Stability margins of nonlinear receding-horizon control via inverse optimality

Using the nonlinear analog of the Fake Riccati equation developed for linear systems, we derive an inverse optimality result for several receding-horizon control schemes. This inverse optimality result unifies stability proofs and shows that receding-horizon control possesses the stability margins of optimal control laws.     

Locally optimal and robust backstepping design

For a class of nonlinear systems, a robust backstepping design achieves both local optimality and global inverse optimality. The design is robust in the sense that it achieves a prescribed level of disturbance attenuation with stability margins. An analytic example illustrates the performance of the locally optimal control design     

Inverse optimal design of input-to-state stabilisation for affine nonlinear systems with input delays

We establish robustness of the predictor feedback control law to perturbations appearing at the system input for affine nonlinear systems with time-varying input delay and additive disturbances. Furthermore, it is shown that it is inverse optimal with respect to a differential game problem. All of the stability and inverse optimality proofs are based on the infinite-dimensional backstepping transformation and an appropriate Lyapunov functional. A single-link manipulator subject to input delays and disturbances is given to illustrate the validity of the proposed method.     

On Stability and Inverse Optimality for a Class of Multi-agent Linear Consensus Protocols

International Journal of Control, Automation and Systems - This paper studies stability and inverse optimality for a class of linear consensus protocols applied to the identical linear...     

Inverse optimal stabilization of a rigid spacecraft

The authors present an approach for constructing optimal feedback control laws for regulation of a rotating rigid spacecraft. They employ the inverse optimal control approach which circumvents the task of solving a Hamilton-Jacobi equation and results in a controller optimal with respect to a meaningful cost functional. The inverse optimality approach requires the knowledge of a control Lyapunov function and a stabilizing control law of a particular form. For the spacecraft problem, they are both constructed using the method of integrator backstepping. The authors give a characterization of (nonlinear) stability margins achieved with the inverse optimal control law     

Partial‐state stabilization and optimal feedback control

In this paper, we develop a unified framework to address the problem of optimal nonlinear analysis and feedback control for partial stability and partial‐state stabilization. Partial asymptotic stability of the closed‐loop nonlinear system is guaranteed by means of a Lyapunov function that is positive definite and decrescent with respect to part of the system state, which can clearly be seen to be the solution to the steady‐state form of the Hamilton–Jacobi–Bellman equation and hence guaranteeing both partial stability and optimality. The overall framework provides the foundation for extending optimal linear‐quadratic controller synthesis to nonlinear nonquadratic optimal partial‐state stabilization. Connections to optimal linear and nonlinear regulation for linear and nonlinear time‐varying systems with quadratic and nonlinear nonquadratic cost functionals are also provided. Finally, we also develop optimal feedback controllers for affine nonlinear systems using an inverse optimality framework tailored to the partial‐state stabilization problem and use this result to address polynomial and multilinear forms in the performance criterion. Copyright © 2015 John Wiley & Sons, Ltd.     

Stochastic differential games and inverse optimal control and stopper policies

In this paper, we consider a two-player stochastic differential game problem over an infinite time horizon where the players invoke controller and stopper strategies on a nonlinear stochastic differential game problem driven by Brownian motion. The optimal strategies for the two players are given explicitly by exploiting connections between stochastic Lyapunov stability theory and stochastic Hamilton–Jacobi–Isaacs theory. In particular, we show that asymptotic stability in probability of the differential game problem is guaranteed by means of a Lyapunov function which can clearly be seen to be the solution to the steady-state form of the stochastic Hamilton–Jacobi–Isaacs equation, and hence, guaranteeing both stochastic stability and optimality of the closed-loop control and stopper policies. In addition, we develop optimal feedback controller and stopper policies for affine nonlinear systems using an inverse optimality framework tailored to the stochastic differential game problem. These results are then used to provide extensions of the linear feedback controller and stopper policies obtained in the literature to nonlinear feedback controllers and stoppers that minimise and maximise general polynomial and multilinear performance criteria.     

