约束非线性系统多变量最优控制研究

近年来，非线性规划算法在最优控制领域中正受到越来越多的关注。该文深人研究并实现了一种新的非线性规划算法——FSQP算法，该算法具有所有迭代点均处于可行域之内、收敛速度较快的特点。提出了一种基于FSQP算法的约束非线性系统最优控制方法。然后，运用该方法解决了带有约束的复杂非线性系统的多变量时间最优控制问题，并通过计算机仿真表明了该控制算法的可行性和良好的控制效果。     

Lossless convexification of a class of optimal control problems with non-convex control constraints

We consider a class of finite time horizon optimal control problems for continuous time linear systems with a convex cost, convex state constraints and non-convex control constraints. We propose a convex relaxation of the non-convex control constraints, and prove that the optimal solution of the relaxed problem is also an optimal solution for the original problem, which is referred to as the lossless convexification of the optimal control problem. The lossless convexification enables the use of interior point methods of convex optimization to obtain globally optimal solutions of the original non-convex optimal control problem. The solution approach is demonstrated on a number of planetary soft landing optimal control problems.     

求解最优控制问题的Chebyshev-Gauss伪谱法

提出了一种求解最优控制问题的Chebyshev-Gauss伪谱法, 配点选择为Chebyshev-Gauss点. 通过比较非线性规划问题的Kaursh-Kuhn-Tucker条件和伪谱离散化的最优性条件, 导出了协态和Lagrange乘子的估计公式. 在状态逼近中, 采用了重心Lagrange插值公式, 并提出了一种简单有效的计算状态伪谱微分矩阵的方法. 该法的独特优势是具有良好的数值稳定性和计算效率. 仿真结果表明, 该法能够高精度地求解带有约束的复杂最优控制问题.     

Optimal worst case estimation for LPV–FIR models with bounded errors

This work presents an approximate solution method for the infinite-time nonlinear quadratic optimal control problem. The method is applicable to a large class of nonlinear systems and involves solving a Riccati equation and a series of algebraic equations. The conditions for uniqueness and stability of the resulting feedback policy are established. It is shown that the proposed approximation method is useful in determining the region in which the constrained and unconstrained optimal control problems are identical. A reactor control problem is used to illustrate the method.     

Three-dimensional trajectory optimization of soft lunar landings from the parking orbit with considerations of the landing site

Minimum fuel, three-dimensional trajectory optimization from a parking orbit considering the desired landing site is addressed for soft lunar landings. The landing site is determined by the final longitude and latitude; therefore, a two-dimensional approach is limited and a three-dimensional approach is required. In addition, the landing site is not usually considered when performing lunar landing trajectory optimizations, but should be considered in order to design more accurate and realistic lunar landing trajectories. A Legendre pseudospectral (PS) method is used to discretize the trajectory optimization problem as a nonlinear programming (NLP) problem. Because the lunar landing consists of three phases including a de-orbit burn, a transfer orbit phase, and a powered descent phase, the lunar landing problem is regarded as a multiphase problem. Thus, a PS knotting method is also used to manage the multiphase problem, and C code for Feasible Sequential Quadratic Programming (CFSQP) using a sequential quadratic programming (SQP) algorithm is employed as a numerical solver after formulating the problem as an NLP problem. The optimal solutions obtained satisfy all constraints as well as the desired landing site, and the solutions are verified through a feasibility check.     

月球软着陆轨道快速优化

月球软着陆轨道是登月飞行器下降到月球表面轨道中很重要一段的轨道,为了实现飞行器自主软着陆,需要进行快速轨道优化设计.文中根据软着陆轨道的特征和优化算法的特点,对软着陆轨道状态方程做合理的简化处理,优化计算量减少,且更适合优化数值解法求解.在此基础上,使用乘子法处理软着陆终端约束条件,然后利用共轭梯度法求解软着陆轨道.在不同初始条件和终端约束条件下,计算机时小于3秒.仿真结果验证该算法具有收敛速度快、对初始控制量不敏感等优点,易于工程实现.     

Sensor scheduling in continuous time

Let N be the number of available sensor sources. Noisy observations of an underlying state process are available for these N sources. We consider the continuous time sensor scheduling problem in which N₁ of these N sources are to be chosen to collect data at each time point. This sensor scheduling problem (with switching costs and switching constraints) is formulated as a constrained optimal control problem. In this framework, the controls represent the sensors that are chosen at a particular time. Thus, the control variables are constrained to take values in a discrete set, and switchings between sensors can occur in continuous time. By incorporating recent results on discrete valued optimal control, we show that this problem can be transformed into an equivalent continuous optimal control problem. In this way, we obtain the sensor scheduling policy as well as the associated switching times.     

