Robust eigenstructure assignment with minimum subspace separation

A computation method for delivering a state feedback controller that assigns closed‐loop eigenstructure with a minimum subspace separation and is robust towards unstructured perturbations is presented. The subspace separation is measured via the difference between two subspace projections, while the eigenstructure robustness is achieved by maximizing an upper bound on the allowable perturbation size such that the closed‐loop system remains stable. By exploiting the Sylvester equation parametrization, the constraint of pole assignment can be satisfied intrinsically and the robust eigenstructure assignment design essentially becomes an unconstrained optimization task. Analytical gradient formulas of the objective functions to be handled are developed. The design approach is demonstrated through numerical examples. Copyright © 2004 John Wiley & Sons, Ltd.     

基于混合灵敏度优化的鲁棒特征结构配置

将H_∞理论的混合灵敏度指标融入到特征结构配置中, 将混合灵敏度指标转换为特征结构设计参数的函数关系, 建立了特征结构配置的频率域指标约束. 这两种方法的结合各自利用了其在时域和频率域的设计优势. 通过优化算法所获得的鲁棒特征结构配置, 既保证了闭环特征值/特征向量, 又使反馈回路具有一定的频率域特性. 计算实例表明这种综合设计方法的有效性.     

Insensitive and robust control design via output feedback eigenstructure assignment

Insensitive and robust control design using output‐feedback eigenstructure assignment for linear multivariable systems is considered in this paper. A parametric expression of closed‐loop eigenvectors and generalized eigenvectors is developed. It can cope with the case where the closed‐loop eigenvalues are multiple and/or the same as the open‐loop ones so that the system to be designed can be uncontrollable and/or unobservable. The controller designed via output‐feedback eigenstructure assignment is expressed by proposed parameter vectors. The freedom provided by output‐feedback eigenstructure assignment is used to optimize some performance functions which are used to measure the sensitivity of the closed‐loop matrix and the robustness of the closed‐loop system. Copyright © 2000 John Wiley & Sons, Ltd.     

Mixed time- and frequency-domain robust eigenstructure assignment

This paper considers robust eigenstructure assignment for multivariable systems. The time-domain performance specifications provided by eigenstructure assignment and robust performance specifications in the frequency domain considered by H infinity control are combined to realize joint optimal robust control design. A parametric expression for state-feedback eigenstructure assignment is introduced on the basis of a set of free parameters. The mixed performance index used as the cost function consists of two parts: robustness in the frequency domain; and system eigenstructure constraints in the time domain. The analytical gradient calculation of the cost function with respect to the free parameters is derived for optimal robust eigenstructure assignment using gradientbased optimization. The mixed time- and frequency-domain robust eigenstructure assignment is demonstrated in the design of a mode-decoupled roll-yaw autopilot.     

广义二阶动力学系统的鲁棒特征结构配置

考虑了具有摄动的广义二阶动力学系统的鲁棒特征结构配置问题, 提出了一种优化算法. 该算法的目标是使得特征结构配置的误差最小. 根据广义二阶动力学系统基于状态加微分反馈的参数化特征结构配置结果, 给出了优化指标的完全参数形式. 该算法不包含“返回”过程, 允许特征值在一定范围内参与优化, 能够给出鲁棒性较强的控制系统. 数值例子说明了算法的有效性.     

Dominant pole and eigenstructure assignment for positive systems with state feedback

In this paper, the dominant pole assignment problem, the dominant eigenstructure assignment problem and the robust dominant pole assignment problem for linear time-invariant positive systems with state feedback are considered. The dominant pole assignment problem is formulated as a linear programming problem, and the dominant eigenstructure problem is formulated as a quasiconvex optimisation problem with linear constraints. The robust dominant pole assignment problem is formulated as a non-convex optimisation problem with non-linear constraints which is solved using particle swarm optimisation (PSO) with an efficient scheme which employs the dominant eigenstructure assignment technique to accelerate the convergence of the PSO procedure. Each of the three problems can be further constrained by requiring that the controller has a pre-specified structure, or the gain matrix have both elementwise upper and lower bounds. These constraints can be incorporated into the proposed scheme without increasing the complexity of the algorithms. Both the continuous-time case and the discrete-time case are treated in the paper.     

