Weight Selection in Feedback Design With Degree Constraints

We present an approach for feedback design which is based on recent developments in analytic interpolation with a degree constraint. Performance is cast as an interpolation problem with bounded analytic functions. Minimizers of a certain weighted-entropy functional provide interpolants having degree less than the number of constraints. The choice of weight parameterizes all such bounded degree solutions. However, the relationship between the weights and the shape of corresponding transfer functions is not direct. Thus, in this paper we develop a formalism that guides weight selection.     

Stability-Preserving Rational Approximation Subject to Interpolation Constraints

A quite comprehensive theory of analytic interpolation with degree constraint, dealing with rational analytic interpolants with an a priori bound, has been developed in recent years. In this paper, we consider the limit case when this bound is removed, and only stable interpolants with a prescribed maximum degree are sought. This leads to weighted $H_2$ minimization, where the interpolants are parameterized by the weights. The inverse problem of determining the weight given a desired interpolant profile is considered, and a rational approximation procedure based on the theory is proposed. This provides a tool for tuning the solution to specifications. The basic idea could also be applied to the case with bounded analytic interpolants.      

New results on the rational covariance extension problem with degree constraint

A new theory for the rational covariance extension problem (with degree constraint), or simply the RCEP, has recently emerged with applications in high-resolution spectral estimation, speech synthesis and possibly new applications in time series analysis and system identification. This paper establishes some new theoretical results on the RCEP via an alternative analysis. In one result we show the bijective correspondence between denominator polynomials of non-strictly-positive solutions of the RCEP and the minimizers of a class of (strictly) convex functionals, associated with non-strictly-positive pseudopolynomials, defined on a subset of a finite dimensional space. This result leads to an alternative constructive proof of a theorem of Georgiou on complete parametrization of all solutions of the RCEP and a new geometric proof, with an extension to non-real interpolators, of a homeomorphism derived previously by Blomqvist, Fanizza and Nagamune. We then generalize this homeomorphism to also allow for variation in the covariance data. Our contribution in the generalization is allowing for non-strictly-positive pseudopolynomials. For the special case of strictly positive pseudopolynomials, a stronger property of diffeomorphism (in the context of the Nevanlinna–Pick interpolation with degree constraint) has been shown earlier by Byrnes and Lindquist. The results of this paper have direct analogues to Nevanlinna–Pick interpolation with degree constraint which is of interest to the robust control community.     

Matrix-valued Nevanlinna-Pick interpolation with complexity constraint: an optimization approach

Over the last several years, a new theory of Nevanlinna-Pick interpolation with complexity constraint has been developed for scalar interpolants. In this paper we generalize this theory to the matrix-valued case, also allowing for multiple interpolation points. We parameterize a class of interpolants consisting of "most interpolants" of no higher degree than the central solution in terms of spectral zeros. This is a complete parameterization, and for each choice of interpolant we provide a convex optimization problem for determining it. This is derived in the context of duality theory of mathematical programming. To solve the convex optimization problem, we employ a homotopy continuation technique previously developed for the scalar case. These results can be applied to many classes of engineering problems, and, to illustrate this, we provide some examples. In particular, we apply our method to a benchmark problem in multivariate robust control. By constructing a controller satisfying all design specifications but having only half the McMillan degree of conventional H/sup /spl infin// controllers, we demonstrate the advantage of the proposed method.     

Practical optimal state feedback control law for continuous-time switched affine systems with cyclic steady state

In this article, a method for computing an optimal state feedback control law for continuous-time switched affine systems exhibiting cyclic behaviour in steady state is presented. The hybrid solutions are deduced from the Fillipov solutions. It is shown that the optimal trajectory synthesis implies to determine singular arcs. Algebraic conditions are given to obtain these particular arcs of the trajectory. A numerical procedure is then proposed to generate optimal trajectories on a given state space area avoiding the classical two-point boundary value problem occurring in optimal control synthesis. The interpolation of the solutions set, through a neural network, yields a state feedback control law. Several examples in the power converters field show the feasibility and the efficiency of the method.     

一种改进的同伦算法与H∞鲁棒控制器设计

提出了一种具有阶次限制的鲁棒控制器设计方法, 该算法将控制系统的性能指标转化为灵敏度函数问题, 并利用Nevanlinna-Pick插值算法进行求解. 提出了一种改进的同伦算法, 将其用于求解由灵敏度函数产生的非线性方程. 基于改进同伦算法设计的鲁棒控制器 不仅避免了传统H∞控制中加权函数的选择问题, 而且克服了鲁棒控制器阶次较高的缺陷. 最后,文章以4阶系统为例, 设计了具有阶次限制的H∞鲁棒控制器, 通过与传统鲁棒控制器的比较可以看出, 基于本文方法设计的控制器不仅具有较低的阶次, 而且其控制性能也具有明显的优越性.     

