PID SELF-TUNING CONTROLLERS USED IN DIFFERENT STRUCTURES OF THE CONTROL LOOPS

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1 FASCICLE III, Vol.31, No., ISSN X ELECTROTECHNICS, ELECTRONICS, AUTOMATIC CONTROL, INFORMATICS PID SELF-TUNING CONTROLLERS USED IN DIFFERENT STRUCTURES OF THE CONTROL LOOPS Viorel DUGAN a, Adria DAN a, Clara IONESCU b, Mirela COTRUMBA c a Duarea de Jos Uiversity of Galati b Uiversity of Ghet, Belgium, EeSA Departmet c Ovidius Uiversity of Costata Abstract: The paper ivestigates the behavior of digital PID self-tuig cotrollers (STC) i differet structures of the cotrol loops used i adaptive systems. I the two phases of this type of systems which use a STC-PID, the first phase, i.e. the task of recursive idetificatio of the plat model parameters, is used a regressio (ARX) model with the recursive least squares method. Because the quality of process model depeds o the order of the ARX model ad of the sample period (T s ), the digital PID parameters are fuctios of these variables ad, supplemetary, of cotiuous-time PID parameters ad of the cotrol loop structures used i the adaptive system (although, ot so largely as the firstly three variables). To see the latter ifluece, i this paper are cosidered two cotrol loop block diagram, ad for simulatios - three differet processes (stable; with o miimum phase; ustable), ad some simulatios with differet T,!, " s. The PID cotroller desig method used to obtai the specs desired for cotrol loop dyamic behavior was the pole assigmet method of the loop. Keywords: STC- PID adaptive systems, differet feedback loop structures, simulatio, specs comparatio. 1. INTRODUCTION I recet years, the theory of adaptive cotrol has made more ad sigificat developmets. After [1], the basic approaches to the problem of adaptive cotrol are gai-schedulig (GS), model-referece adaptive cotrol (MRAC), self-tuig cotrollers (STC) ad dual cotrol (DC). If the estimates of the process parameters are adjusted ad the cotroller parameters are obtaied from the solutio of a desig problem usig the estimated parameters, the system is viewed as a automatio of process modelig ad desig. I this case the process model ad cotrol desig are updated at each samplig period. This cotroller is called self-tuig cotroller (STC) or self-tuig regulator (STR) because it idetifies ukow processes firstly, ad the sythesies the cotrol (i.e. adaptive cotrol with recursive idetificatio). Hitherto, the most useful STC results have bee achieved, maily, i SISO systems, for which were desiged some stable algorithms with differet complexity. Depedig o the ature of the cotrolled process ad o the geeral task of optimal adaptive cotrol with recursive idetificatio, the STCs ca be implicit or explicit. I the STCs where the idetificatio process does ot serve to determie This paper was recommeded for publicatio by Adria Filipescu 1

2 FASCICLE III, Vol.31, No., ISSN X estimates of the process model parameters, but are used recursively to estimate the cotroller parameters directly, they are referred as beig implicit (Fig. ). Cosequetly, whe the STCs use a sythesis from estimates of the process model parameters (idirect idetificatio), these are called explicit. (Fig. 1). I ay adaptive system, both phases represetig recursive idetificatio ad cotroller parameter calculatio are valid oly at the momet whe the STC is beig set up, i.e. durig the adjustmet phase. After the STC has bee adjusted, these are o-valid because the idetificatio is switched off ad the system is cotrolled