Robust eigenstructure clustering by non-smooth optimisation

We extend classical eigenstructure assignment to more realistic problems, where additional performance and robustness specifications arise. Our aim is to combine time-domain constraints, as reflected by pole location and eigenvector structure, with frequency-domain objectives such as the H₂, H_∞ or Hankel norms. Using pole clustering, we allow poles to move in polydisks of prescribed size around their nominal values, driven by optimisation. Eigenelements, that is poles and eigenvectors, are allowed to move simultaneously and serve as decision variables in a specialised non-smooth optimisation technique. Two aerospace applications illustrate the power of the new method.     

Increasing eigenstructure assignment design degree of freedom using lifting

This paper presents the exposition of an output-lifting eigenstructure assignment (EA) design framework, wherein the available EA design degrees of freedom (DoF) is significantly increased, and the desired eigenstructure of a single-rate full state feedback solution can be achieved within an output feedback system. A structural mapping is introduced to release the output-lifting causality constraint. Additionally, the available design DoF can be further enlarged via involving the input-lifting into the output-lifting EA framework. The newly induced design DoF can be utilised to calculate a structurally constrained, causal gain matrix which will maintain the same assignment capability. In this paper, the robustification of the output-lifting EA is also proposed, which allows a trade-off between performance and robustness in the presence of structured model uncertainties to be established. A lateral flight control benchmark in the EA literature and a numerical example are used to demonstrate the effectiveness of the design framework.     

An extension to output-feedback eigenstructure assignment: Structuring controllers by exploiting unused design freedom

A recent algorithm for output-feedback eigenstructure assignment uses a multi-stage design process to assign a combination of left and right eigenvectors. It is possible, depending on the distribution of design freedom between the stages, that not all of the available degrees of freedom will be used by the assignment process. In this paper, an algorithm is presented which exploits unused design freedom to introduce structure to the resulting gain matrix without affecting the assigned eigenstructure. This process will inevitably have an effect on the overall magnitude of the gains, and consideration is given to this.     

Partial eigenstructure assignment problem and its application to the constrained linear problem

This article presents a new partial eigenstructure assignment method. This technique keeps the open-loop stable eigenvalues and the corresponding eigenspace unchanged. The remaining undesirable eigenvalues are replaced by other chosen values. This methodology is easy and permits to overcome some limitations encountered in the previous methods. Furthermore, our method is applied to solve the constrained control problem for linear invariants continuous-time systems. Indeed, the problem of finding a stabilising regulator matrix gain taking into account the asymmetrical control constraints is transformed to a Sylvester equation resolution. Examples are given to illustrate the obtained results.     

Multistage output-lifting eigenstructure assignment: a multirate ball and plate example

By exploiting both the left and the right allowable subspaces in consecutive stages, this paper extends a recently developed output-lifting eigenstructure assignment approach into a multistage eigenstructure assignment scheme. In this scheme, design degrees of freedom, enlarged via output-lifting, are further exploited to improve eigenvector assignment. To mitigate the inherent conflicts between the theoretical development of eigenstructure assignment and inherent physical system characteristics, the paper also clearly demonstrates how to derive an ideal eigenstructure, particularly the desired eigenvectors, to distribute and decouple the natural modes among appropriate states or outputs, based upon an example: a novel multirate ball and plate system. The design and simulation results show the efficacy of the scheme.     

An explicit solution to right factorization with application in eigenstructure assignment

Based on the well-known Leverrier algorithm, a simple explicit solution to right factorization of a linear system is established.This solution is expressed by the controllability matrix of the given system and a symmetric operator matrix.Applications of this solution to a type of generalized Sylvester matrix equations and the problem of parametric eigenstructure assignment by state feedback are investigated,and general complete parametric solutions to these two problems are deduced. These new solutions are simple,and possess desirable structural properties which render the solutions readily implementable.An example demonstrates the effect of the proposed results.     

