ECM Control Engineering Dr Mustafa M Aziz (2013) SYSTEM RESPONSE

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1 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) SYSTEM RESPONSE. Itroductio. Repoe Aalyi of Firt-Order Sytem 3. Secod-Order Sytem 4. Siuoidal Repoe of the Sytem 5. Bode Diagram 6. Baic Fact About Egieerig Sytem. Itroductio The order of a ytem i defied a beig the highet power of derivative i the differetial equatio, or beig the highet power of i the deomiator of the trafer fuctio. A firt-order ytem oly ha to the power oe i the deomiator, while a ecod-order ytem ha the highet power of i the deomiator beig two. Type of the iput fuctio (or tet iput igal) commoly ued are: Impule fuctio: I the time domai, u(t) cδ(t). I the domai, U() c. Step fuctio: I the time domai, u(t) c. I the domai, U() c/. Ramp fuctio: I the time domai, u(t) ct. I the domai, U() c/. Siuoidal fuctio: I the time domai, u(t) ci(ωt). I the domai, U() cω/( +ω ). where c i a cotat i all the above. With thee tet igal, mathematical ad experimetal aalye of cotrol ytem ca be carried out eaily ice the igal are very imple fuctio of time. Which of thee typical igal to ue for aalyig ytem characteritic may be determied by the form of the iput that the ytem will be ubjected to mot frequetly uder ormal operatio. If the iput to a cotrol ytem are gradually chagig fuctio of time, the a ramp fuctio of time may be a good tet igal. Similarly, if a ytem i ubjected to udde diturbace, a tep fuctio of time may be a good tet igal, ad for a ytem ubjected to a hock iput, a pule or a impule fuctio may be bet. Exercie: What are the order of the ytem decribed by the followig trafer fuctio: a) G() m + b + k b) G() RC + c) G() LC + RC +

2 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) The time repoe of a cotrol ytem coit of two part: the traiet repoe ad the teadytate repoe. The traiet repoe i defied a the part of the time repoe which goe from the iitial tate to the fial tate ad reduce to zero a time become very large. The teady-tate repoe i defied a the behaviour of the ytem a t approache ifiity after the traiet have died out. Thu the ytem repoe may be writte a: y t (t) + y (t) where y t (t) deote the traiet repoe, ad y (t) deote the teady-tate repoe.. Repoe Aalyi of Firt-Order Sytem May ytem are approximately firt-order. The importat feature i that the torage of ma, mometum ad eergy ca be captured by oe parameter. Example of firt-order ytem are velocity of a car o the road, cotrol of the velocity of a rotatig ytem, electric ytem where eergy torage i eetially i oe capacitor or oe iductor, icompreible fluid flow i a pipe, level cotrol of a tak, preure cotrol i a ga tak, temperature i a body with eetially uiform temperature ditributio (e.g. team filled veel). Next we will preet everal example to how how to obtai the dyamic equatio of firt-order ytem. Example : Mechaical ytem m i the ma, u(t) i the exteral force, i the velocity ad b i the frictio coefficiet. By Newto law, we have the followig differetial equatio: u(t) m d m + b u(t) dt b Example : Electrical ytem R i the reitace, C i the capacitace, u(t) i the iput voltage ad i the output voltage. By Kirchhoff law: R u(t) Ri(t) + ad i(t) Cd/dt u(t) C Thu d RC + u(t) dt A geeral form of a firt-order ytem ca be repreeted by the block diagram. R() + _ /T Y() R() T + Y()

3 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) Tak: Write the ytem output or repoe to iput uch a the uit-tep, uit-ramp, ad uitimpule fuctio, repectively. The iitial coditio are aumed to be zero. Draw the repoe curve. T i the time cotat of the ytem... Uit-tep repoe of firt-order ytem R() /, ad therefore the uit-tep repoe i: Y() (T + ) T Expadig Y() ito partial fractio: Y() T + + / T Take the ivere Laplace traform: - e -t/t, t. The olutio ha two part: a teady-tate repoe: y (t), ad a traiet repoe: t / T y (t) e, which decay to zero a t. t.9.8 Slope /T Uit-tep repoe, T T r t / T T The lope of the taget lie at t i /T. Pole locatio i the plae: -/T. At t T, y(t) e T i called the time cotat, ad it i the time it take for the tep repoe to rie to 63.% of it fial value. y(t).865; y(3t).95; y(4t).98; y(5t) It ca be ee that for t 4T, the repoe remai withi % of the fial value; thi time i kow a the ettlig time, T. 3

