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Noboru Sakamoto

Publications and source records attributed to Noboru Sakamoto.

10 recordsLinked to original sources
arXiv2024

Maximally Degenerate Floquet Structure and Possible Nonexistence of Optimal Control in a Pendulum Swing-Up Problem

This paper studies an infinite-horizon optimal control problem for a pendulum with quadratic control cost via its associated Hamiltonian system. The problem is strongly degenerate, as the linearization at the upright equilibrium has purely imaginary eigenvalues with multiplicity, making standard linear analysis inconclusive. Using Lie series and Chetaev's instability theorem, we show that the equilibrium is weakly unstable in a non-hyperbolic sense. We further identify a family of periodic orbits forming an invariant cylinder, whose monodromy matrix exhibits a maximally degenerate Floquet structure consisting of a single Jordan block at the unit multiplier. This structure implies slow, non-exponential dynamics and provides a mechanism for long-time transitions with arbitrarily small control effort. Although derived for a pendulum, this phenomenon arises from degeneracy in the optimal control formulation and may occur more broadly, suggesting the possible nonexistence of optimal control despite stabilizability.

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arXiv2022

Optimal Stabilization of Periodic Orbits: A Symplectic Geometry Approach

In this contribution, the optimal stabilization problem of periodic orbits is studied via invariant manifold theory and symplectic geometry. The stable manifold theory for the optimal point stabilization case is generalized to the case of periodic orbit stabilization, where a normally hyperbolic invariant manifold plays the role of a hyperbolic equilibrium point. A sufficient condition for the existence of an NHIM of an extended Hamiltonian system is derived in terms of a periodic Riccati differential equation. It is shown that the problem of optimal orbit stabilization has a solution if a linearized periodic system is stabilizable and detectable. A moving orthogonal coordinate system is employed along the periodic orbit, which is a natural framework for orbital stabilization and linearization along the orbit. Two illustrative examples are presented: the first involves stabilizing a spring-mass oscillator at a target energy level, and the second addresses an orbit transfer problem for a satellite-a classic scenario in orbital mechanics. In both cases, we show that the proposed nonlinear feedback controller outperforms traditional linear control.

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arXiv2021

Optimal Sampled-Data Control of a Nonlinear System

Optimal sampled-data control of a nonlinear system is considered with the stable-manifold approach and extensive use of numerical techniques. The idea is to notice the Hamiltonian system associated with the considered optimal control problem and to compute trajectories on its stable manifold. Since the control input accompanied with those trajectories is proved to be optimal, the optimal control law can be obtained through interpolation. The stable-manifold approach was originally proposed for continuous-time optimal control and here it is adapted for sampled-data control based on the works of Navasca. In the case of sampled-data control, the approach requires the state transition of the controlled plant during one sampling period together with its derivatives with respect to the state and the input. Their computation is achieved by numerical techniques. Moreover, a shooting method is proposed for systematic generation of the trajectories and extension is considered for the intersample behavior to be taken into account. The proposed method is applied to tracking control of a wheeled mobile robot. It works successfully with a rather long sampling period.

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arXiv2021

When does stabilizability imply the existence of infinite horizon optimal control in nonlinear systems?

The paper addresses an existence problem for infinite horizon optimal control when the system under control is exponentially stabilizable or stable. Classes of nonlinear control systems for which infinite horizon optimal controls exist are identified in terms of stability, stabilizability, detectability and growth conditions. The result then applies to estimate the existence region of stable manifolds in the associated Hamiltonian systems. Applications of the results also include the analysis for turnpike property in nonlinear finite horizon optimal control problems by a geometric approach.

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arXiv2021

The turnpike property in nonlinear optimal control -- A geometric approach

This paper presents, using dynamical system theory, a framework for investigating the turnpike property in nonlinear optimal control. First, it is shown that a turnpike-like property appears in general dynamical systems with hyperbolic equilibrium and then, apply it to optimal control problems to obtain sufficient conditions for the turnpike occurs. The approach taken is geometric and gives insights for the behaviors of controlled trajectories, allowing us to find simpler proofs for existing results on the turnpike properties. Attempts to remove smallness restrictions for initial and target states are also discussed based on the geometry of (un)stable manifold and exponential stabilizability of control systems.

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arXiv2020

Rotor imbalance suppression by optimal control

An imbalanced rotor is considered. A system of moving balancing masses is given. We determine the optimal movement of the balancing masses to minimize the imbalance on the rotor. The optimal movement is given by an open-loop control solving an optimal control problem posed in infinite time. By methods of the Calculus of Variations, the existence of the optimum is proved and the corresponding optimality conditions have been derived. Asymptotic behavior of the control system is studied rigorously. By Łojasiewicz inequality, convergence of the optima as time $t\to +\infty$ towards a steady configuration is ensured. An explicit estimate of the convergence rate is given. This guarantees that the optimal control stabilizes the system. In case the imbalance is below a computed threshold, the convergence occurs exponentially fast. This is proved by the Stable Manifold Theorem applied to the Pontryagin optimality system. Moreover, a closed-loop control strategy based on Reinforcement Learning is proposed. Numerical simulations have been performed, validating the theoretical results.

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arXiv2020

The turnpike with lack of observability

We obtain turnpike results for optimal control problems with lack of stabilizability in the state equation and/or detectability in the state term in the cost functional. We show how, under weakened stabilizability/detectability conditions, terminal conditions may affect turnpike phenomena. Numerical simulations have been performed to illustrate the theoretical results.

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arXiv2020

The turnpike property in the maximum hands-off control

This paper presents analyses for the maximum hands-off control using the geometric methods developed for the theory of turnpike in optimal control. First, a sufficient condition is proved for the existence of the maximum hands-off control for linear time-invariant systems with arbitrarily fixed initial and terminal points using the relation with $L^1$ optimal control. Next, a sufficient condition is derived for the maximum hands-off control to have the turnpike property, which may be useful for approximate design of the control.

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arXiv2019

Model Reduction of Converter-Dominated Power Systems by Singular Perturbation Theory

The increasing integration of power electronic devices is driving the development of more advanced tools and methods for the modeling, analysis, and control of modern power systems to cope with the different time-scale oscillations. In this paper, we propose a general methodology based on the singular perturbation theory to reduce the order of systems modeled by ordinary differential equations and the computational burden in their simulation. In particular, we apply the proposed methodology to a simplified power system scenario comprised of three inverters in parallel---controlled as synchronverters---connected to an ideal grid. We demonstrate by time-domain simulations that the reduced and decoupled system obtained with the proposed approach accurately represents the dynamics of the original system because it preserves the non-linear dynamics. This shows the efficiency of our technique even for transient perturbations and has relevant applications including the simplification of the Lyapunov stability assessment or the design of non-linear controllers for large-scale power systems.

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