The Center Manifold Theorem. The Center Manifold Theorem. - Motivation. Step 1 : . . Step 2 : . k. m. k. m. Lower Dimensional part. Question : How do we isolate this lower dimensional part ?. ( times continuously differentiable). k. m. Lower Dimensional part (Continued).
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Theorem (Center Manifold Theorem)
Consider the system defined by .
Step 4 : Solve for h and evaluate the stability of the reduced system.
Remark : One does not need an exact solution of the P.D.E. in order to perform
Step 1 :
Step 2 :
Step 3 :
Too hard to solve, in general, so seek an approximated solution.
Plug this into the reduced order equation
Then the stability of the above system
In general, one would seek to add more terms to h(y), i.e.,
How can this tuning be generalized for nonlinear systems ?
normed linear space
H may be unstable, so y(t) might not have the same norm.
A few examples of gains
1) H: linear time invariant system described by G(s).
2) H: static nonlinearity in the sector [a, b]
Proof : See Nonlinear systems : vol. I