線性系統理論4 -化輸入-輸出描述爲狀態空間描述

原創 tanga_cc 2019-01-12 16:10

由輸入輸出描述導出狀態空間描述

對於單輸入，單輸出線性時不變系統,其微分方程描述

{y^{(n)}} + {a_{n - 1}}{y^{(n - 1)}} + \cdots + {a_1}{y^{(1)}} + {a_0}y = {b_m}{u^{(m)}} + {b_{m - 1}}{u^{(m - 1)}} + \cdots + {b_1}{u^{(1)}} + {b_0}u

其傳遞函數描述

g(s) = \frac{{Y(s)}}{{U(s)}} = \frac{{{b_m}{s^m} + {b_{m - 1}}{s^{m - 1}} + \cdots + {b_1}{s^1} + {b_0}}}{{{s^n} + {a_{n - 1}}{s^{n - 1}} + \cdots + {a_1}s + {a_0}}}

可以導出其狀態空間描述爲

\begin{array}{*{20}{c}} {\dot x = Ax + bu\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; ;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \\ \begin{array}{l} y = cx + du \\ \begin{array}{*{20}{c}} {x \in {R^n}} & {A \in {R^{n \times n}}} & {b \in {R^{n \times 1}}} & {c \in {R^{1 \times n}}} & {d \in {R^{1 \times 1}}} \\ \end{array} \\ \end{array} \\ \end{array}

結論1 :給定單輸入,單輸出線性時不變系統的輸入輸出描述

  • m=n，即系統爲真情形

\begin{array}{l} \dot X = \left[ {\begin{array}{*{20}{c}} 0 & 1 & 0 & \cdots & 0 \\ \vdots & {} & \ddots & {} & \vdots \\ 0 & {} & {} & \ddots & 0 \\ 0 & 0 & 0 & \cdots & 1 \\ { - {a_0}} & { - {a_1}} & { - {a_2}} & \cdots & { - {a_{n - 1}}} \\ \end{array}} \right]x + \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \\ \end{array}} \right]u \\ \\ y = \left[ {\begin{array}{*{20}{c}} {({b_0} - {b_n}{a_0}),} & {({b_1} - {b_n}{a_1}),} & { \cdots ,} & {({b_{n - 1}} - {b_n}{a_{n - 1}})} \\ \end{array}} \right]x + {b_n}u \\ \end{array}

\begin{array}{l} \left[ {\begin{array}{*{20}{c}} {{{\dot x}_1}} \\ {{{\dot x}_2}} \\ \vdots \\ \begin{array}{l} {{\dot x}_{n - 1}} \\ {{\dot x}_n} \\ \end{array} \\ \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ {} & {} & \cdots & {} & {} \\ 0 & 0 & 0 & \cdots & 1 \\ { - {a_0}} & { - {a_1}} & { - {a_2}} & \cdots & { - {a_{n - 1}}} \\ \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}} \\ {{x_2}} \\ \vdots \\ \begin{array}{l} {x_{n - 1}} \\ {x_n} \\ \end{array} \\ \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \\ \end{array}} \right]u \\ y = \left[ {\begin{array}{*{20}{c}} {{b_0} - {a_0}{b_n}} & {{b_1} - {a_1}{b_n}} & \cdots & {{b_{n - 2}} - {a_{n - 2}}{b_n}} & {{b_{n - 1}} - {a_{n - 1}}{b_n}} \\ \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}} \\ {{x_2}} \\ \vdots \\ \begin{array}{l} {x_{n - 1}} \\ {x_n} \\ \end{array} \\ \end{array}} \right] + {b_n}u \\ \end{array}

  • m<n，即系統爲嚴真情形

\begin{array}{l} \dot X = \left[ {\begin{array}{*{20}{c}} 0 & 1 & 0 & \cdots & 0 \\ \vdots & {} & \ddots & {} & \vdots \\ 0 & {} & {} & \ddots & 0 \\ 0 & 0 & 0 & \cdots & 1 \\ { - {a_0}} & { - {a_1}} & { - {a_2}} & \cdots & { - {a_{n - 1}}} \\ \end{array}} \right]x + \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \\ \end{array}} \right]u \\ \\ y = \left[ {\begin{array}{*{20}{c}} {{b_0}} & {{b_1}} & \cdots & {{b_m}} & 0 & \cdots & 0 \\ \end{array}} \right]x \\ \end{array}