4 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) The rie time, T r, i defied a the time for the waveform to go from % to 9% of it fial value. The teady-tate error i the error after the traiet repoe ha decayed leavig oly the cotiuou repoe. The error igal: e(t) r(t) e -t/t e -t/t A t approache ifiity, e -t/t approache zero ad the teady-tate error i: e e( ) lim r(t) t [ ] The larger the time cotat T i, the lower the ytem repoe i. It i oted that the traiet repoe domiate the repoe of the ytem at time immediately after the iput i applied ad ca make igificat cotributio to the ytem repoe whe the time cotat i large. Y() Exercie: A RC circuit ha the followig trafer fuctio: R() + 4 For a tep iput r(t) V, what i the time take for the output of the RC circuit to reach 95% of it teady-tate repoe? Y() 5 Exercie: A ytem ha trafer fuctio: R() + 5 Fid the time cotat, T, the ettlig time, T, ad the rie time, T r for a uit-tep iput. 4

5 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3).. Uit-ramp repoe of firt-order ytem The iput, r(t) t for t. 6 Uit-ramp repoe, T Laplace traform: R() /. 5 The output traform: () Y T + 4 Steady-tate error Expadig Y() ito partial fractio: T T Y() + T + 3 r(t) t Takig the ivere Laplace traform: t - T + Te -t/t, t. Steady-tate error: e e( ) lim[ r(t) ] t T t / T.3. Uit-impule repoe of firt-order ytem The uit-impule iput, r(t) δ(t), t t δ( t) t. /T Uit-impule repoe, T Laplace traform: R()..8 The output traform: Y() T + Takig the ivere Laplace traform: e -t/t /T, t. Note that the impule iput yield the trafer fuctio of the ytem a output t / T 5

6 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) 3. Secod-Order Sytem Example : Mechaical ytem For the mechaical ytem how i the figure, m i the ma, k i the prig cotat, b i the frictio coefficiet, u(t) i the exteral force ad i the diplacemet. From Newto ecod law force ma: d d + b + k u(t) dt dt m k b m u(t) Example : Electrical ytem: RLC circuit Uig Kirchhoff law: di(t) u (t) Ri(t) + L + dt d where i (t) L dt Hece: d LC dt + u(t) R d RC + u(t) dt L C A geeral form of a ecod order ytem i: d + ζω + ω dt dt d kω u(t) Trafer fuctio: Y() R() kω + ζω + ω k: the gai of the ytem ζ: the dampig ratio of the ytem ω : the (udamped) atural frequecy of the ytem Solutio (root, or pole of the ytem) of the characteritic equatio are: Three cae: ζω ω ζ ad ζω + ω ζ ζ, critically damped cae ζ >, overdamped cae < ζ <, uderdamped cae We will tudy the above cae whe k for implicity. 6

7 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) 3.. Step Repoe of Secod-Order Sytem 3... Critically damped cae (ζ ) Two equal pole: -ζω ω For a uit-tep iput R() /, the output i: Y() ( + ω ) ω Expadig Y() ito partial fractio: Y() + ω ( + ω ) + t Takig the ivere Laplace traform: e ( t) Steady-tate error: e( ) Uit-tep repoe of d-order ytem (critically damped cae) -plae jω ω 3 ω -ζω σ Exercie: A ytem ha the followig trafer fuctio: Y() R() t (ec) What i the tate of dampig of the ytem whe it i ubjected to a uit-tep iput? Determie the atural frequecy of the ytem. 7

