Controlling the level of sparsity in MPC

In optimization algorithms used for on-line Model Predictive Control (MPC), linear systems of equations are often solved in each iteration. This is true both for Active Set methods as well as for Interior Point methods, and for linear MPC as well as for nonlinear MPC and hybrid MPC. The main computational effort is spent while solving these linear systems of equations, and hence, it is of greatest interest to solve them efficiently. Classically, the optimization problem has been formulated in either of two ways. One leading to a sparse linear system of equations involving relatively many variables to compute in each iteration and another one leading to a dense linear system of equations involving relatively few variables. In this work, it is shown that it is possible not only to consider these two distinct choices of formulations. Instead it is shown that it is possible to create an entire family of formulations with different levels of sparsity and number of variables, and that this extra degree of freedom can be exploited to obtain even better performance with the software and hardware at hand. This result also provides a better answer to a recurring question in MPC; should the sparse or dense formulation be used.     

粒子群算法的递推分析及改进

对带速度项的PSO算法和不具速度项的动态概率PSO算法进行了随机递推分析,给出了保证收敛的算法的参数取值依据以及相关条件,并基于此提出了改进的动态概率PSO算法（RSPSO）。数值实验分析结果表明,改进的PSO算法能有效避免过早收敛,具有较强的全局搜索能力,且优化能力有了进一步提升。     

Max-plus representation for the fundamental solution of the time-varying differential Riccati equation

Using the tools of optimal control, semiconvex duality and max-plus algebra, this work derives a unifying representation of the solution for the matrix differential Riccati equation (DRE) with time-varying coefficients. It is based upon a special case of the max-plus fundamental solution, first proposed in Fleming and McEneaney (2000). Such a fundamental solution can extend a particular solution of certain bivariate DREs into the general solution, and the DREs can be analytically solved from any initial condition.This paper also shows that under a fixed duality kernel, the semiconvex dual of a DRE solution satisfies another dual DRE, whose coefficients satisfy the matrix compatibility conditions involving Hamiltonian and certain symplectic matrices. For the time-invariant DRE, this allows us to make dual DRE linear and thereby solve the primal DRE analytically. This paper also derives various kernel/duality relationships between the primal and time shifted dual DREs, which lead to an array of DRE solution methods. Time-invariant analogue of one of these methods was first proposed in McEneaney (2008).     

On the discrete and continuous time infinite-dimensional algebraic Riccati equations

The standard state space solution of the finite-dimensional continuous time quadratic cost minimization problem has a straightforward extension to infinite-dimensional problems with bounded or moderately unbounded control and observation operators. However, if these operators are allowed to be sufficiently unbounded, then a strange change takes place in one of the coefficients of the algebraic Riccati equation, and the continuous time Riccati equation begins to resemble the discrete time Riccati equation. To explain why this phenomenon must occur we discuss a particular hyperbolic PDE in one space dimension with boundary control and observation (a transmission line) that can be formulated both as a discrete time system and as a continuous time system, and show that in this example the continuous time Riccati equation can be recovered from the discrete time Riccati equation. A particular feature of this example is that the Riccati operator does not map the domain of the generator into the domain of the adjoint generator, as it does in the standard case.     

New matrix bounds and iterative algorithms for the discrete coupled algebraic Riccati equation

The discrete coupled algebraic Riccati equation (DCARE) has wide applications in control theory and linear system. In general, for the DCARE, one discusses every term of the coupled term, respectively. In this paper, we consider the coupled term as a whole, which is different from the recent results. When applying eigenvalue inequalities to discuss the coupled term, our method has less error. In terms of the properties of special matrices and eigenvalue inequalities, we propose several upper and lower matrix bounds for the solution of DCARE. Further, we discuss the iterative algorithms for the solution of the DCARE. In the fixed point iterative algorithms, the scope of Lipschitz factor is wider than the recent results. Finally, we offer corresponding numerical examples to illustrate the effectiveness of the derived results.     

On the existence of solutions of the Riccati differential equation

We generalize a technique given in C. Martin [1], to obtain a characterization of finite escape times for time-varying Riccati equations which also apply to the non-definite case.     

Matrix Riccati inequality: Existence of solutions

We consider the various criteria (including Hamiltonian and frequency-domain ones) for solvability of the matrix Riccati inequality, arising from control problems. Some criteria are given in terms of symplectic algebra, which is also used as an important technical tool.     

A line search improvement of efficient MPC

A recent efficient Model Predictive Control (MPC) strategy uses a univariate Newton-Raphson procedure to solve a dual problem, but is not amenable to warm starting or early termination. By solving a primal problem, the current note proposes a strategy which is more efficient than the Newton-Raphson method and which enables warm starting and early termination. Performance improvements are demonstrated over the Newton-Raphson method and alternative approaches based on quadratic programming or semidefinite programming.     