  • 例1：給定的系統的輸入-輸出描述爲：m<n

                   {y^{(3)}} + {16}{y^{(2)}} + {194}{y^{(1)}} + {640}y = {160}{u^{(1)}} + {720}u

則系統的狀態空間描述爲

                   \begin{array}{l} \left[ {\begin{array}{*{20}{c}} {{{\dot x}_1}} \\ {{{\dot x}_2}} \\ \begin{array}{l} {{\dot x}_{3}} \\ \end{array} \\ \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0 & 1 & 0 \\ 0 & 0 & 1 \\ { - 640} & { - 194} & { - 16} \\ \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}} \\ {{x_2}} \\ \begin{array}{l} {x_{3}} \\ \end{array} \\ \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \\1 \\ \end{array}} \right]u \\ y = \left[ {\begin{array}{*{20}{c}} 720 & 160 & 0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}} \\ {{x_2}} \\ \begin{array}{l} {x_{3}} \\ \end{array} \\ \end{array}} \right] \\ \end{array}

  • 例2：給定的系統的輸入-輸出描述爲：m=n

            {y^{(3)}} + {16}{y^{(2)}} + {194}{y^{(1)}} + {640}y = {4}{u^{(3)}} +{160}{u^{(1)}} + {720}u

則系統的狀態空間描述爲

                   \begin{array}{*{20}{c}} {\left[ {\begin{array}{*{20}{c}} {{{\dot x}_1}} \\ {{{\dot x}_2}} \\ {\begin{array}{*{20}{c}} {{{\dot x}_3}} \\ \end{array}} \\ \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0 & 1 & 0 \\ 0 & 0 & 1 \\ { - 640} & { - 194} & { - 16} \\ \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}} \\ {{x_2}} \\ {\begin{array}{*{20}{c}} {{x_3}} \\ \end{array}} \\ \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \\ 1 \\ \end{array}} \right]u} \\ {y = \left[ {\begin{array}{*{20}{c}} { - 1840} & { - 616} & { - 64} \\ \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}} \\ {{x_2}} \\ {\begin{array}{*{20}{c}} {{x_3}} \\ \end{array}} \\ \end{array}} \right] + 4u} \\ \end{array}

結論2: 給定單輸入、單輸出線性時不變系統的輸入輸出描述，其對應的狀態空間描述可按如下兩類情況導出

  • m=0情形

\begin{array}{l} \dot x = \left[ {\begin{array}{*{20}{c}} 0 & 1 & 0 & \cdots & 0 \\ \vdots & {} & \ddots & {} & \vdots \\ 0 & {} & {} & \ddots & 0 \\ 0 & 0 & 0 & \cdots & 1 \\ { - {a_0}} & { - {a_1}} & { - {a_2}} & \cdots & { - {a_{n - 1}}} \\ \end{array}} \right]x + \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \\ \vdots \\ 0 \\ {{b_0}} \\ \end{array}} \right]u \\ y = \left[ {\begin{array}{*{20}{c}} {1,} & {0,} & { \cdots ,} & 0 \\ \end{array}} \right]x \\ \end{array}

  • m≠0情形

\begin{array}{l} \dot x = \left[ {\begin{array}{*{20}{c}} 0 & 1 & 0 & \cdots & 0 \\ \vdots & {} & \ddots & {} & \vdots \\ 0 & {} & {} & \ddots & 0 \\ 0 & 0 & 0 & \cdots & 1 \\ { - {a_0}} & { - {a_1}} & { - {a_2}} & \cdots & { - {a_{n - 1}}} \\ \end{array}} \right]x + \left[ {\begin{array}{*{20}{c}} {{\beta _1}} \\ {{\beta _2}} \\ \vdots \\ {{\beta _{n - 1}}} \\ {{\beta _n}} \\ \end{array}} \right]u \\ \\ \begin{array}{*{20}{c}} {} & {} \\ \end{array}y = \left[ {\begin{array}{*{20}{c}} {1,} & {0,} & { \cdots ,} & 0 \\ \end{array}} \right]x + {\beta _0}u \\ \end{array}