8 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) 3... Overdamped cae (ζ > ) We ca write the trafer fuctio of a ecod-order ytem by factorig the deomiator a: Y() ω ω R() ( + ζω + ω ζ )( + ζω ω ζ ) ( )( ) Two pole i the -plae: ζω ω ζ ad ζω + ω ζ For a uit-tep iput, the output i: ω Y() ( )( Takig the ivere Laplace traform yield the time repoe (prove thi time repoe a a exercie): t t e e Whe ζ i much greater tha uity, i.e. ζ >>, the >> ad the term ivolvig i the time repoe will decay fater tha the term ivolvig. The term ivolvig ca therefore be eglected ad the ytem become firt-order decided maily by the pole : Y() / >> ω ω ω / R() ( )( ) ( / )( ) ) Sice ω, the trafer fuctio become: Y() R() The uit-tep time repoe i: ( )t e, t Uit-tep repoe of d order ytem (overdamped cae), ζ, ω.9.8 e t -plae jω ω ( ζ ω ( ζ + ζ ζ ) ) σ e t e t ζ>>, ca be eglected. t (ec) 8

9 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) Uit-tep repoe of d order ytem (overdamped cae), ζ., ω e t e t e t t (ec) ζ, caot be eglected Uderdamped cae ( < ζ < ) Trafer fuctio: Y() R() ( + ζω ω + jω )( + ζω d jω d ) where ω d ω ζ i called the damped atural frequecy. The two pole are: ζω jω ζ ad ζω + ω j ζ -ζω +jω d jω ω d -plae ζ θ ta ζ co(θ) ζ θ -ζω ω σ For uit-tep iput R() /, the output i: -ζω -jω d -ω d Y() + ζω + ζω + ω + ζω ( + ζω ) + ω d ζω ( + ζω ) + ω d Takig the ivere Laplace traform uig the table of Laplace traform yield: - + ζω ζωt e co( ω ( + ζω ) + ωd - dt) ζω ζωt e i( ω ( + ζω ) + ωd d t) 9

10 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) The time repoe i: e where ζ θ ta. ζ ζω t e ζω t ζ co( ω d i( ω t) + d t + θ) ζ ζ i( ω d t) Whe ζ, the repoe become udamped ad ocillatio cotiue idefiitely at frequecy ω. The time repoe i thi cae become: co(ω t) The atural udamped frequecy, ω, i the frequecy of ocillatio of the ytem without dampig Uit-tep repoe of d order ytem (uderdamped cae), ω 3 ζ ζ. ζ.5 ζ t (ec)

11 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) 3.. Traiet ytem pecificatio.4 Uit-tep repoe of a d order ytem, ζ.9, ω 3. Maximum overhoot T r T p T t (ec) 3... Maximum overhoot The maximum amout by which the ytem output repoe proceed beyod the deired repoe. Let y max deote the maximum value of, ad y y( ) the teady-tate value of, the the maximum overhoot of i defied a: maximum overhoot y max - y The maximum overhoot i ofte repreeted by a percetage of the fial value of the tep repoe: y max y ζπ PO ( percet overhoot) % exp y ζ 3... Peak time, T p The time required for the repoe to reach the firt peak of the overhoot: π Tp ω ζ Rie time, T r The time required for the tep repoe to rie from % to 9% of it fial value for critical ad overdamped cae, ad from % to % for uderdamped cae. For the uderdamped cae: T r π θ ω d π ta ω ( ζ ζ / ζ)

12 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) Settlig time, T The time required for the tep repoe to ettle withi a certai percetage of it fial value. A frequetly ued figure i % i which cae the ettlig time i approximately: 4 T ζω 3 For varyig withi 5% of the fial value, the ettig time i: T ζω Note that NOT all thee pecificatio ecearily apply to ay give cae. For example, for a overdamped ytem, the term peak time ad maximum overhoot do ot apply. The traiet behaviour of a ecod-order ytem ca be decribed by: the wifte of the repoe, a repreeted by T r ad T p the cloee of the repoe to the deired repoe, a repreeted by PO ad T. From the deig requiremet, the wifter ad cloer, the better. However, whe ω i fixed, mall T r ad T p require a mall ζ, while mall T ad PO require a large ζ. Note that ( ζπ / ζ ) PO exp, T π/[ ω ζ ], T [ π ta ( ζ / ζ)]/[ ω ζ ]. p Thee lead to coflictig requiremet. A compromie mut be obtaied ometime. Exercie: A ecod-order ytem i uderdamped with a dampig ratio of.4 ad a atural frequecy of Hz. Fid: a) the trafer fuctio b) the time repoe whe it i ubjected to a uit-tep iput c) the percetage overhoot with uch a iput d) the rie time r