Solution for state constrained optimal control problems applied to power split control for hybrid vehicles

This paper presents a numerical solution for scalar state constrained optimal control problems. The algorithm rewrites the constrained optimal control problem as a sequence of unconstrained optimal control problems which can be solved recursively as a two point boundary value problem. The solution is obtained without quantization of the state and control space. The approach is applied to the power split control for hybrid vehicles for a predefined power and velocity trajectory and is compared with a Dynamic Programming solution. The computational time is at least one order of magnitude less than that for the Dynamic Programming algorithm for a superior accuracy.     

Suboptimal local feedback control for a class of constrained discrete time nonlinear control problems

In this paper, we consider a class of constrained discrete time optimal control problems involving general nonlinear dynamics with fixed terminal time. A method to solve the feedback control problem for a class of unconstrained continuous time nonlinear systems has been proposed previously. In that work, the solution is based on synthesizing an approximate suboptimal feedback controller locally in the neighbourhood of a certain nominal optimal trajectory. This paper expands on the same theme by considering problems involving discrete time systems. Taking advantage of the nature of discrete time systems, a further reduction on the computational effort of synthesising the feedback controller is made possible. Also, this paper extends the applicability of the method to constrained systems. For illustration, a numerical example is solved using the proposed method.     

Viscosity solutions and optimal control problems with integral constraints

We discuss optimal control problems with integral state-control constraints. We rewrite the problem in an equivalent form as an optimal control problem with state constraints for an extended system, and prove that the value function, although possibly discontinuous, is the unique viscosity solution of the constrained boundary value problem for the corresponding Hamilton–Jacobi equation. The state constraint is the epigraph of the minimal solution of a second Hamilton–Jacobi equation. Our framework applies, for instance, to systems with design uncertainties.     

Optimal traffic control synthesis for an isolated intersection

A continuous dynamical model of a simplified controlled isolated intersection is derived in order to find and analyze an optimal control policy to minimize total delay. An analytical solution of the optimal control problem with constrained signal light control is presented. The optimal synthesis is found for the four principal control constraint cases. Previous results from the 1960s and 70s are discussed.     

Optimal design of continuous time irrational filter with a set of fractional order gammatone components via norm relaxed sequential quadratic programming approach

This paper proposes a continuous time irrational filter structure via a set of the fractional order Gammatone components instead of via a set of integer order Gammatone components. The filter design problem is formulated as a nonsmooth and nonconvex infinite constrained optimization problem. The nonsmooth function is approximated by a smooth operator. The domain of the constraint functions is sampled into a set of finite discrete points so the infinite constrained optimization problem is approximated by a finite constrained optimization problem. To find a near globally optimal solution, the norm relaxed sequential quadratic programming approach is applied to find the locally optimal solutions of this nonconvex optimization problem. The current or the previous locally optimal solutions are kicked out by adding the random vectors to them. The locally optimal solutions with the lower objective functional values are retained and the locally optimal solutions with the higher objective functional values are discarded. By iterating the above procedures, a near globally optimal solution is found. The designed filter is applied to perform the denoising. It is found that the signal to noise ratio of the designed filter is higher than those of the filters designed by the conventional gradient descent approach and the genetic algorithm method, while the required computational power of our proposed method is lower than those of the conventional gradient descent approach and the genetic algorithm method. Also, the signal to noise ratio of the filter with the fractional order Gammatone components is higher than those of the filter with the integer order Gammatone components and the conventional rational infinite impulse response filters.     

real-time motion planning for multibody systems

The solution of constrained motion planning is an important task in a wide number of application fields. The real-time solution of such a problem, formulated in the framework of optimal control theory, is a challenging issue. We prove that a real-time solution of the constrained motion planning problem for multibody systems is possible for practical real-life applications on standard personal computers. The proposed method is based on an indirect approach that eliminates the inequalities via penalty formulation and solves the boundary value problem by a combination of finite differences and Newton–Broyden algorithm. Two application examples are presented to validate the method and for performance comparisons. Numerical results show that the approach is real-time capable if the correct penalty formulation and settings are chosen.     

A new approach to Lipschitz continuity in state constrained optimal control

For a linear-quadratic state constrained optimal control problem, it is proved in [11] that under an independence condition for the active constraints, the optimal control is Lipschitz continuous. We now give a new proof of this result based on an analysis of the Euler discretization given in [9]. There we exploit the Lipschitz continuity of the control to estimate the error in the Euler discretization. Here we show that the theory developed for the Euler discretization can be used to derive the Lipschitz continuity of the optimal control.     