with fixed parameters. I adaptive cotrol system both task of idetificatio ad cotrol sythesis have with the same level of importace. the loop. Usig suitable pole cofiguratios, it is possible to fulfill the specs for stability ad a desired closed loop-respose (as overshoot, dampig factor, rise-time etc.). w(k) e(k) u(k) y(k) Fig. 3: The first block diagram of a stadard cotrol loop with STC-PID cotroller, amed with a oe degree of freedom (1DOF) Q s Cotroller parameters computatio Recursive idetificatio of process parameters w(k) e(k) u(k) y(k) q v " Q ( -1 ) w e Cotroller u Process y Fig. 4: The secod block diagram of a cotrol loop with STC-PID cotroller, derived from the structure with two degree of freedom (DOF) [1] Fig. 1: Block diagram of a explicit STC (with direct idetificatio) Q s w Recursive idetificatio cotroller parameters e q Cotroller u v Process Fig. : Block diagram of a implicit STC (with idirect idetificatio) As a result, i the majority of practical cases where STCs are desiged ad used to estimate their parameters is used a ARX regressio model. Much more, the least square method described i [13, 8], has give the best result i o-lie estimatio of the ARX parameters [1, 3, 1, 6]. I the followig, the paper has the itetio to asses the differece betwee the behavior of a STC derived with a stadard block diagram of a closed loop (Fig.3, with 1DOF) ad the behavior obtaied with a differet (better) block diagram (see Fig. 4, with DOF). I the above both cases, the STCs ad closed feedback loops are based o the poles assigmet of y. STANDARD CONTROL LOOP WITH STC PID DIGITAL CONTROLLERS Usually, the cotiuous-time PID cotroller equatio from Fig. 3 or Fig. 4 is, [1,, ad 3]: & 1 # = $ + ' t de( t) u( t) K P e( t) e ( ) d( + T (1) D! % T I dt " u(t) beig the cotroller output, y(t) - process output (cotrolled/ maipulated variable), e(t)- trackig error ad w(t)- the referece sigal (set poit); K P, T I ad T D are the cotroller parameters, respectively proportioal gai, itegral time costat ad derivative time costat. Usig the Laplace trasformatio i (1), we obtai the trasfer fuctio of this cotroller: H C U ( s) & 1 # ( s) K P $ 1+ + TDs! () E( s) % TI s " As i [9, 5, 4, ad 6], usig a (small) samplig period to discretie the itegral ad derivative compoets of (1), ca be obtaied the two digital forms of the PID algorithm: i) the positio algorithms of the PID cotrollers (such amed because it fial output is the maipulatio, also kow as the cotrol actuator positio): 11

3 FASCICLE III, Vol.31, No., ISSN X " T $ () + ( ) % # ( ) + ' + / k s e e k e k e( i) + ( & TI ) i= 1 * & P D u( k) = K +, & T + [ e ( k )! e ( k ) ] & & - T & s. (3) where T s is the samplig time (period) of discretisig the itegral ad derivative of cotiuous-time error e(t), at poits e (kt s ), k =,1,,3,. To discretie the derivative compoet of (3), the most algorithms use a differece of the first order (two-poit, backward differece) ad to approximate the itegral (by simple summig) are used three methods: (a) forward rectagular; (b) backward rectagular ad (c) trapeoidal oe (more accurate). However, all three algorithms are called orecurret algorithms, because all previous error values e (k-1), i = 1,, k, have to be kow to calculate the itegral, ad after the cotroller actio. ii) velocity algorithms or icremetal algorithms for the PID cotrollers are algorithms which calculate the icremet (the chage) Δu (k). They ca be determied from (3) by obtaiig u (k-1) from u (k), ad substractig the resultig expressios. The recurret