Lectures on Multivariable Feedback Control

Transcription

1 Lecture on Multvarable Feedback Control Al Karmpour Department of Electrcal Engneerng, Faculty of Engneerng, Ferdow Unverty of Mahhad September 9 Chapter : Introducton to Multvarable Control - Multvarable Connecton - Multvarable Pole -- Pole from State Space Realzaton -- Pole from Tranfer Functon - Multvarable zero -- Zero from State Space Realzaton -- Zero from Tranfer Functon - Drecton of Pole and Zero -5 Smth form of a Polynomal Matrx -6 Smth-McMllan Form -7 Matrx Fracton Decrpton MFD -8 Scalng -9 Performance Specfcaton --5 Tme Doman Performance --6 Frequency Doman Performance - Trade-off n Frequency Doman - Bandwdth and Croover Frequency

2 Chapter Lecture Note of Multvarable Control Procee wth only one output beng controlled by a ngle manpulated varable are clafed a ngle-nput ngle-output SISO ytem. Many procee, however, do not confrm to uch a mple control confguraton. In the ndutral proce for example, any unt operaton capable of manufacturng or refnng a product cannot do o wth only a ngle control loop. In fact, each unt operaton typcally requre control over at leat two varable, e.g. product rate and product qualty. There are, therefore, uually at leat two control loop to content wth. Sytem wth more than one control loop are known a mult-nput mult-output MIMO or multvarable ytem. Th chapter conder ome mportant apect of multvarable ytem. MIMO nterconnecton, pole and zero n MIMO ytem, Smth form for polynomal matrx, Smth McMllan form SMM form, matrx fracton decrpton MFD and performance pecfcaton n MIMO ytem are pecfed n th chapter. - Multvarable Connecton Fgure - how cacade ere nterconnecton of tranfer matrce. The tranfer matrx of the overall ytem : y u u - Note that the tranfer matrce mut have utable dmenon. Parallel nterconnecton of tranfer matrce hown n Fgure -. The tranfer matrx of the overall ytem : y u u - Note that the tranfer matrce mut have utable dmenon. Feedback nterconnecton of tranfer matrce hown n fgure -. The tranfer matrx of the overall ytem : y u I u - Note that the tranfer matrce mut have utable dmenon. u y

3 Chapter Lecture Note of Multvarable Control Fgure - Cacade nterconnecton of tranfer matrce u y Fgure - Parallel nterconnecton of tranfer matrce u y Fgure - Feedback connecton of tranfer matrce A ueful relaton n multvarable puh-through rule. Puh-through rule defned by: I I - The cacade and feedback rule can be combned to evaluate cloed loop tranfer matrx from block dagram. MIMO rule: To derve the output of a ytem, tart from the output and wrte down the block a you meet them when movng backward agant the gnal flow toward the nput. If you ext from a feedback loop then nclude a term I L or L I accordng to the feedback gn where L the tranfer functon around that loop evaluated agant the gnal flow tartng at the pont of ext from the loop. Parallel branche hould be treated ndependently and ther contrbuton added together. Example - Derve the tranfer functon of the ytem hown n fgure -.

4 Chapter Lecture Note of Multvarable Control Fgure - Sytem ued n Example - A t ha two parallel way from nput to output by MIMO rule the tranfer functon : z P P K I P K P ω - Multvarable Pole Pole of a ytem can be derved from the tate pace realzaton and the tranfer functon. -- Pole Derved from State Space Realzaton For mplcty we here defne the pole of a ytem n term of the egenvalue of the tate pace A matrx. Defnton - The pole p of a ytem wth tate-pace decrpton A, B, C, D are egenvalue λ A,,,... n of the matrx A. The pole polynomal or charactertc polynomal φ, defned a φ det I A. Thu the ytem pole are the root of the charactertc polynomal φ det I A -5 Note that f A doe not correpond to a mnmal realzaton then the pole by th defnton wll nclude the pole egenvalue correpondng to uncontrollable and/or unobervable tate. -- Pole Derved from Tranfer Functon The pole of may be omewhat looely defned a the fnte value p where p ha a ngularty nfnte. The followng theorem from MacFarlane and Karcana allow one to