稳定化的有限频段H∞控制及有限频段跟踪问题

基于GKYP引理的动态输出反馈设计,未保证设计后闭环系统的稳定性。针对以小增益作为指标的有限频段动态输出反馈问题,在不增加新变量的前提下,增加稳定性约束,使得设计后的闭环系统渐近稳定且满足有限频段性能指标。针对增加约束后难以找到可行解的情况,基于零空间条件的不惟一性,补充了另一种零空间条件,从而扩大了问题的可行域。将改进后的方法应用于有限频段跟踪问题的研究,通过仿真例子验证,有限频段动态输出反馈虽然存在保守性,但在合理选择基矩阵R的情况下,仍然可以使得其保守性小于传统的全频段最优H∞控制的保守性。     

虚拟装配中基于导纳控制的力觉渲染技术

传统的力触觉渲染多采用阻抗控制,不能很好地满足虚拟装配的应用要求,相比之下导纳控制模式更适用这一领域.为此提出一种基于导纳控制的双线程力觉渲染构架,并给出相应的力觉渲染算法.首先建立用于导纳控制的动力学模型,并讨论了碰撞和约束这2个状态下的力觉渲染;为了使用力觉交互接口进行虚拟装配中的小间隙装配,提出物理约束与几何约束结合的力觉渲染方法;最后针对物理计算和力反馈循环2个线程刷新频率不匹配的问题,利用二次拉格朗日多项式进行数值插值,实现了力觉交互接口的平稳输出.通过力反馈设备与自主开发的虚拟装配原型系统VAPP的连接与应用,验证了所提出的算法满足虚拟装配系统中力觉交互的应用要求.     

A new algorithm for a recursive construction of the minimal interpolation space

In this work, we introduce a new interpolation algorithm, based on a recursive method for computing Lagrange interpolants. This algorithm allows to construct recursively the minimal interpolation space (see [1]) with respect to a finite set of points. We also extend this recursive method to the osculatory interpolation problem.     

A new algorithm for a recursive construction of the minimal interpolation space

In this work, we introduce a new interpolation algorithm, based on a recursive method for computing Lagrange interpolants. This algorithm allows to construct recursively the minimal interpolation space (see [1]) with respect to a finite set of points. We also extend this recursive method to the osculatory interpolation problem.     

A generalized entropy criterion for Nevanlinna-Pick interpolationwith degree constraint

We present a generalized entropy criterion for solving the rational Nevanlinna-Pick problem for n+1 interpolating conditions and the degree of interpolants bounded by n. The primal problem of maximizing this entropy gain has a very well-behaved dual problem. This dual is a convex optimization problem in a finite-dimensional space and gives rise to an algorithm for finding all interpolants which are positive real and rational of degree at most n. The criterion requires a selection of a monic Schur polynomial of degree n. It follows that this class of monic polynomials completely parameterizes all such rational interpolants, and it therefore provides a set of design parameters for specifying such interpolants. The algorithm is implemented in a state-space form and applied to several illustrative problems in systems and control, namely sensitivity minimization, maximal power transfer and spectral estimation     

Discontinuous stabilization of nonlinear systems: Quantized and switching controls

In this paper we consider the classical problem of stabilizing nonlinear systems in the case the control laws take values in a discrete set. First, we present a robust control approach to the problem. Then, we focus on the class of dissipative systems and rephrase classical results available for this class taking into account the constraint on the control values. In this setting, feedback laws are necessarily discontinuous and solutions of the implemented system must be considered in some generalized sense. The relations with the problems of quantized and switching control are discussed.     

A characterization of convex problems in decentralized control

We consider the problem of constructing optimal decentralized controllers. We formulate this problem as one of minimizing the closed-loop norm of a feedback system subject to constraints on the controller structure. We define the notion of quadratic invariance of a constraint set with respect to a system, and show that if the constraint set has this property, then the constrained minimum-norm problem may be solved via convex programming. We also show that quadratic invariance is necessary and sufficient for the constraint set to be preserved under feedback. These results are developed in a very general framework, and are shown to hold in both continuous and discrete time, for both stable and unstable systems, and for any norm. This notion unifies many previous results identifying specific tractable decentralized control problems, and delineates the largest known class of convex problems in decentralized control. As an example, we show that optimal stabilizing controllers may be efficiently computed in the case where distributed controllers can communicate faster than their dynamics propagate. We also show that symmetric synthesis is included in this classification, and provide a test for sparsity constraints to be quadratically invariant, and thus amenable to convex synthesis.     

A characterization of convex problems in decentralized Control/sup */

We consider the problem of constructing optimal decentralized controllers. We formulate this problem as one of minimizing the closed-loop norm of a feedback system subject to constraints on the controller structure. We define the notion of quadratic invariance of a constraint set with respect to a system, and show that if the constraint set has this property, then the constrained minimum-norm problem may be solved via convex programming. We also show that quadratic invariance is necessary and sufficient for the constraint set to be preserved under feedback. These results are developed in a very general framework, and are shown to hold in both continuous and discrete time, for both stable and unstable systems, and for any norm. This notion unifies many previous results identifying specific tractable decentralized control problems, and delineates the largest known class of convex problems in decentralized control. As an example, we show that optimal stabilizing controllers may be efficiently computed in the case where distributed controllers can communicate faster than their dynamics propagate. We also show that symmetric synthesis is included in this classification, and provide a test for sparsity constraints to be quadratically invariant, and thus amenable to convex synthesis.     

