Real-time computation of optimal control

A new approach to the real-time implementation of time-optimal control for linear systems with a bounded control is proposed. The computational costs are separated between preliminary computations and computations in the course of the control process. The preliminary computations are independent of the particular initial condition and are based on the approximation of sets reachable in different times by a collection of hyperplanes. Methods for constructing hyperplanes and selecting a supporting hyperplane are described. Methods are proposed for approximately finding the normalized vector of initial conditions of the adjoint system, the driving time, and the switching times of the time-optimal control, and an iterative method for their refinement is developed. The computational complexity of the method is estimated. The computational algorithm is described, and simulation and numerical results are presented.     

Optimal control with connected initial and terminal conditions

An optimal control problem with linear dynamics is considered on a fixed time interval. The ends of the interval correspond to terminal spaces, and a finite-dimensional optimization problem is formulated on the Cartesian product of these spaces. Two components of the solution of this problem define the initial and terminal conditions for the controlled dynamics. The dynamics in the optimal control problem is treated as an equality constraint. The controls are assumed to be bounded in the norm of L₂. A saddle-point method is proposed to solve the problem. The method is based on finding saddle points of the Lagrangian. The weak convergence of the method in controls and its strong convergence in state trajectories, dual trajectories, and terminal variables are proved.     

Real-time calculation of the current state estimates for a class of delay systems

The linear optimal observation problem is examined for one type of nonstationary delay system with an uncertainty in the initial state. A fast implementation of the dual method is proposed for calculating estimates of the initial state. This implementation is based on the quasi-reduction of the fundamental matrix of solutions to the mathematical model of delay systems. It is shown that an iteration step of the dual method only requires that auxiliary systems of ordinary differential equations be integrated on small time intervals. An algorithm is described for the real-time calculation of current state estimates. The results are illustrated by the optimal observation problem for a third-order stationary delay system.     

Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique

In this paper, the problem of finite-time chaos synchronization between two different chaotic systems with fully unknown parameters is investigated. First, a new nonsingular terminal sliding surface is introduced and its finite-time convergence to the zero equilibrium is proved. Then, appropriate adaptive laws are derived to tackle the unknown parameters of the systems. Afterwards, based on the adaptive laws and finite-time control idea, an adaptive sliding mode controller is proposed to ensure the occurrence of the sliding motion in a given finite time. It is mathematically proved that the introduced sliding mode technique has finite-time convergence and stability in both reaching and sliding mode phases. Finally, some numerical simulations are presented to demonstrate the applicability and effectiveness of the proposed technique.     

Real-Time Computation of Resource Optimal Control

Computational Mathematics and Mathematical Physics - For linear systems with bounded control, a new approach to real-time implementation of a resource-optimal control is proposed. The computational...     

Robust finite-time convergence of chaotic systems via adaptive terminal sliding mode scheme

This paper investigates robust finite-time stabilization of a class of uncertain chaotic systems. A new terminal sliding mode (TSM) algorithm is proposed to steer the plant fast to zero within finite time. In particular, a new form of TSM is developed for multi-input and multi-output systems, and some criteria are presented to facilitate its control design. With adaption laws to identify uncertain parameters and unknown bounds on disturbances, the proposed terminal sliding mode controllers get rid of uncertainties and nonlinearities successfully. The closed-loop systems are provided with fast finite-time stability and strong robustness against uncertainties. Finally, numerical simulation of Lorenz system illustrates the effectiveness of this proposed control scheme.     

Terminal control of boundary models

A terminal optimal control problem for finite-dimensional static boundary models is formulated. The finite-dimensional models determine the initial and terminal states of the plant. The choice of an optimal control drives the plant from one state to another. A saddle-point method is proposed for solving this problem. The convergence of the method in a Hilbert space is proved.     

Finite-time control of discrete-time systems with time-varying exogenous disturbance

This paper deals with some finite-time control problems for uncertain discrete-time linear systems subject to exogenous disturbance. Sufficient conditions are presented for finite-time stabilization via state feedback. These conditions can be reduced to feasibility problems involving linear matrix inequality (LMI). A detailed solving method is proposed for the restricted linear matrix inequalities. Finally, an example illustrates the proposed methodology.     