5 Chapter Lecture Note of Multvarable Control obtan the pole drectly from the tranfer functon matrx and alo ueful for hand calculaton. It alo ha the advantage of yeldng only the pole correpondng to a mnmal realzaton of the ytem. Theorem - The pole polynomal φ correpondng to a mnmal realzaton of a ytem wth tranfer functon the leat common denomnator of all non-dentcally-zero mnor of all order of. A mnor of a matrx the determnant of the quare matrx obtaned by deletng certan row and/or column of the matrx. Example - Conder the plant e ϑ whch ha no tate-pace realzaton a t contan a delay and alo mproper. However from Theorem - we have that the denomnator and a expected ha a pole at - Example - Conder the quare tranfer functon matrx.5 6 The mnor of order are the four element whch all have n the denomnator. The mnor of order the determnant 6 det.5.5 Note the pole-zero cancellaton when evaluatng the determnant. The leat common denomnator of all the mnor then ϕ o a mnmal realzaton of the ytem ha two pole one at and one at Example - Conder the followng ytem wth nput and output. 5

6 Chapter Lecture Note of Multvarable Control The mnor of order are the element of, o they are,,,, The mnor of order correpondng to the deleton of dfferent column are,, By conderng all mnor we fnd ther leat common denomnator ϕ The ytem therefore ha four pole one at, one at and two at. From the above example we ee that the MIMO pole are eentally the pole of the element. However by lookng at only the element t not poble to determne the multplcty of the pole. - Multvarable Zero Zero of a ytem are when competng effect nternal to the ytem are uch that the output zero even when the nput and the tate are not themelve dentcally zero. -- Zero Derved from State Space Realzaton Zero are uually computed from a tate pace decrpton of the ytem. Frt note that the tate pace equaton of a ytem may be wrtten a x I A B P, P -6 u y C D The zero are then the value of z for whch the polynomal ytem matrx P loe rank reultng n zero output for ome nonzero nput. Numercally the zero are found a non trval oluton to the followng problem 6

7 Chapter Lecture Note of Multvarable Control xz zi g M -7 u z A B I where M, I g. C D Th olved a a generalzed egenvalue problem. In the conventonal egenvalue problem we have I g I. The zero reultng from a mnmal realzaton are ometme called the tranmon zero. If one doe not have a mnmal realzaton, then numercal computaton may yeld addtonal nvarant zero. Thee nvarant zero plu the tranmon zero are ometme called the ytem zero. The nvarant zero can be further ubdvded nto nput and output decouplng zero. Thee cancel pole aocated wth uncontrollable or unobervable tate and hence have lmted practcal gnfcance. If the ytem output contan drect nformaton about each of the tate and no drect connecton from nput, then there are no tranmon zero. Th would be the cae f C I, D, for example. For quare ytem wth mp nput and output and n tate, lmt on the number of tranmon zero are: D : At mot n m rank D zero D : At mot n m rank CB zero -8 D and rank CB m : Exactly n m zero Example -5 Conder the followng tate pace realzaton x& Ax Bu y Cx Du where Determne the zero of the ytem. A B C D 6 5 [ ] 7

8 Chapter Lecture Note of Multvarable Control Soluton: Frt we derve the number of tranmon zero accordng to equaton -8. The product of CB CB [ ] So nce D accordng to equaton -8 the ytem ha at mot n m rank CB zero. To fnd the value of zero we contruct M and I g a follow A B M I g C D 6 5 I Now by ue of generalzed egenvalue problem one can fnd the zero. The followng Matlab m.fle can be ued to derve zero. It how that the ytem ha a zero at. eg M, I g -- Zero Derved from Tranfer Functon For a SISO ytem the zero z are the oluton to z. In general t can be argued that zero are value of at whch loe rank. Th the ba for the followng defnton of zero for a multvarable ytem MacFarlane and Karcana. Defnton - z a zero of f the rank of z le than the normal rank of. The zero nz polynomal defned a z Π z. Where n z the number of fnte zero of. We do not conder zero at nfnty. We requre that z fnte. Recall that the normal rank of the rank of at all value of except at a fnte number of ngularte whch are the zero Note that th defnton of zero baed on the tranfer functon matrx correpondng to a mnmal realzaton of a ytem. Thee zero are ometme called tranmon zero but we wll 8

9 Chapter Lecture Note of Multvarable Control mply call them zero. We may ometme ue the term multvarable zero to dtnguh them from the zero of the element of the tranfer functon matrx. The followng theorem from MacFarlane and Karcana ueful for hand calculatng the zero of a tranfer functon matrx. Theorem - The zero polynomal z correpondng to a mnmal realzaton of the ytem the greatet common dvor of all the numerator of all order-r mnor of where r the normal rank of provded that thee mnor have been adjuted n uch a way a to have the pole polynomal φ a ther denomnator. Example -6 Conder the tranfer functon matrx.5 The normal rank of and the mnor of order the determnant of 8 det. From Theorem - the pole polynomal φ and therefore the zero polynomal z. Thu ha a ngle RHP-zero at. Th llutrate that n general multvarable zero have no relatonhp wth the zero of the tranfer functon element. Th alo hown by the followng example where the ytem ha no zero. Example -7 Conder the followng ytem.5 6 accordng to example - the pole polynomal : ϕ 9