A recursive method for solving unconstrained tangentialinterpolation problems

An efficient recursive solution is presented for the one-sided unconstrained tangential interpolation problem. The method relies on the triangular factorization of a certain structured matrix that is implicitly defined by the interpolation data. The recursive procedure admits a physical interpretation in terms of discretized transmission lines. In this framework the generating system is constructed as a cascade of first-order sections. Singular steps occur only when the input data is contradictory, i.e., only when the interpolation problem does not have a solution. Various pivoting schemes can be used to improve numerical accuracy or to impose additional constraints on the interpolants. The algorithm also provides coprime factorizations for all rational interpolants and can be used to solve polynomial interpolation problems such as the general Hermite matrix interpolation problem. A recursive method is proposed to compute a column-reduced generating system that can be used to solve the minimal tangential interpolation problem     

Generalized Hermitian Radial Basis Functions Implicits from polygonal mesh constraints

In this work we investigate a generalized interpolation approach using radial basis functions to reconstruct implicit surfaces from polygonal meshes. With this method, the user can define with great flexibility three sets of constraint interpolants: points, normals, and tangents; allowing to balance computational complexity, precision, and feature modeling. Furthermore, this flexibility makes possible to avoid untrustworthy information, such as normals estimated on triangles with bad aspect ratio. We present results of the method for applications related to the problem of modeling 2D curves from polygons and 3D surfaces from polygonal meshes. We also apply the method to problems involving subdivision surfaces and front-tracking of moving boundaries. Finally, as our technique generalizes the recently proposed HRBF Implicits technique, comparisons with this approach are also conducted.     

集成几何约束与物性仿真技术的虚拟装配

针对传统装配定位导航中零部件位姿的突变问题,利用线性插值与四元数球形插值 的方法,求解零部件约束定位变换过程的渐变运动;同时利用导纳式力觉交互控制模式,解决了 约束定位变换后的控制信号分离问题;在装配运动导航阶段,将约束几何元素具象化为力觉约束 模型,以满足虚拟装配对力觉反馈的需求。针对虚拟装配对物性仿真的需求,将开源物理引擎 Bullet 集成到仿真系统中,提出几何约束与物性仿真技术相结合的装配流程,最后在自主研发的 虚拟装配原型系统中进行了应用实例验证。     

平面三次PH曲线偶C1Hermite插值解的构造和形状分析

一般情况下,三次PH曲线偶的C^1 Hemite插值问题有四个不同的解。在这四个解中,只有一条曲线能很好地满足几何设计的要求。已有的插值算法都是依赖于构造出所有四个解,利用绝对旋转指标或弹性弯曲能量来找出这条“好”的插值曲线。本文提出一种新的方法以区分这些解,即用由三次PH曲线偶和惟一经典三次插值曲线的速端曲线形成的闭环的弯曲数来区分。对于“合理”的Hemite数据,本文还给出了不需计算和比较所有的四个解便可直接构造“好”的三次PH曲线偶的方法。     

Scheduled control for robust attenuation of non-stationary sinusoidal disturbances with measurable frequencies

Attenuation of sinusoidal disturbances with uncertain yet online measurable frequencies is considered. The disturbances are modeled as the outputs of an undisturbed parameter-dependent exogenous system with a skew-symmetric system matrix, obtained in response to nonzero initial conditions. The problem is formulated for a parameter-dependent plant as the synthesis of a parameter-dependent controller in a way to ensure internal stability as well as a desired level of steady-state disturbance attenuation in the face of all admissible parameter variations. The solvability of this problem is first related to the existence of bounded solutions to a matrix differential regulator equation subject to an asymptotic norm constraint. Reformulating this as a parameter-dependent state-feedback like synthesis, based on which suitable solutions to the differential regulator equation can be obtained online, tractable solvability conditions are then provided in the form of parameter-dependent matrix inequalities. Controllers that solve the generalized asymptotic regulation problem are also parameterized in terms of the suitable solutions of the differential regulator equation and some free parameter-dependent matrices that are to be designed off-line to ensure stability. A procedure is then developed to design the free parameters in a way to achieve desirable transient behavior. The use of the developed synthesis procedure is illustrated on a simplified version of the course control problem in ship steering.     

A new result on passivity preserving model reduction

The problem of model reduction with preservation of passivity is investigated. The approach is based on positive real interpolation, and is inspired by the similarity between Löwner and Pick matrices. The former are important in problems of general rational interpolation while the latter in problems of interpolation by positive real functions. It follows that interpolation of the original set of data together with an appropriately defined mirror-image set of data yields automatically positive real interpolants. Subsequently, we show how this result can be implemented using a Krylov projection procedure. The ensuing model reduction method preserves stability and passivity and can be implemented efficiently for the large-scale systems.