Nonlinear–nonquadratic optimal and inverse optimal control for stochastic dynamical systems

In this paper, we develop a unified framework to address the problem of optimal nonlinear analysis and feedback control for nonlinear stochastic dynamical systems. Specifically, we provide a simplified and tutorial framework for stochastic optimal control and focus on connections between stochastic Lyapunov theory and stochastic Hamilton–Jacobi–Bellman theory. In particular, we show that asymptotic stability in probability of the closed‐loop nonlinear system is guaranteed by means of a Lyapunov function that can clearly be seen to be the solution to the steady‐state form of the stochastic Hamilton–Jacobi–Bellman equation and, hence, guaranteeing both stochastic stability and optimality. In addition, we develop optimal feedback controllers for affine nonlinear systems using an inverse optimality framework tailored to the stochastic stabilization problem. These results are then used to provide extensions of the nonlinear feedback controllers obtained in the literature that minimize general polynomial and multilinear performance criteria. Copyright © 2017 John Wiley & Sons, Ltd.     

一类非线性耗散系统的逆最优控制

首先研究一类单输入非仿射非线性系统的逆最优控制问题, 其代价泛函为非线性-非二次型, 设计出一族参数化的状态反馈逆最优控制器;然后讨论当该系统为耗散系统时, 在供给率为二次型的耗散性理论框架下,给出使系统渐近稳定的李雅普诺夫函数和镇定控制律, 并通过适当选取代价泛函中的参数,使得李雅普诺夫函数也是最优值函数,进而揭示出耗散系统在线性输出反馈意义下稳定性与最优性之间的等价关系.     

A local solution with global characteristics for the joint torque optimization of a redundant manipulator

In this article, a stable local solution with global characteristics is developed for the joint torque optimization problem in redundant robotic manipulators. It is shown that the local optimization of the inertia inverse weighted dynamic torque corresponds to the global kinetic energy minimization problem. The proposed local-global alternative to the joint torque optimization problem is compared for stability and torque optimality with five different methods used for redundancy resolution of robotic manipulators at the acceleration level. The proposed local-global solution has been implemented and tested on a planar four-DOF kinematically redundant lab robot which was designed and built at Southwest Research Institute (SWRI). Several numerical simulations confirm the positive advantages of solutions which have a local as well as a global interpretation. In addition, a “dynamic manipulation index” is introduced to monitor the stability of an optimization problem in a kinematically redundant robot.     

Implications of dissipativity,inverse optimal control,and stability margins for nonlinear stochastic feedback regulators

In this paper, we derive stability margins for optimal and inverse optimal stochastic feedback regulators. Specifically, gain, sector, and disk margin guarantees are obtained for nonlinear stochastic dynamical systems controlled by nonlinear optimal and inverse optimal Hamilton‐Jacobi‐Bellman controllers that minimize a nonlinear‐nonquadratic performance criterion with cross‐weighting terms. Furthermore, using the newly developed notion of stochastic dissipativity, we derive a return difference inequality to provide connections between stochastic dissipativity and optimality of nonlinear controllers for stochastic dynamical systems. In particular, using extended Kalman‐Yakubovich‐Popov conditions characterizing stochastic dissipativity, we show that our optimal feedback control law satisfies a return difference inequality predicated on the infinitesimal generator of a controlled Markov diffusion process if and only if the controller is stochastically dissipative with respect to a specific quadratic supply rate.     

Multi-agent consensus algorithm with obstacle avoidance via optimal control approach

Multi-agent consensus problem in an obstacle-laden environment is addressed in this study. A novel optimal control approach is proposed for the multi-agent system to reach consensus as well as avoid obstacles with a reasonable control effort. An innovative nonquadratic penalty function is constructed to achieve obstacle avoidance capability from an inverse optimal control perspective. The asymptotic stability and optimality of the consensus algorithm are proven. In addition, the optimal control law only requires local information from the communication topology to guarantee the proposed behaviour, rather than all agents’ information. The consensus and obstacle avoidance are validated through various simulations.     

An optimal regulation strategy with disturbance rejection for energy management of hybrid electric vehicles

The energy management problem of finding the optimal split between the different sources of energy in a charge-sustaining parallel HEV, ensuring stability and optimality with respect to a performance objective (fuel consumption minimization over a driving cycle), is addressed in this paper. The paper develops a generic stability and optimality framework within which the energy management problem is cast in the form of a nonlinear optimal regulation (with disturbance rejection) problem and a control Lyapunov function is used to design the control law. Two theorems ensuring optimality and asymptotic stability of the energy management strategy are proposed and proved. The sufficient conditions for optimality and stability are used to derive an analytical expression for the control law as a function of the battery state of charge/state of energy and system parameters. The control law is implemented in a simplified backward vehicle simulator and its performance is evaluated against the global optimal solution obtained from dynamic programming. The strategy performs within 4% of the benchmark solution while guaranteeing optimality and stability for any driving cycle.     

