非奇异终端滑模控制(NTSM)

非奇异终端滑模控制(Nonsingular Terminal Sliding Mode Contral，NTSM)

在终端滑膜控制中，最后的部分我们抛出了奇异性的问题，所以在此基础上，我们提出非奇异终端滑膜控制，非奇异终端滑膜控制主要解决了终端滑模控制中的奇异性问题。

非奇异终端滑模的滑模面

s=x1+1βx2pqs = x_1 + \frac{1}{\beta}x_2^{\frac {p} {q}} s=x1+β1x2qp

其中，β>0\beta>0β>0,p,q>0,(p>q)p,q>0,(p>q)p,q>0,(p>q) 为正奇数，且 1<pq<21 < \frac{p}{q} < 21<qp<2。

控制器设计

考虑二阶不确定非线性系统
{x˙1=x2x˙2=f(x)+g(x)u+d(x)\begin{cases} \dot x_1 = x_2 \\ \dot x_2 = f(x) + g(x)u + d(x) \end{cases} {x˙1=x2x˙2=f(x)+g(x)u+d(x)
其中x=[x1,x2]Tx = [x_1,x_2] ^ {T}x=[x1,x2]T，d(x)d(x)d(x) 代表不确定的外部干扰，且有 d(x)≤Dd(x) \le Dd(x)≤D，即干扰有上界

对于二阶系统我们可以将滑膜面设计为
s=x1+1βx2pqs = x_1 + \frac{1}{\beta} x_2 ^ {\frac {p} {q}} s=x1+β1x2qp
非奇异滑膜控制器设计为
u=−g−1(x)(f(x)+βqpx22−qp+(D+ε)sgn(x))u = -g^{-1}(x)(f(x)+\beta \frac{q}{p} x_2^{2 - \frac {q} {p}} + (D+\varepsilon)sgn(x)) u=−g−1(x)(f(x)+βpqx22−pq+(D+ε)sgn(x))
稳定性分析，设李雅普诺夫函数 V=12s2V = \frac {1} {2} s^2V=21s2，所以有 V˙=ss˙\dot V = s \dot sV˙=ss˙，将 uuu 带入可得
V˙=s(x˙1+β−1pqx2qp−1x˙2)=s(x2+β−1pqx2qp−1(f(x)+g(x)u+d(x)))=s(x2+β−1pqx2qp−1(f(x)+g(x)(−g−1(x)(f(x)+βqpx22−qp+(D+ε)sgn(x)))+d(x)))=s(x2+β−1pqx2qp−1(f(x)−(f(x)+βqpx2qp+(D+ε)sgn(x))+d(x)))=s(x2+β−1pqx2qp−1(−βqpx22−qp−(D+ε)sgn(x)+d(x)))=s(β−1pqx2qp−1(−(D+ε)sgn(x)+d(x))=β−1pqx2qp−1(−(D+ε)∣s∣+sd(x))≤β−1pqx2qp−1(−ε∣s∣)\begin{align} \dot V &= s(\dot x_1 + \beta^{-1} \frac {p} {q} x_2^{\frac {q} {p} - 1}{\dot x_2}) \\ &= s(x_2 + \beta^{-1} \frac {p} {q} x_2^{\frac {q} {p} - 1}(f(x) + g(x)u + d(x))) \\ &= s(x_2 + \beta^{-1} \frac {p} {q} x_2^{\frac {q} {p} - 1}(f(x) + g(x)(-g^{-1}(x)(f(x)+\beta \frac{q}{p} x_2^{2 - \frac {q} {p}} + (D+\varepsilon)sgn(x))) + d(x))) \\ &= s(x_2 + \beta^{-1} \frac {p} {q} x_2^{\frac {q} {p} - 1}(f(x) - (f(x)+\beta \frac{q}{p} x_2^{\frac {q} {p}} + (D+\varepsilon)sgn(x)) + d(x))) \\ &= s(x_2 + \beta^{-1} \frac {p} {q} x_2^{\frac {q} {p} - 1}(- \beta \frac{q}{p} x_2^{2-\frac {q} {p}} - (D+\varepsilon)sgn(x) + d(x))) \\ &= s(\beta^{-1} \frac {p} {q} x_2^{\frac {q} {p} - 1}(- (D+\varepsilon)sgn(x) + d(x)) \\ &= \beta^{-1} \frac {p} {q} x_2^{\frac {q} {p} - 1}(- (D+\varepsilon)|s| + sd(x)) \\ &\le \beta^{-1} \frac {p} {q} x_2^{\frac {q} {p} - 1} (-\varepsilon |s|) \\ \end{align} V˙=s(x˙1+β−1qpx2pq−1x˙2)=s(x2+β−1qpx2pq−1(f(x)+g(x)u+d(x)))=s(x2+β−1qpx2pq−1(f(x)+g(x)(−g−1(x)(f(x)+βpqx22−pq+(D+ε)sgn(x)))+d(x)))=s(x2+β−1qpx2pq−1(f(x)−(f(x)+βpqx2pq+(D+ε)sgn(x))+d(x)))=s(x2+β−1qpx2pq−1(−βpqx22−pq−(D+ε)sgn(x)+d(x)))=s(β−1qpx2pq−1(−(D+ε)sgn(x)+d(x))=β−1qpx2pq−1(−(D+ε)∣s∣+sd(x))≤β−1qpx2pq−1(−ε∣s∣)
根据上述分析可知 x2≠0x_2 \ne 0x2=0 时，满足李雅普诺夫稳定条件。接下来需要分析的就是在之前的奇异点的问题，我们发现，将控制量带入到系统模型方程可得
x˙2=−βqpx22−pq+d(x)−(D+ε)sgn(s)\dot x_2 = -\beta \frac{q}{p}x_2^{2-\frac{p}{q}} + d(x) - (D+\varepsilon)sgn(s) x˙2=−βpqx22−qp+d(x)−(D+ε)sgn(s)
当 x2=0x_2 = 0x2=0 有：
x˙2=d(x)−(D+ε)sgn(s)\dot x_2 = d(x) - (D+\varepsilon)sgn(s) x˙2=d(x)−(D+ε)sgn(s)
分析可知，此时当 s>0s>0s>0 则 x2x_2x2 快速减小，此时当 s<0s<0s<0 则 x2x_2x2 快速增加。所以可以在有限时间内实现 s=0s=0s=0。

收敛速度问题

该方法解决了奇异性问题，但是收敛速度问题并没有解决，在接近滑模面时的收敛速度仍然较慢。所以在此基础上提出了非奇异快速终端滑膜控制