Elecronics 3,, 46-79; doi:.339/elecronics346 Aricle OPEN ACCESS elecronics ISSN 79-99 www.mdpi.com/jornal/elecronics A Simlaive Sd on Acive Disrbance Rejecion Conrol (ADRC) as a Conrol Tool for Praciioners Gerno Herbs Siemens AG, Clemens-Winkler-Sraße 3, Chemniz 96, German; E-Mail: gerno.herbs@siemens.com; Tel.: +49-37-475-883. Received: Ma 3; in revised form: 7 Jne 3 / Acceped: Jl 3 / Pblished: 5 Ags 3 Absrac: As an alernaive o boh classical PID-pe and modern model-based approaches o solving conrol problems, acive disrbance rejecion conrol (ADRC) has gained significan racion in recen ears. Wih is simple ning mehod and robsness agains process parameer variaions, i ps iself forward as a valable addiion o he oolbox of conrol engineering praciioners. This aricle aims a providing a single-sorce inrodcion and reference o linear ADRC wih his adience in mind. A simlaive sd is carried o sing generic firs- and second-order plans o enable a qick visal assessmen of he abiliies of ADRC. Finall, a modified form of he discree-ime case is inrodced o speed p real-ime implemenaions as necessar in applicaions wih high dnamic reqiremens. Kewords: acive disrbance rejecion conrol (ADRC); exended sae observer (ESO). Inrodcion Acive disrbance rejecion conrol (ADRC) [ 4] has emerged as an alernaive ha combines eas applicabili known from classical PID-pe conrol mehods wih he power of modern model-based approaches. The fondaion for ADRC is an observer who joinl reas acal disrbances and modeling ncerainies, sch ha onl a ver coarse process model is necessar in order o design a conrol loop, which makes ADRC an aracive choice for praciioners and promises good robsness agains process variaions. Presen applicaions range from power elecronics [5], moion conrol [6] and spercondcing radio freqenc caviies [7] o ension and emperare conrol [8]. While i can be shown (cf. Secion.3) ha he linear case of ADRC is eqivalen o a special case of classical sae space conrol wih disrbance esimaion and compensaion based on he inernal model
Elecronics 3, 47 principle [9], here is an imporan difference o model-based approaches, sch as model predicive conrol [] or embedded model conrol []: for he laer, an explici model of he process o be conrolled is necessar. ADRC, on he oher hand, does onl assme a cerain canonical model regardless of he acal process dnamics and leaves all modeling errors o be handled as a disrbance. Of corse his ma come a he price of performance losses compared o a conroller bil arond a precise process model or a model of he reference rajecor. Therefore, while emploing he same mahemaical ools, ADRC s nified view and reamen of disrbances can be seen as a cerain deparre from he model-based conrol school [], shifing back he focs from modeling o conrol. This ma be ke o is appeal for praciioners. The remainder of his aricle is organized as follows: Afer providing a sep-b-sep inrodcion o he linear case of ADRC in he following secion, a series of simlaive experimens is carried o o demonsrae he abiliies of ADRC when being faced wih varing process parameers or srcral ncerainies and o visall provide insighs ino he effec of is ning parameers. For he discree-ime case, which is inrodced aferwards, an opimized formlaion is presened, which enables a conroller implemenaion wih ver low inp-op dela.. Linear Acive Disrbance Rejecion Conrol The aim of his secion is o repea and presen he linear case of ADRC in a self-conained manner, following [3,3]. While he majori of aricles inrodces ADRC wih a second-order process, here he firs-order case will be considered firs and explicil, de o is pracical imporance, since here are man ssems albei echnicall nonlinear and higher-order which exhibi a dominaing firs-order-like behavior, a leas in cerain operaing poins. The second-order case will be developed sbseqenl wih a similar se case in mind. Addiionall, i will be shown in Secion.3 ha linear ADRC can be seen as a special case from he perspecive of classical sae space conrol wih disrbance compensaion based on he inernal model principle... Firs-Order ADRC Consider a simple firs-order process, P(s), wih a DC gain, K, and a ime consan, T : P(s) = (s) (s) = K T s + We add an inp disrbance, d(), o he process, abbreviae b = K T T ẏ() + () = K () () and rearrange: ẏ() = T () + T d() + K T () = T () + T d() + b () As or las modeling sep, we sbsie b = b + b, where b shall represen he known par of b = K T and b, an (nknown) modeling error, and, finall, obain Eqaion (). We will see soon ha all ha we need o know abo or firs-order process o design an ADRC is b b, i.e., an approximae vale of K T. Modeling errors or varing process parameers are represened b b and will be handled inernall. ẏ() = ( T () + T ) d() + b () +b () = f () + b () () }{{} generalized disrbance f ()
Elecronics 3, 48 B combining T (), he disrbance d(), and he nknown par b () o a so-called generalized disrbance, f (), he model for or process changed from a firs-order low-pass pe o an inegraor. The fndamenal idea of ADRC is o implemen an exended sae observer (ESO) ha provides an esimae, ˆf (), sch ha we can compensae he impac of f () on or process (model) b means of disrbance rejecion. All ha remains o be handled b he acal conroller will hen be a process wih approximael inegraing behavior, which can easil be done, e.g. b means of a simple proporional conroller. In order o derive he esimaor, a sae space descripion of he disrbed process in Eqaion () is necessar: ( ) ( ) ( ) ( ) ( ) ẋ () x () b = + () + ḟ () ẋ () x () }{{}}{{} A B ( ) (3) ( ) x () () = }{{} x () C Since he viral inp, ḟ (), canno be measred, a sae observer for his kind of process can, of corse, onl be bil sing he inp, (), and op, (), of he process. An esimaed sae, ˆx (), however, will provide an approximae vale of f (), i.e., ˆf (), if he acal generalized disrbance, f (), can be considered piecewise consan. The eqaions for he exended sae observer (inegraor process exended b a generalized disrbance) are given in Eqaion (4). Noe ha for linear ADRC, a Lenberger observer is being sed, while in he original case of ADRC, a nonlinear observer was emploed [4]. ( ) ( ) ( ˆx () = ˆx () ( ) l = l }{{} A LC ) ( ) ( ) ˆx () b l + () + (() ˆx ()) ˆx () l ( ) ( ) ( ) ˆx () b l + () + () ˆx () }{{} B l }{{} L One can now se he esimaed variables, ˆx () = ŷ() and ˆx () = ˆf (), o implemen he disrbance rejecion and he acal conroller. () = () ˆf () b wih () = K P (r() ŷ()) (5) The according srcre of he conrol loop is presened in Figre. Since K P acs on ŷ(), raher han he acal op (), we do have a esimaion-based sae feedback conroller, b he resemblance o a classical proporional conroller is sriking o praciioners. In Eqaion (5), () represens he op of a linear proporional conroller. The remainder of he conrol law in () is chosen sch ha he linear conroller acs on a normalized inegraor process if ˆf () f () holds. The effec can be seen b ping Eqaion (5) in Eqaion (): ẏ() = f () + b () ˆf () b = ( f () ˆf () ) + () () = K P (r() ŷ()) (4)
