Invariant-Driven Strategies for Maude

We propose generic invariant-driven strategies that control the execution of systems by guaranteeing that the given invariants are satisfied. Our strategies are generic in the sense that they are parameterized by the system whose execution they control, by the logic in which the invariants are expressed, and by the invariants themselves. We illustrate the use of the strategies in the case of invariants expressed in propositional logic. However, the good properties of Maude as a logical and semantic framework, in which many different logics and formalisms can be expressed and executed allow us to use other logics as parameter of our strategies.     

Spectral Methods for Substantial Fractional Differential Equations

In this paper, a non-polynomial spectral Petrov–Galerkin method and its associated collocation method for substantial fractional differential equations are proposed, analyzed, and tested. We modify a class of generalized Laguerre polynomials to form our trial basis and test basis. After a proper scaling of these bases, our Petrov–Galerkin method results in diagonal and well-conditioned linear systems for certain types of fractional differential equations. In the meantime, we provide superconvergence points of the Petrov–Galerkin approximation for associated fractional derivative and function value of true solution. Additionally, we present explicit fractional differential collocation matrices based upon Laguerre–Gauss–Radau points. It is noteworthy that the proposed methods allow us to adjust a parameter in the basis according to different given data to maximize the convergence rate. All these findings have been proved rigorously in our convergence analysis and confirmed in our numerical experiments.     

随机多变量系统的结构与参数不变量辨识

引入多变量系统Markov参数矩阵相关分析下的最优输入设计向量,得到Markov 参数矩阵估汁.在此基础上,推导出有色噪声干扰下多变量线性系统Kronecker结构不变量 与参数不变量的递推辨识算法.     

Stability and passivity preserving Petrov–Galerkin approximation of linear infinite-dimensional systems

This contribution presents two approximation methods for linear infinite-dimensional systems that ensure the preservation of stability and passivity. The first approach allows one to approximate internal source free infinite-dimensional systems such that the resulting approximation is a port-controlled Hamiltonian system with dissipation. The second method deals with the class of systems that are not required to have conjugated outputs but only a dissipative system operator. It yields approximations with a dissipative system matrix for which bounds of their stability margin are provided. Both approaches are based on a state space formulation of the infinite-dimensional system. This makes it possible to use the Petrov–Galerkin approximation whose free parameters are partly used for achieving the structure preservation. Since still free parameters remain, further application specific objectives, such as, e.g., moment matching, can be achieved. Both approaches are applied to the approximation of an Euler–Bernoulli beam.     

Adaptive H_∞ Control for Nonlinear Hamiltonian Systems with Time Delay and Parametric Uncertainties

This paper addresses the adaptive H∞ control problem for a class of nonlinear Hamiltonian systems with time delay and parametric uncertainties. The uncertainties under consideration are some small parameter perturbations involved in the structure of the Hamiltonian system. Both delay-independent and delay-dependent criteria are established based on the dissipative structural properties of the Hamiltonian systems and the Lyapunov-Krasovskii functional approach. In order to construct the adaptive H∞controller, the situation that the parameter perturbation is inexistent in the system is also studied and the controller is designed.The adaptive H∞ control problem is solved under some sufficient conditions which ensure the asymptotic stability and the L2 gain performance of the resulted closed-loop system. Numerical example is given to illustrate the applicability of the theoretical results.     

Transition to an optimal periodic gait by simultaneous input and parameter optimization method of Hamiltonian systems

This paper is concerned with a gait transition to an optimal periodic gait by a simultaneous input and parameter optimization technique of Hamiltonian systems. First, a continuous-time dynamics of a passive walking/running robot between the touchdown and lift-off is considered as a Hamiltonian system. Then, the control input and some robot parameters, such as the mass, inertia, link length, and so on, are optimized using learning optimal control of Hamiltonian systems, which has been developed by the authors. This method allows one to simultaneously obtain an optimal feedforward input and optimal parameters, which (at least locally) minimize a given cost function. The main advantage is that the precise model of the dynamics of the plant system is not required using a symmetric property of Hamiltonian systems, called variational symmetry. We formulate an optimal gait generation scheme via the learning optimal control, where the robot keeps walking and the gait is optimized with respect to the control input and some adjustable robot parameters simultaneously. As a result, the gait transition to an optimal periodic one is achieved.     

Automatic Generation of Invariants

When proving invariance properties of programs, one is faced with two problems. The first problem is related to the necessity of proving tautologies of the considered assertion language, whereas the second manifests itself in the need of finding sufficiently strong invariants. This paper focuses on the second problem and describes techniques for the automatic generation of invariants. The first set of these techniques is applicable to sequential transition systems and allows deriving so-called local invariants, i.e., predicates which are invariant at some control location. The second is applicable on networks of transition systems and allows combining local invariants of the sequential components to obtain local invariants of the global system.     

