Cotets Maual for K-Notes... Basics of Cotrol Systems... 3 Sigal Flow Graphs... 7 Time Respose Aalysis... 0 Cotrol System Stability... 6 Root locus Techique... 8 Frequecy Domai Aalysis... Bode Plots... 5 Desigs of Cotrol systems... 8 State Variable Aalysis... 3 04 Kreatryx. All Rights Reserved.
Maual for K-Notes Why K-Notes? Towards the ed of preparatio, a studet has lost the time to revise all the chapters from his / her class otes / stadard text books. This is the reaso why K-Notes is specifically iteded for Quick Revisio ad should ot be cosidered as comprehesive study material. What are K-Notes? A 40 page or less otebook for each subject which cotais all cocepts covered i GATE Curriculum i a cocise maer to aid a studet i fial stages of his/her preparatio. It is highly useful for both the studets as well as workig professioals who are preparig for GATE as it comes hady while travelig log distaces. Whe do I start usig K-Notes? It is highly recommeded to use K-Notes i the last moths before GATE Exam (November ed owards). How do I use K-Notes? Oce you fiish the etire K-Notes for a particular subject, you should practice the respective Subject Test / Mixed Questio Bag cotaiig questios from all the Chapters to make best use of it. 04 Kreatryx. All Rights Reserved.
Basics of Cotrol Systems The cotrol system is that meas by which ay quatity of iterest i a machie, mechaism or other equatio is maitaied or altered i accordace which a desired maer. Mathematical Modelig The Differetial Equatio of the system is formed by replacig each elemet by correspodig differetial equatio. For Mechaical systems () dx F Md dt M dt t () F K x x K v v dt dx (3) dx F f v v f dt dt 3
(4) Jd T dt Jd dt (5) Jd T dt Jd dt (6) T K K dt t Aalogy betwee Electrical & Mechaical systems Force (Torque) Voltage Aalogy Traslatio system Rotatioal system Electrical system Force F Torque T Voltage e Mass M Momet of Iertia J Iductace L Visuals Frictio coefficiet f Viscous Frictio coefficiet f Resistat R Sprig stiffess K Tesioal sprig stiffess K Reciprocal of capacitace C Displacemet x Agular Displacemet Charge q Velocity Agular velocity Curret i Force (Torque) curret Aalogy Traslatio system Rotatioal system Electrical system Force F Torque T Curret i Mass M Momet of Iertia J Capacitace C Visuals Frictio coefficiet f Viscous Frictio coefficiet f Reciprocal of Resistat /R Sprig stiffess K Tesioal sprig stiffess K Reciprocal of Iductace L Displacemet x Agular Displacemet Magetic flux likage Velocity Agular Velocity Voltage e 4
Trasfer fuctio The differetial equatio for this system is d x dx F M f kx dt dt Take Laplace Trasform both sides F(s) = Ms Xs fsx s kx s X Fs Ms fs k [Assumig zero iitial coditios] Trasfer fuctio of the system Trasfer fuctio is ratio of Laplace Trasform of output variable to Laplace Trasform of iput variable. The steady state-respose of a cotrol system to a siusoidal iput is obtaied by replacig s with jw i the trasfer fuctio of the system. X jw M jw f jw k F jw w M jwf+k Block Diagram Algebra The system ca also be represeted graphical with the help of block diagram. Various blocks ca be replaced by a sigal block to simplify the block diagram. => => 5
=> => => => 6
Sigal Flow Graphs Node: it represets a system variable which is equal to sum of all icomig sigals at the ode. Outgoig sigals do ot affect value of ode. Brach: A sigal travels alog a brach from oe ode to aother i the directio idicated by the brach arrow & i the process gets multiplied by gai or trasmittace of brach Forward Path: Path from iput ode to output ode. No-Touchig loop: Loops that do ot have ay commo ode. Maso s Gai Formula Ratio of output to iput variable of a sigal flow graph is called et gai. T P k P k P mr K K k = path gai of k th forward path = determiat of graph = (sum of gai of idividual loops) + (sum of gai product of o touchig loops) (sum of gai product of 3 o touchig loops) + P P P... K m m m3 = gai product of all r o touchig loops. = the value of T = overall gai Example : for the part of graph ot touchig k th forward path. Forward Paths: P a a a a45 3 34 P aa3 a35 Loops : P a3a3 P a3a34 a4 P3 a 44 P a a a a 4 3 34 45 5 P5 a3a35 a5 -No Touchig loops P a a a P a a a 3 3 44 ; 3 3 44 7
