16.322 Stochastic Estimation and Control, Fall 2004 Prof vander Velde We started with a closed-loop configuration C(s) F(s) B(s) H(s) 1+C(SF(SB(s) 1-H(SF(SB(s) The loop will be stable, but C(s)may be unstable Special comments about the application of these formulae: a) Unstable F(s) cannot be treated because the Fourier transform of w(t) does not converge in that case. To treat this system, first close a feedback loop around F(s) to create a stable"fixed"part and work with this stable feedback system as F(s). When the optimum compensation is found, it can be collected with the original compensation if desired b)An F(s) which has poles on the jo axis is the limiting case of functions for which the Fourier transform converges. You can move the poles just into the LHP by adding a real part +e to the pole locations. Solve the problem with this 8 and at the end set it to zero Zeroes of F(s)on j@ axis can be included in either factor and the result will be the same. This will permit cancellation compensation of poles of F(s)on the jo axis, including poles at the origin c)In factoring Si(s)into S (S)LS(S)R, any constant factor in Si(s) can be divided between S, (S) and S (s)e in any convenient way. The same true of F(s)and F(s) d)Problems should be well-posed in the first place. Avoid combinations of D(s)and S(o)which imply infinite d(t) because that may assume infinite e- for any realizable filter. Such as a differentiator on a sign which falls off as The point at (=0 was left hanging in several steps of the derivation of the solution formula. Don t bother checking the individual steps; just check the final solution to see if it satisfies the necessary conditions Page 3 of 716.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 3 of 7 We started with a closed-loop configuration: ” ( ) ( ) 1 () () () ( ) ( ) 1 () () () C s H s CsFsBs H s C s H sFsBs = + = − The loop will be stable, but C s( ) may be unstable. Special comments about the application of these formulae: a) Unstable F s( ) cannot be treated because the Fourier transform of ( ) w t F does not converge in that case. To treat this system, first close a feedback loop around F s( ) to create a stable “fixed” part and work with this stable feedback system as F s( ) . When the optimum compensation is found, it can be collected with the original compensation if desired. b) An F s( ) which has poles on the jω axis is the limiting case of functions for which the Fourier transform converges. You can move the poles just into the LHP by adding a real part +ε to the pole locations. Solve the problem with this ε and at the end set it to zero. Zeroes of F s( ) on jω axis can be included in either factor and the result will be the same. This will permit cancellation compensation of poles of F s( ) on the jω axis, including poles at the origin. c) In factoring ( ) ii S s into () () ii L ii R SsSs , any constant factor in ( ) ii S s can be divided between ( ) ii L S s and ( ) ii R S s in any convenient way. The same is true of F s( ) and F s ( ) − . d) Problems should be well-posed in the first place. Avoid combinations of D s( ) and ( ) ss S ω which imply infinite 2 d t( ) because that may assume infinite 2 e for any realizable filter. Such as a differentiator on a signal which falls off as 2 1 ω . e) The point at t = 0 was left hanging in several steps of the derivation of the solution formula. Don’t bother checking the individual steps; just check the final solution to see if it satisfies the necessary conditions