Stability regions in controller parameter space of DC motor speed control system with communication delays

Transcription

1 Stability region in controller parameter pace of DC motor peed control ytem with communication delay Şahin Sönmez, Saffet Ayaun Department of Electrical and Electronic Engineering, Nigde Univerity, 5124, Nigde, Turkey Abtract Thi paper preent a comprehenive tability analyi of a DC motor peed control ytem with communication delay. An effective and imple graphical method i propoed to compute all tabilizing roportional ntegral () controller gain of a DC motor peed control ytem with communication time delay. The approach i baed on extracting tability region and the tability boundary locu in the controller parameter pace having uer defined gain and phae margin, and relative tability. The time-domain imulation tudie indicate that the propoed cheme give better deired dynamic performance a compared to the recently developed cheme for DC motor peed control with communication delay. eyword: Control Deign, Delay Sytem, DC Motor Speed Control, Gain and hae Margin, Stability region. 1. ntroduction The iue of time delay in feedback control ha been received coniderable attention in recent year with the proliferation of networked control ytem (NCS) [1-3]. The time delay in NCS known a network-induced delay conit of enor-to-controller delay and controller-to-actuator delay. Even though the advance in communication network have reduced the magnitude of the networked-induced delay ignificantly, it till cannot be ignored when deigning a control ytem. t i well known that time delay can degrade the performance of control ytem and can even make cloed-loop ytem untable [4-7]. The DC motor control ytem i a typical example of control ytem in which the undeirable impact of time delay on the ytem dynamic are oberved [3]. DC motor control ytem are table ytem in general when time delay are not conidered. However, inevitable time delay may detabilize the cloed-loop ytem when the DC motor i controlled through a network. Therefore, communication and meaurement delay for tability analyi of a networked-controlled DC motor mut be taken into account in the proce of a controller deign, and method need to be developed to compute the tability boundarie in term of time delay and controller parameter. [8-9]. The decription of the ytem tability boundary in term of time delay for a given et of controller parameter include the computation of maximum time delay, known a delay margin, uch that the ytem will be table. On the other hand, one alo need to identify all poible controller parameter, known a tability region, for a given time delay that guarantie a table operation. Such tability boundarie help u deign an appropriate controller for cae in which uncertainty in network-induced delay i unavoidable. To the bet of our knowledge, the tability of networked control DC motor peed control ytem ha not been comprehenively analyzed, and in particular, the decription of the tability boundary in term of the controller parameter for a given time delay ha not been reported in the literature. The exiting tudie in the tability analyi of timedelayed ytem mainly focu on the tability delay margin computation for a given et of controller parameter. Delay margin computation method could be grouped into two main type, namely frequency-domain direct and time-domain indirect method. The main goal of frequency domain approache i to compute all critical purely imaginary root of the characteritic equation for which the ytem will be marginally table. The following three method are the one commonly ued in delay margin computation: i) Schur-Cohn method [1]; ii) Elimination of exponential term in the characteritic equation [11]; iii) Rekaiu ubtitution [12]. Among thee direct method, the method baed on the 86