New Upper Matrix Bounds with Power Form for the Solution of the Continuous Coupled Algebraic Riccati Matrix Equation

In this paper, using the structure and coefficient matrix of the continuous coupled algebraic Riccati matrix equation (CCARE), we firstly construct positive definite matrices with power form. Then, applying the variant of the CCARE and inequalities of positive definite matrices, utilizing the characteristics of special matrices and eigenvalue inequalities, we propose new upper matrix bounds with power form for the solution of the CCARE, which improve and extend some of the recent results. Finally, we give corresponding numerical examples to illustrate the effectiveness of the derived results.     

Solution of matrix Riccati differential equation for nonlinear singular system using genetic programming

In this paper, we propose a novel approach to find the solution of the matrix Riccati differential equation (MRDE) for nonlinear singular systems using genetic programming (GP). The goal is to provide optimal control with reduced calculation effort by comparing the solutions of the MRDE obtained from the well known traditional Runge Kutta (RK) method to those obtained from the GP method. We show that the GP approach to the problem is qualitatively better in terms of accuracy. Numerical examples are provided to illustrate the proposed method.      

Existence of Solution for Generalized Coupled Differential Riccati Equation

By means of the recursive method, the existence of solution is obtained for the generalized coupled differential Riccati equation. As an application, we apply the existence results to consider the optimal control of Markovian jump linear singular system, and obtain the desired explicit representation of the optimal controller for the optimal control problem with the finite horizon.     

代数Riccati方程解的存在性条件

首先综述代数 Riccati 方程解的存在性条件, 然后对于该方程存在唯一正定最优解的充分必要条件给出严格证明. 最后利用这一条件, 纠正了鲁棒分散控制器设计中的一些错误结果.     

区域极点约束下线性离散系统的Riccati鲁棒控制

讨论区域极点约束下,含结构参数扰动的不确定线性离散系统的鲁棒控制问题,即设计一鲁棒状态反馈控制器,使线性离散系统在可允许的参数扰动下,闭环矩阵极点始终位于一预先给定的圆形区域中,从而闭环系统具有期望的动态性能.上述控制目的可通过求解一含参数的代数离散Riccati方程达到.     

Curvature properties of the algebraic Riccati equation

We prove that the solution to the algebraic Ricatti equation (ARE) is concave with respect to a nonnegative-definite symmetric state weighting matrix Q when the input weighting matrix R = R^T > 0. We also prove that the solution to the ARE is concave with respect to a positive-definite diagonal input weighting matrix R when Q = Q^T ≥ 0.     

带有随机丢包的最优控制中Riccati方程解存在的条件

讨论了带有随机丢包的最优控制. 在传感器网络中,控制器与被控对象通过不可靠无线网络通信,因此代数Riccati方程由于通信链路的随机丢包产生了新的参数. 证明了当丢包率大于一临界值时,此Riccati 方程的解不存在. 通过解线性矩阵不等式,得到了这一临界值.     

Bounds on the solution of the algebraic Riccati equation under perturbations in the coefficients

This paper is concerned with bounds on symmetric solutions of the continuous algebraic matrix Riccati equation under perturbations on its coefficients. Special emphasis is given to the question of estimating the ‘size’ of the perturbations on the stabilizing solution of the Riccati equation when one or all its coefficients are subject to small perturbations. Upper bounds on the norm of the perturbations on the stabilizing solution are presented. Moreover, it is shown that these perturbations can be determined as a single-valued continuous function of the perturbations on the coefficient of the Riccati equation.     

Riccati equation solution for controllers with continuous delays

We reduce the solution of a Riccati equation for infinite-time linear quadratic controllers with continuous delays and n state variables to the problem of finding scalar parameter values for an integral kernel whose form is completely specified. To simplify the exposition, the reduction is described only for a special case involving 2-dimensional state variables, but the method is entirely general. An abstract formulation of Vinter and Kwong is used throughout.     

Parallel solvers for discrete-time algebric Riccati equations

We investigate the numerical solution of discrete-time algebraic Riccati equations on a parallel distributed architecture. Our solvers obtain an initial solution of the Riccati equation via the disc function method, and then refine this solution using Newton's method. The Smith iteration is employed to solve the Stein equation that arises at each step of Newton's method. The numerical experiments on an Intel Pentium-II cluster, connected via a Myrinet switch, report the performance and scalability of the new algorithms. Copyright © 2001 John Wiley & Sons, Ltd.