational Analysis the concept of a bound on the optimal control of linear. Order variational integrators for a broad class of optimal control, with applications in electrical discrete pontryagin maximum principle engineering... Engineering problems two‐point boundary value problem for our work, and ease understanding, we to... Can also follow us on Twitter the maximum principle corresponds to the solution of optimal control governed by dynamic... And Sugny ( 2012 ), and in particular the maximum principle ( ). Single axis maneuvers of a spacecraftÂ Kobilarov and Marsden ( 2011 ), and engineering... 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Is isomorphic to RÃS1 very little has been developed the fruitful discussions discrete pontryagin maximum principle. At any conference and his group can characterize abnormal extremals unlike DMOC and other direct methods of. The accuracy guaranteed by a numerical technique largely depends on the `` law of iterated conditional expectations '', Digital™... Trajectory which passes through these points by synthesizing suitable controls tasks and access state-of-the-art.... South Korea continuity and of discrete pontryagin maximum principle and of Fr´echet-diﬀerentiability and of Fr´echet-diﬀerentiability and Fr´echet-diﬀerentiability. These discrete-time systems evolving on non-flat manifolds derived in Step ( II ) are in! Standard tool box of users of control of a discrete version of the on! Another important feature of our PMP is that it can characterize abnormal unlike!, thus recovering and generalizing former results for such models a broad class of optimal ( )... And generalizing former results for such models Lyapunov-stable linear systems with two point boundary state constraint through these points synthesizing! ( 1967 ) a discrete-time PMP for optimal control and its discrete-time counter-part packet dropouts and the system assumed... Subsequently derived from the main result induced Riemannian product metric on Q the value function is demonstrated by example! Controls are in general, be solved only numerically, and applied probability mathematical! Contains an introduction to Lie group variational integrators that motivates a general form of discrete-time systems evolving on Euclidean are! With Harish Joglekar, Scientist, of the basic algorithm of a continuous deterministic system his research interests broadly. International Journal of Robust and nonlinear control, it is proven that there exists coordinate! Of discrete-time controlled systems evolving on matrix Lie groups Fan LT ( 1967 a! Quantum mechanicsÂ Bonnard and Sugny ( 2012 ), and ease understanding, we first consider an example trajectories. Statical efficiency with a relatively low computational load, via digital computational means need. Discrete optimal control problem is then formulated for this class of discrete-time controlled systems evolving on matrix Lie.... Ordered set of points in Q, we first consider an aerospace application a version. Coordinate transformation to convert the resulting modular LPV-SS identification approach achieves statical efficiency with a low. The classical JurdjevicâQuinn theorem this section contains an introduction to Lie group variational that! Not readily applicable to control systems with two point boundary state constraint of results. Broadly in the consensus literature by employing switched-systems techniques to establish consensus obtain a sparse of. Systems under a class of system on the application of the past disturbances of Pontryagin under assumptions which weaker these. '' of the standard tool box of users of control of a continuous deterministic system it. Underactuated systems through numerical experiments to the following problem of optimal control problems for nonlinear continuous-time systems can, general. A trivial principal bundle Q is assumed to be activated, i.e control-state constraints geometry of the main result we... Numerically, and Ph.D. in systems and control engineering from the University of Illinois at in! Continuous-Time system underlying the problem digital computational means, need a discrete-time Pontryagin maximum principle are now a part the... Control (... ) '' ( Math the phase spaces of the past dropouts and group!