2 elimination of exponential term ha been implemented into delay margin computation of DC motor peed control ytem with contant communication delay [13]. The indirect time-domain method that utilize Lyapunov tability theory and linear matrix inequalitie (LM) technique have been ued to etimate delay margin of the timedelayed dynamical ytem with a controller [14-15]. Both frequency and time-domain method dicued above aim to compute delay margin for a given et of controller. However, for a complete picture of the tability of DC motor control ytem, it i alo eential to determine all poible value of controller parameter that enure a table operation when a certain amount of time delay i oberved in the ytem. Becaue, in practical networkedcontrolled ytem, the maximum value of time delay that might be oberved i known and on need to determine the et of controller parameter that make the cloed-loop ytem table. n one of our earlier work [16], a graphical method baed on tability boundary [17-18] ha been applied to tability analyi of time-delayed load frequency control ytem, and a tability region for a given time delay ha been determined in the controller parameter pace, (, ) -plane. However, for the analyi and deign of practical control ytem, beide tability feature, gain margin (GM) and phae margin (M) are two important deign pecification that mut be taken into account in computing tability region [19]. A et of controller parameter that guarantee not only the tability but alo deired gain-phae margin of the time-delayed DC motor peed control ytem mut be determined. For that purpoe, the time-delayed DC motor peed control ytem [13] i modified to include a frequency independent Gain hae Margin Teter (GMT) a a virtual compenator in the feedforward part of the DC motor peed control ytem model. Thi paper extend our earlier work [13] to compute all tabilizing controller parameter that enure a deired dynamic performance of time-delayed DC motor peed control ytem with uer defined phaegain margin. The approach i baed on the tability boundary locu, which can be eaily obtained by equating the real and imaginary part of the characteritic equation to zero. The propoed method ha been effectively applied to controller deign and ynthei of time-delayed integrating ytem [2] and large wind turbine ytem [21]. The impact of the uer-defined phae and gain margin on the tability region i analyzed. t ha been oberved that the tability region become maller a the gain and phae margin increae. n addition to gain-phae margin pecification, in the analyi and deign of control ytem, it i alo important to hift all pole of the characteritic equation of control ytem to a deired region in the complex plane, for example, to a hifted half plane that guarantee a pecific ettling time of the ytem repone. n thi paper, a relative tability region (, ) -plane i alo determined. All value of (, ) will put all the cloed loop pole to the left of a deired = σ ( σ i contant). Finally, the accuracy and effectivene of the propoed method are alo verified by uing time-domain imulation capabilitie of Matlab/Simulink [22]. 2. Modified DC Motor Sytem Model With GMT The block diagram of a DC motor ytem with a communication delay into the control loop i hown in Figure 1. The dynamic of a DC motor driving a load i decribed by a differential equation of the mechanical ytem and volt-ampere equation of the armature circuit [3], [6]. dω J m + Bωm + Tl = Te = ia dt u() t = v = e + R i + L di dt a a aa a a (1) where u = va i the armature winding input voltage, i a, R a and L a are the current, reitance, and inductance of the armature circuit; repectively; ea = aωm i the back-electromotive-force (EMF) voltage (i.e., generated peed voltage); ω m i the angular peed of the motor; T e and T l are the electromagnetic torque developed by the motor and the mechanical load torque oppoing direction; J i the combined moment of inertia of the load and the rotor, B i the equivalent vicou friction contant of 87

3 the load and the motor; and a are the torque contant and the back-emf contant, repectively. The tranfer function of the DC motor could be eaily obtained a JL G () = a 2 R B R B La J JL a a a. (2) The tranfer function of proportional-integral () controller i given a follow: G () c = + (3) where and are the proportional and integral gain, repectively. The proportional term affect the rate of angular peed rie after a tep change. The integral term affect the angular peed ettling time after initial overhoot. The integral controller add a pole at origin and increae the ytem type by one and reduce the teady-tate error. The combined effect of the controller will hape the repone of the peed control ytem to reach the deired performance. ω ref + Reference - peed Controller G c () roceing delay F e τ e τ B Meaurement and communication delay DC motor G() Motor peed ω Fig. 1. Block diagram of a DC motor control ytem with time delay. ωref + Reference - peed GMT C( Aφ, ) Controller G c () roceing delay F e τ e τ B Meaurement and communication delay DC motor G() Motor peed ω Fig. 2. Modified block diagram of a DC motor control ytem with a GMT. A illutrated in Figure 1, all time delay in the feedback loop that are lumped together into a feedback delay ( τ B ) between the output and the controller. Thi delay repreent the meaurement and communication delay (enor-to-controller delay). The controller proceing and communication delay ( τ F ) (controller-to-actuator delay) i placed in the feedforward part between the controller and DC motor. For the tability analyi, the characteritic equation given below of DC motor control ytem with time delay i required [13]. (, τ ) = () + Qe () τ = (4) where τ = τb + τf i the total time delay. (), Q () are polynomial in with real coefficient given in Eq. (2). τ (, τ ) = p 3 + p2 + p 1 + ( q 1 + q) e = R B R B + p p p 3 = 1, 2 = a +, a a 1 = La J JLa = = JL 1, q JLa q a (5) The uer defined gain and phae margin teter (GMT) a a virtual compenator i added to the feedforward path in control loop of the DC motor ytem a hown in Figure 2. The frequency independent GMT i given in the form: j C( A, φ) = Ae φ (6) where A and φ repreent gain and phae margin, repectively. To find the controller parameter for a given value of gain margin A of the DC motor peed control ytem given in Figure 2, one need to et φ = in Eq. (6). On the other hand, etting A = 1 in Eq. (6), one can obtain the controller parameter for a given phae margin φ. The characteritic equation of the DC motor peed control ytem with a GMT given in Figure 2 i firt obtained. τ jφ (, τ ) = () + Q () e e = (7) τ = () + Q () e = Where 88

