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.     

An LMI approach for robust iterative learning control with quadratic performance criterion

This paper presents the design of iterative learning control based on quadratic performance criterion (Q-ILC) for linear systems subject to additive uncertainty. The robust Q-ILC design can be cast as a min–max problem. We propose a novel approach which employs an upper bound of the worst-case performance, then formulates a non-convex quadratic minimization problem to get the update of iterative control inputs. Applying Lagrange duality, the Lagrange dual function of the non-convex quadratic problem is equivalent to a convex optimization over linear matrix inequalities (LMIs). An LMI algorithm with convergence properties is then given for the robust Q-ILC design. Finally, we provide a numerical example to illustrate the effectiveness of the proposed method.     

MISO系统中上下行链路对偶性研究

TDD系统中,利用上下行链路的对偶性可以将非凸的下行优化问题转换到上行链路,从而极大简化问题的分析和数值求解。上下行链路在波束成形和容量域两方面都存在对偶性。本文在统一的系统模型下,针对总功率约束和每天线功率约束两种情况,利用拉格朗日对偶法分别对波束成形对偶、容量域对偶做了推导、分析。此外,用凸优化软件包CVX简化了问题的建模。分析表明,上下行对偶等价于拉格朗日对偶,对偶问题可更一般地表示为极大极小问题。针对每天线功率约束下的波束成形问题,用迭代法和内点法分别对其进行了MATLAB仿真。仿真结果表明,非凸的下行优化问题可以利用对偶性转化为上行链路中的凸优化问题,从而得以解决。     

Explicit predictive control with non-convex polyhedral constraints

This paper proposes an explicit solution to the model predictive control of linear systems subject to non-convex polyhedral constraints. These constraints are modeled as the union of a finite number of convex polyhedra. The algorithm is based on calculating the explicit solution to a modified problem with linear constraints defined as the convex hull of the original ones and classifying its regions by their relation with the regions of the explicit solution to the original problem. Some of the regions are divided, and a procedure based on sum-of-squares programming is designed to determine which of the possible solutions are in fact optimal. Finally, the online algorithm is shown to be better in terms of computational cost and memory requirements than an algorithm based on obtaining and comparing the solutions of the problem using as constraints the polyhedra whose union forms the non-convex regions, both theoretically and by the results of an example.     

Primal explicit max margin feature selection for nonlinear support vector machines

Embedding feature selection in nonlinear support vector machines (SVMs) leads to a challenging non-convex minimization problem, which can be prone to suboptimal solutions. This paper develops an effective algorithm to directly solve the embedded feature selection primal problem. We use a trust-region method, which is better suited for non-convex optimization compared to line-search methods, and guarantees convergence to a minimizer. We devise an alternating optimization approach to tackle the problem efficiently, breaking it down into a convex subproblem, corresponding to standard SVM optimization, and a non-convex subproblem for feature selection. Importantly, we show that a straightforward alternating optimization approach can be susceptible to saddle point solutions. We propose a novel technique, which shares an explicit margin variable to overcome saddle point convergence and improve solution quality. Experiment results show our method outperforms the state-of-the-art embedded SVM feature selection method, as well as other leading filter and wrapper approaches.     

D.C. optimization approach to robust control: Feasibility problems

The feasibility problem for constant scaling in output feedback control is considered. This is an inherently difficult problem since the set of feasible solutions is non-convex and may be disconnected. Nevertheless, we show that this problem can be reduced to the global minimization of a concave function over a convex set, or alternatively, to the global minimization of a convex program with an additional reverse convex constraint. Thus this feasibility problem belongs to the realm of d.c. optimization, a new field which has recently emerged as an active promising research direction in nonconvex global optimization. By exploiting the specific d.c. structure of the problem, several algorithms are proposed which at every iteration require solving only either convex or linear subproblems. Analogous algorithms with new characterizations are proposed for the bilinear matrix inequality (BMI) feasibility problem.     

