Novel Analytic Technique for PID and PIDA Controller Design. Seul Jung and Richard C. Dorf. Department of Electrical and Computer Engineering

Transcription

1 Novel Analytic Technique fo PID and PIDA Contolle Deign Seul Jung and Richad C. Dof Depatment of Electical and Compute Engineeing U niveity of Calif onia; Davi Davi, CA jung@ece.ucdavi.edu, dof@ece.ucdavi.edu Abtact : In thi pape a new deign technique fo a popotional, integal, deivative (PID) and a popotional, integal, deivative and acceleation(pida) contolle to atify pecication fo tanient and teady tate epone of a thid ode plant ae peented. Baed on the -plane technique, the cloed loop oot ae foced to line up on one vetical line to atify the given pecication. The PID o the PIDA contolle value ae analytically calculated by equating two chaacteitic equation: one i fomed fom deied oot location with pecication baed on the deign citeia, and anothe one i fomed fom the nominal contol tuctue a hown in Figue. Thu the calculation of value of contolle i o taight fowad that one can eadily geneate a PIDA contolle to atify given pecication. Key wod : -plane, PID, PIDA, contolle deign Intoduction The taditional oot locu technique decibed in leading textbook involve placing a pai of oot at a location which will poduce the deied epone, a pecibed by ettling time and ovehoot (Dof, 995;Ogata,99;Kuo, 99). Thee two oot ae aumed to be entiely dominant in the conventional deign pocedue. In many cae (e.g. a thid ode plant), thee two oot do not tun out to be completely dominant and the deied epone i not obtained (Dof, et al., 995). The PID(popotional-integal-deivative) contolle i widely ued in the induty becaue it can be implemented eaily fo a typical econd ode plant. The deign of a minimum phae PID contolle fo the econd ode plant eult alway in a table ytem and give zeo teady tate eo fo a tep input. The pope choice of the contolle gain can make the oot at the deied location be entiely dominant. Fo the thid ode plant, howeve, it equie an iteative tial eo method to deign a PID contolle to atify the given pecication. Theefoe the oot locu deign method G () C D() R() Σ Contolle Σ Plant - U() G() Figue : The Contol Stuctue C() may lack analytical meit. One way of deigning the obut PID contolle i to pecify quantitatively the equied ytem pefomance by minimizing pecied pefomance indice (Dof, 995). In thi method the pefomance i et to one typical pecication which can not be eaily changed. In ode to achieve a deied epone moe peciely, the novel popotional-integal-deivativeacceleation(pida) contolle i popoed fo a thid ode plant. The PIDA contolle conit of thee zeo and thee pole, but two pole may be neglected in the deign poce. The intoduction of an exta zeo to the PID contolle i to change the oot locu of the thid ode plant in ode to eliminate the eect of non-dominant oot. Locating two dominant oot to povide the deied epone and one oot o two oot on the eal axi jut below dominant oot with o without one negligible oot fa fom the oigin on the eal axi (depending on the type of the plant) yield a deied chaacteitic equation. The chaacteitic equation of the cloed loop tanfe function fo the contolle and the plant i alo found fom the contol tuctue hown in Figue. Thee two chaacteitic equation ae equated. The ytem of equation i then complete and can be algebaically olved. Value fo the contolle ae found fom the olution. The eulting contolle will poduce the deied epone by atifying all the equiement automatically. Vaiou plant example ae teted fo the popoed method, and thei pefomance