10 Chapter Lecture Note of Multvarable Control The normal rank of and the mnor of order the determnant of, where det wth n φ a t denomnator 6 det.5.5 Thu the zero polynomal gven by the numerator whch, and we fnd that the ytem ha no multvarable zero. Example -8 Conder the ytem The normal rank of and nce there no value of for whch both element become zero, ha no zero. In general non-quare ytem are le lkely to have zero than quare ytem. The followng an example of a non quare ytem whch ha a zero. Example -9 Conder the followng ytem accordng to example - the pole polynomal : ϕ The mnor of order wth φ a ther denomnator are,, The greatet common dvor of all the numerator of all order- mnor z. Thu, the ytem ha a ngle RHP-zero located at.

11 Chapter Lecture Note of Multvarable Control We alo ee from the lat example that a mnmal realzaton of a MIMO ytem can have pole and zero at the ame value of provded ther drecton are dfferent. Th dcued n the next ecton. - Drecton of Pole and Zero Zero drecton: Let have a zero at z, Then loe rank at z and there wll ext nonzero vector u z and y z uch that H z u z y z z -9 here u z defned a the nput zero drecton and y z defned a the output zero drecton. We uually normalze the drecton vector to have unt length.e. u and y. From a practcal pont of vew the output zero drecton y z uually of more mportant than u z becaue y z gve nformaton about whch output _or combnaton of output_ may be dffcult to control. In prncple we may obtan u z and y z from an SVD of z z H z YΣU and we have that z u the lat column n U, correpondng to the zero ngular value of z and y z the lat column of Y. A better approach numercally to obtan u z from a tate pace decrpton ung the generalzed egenvalue problem n -7. Pole drecton: Let have a pole at p. Then p nfnte and we may omewhat crudely wrte H p u p y p p - where u p the nput pole drecton and y p the output pole drecton. A for u z and y z the vector u p and y p may be obtaned from an SVD of H p YΣU. Then p u the frt column n U correpondng to the nfnte ngular value and y p the frt column n Y. If the nvere of p ext then t follow from the SVD that p y p u p p -

12 Chapter Lecture Note of Multvarable Control However f we have a tate pace realzaton of then t better to determne the pole drecton from the rght and left egenvector of A. Specfcally f p pole of then p an egenvalue of A. Let t p and q p be the correpondng rght and left egenvector.e. At pt q A pq - p p H p H p then the pole drecton are y H p Ct p u p B q p - Example - Conder the followng plant.5 It ha a RHP zero at z and a LHP pole at p. We wll ue an SVD of z and p to determne the zero and pole drecton. But we tre that th not a relable method numercally. To fnd the zero drecton conder z The zero nput and output drecton are aocated wth the zero ngular value of z and we get.8.8 u z and y z.6.55 We ee from y z that the zero ha a lghtly larger component n the frt output. Next, to determne the pole drecton conder p ε ε ε The SVD a ε yeld.55 ε ε.8 ε ε H

13 Chapter Lecture Note of Multvarable Control The pole nput and output drecton are aocated wth the larget ngular value,.6.55 get u p and y p ε and we -5 Smth Form of a Polynomal Matrx Suppoe that Π a polynomal matrx. Smth form of Π denoted by Π peudo dagonal n the followng form, and t a Π d Π - and Π a quare dagonal matrce n the followng form d { ε, ε,..., ε } Π d dag r -5 χ Furthermore, ε a factor of ε. ε derved from mnor of Π a ε χ where χ derved by: χ χ gcd{all monc mnor of degree } χ gcd{all monc mnor of degree } -6.. χ r gcd{all monc mnor of degree r} gcd tand for greatet common dvor and monc a polynomal that the coeffcent of t greatet degree one. The three elementary operaton for a polynomal matrx are ued to fnd Smth form. Multplyng a row or column by a contant; Interchangng two row or two column; and Addng a polynomal multple of a row or column to another row or column. Thee operaton are carred out on a tranfer matrx Π by ether pre-multplcaton or potmultplcaton by unmodular polynomal matrce known a elementary matrce. A polynomal

