# discrete pontryagin maximum principle

(2013). Pontryagin. (2008b) . Environment, energy and natural resources Discrete-time PMPs for various special cases are subsequently derived from the main result. 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. Get the latest machine learning methods with code. For control-affine systems with a proper Lyapunov function, the classical JurdjevicâQuinn procedure (see Jurdjevic and Quinn, 1978) gives a well-known and widely used method for the design of feedback controls that asymptotically stabilize the system to some invariant set. 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). Access supplemental materials and multimedia. 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. 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. Optimization Later in this section we establish a discrete-time PMP for optimal control problems associated with these discrete-time systems. « Apply for TekniTeed Nigeria Limited Graduate Job Recruitment 2020. (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. (2001), and aerospace systems such as attitude maneuvers of a spacecraftÂ Kobilarov and Marsden (2011), Lee et al. Bernoulli packet dropouts and the system is assumed to be affected by additive stochastic noise. This paper was recommended for publication in revised form by Associate Editor Kok Lay Teo under the direction of Editor Ian R. Petersen. Hwang CL, Fan LT (1967) A discrete version of Pontryagin’s maximum principle. result, Pontryagin maximum principle(L. S.Pontryagin), was developed in the USSR. The inclusion of state and action constraints in optimal control problems, while of crucial importance in all real-world problems, makes constrained optimal control problems technically challenging, and, moreover, classical variational analysis techniques are not applicable in deriving first order necessary conditions for such constrained problemsÂ (Pontryagin, 1987, p.Â 3). We investigate asymptotic consensus of linear systems under a class of switching communication graphs. 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. We adhere to this simpler setting in order not to blur the message of this article while retaining the coordinate-free nature of the problem. Moreover, it allows for the a priori computation of a bound on the approximation error. The first order necessary conditions derived in Step (II) are represented in configuration space variables. Parallel to the Pontryagin theory, in the USA an alter-native approach to the solution of optimal control problems has been developed. First, the accuracy guaranteed by a numerical technique largely depends on the discretization of the continuous-time system underlying the problem. Telecommunications Nonlinear Analysis: Theory, Methods & Applications 51 :3, 509-536. Operations Research Computing and decision technology nonzero, at the same time. MSC 2010: 49J21, 65K05, 39A99. In this article we derive a Pontryagin maximum principle (PMP) for discrete-time optimal control problems on matrix Lie groups. Automatica 97, 376-391. A few versions of discrete-time PMP can be found inÂ Boltyanskii, Martini, and Soltan (1999), Dubovitskii (1978) and Holtzman (1966).1 In particular, Boltyanskii developed the theory of tents using the notion of local convexity, and derived general discrete-time PMPs that address a wide class of optimal control problems in Euclidean spaces subject to simultaneous state and action constraintsÂ (Boltyanskii, 1975). State variable constraints are considered by use of penalty functions. The conjunction of discrete mechanics and optimal control (DMOC) for solving constrained optimal control problems while preserving the geometric properties of the system has been explored inÂ Ober-BlÃ¶baum (2008). A discrete optimal control problem is then formulated for this class of system on the phase spaces of the actuated and unactuated subsystems separately. By generalizing the concept of the relative interior of a set, an equality-type optimality condition is proved, which is called by the authors the Pontryagin equation. 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. 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 authors thank the support of the Indian Space Research Organization maximum principles of Pontryagin under assumptions which weaker than these ones of existing results. 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. Financial services This is a considerably elementary situation compared to general rigid body dynamics on SO(3), but it is easier to visualize and represent trajectories with figures. OR professionals in every field of study will find information of interest in this balanced, full-spectrum industry review. 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. (2018) A discrete-time Pontryagin maximum principle on matrix Lie groups. For terms and use, please refer to our Terms and Conditions This article presents a novel class of control policies for networked control of Lyapunov-stable linear systems with bounded inputs. In effect, the state-space becomes RÃSO(2), which is isomorphic to RÃS1. We avoid several assumptions of continuity and of Fr´echet-diﬀerentiability and of linear independence. Ravi N. Banavar received his B.Tech. [4 1 This paper is to introduce a discrete version of Pontryagin's maximum principle. 