By T P Leung, Hua Shu Qin

During the last 50 years or so, a few textbooks, monographs or even renowned books were released on nonlinear keep watch over concept and layout tools. within the region of classical keep an eye on, for instance, there exist books all in favour of phase-plane research, describing functionality technique, absolute balance and so forth. within the zone of contemporary keep watch over there are these regarding optimum keep watch over, utilizing differential geometry and the differential algebra procedure, variable structural keep an eye on, H-infinite keep an eye on etc. those books were helpful in selling the advance of automated keep watch over technology and expertise. considering 1990 there were many new effects and contributions within the quarter of nonlinear keep watch over. This ebook introduces these subject matters to readers. it may additionally profit automation engineers, researchers and students in comparable fields.

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10) holds iff JeMJTe = M . 12) 7 M$ We consider the following example. 14) 36 D. 13). • Before end, we leave searching structure-preserving algorithm for further study. But we give some examples to show that some algorithms preserving symplectic structure (Feng et. al 1994, Wang 1994) also preserves other structures. 7 Assume the system x = XH has an invariant structure determined by N. 14) for any N. 1. 15) where s is the step length. Then it preserves any structure. 16) follows immediately. 2.

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2. A straightforward computation shows that GN is a subgroup of GL{n,R) . It is obvi'jus that GN is a topologically closed. So it is a closed subgroup of GL(n, R). Hence it is a Lie subgroup (Varadarajan 1984). 3. We already know that gN is a sub-algebra of gl(n,R). So we have only to show that X e gN iff E = e x Since e etx sGNyteR. e GN . e GN , X e g{GN) , which is the Lie algebra of GN . Therefore That means (e'x)TNetX = etxT NetX =N,teR Differentiating both sides with respect to t yields XTe'xT NetX + e'xT Ne'X X = 0 Setting t = 0, we have XTN + NX = 0, which means X e gN.