Performance limitations of nonlinear periodic sampled-data controllers for Lp disturbance rejection

In this case, the equivalent open-loop cheap controller is given by Q(s) = (s + 1)(s + 2) 2(1 + s)(1 + s) : (32) It then follows that q1 = 1=2, and Q(s) 01 0 q 01 1 = 2[(+ 0 3)s + (1 0 2)] (s + 1)(s + 2) : (33) Using (33) in (20) we have that, as ! 0 tsat ! ln The tracking error on the interval [0 + tsat) is e(t) The value of k t sat given by kt sat = 1 1 0 2(1 +)e 0t + (1 + 2)e 02t t (36) and t star is given in (26). Using (34)–(36), (25), and (26), we can evaluate the cost. The results are shown in Fig. 4. Note that as 1 ! 1 the limiting cost approaches 2, which is consistent with the results of unconstrained cheap control. The results shown in Fig. 4 give insight into the effect of the input constrained achievable performance. It is interesting, for instance, to note that a constraint 1 = 5 (which is five times the steady state input necessary in this case) changes the performance limit associated with a non minimum phase zero at 10 (= 0:1) to be equivalent to the performance limit achieved without constraints for a non minimum phase zero at five. This illustrates the fact that, depending on conditions, the effect of input constraints can swamp linear effects due to right-half plane zeros. This is in accord with intuition. IV. CONCLUSION In this note, we have presented a method to evaluate the degradation in performance of the closed-loop system when constraints are added. We have focused our analysis on open-loop stable SISO systems tracking step references, and used as performance index the value of the L 2-norm of the error. We have obtained analytical expressions for the cost (performance index) that show how system dynamics and constraints interact to deteriorate the performance. Even for simple cases, the obtained expressions indicate that this interaction is far from trivial. In addition, for some cases, tight constraints can swamp limitations associated with system dynamics. The information provided by the analysis has implications on the choice of actuator authority and on the need for using tactical strategies to address the problem of constraint handling. Indeed, it is a common practice to ignore constraints in previous stages of control design and then evaluate the …