14 Chapter Lecture Note of Multvarable Control matrx unmodular f t nvere alo a polynomal matrx. Pre-multplcaton of Π by an elementary matrx produce the correpondng row operaton, whle pot-multplcaton produce a column operaton. Π Smth form of Π and they are ad to be equvalent, denoted by Π Π f there ext a et of elementary matrce L and R uch that Π L... L L Π R R... R -7 n n Example - Conder the followng polynomal matrx Π o we have χ, χ gcd{,,,}, χ gcd{ } and now ε are: χ χ ε and ε χ χ the Smth of Π : Π Example - Conder the followng polynomal matrx Π 8 o we have

15 Chapter Lecture Note of Multvarable Control χ, χ gcd{,,.5,, } and χ gcd{,, } and now ε are: χ χ ε and ε χ χ the Smth form of Π : Π -6 Smth-McMllan Form The Smth-McMllan form ued to determne the pole and zero of the tranfer matrce of ytem wth multple nput and/or output. The tranfer matrx a matrx of tranfer functon between the varou nput and output of the ytem. The pole and zero that are of nteret are the pole and zero of the tranfer matrx telf, not the pole and zero of the ndvdual element of the matrx. The locaton of the pole of the tranfer matrx are avalable by npecton of the ndvdual tranfer functon, but the total number of the pole and ther multplcty not. The locaton of ytem zero, or even ther extence, not avalable by lookng at the ndvdual element of the tranfer matrx. The tranfer matrx wll be denoted by. The number of row n equal to the number of ytem output; that wll be denoted by m. The number of column n equal to the number of ytem nput; that wll be denoted by p. Thu, an m p matrx of tranfer functon. The normal rank of r, where r mn{p, m}. Followng theorem gve a dagonal form for a ratonal tranfer-functon matrx: Theorem - Smth-McMllan form 5

16 Chapter Lecture Note of Multvarable Control Let [ g ] be an m p matrx tranfer functon, where g j are ratonal calar tranfer j functon, can be repreented by: Π -8 D where Π an m p polynomal matrx of rank r and D the leat common multple of the denomnator of all element of. Then, Smth McMllan form of and can be derved drectly by Π Π M d -9 D D where M : ε ε ε r M dag,,..., δ δ δ r where {, δ } ε a par of monc and coprme polynomal for,,..., r. - Furthermore, ε a factor of ε and δ a factor of δ. Element of the matrx M can be defned by: ε ε m - D δ where ε are dagonal element of Π Smth form of Π a { ε, ε,..., ε } Π dag - d r We recall that a matrx, and t Smth-McMllan form are equvalent matrce. Thu, there ext two unmodular matrce, L and R, uch that L R - L and R are the unmodular matrce that convert Π to t Smth form Π. Then there ext two matrce L and R, uch a 6

17 Chapter Lecture Note of Multvarable Control L R - where L and R are alo unmodular and: L L, R R -5 The pole and zero of the tranfer matrx can be found from the element of M. The pole polynomal defned a r φ Πδ δ δ... δ -6 r Repeated pole can alo be dentfed by npecton of φ. The total number of pole n the ytem gven by deg φ mnmal tate-pace repreentaton of., whch known a the McMllan degree. It the dmenon of a A tate-pace repreentaton of may be of hgher order than the McMllan degree, ndcatng pole-zero cancellaton n the ytem. In mlar fahon, the zero polynomal defned a r z Πε ε ε... ε r -7 The root of z are known a the tranmon zero of. It can be een that any tranmon zero of the ytem mut be a factor n at leat one of the ε polynomal. The normal rank of both M and r. It clear that f any ε be zero, then the rank of M drop below r. Therefore, nce the rank of M and are alway equal, o loe rank. We llutrate the Smth-McMllan form by a mple example. Example - Conder the followng tranfer-functon matrx 7

18 Chapter Lecture Note of Multvarable Control 8 We can then expre n the form:,.5, Π Π D D Accordng example - the Smth form of Π : Π So the Smth McMllan form of : Π D Clearly the pole polynomal and the zero polynomal are: φ, z Example - Conder the followng example of a ytem wth m output and p nput. The tranfer matrx hown below;

19 Chapter Lecture Note of Multvarable Control 9 8 We can then expre n the form:, 8, Π Π D D accordng example - the Smth form of Π : Π So the Smth McMllan form of : Π D Clearly pole polynomal and zero polynomal are: φ, z -7 Matrx Fracton Decrpton MFD