a a a a a a a a a a a a a a a a a a a 3 3 3 4 44 3 34 45 5 3 35 5 3 3 44 3 35 5 44 First forward path is i touch with all loops Secod forward path does ot touch oe loop a 44 P P T a a a a a a a a 3 34 45 3 35 44 Effect of Feed back System before feedback System after feedback Effect o Gai Positive feedback Gai = G GH G (gai icreases) Negative feedback Gai = G GH G (gai decreases) Effect o Stability Feedback ca improve stability or be harmful to stability if ot applied properly. Eg. Gai = G GH & GH =, output is ifiite for all iputs. 8
Effect o sesitivity Sesitivity is the ratio of relative chage i output to relative chage i iput T T T LT SG G LG G For ope loop system T = G T G T G SG T G G For closed loop system G T GH T G T G GH SG T G G GH GH (Sesitivity decreases) Effect o Noise Feedback ca reduce the effect oise ad disturbace o system s performace. Ope loop system G Y S NS Closed loop system G YS G N S G G H (Effect of Noise Decreases) 9
Positive feedback is mostly employed i oscillator whereas egative feedback is used i amplifiers. Time Respose Aalysis Stadard Test sigals Step sigal r(t) = Au(t) u(t) = ; t > 0 = 0; t < 0 R(s) = A s Ramp Sigal r(t) = At, t > 0 R(s) = = 0, t < 0 A s Parabolic sigal r(t) = At R(s) = = 0 ; t < 0 A s ; t > 0 3 Impulse t 0 ; t 0 t dt 0
Time Respose of first order systems Uit step iput R(s) = S C(s) = T S Ts S Ts C(t) = e t T Uit Ramp iput R(s) = S C(s) = S Ts C(t) = t T t e T Type of system ss Steady state error of system e depeds o umber of poles of G(s) at s = 0. This umber is kow as types of system Error Costats For uity feedback cotrol systems K P (positio error costat) = lim Gs s 0
K v (Velocity error costat) = lim s G s s 0 K a (Acceleratio error costat) = lim s G s s 0 Steady state error for uity feedback systems Type of Error Step iput Ramp iput Parabolic Iput system cos tats R R R j K K K K K K P K 0 K v 0 K 0 0 3 a p v R K R 0 K R 0 0 K 0 0 0 a For o-uity feedback systems, the differece betwee iput sigal R(s) ad feedback sigal B(s) actuatig error sigal Ea s R s G s H s e a ss lim sr s s 0 G s H s a E s. Trasiet Respose of secod order system Gs w S S+w w Y s R s s w s w Characteristic Equatio: For uit step iput Y s s s w s w 0 w s s w s w
if < (uder damp) e wt y(t) = siw t d w w ; if = (critical damp) y(t) = ( + w t) d =ta e wt - if > (over damp) y(t) = cos h w t si h w t e wt Roots of characteristic equatio are s,s w jw w is dampig costat which govers decay of respose for uder damped system. = cos = 0, imagiary axis If correspods to udamped system or sustaied oscillatios 3
Pole zero plot Step Respose 4
Importat Characteristic of step Respose / Maximum overshoot : 00e % ta Rise Time : w Peak Time : w Settlig Time : t t s s 4 w 3 w (for % margi) (for 5% margi) Effect of Addig poles ad zeroes to Trasfer Fuctio. If a pole is added to forward trasfer fuctio of a closed loop system, it icreases maximum overshoot of the system.. If a pole is added to closed loop trasfer fuctio it has effect opposite to that of case. 3. If a zero is added to forward path trasfer fuctio of a closed loop system, it decreases rise time ad icreases maximum overshoot. 4. If a zero is added to closed loop system, rise time decreases but maximum overshoot decreases tha icreases as zero added moves towards origi. 5
Cotrol System Stability A liear, time-ivariat system is stable if followig otios of stability are satisfied: Whe system is excited by bouded iput, the output is bouded. I absece of iputs, output teds towards zero irrespective of iitial coditios. For system of first ad secod order, the positive ess of coefficiets of characteristic equatio is sufficiet coditio for stability. For higher order systems, it is ecessary but ot sufficiet coditio for stability. Routh stability criterio If is ecessary & sufficiet coditio that each term of first colum of Routh Array of its characteristic equatio is positive if a0 0. Number of sig chages i first colum = Number of roots i Right Half Plae. Example : s 0 a s a s... a 0 a 0 a a 4 s s 3 s a a 3 a 5. a a a a a a a a.......... 0 s 0 3 4 0 5 a a a Special Cases Whe first term i ay row of the Routh Array is zero while the row has at least oe ozero term. Solutio : substitute a small positive umber for the zero & proceed to evaluate rest of Routh Array 6
eg. s 5 s 4 s 3 s 3s 5 0 5 s 3 4 s 5 3 s s 5 4 4 5 s 0 s 5 0, ad hece there are sig chage ad thus roots i right half plae. Whe all the elemets i ay oe row Routh Array are zero. Solutio : The polyomial whose coefficiets are the elemets of row just above row of zeroes i Routh Array is called auxiliary polyomial. o The order of auxiliary polyomial is always eve. o Row of zeroes should be replaced row of coefficiets of polyomial geerated by takig first derivative of auxiliary polyomial. 6 5 4 3 Example : s s 8s s 0s 6s 6 0 6 s 8 0 6 5 s 6 5 s 6 8 4 s 6 4 s 6 8 3 s 0 0 Auxiliary polyomial : A(s) = 6 s 8 0 6 5 s 6 8 4 s 6 8 3 s 4 3 s 3 s 3 8 s 0 3 s 8 4 s 6s 8 0 7
