基于ELM的切换非线性动态系统神经网络控制

贵州大学硕士学位论文

注4.7：四次形式李雅普洛夫函数能够轻松地简化引理1中的单项式

1?T?2V?，所以文献[7-8]中选用四次形式李雅普洛夫函数而不是二次形Tr?gg?2??x2?式李雅普洛夫函数。为此，式(4.32)中也选用了四次形式李雅普洛夫函数单项式

1T?1hk21n4式(4.32)中使用单项式?k?k?k和单项式抵消被backstepping技zi。?24i?12?k术引入的多余项。为了在计算中消去时滞项，把单项式入李雅普洛夫函数(4.32)中。

11?d?tt?d(t)?(x,x(s))ds引

根据定理2.2，李雅普洛夫函数(4.32)沿着切换系统(4.29)的第k个子系统的轨迹的无穷小算子LVk应为

n?1????(n?1)ku?f(x)?m(x,x(t?d(t)))?f(x)?m(x,x(t?d(t)))???nknnlkllkll?knkn??xl?1l?3?LVk?zn?n?1? n?1?2???1(n?1)k(n?1)k???x?nek(xe,xe(t?d(t)))nT??l?1fk(xf,xf(t?d(t)))?l?1?xl?2e,f?1?xe?xf??i?1????(i?1)kx?f(x)?m(x,x(t?d(t)))?f(x)?m(x,x(t?d(t)))?lkllkll???ikii?i?1ikin?1?xl?1l?? ??zi3?i?1? i?1?2???1i?2(i?1)k(i?1)k???x?nek(xe,xe(t?d(t)))nT??l?1fk(xf,xf(t?d(t)))?l?1?xl?2e,f?1?xe?xf???y3?x2?f1k(y)?m1k(y,y(t?d(t)))??32yn1k(y,y(t?d(t)))n1Tk(y,y(t?d(t))) 2Ti?1??i?1?????3n2?(i?1)k(i?1)k??zi?nik(xi,xi(t?d(t)))??nlk(xl,xl(t?d(t)))??nik(xi,xi(t?d(t)))??nlk(xl,xl(t?d(t)))? 2i?2?l?1?xll?1?xl????11?d?(t)? (4.33) ?T??1??hkh?（x,x）-?(x,x(t?d(t)))??kkk1?d1?d?kk注4.8：不同于4.2节，为了把ELM算法更方便地运用到伪神经控制机制中，

1?1?k沿着闭环系统(4.29)的第k个子系统的轨迹的无穷李雅普洛夫单项式?kT?k2T?1??T??1?而非?T??1???T?1小算子计算为?kkkkkk。根据矩阵知识，实际上?k?k?k与?k?k?k是

相等的。

46

贵州大学硕士学位论文

利用坐标变换(4.30)，式(4.33)变为

n?1??i?1??n?1??n?13??3(n?1)k(i?1)kLVk?z?uk??xl?1???zi??ik??xl?1??y?1k??zi3zi?1

?xl?xll?1l?1i?1??i?2??3nnni?1??z?fik(xi)?mik(xi,xi(t?d(t)))???z3ii?1i?23i?l?1??(i?1)k?xl?flk(xl)?mlk(xl,xl(t?d(t)))?

1n3i?1??(i?1)k??zi?nek(xe,xe(t?d(t)))nTfk(xf,xf(t?d(t))) 2i?2e,f?1?xe?xfi?1??i?1?????3n2?(i?1)k(i?1)k??zi?nik(xi,xi(t?d(t)))??nlk(xl,xl(t?d(t)))??nik(xi,xi(t?d(t)))??nlk(xl,xl(t?d(t)))? 2i?1?l?1?xll?1?xl???T2 ?11?d?(t)? (4.34) ?T??1??hkh?（x,x）-?(x,x(t?d(t)))??kkk1?d1?d?kk根据引理2.2(Young不等式),可以获得如下不等式去简化上述等式(4.34)。

?zzi?1n3in?13ii?14nz3n?1414???k31izi??4i (4.35) 4i?14i?2?k1(i?1)?z?fi?1ik(xi)?mik(xi,xi(t?d(t)))?

4?41n13n?41n14433????k2i??k3i?zi??4fik(xi)??4mik(xi,xi(t?d(t)))(4.36) 4i?1?4i?1?k2i4i?1?k3i???zi?2n3i?l?1i?1??(i?1)k?xl?flk(xl)?mlk(xl,xl(t?d(t)))?

4n?3n?4114????k34i??k35i?zi??44i?2?4i?2?k4i??l?1i?1??(i?1)k?xl44flk(xl)

?11?4i?2?k45in?l?1i?1??(i?1)k?xlmlk(xl,xl(t?d(t))) (4.37)

1n3i?1??(i?1)k??zi?nek(xe,xe(t?d(t)))nTfk(xf,xf(t?d(t))) 2i?2e,f?1?xe?xf??2?(i?1)k126 ???k6izi???4i?2e,f?1??xe?xfni?12? ???247