Antenna array synthesis with clusters of unmanned aerial vehicles

In this paper, we investigate the use of formations of Unmanned Aerial Vehicles (UAVs) as phased antenna arrays. This will help to improve communications with clusters of small unmanned aerial vehicles which are currently constrained by on-board power limitations. The problem of maximizing the power output from the array in the direction of the receiver is posed as an optimization problem which happens to be non-convex; a relaxation of this problem is then solved as a computationally tractable (convex) Second-Order Conic Program (SOCP). The performance obtained by the simplified approach is then tested against rigorous numerical bounds obtained using Semidefinite Programming (SDP) duality theory; these bounds are of independent interest in antenna theory. In order to maintain the objective value close to the optimal when the vehicles deviate from their positions (due to wind gusts, for example), a simple linear control law is proposed. Simulation results are given to show the effectiveness of the proposed approaches.     

Robust constrained predictive control using comparison model

This paper proposes a quadratic programming (QP) approach to robust model predictive control (MPC) for constrained linear systems having both model uncertainties and bounded disturbances. To this end, we construct an additional comparison model for worst-case analysis based on a robust control Lyapunov function (RCLF) for the unconstrained system (not necessarily an RCLF in the presence of constraints). This comparison model enables us to transform the given robust MPC problem into a nominal one without uncertain terms. Based on a terminal constraint obtained from the comparison model, we derive a condition for initial states under which the ultimate boundedness of the closed loop is guaranteed without violating state and control constraints. Since this terminal condition is described by linear constraints, the control optimization can be reduced to a QP problem.     

基于粒子群优化的有约束模型预测控制器

研究了模型预测控制(MPC)中解决带约束的优化问题时所用到的优化算法,针对传统的二次规划(QP)方法的不足,引入了一种带有混沌初始化的粒子群优化算法(CPSO),将其应用到模型预测控制中,用十解决同时带有输入约束和状态约束的控制问题.最后,引入了一个实际的带有约束的线性离散系统的优化控制问题,分别用二次规划和粒子群优化两种算法去解决,通过仿真结果的比较,说明了基于粒子群优化(PSO)的模型预测控制算法的优越性.     

Proportioning with second-order information for model predictive control

We propose an algorithm for the effective solution of quadratic programming (QP) problems arising from model predictive control (MPC). MPC is a modern multivariable control method which gives the solution for a QP problem at each sample instant. Our algorithm combines the active-set strategy with the proportioning test to decide when to leave the actual active set. For the minimization in the face, we use a direct solver implemented by the Cholesky factors updates. The performance of the algorithm is illustrated by numerical experiments, and the results are compared with the state-of-the-art solvers on benchmarks from MPC.     

Optimizing model predictive control of an industrial distillation column

The main scope of this work is the implementation of an MPC that integrates the control and the economic optimization of the system. The two problems are solved simultaneously through the modification of the control cost function that includes an additional term related to the economic objective. The optimizing MPC is based on a quadratic program (QP) as the conventional MPC and can be solved with the available QP solvers. The method was implemented in an industrial distillation system, and the results show that the approach is efficient and can be used, in several practical cases.     

Semidefinite relaxation for linear programs with equilibrium constraints

In this paper, we present a semidefinite programming (SDP) relaxation for linear programs with equilibrium constraints (LPECs) to be used in a branch‐and‐bound (B&B) algorithm. The procedure utilizes the global optimal solution of LPECs and was motivated by the B&B algorithm proposed by Bard and Moore for linear/quadratic bilevel programs, where complementarities are recursively enforced. We propose the use of the SDP relaxation to generate bounds at the nodes of the B&B tree. Computational results compare the quality of the bounds given by the SDP relaxation with the ones given by conventional linear programming relaxations.     

A global piecewise smooth Newton method for fast large-scale model predictive control

In this paper, the strictly convex quadratic program (QP) arising in model predictive control (MPC) for constrained linear systems is reformulated as a system of piecewise affine equations. A regularized piecewise smooth Newton method with exact line search on a convex, differentiable, piecewise-quadratic merit function is proposed for the solution of the reformulated problem. The algorithm has considerable merits when applied to MPC over standard active set or interior point algorithms. Its performance is tested and compared against state-of-the-art QP solvers on a series of benchmark problems. The proposed algorithm is orders of magnitudes faster, especially for large-scale problems and long horizons. For example, for the challenging crude distillation unit model of Pannocchia, Rawlings, and Wright (2007) with 252 states, 32 inputs, and 90 outputs, the average running time of the proposed approach is 1.57 ms.     

小型无人直升机的模型预测控制算法研究

针对无人直升机在阵风干扰环境中的姿态控制精度低的问题.本文将非线性刚体动力学模型在悬停点应用小扰动理论得到了线性化数学模型.考虑系统输入输出和控制量约束,采用模型预测控制将控制器的设计问题转化为每个采样时刻求解一个带不等式和等式约束的凸二次规划问题.通过设计终端状态约束解决了有限时域模型预测控制(model predictive control, MPC)算法的稳定性问题,并通过引入松弛变量使得约束优化问题更容易求解.随机和常值阵风干扰下无人机悬停仿真验证了本文MPC预测控制器具有幅度不超过0.25 m/s的良好干扰抑制能力,性能明显优于线性二次型调节器(linear-quadratic regulator, LQR).     

