Let h>0 be. Copyright Â© 2020 Elsevier B.V. or its licensors or contributors. The material in this paper was not presented at any conference. Second, classical versions of the PMP are applicable only to optimal control problems in which the dynamics evolve on Euclidean spaces, and do not carry over directly to systems evolving on more complicated manifolds. (2017) Prelimenary results on the optimal control of linear complementarity systems. 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. It is at-tributed mainly to R. Bellman. Part 1 of the presentation on "A contact covariant approach to optimal control (...)'' (Math. Â© 1967 INFORMS First order necessary conditions for the optimal control problem defined in local coordinates are derived using the method of tentsÂ (Boltyanskii et al., 1999). Browse our catalogue of tasks and access state-of-the-art solutions. 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 PMPs for discrete-time systems evolving on Euclidean spaces are not readily applicable to discrete-time models evolving on non-flat manifolds. 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). (2017) A nonlinear plate control without linearization. In this article we derive a Pontryagin maximum principle (PMP) for discrete-time optimal control problems on matrix Lie groups. The proposed approach is then demonstrated on two benchmark underactuated systems through numerical experiments. The authors thank the support of the Indian Space Research Organization 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. (2002) Pontryagin maximum principle of optimal control governed by fluid dynamic systems with two point boundary state constraint. The Pontrjagin maximum principle Pontryagin et al. 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 This shall pave way for an alternative numerical algorithm to train (2) and its discrete-time counter-part. Abstract By introducing the concept of a γ-convex set, a new discrete analogue of Pontryagin’s maximum principle is obtained. Stochastic models 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. It was first formulated in 1956 by L.S. Pontryagin. Mixing it up: Discrete and Continuous Optimal Control for Biological Models Optimal Control of PDEs There is no complete generalization of Pontryagin’s Maximum Principle in the optimal control of PDEs. 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. 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). Telecommunications 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. 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. 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. Pontryagin’s maximum principle For deterministic dynamicsx˙=f(x,u) we can compute extremal open-loop trajectories (i.e. (2011). However, there is still no PMP that is readily applicable to control systems with discrete-time dynamics evolving on manifolds. Pontryagin’s Maximum Principle, in discrete time, is used to characterize the optimal controls and the optimality system is solved by an iterative method. This article presents a novel class of control policies for networked control of Lyapunov-stable linear systems with bounded inputs. INFORMS promotes best practices and advances in operations research, management science, and analytics to improve operational processes, decision-making, and outcomes through an array of highly-cited publications, conferences, competitions, networking communities, and professional development services. More precisely, the underlying assumption in calculus of variations that an extremal trajectory admits a neighborhood in the set of admissible trajectories does not necessarily hold for such problems due to the presence of the constraints. [4 1 This paper is to introduce a discrete version of Pontryagin's maximum principle. IFAC-PapersOnLine 50:1, 2977-2982. 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. « Apply for TekniTeed Nigeria Limited Graduate Job Recruitment 2020. 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. He is currently a Postdoctoral researcher at KAIST, South Korea. 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. 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. discrete The Pontryagin maximum principle for discrete-time control processes. 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. This paper was recommended for publication in revised form by Associate Editor Kok Lay Teo under the direction of Editor Ian R. Petersen. Here we establish a PMP for a class of discrete-time controlled systems evolving on matrix Lie groups. Key words: inﬁnite-horizon optimal control, discrete time … Later in this section we establish a discrete-time PMP for optimal control problems associated with these discrete-time systems. MSC 2010: 49J21, 65K05, 39A99. The discrete maximum principle Propoj [1973] solves the problem of optimal control of a discrete time deterministic system. [1962], Boltjanskij [1969] solves the problem of optimal control of a continuous deterministic system. 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. All Rights Reserved. This section contains an introduction to Lie group variational integrators that motivates a general form of discrete-time systems on Lie groups. In the nite element literature, the maximum principle has attracted a lot of attention; see [7,8,24,29,30], to mention a few. 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 result is applied to generate a trajectory for the generalized Purcellâs swimmer - a low Reynolds number microswimming mechanism. 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. 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. Our results rely solely on asymptotic properties of the switching communication graphs in contrast to classical average dwell-time conditions. (2008b), and quantum mechanicsÂ Bonnard and Sugny (2012), Khaneja et al. 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). Read your article online and download the PDF from your email or your account. An example is solved to illustrate the use of the algorithm. We investigate asymptotic consensus of linear systems under a class of switching communication graphs. Discrete-time PMPs for various special cases are subsequently derived from the main result. We further consider a regularization term in a quadratic performance index to promote sparsity in control. Debasish Chatterjee received his Ph.D. in Electrical & Computer Engineering from the University of Illinois at UrbanaâChampaign in 2007. 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. Transportation. In effect, the state-space becomes RÃSO(2), which is isomorphic to RÃS1. (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. 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. 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. It has been shown in [4, 5] that the consistency condition in (a) is essential for the validity of … I It does not apply for dynamics of mean- led type: The authors acknowledge the fruitful discussions with Harish Joglekar, Scientist, of the Indian Space Research Organization. 