Study on H_2/H_* Optimal Control Using Descriptor Riccati Equation

研究一类广义系统的H_2/H_*最优控制和滤波问题。以下是过去两年取得的成果1.对于离散广义系统，我们提出了一种新的求解H_2控制问题的算法和相关的谱分解。基于广义特征问题(GEP)方法，我们推导了一种计算最优控制和谱因子的数值方法。推广了上述结果，给出了离散广义系统的(J，J‘)-谱分解算法。对于连续时间广义系统，我们提出了一个新的广义Riccati方程来解决LQ调节器问题。并在GEP.4的基础上开发了一个数值算法。给出了连续时间广义系统的谱分解和J-谱分解算法。基于广义系统的J-谱分解和Youla参数化，我们发展了一种连续时间广义系统H_*控制的求解方法。目前，我们仍在沿着这条线广泛研究H_*控制问题。应用模型匹配技术，给出了所有稳定的无偏H_*估计的参数化形式。给出了加权H_*误差性能下的全阶估计器的设计方法。虽然H_*估值器的结果是针对状态空间模型得到的，但我们将能够将这些结果推广到广义系统的H_*估计。

This research is concerned with the H_2/H_* optimal control and filtering problems for a descriptor system. The following are the results obtained for the past two years.1. For a discrete-time descriptor system, we have developed a new algorithm for solving the H_2 control problem and a related spectral factorization. We have derived a numerical method for computing optimal controls and spectral factors based on the generalized eigenproblem (GEP) approach.2. Extending the above result, we also developed a (J, J')-spectral factorization algorithm for a discrete-time descriptor system.3. For a continuous-time descriptor system, we have presented a new generalized Riccati equation to solve the LQ regulator problem. A numerical algorithm is also developed based on the GEP.4. We have developed spectral and J-spectral factorization algorithms for a continuous-time descriptor systems.5. Based on the J-spectral factorization and a Youla parametrization for descriptor systems, we have developed a solution method for H_* control for continuous-time descriptor systems. At present, we are still extensively studying H_* control problem along this line.6. Applying model matching technique, the parametrization of all stable unbiased H_* estimators is derived. Also, we have presented a design method of full-order estimator under the weighted H_* error performance. Although the results for H_* estimators are obtained for state-space models, we will be able to extend these results to the H_* estimation for descriptor systems.