Necessary and sufficient conditions for global optimality of eigenvalue optimization problems

The necessary and sufficient conditions for global optimality are derived for an eigenvalue optimization problem. We consider the generalized eigenvalue problem where real symmetric matrices on both sides are linear functions of design variables. In this case, a minimization problem with eigenvalue constraints can be formulated as Semi-Definite Programming (SDP). From the Karush-Kuhn-Tucker conditions of SDP, the necessary and sufficient conditions are derived for arbitrary multiplicity of the lowest eigenvalues for the case where important lower bound constraints are considered for the design variables. Received May 18, 2000     

Steady-state target optimization designs for integrating real-time optimization and model predictive control

In industrial practice, the optimal steady-state operation of continuous-time processes is typically addressed by a control hierarchy involving various layers. Therein, the real-time optimization (RTO) layer computes the optimal operating point based on a nonlinear steady-state model of the plant. The optimal point is implemented by means of the model predictive control (MPC) layer, which typically uses a linear dynamical model of the plant. The MPC layer usually includes two stages: a steady-state target optimization (SSTO) followed by the MPC dynamic regulator. In this work, we consider the integration of RTO with MPC in the presence of plant-model mismatch and constraints, by focusing on the design of the SSTO problem. Three different quadratic program (QP) designs are considered: (i) the standard design that finds steady-state targets that are as close as possible to the RTO setpoints; (ii) a novel optimizing control design that tracks the active constraints and the optimal inputs for the remaining degrees of freedom; and (iii) an improved QP approximation design were the SSTO problem approximates the RTO problem. The main advantage of the strategies (ii) and (iii) is in the improved optimality of the stationary operating points reached by the SSTO-MPC control system. The performance of the different SSTO designs is illustrated in simulation for several case studies.     

Quadratic programming algorithms for large-scale model predictive control

Quadratic programming (QP) methods are an important element in the application of model predictive control (MPC). As larger and more challenging MPC applications are considered, more attention needs to be focused on the construction and tailoring of efficient QP algorithms. In this study, we tailor and apply a new QP method, called QPSchur, to large MPC applications, such as cross directional control problems in paper machines. Written in C++, QPSchur is an object oriented implementation of a novel dual space, Schur complement algorithm. We compare this approach to three widely applied QP algorithms and show that QPSchur is significantly more efficient (up to two orders of magnitude) than the other algorithms. In addition, detailed simulations are considered that demonstrate the importance of the flexible, object oriented construction of QPSchur, along with additional features for constraint handling, warm starts and partial solution.     

Who needs QP for linear MPC anyway?

Conventional MPC uses quadratic programming (QP) to minimise, on-line, a cost over n linearly constrained control moves. However, stability constraints often require the use of large n thereby increasing the on-line computation, rendering the approach impracticable in the case of fast sampling. Here, we explore an alternative that requires a fraction of the computational cost (which increases only linearly with n), and propose an extension which, in all but a small class of models, matches to within a fraction of a percent point the performance of the optimal solution obtained through QP. The provocative title of the paper is intended to point out that the proposed approach offers a very attractive alternative to QP-based MPC.     

Reduction of the QP-MPC cascade structure to a single layer MPC

Some commercial MPC packages are implemented in two layers, the QP static layer and the MPC dynamic layer. In the absence of an upper Real Time Optimization layer, the static layer solves a simplified economic optimization problem, which defines optimum feasible targets for the dynamic layer. Since the LP/QP static layer and the MPC dynamic layer are usually executed within the same sampling period, it is not trivial to guarantee that the interaction between the two layers will not disrupt the stability of the whole structure. In this paper, it is proposed an approach to reduce the two-layer structure of some commercial MPCs to a single dynamic layer where the control cost function is extended to include the economic objective. In the proposed approach the convergence and stability of the closed-loop system can be obtained if the economic term of the cost function is properly weighted. A simulation example of a simple industrial system shows the efficiency of the proposed strategy.     

Price-driven coordination method for solving plant-wide MPC problems

In large-scale model predictive control (MPC) applications, such as plant-wide control, two possible approaches to MPC implementation are centralized and decentralized MPC schemes. These represent the two extremes in the “trade-off” among the desired characteristics of an industrial MPC system, namely accuracy, reliability and maintainability. To achieve optimal plant operations, coordination of decentralized MPC controllers has been identified as both an opportunity and a challenge. Typically, plant-wide MPC problem can be formulated as a large-scale quadratic program (QP) with linking equality constraints. Such problems can be decomposed and solved with the price-driven coordination method and on-line solutions of these structured large-scale optimization problems require an efficient price-adjustment strategy to find an “equilibrium price”. This work develops an efficient price-adjustment algorithm based on Newton’s method, in which sensitivity analysis and active set change identification techniques are employed. With the off-diagonal element abstraction technique and the enhanced priced driven coordination algorithm, a coordinated, decentralized MPC framework is proposed. Several case studies show that the proposed coordination-based decentralized MPC scheme is an effective approach to plant-wide MPC applications, which provides a high degree of reliability and accuracy at a reasonable computational load.     