Multiobjective controller synthesis via eigenstructure assignment with state feedback

A general parameter scheme for multiobjective controller synthesis via eigenstructure assignment with state feedback is proposed. The scheme provides total pole configurability, that is, pole assignment constraints, partial pole assignment constraints, generalised regional pole assignment constraints can be dealt with simultaneously without introducing essential conservatism. The scheme is derived from the pole assignment approach using Sylvester equations, and the parameter space is the Cartesian product of some subspaces characterising the free parameters. Under the scheme, the controller design problems are formulated as nonlinear optimisation problems with both objectives and constraints being differentiable and can be solved by derivative-based nonlinear programming technique. Numerical examples are given to illustrate the efficiency of the proposed method.     

An explicit solution to right factorization with application in eigenstructure assignment

1Proble mfor mulationRight factorization oflinear systems is a basic problemincontrol systems theory,and has important applications inmany analysis and design problems ,such as , robuststabilization [1] ,minimal state space realization [2,3] ,speed servo system design [4] ,synthesis of H_infinitycontrollers [5] ,etc .Furthermore ,it has been shown byDuan and his co_authors that a factorization can be usedtoparameterize all the solutions to a generalized_type ofSylvester matrix equations [6 ~8…     

不确定时滞系统鲁棒镇定的优化方法

对一类含有时滞不确定的线性系统,考虑了鲁棒镇定问题。系统中的不确定性假设满足范数有界备件,通过用改进的Razmikhin-type定理,给出了鲁棒镇定控制器存在的充分条件。通过对Lyapunov矩阵方程的完全参数化求解,得到了鲁棒镇定控制器的参数化形式,鲁棒镇定问题转化为一类优化问题。基于这一条件与参数化特征结构配置结果,得到了状态反馈镇定控制器的设计算法。数值例子说明了该方法的有效性。     

一种改进的输出反馈特征结构配置方法在飞翼飞机增稳系统设计中的应用

针对常规输出反馈特征结构配置方法不能完全保证闭环系统稳定性的问题, 提出一种改进的输出反馈特征结构配置方法. 该改进的算法通过引入二次型性能指标, 将闭环系统的稳定性问题转化为线性二次型最优控制问题,从而保证闭环系统的稳定性; 考虑到采用特征结构配置方法所设计的闭环系统鲁棒性不强, 给出保证系统鲁棒性的条件, 以解决系统鲁棒性问题. 最后, 通过在飞翼飞机上的仿真结果验证了所提出算法的有效性.

     

FIR filters for online trajectory planning with time- and frequency-domain specifications

In this paper, the use of FIR (Finite Impulse Response) filters for planning minimum-time trajectories for robots or automatic machines under constraints of velocity, acceleration, etc. is presented and discussed. In particular, the relationship between multi-segment polynomial trajectories, i.e. trajectories composed of several polynomial segments, each one possibly characterized by constraints on one or more specific derivatives (i.e. velocity, acceleration, jerk, etc.), and FIR filters disposed in a cascade configuration is demonstrated and exploited in order to design a digital filter for online trajectory planning. The connection between analytic functions and dynamic filters allows a generalization of these trajectories, usually obtained by second- or third-order polynomial functions (e.g. trapezoidal velocity and double S velocity trajectories), to a generic order with only a modest increase of the complexity. As a matter of fact, the computation of trajectories with higher degree of continuity simply requires additional FIR filters in the chain. Moreover, the modular structure of the planner provides a direct frequency characterization of the motion law. In this way, it is possible to define the trajectories by considering constraints expressed in the frequency-domain besides the classical time-domain specifications, such as bounds on velocity, acceleration, and so on. Two examples illustrate the main features of the proposed trajectory planner, in particular with respect to the problems of multi-point trajectories generation and residual vibrations suppression.