Elecronics 3, 49 Figre. Conrol loop srcre wih ADRC for a firs-order process. r K P b Process ŷ = ˆx ˆf = ˆx Observer If ŷ() () holds, we obain a firs-order closed loop behavior wih a pole, s CL = K P : K P ẏ() + ŷ() K P ẏ() + () r() If he sae esimaor and disrbance rejecion work properl, one has o design a proporional conroller onl one single ime o obain he same closed loop behavior, regardless of he parameers of he acal process. For example, one can calclae K P from a desired firs-order ssem wih %-seling ime: K P 4 (6) T sele In order o work properl, observer parameers, l and l, in Eqaion (4) sill have o be deermined. Since he observer dnamics ms be fas enogh, he observer poles, s ESO /, ms be placed lef of he closed loop pole, s CL. A simple rle of hmb sggess for boh poles: s ESO / = seso (3...) s CL wih s CL = K P 4 T sele (7) Placing all observer poles a one locaion is also known as bandwidh parameerizaion [3]. Since he marix (A LC) deermines he error dnamics of he observer, we can compe he necessar observer gains for he common pole locaion, s ESO, from is characerisic polnomial: de(si (A LC)) = s + l s + l! = ( s s ESO) = s s ESO s + ( s ESO) From his eqaion, he solions for l and l can immediael be read off: l = s ESO and l = ( s ESO) (8) To smmarize, in order o implemen a linear ADRC for a firs-order ssem, for seps are necessar:. Modeling: For a process wih (dominaing) firs-order behavior, P(s) = K, all ha needs o T s + be known is an esimae b K T.. Conrol srcre: Implemen a proporional conroller wih disrbance rejecion and an exended sae observer, as given in Eqaions (4) and (5): ( ) ( ) ( ) ( ) ( ) ˆx () l ˆx () b l = + () + () ˆx () l ˆx () () = K P (r() ŷ()) ˆf () b l = K P (r() ˆx ()) ˆx () b
Elecronics 3, 5 3. Closed loop dnamics: Choose K P, e.g. according o a desired seling ime Eqaion (6): K P 4 T sele. 4. Observer dnamics: Place he observer poles lef of he closed loop pole via Eqaions (7) and (8): l = s ESO, l = ( s ESO) wih s ESO (3...) s CL and s CL = K P I shold be noed ha he same conrol srcre can be applied o a firs-order inegraing process: P(s) = (s) (s) = K I s () = K I () Wih an inp disrbance, d(), and a sbsiion, K I = b = b + b, wih b represening he nknown par of K I, we can model he process in an idenical manner as Eqaion (), wih all differences hidden in he generalized disrbance, f (): ẏ() = (d() + b ()) +b }{{} () = f () + b () gen. disrbance f () Therefore, he design of he ADRC for a firs-order inegraing process can follow he same for design seps given above, wih he onl disincion ha b ms be se o b K I in sep... Second-Order ADRC Following he previos secion, we now consider a second-order process, P(s), wih a DC gain, K, damping facor, D, and a ime consan, T. P(s) = (s) (s) = K T s + DT s + T ÿ() + DT ÿ() + () = K () (9) As for he firs-order case, we add an inp disrbance, d(), abbreviae b = K and spli b ino a T known and nknown par, b = b + b: ( ÿ() = D T ẏ() T () + ) d() + b () +b T () = f () + b () () }{{} generalized disrbance f () Wih everhing else combined ino he generalized disrbance, f (), all ha remains of he process model is a doble inegraor. The sae space represenaion of he disrbed doble inegraor is: ẋ () x () ẋ () = x () + b () + ḟ () ẋ 3 () x 3 () }{{}}{{} A B () ( ) x () () = x () }{{} x C 3 ()
Elecronics 3, 5 Figre. Conrol loop srcre wih acive disrbance rejecion conrol (ADRC) for a second-order process. r K P b Process K ˆf = ˆx 3 D ŷ = ˆx ŷ = ˆx Observer In order o emplo a conrol law similar o he firs-order case, an exended sae observer is needed o provide an esimaion, ˆx () = ŷ(), ˆx () = ŷ() and ˆx 3 () = ˆf (): ˆx () ˆx () l ˆx () = ˆx () + b () + l (() ˆx ()) ˆx 3 () ˆx 3 () l 3 l ˆx () l = l ˆx () + b () + l () () l 3 }{{ } ˆx 3 () }{{} l 3 }{{} A LC B L Using he esimaed variables, one can implemen he disrbance rejecion and a linear conroller for he remaining doble inegraor behavior, as shown in Figre. A modified PD conroller (wiho he derivaive par for he reference vale r()) will lead o a second-order closed loop behavior wih adjsable dnamics. Again, his acall is an esimaion-based sae feedback conroller. () = () ˆf () b wih () = K P (r() ŷ()) K D ŷ() (3) Provided he esimaor delivers good esimaes, ˆx () = ŷ() (), ˆx () = ŷ() ẏ() and ˆx 3 () = ˆf () f (), one obains afer insering Eqaion (3) ino Eqaion (): ÿ() = ( f () ˆf () ) + () () K P (r() ()) K D ẏ() Under ideal condiions, his leads o: K P ÿ() + K D K P ẏ() + () = r() While an second-order dnamics can be se sing K P and K D, one pracical approach is o ne he closed loop o a criicall damped behavior and a desired % seling ime T sele, i.e., choose K P and K D o ge a negaive-real doble pole, s CL / = scl : K P = ( s CL) and K D = s CL wih s CL 6 T sele (4) Similar o he firs order case, he placemen of he observer poles can follow a common rle of hmb: s ESO //3 = seso (3...) s CL wih s CL 6 T sele (5)