4 1 () = () = p + p + p ( ) ( ) Q = A JL + A JL = q + q () a a 1 Note that in Eq. (7) we have an exponential term e τ rather than e τ a in Eq. (1). Thi i obtained by combining e τ j and e φ into a ingle exponential term for = jωc which i the root of Eq. (7) on the imaginary axi. The relationhip between and i given a φ τ = τ + (8) ω c 3. Gain and hae Margin Baed Stability Region n order to obtain the boundary of the tability region, we ubtitute = jωc and ω c > into the characteritic equation in Eq. (7) a follow jωτ c ( ) ( ω ) ( jω, τ ) = A JL e ( j ) + c a c + p3 jωc + p2 jωc + p1 jωc = ( ) ( ) ( ) (9) jωτ Subtituting e = co( ωτ ) jin( ωτ ) into Eq. (9) and eparating into the real and imaginary part, we obtain the following characteritic equation. ( a) c c 2 ( a) co( ωτ c ) 2ωc ( ( a) ωcco( ωτ c ) 3 ( a) ωτ c 3ωc 1ωc) ( jωτ, ) = A JL ω in( ωτ ) + A JL p + j A JL + A JL in( ) p + p = (1) Equating the real and imaginary part of ( jωc, τ ) in Eq. (1) to zero, we get the following equation. where A1( ωc) + B1( ωc) + C1( ωc) = A ( ω ) + B ( ω ) + C ( ω ) = 2 c 2 c 2 c (11) ( ) ( ) ( ) A( JL ) A ( ω ) = A JL ω in( ωτ ); 1 c a c c 2 1 ωc = a ωτ c 1 ωc = 2ωc B ( ) A JL co( ); C ( ) p ; A2 ( ωc) = A JLa ωcco( ωτ c ); B ( ω ) = in( ωτ ); 2 c a c 3 2( ωc) = 3ωc + 1ωc C p p Solving two equation in Eq. (11) imultaneouly, the tability boundary locu (,, ωc) in the (, )-plane i obtained. B1( ωc) C2( ωc) B2( ωc) C1( ω ) = c A1( ωc) B2( ωc) A2( ωc) B1( ωc) A2( ω) C1( ω) A1( ω) C2( ω) = A ( ω ) B ( ω ) A ( ω ) B ( ω ) 1 c 2 c 2 c 1 c (12) Note that the line = i alo in the boundary locu ince a real root of (, τ ) = given in Eq. (9) can cro the imaginary axi at = jω c = for =. Therefore, the tability boundary locu, (,, ωc) and the line = divide (, ) -plane into table and untable region. Thi part of the boundary locu i known a the Real Root Boundary (RRB) and the one obtained from Eq. (12) i defined a the Complex Root Boundary (CRB) of the tability region [17-18]. 4. Relative Stability Region n thi ection, the graphical approach i further developed to obtain relative tability region. n the analyi and deign of control ytem, it i important to hift all pole of the characteritic equation of time-delayed DC motor control ytem to a deired region in the complex plane, for example, to a hifted half plane that guarantee a pecific ettling time of the ytem peed repone. The objective of thi ection i to compute all value of (, ) that will put all the cloed loop pole to the left of a deired = σ (σ i negative contant). Uing + σ intead of in Equation (4)-(5) we obtain the correponding characteritic equation a 89

