TUTORIAL 7 STABILITY ANALYSIS

TUTORIAL 7 STABILITY ANALYSIS Ths tutral s specfcally wrtten fr students studyng the EC mdule D7 Cntrl System Engneerng but s als useful fr any student studyng cntrl. On cmpletn f ths tutral, yu shuld be able t d the fllwng. Explan the basc defntn f system nstablty. Explan and plt Nyqust dagrams. Explan and calculate gan and phase margns. Explan and prduce Bde plts. The next tutral cntnues the study f stablty. If yu are nt famlar wth nstrumentatn used n cntrl engneerng, yu shuld cmplete the tutrals n Instrumentatn Systems. In rder t cmplete the theretcal part f ths tutral, yu must be famlar wth basc mechancal and electrcal scence. Yu must als be famlar wth the use f transfer functns and the Laplace Transfrm (see maths tutrals). D.J.Dunn

. INSTABILITY Cnsder a system n whch the sgnals are added nstead f subtracted by the summer. Ths s pstve feed back. The electrnc amplfer shwn s an example f ths. Fgure We may derve the clsed lp transfer functn as fllws. G e + G G G G ( + G ) e - G G G G + G G ( G G ) G cl + G G G G If G G then G cl and the system s unstable. If G G then G cl has a fnte value whch s small r large dependng n the values. In the electrnc amplfer, the gan can be cntrlled by adjustng the feed back resstr (attenuatr). Suppse G and G 0.8 G cl /(- 0.8) 5 Suppse G and G 0.99 G cl /(- 0.99) 00 The clser the value f G s t the hgher the verall gan. It s essental that G s an attenuatr f the system s nt t be unstable. A system desgned fr negatve feed back wth a summer that subtracts shuld be stable but when the sgnals vary, such as wth a snusdal sgnal, t s pssble fr them t becme unstable. Cnsder an autmatc cntrl system such as the stablsers n a shp. When the shp rlls, the stablsers change angle t brng t back befre the rll becmes uncmfrtably large. If the stablsers mved the wrng way, the shp wuld rll further. Ths shuld nt happen wth a well desgned system but there are reasns and causes that can make such a thng happen. Suppse the shp was rllng back and frth at ts natural frequency. All shps shuld have a rghtng frce when they rll because f the buyancy and wll rll at a natural frequency defned by the weght, sze and dstrbutn f mass and s n. The sensr detects the rll and the hydraulc system s actvated t mve the stablser fns. Suppse that the hydraulcs mve t slw (perhaps a leak n the lne) and by the tme the stablsers have respnded, the shp has already rghted tself and started t rll the ther way. The stablsers wll nw be n the wrng pstn and wll make the shp rll even further. The tme delay has made matters wrse nstead f better and ths s bascally what nstablty s abut. Anther example f pstve feed back s when yu place a mcrphne n frnt f a lud speaker and get a lud scllatn. Anther example s when yu push a chld n swng. If yu gve a small push at the start f each swng, the swng wll mve hgher and hgher. Yu are addng energy t the system, the ppste affect f dampng. If yu stp pushng, frctn wll slwly brng t back t rest. Wth pstve feedback, energy s added t the system makng the utput grw ut f cntrl. A system wuld nt be desgned wth pstve feedback but when a snusdal sgnal s appled, t s pssble fr the negatve feedback t be cnverted nt pstve feedback. Ths ccurs when the phase shft f the feedback s 80 t the nput and the gan f the system s ne r mre. When ths happens, the feedback renfrces the errr nstead f reducng t. D.J.Dunn