Optimization strategies for discrete multi-material stiffness optimization

Design of composite laminated lay-ups are formulated as discrete multi-material selection problems. The design problem can be modeled as a non-convex mixed-integer optimization problem. Such problems are in general only solvable to global optimality for small to moderate sized problems. To attack larger problem instances we formulate convex and non-convex continuous relaxations which can be solved using gradient based optimization algorithms. The convex relaxation yields a lower bound on the attainable performance. The optimal solution to the convex relaxation is used as a starting guess in a continuation approach where the convex relaxation is changed to a non-convex relaxation by introduction of a quadratic penalty constraint whereby intermediate-valued designs are prevented. The minimum compliance, mass constrained multiple load case problem is formulated and solved for a number of examples which numerically confirm the sought properties of the new scheme in terms of convergence to a discrete solution.     

罚函数凸优化迭代算法及在无人机路径规划中的应用

针对无人机路径规划问题,建立了具有定常非线性系统、非仿射等式约束、非凸不等式约束的非凸控制问题模型,并对该模型进行了算法设计和求解。基于迭代寻优的求解思路,提出了凸优化迭代求解方法和罚函数优化策略。前者利用凹凸过程(CCCP)和泰勒公式对模型进行凸化处理,后者将经处理项作为惩罚项施加到目标函数中以解决初始点可行性限制。经证明该方法严格收敛到原问题的Karush-Kuhn-Tucker(KKT)点。仿真实验验证了罚函数凸优化迭代算法的可行性和优越性,表明该算法能够为无人机规划出一条满足条件的飞行路径。     

Waste minimization in irregular stock cutting

This paper addresses a category of two dimensional NP-hard knapsack problem in which a given convex/non-convex planner items (polygons) have to be cut out of a single convex/non-convex master surface (stock). This cutting process is found in many industrial applications such as sheet metal processes, home-textile, garment, wood, leather and paper industries. An approach is proposed to solve this problem, which depends on the concept of the difference between the area of a collection of polygons and the area of their convex hull. The polygon assignment inside the stock is subjected to feasibility tests to avoid overlapping, namely, angle test, bound test, point inclusion and polygon intersection test. An iterative scheme is used to generate different polygon placements while optimizing the objective function. Computer software is developed to solve and optimize the problem under consideration. Few examples are conducted for different combinations of convex, non-convex items and stocks. Well-known benchmark problems from the literature are tested and compared with our approach. The results of our algorithm have an interesting computational time and can compete with the results of previous work in some particular problems. The computational performance of the developed software indicates the efficiency of the algorithm for solving 2-D irregular cutting of non-convex polygons out of non-convex stock.     

Non-smooth DC-constrained optimization: constraint qualification and minimizing methodologies

This work concerns the study of a constraint qualification for non-smooth DC-constrained optimization problems, as well as the design and convergence analysis of minimizing algorithms to address the task of computing a stationary/critical point for problems of this class. Specialized algorithms for DC programming approximate the non-convex optimization problem by a sequence of convex subproblems, obtained by linearizing the second components of the involved DC (difference of convex) functions. We propose new approaches that define trial points as inexact solutions of such convex subproblems. This is a property of practical interest that substantially reduces the computational burden to compute a stationary/critical point of non-smooth DC-constrained optimization problems. One variant of the proposed algorithmic patterns is numerically assessed on a DC reformulation of an energy management problem considering a smart-grid controlled by a local actor (follower) and its interaction with a global actor (leader) in the power system.     

The dilemma of PID tuning

A lot of automatic feedback control and learning tasks carried out on many dynamical systems still fundamentally rely on a form of proportional–integral–derivative (PID) control law. The PID law is often viewed as a simplistic computational control algorithm. However just like all non-convex optimization problems, tuning the PID algorithm for accurate and stable closed-loop control becomes a NP-Hard Problem. This leads to a dilemma, for both users and designers, most especially in practise. It is then no wonder that tuning software is a big business in the industrial automation sector. In this review, we present and classify PID tuning methods till date into three general areas. Finally, we then present a proposal to minimize the dilemma of complexity and cost that has become associated with tuning the three main parameters of the PID control law. Hopefully, continuous attempts at the minimization of this dilemma can lead to both a money-savings investment and a significant improvement in the field of PID control design.     