2 ae compaed. jw Deign Pefomance Specication Conide the contol tuctue hown in Figue whee a unity feedback i aumed. The PID compenato i dened a C() = K P + K I + K D ( + a)( + b) = K ( + c) ( + c) () And a; b c, whee a; b and c ae zeo and a pole of the PID contolle, epectively and whee c i negligible. The PIDA compenato i dened a * * R - * * q M q^ θ C() = K P + K I + K D ( + d) + K A ( + d)( + e) ( + a)( + b)( + z) = K ( + d)( + e) () Figue : A deied dominant oot egion and deied cloed loop oot location (*) in -plane And a; b; z d; e, whee a; b; z and d; e ae zeo and pole of the PIDA contolle, epectively. Since a; b; z d; e the pole d; e ae conideed to be negligible. The cloed loop tanfe function T () i T () = G()C() + G()C() () whee + G()C() = i the chaacteitic equation. The pefomance of contol ytem hown in Figue i often dened by the tanient epone which include paamete uch a pecent ovehoot(p:o), ettling time(t ), and peak time(t p ). Hee we have et the deied pecication a follow: P:O L () T M (5) Output epone jcj maxj j = max Ditubance ignal < W (6) The value of L; M and W ae elected by the deigne. We know that the teady tate eo e = with the PID o PIDA contolle fo a tep input. Thee pecication detemine how fat, accuate, and obut the ytem i. Dening L and M detemine the deied dominant oot location egion in an -plane a hown in Figue whee = co? (ln ( L ) ). Since the +(ln L ) deigned contolle afte the t attempt uually doe not atify all the pecication, epeated attempt ae needed until it meet all the pecication. Thi poce i quite time conuming. Futhemoe, thi deign technique become moe dicult fo a thid ode plant. The pefomance of a thid ode plant depend on how the othe cloed loop oot beide the two dominant oot contibute to the pefomance, and it often happen that the othe oot caue moe ovehoot and ettling time than deied. Thu the pupoe of thi pape i to povide a new analytical method of deigning a PID contolle fo a econd ode plant and a PIDA contolle fo a thid ode plant atifying the given pecication. Analytic PID Contolle Deign Technique The oot locu method of a PID contolle deign fo a econd ode plant i well developed in (Dof, 995) and faily taight fowad. Hee we ae popoing the analytic deign technique fo a econd ode plant in - plane by putting two dominant cloed loop oot at the deied location and one eal oot at a point fa fom the oigin. Step by tep pocedue ae given below: Step. Detemine! n of the dominant oot fom T pecication whee T =! n. The dominant oot p ae = q and ^q whee q =?! n + j! n?. Step. Detemine of dominant oot fom P.O peci- cation whee = (ln L ) +(ln L ). Step. Select the eal oot o that?! n : Step. Wite the chaacteitic equation fo +GC = and et ( + )( + q)( + ^q) = Step 5. Equate the chaacteitic equation.

3 Step 6. Solve the imultaneou et of equation: equation fo a nd ode G() Step 7. Plot the epone: a. Fo c(t) given (t) = unity tep. b. Fo c(t) given (t) = and d(t) = unity tep. If the P.O i not atied inceae K to minimize the ovehoot. Baed on the pocedue given above conide thee type of the econd ode plant The deied pefomance ae ame fo all the plant a P:O 5%; T ec, e = ; and jcj :5. Type Plant : G() = (+)(+) Following the pocedue tep given above we have Step, and.! n? and :77 fo the egion of the dominant oot location. Then elect the deied dominant oot at?: :j and one negligible oot,, at. Step. The deied chaacteitic equation i + G()C() = ( + )( + : :j) (7) Step 5. Equating two chaacteitic equation yield + ( + K) + ( + K(a + b)) + Kab = + : + : + 5: (8) Step 6. So fa we have thee equation fo thee unknown value. Solving thee equation yield a o b = :957 :89j and K = : C() = : + :9 + 8:9 (9) Step 7. Plot a tep epone to ee if the epone ati- e the pecication. Step 8. The pefomance of thi contolle i lited in Table, and the tep epone i plotted in Figue. All the pecication ae atied. Type Plant : G() = (+) Step,, and. Same a above. Step 5. Equating two chaacteitic equation yield + ( + K) + K(a + b) + Kab = + : + : + 5: () Step 6. Solving thee equation yield a o b = :87 :865j and K = : C() = : + :7 + 7:85 () Step 7. Plot a tep epone to ee if the epone ati- e the pecication. The P.O i 6:6% which i geate than pecied. In ode to educe the P.O we inceae K to. Step 8. The pefomance of thi contolle i lited in Table, and the tep epone i plotted in Figue. Now all the pecication ae atied. Type Plant : G() = Step,, and. Same a above. Step 5. Equating two chaacteitic equation yield + K + (a + b) + Kab = + : + : + 5: () Step 6. Solving thee equation yield a o b = :965 :875j and K = : C() = : + :9 + 7:77 () Step 7. Plot a tep epone to ee if the epone ati- e the pecication. The P.O i 9:% which i geate than pecied. In ode to educe the P.O we inceae K to 7. Step 8. The pefomance of thi contolle i lited in Table, and the tep epone i plotted in Figue 5. Now all the pecication ae atied. Table. Pefomance fo econd ode plant Plant (+)(+) (+) Spec P:O(%) T p (ec)..7. AFAP T (ec).9.6. : max jcj.5.. < :5 e tep(%) e amp(%).8 o 5 Analytic PIDA Contolle Deign Technique The deign of PIDA contolle i dieent fo the diffeent type of the plant (zeo, one, two) ince they have dieent oot locu plot. The oot locu plot of a Type (no pole at the oigin) thid ode plant with a PID contolle i hown a Figue 6. The oot locu of a Type plant(one pole at the oigin) i imila to that of Type plant except fo the location of the cloed loop pole. Fo a Type thid ode plant(two pole at the oigin), the oot locu plot i dieent fom thoe of Type o plant a hown Figue 7 which i untable. It i quite dicult to deign a PID contolle fo a thid ode plant. The PIDA which incopoate an additional zeo i ued hee. The idea behind the PIDA deign technique on -plane i to modify the oot locu plot by adding an additional zeo appopiately o that the dominant oot ae entiely dominant. Fo a Type o Type plant fou oot excluding two negligible

