Module 4: Time Response of discree ime sysems Lecure Noe 2 1 Prooype second order sysem The sudy of a second order sysem is imporan because many higher order sysem can be approimaed by a second order model if he higher order poles are locaed so ha heir conribuions o ransien response are negligible. A sandard second order coninuous ime sysem is shown in Figure 1. We can wrie + _ E(s) ω 2 n s(s+2ξω n) C(s) Figure 1: Block Diagram of a second order coninuous ime sysem G(s) = ω 2 n s(s+2ξω n ) ω 2 n Closed loop: G c (s) = s 2 +2ξω n s+ωn 2 where, ξ = damping raio ω n = naural undamped frequency Roos: ξω n ±jω n 1 ξ 2 1.1 Comparison beween coninuous ime and discree ime sysems The simplified block diagram of a space vehicle conrol sysem is shown in Figure 2. The objecive is o conrol he aiude in one dimension, say in pich. For simpliciy vehicle body is considered as a rigid body. Posiion c() and velociy v() are fedback. The open loop ransfer funcion can be calculaed I. Kar 1
+ _ E(s) K K p + 1 J vs 1 s C(s) K R Figure 2: Space vehicle aiude conrol as Closed loop ransfer funcion is G(s) = C(s) E(s) = KK P = KK P s(j v s+k R ) 1 K R +J v s 1 s G c s = = G(s) 1+G(s) KK p J v s 2 +K R s+kk P Wih he above parameers, K P = Posiion Sensor gain = 1.65 10 6 K R = Rae sensor gain = 3.71 10 5 K = Amplifier gain which is a variable J v = Momen of ineria = 41822 G(s) = C(s) = 39.45K s(s+8.87) 39.45K s 2 +8.87s+39.45K Characerisics equaion s 2 +8.87s+39.45K = 0 ω n = 39.45K rad/sec, ξ = 8.87 2ω n Sincehesysemisof2 nd order,heconinuousimesysemwillalwaysbesableifk P, K R, K, J v are all posiive. Now, consider ha he coninuous daa sysem is subjec o sampled daa conrol as shown in Figure 3. I. Kar 2
E(s) E (s) 1 1 C(s) + ZOH KK P + J vs s K R Figure 3: Discree represenaion of space vehicle aiude conrol For comparison purpose, we assume ha he sysem parameers are same as ha of he coninuous daa sysem. G(s) = C(s) E (s) = G ho (s)g p (s) = 1 e Ts s G(z) = (1 z 1 ) KKp K R = (1 z 1 ) KKp = KKp K 2 R KKp/J v. s(s+k R /J v ) [ 1 Z s J ] v 2 K R s + J v K R (s+k R /J v ) [ ] Tz K R (z 1) J v z 2 K R (z 1) + J v z K R (z e K RT/Jv ) [ ] (TKR J v +J v e KRT/Jv )z (TK R +J v )e KRT/Jv +J v (z 1)(z e K RT/Jv ) Characerisic equaion of he closed loop sysem: z 2 +α 1 z +α 0 = 0, where Subsiuing he known parameers: For sabiliy α 1 = f 1 (K,Kp,K R,Jv) α 0 = f 0 (K,Kp,K R,Jv) α 1 = 0.000012K(3.71 10 5 T 41822+41822e 8.87T ) 1 e 8.87T α 0 = e 8.87T +0.000012K [ 41822 (3.71 10 5 T +41822)e 8.87T] (1) α 0 < 1 (2) P(1) = 1+α 1 +α 0 > 0 = 1 e 8.87T > 0 always saisfied since T is posiive (3) P( 1) = 1 α 1 +α 0 > 0 Choice of K and T: If we plo K versus T hen according o condiions (1) and (3) he sable region is shown in Figure 4. Pink region represens he siuaion when condiion (1) is saisfied bu he (3) is no. Red region depics he siuaion when condiion (1) is saisfied, no he (3). Yellow is he sable region where boh he condiions are saisfied. If we I. Kar 3