Elecronics 3, 5 Once he pole locaions are chosen in his manner, he observer gains are comped from he characerisic polnomial of (A LC): de(si (A LC)) = s 3 + l s + l s + l 3! = ( s s ESO) 3 = s 3 3s ESO s + 3 (s ESO) s ( s ESO ) 3 The respecive solions for l, l and l 3 are: l = 3 s ESO, l = 3 (s ESO) and l 3 = ( s ESO) 3 (6) To smmarize, ADRC for a second-order ssem is designed and implemened as follows: K. Modeling: For a process wih (dominaing) second-order behavior, P(s) = T s + DT s +, one onl needs o know an approximae vale b K. T. Conrol srcre: Implemen a proporional conroller wih disrbance rejecion and an exended sae observer, as given in Eqaions () and (3): ˆx () l ˆx () l ˆx () = l ˆx () + b () + l () ˆx 3 () l 3 ˆx 3 () ( KP (r() ŷ()) K D ŷ() ) ˆf () () = = (K P (r() ˆx ()) K D ˆx ()) ˆx 3 () b b 3. Closed loop dnamics: Choose K P and K D, e.g. according o a desired seling ime as given in Eqaion (4): K P = ( s CL), KD = s CL wih s CL 6 T sele 4. Observer dnamics: Place he observer poles lef of he closed loop poles via Eqaions (5) and (6): l = 3 s ESO, l = 3 (s ESO), l3 = ( s ESO) 3 wih s ESO (3...) s CL.3. Relaion o Linear Sae Space Conrol wih Disrbance Esimaion and Compensaion Given ha linear ADRC onl emplos ools known from classical linear sae space conrol, how can i be compared o exising approaches? In his secion, we will demonsrae ha linear ADRC can be relaed o sae space conrol wih disrbance esimaion and compensaion based on he inernal model principle [9]. We will sar wih a linear sae space model of a process disrbed b d(), as follows: l 3 ẋ() = Ax() + B() + Ed(), () = Cx() (7) Frher, we assme o possess a model for he generaion of he disrbance, d(): ẋ d () = A d x d (), d() = C d x d () (8) Noe ha A d and C d in Eqaion (8) refer o modeling he disrbance generaor in his secion onl and shold no be misaken for he discree-ime versions of A and C sed in Secion 4. The process model is now being exended b incorporaing he inernal sae variables, x d (), of he disrbance
Elecronics 3, 53 generaor, resling in an agmened process model in Eqaion (9), for which an observer given in Eqaion () can be se p [4]: ( ) ( ) ( ) ( ) ( ) ẋ() A EC d x() B ( ) x() = + (), () = C (9) ẋ d () A d x d () x d () ( ) ( ) ( ) ( ) ( ) ( ( )) ˆx() A EC d ˆx() B L ( ) ˆx() = + () + () C () ˆx d () A d ˆx d () L d }{{} ˆx d () }{{}}{{}}{{} C Ã B L Accordingl, he sandard sae space conrol law can now be enhanced b he esimaed sae variables of he disrbance generaor in order o compensae or minimize he impac of he disrbance on he process, if a siable feedback marix, K d, can be fond: () = G r() K ˆx() K d ˆx d () () Afer insering Eqaion () ino Eqaion (7), i becomes apparen ha provided an accrae esimaion, ˆx d (), is available he disrbance ma be compensaed o he exen ha BK d = EC d can be saisfied [4]. We will now compare he combined sae and disrbance observer based on he agmened process model, as well as he conrol law o linear ADRC presened before. The firs- and second-order case will be disingished b and : Process model and disrbance generaor: When comparing Eqaions () o (4) and (), respecivel, one obains he (doble) inegraor process wih a consan disrbance model. The respecive marices E can be fond sing Eqaions () and (): A =, B = b, C =, E = T, A d =, C d = T = b K ( ) ( ) ( ) ( ) A =, B =, C =, E =, A d =, C d = T = b K b Conrol law: The comparison of Eqaions (5) and () or (3) can be made wih ˆf = ˆx d being he esimaed sae of he disrbance generaor: (K P (r() ŷ()) ˆf () ) = K P r() K P ˆx () ˆf () b b b b gives: K d =, K = K = K P, G = K P = K b b b (K P (r() ŷ()) K D ŷ() ˆf () ) = K P r() K P ˆx () K D ˆx () ˆf () b b b b b gives: K d = ), K = (K b K wih K = K P, K = K D, G = K P = K b b b One can see ha he observer and conrol law are eqivalen in srcre for boh linear ADRC and a sae space approach based on he inernal model principle, and he model of he disrbance generaor in ADRC cold be made more visible b his comparison. If he sandard design procedre of a sae space observer and conroller wih disrbance compensaion will lead o he same parameer vales as ADRC will be verified sbseqenl b following he necessar seps o design K, K d, G and L based on he same design goals sed in linear ADRC before: T
Elecronics 3, 54 Feedback gain K: The closed loop dnamics are deermined b he eigenvales of (A BK). For ADRC, all poles were placed on one locaion, s CL. Wih his design goal, one obains for K: de(si (A BK)) =! s s CL gives K = scl b de(si (A BK)) =! ( ( s s CL) s CL ) gives K = and K = scl b b Gain compensaion G: In order o eliminae sead sae racking errors, G ms be chosen o (, G = C (A BK) B) which gives G = K for boh he firs- and second-order case. Observer gain L: The dnamics of ) he observer for he agmened ssem are deermined b placing he eigenvales of (Ã L C as desired, which is he idenical procedre, as in Eqaions (8) and (6). Disrbance compensaion gain K d : As menioned above, K d shold be chosen o achieve! BK d = EC d if possible:! BK d = b K d = EC d = gives K d = ( ) ( ) b! BK d = = EC d gives K d = b K d b Obviosl, boh designs deliver he same parameers. Based on his comparison, one ma view linear ADRC and is conroller design as a special case of classical sae space conrol wih an observer sing a ssem model agmened b a cerain disrbance generaor model (following he inernal model principle) and disrbance compensaion. However, a sble, b imporan, disincion has o be made: while he laer relies on a model of he plan (like all modern model-based conrol approaches), ADRC does alwas deliberael assme an inegraor model for he plan and leaves all modeling errors o be handled b he disrbance esimaion. Therefore, ADRC can be applied wiho accrael modeling he process, which presens a deparre from he model-based conrol school []. 3. Simlaive Experimens In he ideal case wih noise-free measremens and nlimied, ideal acaors in he conrol loop ADRC wold be able o sppress basicall all effecs of disrbances and parameer variaions of he process. In pracice, however, we have o live wih consrains, sch as limied observer dnamics or saraed conroller ops, leading o a compromise when choosing he parameers of he conroller. This secion is mean o provide insighs ino he abiliies and limiaions of coninos-ime ADRC in a visal manner. To ha end, he conroller designed once and hen lef nchanged will be confroned wih a heavil varing process. The inflence of he ESO pole placemen (relaive o he closed loop poles) will also be examined, as well as limiaions of he acaor. The experimens are carried o b means of Malab/Simlink-based simlaions. In Secion 4., frher simlaions will be performed for he discree-ime implemenaion of ADRC, also addressing he effec of noise and sampling ime.