\begin{array}{l} {\beta _0} = {b_n} \\ {\beta _1} = {b_{n - 1}} - {a_{n - 1}}{\beta _0} \\ {\beta _2} = {b_{n - 2}} - {a_{n - 2}}{\beta _1} - {a_{n - 1}}{\beta _0} \\ \begin{array}{*{20}{c}} {} & {} & {} \\ \end{array} \vdots \\ {\beta _n} = {b_0} - {a_{n - 1}}{\beta _{n - 1}} - {a_{n - 2}}{\beta _{n - 2}} - \cdots - {a_1}{\beta _1} - {a_0}{\beta _0} \\ \end{array}

結論3 :給定單輸入單輸出線性時不變系統的傳遞函數描述爲：

g(s) = \frac{{{b_m}{s^m} + {b_{m - 1}}{s^{m - 1}} + \cdots + {b_1}s + {b_0}}}{{{s^n} + {a_{n - 1}}{s^{n - 1}} + \cdots + {a_1}s + {a_0}}}

其極點即分母方程的根 {\lambda _1},{\lambda _2}, \cdots ,{\lambda _n}爲兩兩互異實數，則對應的狀態空間描述可按如下兩類情形導出：

  • m<n，即系統爲嚴真情形

              \begin{array}{l} g(s) = \frac{{{k_1}}}{{s - {\lambda _1}}} + \frac{{{k_2}}}{{s - {\lambda _2}}} + \cdots + \frac{{{k_n}}}{{s - {\lambda _n}}} \\ {k_i} = \mathop {\lim }\limits_{s \to \infty } g(s)(s - {\lambda _i}),\begin{array}{*{20}{c}} {} & {} \\ \end{array}i = 1,2, \cdots ,n \\ \end{array}

對應的狀態空間描述爲

       \begin{array}{l} \dot x = \left[ {\begin{array}{*{20}{c}} {{\lambda _1}} & {} & {} & {} \\ {} & {{\lambda _2}} & {} & {} \\ {} & {} & \ddots & {} \\ {} & {} & {} & {{\lambda _n}} \\ \end{array}} \right]x + \left[ {\begin{array}{*{20}{c}} {{k_1}} \\ {{k_2}} \\ \vdots \\ {{k_n}} \\ \end{array}} \right]u \\ y = \left[ {\begin{array}{*{20}{c}} {1,} & {1,} & { \cdots ,} & 1 \\ \end{array}} \right]x \\ \end{array}

  • m=n，即系統爲真情形

​​​​​​​            \begin{array}{l} g(s) = \frac{{{b_n}{s^n} + {b_{n - 1}}{s^{n - 1}} + \cdots + {b_1}s + {b_0}}}{{{s^n} + {a_{n - 1}}{s^{n - 1}} + \cdots + {a_1}s + {a_0}}} = {b_n} + \bar g(s) \\ g(s) = \frac{{\left( {{b_{n - 1}} - {b_n}{a_{n - 1}}} \right){s^{n - 1}} + \cdots + \left( {{b_0} - {b_n}{a_0}} \right)}}{{{s^n} + {a_{n - 1}}{s^{n - 1}} + \cdots + {a_1}s + {a_0}}} \\ {{\bar k}_i} = \mathop {\lim }\limits_{s \to \infty } \bar g(s)(s - {\lambda _i}),\begin{array}{*{20}{c}} {} & {} \\ \end{array}i = 1,2, \cdots ,n \\ \end{array}

對應的狀態空間描述爲

          \begin{array}{l} \dot x = \left[ {\begin{array}{*{20}{c}} {{\lambda _1}} & {} & {} & {} \\ {} & {{\lambda _2}} & {} & {} \\ {} & {} & \ddots & {} \\ {} & {} & {} & {{\lambda _n}} \\ \end{array}} \right]x + \left[ {\begin{array}{*{20}{c}} {{{\bar k}_1}} \\ {{{\bar k}_2}} \\ \vdots \\ {{{\bar k}_n}} \\ \end{array}} \right]u \\ y = \left[ {\begin{array}{*{20}{c}} {1,} & {1,} & { \cdots ,} & 1 \\ \end{array}} \right]x + {b_n}u \\ \end{array}

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