relatio u(k) = Δu(k) + u(k-1) used i (3) is! Ts " u( k) = KP $ e( k) # e( k # 1) + e( k) + % TI T & D + [ e ( k ) # e ( k # 1) + e ( k # ) ]' Ts ( ad, i geeral form: (4) u k) = q e( k) + q e( k ) + q e( k! ) + u( k 1) (5) ( 1! Depedig o the three methods used to approximate the cotiuous-time fuctio by samplig periods T s of the costat fuctio (step, rectagle, i.e. forward rectagle, backward rectagular or trapeoidal method), ca be obtaied three differet icremetal cotroller parameters q o, q 1, q, as fuctios of K P, T I, T D ad T s parameters: q k = f (K p, T I, T D, T s ), k =, 1,. (6) For example, i the recurret relatio (5) obtaied from (4), the cotroller parameters are: q = K p (1+T s /T I, + T D / T s ), q 1 = - K p (1+T D /Ts), (7) q =K p T D /Ts Much more, these parameters are fuctios ot oly as i (6), but deped of the discretiatio method, ad of the closed loop block diagram. The characteristic polyomial of the stadard cotrol loop with a STC The discrete trasfer fuctios of the cotrolled process ad PID cotroller are (see Fig. 3, -1 = backward time-shift operator, i.e. x (k-1) = -1 x (k)): Y ( ) B( ) b + b H (8) P ad! 1 ( ) =! U ( ) A( ) 1+ a1 + a! U ( ) Q( ) q + q1 + q H R ( ) = E( ) ) (1! )(1 + " ) (9) I (9), the PID cotroller discrete trasfer fuctio stadard form, i.e.! Q( ) q + q1 + q H R ( ) ) (1! ), is used together with a serially coected digital filter: H 1 ( s) = (1) F 1+ " filter used to compesate uwated iterferece of the expressio " 1 (1! ) (! b / b ) 1 + =. From (9) ca be obtaied the cotrol (commad) actio Q( ) U ( ) = E( ), (11) ) Isertig here the polyomials of Q ( -1 ) ad -1 ), the cotroller output usig differece equatios is u( k) = q e( k) + q e( k ) + q e( k! ) (1!" ) u( k ) + " u( k! ) (1) From the stadard block diagram show i Fig. 3, the closed loop trasfer fuctio is ( Y ( ) B( ) Q( ) H ) (13)! W ( ) A( ) ) + B( ) Q( 1 ) To fix the desired pole assigmet for above closed loop trasfer fuctio is ecessary to choose the characteristic polyomial as 1 (! ) = 1 + # d! i i, d " 4 (, 1,, ) i= 1 D d q q q $ (14) 1

4 FASCICLE III, Vol.31, No., ISSN X for the deomiator of (13), i. e. for H! d B( ) [( b + b1 )] )! A( ) (1 + a + a ( P 1 ) (19) A ( ) ) + B( ) Q( ) = D( ) (15) I other words, to achieve the specs is ecessary to select the correct parameters for cotroller polyomials (11), which are the solutios of polyomial (15). We kow from [6], that the most frequetly used method of pole assigmet to obtai a required cotrol respose of a closed loop, is doe by selectig atural frequecy! ad dampig factor! i the characteristic equatio for a secod order plat, as the polyomial D(s) = + s s "! +! =. For the polyomial form of D ( -1 ) have bee chose D ( -1 ) =1+d 1-1 +d - (16) For a samplig period T s, the coefficiets of (16) are, i this case: d1 = # exp( #!" T )cos(" T 1#! ), if" ; d d = # exp( #!" T )cosh(" T 1# ), if! > 1; 1! = exp( #!" ). T Equatig (16) i the right side of (15), the four ukow cotroller parameters ca be obtaied [9], as fuctios of the cotrolled system parameters ad of the umber of poles ad their desired positio i the complex plae. 3. THE SECOND STRUCTURE OF THE CONTROL LOOP WITH A STC PID CONTROLLER Comparig the first stadard loop with the secod closed-loop block diagram from Fig. 4 desiged i [3] ad [4], the polyomial P ( -1 ) has the same form as polyomial P ( -1 ) of the PID cotroller deomiator discrete trasfer fuctio, i.e. P ( ) = (1! )(1 + " ) (17) I Fig. 3, the cotroller trasfer fuctio is! Q( ) q + q1 + q H ( ) (18) R ) (1! ) ad the discrete trasfer fuctio of the process is where b! ad d > is the umber of the time delay steps. I the cotroller equatio from the secod closed loop structure i Fig. 4, i.e. i ( 1 U ) = [" E( )! Q'( ) Y ( )] ()! 