Elecronics 3, 55 3.. Firs-Order ADRC wih a Firs-Order Process In a firs series of experimens, a coninos-ime ADRC (as inrodced in Secion.) operaing on a firs-order ssem will be examined. The process srcre and nominal parameers sed o design he conroller are: P(s) = (s) (s) = K T s + wih K = and T = The ADRC is designed following he for seps described in Secion.. For now, we assme perfec knowledge of or process and se b = K T =, b will leave his vale nchanged for he res of he experimens in his secion. The desired closed loop seling ime is chosen, T sele =. To his end, he proporional gain of he linear conroller is, according o Eqaion (6), se o K P = T 4 sele = 4. Unless oherwise noed in individal experimens, he observer poles are chosen be s ESO / = seso = s CL = ( K P ) = 4. The respecive vales of he observer gains are obained via Eqaion (8). Throgho Secion 3., noise-free conrol variables and ideal measremens will be assmed. Reacion on nois measremens will be par of frher experimens in Secion 4.. 3... Sensiivi o Process Parameer Variaions Given he explici feare of ADRC o cope wih modeling errors, or firs goal will be o provide visal insighs ino he conrol loop behavior nder (heav) variaions of process parameers. A series of simlaions was rn wih fixed ADRC parameers as given above and a firs-order process wih varing parameers, K (DC gain) and T (ime consan). In Figre 3, he closed loop sep responses are displaed on he lef- and he according conroller ops on he righ-hand side. For boh K and T, vales were chosen from an inerval reaching from % of he nominal vale o %, i.e., a decrease and increase b facors p o en. The resls are qie impressive. One can see ha he closed loop sep response remains similar or nearl idenical o he desired behavior (seling ime of one second) for mos process parameer seings in Figre 3. Almos onl for he five- and en-fold increased ime consan, larger overshoos become visible. In heor, ideal behavior (i.e., almos complee ignorance of parameer variaions) can be obained b placing he observer poles far enogh o he lef of he closed loop poles, as will be shown in Secion 3... Noe ha we are no consrained b acaor saraion here; his case will be examined in Secion 3..3. 3... Effec of Observer Pole Locaions In he previos Secion 3.., he observer poles were placed en imes faser han he closed loop pole (s ESO / = seso = s CL ). How does he choice of his facor inflence he behavior and abiliies regarding process parameer variaions? We will repea he simlaions wih varing ime consan, T, of process boh wih slower and faser observers. To demonsrae a raher exreme case, as well, we will, for he faser seing, appl a facor of s ESO = s CL. The resls in Figre 4 are ordered from fases (op) o slowes observer seing (boom). For he fases seing, he heoreical abili o almos compleel ignore an modeling error is confirmed. In order o achieve his behavior in pracice, he acaor ms be fas enogh and ms no sarae wihin
Elecronics 3, 56 Figre 3. Experimen 3..: fixed firs-order ADRC conrolling firs-order process wih varing parameers. Nominal process parameers: K =, T =. ADRC parameers: b = K T =, T sele =, s ESO = s CL. Variaion of K, closed loop sep response; Conroller op for ; (c) Variaion of T, closed loop sep response; (d) Conroller op for (c)..4..8 K = K =..6 K =. K =.5.4 K =. K = 5 K =.5.5.5 3.4..8 T = T =..6 T =. T =.5.4. T = 5 T =.5.5 (c) 5 5 K = K =. K =. K =.5 K = K = 5 K = 5.5.5.5 3 5 5 T = T =. T =. T =.5 T = 5 T = 5.5.5 he desired range of parameer variaions. For slower observer seings, he acal closed loop dnamics increasingl differ from he desired dnamics, noiceable especiall from larger overshoos for process variaions wih sronger low-pass characer. 3..3. Effec of Acaor Saraion In pracice, he abiliies of an conroller are ied o limiaions of he acaor, i.e., is dnamics and he realizable range of vales of he acaing variable. While he acaor dnamics can be viewed as par of he process dnamics dring conroller design, one has o ake possible effecs of acaor saraion ino accon. If parameers of or example process change, he conrol loop behavior ma be inflenced or limied b acaor saraion in differen was: If he process becomes slower (i.e., he ime consan T increases), acaor saraion will increase he seling ime. If, on he oher hand, he DC gain of (d)
Elecronics 3, 57 Figre 4. Experimen 3..: effec of pole locaions for he exended sae observer (ESO) on a fixed firs-order ADRC conrolling firs-order process wih varing parameer, T. Nominal process parameers: K =, T =. ADRC parameers: b = K T =, T sele =, s ESO varing. Closed loop sep response, s ESO = s CL ; Conroller op for ; (c) Closed loop sep response, s ESO = s CL ; (d) Conroller op for (c); (e) Closed loop sep response, s ESO = 5 s CL ; (f) Conroller op for (e)..4..8 T = T =..6 T =. T =.5.4. T = 5 T =.5.5.4..8 T = T =..6 T =. T =.5.4. T = 5 T =.5.5.4. (c).8 T = T =..6 T =. T =.5.4. T = 5 T =.5.5 (e) 4 3 T = T =. T =. T =.5 T = 5 T =.5.5 5 5 T = T =. T =. T =.5 T = 5 T = 5.5.5 5 5 (d) T = T =. T =. T =.5 T = 5 T = 5.5.5 (f)
Elecronics 3, 58 Figre 5. Conrol loop srcre of firs-order ADRC considering acaor saraion. r K P b Acaor model lim Process Observer ŷ = ˆx ˆf = ˆx Figre 6. Experimen 3..3: effec of acaor saraion, lim 5, on fixed a firs-order ADRC conrolling firs-order process wih varing parameers. Nominal process parameers: K =, T =. ADRC parameers: b = K T =, T sele =, s ESO = s CL. On he righ-hand side, he conroller ops are shown before he limiaion (dashed line) akes effec. Variaion of K, closed loop sep response; Conroller op for ; (c) Variaion of T, closed loop sep response; (d) Conroller op for (c)..4..8 K = K =..6 K =. K =.5.4 K =. K = 5 K =.5.5.5 3.4..8 T = T =..6 T =. T =.5.4. T = 5 T =.5.5 (c) 8 6 4 K = K =. K =. K =.5 K = K = 5 K =.5.5.5 3 8 6 4 T = T =. T =. T =.5 T = 5 T =.5.5 he process decreases, he conrol loop ma be no be able o reach he reference vale nder acaor saraion. In classical PID-pe conrol, some sor of ani-windp sraeg wold be reqired o preven he side-effecs of acaor saraion. We will see in he experimens of his secion ha for ADRC, hose effecs can be overcome ver simpl b feeding he sae observer wih he limied acaing variable, lim (), insead of (), eiher b a measred vale or b an acaor model, cf. Figre 5. (d)
Elecronics 3, 59 Figre 7. Experimen 3..4: effec of dead ime,, on a firs-order ADRC conrolling a process wih firs-order dnamics (K =, T = ) and nknown dead ime. ADRC parameers: b = K T =, T sele =, s ESO = s CL. In (c) and (d), a fixed dead ime, Tdead ESO, was incorporaed ino he observer o improve he conroller behavior. Variaion of, closed loop sep response; Conroller op for ; (c) Closed loop sep response, observer wih Tdead ESO =.5; (d) Conroller op for (c)..4..8.6.4. = =. =.5 =..5.5.4..8.6.4..5.5 (c) = =. =.5 =. 5 4 3 = =. =.5 =..5.5 5 4 3 = =. =.5 =..5.5 To demonsrae he behavior of ADRC nder acaor saraion, he experimens of Secion 3.. will now be repeaed nder an (arbiraril chosen) limiaion of he acaing variable, lim 5. Figre 6 shows he conrol loop behavior wih varing process parameers, K and T. On he righ-hand side of Figre 6, he conroller ops are shown before being fed ino he acaor model, i.e., before he limiaion becomes effecive. One can see ha for redced process gains, K., he reference vale canno be reached anmore. From he respecive conroller ops, i becomes apparen ha () does no wind p when acaor saraion akes effec, b converges o a sead-sae vale. For slower process dnamics, T = 5 and T =, he seling ime increases considerabl, e here is almos no overshoo visible. Since he acaor is saraed, his alread is he fases possible sep response. Obviosl, he conroller recovers ver well from periods of acaor saraion. To smmarize, for pracical implemenaions of ADRC, his means ha apar from he small modificaion shown in Figre 5, here are no frher ani-windp measres necessar. (d)