13 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) Exercie: Give the trafer fuctio: fid T p, PO, T ad T r. G () Ramp repoe of a ecod-order ytem The Laplace traform of a uit-ramp iput i R() / The output i: kω Y() ( + ζω + ω ) Agai, there are three cae: ζ, critically damped cae ζ >, overdamped cae < ζ <, uderdamped cae 3

14 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) 3.4. Impule repoe of a ecod-order ytem The Laplace traform of a uit-impule iput i R(). The output traform i therefore equal to the trafer fuctio of the ytem, i.e.: kω Y() + ζω + ω Fact: the uit-impule fuctio i the time derivative of the uit-tep fuctio. Therefore, the impule repoe of a LTI ytem ca be foud from the time derivative of the tep repoe for a give dampig. Takig the example of a critically damped ytem where ζ, the uit-tep repoe i give by: t e ( + t) The uit-impule repoe i therefore: d dt ω te t Note, agai, that the impule iput give the trafer fuctio of the ytem. 4

15 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) 4. Siuoidal Repoe of the Sytem Although tep repoe are commoly ued i both imulatio ad experimetal tet, it i alo commo to udertake frequecy repoe tet o the ytem. The frequecy repoe of a ytem i defied a the teady-tate repoe of the ytem to a iuoidal iput igal. u(t) U o i(ωt) y (t) U o G(jω) i(ωt+φ) U o G(jω) U o φ t The liear, time-ivariat ytem G() ubjected to a iuoidal iput of amplitude U o ad frequecy ω decribe by: u(t) U o i(ωt) will, at teady tate, have a iuoidal output of the ame frequecy a the iput but, geerally, with differet amplitude ad phae give by: y (t) U o G(jω) i(ωt + φ(ω)) where U o G(jω) i the amplitude of the output ie wave: G(jω ) {Re[G( jω)]} + {Im[G( jω ad φ(ω) i the phae hift i radia or degree give by: Im[G( jω)] φ( ω) ta G( jω) Re[G( jω)] The iuoidal trafer fuctio of ay liear ytem i obtaied by ubtitutig jω for i the trafer fuctio of the ytem. )]} Proof: Coider a ytem decribed by: Y() G() U() u(t) U() G() y() Y() The iput u(t) i a ie wave with ad amplitude U o ad frequecy ω: u(t) U o i(ωt) U oω The Laplace traform of u(t) i: U() + ω 5

16 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) With zero iitial coditio, the Laplace traform of the output i: U oω Y() G() + ω A partial fractio expaio of a geeral ytem (aumig the pole of G() are ditict) yield: * c c c c c Y() + + L p + p + p + jω jω Partial fractio term from G() * where c are cotat ad c ad c are a complex cojugate pair that ca be obtaied uig the cover-up rule: G( jω)u o * G(jω)U o c c j j Sice G(jω) i a complex quatity, it ca be writte i the form: G(jω) G(jω) e jφ where G(jω) i the magitude ad φ i the phae give repectively by: Im[G( jω)] G(jω ) {Re[G( jω)]} + {Im[G( jω)]} φ( ω) ta G( jω) Re[G( jω)] Similarly: G(-jω) G(-jω) e -jφ G(jω) e -jφ Therefore: c G( jω) e j jφ U o c * G(jω) e j jφ U o The time repoe that correpod to Y() i: p t pt c e + c e + L + c pt jωt * j e + ce + c e ω pt If all the pole of the ytem repreet a table behaviour, the atural uforced repoe ( c e decay to zero at t ) will die out evetually ad therefore the teady-tate repoe of the ytem will be due olely to the iuoidal term which i caued by the iuoidal excitatio, i.e. jωt * jωt y (t) lim c e + c e Subtitutig for c ad t * c ad otig that i(x) (e jx e -jx )/(j), give the teady-tate output: y (t) U o G(jω) i(ωt + φ) t Advatage of frequecy domai aalyi: The trafer fuctio of complicated compoet ca be determied experimetally by frequecy repoe tet (without derivig their mathematical model) uig available igal geerator ad precie meauremet equipmet (e.g. pectrum aalyer). 6

