If p22 = 0, an improvement in performance of steady-state operation is feasible only in the range 0 < r) < 772 [Figure 7.3(a)]. If p22 < 0, a positive SJ occurs in the range rji < 77 < r/2 [t) — fj/rt, Figure 7.3(a)], with m

2 - P11P22

(-P11) provided /2 > pwpn-Fixed ri, variable r2.

The necessary and sufficient condition for occurrence of a local maximum in SJ and its value are

SJ - (pn -—) at C = - = - — if P22 < 0. (7.92) 2 V P22J ri p22

The upper bound on 8J in Eq. 7.92 corresponds to / = |p2i| (defined in Eq. 7.89). The necessary condition for a positive maximum in 5J is that \P2l\2 > P11P22-

If weak perturbations in u\ alone lead to improved process performance (pn > 0), then it is evident from Eq. 7.92 that periodic variations in u2 as well as u\ lead to further improvement in the process performance. When pn > 0, forced periodic operation involving variations in m and u2 is superior to steady-state operation only in the range 0 < C < C2 [C = ri/r2, Figure 7.3(a)], (2 being

If P11 = 0, a forced periodic operation involving variations in u\ and u2 is superior to steady-state operation (SJ > 0 in Eq. 7.88) only in the range

Figure 7.3. (a) Portraits of G\ and r/ for p22 < 0 and portraits of G2 and C for pn <0. G1 = pn + 2frj + p22V2, G2 = P22 + 2/C + PuC2-[(a:, y) = (77, G1) and (x,y) = ((, G2).] The profiles 1, 2 and 3 correspond to pn > 0, pn — 0 and pn < 0, respectively, when (x,y) = (rj, G1) and to P22 > 0, P22 = 0 and p22 < 0, respectively, when (x,y) = (C, G2)-(b) Profiles of SJ = c (c an arbitrary constant) when min(/dn, P22) > 0, max(pu, P22) > 0 and / > 0 [449].

Figure 7.3. (a) Portraits of G\ and r/ for p22 < 0 and portraits of G2 and C for pn <0. G1 = pn + 2frj + p22V2, G2 = P22 + 2/C + PuC2-[(a:, y) = (77, G1) and (x,y) = ((, G2).] The profiles 1, 2 and 3 correspond to pn > 0, pn — 0 and pn < 0, respectively, when (x,y) = (rj, G1) and to P22 > 0, P22 = 0 and p22 < 0, respectively, when (x,y) = (C, G2)-(b) Profiles of SJ = c (c an arbitrary constant) when min(/dn, P22) > 0, max(pu, P22) > 0 and / > 0 [449].

0 < £ < C2 [Figure 7.3(a)]. If p\\ < 0, a positive 6 J occurs in the range Ci < C < C2 [C = r2/ru Figure 7.3(a)], with

f-Vf2

P\lP22

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