Frequency response algorithms for H∞optimization with time domain constraints

A very broad framework for control system design is considered that encompasses frequency-response methodologies for H_∞ optimization that solve various aspects of the control design problem and that are less well known that state-space methods. The focus is on linear programming, Lawson's algorithm, and Trefethen's algorithm. A modified Lawson's algorithm is proposed and related to Trefethan's method. The modified algorithm is shown to be significantly faster than linear programming and Lawson's algorithm. It is also shown how to extend the modified Lawson's algorithm so as to handle time-domain constraints in addition to frequency-domain specifications, which distinguishes it from other H_∞ optimization methods. Some steps are taken toward dealing with time domain constraints within an H_∞ optimization framework     

单变量离散混合l1/H2问题

本文描述了Ｈ２性能约束下的离散控制系统ｌ１设计问题，并提出了该无穷维优化问题的一种上逼近解法，通过求解一系列的截断问题能够得到无穷维问题的次优解，混合ｌ１／Ｈ２问题最优值对Ｈ２约束连续依赖的性质也在本文得到研究。     

H∞ optimal control as a weightedWiener-Hopf problem

It is shown that H_∞ optimization is equivalent to weighted H₂ optimization in the sense that the solution of the latter problem also solves the former. The weighting rational matrix that achieves this equivalence is explicitly computed in terms of a state-space realization. The authors do not suggest transforming H_∞ optimization problems to H₂ optimization problems as a computational approach. Rather, their results reveal an interesting connection between H_∞ and H₂ optimization problems which is expected to offer additional insight. For example, H₂ optimal controllers are known to have an optimal observer-full state feedback structure. The result obtained shows that the minimum entropy solution of H_∞ optimal control problems can be obtained as an H₂ optimal solution. Therefore, it can be expected that the corresponding H_∞ optimal controller has an optimal observer-full state feedback structure     

A two-sided interpolation approach to H∞optimization problems

A solution to the two-sided interpolation problem which arises in H_∞ optimization theory is obtained. This solution is found in closed form, explicitly in terms of the required interpolation directions. It is simple to obtain and it does not require the application of the relatively complicated matrix Pick-Nevanlinna theory. The solution obtained is of minimum order; due to its simplicity, the order reduction, which occurs at the minimum value of the H_∞-norm, is clearly explained     

LQG control with an H∞ performance bound:a Riccati equation approach

An LQG (linear quadratic Gaussian) control-design problem involving a constraint on H^∞ disturbance attenuation is considered. The H^∞ performance constraint is embedded within the optimization process by replacing the covariance Lyapunov equation by a Riccati equation whose solution leads to an upper bound on L₂ performance. In contrast to the pair of separated Riccati equations of standard LQG theory, the H^∞-constrained gains are given by a coupled system of three modified Riccati equations. The coupling illustrates the breakdown of the separation principle for the H^∞ -constrained problem. Both full- and reduced-order design problems are considered with an H^∞ attenuation constraint involving both state and control variables. An algorithm is developed for the full-order design problem and illustrative numerical results are given     

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.     

Robust stabilization of uncertain linear systems: quadraticstabilizability and H∞ control theory

The problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time-domain results on the quadratic stabilization of uncertain systems and the frequency-domain results on H^∞ optimization. A complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given. Relations between these robust stabilization problems and H^∞ control theory are explored. It is also shown that in a number of cases, if a robust stabilization problem can be solved via Lyapunov methods, then it can be also be solved via H ^∞ control theory-based methods     

On the dead-time compensation from L/sup 1/ perspectives

It is known that both H/sup 2/ and H/sup /spl infin// optimization problems for dead-time systems are solved by controllers having the so-called modified Smith predictor (dead-time compensator) structure. This note shows that this is also true for the L/sup 1/ control problem. More precisely, it is demonstrated that the use of the modified Smith predictor enables one to reduce the standard L/sup 1/ problem for systems with a single loop delay to an equivalent delay-free problem. The (sub)optimal solution therefore always contains the modified Smith predictor.     

Direct state-space solution to the discrete-timeH∞ one-block problem

A general state-space representation is used to allow a complete formulation of the H_∞ optimization problem without any invertibility condition on the system matrix, unlike existing solutions. A straightforward approach is used to solve the one-block H_∞ optimization problem. The parameterization of all solutions to the discrete-time H_∞ suboptimal one-block problem is first given in transfer function form in terms of a set of functions in H _∞ that satisfy a norm bound. The parameterization of all solutions is also given as a linear fractional representation     

MIMO optimal control design: the interplay between the ℋ2 and the l1 norms

We consider controller design methods which can address directly the interplay between the ℋ2 and l₁ performance of the closed loop. The development is devoted to multi-input/multi-output (MIMO) systems. Two relevant multi-objective performance problems are considered each being of interest in its own right. In the first, termed as the combination problem, a weighted sum of the l₁ norms and the square of the ℋ₂ norms of a given set of input-output transfer functions constituting the closed loop is minimized. It is shown that, in the one-block case, the solution can be obtained via a finite-dimensional quadratic optimization problem which has an a priori known dimension. In the four-block case, a method of computing approximate solutions within any a priori given tolerance is provided. In the second, termed as the mixed problem, the ℋ₂ performance of the closed loop is minimized subject to an l₁ constraint. It is shown that approximating solutions within any a priori given tolerance can be obtained via the solution to a related combination problem     

l/sub 1/-suboptimal pole placement design

We incorporate an arbitrary number of "candidate" positive distinct poles into the scalar l/sub 1/ optimization framework in order to obtain low-order rational suboptimal solutions to constrained l/sub 1/ optimization problems. Our approach uses a single linear program to minimize an upper bound on the l/sub 1/ norm and gives a solution which uses only the best out of the prespecified poles. Rational suboptimal solutions to a scalar two-block problem are obtained similarly.     