History of spacecraft control systems

The paper discusses development of spacecraft control systems from the first vehicles to up-to-date space complexes. Academician B.N. Petrov was one the founders of the spacecraft control theory. Advances in automatic systems, control system instrumentation, computers, actuators, and control system techniques which allowed of development of multi-purpose and multi-mode systems with long-term active life cycle and systems with the maximum endurance and control process automation are examined.     

On the Conservation of Fractional Nonlinear Schrödinger Equation’s Invariants by the Local Discontinuous Galerkin Method

Using the primal formulation of the Local Discontinuous Galerkin (LDG) method, discrete analogues of the energy and the Hamiltonian of a general class of fractional nonlinear Schrödinger equation are shown to be conserved for two stabilized version of the method. Accuracy of these invariants is numerically studied with respect to the stabilization parameter and two different projection operators applied to the initial conditions. The fully discrete problem is analyzed for two implicit time step schemes: the midpoint and the modified Crank–Nicolson; and the explicit circularly exact Leapfrog scheme. Stability conditions for the Leapfrog scheme and a stabilized version of the LDG method applied to the fractional linear Schrödinger equation are derived using a von Neumann stability analysis. A series of numerical experiments with different nonlinear potentials are presented.     

Invariant based modeling and control of multi-phase reactor systems

We propose a modeling framework for stable simulation of multi-phase reactor systems operating at thermodynamic equilibrium. The model framework can be used to determine system characteristics, explore parameter sensitivity and test control system strategies. The thermodynamic equilibrium assumption and the use of reaction invariants make it computationally inexpensive. We show that the feedback control approach based on the overall inventories of the system can be effectively used for improved performance of such reactor systems. Two multi-phase reactor systems – the vapor recovery reactor used in carbothermic aluminum reduction and the gasification reactor used in IGCC are considered to demonstrate the efficacy of the proposed modeling and control approach.     

Energy-shaping for Hamiltonian control systems with time delay

This paper investigates the asymptotical stabilization of Hamiltonian control systems with time delay. First, Hamiltonian control systems with time delay are proposed. Second, a two-to-one (TTO) principle is introduced that two different Hamiltonian functions are simultaneously energy-shaping by one desired energy function. Third, a novel matching equation is built via the TTO principle for the Hamiltonian control systems with time delay, which generates an effective control law for the Hamiltonian control systems with time delay. Finally, a numerical example shows the effectiveness of proposed method.     

Robust Adaptive Control of Uncertain Stochastic Hamiltonian Systems with Time Varying Delay

This paper investigates the robust adaptive control problem for a class of time‐delay stochastic Hamiltonian systems. The system under study involves stochastics, parameter uncertaintiess, and time varying delay. The aim of this study is to design an uncertainty‐independent adaptive control law such that, for all admissible uncertainties, as well as stochastics, the closed‐loop Hamiltonian system is robustly asymptotically stable in mean square. Sufficient conditions are proposed to guarantee the rationality and validity of the proposed control laws, which are derived based on Lyapunov functional method. The performance of the controllers is validated through digital simulations.     

Quantifier-free encoding of invariants for hybrid systems

Hybrid systems are a clean modeling framework for embedded systems, which feature integrated discrete and continuous dynamics. A well-known source of complexity comes from the time invariants, which represent an implicit quantification of a constraint over all time points of a continuous transition. Emerging techniques based on Satisfiability Modulo Theory (SMT) have been found promising for the verification and validation of hybrid systems because they combine discrete reasoning with solvers for first-order theories. However, these techniques are efficient for quantifier-free theories and the current approaches have so far either ignored time invariants or have been limited to hybrid systems with linear constraints. In this paper, we propose a new method that encodes a class of hybrid systems into transition systems with quantifier-free formulas. The method does not rely on expensive quantifier elimination procedures. Rather, it exploits the sequential nature of the transition system to split the continuous evolution enforcing the invariants on the discrete time points. This way, we can encode all hybrid systems whose invariants can be expressed in terms of polynomial constraints. This pushes the application of SMT-based techniques beyond the standard linear case.     