基于动态规划的约束优化问题多参数规划求解方法及应用

结合动态规划和单步多参数二次规划, 提出一种新的约束优化控制问题多参数规划求解方法. 一方面能得到约束线性二次优化控制问题最优控制序列与状态之间的显式函数关系, 减少多参数规划问题求解的工作量; 另一方面能够同时求解得到状态反馈最优控制律. 应用本文提出的多参数二次规划求解方法, 建立无限时间约束优化问题状态反馈显式最优控制律. 针对电梯机械系统振动控制模型做了数值仿真计算.     

Optimization of airplane wing structures under landing loads

A methodology is presented for the optimum design of aircraft wing structures subjected to landing loads. The stresses developed in the wing during landing are computed by considering the interaction between the landing gear and the flexible airplane structure. The landing gear is assumed to have nonlinear characteristics typical of conventional gears, namely, velocity squared damping, polytropic air-compression springing and exponential tire force-deflection characteristics. The coupled nonlinear differential equations of motion that arise in the landing analysis are solved by using a step-by-step numerical integration technique. In order to find the behavior of the wing structure under landing loads and also to obtain a physical insight into the nature of the optimum solution, the design of the typical section (symmetric double-wedge airfoil) is studied by using a graphical procedure. Then a more realistic wing optimization problem is formulated as a constrained nonlinear programming problem based on finite element modeling. The optimum solutions are found by using the interior penalty function method. A sensitivity analysis is conducted to find the effect of changes in design variables about the optimum point on the various response parameters on the wing structure.     

Consistent approximation of a nonlinear optimal control problem with uncertain parameters

This paper focuses on a non-standard constrained nonlinear optimal control problem in which the objective functional involves an integration over a space of stochastic parameters as well as an integration over the time domain. The research is inspired by the problem of optimizing the trajectories of multiple searchers attempting to detect non-evading moving targets. In this paper, we propose a framework based on the approximation of the integral in the parameter space for the considered uncertain optimal control problem. The framework is proved to produce a zeroth-order consistent approximation in the sense that accumulation points of a sequence of optimal solutions to the approximate problem are optimal solutions of the original problem. In addition, we demonstrate the convergence of the corresponding adjoint variables. The accumulation points of a sequence of optimal state-adjoint pairs for the approximate problem satisfy a necessary condition of Pontryagin Minimum Principle type, which facilitates assessment of the optimality of numerical solutions.     

Robust optimal parametric LQ control with a guaranteed cost bound and applications

For an optimal parametric linear quadratic (LQ) control problem, a design objective is to determine a controller of constrained structure such that the closed-loop system is asymptotically stable and an associated performance measure is optimized. In the presence of system uncertainty, the system via a parametric LQ design is further required to be robust in terms of maintaining the closed-loop stability with a guaranteed cost bound. This problem is referred to as ‘robust optimal parametric LQ control with a guaranteed cost bound’ and is addressed in this work. A new design method is proposed to find an optimal controller for simultaneously guaranteeing robust stability and performance over a specified range of parameter variations. The results presented generalize some previous work in this area. A versatile numerical algorithm is also given for computing the robust optimal gains. The usefulness of the design method is demonstrated by numerical examples and a design of the robust control of a VTOL helicopter.     

A new approach to Lipschitz continuity in state constrained optimal control

For a linear-quadratic state constrained optimal control problem, it is proved in [11] that under an independence condition for the active constraints, the optimal control is Lipschitz continuous. We now give a new proof of this result based on an analysis of the Euler discretization given in [9]. There we exploit the Lipschitz continuity of the control to estimate the error in the Euler discretization. Here we show that the theory developed for the Euler discretization can be used to derive the Lipschitz continuity of the optimal control.     

A Hamiltonian approach to compute an energy efficient trajectory for a servomotor system

This paper considers a nonlinear constrained optimal control problem (NCOCP) originated from energy optimal trajectory planning of servomotor systems. Solving the exact optimal solution is challenging because of the nonlinear and switching cost function, and various constraints. This paper proposes a method to manage the switching cost function to establish a set of necessary conditions of an NCOCP. Specifically, a concept “sub-trajectory” is introduced to match multiple Hamiltonian due to switches in the cost function. Necessary conditions on the optimal trajectory are established as a union of conditions for all sub-trajectories and Weierstrass–Erdmann corner conditions between sub-trajectories. The set of feasible structures of optimal trajectories is further identified and represented by various state transition diagrams for the servomotor application. A decomposition-based shooting method is proposed to compute an optimal trajectory by solving multi-point boundary value problems. Simulations and experiments validate the effectiveness of the methodology and the energy saving benefit.