A combined shape control procedure of cable mesh reflector antennas with optimality criterion and integrated structural electromagnetic concept

A combined shape control procedure with optimality criterion and integrated structural electromagnetic concept for cable mesh reflector antennas is presented in this study. Using the optimality criterion, the shape control algorithm drives the distorted surface towards the ideal shape. The optimality criterion is implemented by pseudo inverse of sensitivity matrix of surface nodal displacements with respect to cable member dimensions to accelerate the iterative convergence. The following integrated structural electromagnetic design is performed to make good electromagnetic performance by a sequential quadratic programming optimization model. A distorted offset cable mesh reflector antenna is employed to show its effectiveness.     

Uniting local and global controllers for uncertain nonlinear systems: beyond global inverse optimality

This paper provides a solution to a new problem of global robust control for uncertain nonlinear systems. A new recursive design of stabilizing feedback control is proposed in which inverse optimality is achieved globally through the selection of generalized state-dependent scaling. The inverse optimal control law can always be designed such that its linearization is identical to linear optimal control, i.e. optimal control, for the linearized system with respect to a prescribed quadratic cost functional. Like other backstepping methods, this design is always successful for systems in strict-feedback form. The significance of the result stems from the fact that our controllers achieve desired level of ‘global’ robustness which is prescribed a priori. By uniting locally optimal robust control and global robust control with global inverse optimality, one can obtain global control laws with reasonable robustness without solving Hamilton–Jacobi equations directly.     

Design optimization with respect to ergonomic properties

This paper introduces the idea of engineering design optimization with respect to ergonomic properties. The optimization is based on a detailed inverse dynamic analysis of the motion and forces of the human body. The problem of muscle recruitment calculation in inverse dynamics is introduced and solved via a min/max optimality criterion. The implementation of this criterion in the body modelling system AnyBody is described, and the use of the system is demonstrated on the sample problem of designing a hand saw. We conclude that the methodology is applicable but dependent on anthropometrically accurate body models.     

Differential games,finite-time partial-state stabilisation of nonlinear dynamical systems,and optimal robust control

Two-player zero-sum differential games are addressed within the framework of state-feedback finite-time partial-state stabilisation of nonlinear dynamical systems. Specifically, finite-time partial-state stability of the closed-loop system is guaranteed by means of a Lyapunov function, which we prove to be the value of the game. This Lyapunov function verifies a partial differential equation that corresponds to a steady-state form of the Hamilton–Jacobi–Isaacs equation, and hence guarantees both finite-time stability with respect to part of the system state and the existence of a saddle point for the system's performance measure. Connections to optimal regulation for nonlinear dynamical systems with nonlinear-nonquadratic cost functionals in the presence of exogenous disturbances and parameter uncertainties are also provided. Furthermore, we develop feedback controllers for affine nonlinear systems extending an inverse optimality framework tailored to the finite-time partial-state stabilisation problem. Finally, two illustrative numerical examples show the applicability of the results proven.     

Optimal consensus algorithm integrated with obstacle avoidance

This article proposes a new consensus algorithm for the networked single-integrator systems in an obstacle-laden environment. A novel optimal control approach is utilised to achieve not only multi-agent consensus but also obstacle avoidance capability with minimised control efforts. Three cost functional components are defined to fulfil the respective tasks. In particular, an innovative nonquadratic obstacle avoidance cost function is constructed from an inverse optimal control perspective. The other two components are designed to ensure consensus and constrain the control effort. The asymptotic stability and optimality are proven. In addition, the distributed and analytical optimal control law only requires local information based on the communication topology to guarantee the proposed behaviours, rather than all agents’ information. The consensus and obstacle avoidance are validated through simulations.     

An Efficient Algorithm to Computing Max–Min Inverse Fuzzy Relation for Abductive Reasoning

This paper provides an alternative formulation to computing the max-min inverse fuzzy relation by embedding the inherent constraints of the problem into a heuristic (objective) function. The optimization of the heuristic function guarantees maximal satisfaction of the constraints, and consequently, the condition for optimality yields solution to the inverse problem. An algorithm for computing the max-min inverse fuzzy relation is proposed. An analysis of the algorithm indicates its relatively better computational accuracy and higher speed in comparison to the existing technique for inverse computation. The principle of fuzzy abduction is extended with the proposed inverse formulation, and the better relative accuracy of the said abduction over existing works is established through illustrations with respect to a predefined error norm.