17 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) The amplitude ad phae of the frequecy repoe ca be ued to predict both time-domai traiet ad teady-tate ytem performace. Sytem may be deiged to achieve traiet ad teady-tate requiremet uig frequecy repoe aalyi, ad uch aalyi ad deig may be exteded to certai oliear cotrol ytem. Exercie: For the iuoidal iput u(t) i(t) applied to the ytem: determie the teady-tate output of the ytem. G(), +. Liear Simulatio Reult Amplitude Time (ec.) 7

18 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) There are two commoly ued repreetatio of iuoidal trafer fuctio: ) Nyquit or polar plot, ad ) Bode diagram We hall focu o the more popular Bode aalyi ad how how we ca ue MATLAB to produce thee plot. 5. Bode Diagram (Aymptotic Approximatio) The frequecy repoe of a ytem G() ca be decribed a: G() jω G( jω) Re[G( jω)] + Im[G( jω)] R( ω) + jx( ω) A Bode diagram coit of two graph: - plot of the logarithm of the magitude of the a iuoidal trafer fuctio, G(jω) - plot of the phae agle, φ(ω) both are plotted agait the frequecy ω (rad/) o a logarithmic (bae ) cale. Logarithmic magitude (alo called gai) of G(jω), M log G(jω) (uit i decibel, db) Phae: φ( ω) ta X( ω) R( ω) Advatage of Bode diagram: Bode plot of ytem i erie imply add, which i quite coveiet. For example, coider the trafer fuctio: b m ( z)( z ) L( z m ) G() ( p )( p ) L( p ) The magitude of the frequecy repoe of the ytem i give by: b m ( z) ( z ) L ( z m ) G(jω) ( p ) ( p ) L ( p ) Takig the logarithm yield: jω log G(jω) log b m + log ( - z ) + log ( z ) + - log ( - p ) - log ( p ) - jω Kowig the repoe of each term, the algebraic um would give the total repoe i db. Bode' importat phae-gai relatiohip i plotted uig aymptotic approximatio o a logarithmic cale, which mea a wider rage of ytem behaviour, from low to high frequecie, ca be diplayed o a igle plot. Dyamic compeator (cotroller) deig ca be baed etirely o Bode plot. 8

19 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) Example: Fid the Bode plot for the followig RC filter with: G(jω ) RC( jω) + jωt + R Solutio: u(t) C Magitude: G(jω ) + ω T / T / T + ω Logarithmic Gai (db) log ω / G(jω) log (/ T) log (/ T + ) At low frequecie (ω ): gai (db) log (/ T) log (/ T) db At high frequecie (ω ): gai (db) log (/ T) log ( ) ω If we plot the logarithmic gai (db) agait log ( ω ), the the above equatio become the traight lie: y c + mx where y log G(jω ), c itercept, m lope, ad x log ( ω). Thu the logarithmic gai reduce by db (egative lope) per factor of (decade) icreae i frequecy ω (db/decade). The traitio betwee the high ad low frequecy aymptote i foud by equatig the low ad high frequecy limit thi i kow a the corer or cut-off frequecy: ω c T Sice G jωt (jω) R( ω) + jx( ω) + jωt + ω T + ω T X R Phae: φ( ω) ta ta ( ωt) (rad) At low frequecie (ω ): φ() ta ( ) rad At high frequecie (ω ): φ( ) ta ( ) π / At corer frequecy (ω /T): φ(/ T) ta ( ) π / 4 rad rad 9

20 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) 5 Aymptotic curve of G(jω) Aymptote Corer frequecy Aymptote Aymptotic curve of G(jω) Exact curve db -5 Exact curve φ -π/4 Aymptote /T./T /T /T ω -π/./t./t /T /T ω Exercie: Sketch the Bode plot for the followig trafer fuctio: G() 4 +

