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本文参考：
B站：DR_CAN
Dr. CAN学习笔记-自动控制原理Ch1-8Lag Compensator滞后补偿器
从稳态误差入手（steady state Error）
误差 Error ： E ( s ) = R ( s ) − X ( s ) = R ( s ) − E ( s ) ⋅ K G ( s ) ⇒ E ( s ) ( 1 + K G ( s ) ) = R ( s ) ⇒ E ( s ) = 1 1 + K G ( s ) R ( s ) = R ( s ) 1 1 + K N ( s ) D ( s ) = 1 s 1 1 + K N ( s ) D ( s ) E\left( s \right) =R\left( s \right) -X\left( s \right) =R\left( s \right) -E\left( s \right) \cdot KG\left( s \right) \Rightarrow E\left( s \right) \left( 1+KG\left( s \right) \right) =R\left( s \right) \Rightarrow E\left( s \right) =\frac{1}{1+KG\left( s \right)}R\left( s \right) =R\left( s \right) \frac{1}{1+K\frac{N\left( s \right)}{D\left( s \right)}}=\frac{1}{s}\frac{1}{1+K\frac{N\left( s \right)}{D\left( s \right)}} E(s)=R(s)−X(s)=R(s)−E(s)⋅KG(s)⇒E(s)(1+KG(s))=R(s)⇒E(s)=1+KG(s)1R(s)=R(s)1+KD(s)N(s)1=s11+KD(s)N(s)1
单位阶跃unit step ： R ( s ) = 1 s R\left( s \right) =\frac{1}{s} R(s)=s1
稳态误差Steady State Error——FVT终值定理
e s s = lim t → ∞ e ( t ) = lim s → o s E ( s ) = lim s → o s ⋅ 1 s 1 1 + K N ( s ) D ( s ) = 1 1 + K N ( 0 ) D ( 0 ) = D ( 0 ) D ( 0 ) + K N ( 0 ) ess=\underset{t\rightarrow \infty}{\lim}e\left( t \right) =\underset{s\rightarrow o}{\lim}sE\left( s \right) =\underset{s\rightarrow o}{\lim}s\cdot \frac{1}{s}\frac{1}{1+K\frac{N\left( s \right)}{D\left( s \right)}}=\frac{1}{1+K\frac{N\left( 0 \right)}{D\left( 0 \right)}}=\frac{D\left( 0 \right)}{D\left( 0 \right) +KN\left( 0 \right)} ess=t→∞lime(t)=s→olimsE(s)=s→olims⋅s11+KD(s)N(s)1=1+KD(0)N(0)1=D(0)+KN(0)D(0)