Time-optimal control for discrete-time hybrid automata

In this paper, the problem of time-optimal control for hybrid systems with discrete-time dynamics is considered. The hybrid controller steers all trajectories starting from a maximal set to a given target set in minimum time. We derive an algorithm that computes this maximal winning set. Also, algorithms for the computation of level sets associated with the value function rather than the value function itself are presented. We show that by solving the reachability problem for the discrete time hybrid automata we obtain the time optimal solution as well. The control synthesis is subject to hard constraints on both control inputs and states. For linear discrete-time dynamics, linear programming and quantifier elimination techniques are employed for the backward reachability analysis. Emphasis is given on the computation of operators for non-convex sets using an extended convex hull approach. A two-tank example is considered in order to demonstrate the techniques of the paper.     

Rank-one LMI approach to simultaneous stabilization of linear systems

Following a polynomial approach to control design, the simultaneous stabilization by a controller of given fixed order of a family of SISO linear systems is interpreted as an NP-hard BMI feasibility problem. Upon formulating this BMI problem as an LMI problem with an additional non-convex rank constraint, two simultaneous stabilization methods are then proposed. The first method is a heuristic algorithm performing rank minimization by potential reduction. The second method hinges upon necessary conditions and sufficient conditions for simultaneous stabilization derived from geometric properties of the intersection of a set of ellipsoids. Both methods are then illustrated by numerical examples.     

典型工业过程鲁棒PID控制器的整定

提出一阶迟延过程、积分迟延过程、不稳定迟延过程和二阶迟延过程等典型工业过程的鲁棒PID整定公式.本文从抗干扰性能和鲁棒性能两方面综合考虑,把鲁棒PID控制器的设计问题转化为求解一个带鲁棒性能约束的绝对误差积分指标(IAE)优化问题.鉴于该问题是非凸的,本文采用遗传算法来求解,并通过曲线拟合得到典型工业过程的PID控制器的整定公式.仿真结果表明本文的PID整定公式有效,且控制器具有良好的抗干扰能力和频域鲁棒性.     

Mixed coordination method for long-horizon optimal control problems

We present a new approach to solving long-horizon, discrete-time optimal control problems using the mixed coordination method. The idea is to decompose a long-horizon problem into subproblems along the time axis. The requirement that the initial state of a subproblem equal the terminal state of the preceding subproblem is relaxed by using Lagrange multipliers. The Lagrange multipliers and initial state of each subproblem are then selected as high-level variables. The equivalence of the two-level formulation and the original problem is proved for both convex and non-convex cases. The low-level subproblems are solved in parallel using extended differential dynamic programming (DDP). An efficient way to find the gradient and hessian of a low-level objective function with respect to high-level variables is developed. The high-level problem is solved using the modified Newton method. An effective procedure is developed to select initial values of multipliers based on the initial trajectory. The method can convexify the high-level problem while maintaining the separability of an originally non-convex problem. The method performs better and faster than one-level DDP for both convex and non-convex test problems.     

基于L1/2正则化的三维人体姿态重构

针对从给定2D特征点的单目图像中重构对象的3D形状问题,本文在形状空间模型的基础上,结合L_1/2正则化和谱范数的性质提出一种基于L_1/2正则化的凸松弛方法,将形状空间模型的非凸求解问题通过凸松弛方法转化为凸规划问题;在采用ADMM算法对凸规划问题进行优化求解过程中,提出谱范数近端梯度算法保证解的正交性与稀疏性.利用所提的优化方法,基于形状空间模型和3D可变形状模型在卡内基梅隆大学运动捕获数据库上进行3D人体姿态重构,定性和定量对比实验结果表明本文方法均优于现有的优化方法,验证了所提方法的有效性.     