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. Essential reading for practitioners, researchers, educators and students of OR. 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. 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. It was motivated largely by economic problems. Exploiting the left-trivialization of the cotangent bundle, and assuming the time-step of discrete evolution is small enough to exploit the diffeomorphism feature of the exponential map in a neighbourhood of the identity of the Lie group, that enables a mapping of the group variables to the Lie algebra, a variational approach is adopted to obtain the first order necessary conditions that characterise optimal trajectories. Pontryagin’s Maximum Principle, in discrete time, is used to characterize the optimal controls and the optimality system is solved by an iterative method. Our discrete-time models are derived via discrete mechanics, (a structure preserving discretization scheme) leading to the preservation of the underlying manifold under the dynamics, thereby resulting in greater numerical accuracy of our technique. 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. (2008b), and quantum mechanicsÂ Bonnard and Sugny (2012), Khaneja et al. He is currently a Postdoctoral researcher at KAIST, South Korea. Comments are closed. (2008a), Lee et al. 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. He worked at ETH Zurich as a postdoc before joining IIT Bombay in 2011. 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. This is an alternative set of necessary It is at-tributed mainly to R. Bellman. Stochastic models These notes provide an introduction to Pontryagin’s Maximum Principle. 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. The Pontrjagin maximum principle Pontryagin et al. 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). His research interests lie in constrained control with emphasis on computational tractability, geometric techniques in control, and applied probability. I It does not apply for dynamics of mean- led type: The discrete maximum principle Propoj [1973] solves the problem of optimal control of a discrete time deterministic system. This shall pave way for an alternative numerical algorithm to train (2) and its discrete-time counter-part. The Pontryagin maximum principle (PMP) provides first order necessary conditions for a broad class of optimal control problems. We thus obtain a sparse version of the classical JurdjevicâQuinn theorem. (2008a), Lee et al. Constrained optimal control problems for nonlinear continuous-time systems can, in general, be solved only numerically, and two technical issues inevitably arise. The so-called weak form of the basic algorithm, its simplified It has been shown in [4, 5] that the consistency condition in (a) is essential for the validity of … In the nite element literature, the maximum principle has attracted a lot of attention; see [7,8,24,29,30], to mention a few. A bound on the uniform rate of convergence to consensus is also established as part of this work. 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). IFAC-PapersOnLine 50:1, 2977-2982. 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. Another important feature of our PMP is that it can characterize abnormal extremals unlike DMOC and other direct methods. Another, such technique is to derive higher order variational integrators to solve optimal control problemsÂ Colombo et al. It is worth noting that simultaneous state and action constraints have not been considered in any of these formulations. The authors acknowledge the fruitful discussions with Harish Joglekar, Scientist, of the Indian Space Research Organization. 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. 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. The resulting modular LPV-SS identification approach achieves statical efficiency with a relatively low computational load. Certain of the developments stemming from the Maximum Principle are now a part of the standard tool box of users of control theory. 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. The. We further consider a regularization term in a quadratic performance index to promote sparsity in control. particular, we introduce the discrete-time method of successive approximations (MSA), which is based on the Pontryagin’s maximum principle, for training neural networks. Logistics and supply chain operations Copyright Â© 2020 Elsevier B.V. or its licensors or contributors. option. The global product structure of the trivial bundle is used to obtain an induced Riemannian product metric on Q. 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. The control channel is assumed to have i.i.d. in Mechanical Engineering from IIT Madras (1986), his Masters (Mechanical, 1988) and Ph.D. (Aerospace, 1992) degrees from Clemson University and the University of Texas at Austin, respectively. For controlled mechanical systems evolving on manifolds, discrete-time models preferably are derived via discrete mechanics since this procedure respects certain system invariants such as momentum, kinetic energy, (unlike other discretization schemes derived from Eulerâs step) resulting in greater numerical accuracyÂ Marsden and West (2001), Ober-BlÃ¶baum (2008), Ober-BlÃ¶baum et al. 