Elecronics 3, 6 Figre 8. Experimen 3..5: effec of srcral ncerainies on a firs-order ADRC conrolling a process wih dominaing firs-order behavior (K =, T = ) and higher-order dnamics cased b an nknown second pole a s = /T. ADRC parameers: b = K T =, T sele =, s ESO = s CL. Variaion of T, closed loop sep response; Conroller op for..4..8.6.4. T =. T =. T =.5 T =..5.5 3..4. Effec of Dead Time 4 3 T =. T =. T =.5 T =..5.5 Man pracical processes wih dominaing firs-order behavior do exhibi a dead ime. While here are man specialized model-based approaches o conrol sch processes, we are ineresed in how ADRC will handle an nknown albei small amon of dead ime in he conrol loop. In Figre 7, simlaions were performed as in previos experimens, and a dead ime,, was added o he process wih., i.e., p o %, compared o he ime consan, T, of he process. As expeced, oscillaions are ineviabl saring o appear wih increasing, especiall in he conroller op. However, his siaion can be improved if he dead ime of he process is a leas approximael known. An eas wa of incorporaing small dead imes ino ADRC can be fond b delaing he conroller op fed ino he observer b Tdead ESO, i.e., sing ( T ESO dead ) insead of () in Eqaion (4). In Figre 7(c) and (d), his approach was implemened sing T ESO dead oscillaions in he conroller op are less prominen, even if T ESO dead 3..5. Effec of Srcral Uncerainies =.5. Clearl, he does no mach he acal dead ime. In he experimens carried o so far in his secion, i was assmed ha or process cold be reasonabl well described b a firs-order model. In pracice, sch a firs-order model almos alwas resls from neglecing higher-order dnamics, e.g. of he acaor. While i cold alread be seen ha ADRC can handle variaions of parameers ver well, how does i behave if higher-order dnamics become nexpecedl visible in he process? To demonsrae his behavior, a second pole was added o he process in he simlaions from Figre 8, resling in a second ime consan, T., i.e., p o % of he dominan ime consan, T. One can see ha some oscillaions sar o appear dring he ransien as he higher-order pole approaches he dominan pole. While hese resls are accepable, frher simlaions showed ha ADRC did no
Elecronics 3, 6 Figre 9. Experimen 3..6: comparison of firs-order ADRC and PI conroller faced wih varing process parameers, K and T. Nominal process and ADRC parameers are as hrogho Secion 3.. For each combinaion, he closed loop sep response is shown. Variaion of K, ADRC; Variaion of K, PI; (c) Variaion of T, ADRC; (d) Variaion of T, PI..4..8 K = K =..6 K =. K =.5.4 K =. K = 5 K =.5.5.5 3.4..8 T = T =..6 T =. T =.5.4. T = 5 T =.5.5 (c).4..8 K = K =..6 K =. K =.5.4 K =. K = 5 K =.5.5.5 3.4..8 T = T =..6 T =. T =.5.4. T = 5 T =.5.5 provide an advanage comparing o sandard PI conrollers as large as i does in he case of parameer robsness, cf. Secion 3..6. 3..6. Comparison o PI Conrol Given he biqi of PID-pe conrollers in indsrial pracice, how does he sandard approach keep p agains ADRC? To ha end, we will repea he experimen regarding sensiivi owards process parameer variaions and compare he ADRC resls o a PI conroller. For he firs-order process, a PI conroller is sfficien o achieve an desired firs-order closed loop behavior. In order o obain comparable resls, he PI conroller is designed for nearl idenical closed loop dnamics as he ADRC b aiming for he same seling ime and placing a zero on he pole of he firs-order process: (d) C PI (s) = K P + K I s wih K I = 3.85 K T sele = 3.85 and K P = K I T = 3.85
Elecronics 3, 6 Figre. Experimen 3..7: comparison of disrbance rejecion behavior (firs-order ADRC vs. PI conroller). Process and ADRC parameers are as hrogho Secion 3.. Inp disrbance, d =, was effecive from = nil = 4. Sep response and reacion on disrbance; Conroller op for..4..8.6.4. ADRC PI 3 4 5 6 4 3 ADRC PI 3 4 5 6 The simlaion resls in Figre 9 clearl demonsrae he abili of he ADRC approach o keep he closed loop dnamics similar, even nder major parameer variaions. The PI conroller delivers dnamics ha var heavil, as do he process parameers. 3..7. Disrbance Rejecion of ADRC and PI As a final experimen for he coninos-ime firs-order ADRC, we wan o examine he disrbance rejecion abiliies b injecing an inp disrbance ino he process for boh ADRC and he PI conroller from Secion 3..6. On he lef-hand side of Figre, a closed loop sep response is shown. The inp disrbance is effecive dring he period from = nil = 4. While boh conrollers were ned for he same reacion on sepoin changes, he impac of he disrbance is compensaed for mch faser b ADRC compared o he PI conroller. In classical conrol, one wold need o ne he PI conroller mch more aggressivel and add a sepoin filer or follow anoher DOF approach in order o obain similar resls [5]. 3.. Second-Order ADRC wih a Second-Order Process The second-order ADRC will be examined sing a second-order process wih he following nominal parameers: P(s) = (s) (s) = K T s + DT s + wih K =, D = and T = Again, perfec knowledge of he process is assmed, b = K T =, b hen b is lef nchanged hrogho Secion 3.. The desired closed loop seling ime is T sele = 5. The parameers of he PD conroller are, following Eqaion (4), se o K P = ( 6 T sele ) =.44 and KD = T sele =.4. Unless oherwise noed in individal experimens, he observer poles are chosen o be s ESO //3 = seso = s CL =
Elecronics 3, 63 ( ) T 6 sele =. The respecive vales of he observer gains, l //3, are obained via Eqaion (6). As in Secion 3., hrogho his secion, noise-free conrol variables and ideal measremens will be assmed. 3... Sensiivi o Process Parameer Variaions For he second-order ADRC, sensiivi o variaions of he process parameers will be examined firsl. Deviaions from he original DC gain (K = ) were made in a range of % o 5% b seing K o one of he vales.,.,.5, and 5. The damping facor, D, of he process was varied wihin a range of % o % of he original vale, D =, sing seings.,.,.5,, 5 and. Finall, he ime consan, T, of he second-order process was varied in a range of 5% o 35% of he original vale, T =, sing he vales.5,, 3 and 3.5. From he resls in Figre, one can see ha he closed loop behavior is almos no a all affeced, even b relaivel large changes of K and D. Onl for ver small vales of K, he sep response becomes slower; and for ver large vales of he damping, D, some overshoo becomes visible. Changes in he ime consan, T, do, for larger vales, increase overshoo and oscillaions more han in he firs-order case. In order o fll appreciae he robsness owards parameer changes of he process, one has o compare hese resls agains sandard conrollers, sch as PID, as will be done in Secion 3..6. 3... Effec of Observer Pole Locaions In he previos secion, we saw ha changes in he process ime consan, T, affeced he closed loop behavior sronger han changes in K and D. We will herefore demonsrae he inflence of he observer poles on he sensiivi b repeaing he experimens wih varing T for differen observer pole locaions. In he previos experimens, he observer poles were se en imes o he lef of he closed loop poles in he s-plane, i.e., s ESO = s CL. Here, boh slower and faser observers will be examined, as well, b seing s ESO = s CL and s ESO = 5 s CL. As visible in he simlaion resls in Figre, almos ideal behavior can a leas in heor be obained b placing he observer poles far enogh lef in he s-plane, i.e., choosing large facors k in s ESO = k s CL. Of corse, his comes a he price of he need for faser acaors and larger conroller ops. Frhermore as we will see in Secion 4. faser observers are more sensiive o measremen noise and will be limied b sample ime resricions in discree ime implemenaions. 3..3. Effec of Acaor Saraion Analogosl o Secion 3..3, he experimens were carried o b exending he conroller srcre, sch ha he saraed conroller op, lim (), was fed back o he observer insead of () (compare Figre 5 for he firs-order case). The resling sep responses and conroller ops for varing process parameers can be fond in Figre 3. As expeced, lowering he DC gain, K, of he process forced he conroller ino saraion, sch ha he desired process op cold no be reached anmore. I has o be sressed again ha he conroller op did no wind p, as visible in Figre 3.