l1 optimal robust controller for SISO plant undercoprime factor perturbations

The problem of synthesis of l₁ optimal robust controller for SISO plant under coprime factor perturbations and bounded external disturbance is considered. A geometric interpretation of l₁ optimal robust controller, standard l₁ optimal controller, and two other optimal robust controllers is presented. The existence of l₁ optimal robust linear controller is proved and an algorithm for approximate solution of the problem is proposed. The algorithm is reduced to approximate solution of finite family of standard l₁ optimization problems     

Data reduction in the mixed-sensitivityH∞ design problem

The mixed-sensitivity H^∞ design problem is discussed in the two-block case. The state-space approach is used to investigate the internal structure of this design problem, which inherits many pole-zero cancellations. The complete state-space solution is provided. Efficient mathematical software can then be implemented     

∞ optimal interconnections

In this paper a general _∞ problem for continuous time, linear time systems is formulated and solved in a behavioral framework. This general formulation, which includes standard _∞ optimization as a special case, provides added freedom in the design of sub-optimal compensators, and allows for singular interconnections, which naturally occur when interconnecting first principles models. It is shown that a necessary condition for the existence of a solution is the existence of a partition of the control variables which allows one to express the given system by a possibly non-proper, non-standard, input-state-output map. A slight modification of the DGKF Riccati-based solution may then be employed to provide a complete solution. It is shown, with the aid of an example, that the optimal solution may lead to some counter-intuitive constraints which may require novel mechanisms for implementation; thus, the formal optimization techniques being considered in this paper may provide guidelines for design.     

The application of scheduled H∞controllers to a VSTOL aircraft

The authors apply H_∞-designed controllers to a generic VSTOL (vertical and short takeoff and landing) aircraft model GVAM. The design study motivates the use of H _∞ techniques, and addresses some of the implementation issues which arise for multivariable and H_∞-designed controllers. An approach for gain scheduling H_∞ controllers on the basis of the normalized comprime factor robust stabilization problem formulation used for the H_∞ design is developed. It utilizes the observer structure unique to this particular robustness optimization. A weighting selection procedure, has been developed for the associated loop-shaping technique used to specify performance. Multivariable controllers pose additional problems in the event of actuator saturations, and a desaturation scheme which accounts for this is applied to the GVAM. A comprehensive control law was developed and evaluated using the Royal Aerospace Establishment piloted simulation facility     

0/1背包问题及其解法研究

0/1背包问题是实际当中经常遇到的一类经典NP-hard组合优化问题之一。本文分别从贪心方法、动态规划、回溯法、分枝-限界法,遗传算法这五种算法设计方法入手,概述了各种设计方法的基本原理,提出了求解0/1背包问题的算法思想,并对算法进行分析,提出了改进方法。     

H2-norm optimization with constrained dynamic outputfeedback controllers: decentralized and reliable control

This paper addresses some open problems in the area of dynamic output feedback control design. The first one is the structurally constrained decentralized control problem, and the second is related to robust and reliable control. A necessary and sufficient condition for decentralized and quadratic stabilizability is given and used to provide a solution to the H₂-norm optimization problem. A numerical cross decomposition algorithm is developed and satisfactorily applied to find the solution of some illustrative examples. A comprehensive benchmark validates the results     

Minimum H∞-norm regulation of lineardiscrete-time systems and its relation to linear quadratic discretegames

A solution is derived to the H_∞-optimization problem that arises in multivariable discrete-time regulation when the controller has full access to the state vector. The solution method is based on the close relations that exist between linear quadratic differential game theory and H_∞-optimization. The existing theory of discrete-time quadratic games is readily applied in order to derive the solution to a finite-time horizon version of the H_∞ -optimization problem. The solution of the infinite-time horizon H_∞-optimization problem is obtained by formally taking the limit of the number of stages to infinity     

SISO多率离散系统L1优化问题求解研究

采样系统ｌ＾１优化设计的关键在于多率离散系统Ｌ＾１翁化问题的解决，本文讨论了其基本问题的性质，证明了对于基于ＳＩＳＯ多率Ｌ＾１优化问题，总可以通过构造相应的单变量Ｌ＾１优化问题或多变量易求解Ｌ＾１优化问题，求得次优解。     

Finite horizon H∞ control problems withterminal penalties

The standard H^∞ control problem is generalized to the finite horizon case with two (possibly singular) terminal penalties at the initial and final times. The major objective of the generalization is to increase flexibility of H^∞ controls. Transients in the finite horizon and the terminal penalties are taken into account within the framework of H^∞ control problems. A complete solution, a necessary and sufficient condition, and a parameterization to the finite horizon H^∞ control problem are given. The solution is a natural extension of the Riccati equation solution. In the special case, when all the terminal penalties vanish, the solution is reduced to the existing one and to the finite horizon standard H^∞ control problem. This approach to the problem is based on completing the square argument of a particular quadratic form, which is at least technically different from the previous ones