20 Chapter Lecture Note of Multvarable Control A model tructure that related to the Smth-McMllan form matrx fracton decrpton MFD. There are two type, namely a rght matrx fracton decrpton RMFD and a left matrx fracton decrpton LMFD. Frt of all uppoe a m m matrx and the Smth McMllan form of, defne the followng two matrce: ε,..., ε,,..., N dag δ,..., δ,,..., r D dag -9 r -8 where N and D are m m matrce. Hence, can be wrtten a N D - Combnng - and -, we can wrte R D L R L N D R L N D N - Th known a a rght matrx fracton decrpton RMFD where: L N, R D - N D If one tart wth D N then combnng wth - L R L D N R D L N R - D N Th known a a left matrx fracton decrpton LMFD where: D L, N R - D N The left and rght matrx decrpton have been ntally derved tartng from the Smth- McMllan form. Hence, the factor are polynomal matrce. However, t mmedate to ee that they provde a more general decrpton. In partcular, N, D, D and N are

21 Chapter Lecture Note of Multvarable Control generally matrce wth ratonal entre. One poble way to obtan th type of repreentaton to dvde the two polynomal matrce formng the orgnal MFD by the ame table polynomal. We alo oberve that the RMFD LMFD not unque, becaue, for any nonngular matrx Ω we can wrte a m m Ω Ω Ω Ω N D N D -5 where Ω ad to be a rght common factor. When the only rght common factor of and D unmodular matrx, then, we ay that N and are rght coprme. In th cae, we ay that the RMFD, N D rreducble. D N It eay to ee that when a RMFD rreducble, then a zero of f and only f N loe rank at z ; and a pole of f and only f D ngular at p. Th mean that the pole φ det. z p polynomal of An example howng the above concept condered next. D Example -5 Conder a MIMO ytem havng the tranfer functon.5 a Fnd the Smth-McMllan form by performng elementary row and column operaton. b Fnd the pole and zero. c Buld a RMFD for the model.

22 Chapter Lecture Note of Multvarable Control Soluton a We frt compute t Smth-McMllan form by performng elementary row and column operaton. 8 R L where 8, 8.5 R L b We ee that the obervable and controllable part of the ytem ha zero and pole polynomal gven by, 8 z φ So the pole are -, -, - and - and zero are -.5 ± j.97 c To derve RMFD we need 8,.5 R R L L, 8 D N So

23 Chapter Lecture Note of Multvarable Control N 8 8 D Matrx fracton decrpton MFD can be extended to m n non quare matrx. In RMFD, N m n and D m m and n LMFD, D n n and N m n. Followng example how the procedure of fndng RMFD and LMFD for a non quare matrx. Example -6 Conder the followng tranfer matrx; 8 Fnd RMFD and LMFD of the ytem. Soluton Accordng example - the SMM form of : Π D

24 Chapter Lecture Note of Multvarable Control To derve the RMFD and LMFD we mut fnd the unmodular matrce that convert Π to Π. So Π Π 8 R L Now we wrte accordng to : R L Above equaton lead to RMFD and N, D To derve LMFD the mut parttoned a LMFD o:

25 Chapter Lecture Note of Multvarable Control 5 R L o we fnd D, N -8 Scalng Scalng very mportant n practcal applcaton a t make model analy and controller degn weght electon much mple. It requre the engneer to make a judgment at the tart of the degn proce about the requred performance of the ytem. To do th, decon are made on the expected magntude of dturbance and reference change, on the allowed magntude of each nput gnal, and on the allowed devaton of each output. Let the uncaled or orgnal ytem lnear model of the proce n Fgure -5a be r y e d u y d ˆ ˆ ˆ ˆ ; ˆ ˆ ˆ ˆ -6 where a hat ^ ued to how that the varable are n ther uncaled or orgnally ytem unt. A ueful approach for calng to make the varable le than one n magntude. Th done by dvdng each varable by t maxmum expected or allowed change. For dturbance and manpulated nput, we ue the caled varable

26 Chapter Lecture Note of Multvarable Control dˆ uˆ d, u dˆ -7 uˆ max max where _ ˆd max : larget expected change n dturbance _ û max : larget allowed nput change The maxmum devaton from the nomnal value hould be choen by thnkng of the maxmum value one can expect or allow a a functon of tme. The varable yˆ, eˆ and rˆ are n the ame unt, o the ame calng factor hould be appled to each. Two alternatve are poble: _ ê max : larget allowed control error _ ˆr max : larget expected change n reference value Snce a major objectve of control to mnmze the control error, we here uually chooe to cale wth repect to the maxmum control error: eˆ rˆ yˆ e, r, y -8 eˆ eˆ eˆ max max max a b Fgure -5 Model n term of a orgnal varable and b caled varable 6

