By T P Leung, Hua Shu Qin

ISBN-10: 9810244428

ISBN-13: 9789810244422

Over the past 50 years or so, a couple of textbooks, monographs or even renowned books were released on nonlinear keep watch over idea and layout equipment. within the quarter of classical keep an eye on, for instance, there exist books excited by phase-plane research, describing functionality procedure, absolute balance and so forth. within the region of contemporary keep an eye on there are these relating to optimum keep an eye on, utilizing differential geometry and the differential algebra approach, variable structural keep watch over, H-infinite keep watch over and so forth. those books were valuable in selling the improvement of computerized regulate technological know-how and know-how. on account that 1990 there were many new effects and contributions within the quarter of nonlinear keep watch over. This ebook introduces these themes to readers. it may additionally gain automation engineers, researchers and students in comparable fields.

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**Example text**

Al holds when M(x)is non-singular. (The systems discussed in Maschke et. al 1998, etc. 11) i =0 Proof. Since A is integrable, we can find at each p a local coordinate chart, (x,z), such that A-spam—,i=l,---,dim(A)\. [dxt Note that under such a canonical j coordinate chart the structure matrix M can be expressed as r M{x) = Mx(x,z) 0 M2(x,z)) 0 It is easy to prove that the assumption Al is coordinate free. It implies that M 2 (x, z) = 0 . 11) follows immediately. • The coordinate frame, (x, z), in the above proof is said to be friend to A.

Then dim{Gj^) = n{n + \)l2 ;Particularly, dim(Sp(2n, R)) = n(2n +1) . The following theorem shows that for any N e Mn, GN is nontrivial. 5 For any nxn matrix N(n>l), words: GN is not discrete. In other dim(Gjy)>0. Proof. Decompose N as N =H +K where H is symmetric, HT = H , and K is skew-symmetric, KT =-K . We first assume H is singular. Then there exists a E, ^ 0 such that HE, = 0 . T Consider ££ : Case 1. &TK = 0 : Set X = %f. 4) imply that NX + XTN = 0 Case 2. 5) . 5) too. 5). That is, d i m ^ ) > 0.

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. 11 Stabilization of excitation control As an example of the generalized controlled Hamiltonian systems we consider the excitation system.

### Advanced Topics in Nonlinear Control Systems by T P Leung, Hua Shu Qin

by Christopher

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