Linear Periodic Systems Typical Stability Boundary Consider the system Adjustment of feedforward gain MIT rule Process model d=A(t)中 G(s) s+1 A(t+t)=A(t) Command signal Solution has the form u(t)=sin t Φ(t)=D(t)e where D(t)=D(t+r) Stability condition All eigenvalues of c less than one in magnitude How to compute C? 0 Summary 4. Stability of dstR Adaptive systems are nonlinear Review of Algorithm An Properties of Estimator Equilibria Main Result Local analysis- linearization · Discuss Assumptions e Global behavio Disturbances Difficult Behavior can be very complicated Simulation roximations · Use special structure e more of this to follow e K.J. Astrom and BWittenmarkLinear Periodic Systems Consider the system dΦ dt
A(t)Φ with A(t + τ )
A(t) Solution has the form Φ(t)
D(t)eCt where D(t)
D(t + τ ). Stability condition: All eigenvalues of C less than one in magnitude. How to compute C? Typical Stability Boundary Adjustment of feedforward gain MIT rule Process model G(s)
1 s + 1 Command signal uc(t)
sinωt 100 0 0 1 2 ω γ Stable Summary Adaptive systems are nonlinear! Analysis • Equilibria • Local analysis - Linearization • Global behavior – Difficult – Behavior can be very complicated • Simulation • Approximations • Use special structure • More of this to follow 4. Stability of DSTR • Review of Algorithm • Properties of Estimator • Main Result • Discuss Assumptions • Disturbances c K. J. Åström and B. Wittenmark 5