Optimal resource consumption control of perturbed systems

A method for calculating the optimal resource consumption control of perturbed dynamic systems is developed. This method includes both normal and singular solutions. According to the method proposed, the problem is subdivided into three independent tasks: (1) consideration of the effects of perturbations on the system, (2) computation of the optimal control structure, and (3) computation of the switching instants of the optimal control. The consideration of the influence of perturbations on the system and the transfer to a nonzero final state are reduced to the transformation of the initial and final states of the system. The control structure calculation is based on a specific method of quasi-optimal control formation. The control switching instants are found by using the relationship between deviations in the initial conditions of the conjugate system and deviations of the phase trajectory at the final instant. An iterative algorithm is developed, and its characteristics are considered. Results of modeling and numerical calculations are presented.     

Singular solution to the problem of minimizing resource consumption

An iterative method of finding a singular solution to the problem of minimizing resource consumption has been developed. This method is based on the information about the finite control structure. A condition for existence of a singular solution is obtained. The limit value for transferring the time between the normal and the singular solutions is found. A relation between the variations of the control switching instants and the variations of the initial conditions of the adjoint system is found. A system of linear algebraic equations relating the variations of the initial conditions of the adjoint system to the deviations of the phase coordinates from a given final state of the system is obtained. The calculation algorithm and the results of modeling and numerical calculations are presented.     

Synthesis of linear systems with desired equivalent form

A method for feedback synthesis of linear control systems with desired linearly equivalent form of the closed loop system matrix is proposed. The method is based on the serial canonical form of linear multivariable systems which is an alternative to the Luenberger canonical form. A stable computational algorithm for finding the serial canonical form using orthogonal similarity transformations is described. The algorithm for synthesis involves a simple recurrent procedure and gives the possibility to achieve any attainable equivalent form of the closed loop system. The results obtained are extended to the synthesis of reduced order state observers.     

Improving an estimate of the convergence rate of the seidel method by selecting the optimal order of equations in the system of linear algebraic equations

The Seidel method for solving a system of linear algebraic equations and an estimate of its convergence rate are considered. It is proposed to change the order of equations. It is shown that the method described in Faddeevs’ book Computational Methods of Linear Algebra can deteriorate the convergence rate estimate rather than improve it. An algorithm for establishing the optimal order of equations is proposed, and its validity is proved. It is shown that the computational complexity of the reordering is 2n ² additions and (12)n ² divisions. Numerical results for random matrices of order 100 are presented that confirm the proposed improvement.     

Observer-based finite-time control of time-delayed jump systems

This paper provides the observer-based finite-time control problem of time-delayed Markov jump systems that possess randomly jumping parameters. The transition of the jumping parameters is governed by a finite-state Markov process. The observer-based finite-time H_∞ controller via state feedback is proposed to guarantee the stochastic finite-time boundedness and stochastic finite-time stabilization of the resulting closed-loop system for all admissible disturbances and unknown time-delays. Based on stochastic finite-time stability analysis, sufficient conditions that ensure stochastic robust control performance of time-delay jump systems are derived. The control criterion is formulated in the form of linear matrix inequalities and the designed finite-time stabilization controller is described as an optimization one. The presented results are extended to time-varying delayed MJSs. Simulation results illustrate the effectiveness of the developed approaches.     

Numerical method of solving a linear problem on the minimum of resource consumption

A simple algorithm for developing a quasioptimal control over resource consumption is considered. The control is used as an initial approach to an iterative procedure of computing an optimal control. A system of linear algebraic equations is derived which approximately relate increments of initial conditions of an adjoint system to increments of amplitudes of a quasioptimal control with respect to ultimate values. Local convergence of the computing process with a quadratic rate is proved, and the convergence radius is found. A condition for global convergence of the method is specified.     

Implicit algorithms for solving the Cauchy problem for the equations of the dynamics of mechanical systems