Application of facial reduction to H ∞ state feedback control problem

One often encounters numerical difficulties in solving linear matrix inequality (LMI) problems obtained from H _∞ control problems. For semidefinite programming (SDP) relaxations for combinatorial problems, it is known that when either an SDP relaxation problem or its dual is not strongly feasible, one may encounter such numerical difficulties. We discuss necessary and sufficient conditions to be not strongly feasible for an LMI problem obtained from H _∞ state feedback control problems and its dual. Moreover, we interpret the conditions in terms of control theory. In this analysis, facial reduction, which was proposed by Borwein and Wolkowicz, plays an important role. We show that the dual of the LMI problem is not strongly feasible if and only if there exist invariant zeros in the closed left-half plane in the system, and present a remedy to remove the numerical difficulty with the null vectors associated with invariant zeros in the closed left-half plane. Numerical results show that the numerical stability is improved by applying it.     

Stochastic MPC with inequality stability constraints

Stochastic uncertainty is a common feature of many control engineering problems, and is also present in a wider class of applications, e.g. finance and sustainable development. Recent work proposed a constrained MPC approach that took explicit account of the distributions of uncertain model parameters but used terminal equality constraints to ensure stability. The present paper reformulates the problem in order to relax the stability constraints by invoking appropriate terminal inequalities. The application of the proposed strategy and its advantages over earlier work are illustrated by means of a numerical example.     

A Survey on the Control Lyapunov Function and Control Barrier Function for Nonlinear-Affine Control Systems

This survey provides a brief overview on the control Lyapunov function (CLF) and control barrier function (CBF) for general nonlinear-affine control systems. The problem of control is formulated as an optimization problem where the optimal control policy is derived by solving a constrained quadratic programming (QP) problem. The CLF and CBF respectively characterize the stability objective and the safety objective for the nonlinear control systems. These objectives imply important properties including controllability, convergence, and robustness of control problems. Under this framework, optimal control corresponds to the minimal solution to a constrained QP problem. When uncertainties are explicitly considered, the setting of the CLF and CBF is proposed to study the input-to-state stability and input-to-state safety and to analyze the effect of disturbances. The recent theoretic progress and novel applications of CLF and CBF are systematically reviewed and discussed in this paper. Finally, we provide research directions that are significant for the advance of knowledge in this area.      

Application of distributed predictive control to motion and coordination problems for unicycle autonomous robots

This paper presents a Distributed Predictive Control (DPC) approach for the solution of a number of motion and coordination problems for autonomous robots. The proposed scheme is characterized by a multilayer structure: at the higher layer the reference trajectories of the robots are computed as the solution of suitable optimization problems. It is shown that, at this level, the definition of the cost function to be minimized allows to consider many different problems, such as formation control, coverage and optimal sensing, containment control, inter-robot and obstacle collision avoidance, and patrolling in an unknown environment. At the lower layers of the control structure, proper state and control reference trajectories are defined and a robust Model Predictive Control (MPC) problem is solved by each robot. To reduce the computational burden required by the algorithm, collision and obstacle avoidance constraints are reformulated in linear terms, so that the optimization problem to be solved on-line is a Quadratic Programming (QP) one. A number of experimental and simulation results are reported to witness the flexibility and performances of the method.     

Linear programming and model predictive control

The practicality of model predictive control (MPC) is partially limited by the ability to solve optimization problems in real time. This requirement limits the viability of MPC as a control strategy for large scale processes. One strategy for improving the computational performance is to formulate MPC using a linear program. While the linear programming formulation seems appealing from a numerical standpoint, the controller does not necessarily yield good closed-loop performance. In this work, we explore MPC with an l₁ performance criterion. We demonstrate how the non-smoothness of the objective function may yield either dead-beat or idle control performance.     

基于多参数规划的显式模型预测系统设计的可行域扩张算法

基于约束线性优化控制问题的多参数二次规划求解方法, 提出设计显式模型预测控制系统的可行域逐步扩张算法. 首先建立一种求取优化控制问题输出不变集的迭代算法. 以该输出不变集作为多参数规划问题中状态区域约束限制的初始条件, 通过反复求解多参数规划问题和不断改变状态区域约束限制, 能够逐步扩大显式模型预测控制系统的无限时间可行区域, 直到可行域不再继续扩大. 算法收敛时设计得到的显式模型预测控制系统在其所有的状态分区上都是无限时间可行的. 通过数值仿真计算, 验证本文提出算法的有效性.