Energy-based control for hybrid port-controlled Hamiltonian systems

In this paper we develop an energy-based hybrid control framework for hybrid port-controlled Hamiltonian systems. In particular, we obtain constructive sufficient conditions for hybrid feedback stabilization that provide a shaped energy function for the closed-loop system, while preserving a hybrid Hamiltonian structure at the closed-loop level. Furthermore, an inverse optimal hybrid feedback control framework is developed that characterizes a class of globally stabilizing energy-based controllers that guarantee hybrid sector and gain margins to multiplicative input uncertainty of hybrid Hamiltonian systems.     

Stabilization of Hamiltonian systems with dissipation

Stabilization of generalized Hamiltonian systems with dissipation is investigated. First, we generalize the Casimir submanifold approach with constant controls to the static state feedback control case. Secondly, a direct Hamiltonian function method is proposed and it is shown that this method is equivalent to the Casimir sub-manifold approach. Furthermore, a dynamic state feedback control is proposed. Some sufficient conditions and a constructive process for determining controllers are provided. It is shown that the dynamic state feedback control is more powerful than the static one. Finally, the problem of a dissipative type realization of the general controlled Hamiltonian system is discussed.     

Generalized Hamiltonian realization of time-invariant nonlinear systems

A key step in applying the Hamiltonian function method is to express the system under consideration into a generalized Hamiltonian system with dissipation, which yields the so-called generalized Hamiltonian realization (GHR). In this paper, we investigate the problem of GHR. Several new methods and the corresponding sufficient conditions are presented. A major result is that if the Jacobian matrix of a time-invariant nonlinear system is nonsingular, the system has a GHR whose structure matrix and Hamiltonian function are given in simple forms. Then the orthogonal decomposition method and a sufficient condition for the feedback dissipative realization are proposed.     

Stochastic stability of quasi-integrable Hamiltonian systems with time delay by using Lyapunov function method

The asymptotic Lyapunov stability of one quasi-integrable Hamiltonian system with time-delayed feedback control is studied by using Lyapunov functions and stochastic averaging method.First,a quasi-integrable Hamiltonian system with time-delayed feedback control subjected to Gaussian white noise excitation is approximated by a quasi-integrable Hamiltonian system without time delay.Then,stochastic averaging method for quasi-integrable Hamiltonian system is used to reduce the dimension of the original system,a...     

From nonlinear to Hamiltonian via feedback

Mechanical control systems are an especially interesting and important class of nonlinear control systems. They possess a rich mathematical structure and yet, physical considerations are extremely important for the solution of a large class of control problems. We broaden the applicability of design methodologies developed for mechanical control systems by rendering nonlinear control systems, mechanical by a proper choice of feedback. In particular, we characterize which control systems can be transformed to Hamiltonian control systems by a feedback transformation.     

Nonlinear optimal control as quantum mechanical eigenvalue problems

A novel approach for approximating the nonlinear optimal feedback control of a system with a terminal cost is proposed. To lessen the difficulty due to nonlinearity, we try to treat the system in a framework of linear theories. For this, we assume a quantum mechanical linear wave associated with the system. Since the control system is constrained by state equations, we handle the system according to quantum mechanics of constrained dynamics. A Hamiltonian is represented as a linear operator acting on a function that describes behavior of waves. Subsequently, nonlinear feedback is calculated without any time integration in the backward direction. Using eigenvalues and eigenfunctions of the linear Hamiltonian operator, an optimal feedback law is given as a combination of analytic functions of time and state variables. We take as an example a system described by two scalar variables for state and control input. Simulation studies on the system by the eigenvalue analysis show that the proposed method reduces calculation time to nearly a tenth that of a numerical calculation of a Hamilton-Jacobi equation by a finite difference method.     

Adaptive robust simultaneous stabilization of multiple n-degree-of-freedom robot systems

In this paper, the adaptive robust simultaneous stabilization problem of uncertain multiple n-degree-of-freedom (n-DOF) robot systems is studied using the Hamiltonian function method, and the corresponding adaptive L2 controller is designed. First, we investigate the adaptive simultaneous stabilization problem of uncertain multiple n-DOF robot systems without external disturbance. Namely, the single uncertain n-DOF robot system is transformed into an equivalent Hamiltonian form using the unified partial derivative operator (UP-DO) and potential energy shaping method, and then a high dimensional Hamiltonian system for multiple uncertain robot systems is obtained by applying augmented dimension technology, and a single output feedback controller is designed to ensure the simultaneous stabilization for the higher dimensional Hamiltonian system. On this basis, we further study the adaptive robust simultaneous stabilization control problem for the uncertain multiple n-DOF robot systems with external disturbances, and design an adaptive robust simultaneous stabilization controller. Finally, the simulation results show that the adaptive robust simultaneous stabilization controller designed in this paper is very effective in stabilizing multi-robot systems at the same time.     