Study on the stability of switched dissipative Hamiltonian systems

The hybrid Hamiltonian system is a kind of important nonlinear hybrid systems. Such a system not only plays an important role in the development of hybrid control theory, but also finds many applications in practical control designs for obtaining better control performances. This paper investigates the stability of switched dissipative Hamiltonian systems under arbitrary switching paths. Under a realistic assumption, it is shown that the Hamiltonian functions of all the subsystems can be used as the multiple-Lyapunov functions for the switched dissipative Hamiltonian system. Based on this and using the dissipative Hamiltonian structural properties, this paper then proves that the P-norm of the state of switched dissipative Hamiltonian system converges to zero with the time increasing, and presents two sufficient conditions for the asymptotical stability under arbitrary switching paths. Utilizing these new results, this paper also obtains two useful corollaries for the asymptotical stability of switched nonlinear time-invariant systems. Finally, two examples are studied by using the new results proposed in this paper, and some numerical simulations are carried out to support our new results.     

Stabilization of Hamiltonian systems with nonholonomic constraints based on time-varying generalized canonical transformations

This paper is concerned with the stabilization of nonholonomic systems in port-controlled Hamiltonian formulae based on time-varying generalized canonical transformations. A special class of time-varying generalized canonical transformations are introduced which modify the kinetic energy of the original system without changing the generalized Hamiltonian structure with passivity. Utilizing these transformations, time-varying asymptotically stabilizing controllers for the nonholonomic Hamiltonian systems are derived. Since the proposed method is a natural generalization of passivity based control for conventional holonomic systems, it is expected that the tools developed for conventional systems will be applicable to nonholonomic systems based on the proposed method.     

Constant and switching gains in semi-active damping of vibrating structures

A limit cycle is the stability boundary for linear and non-linear control systems. Hamiltonian mechanics and power flow control are employed to demonstrate this property of limit cycles. The presentation begins with the concept of linear limit cycles which is extended to non-linear limit cycles. Many examples are used to demonstrate these concepts including linear and non-linear oscillators, power engineering, and an extension to a class of plane differential systems. Power flow control based on Hamiltonian mechanics is shown to be applicable to a large class of non-linear systems. Finally, eigenanalysis and flight stability for linear systems are extended to non-linear systems and is referred to as ‘the power flow principle of stability for non-linear systems’.     

Adaptive H ∞ excitation control of multimachine power systems via the Hamiltonian function method

Using the Hamiltonian function method, this paper investigates the adaptive H _∞ excitation control of multimachine power systems with disturbances and parameter perturbations. A key step in applying the Hamiltonian function method to the multimachine system is to express the system as a dissipative Hamiltonian system, i.e. to complete dissipative Hamiltonian realization (DHR). By using pre-feedback technique, this paper expresses the multimachine power system as a dissipative Hamiltonian system. Then, the stability analysis of the achieved dissipative Hamiltonian system is proceeded. Finally, based on the achieved DHR form, the adaptive H _∞ excitation control of the multimachine power system is investigated and a decentralized simple excitation control strategy is obtained. Simulations on a six-machine system show that the adaptive H _∞ excitation control strategy proposed in the paper is more effective than some other control schemes.     

Robust stochastic maximum principle for multi-model worst case optimization

This paper develops a version of the robust maximum principle applied to the minimax Mayer problem formulated for stochastic differential equations with a control-dependent diffusion term. The parametric families of first and second order adjoint stochastic processes are introduced to construct the corresponding Hamiltonian formalism. The Hamiltonian function used for the construction of the robust optimal control is shown to be equal to the sum of the standard stochastic Hamiltonians corresponding to each possible value of the parameter. The cost function is defined on a finite horizon and contains the mathematical expectation of a terminal term. A terminal condition, given by a vector function, is also considered. The optimal control strategies, adapted for available information, for the wide class of multi-model systems given by a stochastic differential equation with parameters from a given finite set are constructed. This problem belongs to the class of minimax stochastic optimization problems. The proof is based on the recent results obtained for deterministic minimax Mayer problem by Boltyanski and Poznyak as well as on the results of Zhou and of Yong and Zhou, obtained for stochastic maximum principle for non-linear stochastic systems with a single-valued parameter. Two illustrative examples, dealing with production planning and reinsurance-dividend management, conclude this study.     

2-自由度强非线性振动系统的参数识别

提出了非线性多自由度系统的一种新的参数识别方法,研究了二次非线性的2-自由度系统.基于保守系统存在能量积分的特点,由系统的运动微分方程导出了哈密尔顿函数,并用它作为参数识别的数学模型.利用系统自由振荡条件下相坐标测量值集合对系统的哈密尔顿函数进行拟合,并用最小二乘法进行参数识别.不管系统非线性度的强弱如何,只要系统是保守的,这种方法就有效.