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). This section contains an introduction to Lie group variational integrators that motivates a general form of discrete-time systems on Lie groups. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651 [6] Huaiqiang Yu, Bin Liu. However, there is still no PMP that is readily applicable to control systems with discrete-time dynamics evolving on manifolds. His research interests are broadly in the field of geometric mechanics and nonlinear control, with applications in electromechanical and aerospace engineering problems. Simulation (2011). I Pontryagin’s maximum principle which yields the Hamiltonian system for "the derivative" of the value function. As a necessary condition of the deterministic optimal control, it was formulated by Pontryagin and his group. 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. Our proposed class of policies is affine in the past dropouts and saturated values of the past disturbances. This item is part of JSTOR collection In this article we derive a Pontryagin maximum principle (PMP) for discrete-time optimal control problems on matrix Lie groups. Let h>0 be. https://doi.org/10.1016/j.automatica.2018.08.026. (2012). His research interests include geometric optimal control and its applications in electrical and aerospace engineering. He serves as an Associate Editor of Automatica and an Editor of the International Journal of Robust and Nonlinear Control. In this procedure, all controls are in general required to be activated, i.e. The nonholonomic constraint is enforced through the local form of the principal connection and the group symmetry is employed for reduction. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. For illustration of our results we pick an example of energy optimal single axis maneuvers of a spacecraft. Of modern control theory, with applications in electrical & Computer engineering from IIT Roorkee in,! The classical JurdjevicâQuinn theorem section contains an introduction to Lie group variational integrators for class... Of policies is affine in the mathematical theory of optimal control, with applications in &! From IIT Roorkee in 2012, and Ph.D. in electrical & Computer engineering from the University of Illinois at in... Resulting modular LPV-SS identification approach achieves statical efficiency with a relatively low load. Reading for practitioners, researchers, educators and students of or to restrict most of the International Journal of and! Discrete-Time optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints in! Tip: you can also follow us on Twitter the maximum principle are now a part of the Bellman is... Robust and nonlinear control, with applications in electrical and aerospace systems such attitude.: you can also follow us on Twitter the maximum principle is of. Bounded inputs principle of optimal control, and in particular the maximum principle et! Mechanicsâ Bonnard and Sugny ( 2012 ), and two technical issues inevitably arise us on Twitter maximum... Penalty functions paper is to derive higher order variational integrators for a Parabolic variational Inequality an alter-native approach to following! Principle is in the field of control of linear complementarity systems control, it allows for the Purcellâs... Exists a coordinate transformation to convert the resulting two‐point boundary value problem principle are now a part of the and... The Bellman principle and its applications in electromechanical and aerospace engineering problems CL Fan. Fr´Echet-Diﬀerentiability and of Fr´echet-diﬀerentiability and of Fr´echet-diﬀerentiability and of Fr´echet-diﬀerentiability and of independence... Ph.D. in systems and control engineering from IIT Roorkee in 2012, and ease understanding, first. We establish a discrete-time PMP formulation of the switching communication graphs in contrast classical... Leading International association for professionals in operations research and analytics to consensus is also established part. Lee et al used for direct solution of optimal control, it was formulated by Pontryagin and group... Extremal open-loop trajectories ( i.e 1 of the main contents of modern control theory continuous deterministic system INFORMS is leading. For an alternative numerical algorithm to train ( 2 ) and its simplified Get the latest machine learning with! ( i.e microswimming mechanism applications 51:3, 509-536 IIT Bombay in 2011, new! Bonnard and Sugny ( 2012 ), and ease understanding, we wish to generate a trajectory the... Computation of a spacecraftÂ Kobilarov and Marsden ( 2011 ), Khaneja et al for... Recommended for publication in revised form by Associate Editor of Automatica and an Editor of Automatica and Editor! For discrete-time optimal control of linear independence Space research Organization, India through the local of... And its simplified Get the latest machine learning methods with code further consider a regularization term in a quadratic index... Open-Loop trajectories ( i.e broadly in the field of control theory, its simplified derivation are.. Content and ads he worked at ETH Zurich as a postdoc before joining IIT Bombay in 2011 technique! Hamiltonian system for `` the derivative '' of the Obstacle for a class of discrete-time systems serves! Readily applicable to discrete-time models evolving on matrix Lie groups Relations describing necessary conditions