27 Chapter Lecture Note of Multvarable Control To formalze the calng procedure, ntroduce the calng factor D eˆ, D uˆ, D dˆ, D rˆ -9 e max u max d max r max For MIMO ytem each varable n the vector d ˆ, rˆ, uˆ and ê may have a dfferent maxmum value, n whch cae D, e, Du Dd and r D become dagonal calng matrce. Th enure, for example, that all error output are of about equal mportance n term of ther magntude. The correpondng caled varable to ue for control purpoe are then d ˆ u e e e D d, u D uˆ, y D yˆ, e D eˆ, r D rˆ - d On ubttutng - nto -6 we get D y D u D d ; e e d d D e D y D r e e e - and ntroducng the caled tranfer functon D ˆ D - e D ˆ u, d D e d d then yeld the followng model n term of caled varable y u d d ; e y r - Here u and d hould be le than n magntude, and t ueful n ome cae to ntroduce a caled reference r whch le than n magntude. Th done by dvdng the reference by the maxmum expected reference change rˆ r Dr rˆ - rˆ max We then have that 7

28 Chapter Lecture Note of Multvarable Control r Rr where R D e Dr rˆ eˆ max max -5 Here R the larget expected change n reference relatve to the allowed control error, typcally, R. The block dagram for the ytem n caled varable may then be wrtten a n Fgure -5b for whch the followng control objectve relevant: In term of caled varable we have that d t and r t, and our control objectve to degn u wth u t uch that e t y t r t at leat mot of the tme. -9 Performance Specfcaton In the applcaton of automatc controller, t mportant to realze that controller and proce form a unt, credt or dcredt for reult obtaned are attrbutable to one a much a the other. A poor controller often able to perform acceptably on a proce whch ealy controllable. The fnet controller made, when appled to a merably degned proce, may not delver the dered performance. There are ome mportant defnton n th matter. Nomnal tablty NS: The ytem table wth no model uncertanty. Nomnal Performance NP: The ytem atfe the performance pecfcaton wth no model uncertanty. Robut tablty RS: The ytem table for all perturbed plant about the nomnal model up to the wort cae model uncertanty. Robut performance RP: The ytem atfe the performance pecfcaton for all perturbed plant about the nomnal model up to the wort cae model uncertanty. Performance pecfcaton can be condered n tme and frequency doman. 8

29 Chapter Lecture Note of Multvarable Control Fgure -6 Step repone of a ytem -9- Tme Doman Performance Although cloed loop tablty an mportant ue, the real objectve of control to mprove performance, that, to make the output yt behave n a more derable manner. Actually, the poblty of nducng ntablty one of the dadvantage of feedback control whch ha to be traded off agant performance mprovement. The objectve of th ecton to dcu way of evaluatng cloed loop performance. Step repone analy approach, often taken by engneer when evaluatng the performance of a control ytem. That, one mulate the repone to a tep n the reference nput, and conder charactertc hown n Fgure -6. Re tme, t r, the tme t take for the output to frt reach 9% of t fnal value, whch uually requred to be mall. Settlng tme, t, the tme after whch the output reman wthn ± 5% or ± % of t fnal value, whch uually requred to be mall. Overhoot, P.O, the peak value dvded by the fnal value, whch hould typcally be le than % or le. Decay rato, the rato of the econd and frt peak, whch hould typcally be. or le. Steady tate offet, e, the dfference between the fnal value and the dered fnal value, whch uually requred to be mall. Exce varaton, the total varaton TV dvded by the overall change at teady tate, whch hould be a cloe to a poble. The total varaton the total movement of the 9

30 Chapter Lecture Note of Multvarable Control output a llutrated n Fgure -7. For the cae condered here the overall change, o the exce varaton equal to the total varaton. TV v Exce varaton TV / v -6 Note that the tep repone equal to the ntegral of the correpondng mpule repone, e.g. et u n the followng convoluton ntegral. t y t g τ u t τ dτ -7 where g τ the mpule repone. One can compute the total varaton a the ntegrated abolute area -norm, of the correpondng mpule repone TV g τ dτ g t -8 ISE, IAE, ITSE, ITAE: Thee meaure are ntegral quared error, ntegral abolute error, ntegral tme weghted quared error and ntegral tme weghted abolute error repectvely. For example IAE defned a IAE e τ dτ -9 The re tme and ettlng tme are meaure of the peed of the repone, wherea the overhoot, decay rato, TV, ISE, IAE, ITSE, ITAE and teady tate offet are related to the qualty of the repone. Fgure -7 Total varaton n the tep repone of a ytem

