Question

5 Problem 4.41 Consider the second-order plant with transfer function: 1 G(s) = (s1)(5s 1) and in a unity feedback structure 1. Determine the system type and error constant with respect to tracking polynomial reference inputs of the system for P (i.e., De = kp), PD (i.e., Dc = kp+kps), and PID (i.e., De=kp +ki/s+ kps) controllers. Let kp = 19,ki 0.5,kp = 4/19. 2. Determine the system type and error constant of the system with respect to disturbance inputs for each of the three regulators in part (1) with respect to rejecting polynomial distur- bance w(t) at the input to the plant 3. Is this system better at tracking reference or rejecting disturbances? Explain your response briefly.

Answer 1

The open loop teansfa function is HU)=C13) x ars) i to (is)= kp Нs) = ke S (S+) (SS) This is a type -o system with Ena Constant position 17 HD) It 19 His)- (ii) to (i)= kpt ky s = 19+ 48 Hissa (19+45 ) (5+1) (SSH) This is also a type to system ega l a L = oo 0.05 Itto) lt 19

I (1) (13); 19+ o.stysa 0.21055+ 195tos s Hisz This is a velocity ef= (0.810557 195 +0.5), ! 5 * +1) (SS+1) typert system, so there will be enos Conspant. bence. t where kur ht Sx His) ود & Ke=kt $x sto (0.21055 +195+0:5) x 1 (S) (55+) a 0.5 z ewa to 32 - 2 wloan 0:5 The ofen Block loop Diagram transfer Ge= with function a Distubance. will be It GC- G = " for cusa kp Gels) = [(+1) (SSH) 1 + (p-1) 1S+) (53)

1 Gels = ISH) (5571) + kpl sustom type-o This is a 28 دن 50026 - لے - اے ۱۹۶۸ ۷۸ (ها ه 4 ا وا ۱۹۰ دی) ) (6) (هي داکا (ا- كما ۱۹۰) ا 11 (SI) (+ی * ( حاله . | کہا + syslem o type a also is This 5005 و ا لے =ع ا 13 11 هامة 11 005+ ۱۹۰ - 2105 00 <يا) + (1) | های)| (S61) (ای ع - 05+ ۱۹۱+یک واد، 2 | 1 (الك) (۱می 0 . 53+ 18+ s 0 . 810 + (اهلها (مکم

This to abo a type zero systém е те т It GO 3) The system is better at hacking the Reference and at the same time Refecting the Disturbanke. with the inhoone in the gain further the enos Con be brough down. Except to the PID Controller