C'est la Vie – comme ci comme ça

January 31, 2012

(SSI-02)动力系统的状态空间模型

Filed under: Study and Work — hitzws @ 5:57 PM

线性系统的状态空间描述是建立在状态和状态空间概念的基础上的。完全地表征系统时间域行为的一个最小内部变量组称为动力学系统的状态,组成这个变量组的变量称为系统的状态变量,由状态变量组成列向量称为系统的状态向量,其取值的向量空间称为状态空间。
现假设状态向量为:
Z(t)=\left( \begin{array}{rcl} X(t)  \\ \dot{X}(t) \end{array} \right)(5)
可以将方程(1)与恒等式\dot{X}=I\dot{X} 一起可以转换为连续时间的状态空间方程:
\left\{ \begin{array}{rcl}\dot{Z}(t)&=&A_cZ(t)+B_cF(t)\\Y(t)&=&C_cZ(t)+D_cF(t) \end{array} \right.(6)

式中A_c=\left( \begin{array}{cc} 0 & I\\-M^{-1}K & -M^{-1}C \end{array} \right)B_c=\left( \begin{array}{cc} 0 \\M^{-1} \end{array} \right)
对动力系统(6)以1/\tau的采样频率采样,可得到以下离散的系统状态空间模型:
\left\{ \begin{array}{rcl} Z_{k+1} & = & AZ_k + BF_k+w_k  \\ Y_k & = & CZ_k+DF_k+v_k \end{array} \right.(7)
该系统的状态和输出分别为:
Z_k=\left( \begin{array}{rcl} X(k \tau)  \\ \dot{X}(k \tau) \end{array} \right)~,~~ Y_k=Y(k \tau)(8)
式(7)中的离散状态矩阵A、离散输入矩阵B以及状态观测矩阵C,如下式:
A=e^{A_c \tau}, B=(A-I)A_c^{-1}B_c(9)
\lambda, \phi_\lambda为状态矩阵F的特征值和特征向量。这样,式(2)所定义的系统模态参数可以从上述两个特征结构中计算得到,如下式所示:
e^{\tau\mu}=\lambda~,~~ L \psi_\mu=\varphi_\lambda = H~\phi_\lambda(10)
系统的频率和阻尼比可由离散的特征值\lambda得到:
频率=\dfrac{a}{2\pi\tau},阻尼=\dfrac{\left|b\right|}{\sqrt{a^2+b^2}}(11)
式中a=\left|\mbox{arctan}\left[\Im(\lambda)/\Re(\lambda)\right]\right|b=\mbox{ln}\lambda

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