Goal: Design a controller K(s so that the system has some desired characteristics. Typical objectives Stabilize the system( Stabilization) Regulate the system about some design point(Regulation Follow a given class of command signals(Tracking) Reduce the response to disturbances(Disturbance Rejection Typically think of closed-loop control > so we would analyze the

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properties of finite dement soltions Noded point equilibrio At a node, the sow of he elemeat nodal forces s in equilibriun with Hhe externed loads

Concept Question How many quadrature points do you need to integrate exactly the stiffness matrix of a lD finite element with quadratic interpolation for the displacements? gauss

cmaulenon o isoparametRic elemets (Bathes book Consider the uadriltersd denat shown in the 8oc nodd coordinates

Numerical integration Consider He 4-D integral =+(s 1 Seek n-point apploxiwisfions G~2M =1 are the weights and

Finite element wodel of a beam(Euler-Bernollia- governing eQuations

The correct global local node mapping for the quadraticelement mesh in the figure is

The reaction on the left end is not exact because 1. The order of interpolation is too low, a higher order of interpolation would give the right reaction 2. The distributed load attributed to node one does not ke it into the solution

The finite element melod I In FEM I We derived basis functions of arbitrary order for Hhe rod Model