It is shown that in the numerical solution of the Cauchy problem for systems of second-order ordinary differential equations, when solved for the highest-order derivative, it is possible to construct simple and economical implicit computational algorithms for step-by-step integration without using laborious iterative procedures based on processes of the Newton-Raphson iterative type. The initial problem must first be transformed to a new argument — the length of its integral curve. Such a transformation is carried out using an equation relating the initial parameter of the problem to the length of the integral curve. The linear acceleration method is used as an example to demonstrate the procedure of constructing an implicit algorithm using simple iterations for the numerical solution of the transformed Cauchy problem. Propositions concerning the computational properties of the iterative process are formulated and proved. Explicit estimates are given for an integration stepsize that guarantees the convergence of the simple iterations. The efficacy of the proposed procedure is demonstrated by the numerical solution of three problems. A comparative analysis is carried out of the numerical solutions obtained with and without parametrization of the initial problems in these three settings. As a qualitative test the problem of the celestial mechanics of the “Pleiades” is considered. The second example is devoted to modelling the non-linear dynamics of an elastic flexible rod fixed at one end as a cantilever and coiled in its initial (static) state into a ring by a bending moment. The third example demonstrates the numerical solution of the problem of the “unfolding” of a mechanical system consisting of three flexible rods with given control input.     

A sliding mode approach to robust stabilisation of Markovian jump linear time-delay systems with generally incomplete transition rates

This paper is devoted to investigating the problem of robust sliding mode control for a class of uncertain Markovian jump linear time-delay systems with generally uncertain transition rates (GUTRs). In this GUTR model, each transition rate can be completely unknown or only its estimate value is known. By making use of linear matrix inequalities technique, sufficient conditions are presented to derive the linear switching surface and guarantee the stochastic stability of sliding mode dynamics. A sliding mode control law is developed to drive the state trajectory of the closed-loop system to the specified linear switching surface in a finite-time interval in spite of the existing uncertainties, time delays and unknown transition rates. Finally, an example is presented to verify the validity of the proposed method.     

A note on the implementation of an interior-point algorithm for nonlinear optimization with inexact step computations

This paper describes an implementation of an interior-point algorithm for large-scale nonlinear optimization. It is based on the algorithm proposed by Curtis et?al. (SIAM J Sci Comput 32:3447?C3475, 2010), a method that possesses global convergence guarantees to first-order stationary points with the novel feature that inexact search direction calculations are allowed in order to save computational expense. The implementation follows the proposed algorithm, but includes many practical enhancements, such as functionality to avoid the computation of a normal step during every iteration. The implementation is included in the IPOPT software package paired with an iterative linear system solver and preconditioner provided in PARDISO. Numerical results on a large nonlinear optimization test set and two PDE-constrained optimization problems with control and state constraints are presented to illustrate that the implementation is robust and efficient for large-scale applications.     

不确定非自治混沌陀螺仪在非线性输入下的有限时间稳定性

陀螺仪是一个非常有趣,又是永恒的非线性非自治动力系统课题,它可以显示出非常复杂的动力学行为,如混沌现象.在一个给定的有限时间内,研究非线性非自治陀螺仪鲁棒稳定性问题.假设陀螺仪系统受到模型不确定的外部扰动而摄动,系统参数并不知道,同时考虑了非线性输入的影响.为未知参数提出了适当的自适应律.以自适应律和有限时间控制理论为基础,提出非连续有限时间控制理论,来研究系统的有限时间稳定性.解析证明了闭循环系统的有限时间稳定性及其收敛性.若干数值仿真结果表明,该文的有限时间控制法是有效的,同时验证了该文的理论结果.     

Optimal Control of Iterative Solution Methods for Linear Systems of Equations

Iterative solution methods for linear systems of equations can be regarded as discrete-time control systems, for which a stabilizing feedback control has to be found. Well known algorithms such as GMRES(m) may exhibit unstable dynamics or sensitive dependence on initial conditions, thus preventing the algorithm to converge to the desired solution. Based on linear system feedback design techniques a new algorithm is proposed that does not suffer under such shortcomings. Global convergence to the desired solution is shown for any initial state. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite-time stabilization of nonlinear impulsive dynamical systems

Finite-time stability involves dynamical systems whose trajectories converge to a Lyapunov stable equilibrium state in finite time. For continuous-time dynamical systems finite-time convergence implies nonuniqueness of system solutions in reverse time, and hence, such systems possess non-Lipschitzian dynamics. For impulsive dynamical systems, however, it may be possible to reset the system states to an equilibrium state achieving finite-time convergence without requiring non-Lipschitzian system dynamics. In this paper, we develop sufficient conditions for finite-time stability of impulsive dynamical systems using both scalar and vector Lyapunov functions. Furthermore, we design hybrid finite-time stabilizing controllers for impulsive dynamical systems that are robust against full modelling uncertainty. Finally, we present a numerical example for finite-time stabilization of large-scale impulsive dynamical systems.