4 oot ae automatically pelocated with thee of them lining up on one vetical line in the left half plane with one additional oot fa fom oigin conideed a not ignicant a hown in Figue. Adding an exta zeo z change the oot locu plot fom Figue 6 to that of Figue 8. Fo a Type plant fou oot ae automatically pelocated. Intead of locating a oot fa fom oigin we foce fou of them to line up on one vetical line in left half plane a hown in Figue 9. The pocedue fo deigning the PIDA contolle in tep i a follow: Step. Detemine! n of the dominant oot fom T pecication whee T =! n. The dominant oot ae p = q and ^q whee q =?! n + j! n?. Step. Detemine of dominant oot fom P.O peci- cation whee = (ln L ) +(ln L ). Step. Select the eal oot equal to the eal pat of the dominant oot: R = Refdominantootg?! n. In ode to minimize the eect fom undominant cloed loop pole it i uggeted to elect R whee it i to the left of the laget open loop pole of a plant not at the oigin in left half - plane. If the plant i Type plant, then elect a double pole at R. Step. Select the eal oot o that?! n if the plant i Type o Type plant. Step 5. Wite the chaacteitic equation fo +GC = and et ( + )( + R)( + q)( + ^q) = fo Type o Type plant o ( + R) ( + q)( + ^q) = fo Type plant. Step 6. Equate the chaacteitic equation. Step 7. Solve the imultaneou et of equation: equation fo a d ode G() Step 8. Plot the epone: a. Fo c(t) given (t) = unity tep. b. Fo c(t) given (t) = and d(t) = unity tep. If P:O i not atied, then inceae the gain K. 5 Contolle Deign Example with Analytical Method Fo a thid ode plant the pocedue i ame a that of the econd ode plant except that we add one zeo z to the chaacteitic equation. The calculation i moe complex becaue we have fou equation to olve. The deied pefomance pecication ae P:O 5%; T ec;and jcj :5. Conide the plant Type Plant : G() = (+)(+)(+6) Following the pocedue tep given above we have Step,, and.! n? and :77 fo dominant oot location egion. Then elect the deied dominant oot at?: :j, one eal oot, R at. and one negligible oot, at. Step 5. + G()C() = ( + :)( + )( + : :j) Step 6. + ( + K) + (K(a + b + z) + 7) + (8 + K(ab + z(a + b))) + Kabz = + 6: + 6: + 5: :8 Step 7. Fou equation fo fou unknown value can be olved. a; b = :9 :5j, z = :88 and K = 6:. C() = 6: ( + : :998)( + :88) () Step 8. Plot a tep epone. The tep epone i hown in Figue and ecoded in Table. The eulting epone meet all pecication. Type Plant : G() = (+)(+7) Step,,, and 5. Same a above. Step 6. +(8+K) +(Kz + K(a+b)+ 7) + K(ab+ z(a + b)) + Kabz = + 6: + 6: + 5: :8 Step 7. a; b = :965 :58j, z = :7 and K = 8:. C() = 8: ( + :59 + 7:65)( + :7) (5) Step 8. Plot a tep epone. The P:O = 6% which i geate than pecied. Inceae K to 5. The tep epone i hown in Figue and ecoded in Table and meet the pecication. Type Plant : G() = (+) Step,,, and 5. Same a above. Step 6. + ( + K) + K(a + b + z) + K(ab + z(a + b)) + Kabz = + 8: + :6 + 5:8 + 7: Step 7. a; b = :7:76j, z = :967 and K = 7:. C() = 7: ( + :6 + :98)( + :967) (6) Step 8. Plot a tep epone. Since the ovehoot i not atied with P:O = 6:8%, inceae the gain K to 55. The tep epone i hown in Figue and ecoded in Table. Table. Pefomance with dieent plant by the analytic method Plant (+)(+)(+6) (+)(+7) (+) P:O% T p (ec) T (ec) max jcj.6.. e tep(%) e amp(%).

