Fal!2001 16.3118-7 Analysis Tools to∪se? Eigenvalues give a definite answer on the stability(or not) of the closed-loop system Problem is that it is very hard to predict where the closed-loop poles will go as a function of errors in the plant model Consider the case were the model of the system is Aoe+ Bu Controller also based on Ao, so nominal closed-loop dynamics A BK Ao- BK BK LC A0-BK-LC A0-LC But what if the actual system has dynamics =(A+△Ax+Bu Then perturbed closed-loop system dynamics are A0+△A BK Ao+△A-BKBK LC A0-BK-LC/ △A A0-LC Transformed Acl not upper-block triangular, so perturbed closed- loop eigenvalues are NoT the union of regulator estimator poles Can find the closed-loop poles for a specific AA, but Hard to predict change in location of closed-loop poles for a range of possible modeling errors pFall 2001 16.31 18—7 Analysis Tools to Use? • Eigenvalues give a definite answer on the stability (or not) of the closed-loop system. — Problem is that it is very hard to predict where the closed-loop poles will go as a function of errors in the plant model. • Consider the case were the model of the system is x˙ = A0x + Bu — Controller also based on A0, so nominal closed-loop dynamics: ∙ A0 −BK LC A0 − BK − LC ¸ ⇒ ∙ A0 − BK BK 0 A0 − LC ¸ • But what if the actual system has dynamics x˙ = (A0 + ∆A)x + Bu — Then perturbed closed-loop system dynamics are: ∙ A0 + ∆A −BK LC A0 − BK − LC ¸ ⇒ ∙ A0 + ∆A − BK BK ∆A A0 − LC ¸ • Transformed A¯cl not upper-block triangular, so perturbed closed￾loop eigenvalues are NOT the union of regulator & estimator poles. — Can find the closed-loop poles for a specific ∆A, but — Hard to predict change in location of closed-loop poles for a range of possible modeling errors