Elecronics 3, 64 Figre. Experimen 3..: a fixed second-order ADRC conrolling second-order process wih varing parameers. Nominal process parameers: K =, D =, T =. ADRC parameers: b = K T =, T sele = 5, s ESO = s CL. Variaion of K, closed loop sep response; Conroller op for ; (c) Variaion of D, closed loop sep response; (d) Conroller op for (c); (e) Variaion of T, closed loop sep response; (f) Conroller op for (e)..4..8 K =.6 K =. K =..4 K =.5. K = K = 5 5 5.4..8 D = D =..6 D =. D =.5.4 D =. D = 5 D = 5 5.4..8 (c).6 T = T =.5.4. T = 3 T = 3.5 5 5 (e) 5 5 K = K =. K =. K =.5 K = K = 5 5 5 5 6 5 4 3 D = D =. D =. D =.5 D = D = 5 D = 5 5 6 4 (d) T = T =.5 T = 3 T = 3.5 5 5 (f)
Elecronics 3, 65 Figre. Experimen 3..: effec of observer pole locaions for a second-order ADRC conrolling second-order processes wih varing parameer, T. Nominal process parameers: K =, D =, T =. ADRC parameers: b = K T =, T sele = 5, s ESO varing. Closed loop sep response, s ESO = s CL ; Conroller op for ; (c) Closed loop sep response, s ESO = s CL ; (d) Conroller op for (c); (e) Closed loop sep response, s ESO = 5 s CL ; (f) Conroller op for (e)..4..8.6 T = T =.5.4. T = 3 T = 3.5 5 5.4..8.6 T = T =.5.4. T = 3 T = 3.5 5 5.4..8 (c).6 T = T =.5.4. T = 3 T = 3.5 5 5 (e) 5 5 T = T =.5 T = 3 T = 3.5 5 5 5 6 4 T = T =.5 T = 3 T = 3.5 5 5 4 3 (d) T = T =.5 T = 3 T = 3.5 5 5 (f)
Elecronics 3, 66 Figre 3. Experimen 3..3: effec of acaor saraion, lim 3, on a fixed second-order ADRC conrolling a second-order process wih varing parameers. Nominal process parameers: K =, D =, T =. ADRC parameers: b = K =, T T sele = 5, s ESO = s CL. The conroller ops are shown before he limiaion (dashed line) akes effec. Variaion of K, closed loop sep response; Conroller op for ; (c) Variaion of D, closed loop sep response; (d) Conroller op for (c); (e) Variaion of T, closed loop sep response; (f) Conroller op for (e)..4..8 K =.6 K =. K =..4 K =.5. K = K = 5 5 5.4..8 D = D =..6 D =. D =.5.4 D =. D = 5 D = 5 5.4..8 (c).6 T = T =.5.4. T = 3 T = 3.5 5 5 (e) 5 4 3 K = K =. K =. K =.5 K = K = 5 5 5 4 3 D = D =. D =. D =.5 D = D = 5 D = 5 5 4 3 (d) T = T =.5 T = 3 T = 3.5 5 5 (f)
Elecronics 3, 67 Figre 4. Experimen 3..4: effec of dead ime,, on a second-order ADRC conrolling a process wih second-order dnamics (K =, D =, T = ) and nknown dead ime. ADRC parameers: b = K =, T T sele = 5, s ESO = 5 s CL. In (c) and (d), a fixed dead ime, Tdead ESO, was incorporaed ino he observer o improve he conroller behavior. Variaion of, closed loop sep response; Conroller op for ; (c) Closed loop sep response, observer wih Tdead ESO =.; (d) Conroller op for..4..8.6.4. = =. =. =.3 5 5.4..8.6.4. 5 5 (c) = =. =. =.3.6.4..8.6 = =. =. =.3.4 5 5.6.4..8.6 = =. =. =.3.4 5 5 When deceleraing he process (increased damping D), he closed loop dnamics sffered when he conroller op ran ino saraion (increased seling ime), b here were no addiional oscillaions afer recovering from saraion, as known from classical PID conrol wiho ani-windp measres. 3..4. Effec of Dead Time Similar o Secion 3..4, he second-order ADRC was confroned wih an nknown dead ime in he process wih.3. One can see from Figre 4 ha he conroller and observer work ver well ogeher, sch ha he op is hardl being affeced b he dead ime, which is also an improvemen compared o he firs-order case in Secion 3..4. In he conroller op in Figre 4, however, oscillaions become increasingl visible wih larger vales of. Again, his siaion can be improved b delaing he inp of he observer, i.e., sing ( Tdead ESO ) insead of () in Eqaion (). For a (d)
Elecronics 3, 68 Figre 5. Experimen 3..5: effec of srcral ncerainies on a second-order ADRC conrolling a process wih dominaing second-order behavior (K =, D =, T = ) and higher-order dnamics cased b an nknown hird pole a s = /T 3. ADRC parameers: b = K T =, T sele = 5, s ESO = 5 s CL. Variaion of T 3, closed loop sep response; Conroller op for..4..8.6.4. T 3 =. T 3 =. T 3 =. T 3 = 5 5.5 T 3 =. T 3 =. T 3 =. T 3 =.5 5 5 fixed vale, Tdead ESO =., he conroller behavior is shown in Figre 4(c) and (d), where he conroller oscillaions are alread significanl redced, even if Tdead ESO does no mach he acal dead ime. 3..5. Effec of Srcral Uncerainies Besides nknown dead imes, he conroller ma be faced wih higher-order dnamics in he process. Therefore, he effec of an nknown hird pole in he process was examined in Figre 5. The ime consan, T 3, of he hird pole was varied wihin. T 3, i.e., he hird pole was idenical o he wo known poles of he plan in he exreme case. In conras o he firs-order ADRC in Secion 3..5, he second-order ADRC proved o have ver good robsness agains an nknown higher-order dnamics, even in he more challenging cases of Figre 5. 3..6. Comparison o PI and PID Conrol To view and assess he abiliies of he second-order ADRC from a perspecive of classical conrol, we will now emplo sandard PI and PID conrollers and afer fixing heir parameers expose hem o he same process parameer variaions, as done in Secion 3... To ensre comparabili, all conrollers are being designed for he same closed loop dnamics (seling ime T sele = 5, no overshoo) sing he nominal process parameers, K =, D =, T =. The PI conroller is given and parameerized as follows: C PI (s) = K P + K I s wih K I =.55 K T sele =.5 and K P = K I.5 D T =.765