Nonlinear decentralized controller design for multimachine power systems using Hamiltonian function method

In this paper, we first express a multimachine power system as a Hamiltonian control system with dissipation. Then, using the Hamiltonian function method a decentralized excitation control scheme, as a static measurable feedback, is proposed to stabilize the multimachine power system. Then, it is shown that the control scheme with properly chosen parameters is also an H_∞ control, which solves the problem of disturbance attenuation simultaneously. Finally, the design technique is demonstrated by a three-machine power system.     

Iterative feedback tuning for Hamiltonian systems based on variational symmetry

This paper proposes a novel iterative feedback tuning (IFT) for Hamiltonian systems, which can describe a practically important class of nonlinear systems. Hamiltonian systems have a special property called variational symmetry, and it can be used to estimate the input‐output mapping of the variational adjoint for certain input‐output mappings of the systems. First, we derive a modified variational symmetry to adapt to the gradient estimation of an optimal control–type cost function with respect to adjustable parameters of a controller. Second, we provide an IFT algorithm based on the property, which generates the optimal parameters minimizing the cost function by iteration of experiments. The proposed algorithm requires less number of experiments to estimate the gradient than the conventional IFT methods for nonlinear systems. We also provide a method to optimize the elements of the dissipation matrix, which does not directly appear in the Hamiltonian function, by equipping a dynamic feedback of the generalized coordinate. Moreover, we provide an IFT algorithm considering parameter constraints so that the parameters can be optimized within a prescribed search range. Finally, a numerical simulation of a two‐link robot manipulator including a comparison with the conventional IFT methods demonstrates the effectiveness of the proposed method.     

New approaches to generalized Hamiltonian realization of autonomous nonlinear systems

The Hamiltonian function method plays an important role in stability analysis and stabilization. The key point in applying the method is to express the system under consideration as the form of dissipative Hamiltonian systems, which yields the problem of generalized Hamiltonian realization. This paper deals with the generalized Hamiltonian realization of autonomous nonlinear systems. First, this paper investigates the relation between traditional Hamiltonian realizations and first integrals, proposes a new method of generalized Hamiltonian realization called the orthogonal decomposition method, and gives the dissipative realization form of passive systems. This paper has proved that an arbitrary system has an orthogonal decomposition realization and an arbitrary asymptotically stable system has a strict dissipative realization. Then this paper studies the feedback dissipative realization problem and proposes a control-switching method for the realization. Finally, this paper proposes several sufficient     

On feedback equivalence to port controlled Hamiltonian systems

In the last few years port controlled Hamiltonian (PCH) systems have emerged as an interesting class of nonlinear models suitable for a large number of physical applications. In this paper we study the question of feedback equivalence of nonlinear systems to PCH systems. More precisely, we give conditions under which a general nonlinear system can be transformed into a PCH system via static state feedback. We consider the two extreme cases where the target PCH system is completely a priori fixed or completely free, as well as the case where it is only partially predetermined. When the energy function is free a set of partial differential equations needs to be solved, on the other hand, if it is fixed we have to deal with a set of algebraic equations. In the former case, we give some verifiable necessary and sufficient conditions for solvability. As a by-product of our analysis we obtain some stabilization results for nonlinear systems.     

Structure-preserving stabilization of Hamiltonian control systems

Sufficient conditions for nonasymptotical stabilization of equilibrium points in Hamiltonian control systems using the Lyapunov technique and feedback transformation which preserve the Hamiltonian structure are presented.     

Non-linear decentralized saturated controller design for multi-machine power systems

A direct method of constructing a decentralized saturated controller is very important in the controller design of multimachine power systems. Among the existing designs the author first constructed a state feedback, then saturated the controller. However, the author could not prove that this saturated controller was valid. In this paper a multi-machine power system as a Hamiltonian control system with dissipation is first represented. Then using a Hamiltonian function a decentralized saturated excitation control scheme, which is a static measurable feedback, is proposed. Last, the design is discussed in detail on a three-machine power system. Simulations show that the saturated, simple control law works well.     

Adaptive H-infinity control of synchronous generators with steam valve via Hamiltonian function method

Based on Hamiltonian formulation, this paper proposes a design approach to nonlinear feedback excitation control of synchronous generators with steam valve control, disturbances and unknown parameters. It is shown that the dynamics of the synchronous generators can be expressed as a dissipative Hamiltonian system, based on which an adaptive H-infinity controller is then designed for the systems by using the structure properties of dissipative Hamiltonian systems. Simulations show that the controller obtained in this paper is very effective.     