Types of stability Limited stable : if o-repeated root of characteristic equatio lies o jw- axis. Absolutely stable: with respect to a parameter if it is stable for all value of this parameter. Coditioally stable : with respect to a parameter if system is stable for bouded rage of this parameter. Relative stability If stability with respect to a lie s is to be judged, the we replace s by z i characteristic equatio ad judge stability based o Routh criterio, applied o ew characteristic equatio. Root locus Techique Root Loci is importat to study trajectories of poles ad zeroes as the poles & zeroes determie trasiet respose & stability of the system. Characteristic equatio +G(s)H(s) =0 Assume G(s)H(s) = KG sh s KG sh s 0 G s H s K Coditio of Roots locus G sh s k k G s H s i K 0 = odd multiples of 80 G s H s i K 0 = eve multiples of 80 8
Coditio for a poit to lie o root Locus The differece betwee the sum of the agles of the vectors draw from the zeroes ad those from the poles of G(s) H(s) to s is o odd multiple of 80 if K > 0. The differece betwee the sum of the agles of the vectors draw from the zeroes & those from the poles of G(s)H(s) to s is a eve multiple of 80 icludig zero degrees. Properties of Roots loci of KG s H s 0. K = 0 poits : These poits are poles of G(s)H(s), icludig those at s =.. K = poit : The K = poits are the zeroes of G(s)H(s) icludig those at s =. 3. Total umbers of Root loci is equal to order of KG s H s 0 equatio. 4. The root loci are symmetrical about the axis of symmetry of the pole- zero cofiguratio G(s) H(s). 5. For large values of s, the RL (K > 0) are asymptotes with agles give by: i 80 i m for CRL(complemetary root loci) (K < 0) i i 80 m where i = 0,,,., m = o. of fiite poles of G(s) H(s) m = o. of fiite zeroes of G(s) H(s) 6. The itersectio of asymptotes lies o the real axis i s-plae. The poit of itersectio is called cetroid ( ) real parts of poles G(s)H(s) real parts of zeroes G(s)H(s) = m 7. Roots locus are foud i a sectio of the real axis oly if total umber of poles ad zeros to the right side of sectio is odd if K > 0. For CRL (K < 0), the umber of real poles & zeroes to right of give sectio is eve, the that sectio lies o root locus. 8. The agle of departure or arrival of roots loci at a pole or zero of G(s) H(s) say s is foud by removig term (s s ) from the trasfer fuctio ad replacig s by s i the remaiig trasfer fuctio to calculate Gs Hs 9
Agle of Departure (oly applicable for poles) = 80 0 + Gs Hs Agle of Arrival (oly applicable for zeroes) = 80 0 - Gs Hs 9. The crossig poit of root-loci o imagiary axis ca be foud by equatig coefficiet of s i Routh table to zero & calculatig K. The roots of auxiliary polyomial give itersectio of root locus with imagiary axis. 0. Break-away & Break-i poits These poits are determied by fidig roots of for breakaway poits : For break i poits : dk 0 ds dk ds 0 dk ds 0. Value of k o Root locus is K G s H s Additio of poles & zeroes to G(s) H(s) Additio of a pole to G(s) H(s) has the effect of pushig of roots loci toward right half plae. Additio of left half plae zeroes to the fuctio G(s) H(s) geerally has effect of movig & bedig the root loci toward the left half s-plae. 0
Frequecy Domai Aalysis Resoat Peak, M r It is the maximum value of M(jw) for secod order system Mr =, 0.707 = dampig coefficiet Resoat frequecy, w r The resoat frequecy w r is the frequecy at which the peak M r occurs. r w w, for secod order system Badwidth, BW The badwidth is the frequecy at which M(jw) drops to 70.7% of, or 3dB dow from, its zero frequecy value. for secod order system, BW = 4 w 4 Note : For > 0.707, w r = 0 ad M r = so o peak. Effect of Addig poles ad zeroes to forward trasfer fuctio The geeral effect of addig a zero to the forward path trasfer fuctio is to icrease the badwidth of closed loop system. The effect of addig a pole to the forward path trasfer fuctio is to make the closed loop less stable, which decreasig the bad width. Nyquist stability criterio I additio to providig the absolute stability like the Routh Hurwitz criterio, the Nyquist criterio gives iformatio o relative stability of a stable system ad the degree of istability of a ustable system.