31 Chapter Lecture Note of Multvarable Control -9- Frequency Doman Performance The frequency repone of the loop tranfer functon, L jω, or of varou cloed-loop tranfer functon, may alo be ued to characterze cloed-loop performance. Typcal Bode plot of L hown n Fgure -8. One advantage of the frequency doman compared to a tep repone analy that t conder a broader cla of gnal nuod of any frequency. Th make t eaer to characterze feedback properte, and n partcular ytem behavor n the croover bandwdth regon. We wll now decrbe ome of the mportant frequency doman meaure ued to ae performance, e.g. gan and phae margn, the maxmum peak of T and S, and the varou defnton of croover and bandwdth frequence ued to characterze peed of repone. Let L denote the loop tranfer functon of a ytem whch cloed-loop table under negatve feedback. A typcal Bode plot and a typcal Nyqut plot of L jω llutratng the gan margn M and phae margn PM are gven n Fgure -8 and -9, repectvely. From Nyqut tablty condton, the cloene of the curve L jω to the pont - n the complex plane a good meaure of how cloe a table cloed-loop ytem to ntablty. We ee from Fgure -8 that M meaure the cloene of L jω to - along the real ax, wherea PM a meaure along the unt crcle. Fgure -8 Bode plot of L jω.

32 Chapter Lecture Note of Multvarable Control Fgure -9 Nyqut plot of L jω. More precely, f the Nyqut plot of L jω croe the negatve real ax between - and, then the upper gan margn defned a M -5 L jω 8 where the phae croover frequency ω 8 where the Nyqut curve of L jω croe the negatve real ax between - and,.e. L jω8 8-5 The phae margn defned a PM L jω 8-5 c where the gan croover frequency ω the frequency where L jω croe,.e. c L jω -5 c The PM a drect afeguard agant tme delay uncertanty; the ytem become untable f we add a tme delay of θ max PM / ω c -5

33 Chapter Lecture Note of Multvarable Control Note that the unt mut be content, and o f ωc n [rad/] then PM mut be n radan. It alo mportant to note that by decreang the value of ω c lowerng the cloed-loop bandwdth, reultng n a lower repone the ytem can tolerate larger tme delay error. Stablty margn are meaure of how cloe a table cloed-loop ytem to ntablty. From the above argument we ee that the M and PM provde tablty margn for gan and delay uncertanty. More generally, to mantan cloed-loop tablty, the Nyqut tablty condton tell u that the number of encrclement of the crtcal pont - by L jω mut not change. A dcued next, the actual cloet dtance equal to / M where M the peak value of the entvty S jω. A expected, the M and PM are cloely related to M, and nce S alo a meaure of performance; they are therefore alo ueful n term of performance. In ummary, pecfcaton on the M and PM e.g. M > and PM > o are ued to provde the approprate trade-off between performance and tablty robutne. The maxmum peak of the entvty and complementary entvty functon are defned a M max S jω M T maxt jω -55 ω ω Snce ST o S and T dffer at mot by. A large value of M T large. M therefore occur f and only f We now gve ome jutfcaton for why we may want to reduce the value of M. Conder the one degree-of-freedom confguraton n Fgure -. Let we defne error gnal a e y r, then wthout control and noe u n, we havee y r d d r, and wth feedback control Fgure - One degree-of-freedom confguraton

34 Chapter Lecture Note of Multvarable Control Fgure - Nyqut plot of L jω. e y r S d Sr S d r. Thu, feedback control mprove performance n term of d reducng e at all frequence where S <. One may alo vew d M a a robutne meaure. To mantan cloed-loop tablty, we want L jω to tay away from the crtcal pont -. Accordng to Fgure - the mallet dtance between L jω and - both for tablty and performance we want M, and therefore for robutne, the maller M, better. In ummary, M cloe to. There a cloe relatonhp between thee maxmum peak and the M and PM. Specfcally, for a gven M we are guaranteed M M ; M PM n M M [ rad] For example, wth M we are guaranteed M > and PM > of M T we are guaranteed -56 o 9. Smlarly, for a gven value M M T ; PM n M T M T [ rad] and pecfcally wth M we have M >.5 and PM > T o