5 τ( + σ) ( + στ, ) = ( + σ) + Q ( + σ) e = 1 ( + στ, ) = p ( + σ) + p ( + σ) + p ( + σ) ( + στ ) ( JL ) e ( σ ) + ( + ) + = a (13) Firt we ubtitute = jωc with ω c > and ( jωc + στ ) e = e [ co( ωτ c ) jin( ωτ c )] into Eq. (13) and equate it real and imaginary part into zero to determine the following equation. A1( ωc) + B1( ωc) + C1( ωc) = (14) A2( ωc) + B2( ωc) + C2( ωc) = where ( JL )( ) A1 ( ωc) = a e σco( ωτ c ) + e ωcin( ωτ c ) ; ( JL ) B1 ( ωc) = a e co( ωτ c ); c ( ) ( ) ( JLa)( c c c ) C1( ωc) = ω 3 σ p3 + p2 + p1σ + p2σ + p3σ ; A2 ( ω) = e ω co( ωτ) e σin( ωτ) ; ( JL ) B2 ( ωc) = a e in( ωτ c ); C2( ωc) = p3ωc + ωc p1+ 2p2σ + 3p3σ ( ) (15) Solving two equation in Eq. (14) imultaneouly, the tability boundary locu (,, ωc) in the (, )-plane i obtained a: B ( ω ) C ( ω ) B ( ω ) C ( ω ) 1 c 2 c 2 c 1 c = A1( ωc) B2( ωc) A2( ωc) B1( ωc) (16) A2( ωc) C1( ωc) A1( ωc) C2( ωc) = A1( ωc) B2( ωc) A2( ωc) B1( ωc) Similar to the previou ection, we need to et = jω c = to determine the real root boundary. t i clear from Equation (14)-(15) that A2, B2, C 2 become all zero for = jω c =. Then, the functional relation between and could be eaily determined from A1( ω) + B1( ω) + C1( ω) = a ( p3σ + p2σ + p1σ) = σ ( JLa ) e στ (17) Note that the real root boundary defined by Eq. (17) i a traight line (, )-plane. 4. Reult The ection preent tability region baed on both gain-phae margin, and location of the root of the characteritic equation. The parameter of the DC motor control ytem are given in [3] and [13]. 4.1 Relative tability region baed on gain and phae margin The time delay i choen a τ =.5 and the croing frequency i elected in the range of ω [,1 ] rad / for tability boundary locu. Our firt main purpoe i to determine the tabilizing value of and uch that the characteritic equation of Eq. (7) hould be Hurwitz table with deired gain and phae margin. Suppoe our deired phae and gain margin are A 2 and φ 1, repectively, then we ubtitute A = 2 and φ = in Equation (1)-(12) and get the tability boundary locu (SBL) for pecific gain margin with a croing frequency of ω = 3.98 rad /. The correponding tability region i labeled a R1 in Figure 3. Similarly, by ubtituting A = 1 and φ = 1 in Equation (1)-(12), we obtain the SBL for pecific phae margin with a croing frequency of ω = rad /. The tability region i denoted by R2. The interection of R1 and R2 region give the tability region in (, ) -plane hown in Figure 3, and enure the DC motor peed control ytem will have a deired gain and phae margin of A 2 and φ 1. Finally, the tability region without conidering phae and gain margin i determined by ubtituting A = 1 and φ = in Equation (1)-(12). Thi region i repreented by R3 in Figure 3. Oberve that tability region with deired gain and/or phae margin, R1 and R2 are much maller than tability region R3. 9