1 ) the polyomial P ( -1 ) has the same form as above polyomial (18) for the first cotroller (Fig. 3). The polyomial Q ( -1 ) from (18), is differet here ad has the form (1) Q '( ) = (1! )( q '! q ' ) (1) Usig P ( -1 ) from (17) ad Q ( -1 ) from (1) i (), ca be obtaied cotroller output as: u( k) = #[( q + " u( k # ) +! w( k) ' +! ) y( k) # ( q ' + q ') y( k # 1) + + q ' y( k # )] # (" # 1) u( k # 1) + () The closed loop trasfer fuctio from Fig. 4 is = Y ( ) H( ) W ( ) " B( ) { A( ) ) + B( )[ Q '( ) + " ]} (3) where the characteristic polyomial equatio (i.e. the deomiator) is: D A P B Q! " 1 " 1 " 1 " 1 " 1 ( ) = ( ) ( ) + ( )[ '( ) + ] (4) If the cotrolled process discrete trasfer fuctio has the same polyomials A ( -1 ) ad B ( - 1 ) as i (19), usig (4) ca be foud the four ukow cotroller parameters q, q, β ad γ (q 1 = i the polyomial Q ( -1 )). As above i Fig. 3, i Fig. 4 was used the same pole assigmet method, by choosig the characteristic polyomial of the form (14) i the polyomial (4), to assig the desired pole placemet for the closed loop trasfer fuctio (13). 4. THE EFFECTS OF THE DIFFERENT DISCRETE STC-PID PARAMETERS I order to observe the effects of the differet discrete STC-PID parameters obtaied with the two differet cotrol loop block diagram, were cosidered three kid cotrolled process (c.p.) with the followig trasfer fuctios: i) A stable c.p.: H 1 (s) = 1/ [(3s+1) (s+1)]; ii) A omiimum phase c.p., H : 13

5 H (s) = (-s + 1)/ [(4s+1) (s+1)]; THE ANNALS OF DUNAREA DE JOS UNIVERSITY OF GALATI FASCICLE III, Vol.31, No., ISSN X iii) A ustable c.p., H 3 (s) = (s + 1)/ [(s+1) (4s-1)]; With T s =1, the discretied trasfer fuctios are: H Y ( ) U ( ) B( A( ) ) ! ! 1( ) =! Y ( ) B( )! H ( ) = U ( ) A( ) !! Y ( ) B( ).84!.145 H ( ) = U ( ) A( ) ! 3! Usig a dampig factor ζ=1 (uperiodic critic), samplig-time T s =1s ad differet values for atural frequecy (to observe the recommeded values from [1], i.e..45 "! T ".9 were obtaied differet s STC-PID parameters ad the particular recurret equatios (1), ad () respectively, for each case. The results obtaied for dyamic behaviors of the three processes (systems) are show below. For each STC-PID cotroller desiged ad used i the stadard cotrol loop Fig. 3 with the three above processes, were simulated ad show: step respose for H 1 (s), ad Simulik results (y, w ad u variables of the H 1 () model), i Fig. 5, 6, 7); the same simulatios for the H (s) ad H () models i Fig. 8, 9, 1); fially, for the H 3 (s) ad H 3 () models, i Fig. 11, 1 ad 13, respectively. The results obtaied with the secod closed loop block diagram for the same three models i the same fashio, are show i the Fig. 14, 15, 16 (H 1 (s), H 1 ()), the Fig. 17, 18, 19 (H (s), H ()), ad the Fig., 1, (H 3 (s), H 3 ()), respectively. Fig. 6: The variables y, w, H 1 () Fig. 7: The commad u, H 1 () Results: the first cotrol loop structure (Fig. 5-13, without Fig. 1, i.e. the y, w variables of H 3 ()): Fig. 5: Step respose, H 1 (s) Fig. 8: Step respose, H (s) 14

6 FASCICLE III, Vol.31, No., ISSN X Fig. 9: The variables y, w, H () Fig. 13: The commad u, H 3 (), the ustable system Simulatios obtaied with the secod loop structure (Fig. 14-, without Fig. 1, i.e. the y, w variables of H 3 ()): Fig.1: The commad u, H () Fig. 14: *Step respose, H 1 (s) Fig. 11: Step respose, H3(s), the ustable system Fig.15:*The variables y, w, H 1 () 15

