【作者】 孟范伟；

【导师】 周荻； 何朕；

【作者基本信息】 哈尔滨工业大学 ， 控制科学与工程， 2013， 博士

【摘要】 控制系统的设计近年来逐渐从基于分析的设计方法转向综合方法，即根据设计要求直接给出控制器。在线性系统方面，这就是H_∞控制；在非线性系统设计方面，除非线性H_∞控制外，近年来兴起的平方和(sum of squares，SOS)法实际上也就是一种综合方法，可以直接计算出所要求的Lyapunov函数和非线性控制律。这些先进的综合方法在处理常规系统的设计方面都有很好的例子，但是在处理一些特殊类型系统的设计方面尚少有讨论或只有一些原理性的讨论。例如对带弱阻尼谐振模态系统的设计、不稳定对象的控制、非线性系统的设计等。这里指的是如何明确这些特殊类型系统的设计要求、设计限制和特点，以及这些先进综合方法在满足特殊设计要求时理论上要进行的一些工作。本论文的工作就是要将这些先进的设计方法推广到一些常见的特殊系统上，为这些系统提供一些实用的设计方法，同时又充实和发展这些新兴的设计理论。本论文的研究主要包括以下几个内容：论文结合弱阻尼挠性系统的设计来研究H_∞回路成形法。指出已有的关于H_∞回路成形法的研究，只谈互质因式摄动可以描述弱阻尼模态摄动，本文则通过对互质因式摄动的详尽分析，给出了弱阻尼模态对互质因式摄动范数以及设计结果的鲁棒性的影响。论文还指出H_∞回路成形法中的H_∞范数表示的是稳定裕度，回路成形设计的H_∞控制器是用来保证稳定裕度满足鲁棒稳定性要求的，而一般H_∞控制理论中，这个H_∞范数则是系统的性能指标。结合磁悬浮控制系统给出了适合不稳定对象的H_∞状态反馈和H_∞输出反馈设计方法，对原有的H_∞状态反馈和H_∞输出反馈在设计上的不足进行了补充和改进。对于H_∞状态反馈设计，提出在求解Riccati方程之上要再增加由Bode积分定理所规定的鲁棒性约束，才是一个完整的设计。对于输出反馈设计，H_∞设计中的权函数一般都是不考虑中频段的要求，但是对不稳定对象来说就不够了，本论文提出不稳定对象输出反馈设计中加权函数的选择应该根据中频段的Bode积分约束再加上H_∞优化设计来解决。对于非线性系统来说，非线性H_∞设计继承了线性系统中的H_∞设计思想，但解析求解HJI不等式是非线性H_∞设计的主要难题，本论文给出一种基于泰勒(Taylar)级数展开的分步求解方法，并将其应用于仿射非线性磁悬浮系统的设计。并通过Hamilton函数的分析对非线性H_∞设计结果进行了验证。近年来出现的SOS方法，是一种以多项式为研究对象的数值求解方法，论文先通过吸引域估计来说明SOS法使用上的特点，并提出了处理集合包含问题的广义S方法，以及决策变量的确定。结合卫星大角度姿态机动控制这一非线性问题提出了对角占优的方法来解决计算误差加速收敛等SOS分析和设计中会经常遇到的一些特殊问题。SOS法是采用数值求解的方法来求解不容易解析求解的非线性问题，类似于线性系统中的LMI法，相信会在非线性系统的分析和设计中有更广阔的应用前景。更多还原

【Abstract】 Recently more control systems are designed based on synthesis rather by usingthe analysis method, i.e., the controller is designed directly to meet thespecifications. For linear systems, this type of design is the well-known H_∞control.For nonlinear systems, besides the nonlinear the H_∞control, so-called sum ofsquares (SOS) method is really a synthesis method, it can give the requiredLyapunov function and nonlinear control law directly. There are already somesuccessful applications of these methods for ordinary systems. However, for somespecial classes of systems, they are still less discussed. For example, for the lightlydamped systems, the unstable systems and nonlinear systems, etc.. This includes thedetermination of the performance specifications, the design limititions and thetheoretical works needed to meet these particular design requirements.The contributions of this dissertation is to generalize the above metionedadvanced synthesis methods to some special classes of systems, to present practicaldesign procedures and in the meantime to develop some new theoretical results. Thisdissertation is organized as follows.The H_∞loop shaping method is studied with the control design of the lightlydamped flexible system. The influences of the lightly damped poles on the norm ofthe coprime factor perturbations and on the resulting robustness of the design aregiven. And it is pointed out that the H_∞-norm in the H_∞loop shaping design is justthe stability margin of the closed-loop system and is used to guarantee the robuststability of the system, though the H_∞-norm is traditionally regarded as theperformance index in the H_∞control theory.The H_∞state feedback control and H_∞output feedback control for the unstablesystems are studied and refined on an electromagnetic suspension system. For the H∞state feedback control, it is shown that the design is completed only when therobustness constraints from the Bode integral theorem are considered besidessolving the Riccati equations. For the output feedback design, the requirements forthe mid-frequency range are alway neglected in the ordinary H_∞design, however, itis different for the unstable systems. It is pointed out that the weighting functionsfor the output feedback control design must be determined by the mid-frequencyrange Bode integral constraint with H_∞optimization.For the nonlinear systems, nonlinear H_∞control inherits the design ideas thatfor the linear systems, and the difficulty is to solve the HJI (Hamilton-Jacobi-Issacs)inequalities implicitly. A Taylor series expansion based method is presented in the design of an affine nonlinear electromagnetic suspension system. This nonlinear H_∞design result is verified by using the Hamilton function.The recently emerged SOS method is a novel nonlinear control design method,which belongs to a kind of numerical methods for polynomials. The SOS method ispresented by estimation the region of attraction of the system. A generalizedS-method and the determination of decision variables for set-inclusion problems areproposed. The SOS control design of large attitude maneuvers for satellites is usedas an example. A diagonal dominant procedure for the SOS problem is proposed toreduce the numerical error and speedup its convergence. Similar to the LinearMatrix Inequalities (LMI) method, SOS method solves the difficult nonlinearproblem with numerical methods, and it may have more applications in thenonlinear control field in future.更多还原