5 . Step Repone. Step Repone when (t) i tep : K =.(--) and K = ( ) time (ec) time (ec) Figue : Tanient epone fo a Type econd ode ytem fo K = : with a PID contolle Figue : Tanient epone fo a Type econd ode ytem with a PID contolle 6 Concluion. Step Repone : K =.(--) and K = 7( ) New tep by tep pocedue fo deigning a PID contolle analytically fo a given econd ode plant with pecication ae peented in thi pape. The deied pecication ae met with accuacy fo vaiou type of plant by undegoing the popoed method tep by tep. We addeed the diculty of deigning PID contolle fo a thid ode plant and povided a olution by utilizing a new PIDA contolle with analytical meit and poved thei pefomance by detemining the epone with dieent type of thid ode plant. Since thee analytical deign technique avoid iteative deign pocedue not only the deign poce i eaie and imple but the eulting pefomance ae alo atied. Theefoe ou popoal hee fo a PID o a PIDA contolle deign i o taight fowad and analytical that a novice deigne can eaily develop a equied contolle fo any given pecication without any dicultie. Refeence Dof, R. C. and R. H. Bihop (995) Moden Contol Sytem, 7th edition, Addion Weley Ogata. K.(99) Moden Contol Engineeing, nd edition, Pentice Hall Kuo, B. C.(99) Automatic Contol Sytem, 6th edition, Pentice Hall Dof, R. C., S. Jung, J. Dawe and L. Ng (995) An -Plane Analytic Technique fo Lead-Lag Contolle Deign, Poc. of Ameican Contol Confeence, pp. 7-8, Seattle time (ec) Figue 5: Tanient epone fo a Type econd ode ytem with a PID contolle Imag Axi 5 - Root Locu Plot fo a Thid Ode Plant (Type ) Real Axi Figue 6: Root locu of Type of a thid ode with a PID contolle

6 Root Locu Plot fo a Thid Ode Plant (Type ). Step Repone when (t) i tep.8 Imag Axi Real Axi Figue 7: Root locu of Type of a thid ode with a PID contolle time (ec) Figue : Tanient epone of Type of a thid ode fo a unit tep input with a PIDA contolle Root Locu Plot fo a Thid Ode Plant (Type ). Step Repone when (t) i tep : K = 8.(--) and K = 5( ).8 Imag Axi Real Axi Figue 8: Root locu of Type of a thid ode with a PIDA contolle time (ec) Figue : Tanient epone of Type of a thid ode fo a unit tep input with a PIDA contolle 5 Root Locu Plot fo a Thid Ode Plant (Type ). Step Repone when (t) i tep : K = 7.(--), K = 55( ). Imag Axi Real Axi. time (ec) Figue 9: Root locu of Type of a thid ode with a PIDA contolle Figue : Tanient epone of Type of a thid ode fo a unit tep input with a PIDA contolle