How to deal with Disturbances Modified Control design Characterize disturbances as filt ers driven Process model (pulse) Ay=B(u+u) Sequences of im pulses(pulses) Disturbance model White noise The only thing that matters is the poles of the filter! think about t his as the H nce disturbance annihil ator which wipes out as much as possible of the dist ur bance BI BR AR+ Ad(ar+ bs Adv=e BS AR+Bs Ad (ar+ bs) ●H。 w will t he dist urbance influence t he system The effect of t he dist urbances can be reduced How s hould the co ntrol system be mo by requiring that Ad is a factor of r The co nt roller s hould co nt ain a model of the the Internal Model principle dist urbances(Internal Model Principle) Details Find a cont roller which gives a specified closed op charact eristic poly nomial such that Ad is ample Integral acti a factor of r Step dist urbance Let ro and so be a solut ion to ARo+BS=A Same Ad for any piece-wise const ant sig nal Poly nomials Choose X=g+zo. the R=XRo+YB (q-1)R'=(q +zo)R+yo B then satisfies Hence AR+ BS=XA Determine ro, so to give A0 as before. Choose The new co nt roller is given by a stable polyno mial X with deg X which represents the addit io nal dynamics R=(q+co)Ro+ yo B equired to deal wit h dist ur bances. Determine (q+so)S-yo A e and x so th R=Adr=Xro+YB C K J. Ast ro m and B.WittenmarkHow to Deal with Disturbances  Characterize disturbances as lters driven by { An impulse (pulse) { Sequences of impulses (pulses) { White noise The only thing that matters is the poles of the lter! Think about this as the disturbance annihilator which wipes out as much as possible of the disturbance! Adv = e  How will the disturbance in uence the system?  How should the control system be modi- ed?  The Internal Model Principle Modied Control Design Process model Ay = B(u + v) Disturbance model Adv = e Hence y = BT AR + BS uc + BR Ad (AR + BS) e u = AT AR + BS uc ￾ BS Ad (AR + BS) e The eect of the disturbances can be reduced by requiring that Ad is a factor of R!! The controller should contain a model of the disturbances (Internal Model Principle) Details Find a controller which gives a specied closed loop characteristic polynomial such that Ad is a factor of R Let R0 and S0 be a solution to AR0 + BS0 = A0 c Polynomials R = XR0 + Y B S = XS0 ￾ Y A then satises AR + BS = XA0 c Determine R0 , S0 to give A0 c as before. Choose a stable polynomial X with deg X = deg Ad which represents the additional dynamics required to deal with disturbances. Determine R0 and X so that R = AdR0 = XR0 + Y B Example Integral action Step disturbance Ad = q ￾ 1 Same Ad for any piece-wise constant signal Choose X = q + x0. Then (q ￾ 1)R0 = (q + x0)R0 + y0B Hence y0 = ￾ (1 + x0)R0 (1) B(1) The new controller is given by R = (q + x0)R0 + y0B S = (q + x0)S0 ￾ y0A c K. J. Åström and B. Wittenmark 3