Elecronics 3, 69 Figre 6. Experimen 3..6: comparison of second-order ADRC, PI and PID conrollers faced wih varing process parameers, K, D and T. Nominal process and ADRC parameers are as hrogho Secion 3.. For each combinaion, he closed loop sep response is shown. Variaion of K, ADRC; Variaion of K, PI; (c) Variaion of K, PID; (d) Variaion of D, ADRC; (e) Variaion of D, PI; (f) Variaion of D, PID; (g) Variaion of T, ADRC; (h) Variaion of T, PI; (i) Variaion of T, PID..4..8 K =.6 K =. K =..4 K =.5. K = K = 5 5 5.4..8 D = D =..6 D =. D =.5.4 D =. D = 5 D = 5 5.4..8 (d).6 T = T =.5.4. T = 3 T = 3.5 5 5 (g).4..8 K =.6 K =. K =..4 K =.5. K = K = 5 5 5.4..8 D = D =..6 D =. D =.5.4 D =. D = 5 D = 5 5.4..8 (e).6 T = T =.5.4. T = 3 T = 3.5 5 5 (h).4..8 K =.6 K =. K =..4 K =.5. K = K = 5 5 5.4. (c).8 D = D =..6 D =. D =.5.4 D =. D = 5 D = 5 5.4..8 (f).6 T = T =.5.4. T = 3 T = 3.5 5 5 (i) For he PID conroller, we will emplo a PIDT -pe conroller in wo-pole-wo-zero form, designed sch ha he zeros of he conroller cancel he process poles. The conroller gain is chosen o mach he closed loop dnamics o he PI and ADRC case: C PID (s) = K I ( + T Z s)( + T Z s) s ( + T s) wih K I = 3 K T sele =.6, T Z/ = T = and T =. The simlaion resls are compiled in Figre 6, where each colmn represens one of he hree conroller pes and each row, a differen process parameer being varied. To smmarize, i has o be saed ha in each case, he ADRC approach srpasses he resls of PI and PID conrol b a large margin wih respec o sensiivi owards parameer variaions, wih a sligh advanage of PID compared o PI conrol.
Elecronics 3, 7 Figre 7. Experimen 3..7: comparison of disrbance rejecion behavior (second-order ADRC vs. PI vs. PID conroller). Process and ADRC parameers are as hrogho Secion 3.. Inp disrbance, d =.5, effecive from = 5 nil = 5. Sep response and reacion on disrbance; Conroller op for..4..8.6.4 ADRC. PI PID 5 5 5 3 35 3..7. Disrbance Rejecion of ADRC, PI and PID 3.5.5.5 ADRC PI PID 5 5 5 3 35 As or final experimen for he second-order case, we will evalae he disrbance rejecion capabiliies of ADRC, PI and PID conrol sing he conroller seings from he previos secion. All of hem are ned for he same closed loop response. For each conrol loop, an inp disrbance, d =.5, is applied dring a en-second inerval afer reaching sead sae; compare Figre 7. One can easil recognize ha ADRC compensaes he disrbance mch faser, sch ha is effec on he conrol variable remains ver low. While boh PI and PID conrollers cold also be ned more aggressivel for a similar disrbance rejecion behavior, one wold have o emplo addiional measres, sch as sepoin filers, o reain a non-oscillaing reference racking behavior. 4. Discree Time ADRC Pracical implemenaions of a conroller wih a sae observer, sch as he ADRC approach, will mos likel be done in discree ime form, e.g. emploing a microconroller. Since he acal conrol law for linear ADRC is based on proporional sae feedback, a discree ime version can alread be obained b onl discreizing he exended sae observer, which will be done in Secion 4.. The qasi-coninos approach will be valid onl for sfficienl fas sampling, of corse. Oherwise, he sae feedback shold be designed explicil for a discreized process model incorporaing sampling delas. In Secion 4., simlaive experimens will be carried o in order o visall assess he inflence of he discreisaion process and measremen noise on he conrol loop performance.
Elecronics 3, 7 4.. Discreisaion of he Sae Observer For a ssem wiho a feed-hrogh erm, he sandard approach for a discree-ime observer is: ˆx(k + ) = A d ˆx(k) + B d (k) + L p ((k) C d ˆx(k)) () If we look a he eqaion for he esimaion error of Eqaion (), we can see ha js as in he coninos case he dnamics of he error deca are deermined b he marix ( A d L p C d ) : e(k + ) = x(k + ) ˆx(k + ) = ( A d L p C d ) (x(k) ˆx(k)) Since he observer gains in L p inflence he pole placemen for he marix ( A d L p C d ), he can be chosen sch ha he esimaion seles wihin a desired ime. Here, A d, B d and C d refer o ime-discree versions of he respecive marices in he sae space process models Eqaions (3) and () obained b a discreisaion mehod. Since here is no marix D d in he observer eqaions, he discreisaion of he model ms deliver D d =, for example, via zero order hold (ZOH) sampling. This observer approach is also being called predicion observer, since he crren measremen, (k), will be sed as a correcion of he esimae onl in he sbseqen ime sep, ˆx(k+). In order o redce nnecessar ime delas (which ma even desabilize he conrol loop), i is advisable [6] o emplo a differen observer approach called filered observer or crren observer [7]. The nderling idea is similar o Kalman filering o spli he pdae eqaion in wo seps, namel a predicion sep o predic ˆx(k k ) based on measremens of he previos ime sep, k, and a correcion sep o obain he final esimae, ˆx(k k), incorporaing he mos recen measremen, (k): ˆx(k k ) = A d ˆx(k k ) + B d (k ) ˆx(k k) = ˆx(k k ) + L c ((k) C d ˆx(k k )) (predicion) (correcion) If we p he predicion ino he correcion eqaion and abbreviae ˆx(k k) = ˆx(k), we obain one pdae eqaion for he observer: ˆx(k) = (A d L c C d A d ) ˆx(k ) + (B d L c C d B d ) (k ) + L c (k) (3) From he esimaion error, we can see ha, in conras o he predicion observer, he error dnamics are deermined b he marix (A d L c C d A d ): e(k + ) = x(k + ) ˆx(k + ) = (A d L c C d A d ) (x(k) ˆx(k)) When comping he observer gains in L c, his has o be aken ino accon, i.e., one ms choose L c, sch ha he eigenvales of (A d L c C d A d ) mach he desired observer pole locaions. The discree ime versions of he marices A, B and C from he he sae space process models, which are necessar for he observer eqaions, can be obained b ZOH discreisaion [7]: A d = I + i= A i T i sample i!, B d = ( i= A i Tsample i ) B, C d = C, D d = D (4) i!