21 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) 3 Bode Diagram Phae (deg) Magitude (db) ω (rad/ec) 5.. Bode plot with MATLAB The MATLAB commad bode(sys) compute the logarithmic gai ad phae agle of the frequecy repoe of the LTI SYStf(um,de), where um ad de are the umerator ad deomiator coefficiet of the ytem, repectively. For example, to plot the Bode diagram how for the trafer fuctio of the previou exercie, we eter o the MATLAB commad lie: um [ ]; de [4 ]; SYS tf(um,de); bode(sys) or bode(um,de)

22 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) 3 Bode Diagram 5 Phae (deg); Magitude (db) Frequecy (rad/ec) 6. Baic Fact About Egieerig Sytem ENGINEERING SYSTEMS OFTEN ARE REQUIRED TO OPERATE IN A STEADY OUTPUT CONDITION with the ytem deiged o the bai of achievig the bet output from the give raw material or power available. The deig of the ytem to produce the deired teady-tate i a problem i it ow right but ot the ubject matter of thi coure. OPERATING CONDITIONS ARE OFTEN CHANGED BY OPERATOR OR COMPUTER INTERVENTION uch a chage i power upplied by a power tatio to the atioal grid due to chage i the overall atioal power coumptio. ENGINEERING SYSTMES HAVE DYNAMICAL BEHAVIOURS they do ot jut produce the deired output temperature, preure, cocetratio, voltage, curret, frequecy, poitio, velocity, acceleratio, force, torque, flow, level, cocetratio, reactio rate,.. etc eve if that output variable i required to be cotat. Thi ca be due to the effect of diturbace (kow or ukow), phyical effect withi the ytem or huma itervetio/iterferece uch a chage required i operatio coditio. ENGINEERING SYSTEMS REQUIRE CONTROL to couteract the dyamic behaviour preferred by the ytem ad replace them by acceptable dyamic repoe, the proce mut be augmeted by a cotrol ytem icorporatig feature of output meauremet (eor), iput variable chage (actuator) ad a (dyamical) data proceig device that procee both eor data ad deired output pecificatio to geerate a deired plat iput igal to the actuator. Thi ytem almot alway ha a FEEDBACK tructure reflectig the fact that it output i ued to create the deired iput i real time. CONTROL SYSTEMS REQUIRE DESIGNING otwithtadig the impreio left by the popular TV ciece programme Tomorrow World, cotrol ytem are deiged rather tha beig available (without thought required from the uer) i a form that imply eed to be hooked

23 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) up to the plat. Eve thoe cotroller that are available commercially require a elemet of deig either off-lie or o-lie ad hece a effective egieer require a appreciatio of the deig proce ad deig tool curretly available. CONTROLLERS DEPEND ON THE DETAILED DYNAMICAL CHARACTERISTICS OF THE PROCESS TO BE CONTROLLED If that were ot the cae the it would oly be eceary to have oe cotrol ytem o ale. Practical experiece ha how that thi i ot feaible it i foud that it i eceary to have appropriate data ad ome udertadig of the proce to be cotrolled. Thi typically take the form of EITHER ) a mathematical model expreed i differetial equatio, trafer fuctio or tate pace form, AND/OR ) data o the behaviour of the plat output() i repoe to kow iput (uch a tep or iuoid) from which (a) the deired parameter for the model (obtaied i ()) ca be etimated or (b) if a phyical model i ot available a model ca be cotructed from a curve fittig or a, o-called, idetificatio procedure. THE DEGREE OF CONTROL OBTAINED DEPENDES ON THE AVAILABILITY OF SUITABLE MEASUREMENTS OF SYSTEM BEHAVIOUR ad the more accurate ad exteive thee meauremet are, the better cotrol will be. YOU CANNOT ALWAYS MEASURE WHAT YOU WANT TO MEASURE if you caot meaure the output variable of iteret (due to extreme phyical coditio of peed, temperature or preure.. etc, it i eceary to create a itelliget device which oberve the (available) meauremet ad ue them to create a ueful etimate of the (uavailable) output. A a example, how i it poible to cotrol the temperature i the cetre of a furace whe the temperature eor are placed o the exteral wall? CONTROLLERS CAN DO AMAZING THINS A cotrol requiremet are a varied a the applicatio ad eed for ew product, eve the virtually impoible ha bee aked for i the earch for the itelliget cotroller. Thi require the developmet of a abtract way of thikig but ha amazig coequece e.g. (a) the developmet of cotrol elemet capable of obervig ad accurately etimatig variable that caot be meaured (b) the developmet of cotrol ytem capable of adaptig to ew ituatio ad learig from experiece. 3