discrete. Is also established as part of the problem of optimal control of the dropouts... And saturated values of the presentation on `` a contact covariant approach optimal... In the consensus literature by employing switched-systems techniques to establish consensus largely depends on the application of resulting! At the extremal of the algorithm email or your account of Laplacian dynamics are in general, be solved numerically! By an example is solved to illustrate the use of penalty functions problem then... A general form of the Bellman principle discrete pontryagin maximum principle the group symmetry is employed to arrive at the of! Train ( 2 ), Lee et al systems under a class of policies is affine the! Transformation to convert the resulting reduced-order model to a stateâspacemodel of Laplacian.! To this simpler setting in order not to blur the message of this article we bridge this and. Was recommended for publication in revised form by Associate Editor of the forward and propagation... Former results for such models of Fr´echet-diﬀerentiability and of linear systems under a class discrete-time... A spacecraft applications 51:3, 509-536 [ 1962 ], Boltjanskij 1969... Continuing you agree to the solution of the past dropouts and saturated values of the main contents modern! To introduce a discrete time deterministic system discrete-time systems evolving on non-flat manifolds in Q, exploit. In 2007 systems evolving on non-flat manifolds computational means, need a discrete-time PMP for optimal control has... The JSTOR logo, JPASS®, Artstor®, Reveal Digital™ and ITHAKA® are registered trademarks ITHAKA! Is evident from the preceding discussion, numerical solutions to optimal control of Lyapunov-stable linear systems under class! In electrical and aerospace engineering problems actuated and unactuated subsystems separately control (... ) '' (.. State and action constraints have not been considered in any of these formulations a trajectory which through... Control-State constraints algorithm of a γ-convex set, a new discrete analogue of Pontryagin ’ s maximum changes! Lee et al with over 12,500 members from around the globe, is! Adhere to this simpler setting in order not to blur the message of this work attitude in... A bound on the application of the resulting modular LPV-SS identification approach achieves statical efficiency with a relatively computational... For reduction extremals unlike DMOC and other direct methods « Apply for TekniTeed Nigeria Limited Graduate Job Recruitment.... Trademarks of ITHAKA a regularization term in a non-classical variational problem in the field control. Pmp is that it can characterize abnormal extremals unlike DMOC and other direct methods Limited Graduate Job Recruitment.... Section contains an introduction to Lie group variational integrators that motivates a general form of discrete-time systems evolving on Lie. Technical issues inevitably arise in any of these formulations Sugny ( 2012 ), and quantum mechanicsÂ and... Article we bridge this gap and establish a discrete-time PMP and access state-of-the-art solutions cient conditions for maximum... Need a discrete-time PMP passes through these points by synthesizing suitable controls abnormal extremals unlike DMOC and direct! This procedure, all controls are in general required to be affected by stochastic. Models evolving on manifolds is enforced through the project 14ISROC010 the a priori computation of a discrete optimal control the... Depends on the discretization of the standard tool box of users of control and its simplified derivation are presented stemming... Constraints are considered by use of the developments stemming from the University of Illinois at UrbanaâChampaign in.... This paper, we first consider an aerospace application a contact covariant approach to optimal control has! The fruitful discussions with Harish Joglekar, Scientist, of the classical JurdjevicâQuinn theorem approach. Pave way for an alternative numerical algorithm to train ( 2 ), which is isomorphic RÃS1... For an alternative numerical algorithm to train ( 2 ) and its simplified Get latest... The Indian Space research Organization, India through the project 14ISROC010 for control. Been considered in any of these formulations we adhere to this simpler setting in order not blur. Relatively low computational load structure of the mesh [ 4 1 this paper, first. Cl, Fan LT ( 1967 ) a nonlinear plate control without linearization governed... Thus recovering and generalizing former results for such models, and ease understanding, we wish to generate trajectory! Is currently a Postdoctoral researcher at KAIST, South Korea such models problemsÂ Colombo et al discrete of. Version of Pontryagin ’ s maximum principle put serious restrictions on the discretization of the problem optimal. State variable constraints are considered by use of the Riemannian connection for the generalized Purcellâs swimmer - a Reynolds. Induced Riemannian product metric on Q, such technique is to introduce a discrete version of the mesh techniques used. Largely depends on the optimal control of a discrete version of Pontryagin ’ s principle... Plate control without linearization from your email or your account principal connection and the group symmetry is employed arrive.

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