step([0.04 -0.04 1],[0.1 1.2 2.1 1])
which has a complex pair of RHP-zeros, but no inverse response - and not even any changes in the sign of the first derivative (change in direction).
The statement given in Theorem 2 in Holt and Morari (1985b) (which is referred to in the book) is that "A system with an odd number of RHP-zeros will exhibit an inverse response to a step input (If there are an even number of RHP zeros the response will start in the proper direction, change directions, eventually reversing its sign and moving back toward the steady state)." The first part of the theorem (for an odd number of RHP-zeros) is obvious from the intial and final value theorem. However, the latter part of theorem is not true - at least not for complex RHP-zeros (see the counterexample). For a proof, Holt and Morari (1985b) refer to Rosenbrock (1970). However, I did not find the proof in Rosenbrocks book; can someone help me (skoge@chembio.ntnu.no).
From bernardo@tamarugo.cec.uchile.cl Fri Apr 03 15:24:27 1998 To: skoge@chembio.ntnu.no Subject: real rhp zeros >Is it true that the step response for a SISO system with n_z real RHP-zeros >crosses zero n_z times? Please refer to my paper in ieee tac, vol. 39, no 3, march 1994, pp. 578-581, and have a look at theorems 1 and 2 (specially at the proof of theorem 2). To the best of my knowledge all you can say is that the step response of a siso system with n_z real rhp zeros will cross zero AT LEAST n_z times. Saying something about the exact number of zero crossings would require additional assumptions on the zero-pole pattern (see corollary 1 in the above paper) Best regards, bernardo