. NYQUIST DIAGRAMS A system wth negatve feed back becmes unstable f the sgnal arrvng back at the summer s larger than the nput sgnal and has shfted 80 relatve t t. Cnsder the blck dagram f the clsed lp system. A snusdal sgnal s put n and the feed back s subtracted wth the summer t prduce the errr. Due t tme delay the feed back s 80 ut f phase wth the nput. When they are summed the result s an errr sgnal larger than the nput sgnal. Ths wll prduce nstablty and the utput wll grw and grw. Fgure A methd f checkng f ths s gng t happen s t dscnnect the feed back at the summer and measure the feed back ver a wde range f frequences. Fgure 3 If t s fund that there s a frequency that prduces a phase shft f 80 and there s a gan n the sgnal, then nstablty wll result. The Nyqust dagram s the lcus f the pen lp transfer functn pltted n the cmplex plane. If a system s nherently unstable, the Nyqust dagram wll enclse the pnt - (the pnt where the phase angle s 80 and unty gan). Cnsder the fllwng system. D.J.Dunn 3 Fgure 4

The transfer functn relatng and s G(s) G x G x G3 /{s(+s)(+s)}. Cnvertng ths nt a cmplex number (s j) we fnd 3 { 3 } ( ) G( j ) j 4 3 4 3 9 + ( ) 9 + ( ) { } { } The plar plt belw (Nyqust Dagram) s shwn fr and 0.4. We can see that at the 80 pstn the radus s less than when s the system wll be stable. When the radus s greater than s the system s unstable. We cnclude that turnng up the gan makes the system becme unstable. Fgure 5 The plt wll crss the real axs when 3 r 0.707 and ths s true fr all frequences. The plt wll enclse the - pnt f { 3 } 4 3 s the lmt s when 3 9 + ( ) 4 3 { 9 + ( ) } Puttng 0.707 the lmtng value f s.5 ALTERNATIVE METHOD There s anther way t slve ths and smlar prblems. The transfer functn s brken dwn nt separate cmpnents s n the abve case we have: G(s) x x s ( + s) ( + s) Each s turned nt plar c-rdnates (see prevus tutral). prduces a radus f and an angle f - 90 fr all rad s - prduces a radus f and angle tan ( ) + s + - prduces a radus f and angle tan + s + 4 When we multply plar crdnates remember that the resultant radus s the prduct f the ndvdual rad and the resultant angle s the sum f the ndvdual angles. The plar crdnates f the transfer functn are then: - - Radus s x x Angles - 90 tan ( ) tan + + 4 Put 0.707 Radus.44 x 0.865 x 0.577 0.667 Angle -90-35.6-54.74-80 If.5 the radus s as stated prevusly. D.J.Dunn 4

3 PHASE MARGIN and GAIN MARGIN 3. PHASE MARGIN Ths s the addtnal phase lag whch s needed t brng the system t the lmt f stablty. In ther wrds t s the angle between the pnt - and the vectr f magntude. 3. GAIN MARGIN Ths s the addtnal gan requred t brng the system t the lmt f stablty. Fgure 6 WORED EXAMPLE N. The pen lp transfer functn f a system s G(s) 00/{(+S)(3+S)(5+S)}. Prduce a plar plt fr 3 t 0. Determne the phase and gan margn. SOLUTION Evaluate the plar crdnates fr 00/( + s), then /(+s) then /(5+s) (See prevus tutral) D.J.Dunn 5

Nw add the three sets f angles and multply the three sets f rad and plt the results. Fgure 7 The regn f nterest s where the plt s -80 and the radus s. Ths wuld requre a much mre accurate plt arund the regn fr 3 t 5 as shwn belw. Fgure 8 The phase margn s 80 66 4 The gan margn s 0.65 0.35 SELF ASSESSMENT EXERCISE N. Determne the steady state gan and prmary tme cnstant fr G(s) 0/(s + 5). Determne the plar crdnates when /T (Gan and Radus.44 and angle -45 ). Determne the steady state gan fr G(s) 0.5/{(s+)(s+0)}. Determne the plar crdnates when 0.5 (Gan 0.05, R 0.043 φ -6.9 ) 3. The pen lp transfer functn f a system s G(s) 80/{(s+)(s+)(s+4)}. Prduce a plar plt fr 3 t 0. Determne the phase and gan margn. (0. and 3.5 ). D.J.Dunn 6