Elecronics 3, 7 ( For) he firs-order process in Eqaion (3), A and B are being discreized as follows, while C d = C = and D d = D = remain nchanged: ( ) ( ) ( ) ( ) b T sample b T sample A =, B = gives A d =, B d = This can be comped via Eqaion (4) easil, since( A i = for ) i. Following he same procedre for he second-order case in Eqaion (), C d = C = and D d = D = remain nchanged, and one obains for A d and B d (since A i = for i 3): T sample Tsample / b Tsample / A =, B = b gives A d = T sample, B d = b T sample ( ) T ) T Now, one can compe he observer gain, L c = l l (firs-order case) or Lc = (l l l 3 (for he second-order ADRC) o obain he desired observer dnamics. The desired pole locaions can, in a firs sep, be formlaed in he s-plane, as in Secions. and., and hen be mapped o he z-plane: z ESO = e seso T sample. The compaion of L c ma eiher be done nmericall or analicall. To demonsrae his, we will derive he eqaions for he firs-order case wih one common pole locaion, z ESO, for boh poles: de de(zi (A d L c C d A d )) =! ( z z ESO) ) l T sample T sample l z + l T sample ( z + l! = z z ESO z ( z ESO) z + ( l + l T sample ) z + ( l )! = z z ESO z ( z ESO) B comparing he coefficiens, one obains for l and l (solions for higher-order observers can be fond in [6]): l = ( z ESO), l = ( ( z ESO) ) (5) T sample As in he coninos ime case in Secion, he implemenaion and design of a discree ime linear ADRC can be smmarized in for seps, where he firs-order and second-order case are disingished b and :. Modeling: For a process wih dominaing firs-order behavior, P(s) = esimae b K T. For a second-order process, P(s) = is sfficien. K, one shold provide an T s + K T s + DT s +, an approximae vale b K T. Conrol srcre: Implemen a discree-ime observer wih wo sae variables, ˆx(k) = firs-order case) or hree sae variables ˆx(k) = as follows: ( ˆx (k) ˆx (k)) T (for he ( ˆx (k) ˆx (k) ˆx 3 (k)) T (for he second-order case),
Elecronics 3, 73 ˆx(k) = (A d L c C d A d ) ˆx(k ) + (B d L c C d B d ) (k ) + L c (k) Using he esimaed sae variables, implemen a proporional sae feedback conroller: (k) = K P (r(k) ŷ(k)) ˆf (k) = K P (r(k) ˆx (k)) ˆx (k) ( b b KP (r(k) ŷ(k)) K D ŷ(k) ) ˆf (k) (k) = b = (K P (r(k) ˆx (k)) K D ˆx (k)) ˆx 3 (k) b 3. Closed loop dnamics: Choose he conroller parameers, e.g. via pole placemen according o a desired seling ime: K P = s CL wih s CL 4 T sele K P = ( s CL), KD = s CL wih s CL 6 T sele 4. Observer dnamics: Place he desired observer poles far enogh lef of he closed loop poles in he s-plane, e.g. b a common locaion, s ESO = k ESO s CL (3...) s CL. Map o he z-plane via z ESO = e seso T sample and compe he necessar observer gains: l = ( z ESO), l = T sample ( z ESO) l = ( z ESO) 3, l = T 3 sample ( z ESO) ( + z ESO ), l 3 = ( z ESO) 3 Tsample 4.. Simlaive Experimens As in he coninos-ime experimens in Secion 3., a firs-order process will be examined in his secion: P(s) = (s) (s) = K wih K = and T = T s + Unless oherwise noed in individal experimens, he ADRC will be designed assming perfec knowledge, b = K/T =, wih k ESO = 5, and a desired % seling ime, T sele =. The discreisaion will be based on a sampling ime T sample =.. Gassian measremen noise will be added wih a variance σ noise =.. 4... Effec of Sample Time When choosing he sampling ime for a discree-ime implemenaion of a conroller wih an observer, one has o consider no onl he process dnamics, b also he dnamics of he observer. If, on he oher hand, resricions o sample imes are presen, he dnamics of he observer will be limied. In he case of ADRC, his wold mean ha he desired behavior of he process ma no be achieved nder all circmsances. In Figre 8, simlaions were performed sing sample imes for ADRC ranging from T sample =. o T sample =.. As expeced, oscillaions in he conroller op are increasingl visible for large sample imes. Conseqenl, he closed loop dnamics differ from he desired firs-order behavior as he sampling inerval becomes oo coarse.
Elecronics 3, 74 Figre 8. Experimen 4..: inflence of differen sample ime seings on discree-ime ADRC. Measremen noise variance: σ noise =.. Observer pole placemen: k ESO = 5. Variaion of T sample, closed loop sep response; Conroller op for...8.6 Ts =..4 Ts =. Ts =.5. Ts =. Ts =..5.5 4 3 Ts =. Ts =. Ts =.5 Ts =. Ts =..5.5 Figre 9. Experimen 4..: inflence of differen levels of measremen noise on discree-ime ADRC. Sample ime: T sample =.. Observer pole placemen: k ESO = 5. Variaion of σnoise, closed loop sep response; Conroller op for...8.6.4 noise = noise =.. noise =. noise = e 5.5.5 4... Effec of Measremen Noise 4 3 noise = noise =. noise =. noise = e 5.5.5 While sae feedback conrollers based on observers raher han direc measremens are less sscepible o measremen noise, oscillaions in he conroller op will sill occr wih higher noise levels. In he simlaions presened in Figre 9, normall disribed noise wih increasing variance ranging from σ noise = o σ noise =. was added o he process op. The resls correspond o hese expecaions. The effec of measremen noise on oscillaions in he conroller op, however, ma be miigaed b designing an observer wih slower dnamics, which will be demonsraed in he following experimen.