45 40 35 30 25 K 20 15 10 5 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 T T=0.08 Figure 4: K vs. T for space vehicle aiude conrol sysem wan a comparaively large T, such as 0.2, he gain K is limied by he range K < 5. Similarly if we wan a comparaively high gain such as 25, we have o go for T as small as 0.08 or even less. From sudies of coninuous ime sysems i is well known ha increasing he value of K generally reduces he damping raio, increases peak overshoo, bandwidh and reduces he seady sae error if i is finie and nonzero. 2 Correlaion beween ime response and roo locaions in s-plane and z-pane The mapping beween s-plane and z-plane was discussed earlier. For coninuous ime sysems, he correlaion beween roo locaion in s-plane and ime response is well esablished and known. A roo in negaive real ais of s-plane produces an oupu eponenially decaying wih ime. Comple conjugae pole pairs in negaive s-plane produce damped oscillaions. Imaginary ais conjugae poles produce undamped oscillaions. Comple conjugae pole pairs in posiive s-plane produce growing oscillaions. Digial conrol sysems should be given special aenion due o sampling operaion. For eample, if he sampling heorem is no saisfied, he folding effec can enirely change he rue response of he sysem. I. Kar 4
The pole-zero map and naural response of a coninuous ime second order sysem is shown in Figure 5. jω ω 1 σ 1 σ (a) (b) Figure 5: Pole zero map and naural response of a second order sysem If he sysem is subjec o sampling wih frequency ω s < 2ω 1, i will generae an infinie number of poles in he s-plane a s = σ 1 ± jω 1 + jnω s for n = ±1,±2,... The sampling operaion will fold he poles back ino he primary srip where ω s /2 < ω < ω s /2. The ne effec is equivalen o having a sysem wih poles a s = σ 1 ± j(ω s ω 1 ). The corresponding plo is shown in Figure 6. jω ω 1 ω s Wih folding ω s/2 ω 1 ω s σ 1 ω 1 +ω s σ ω s/2 Acual ω 1 ω s (a) (b) Figure 6: Pole zero map and naural response of a second order sysem Roo locaions in z-plane and he corresponding ime responses are shown in Figure 7. I. Kar 5
Figure 7: Pole zero map and naural response of a second order sysem 3 Dominan Closed Loop Pole Pairs As in case of s-plane, some of he roos in z-plane have more effecs on he sysem response han he ohers. I is imporan for design purpose o separae ou hose roos and give hem he name dominan roos. In s-plane, he roos ha are closes o jω ais in he lef plane are he dominan roos because he corresponding ime response has slowes decay. Roos ha are for away from jω ais correspond o fas decaying response. In Z-plane dominan roos are hose which are inside and closes o he uni circle whereas insignifican region is near he origin. The negaive real ais is generally avoided since he corresponding ime response is oscillaory in naure wih alernae signs. I. Kar 6
Figure 8 shows he regions of dominan and insignifican roos in s-plane and z-plane. jω j Im z Insignifican roos Dominan roos 01 0011 01 01 00000 11111 01 σ 0 1 0 Dominan roos Re z 1 s-plane z-plane Insignifican roos Figure 8: Pole zero map of a second order sysem In s-plane he insignifican roos can be negleced provided he dc-gain (0 frequency gain) of he sysem is adjused. For eample, 10 (s 2 +2s+2)(s+10) 1 (s 2 +2s+2) In z-plane, roos near he origin are less significan from he maimum overshoo and damping poin of view. However hese roos canno be compleely discarded since he ecess number of poles over zeros has a delay effec in he iniial region of he ime response, e.g., adding a pole a z = 0 would no effec he maimum overshoo or damping bu he ime response would have an addiional delay of one sampling period. The proper way of simplifying a higher order sysem in z-domain is o replace he poles near origin by Poles a z = 0 which will simplify he analysis since he Poles a z = 0 correspond o pure ime delays. I. Kar 7