线性离散系统Delta域控制器设计

Controller design for linear discrete-time system in delta-domain

针对z域内离散化方法无法满足线性系统快速采样时的性能指标,给出了delta域内离散化系统数字比例-积分-微分(PID)控制器设计方法。利用广义Kalman-Yakubovic-Popov(GKYP)引理将delta域按系统频率和采样周期的乘积进行划分,依据模型匹配原则将PID控制器的设计问题转化为求相应delta域区域内H∞范数构成的不等式最优解问题。进一步地,将问题转化为与系统状态空间各参数矩阵相关的线性矩阵不等式(LMI)求可行解问题。最后,通过数值例子证明,该方法提高了控制系统对采样周期取值的鲁棒性,满足预期的性能指标,且能保证系统的稳定性和最小相位特性。

In view of that the z-domain discrete-time controller can not satisfy the performance requirements of the linear system with fast-sampling,a new method is presented to design the digital proportion-integration-differentiation（ PID） controller for the delta-domain discrete-time system. By using the generalized Kalman-Yakubovic-Popov（ GKYP） lemma,the delta-domain is correspondingly divided into certain appropriate areas according to the product scopes of the frequency and sampling period. Based on the principle of approximate model matching,the design of PID controller in the fast-sampling system is converted into solving the optimization problem of H∞norm with the form of inequality for the restricted areas. Furthermore,the inequalities described in the form of coefficient matrices are transformed to solve the linear matrix inequality（ LMI） in the state space realization. Finally,a numerical simulation shows that the proposed method can enhance the robustness to the sample period values,achieve the predictive performance index and guarantee the stability of the systemas well as the minimum phase characteristic.