Scalable anomaly detection in large homogeneous populations

Anomaly detection in large populations is a challenging but highly relevant problem. It is essentially a multi-hypothesis problem, with a hypothesis for every division of the systems into normal and anomalous systems. The number of hypothesis grows rapidly with the number of systems and approximate solutions become a necessity for any problem of practical interest. In this paper we take an optimization approach to this multi-hypothesis problem. It is first shown to be equivalent to a non-convex combinatorial optimization problem and then is relaxed to a convex optimization problem that can be solved distributively on the systems and that stays computationally tractable as the number of systems increase. An interesting property of the proposed method is that it can under certain conditions be shown to give exactly the same result as the combinatorial multi-hypothesis problem and the relaxation is hence tight.     

Evolutionary approach on connectivity-based sensor network localization

The sensor network localization based on connectivity can be modeled as a non-convex optimization problem. It can be argued that the actual problem should be represented as an optimization problem with both convex and non-convex constraints. A two-objective evolutionary algorithm is proposed which utilizes the result of all convex constraints to provide a starting point on the location of the unknown nodes and then searches for a solution to satisfy all the convex and non-convex constraints of the problem. The final solution can reach the most suitable configuration of the unknown nodes because all the information on the constraints (convex and non-convex) related to connectivity have been used. Compared with current models that only consider the nodes that have connections, this method considers not only the connection constraints, but also the disconnection constraints. As a MOEA (Multi-Objective Evolution Algorithm), PAES (Pareto Archived Evolution Strategy) is used to solve the problem. Simulation results have shown that better solution can be obtained through the use of this method when compared with those produced by other methods.     

New optimal controller tuning method for an AVR system using a simplified Ant Colony Optimization with a new constrained Nelder–Mead algorithm

In this paper, an optimal gain tuning method for PID controllers is proposed using a novel combination of a simplified Ant Colony Optimization algorithm and Nelder–Mead method (ACO-NM) including a new procedure to constrain NM. To address Proportional-Integral-Derivative (PID) controller tuning for the Automatic Voltage Regulator (AVR) system, this paper presents a meta-analysis of the literature on PID parameter sets solving the AVR problem. The investigation confirms that the proposed ACO-NM obtains better or equivalent PID solutions and exhibits higher computational efficiency than previously published methods. The proposed ACO-NM application is extended to realistic conditions by considering robustness to AVR process parameters, control signal saturation and noisy measurements as well as tuning a two-degree-of-freedom PID controller (2DOF-PID). For this type of PID, a new objective function is also proposed to manage control signal constraints. Finally, real time control experiments confirm the performance of the proposed 2DOF-PIDs in quasi-real conditions. Furthermore, the efficiency of the algorithm is confirmed by comparing its results to other optimization algorithms and NM combinations using benchmark functions.     

无线携能通信系统中基于人工噪声的波束成形算法研究

在存在多天线窃听者的情况下,研究了无线携能通信系统的物理层安全传输问题。传统上,人工噪声方案是一种保证物理层安全通信的有效策略。因此,提出了一种人工噪声辅助的波束成形算法,以提高信息与能量联合传输的安全性。该算法在满足多项约束（接收端能量采集门限、信息泄漏控制指标和发送功率）的基础上,通过优化波束成形预编码矩阵和人工噪声实现无线携能通信系统的物理层安全通信。在数学上该算法是一个非凸优化问题,不易求解。为此,首先引入最小均方误差接收算法和连续凸近似方法将其转换为凸的二阶锥规划子问题,然后对该凸问题进行迭代求解。仿真结果表明,该算法能够在保证窃听者无法解码信息的同时,实现信息和能量的联合传输,并且算法收敛速度快。