24 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) TUTORIAL PROBLEM SHEET 4. Idetify the gai ad time cotat of the followig firt-order trafer fuctio: 3 G() +. A RC circuit ha the followig trafer fuctio: Y() G() R() + 4 For a tep iput u(t) V for t : a) What i the teady-tate repoe of the circuit? b) What i the time take for the output of the RC circuit to reach 95% of it teady-tate repoe? c) Check your reult with MATLAB. 3. The geeral form of a firt-order ytem i decribed by: d T + ku(t) dt where T i the time cotat, k i the gai, u(t) ad are the iput ad output of the ytem repectively. The uit-tep repoe of a firt-order ytem i how i the followig figure. Determie the parameter k ad T from thi figure..5 Step Repoe Amplitude Time (ec.) 4

25 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) 4. Coider the firt-order ytem, Y() R() T + Obtai the uit-tep repoe curve for T.,.5,., 5. ad. repectively, with MATLAB. 5. Coider the firt-order ytem, Y() k R() + Obtai the uit-tep repoe curve for k.,.5,., 5., ad. repectively, with MATLAB. 6. A geeral ecod-order ytem ha the form: Y() kω R() + ζω + ω What are the value of k, ζ, ad ω for the followig ytem: Y() 3 R() A ecod-order ytem i decribed by the differetial equatio: d d u(t) dt dt a) Write dow the trafer fuctio Y()/U() of the ytem, where U() ad Y() are the Laplace traform of u(t) ad, repectively. b) Obtai the dampig ratio ζ ad the atural frequecy ω of the ytem. c) Calculate the rie time ad percet overhoot of the ytem. d) Evaluate for a uit-tep iput u(t). e) Check your awer of the above with MATLAB. 8. For the cotrol ytem how by the block diagram, the umerical value of J kg-m ad B N-m/(rad/ec). R() + _ K + J + B _ / Y() K a) Fid the trafer fuctio Y()/R(). b) Determie the value of the gai K ad velocity feedback cotat K o that the maximum overhoot i the uit-tep repoe i. ad peak time i ec. c) With thee value of K ad K, obtai the rie time ad ettlig time. d) Obtai the repoe for the uit-tep iput r(t). e) Check the above calculatio with MATLAB. 5

26 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) 9. Whe the ecod-order ytem Y() R() T K + + K i ubjected to a uit-tep iput, the ytem output repod a how i the followig figure. Determie K ad T from the repoe curve. Step Repoe.4.3. Amplitude Time (ec.). A iuoidal iput u(t) i(t) i applied to a ytem with trafer fuctio: Y() U() ( + ) Determie the teady-tate output, y (t), of the ytem.. The figure below how a block diagram of a pace vehicle attitude cotrol ytem where R ad Y are the Laplace traform of the referece (or deired) ad actual attitude agle repectively. Determie the value of K P ad K D to yield a ettlig time of.5 ecod ad % overhoot i the cloe-loop ytem for a uit-tep iput. R() + K P + Y() K D 6

27 ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3). Coider the ecod-order ytem Y() R() + ζ + Obtai the uit-impule repoe curve for ζ.,.3,.5,.7,., ad 4. repectively, with MATLAB. 3. Coider the ecod-order ytem Y() R() kω + ζω + ω Aumig that ω, k, obtai the uit-impule repoe curve for ζ.,.3,.5,.7,., ad 4. repectively, with MATLAB. 7