4. BODE PLOTS These are lgarthmc plts f the magntude (radus f the plar plt) and phase angle f the transfer functn. Frst cnsder hw t express the gan n decbels. Strctly G s a pwer gan and G Pwer ut/pwer In If the pwer n and ut were electrc then we may say G Usng Ohms Law ths wth the same value f Resstance at nput and utput ths becmes V I G r V I Expressng G n decbels V G(db) 0 lg 0 lg V V V V ut V n I I ut n I r 0 lg I Frm ths, t s usual t express the mdulus f G as G 0 lg (/) Nte that the gan n db s the 0 lg R where R s the radus f the plar plt n prevus examples. Cnsder the transfer functn G(s) s - j - j G(j ) G G(db) 0lg 0lg j Plttng ths equatn prduces the fllwng graph. The graph shws a straght lne passng thrugh 0 db at wth a gradent f -0 db per decade. The phase angle s -90 at all values f. Fgure 9 Nw cnsder the fllwng transfer functn G(s) G Ts + j T + j T + G(j ) gan. T j The radus f the plar crdnate s + T + T ( T + ) and ths s the The gan n db s then G(db) 0 lg 0 lg( T + ) 0lg( T + ) ( T ) + The phase angle s tan - (T) D.J.Dunn 7

If we put T as a cnvenent example, and plt the result, we get tw dstnct straght lnes shwn n the left graph. The hrzntal lne s prduced by very small values f and s t s called the LOW FREQUENCY ASYMPTOTE. The slpng straght lne ccurs at hgh values f and s called the HIGH FREQUENCY ASYMPTOTE and has a gradent f -0 db per decade. The tw lnes meet at the breakpnt frequency r natural frequency gven by n/t n ths case. The graph n the rght shws phase angle pltted aganst and t ges frm 0 t -90. The 45 pnt ccurs at the break pnt frequency. Fgure 0 Nw cnsder the fllwng transfer functn. (Standard frst rder respnse t a step nput) G(s) G(j ) s( + st) j ( + j T) Nte there s an easer way t fnd G as fllws. Separate the tw parts and fnd the mdulus f each separately. G j ( + T Takng lgs we get j + j T G Lg G Lg G j + j T + T lg + lg lg lg + + lg lg T + + T ( + T ) T ( + T ) Lg G db 0 lg lg lg There are three cmpnents t ths and we may plt all three separately as shwn. The graph fr the cmplete equatn s the sum f the three cmpnents. The result s that the graph has tw dstnctve slpes f -0 db per decade and -40 db per decade. ( and T were taken arbtrarly as 0 gvng a breakpnt f /T 0.. D.J.Dunn 8

Fgure The plt f phase angle aganst frequency n the lgarthmc scale shws that the phase angle shfts by 90 every tme t passes thrugh a breakpnt frequency. The plt fr the case under examnatn s shwn. A reasnable result s btaned by sketchng the asympttes fr each and addng them tgether. D.J.Dunn 9