Dampening controllers via a Riccati equation approach

An algorithm is presented which computes a state feedback for a standard linear system which not only stabilizes, but also dampens the closed-loop system dynamics. In other words, a feedback gain matrix is computed such that the eigenvalues of the closed-loop state matrix are within the region of the left half-plane where the magnitude of the real part of each eigenvalue is greater than that of the imaginary part, This may be accomplished by solving a damped algebraic Riccati equation and a degenerate Riccati equation. The solution to these equations are computed using numerically robust algorithms, Damped Riccati equations are unusual in that they may be formulated as an invariant subspace problem of a related periodic Hamiltonian system. This periodic Hamiltonian system induces two damped Riccati equations: one with a symmetric solution and another with a skew symmetric solution. These two solutions result in two different state feedbacks, both of which dampen the system dynamics, but produce different closed-loop eigenvalues, thus giving the controller designer greater freedom in choosing a desired feedback     

Feedback control of nonlinear stochastic systems for targeting a specified stationary probability density

In the present paper, an innovative procedure for designing the feedback control of multi-degree-of-freedom (MDOF) nonlinear stochastic systems to target a specified stationary probability density function (SPDF) is proposed based on the technique for obtaining the exact stationary solutions of the dissipated Hamiltonian systems. First, the control problem is formulated as a controlled, dissipated Hamiltonian system together with a target SPDF. Then the controlled forces are split into a conservative part and a dissipative part. The conservative control forces are designed to make the controlled system and the target SPDF have the same Hamiltonian structure (mainly the integrability and resonance). The dissipative control forces are determined so that the target SPDF is the exact stationary solution of the controlled system. Five cases, i.e., non-integrable Hamiltonian systems, integrable and non-resonant Hamiltonian systems, integrable and resonant Hamiltonian systems, partially integrable and non-resonant Hamiltonian systems, and partially integrable and resonant Hamiltonian systems, are treated respectively. A method for proving that the transient solution of the controlled system approaches the target SPDF as t→∞ is introduced. Finally, an example is given to illustrate the efficacy of the proposed design procedure.     

Asymptotic Lyapunov stability with probability one of quasi-integrable Hamiltonian systems with delayed feedback control

The asymptotic Lyapunov stability with probability one of multi-degree-of-freedom (MDOF) quasi-integrable and nonresonant Hamiltonian systems with time-delayed feedback control subject to multiplicative (parametric) excitation of Gaussian white noise is studied. First, the time-delayed feedback control forces are expressed approximately in terms of the system state variables without time delay and the system is converted into ordinary quasi-integrable and nonresonant Hamiltonian system. Then, the averaged Itô stochastic differential equations are derived by using the stochastic averaging method for quasi-integrable Hamiltonian systems and the expression for the largest Lyapunov exponent of the linearized averaged Itô equations is derived. Finally, the necessary and sufficient condition for the asymptotic Lyapunov stability with probability one of the trivial solution of the original system is obtained approximately by letting the largest Lyapunov exponent to be negative. An example is worked out in detail to illustrate the above mentioned procedure and its validity and to show the effect of the time delay in feedback control on the largest Lyapunov exponent and the stability of the system.     

Controlled invariance for hamiltonian systems

A notion of controlled invariance is developed which is suited to Hamiltonian control systems. This is done by replacing the controlled invariantdistribution, as used for general nonlinear control systems, by the controlled invariantfunction group. It is shown how Lagrangian or coisotropic controlled invariant function groups can be made invariant by static, respectively dynamic, Hamiltonian feedback. This constitutes a first step in the development of a geometric control theory for Hamiltonian systems that explicitly uses the given structure.     

Stabilization of an underactuated bottom‐heavy airship via interconnection and damping assignment

This paper focuses on feedback stabilization of a neutrally buoyant and bottom‐heavy airship actuated by only five independent controls (with the rolling motion underactuated). The airship is modelled as an eudipleural submerged rigid body whose dynamics is formulated as a Hamiltonian system with respect to a Lie–Poisson structure. By exploiting the geometrical structure and using the so‐called interconnection and damping assignment (IDA) passivity‐based methodology for port‐controlled Hamiltonian systems, state feedback control laws asymptotically stabilizing two typical motions are designed via La Salle invariance principle and Chetaev instability theorem. Simulation results verify the control laws. Copyright © 2007 John Wiley & Sons, Ltd.     

Noninteracting control with stability for Hamiltonian systems

The problem of noninteraction with stability via dynamic state feedback is addressed and solved for a class of nonlinear Hamiltonian systems. A simple way to check necessary and sufficient conditions is proposed. It is well known that to decide if the problem is solvable, and which class of state feedback has to be used, the stability properties of some special dynamics are to be investigated. For this reason, on the way to the main result, it is shown that such dynamics are not necessarily Hamiltonian. Several examples, clarifying the role of different classes of state-feedback control laws (either static or dynamic) in the solution of the problem, are proposed