Stability coditio Ope loop stability If all poles of G(s) H(s) lie i left half plae. Closed loop stability If all roots of + G(s)H(s) = 0 lie i left half plae. Ecircled or Eclosed A poit of regio i a complex plae is said to be ecircled by a closed path if it is foud iside the path. A poit or regio is said to eclosed by a closed path if it is ecircled i the couter clockwise directio, or the poit or regio lies to the left of path. Nyquist Path If is a semi-circle that ecircles etire right half plae but it should ot pass through ay poles or zeroes of s GsHs & hece we draw small semi-circles aroud the poles & zeroes o jw-axis. Nyquist Criterio. The Nyquist path s is defied i s-plae, as show above.. The L(s) plot (G(s)H(s) plot) i L(s) plae is draw i.e., every poit correspodig value of L(s) = G(s)H(s). s plae is mapped to 3. The umber of ecirclemets N, of the ( + j0) poit made by L(s) plot is observed. 4. The Nyquist criterio is N= Z P N = umber of ecirclemet of the ( + j0) poit made by L(s) plot. Z = Number of zeroes of + L(s) that are iside Nyquist path (i.e., RHP) P = Number of poles of + L(s) that are iside Nyquiest path (i.e., RHP) ; poles of + L(s) are same as poles of L(s).
For closed loop stability Z must equal 0 For ope loop stability, P must equal 0. for closed loop stability N = P i.e., Nyquist plot must ecircle ( + j0) poit as may times as o. of poles of L(s) i RHP but i clockwise directio. Nyquist criterio for Miimum phase system A miimum phase trasfer fuctio does ot have poles or zeroes i the right half s-plae or o axis excludig origi. For a closed loop system with loop trasfer fuctio L(s) that is of miimum phase type, the system is closed loop stable if the L(s) plot that correspods to the Nyquist path does ot ecircle ( + j0) poit it is ustable. i.e. N=0 Effect of additio of poles & zeroes to L(s) o shape of Nyquiest plot If L(s) = K T s Additio of poles at s = 0. Ls K s T s Both Head & Tail of Nyquist plot are rotated by 90 clockwise.. Ls K s T s 3
3. Ls K 3 s T s Additio of fiite o-zero poles L s K T s T s Oly the head is moved clockwise by 90 but tail poit remais same. Additio of zeroes Additio of term T s i umerator of L(s) icreases the phase of L(s) by 90 at w ad hece improves stability. d Relative stability: Gai & Phase Margi Gai Margi Phase crossover frequecy is the frequecy at which the L(jw) plot itersect the egative real axis. L jw 80 or where gai margi = GM = P 0log 0 L jw if L(jw) does ot itersect the egative real axis P L jw 0 GM = db GM > 0dB idicates stable system. GM = 0dB idicates margially stable system. GM < 0dB idicates ustable system. Higher the value of GM, more stable the system is. P 4
Phase Margi It is defied as the agle i degrees through which L(jw) plot must be rotated about the origi so that gai crossover passes through (, j 0) poit. Gai crossover frequecy is w g s.t. g L jw Phase margi (PM) = g L jw 80 Bode Plots Bode plot cosist of two plots 0 log G jw vs log w w vs log w Assumig G s K Ts Ts s Tas s s e Ts d 0 0 G jw 0log G jw 0log K 0log jwt db 0log jwt 0log jw 0log jwt 0 0 0 a w 0log w 0 j w w G jw jwt jwt jw jwta jw / w w jwtd rad w 5
Magitude & phase plot of various factor Factor Magitude plot Phase Plot K P jw jw P jwt a jwt a G jw j w w w w 6
Example : Bode plot for Gs G jw 0s 0 s s s 5 If w = 0. 0 0 jw jw jw jw 5 0 G jw 00 0. 5 ; For 0. < w < G jw For < w < 5 0 0 w 5 w G jw 90 0 0 0 jw jw 5 w G jw G jw 80 For 5 < w < 0 0 0 00 jw jw jw w j 3 G jw For w > 0 G jw 70 0 jw 0 G jw jw jw jw G jw 80 ; slope = 0 db / dec ; G jw w ; G jw ; G jw Slope = 40 db/ dec Slope = 60 db/ dec slope = 40 db/ dec 7