7 FASCICLE III, Vol.31, No., ISSN X Fig.16: *The commad u, H 1 () Fig. 19:*The commad u, H () Fig. 17: *Step respose, H (s) Fig. :*Step respose, H 3 (s), the ustable system Fig. 18:*The variables y, w, H () Fig. :*The commad u, H 3 (), the ustable system 16

8 FASCICLE III, Vol.31, No., ISSN X 5. CONCLUSIONS From the above desig ad simulatio results, the followig coclusios ca be draw: 1. The quality of cotrol with digital STC-PID is affected by some parameters as e.g. samplig period, cotiuous-time PID parameters, cotrol law algorithm, actuator saturatio, iitial parameter estimatio, recursive least squares (RLS) method used to o-lie idetificatio, the type of closed-loop cotrol block diagram (1DOF or DOF).. The two closed-loop cotrol block diagram ad the simulatios for ay of the three processes used i the paper (H i (s), I= 1,, 3), have, each, a differet cotroller equatio, i.e. (1) or (), with differet cotroller parameters. 3. For desig parameters of STC-PID was used pole placemet method via characteristic polyomial (16), i.e. D ( -1 ) =1+d 1-1 +d -, real roots, to obtai a similar dyamic behavior to that of secod-order cotiuous-time systems with a characteristic polyomial D(s) = s + ξω s + ω (where the domiat poles are give by desired dampig factor ξ ad the atural frequecy ω of the closed loop). I all desigs the dampig factor was ξ =1 but differet ω, beig respected the iequality.45 "! T ".9 from the referece [1]. 4. The resposes were compared to see the covetioal specs: percet overshoot (POS); risetime (t r ); settlig-time (t s ) ad steady-state error (ε ss ). 5. As a fial coclusio resulted from all resposes, the secod closed-loop block-diagram (DOF) whe is used with STC-PID, give better results as i the first case, stadard case (1DOF). REFERENCES [1] Astrom, K.J., ad B.Wittemark, (1995), Adaptive Cotrol, Addiso - Wesley Publ. Co., Ic. pp s [] Bobal, V. (1995), Self-Tuig Ziegler-Nichols PID Cotroller, It. J. of Adaptive Cotrol ad Sigal Processig, vol.9, pp [3] Bobal, V., P. Dostal, J. Machacek, ad M. Viteckova, (), Self-Tuig PID Cotrollers Based o Dyamics Iversio Method i Proc. of the IFAC Workshop i Digital Cotrol: Past, Preset ad Future of PID Cotrol, Terrassa, Spai, April 5-7, pp [4] Bobal, V. J. Böhm, J. Fessl, ad J.Machack, (5), Digital Self-Tuig Cotrollers, Spriger-Verlag, Ltd. [5] Bobal, V., Chalupa, P., (3), Self-Tuig Cotrollers Simulik Library, [6] Frakli, G.F., J.D. Powel, ad M. Workma (1998), Digital Cotrol of Dyamic Systems, Addiso Wesley Logma, Ic, Melo Park, CA, pp [7] Levie, W.S. (Ed), (1996), The Cotrol Hadbook, (sectio X, Adaptive Cotrol), CRC Press, Boca Rato, Florida, USA [8] Ljug, L., ad T.Söderstorm, (1983), Theory ad Practice of Recursive Idetificatio, MIT Press, Cambridge, Massachusetts [9] Ljug, L. (1987), System Idetificatio. Theory for the Users, Pretice-Hall, Lodo, UK [1] Perdikaris, G.A. (1991) Computer Cotrolled Systems. Theory ad Applicatios, Kluwer Academic Publishers. [11] Quevedo, J., ad Escobet, T. (Eds), (), Proc. of the IFAC Workshop o Digital Cotrol: Past, Preset ad Future of PID Cotrol, Terrassa, Spai, April 5-7. [1] Ortega, R., ad R.Kelly, (1984), PID Self- Tuers: Some Theoretical ad Practical Aspects, IEEE Tras. Id. Electoics, vol.31, pp [13] Radke, F. ad R. Iserma, (1987), A Parameter Adaptive PID Cotroller with Stepwise Parameter Optimiatio, Automatica, vol. 3, pp [14] Söderstrom, T., ad P. Stoica, (1989), System Idetificatio, Pretice-Hall, Eglewood Cliffs, NJ, USA. 17