WORED EXAMPLE N. A system has a transfer functn G(s) s(ts + ) Where the tme cnstant T s 0.5 secnds. Plt the Bde dagram fr gan and phase angle. Fnd the lw frequency gan per decade, the hgh frequency gan per decade and the break pnt frequency. SOLUTION G(s) s( Ts + ) G(j( j j T + G ( ) + T φ -90 tan j j T + T + G (db) 0 lg lg T 0.00 0.0 0..0 0 00 000 -lg 3 0 - - -3 -½lg(T+) tny tny tny -0.048-0.707 -.707 -.707 Ttal Gan (unts) 3 -.048 -.707-3.707-5.707 Ttal Gan db 60 40 0-0.96-34.4-74.4-4.4 φ degrees -90-90.3-9.9-7 -69-79 - 80 Examnng the table we see that the gan drps by 0 db per decade at lw frequences and by 40 db per decade fr hgh frequences. Plttng the graph n lgarthmc paper reveals a breakpnt frequency f rad/s whch s als fund by n /T /0.5 A quck way f drawng an apprxmate Bde plt s t evaluate the gan n db at the breakpnt frequency and draw asympttes wth a slpe f -0 db per decade prr t t and -40 db per decade after t. The phase angle may be fund by addng the tw cmpnents. G(s) G(j ) + s(ts + ) j j T + The phase angle fr the frst part s the angle f a vectr at pstn -/ n the j axs whch crrespnds t -90. The phase angle f the secnd part s the angle f a vectr at - n the real axs and - T n the j axs. The tw phase angles may be added t prduce the verall result. A quck way t draw the Bde phase plt s t nte that the break pnt frequency ccurs at the md pnt f the phase shft (-35 n ths case) s draw the asympttes such that they change by -90 at each breakpnt frequency. Fgure D.J.Dunn 0

GAIN AND PHASE MARGINS FROM THE BODE PLOT Gan and phase margns may be fund frm Bde plts as fllws. Lcate the pnt where the gan s zer db (unty gan) and prject dwn nt the phase dagram. The phase margn s the margn between the phase plt and -80. Lcate the pnt where the phase angle reaches ±80. Prject ths back t the gan plt and the gan margn s the margn between ths pnt and the zer db level. If the gan s ncreased untl ths s zer, the system becmes unstable. Fgure 3 D.J.Dunn

WORED EXAMPLE N.3 Draw the asympttes f the Bde plts fr the systems havng a transfer functn G(s) T s + (T s + s 0, T s secnds and T s 0. secnds. Fnd the value f whch makes the system stable. SOLUTION ( ) ) The tw break pnt frequences are /T / 0.5 rad/s and /T /0. 5 rad/s. GAIN Lcate the tw frequences and draw the asympttes. The frst ne s 0 db/decade up t 0.5 rad/s. The secnd ne s -0 db per decade untl t ntercepts 5 rad/s. Frm then n t s -40 db per decade. PHASE The phase angle dagram s n s easy t cnstruct frm asympttes. Lcate the break pnt frequences. These mark the md pnts between 0 and 90 (45 ) fr bth functns. ( has zer angle). The resultant phase angle vares frm 0 t -80 reachng -35 half way between the break pnts. Fgure 4 The phase angle reaches 80 at arund 0 rad/s. The gan at 0 rad/s s abut -60 db hence the gan margn s abut 60 db. T make the gan unty (zer db) we need an extra gan f: 60 0 lg G G 000 If the plt s repeated wth 000 t wll be seen that the gan margn s abut zer. D.J.Dunn

SELF ASSESSMENT EXERCISE N.. A system has a transfer functn T s 0. secnds. 4 G(s) s(t s + ) What s the steady state gan? (4) What s the lw frequency gan per decade, the hgh frequency gan per decade and the break pnt frequency? (-0 db, -40 db and 0 rad/s). A system has a transfer functn G(s) T s 0.5 secnds and T s 0.5 secnds. ( T s + )(T s + ) Fnd the lw frequency gan per decade, the hgh frequency gan per decade and the break pnt frequences. (0 db, -40 db and 4 rad/s and 6.7 rad/s) Fnd the gan margn and phase margn. (-80 db and 0 apprxmately) 3. Draw the asympttes f the Bde plts fr the systems havng a transfer functn G(s) T s + (T s + s, T s 0. secnds and T s 0 secnds. ( ) ) Fnd the gan margn and the value f whch makes the system stable. (30 db and 3 x 0 6 apprx) 4. The dagram shws the bde gan and phase plt fr a system. Determne the gan margn and whether r nt the system s stable. (45 db and 0 apprx Unstable) Fgure 5 D.J.Dunn 3