6 Next, we elect three point each region, ( =.2, =.3) in R1, ( =.2, =.6) in R2, and ( =.2, =.85) in R3 region, repectively. Figure 4 how the peed repone of the DC motor control ytem. t i clear that all repone are table. However, the peed deviation for ( =.2, =.85) contain undeirable ocillation a compared to other two repone. From a practical point of view, uch ocillation are not acceptable. t i clear that the diturbance rejection performance of the DC motor peed control ytem i quicker and non-ocillatory for the (, ) value elected from relative tability region R1 and R2 with deired gain and/or phae margin. Moreover, the impact of the time delay on the ize and hape of the tability region for a given gain-and phae margin i invetigated. Figure 5 how tability region for three different time delay, τ =.5, τ = 1., τ = 1.5 and A = 2, φ =. t i clear that the ize tability region decreae and the hape doe not change a the time delay increae. Fig. 4. Speed deviation repone for three different value controller gain. Fig. 5. Stability region of controller for three different time delay and A = 2, φ =. Fig. 3. Stability region of controller for A 2, φ 1, and τ =.5. Fig. 6. Relative tability region for σ =,.1,.2. 91

7 Fig. 7. Speed deviation repone for three different value controller gain. relative tability region i the ame a the gain-phae baed tability region. Figure 8 illutrate how the ize of the region decreae a the time delay increae for σ = Concluion Fig. 8. Relative tability region for three different time delay and.1 σ =. 4.2 Relative tability region baed on the root location Our econd main purpoe i to obtain all tabilizing value of and uch that the characteritic equation of Eq. (13) hould be Hurwitz table and it root hould lie in a deired region with a boundary = σ ( σ < ) in the left-half complex plane. Similar procedure of the gain-phae margin cae i followed and relative tability region are obtained for three different σ value, σ =, σ =.1 and σ =.2. Figure 6 preent the correponding relative tability region for τ =.5. Note that the region for σ = i the ame region R3 in Figure 3, whoe tability boundary i the imaginary axi. Note that R1 region with σ =.2 and R2 region with σ =.1 are much maller that R3 region with σ = a expected. The complex root boundary of R1 and R2 are determined by Eq. (16) while the real root boundary of R3 i determined by Eq. (17). Time-domain imulation are carried out for three different (, ) value one each region, ( =.2, =.4) in R1, ( =.2, =.75) in R2 and ( =.2, =.85) in R3. Figure 7 depict the peed deviation of the time-delayed DC motor control ytem. Once again, the controller gain choen from R1 give better dynamic performance without having any ocillation in the peed deviation and the peed deviation quickly reache to zero. Additionally, the effect of the time delay on the Thi paper ha preented a graphical method baed on the tability boundary locu to determine tability region in the controller parameter pace for uer defined gain-phae margin and for a deired location of the cloed-loop root of DC motor peed control ytem with communication delay. The relative tability region provide u all tabilizing value of controller gain that enure that the time-delayed DC motor peed control ytem will have deired gain and phae margin. Relative tability region hrink a gain and/or phae margin increae. With the help of the tability region, the controller parameter could be properly elected uch that the DC motor peed control containing certain amount of communication delay will be not only table and but alo will have a deired dynamic performance in term of damping, ettling time and non-ocillatory behavior. Reference [1]. Antakli, and J. Baillieul, Special iue on networked control ytem, EEE Tranaction on Automatic Control, Vol. 49, No. 9, 24, pp [2] W. Zhang, M.S. Branicky, and S.M. hillip, Stability of networked control ytem, EEE Control Sytem Magazine, Vol. 21, No. 1, 21, pp [3] M.Y. Chow, and Y. Tipuwan, Gain adaptation of networked DC motor controller baed on QOS variation, EEE Tranaction on ndutrial Electronic, Vol. 5, No. 5, 23, pp [4] D.S. im, Y.S. Lee, W.H. won, and H.S. ark, Maximum allowable delay bound of networked control ytem, Control Engineering ractice, Vol. 11, No. 1, 23, pp [5] C. Tan, L. Li, H. Zhang, Stabilization of networked control ytem with both networkinduced delay and packet dropout, Automatica, Vol. 59, 215, pp

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