Desigs of Cotrol systems P cotroller The trasfer fuctio of this cotroller is K P. The mai disadvatage i P cotroller is that as K P value icreases, hece overshoot icreases. As overshoot icreases system stability decreases. decreases & I cotroller The trasfer fuctio of this cotroller is k i s It itroduces a pole at origi ad hece type is icreased ad as type icreases, the SS error decrease but system stability is affected.. D cotroller It s purpose is to improve the stability. The trasfer fuctio of this cotroller is K DS. It itroduces a zero at origi so system type is decreased but steady state error icreases. PI cotroller It s purpose SS error without affectio stability. Trasfer fuctio = K P K s SK K i P i It adds pole at origi, so type icreases & SS error decreases. It adds a zero i LHP, so stability is ot affected. S Effects: o Improves dampig ad reduces maximum overshoot. o Icreases rise time. o Decreases BW. o Improves Gai Margi, Phase margi & Mr. o Filter out high frequecy oise. 8
PD cotroller Its purpose is to improve stability without affectig stability. Trasfer fuctio: K P K S D It adds a zero i LHP, so stability improved. Effects: o Improves dampig ad maximum overshoot. o Reduces rise time & settig time. o Icreases BW o Improves GM, PM, Mr. o May atteuate high frequecy oise. PID cotroller Its purpose is to improve stability as well as to decrease e ss. Ki Trasfer fuctio = KP skd s o If adds a pole at origi which icreases type & hece steady state error decreases. o If adds zeroes i LHP, oe fiite zero to avoid effect o stability & other zero to improve stability of system. Compesators Lead Compesator G e s c ; < ZS ZS = phase lead = ta w ta w For maximum phase shift ta m w = Geometric mea of corer frequecies = 9
Effect o o o It icreases Gai Crossover frequecy It reduces Badwidth. It reduces udamped frequecy. Lag compesators Ge Ge s jw s s jw jw For maximum phase shift w ; ta m Effect o o o Icrease gai of origial Network without affectig stability. Reduces steady state error. Reduces speed of respose. Lag lead compesator G e s S S S S > ; < G e jw jw jw jw jw / 30
State Variable Aalysis The state of a dyamical system is a miimal set of variables (kow as state variables) such that the kowledge of these variables at t = t 0 together with the kowledge of iput for t t 0 completely determie the behavior of system at t > t 0. State variable x(t) = x x.. x t t t ; y(t) = y y.. yp t t t ; u(t) = u t u t.. um t Equatios determiig system behavior : X(t) = A x (t) + Bu(t) ; State equatio y(t) = Cx (t) + Du (t) ; output equatio State Trasitio Matrix It is a matrix that satisfies the followig liear homogeous equatio. dx t Ax t dt Assumig Properties: t is state trasitio matrix t SI A ) 0 = I (idetity matrix) ) t t At 3 3 t e I At A t A t...! 3! 3) t t t t t t 0 0 4) t kt K for K > 0 Solutio of state equatio State Equatio: X(t) = A x(t) + Bu(t) X(t) = t At At e x 0 e Bud 0 3
Relatioship betwee state equatios ad Trasfer Fuctio X(t) = Ax (t) + Bu(t) Takig Laplace Trasform both sides sx(s) = Ax (s) + Bu(s) (SI A) X(s) = Bu(s) X(s) = (SI A) B u (s) y(t) = Cx(t) + D u(t) Take Laplace Trasform both sides. y(s) = Cx(s) + D u (s) x(s) = (si A) B u (s) y(s) = [C(SI A) B + D] U(s) y s Us C SI A B D = Trasfer fuctio Eige value of matrix A are the root of the characteristic equatio of the system. Characteristic equatio = SI A 0 Cotrollability & Observability A system is said to be cotrollable if a system ca be trasferred from oe state to aother i specified fiite time by cotrol iput u(t). A system is said to be completely observable if every state of system X idetified by observig the output y(t). i t ca be Test for cotrollability Q C = cotrollability matrix = [B AB A B A B] Here A is assumed to be a x matrix B is assumed to be a x matrix If det Q C = 0 system is ucotrollable det Q 0, system is cotrollable C 3
Test for observability Q 0 = observability matrix = C CA CA... C A A is a x matrix C is a ( x ) matrix If det Q0 0, system is uobservable det Q 0, system is observable 0 33