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. Check out using a credit card or bank account with. nonzero, at the same time. The global product structure of the trivial bundle is used to obtain an induced Riemannian product metric on Q. Access supplemental materials and multimedia. 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. OR professionals in every field of study will find information of interest in this balanced, full-spectrum industry review. The control channel is assumed to have i.i.d. 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). 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. Hwang CL, Fan LT (1967) A discrete version of Pontryagin’s maximum principle. 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). maximum principles of Pontryagin under assumptions which weaker than these ones of existing results. The explicit form of the Riemannian connection for the trivial bundle is employed to arrive at the extremal of the cost function. (2001), and aerospace systems such as attitude maneuvers of a spacecraftÂ Kobilarov and Marsden (2011), Lee et al. For terms and use, please refer to our Terms and Conditions 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. Features of the Bellman principle and the HJB equation I The Bellman principle is based on the "law of iterated conditional expectations". in Applied Mathematics from IIT Roorkee in 2012, and Ph.D. in Systems and Control Engineering from IIT Bombay in 2018. Bernoulli packet dropouts and the system is assumed to be affected by additive stochastic noise. Unlike Pontryagin’s continuous theory it For piecewise linear elements … The. The numerical simulation is carried out using Matlab. 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. Operations Research 1. 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. Automatica 97, 376-391. He worked at ETH Zurich as a postdoc before joining IIT Bombay in 2011. (2012). The so-called weak form of the basic algorithm, its simplified Essential reading for practitioners, researchers, educators and students of OR. Environment, energy and natural resources In this article we derive a Pontryagin maximum principle (PMP) for discrete-time optimal control problems on matrix Lie groups. 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. Finally, the feasibility of the method is demonstrated by an example. 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]. As a necessary condition of the deterministic optimal control, it was formulated by Pontryagin and his group. , India through the project 14ISROC010. (2008b), Saccon et al. the maximum principle is in the field of control and process design. Another, such technique is to derive higher order variational integrators to solve optimal control problemsÂ Colombo et al. This item is part of JSTOR collection particular, we introduce the discrete-time method of successive approximations (MSA), which is based on the Pontryagin’s maximum principle, for training neural networks. In this paper, we exploit this optimal control viewpoint of deep learning. 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). 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. 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. Get the latest machine learning methods with code. Constrained optimal control problems for nonlinear continuous-time systems can, in general, be solved only numerically, and two technical issues inevitably arise. Simulation JSTOR®, the JSTOR logo, JPASS®, Artstor®, Reveal Digital™ and ITHAKA® are registered trademarks of ITHAKA. We illustrate our results by applying them to opinion formation models, thus recovering and generalizing former results for such models. (2018) A discrete-time Pontryagin maximum principle on matrix Lie groups. (2013). https://doi.org/10.1016/j.automatica.2018.08.026. 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. 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. The maximum principle changes the problem of optimal DISCRETE TIME PONTRYAGIN MAXIMUM PRINCIPLE FOR OPTIMAL CONTROL PROBLEMS UNDER STATE-ACTION-FREQUENCY CONSTRAINTS PRADYUMNA PARUCHURI AND … He serves as an Associate Editor of Automatica and an Editor of the International Journal of Robust and Nonlinear Control. The nonholonomic constraint is enforced through the local form of the principal connection and the group symmetry is employed for reduction. A basic algorithm of a discrete version of the maximum principle and its simplified derivation are presented. Select the purchase Consequently, the obtained results confirm the performance of the optimization strategy. However, the su cient conditions for discrete maximum principle put serious restrictions on the geometry of the mesh. 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. Tip: you can also follow us on Twitter Moreover, it allows for the a priori computation of a bound on the approximation error. State variable constraints are considered by use of penalty functions. Financial services A bound on the uniform rate of convergence to consensus is also established as part of this work. 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 derive first order necessary conditions bypassing techniques involving classical variational analysis. The effectiveness of the full identification scheme is demonstrated by a Monte Carlo study where our proposed method is compared to existing schemes for identifying a MIMO LPV system. His research interests lie in constrained control with emphasis on computational tractability, geometric techniques in control, and applied probability. This paper studies model order reduction of multi-agent systems consisting of identical linear passive subsystems, where the interconnection topology is characterized by an undirected weighted graph. A discrete optimal control problem is then formulated for this class of system on the phase spaces of the actuated and unactuated subsystems separately. 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. We avoid several assumptions of continuity and of Fr´echet-diﬀerentiability and of linear independence. To illustrate the engineering motivation for our work, and ease understanding, we first consider an aerospace application. 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 $. His research interests are broadly in the field of geometric mechanics and nonlinear control, with applications in electromechanical and aerospace engineering problems. In this article we bridge this gap and establish a discrete-time PMP on matrix Lie groups. (2008a), Lee et al. 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). Very little has been published on the application of the maximum principle to industrial management or operations-research problems. Our proposed class of policies is affine in the past dropouts and saturated values of the past disturbances. Oper Res 15:139–146 CrossRef zbMATH MathSciNet Google Scholar Jordan BW, Polak E (1964) Theory of a class of discrete optimal control systems. 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. Optimization We present a geometric discrete‐time Pontryagin maximum principle (PMP) on matrix Lie groups that incorporates frequency constraints on the control trajectories in addition to pointwise constraints on the states and control actions directly at the stage of the problem formulation.