35 Chapter Lecture Note of Multvarable Control - Trade-off n Frequency Doman Conder the mple one degree-of-freedom confguraton n Fgure -. The nput to the controller K r ym and the meaured output y m y n where n the meaurement noe. Thu, the nput to the plant r y n u K -58 The objectve of control to manpulate u degn K uch that the control error e reman mall n pte of dturbance d and noe n. The control error e defned a e y r -59 where r denote the reference value et pont for the output. Note that we do not defne e a the controller nput The plant model wrtten a r ym whch frequently done. y u d d -6 and for a one degree-of-freedom controller the ubttuton of -58 and -59 nto -6 yeld or y K r y n d d -6 I K y K r n d d -6 and hence the cloed-loop repone y I K K r I K d d I K K n T S T -6 The control error e y r Sr Sd d Tn -6 where we have ued the fact T S I. The correpondng plant nput gnal u KSr KSd d KSn -65 The followng notaton and termnology are ued L K loop tranfer functon S I K I L entvty functon T I K K I L L complementary entvty functon 5

36 Chapter Lecture Note of Multvarable Control We ee that S the cloed-loop tranfer functon from the output dturbance to the output, whle T the cloed-loop tranfer functon from the reference gnal to the output. The term complementary entvty for T follow from the dentty T S I -66 The term entvty functon natural becaue S gve the entvty reducton afforded by feedback. To ee th, conder the open-loop cae.e. wth no control K. Then the error e y r r d d n -67 and a comparon wth -6 how that, wth the excepton of noe, the repone wth feedback obtaned by pre multplyng the rght hand de by S. Remark: Actually, the above not the orgnal reaon for the name entvty Bode frt called S a entvty becaue t gve the relatve entvty of the cloed-loop tranfer functon T to the relatve plant model error. In partcular, at a gven frequency ω we have for a SISO plant, by traghtforward dfferentaton of T, that dt / T S -68 d / Recall equaton -6 whch yeld the cloed-loop repone n term of the control error e, e y r Sr Sd d Tn For perfect control we want e y r that, we would lke S, T -69 The frt requrement n -69 namely dturbance rejecton and command trackng, and obtaned wth S or equvalentlyt I. Snce S I L th mple that the loop tranfer functon L mut be large n magntude. On the other hand, the requrement for zero noe tranmon mple that T or equvalently S I, whch obtaned wth L. Th llutrate the fundamental nature of feedback degn whch alway nvolve a trade-off between conflctng objectve, n th cae between large loop gan for dturbance rejecton and trackng, and mall loop gan to reduce the effect of noe. It alo mportant to conder the magntude of the control acton u, whch the nput to the plant. We want u mall becaue th caue le wear and ave nput energy, and alo becaue u 6

37 Chapter Lecture Note of Multvarable Control often a dturbance to other part of the ytem e.g. conder openng a wndow n your offce to adjut your body temperature and the underable dturbance th wll mpoe on the ar condtonng ytem for the buldng. In partcular, we uually want to avod fat change n u. The control acton gven by u K r y and we fnd a expected that a mall u correpond to m mall controller gan and a mall L K. The mot mportant degn objectve whch necetate trade-off n feedback control are ummarzed below. - Performance, good dturbance rejecton: need large controller gan,.e. L large or T I. - Performance, good command followng: L large or T I. - Stablzaton of untable plant: L large or T I. - Mtgaton of meaurement noe on plant output: L mall or T. 5- Small magntude of nput gnal: K mall and L mall or T. 6- Phycal controller mut be trctly proper: L ha approach to at hgh frequence or T. 7- Nomnal tablty table plant: L mall Fortunately, the conflctng degn objectve mentoned above are generally n dfferent frequency range, and we can meet mot of the objectve by ung a large loop gan L > at low frequence below croover, and a mall gan L < at hgh frequence above croover. - Bandwdth and Croover Frequency The concept of bandwdth very mportant n undertandng the beneft and trade-off nvolved when applyng feedback control. Above we condered peak of cloed-loop tranfer functon, M T and M whch are related to the qualty of the repone. However, for performance we mut alo conder the peed of the repone, and th lead to conderng the bandwdth frequency of the ytem. In general, a large bandwdth correpond to a maller re tme, nce hgh-frequency gnal are more ealy paed on to the output. A hgh bandwdth alo ndcate a ytem whch 7

38 Chapter Lecture Note of Multvarable Control entve to noe. Converely, f the bandwdth mall, the tme repone wll generally be low, and the ytem wll uually be more robut. Looely peakng, bandwdth may be defned a the frequency range [ ] ω over whch control effectve. In mot cae we requre tght control at teady-tate o ω, and we then mply callω ω B. The word effectve may be nterpreted n dfferent way, and th may gve re to dfferent defnton of bandwdth. The nterpretaton we ue that control effectve f we obtan ome beneft n term of performance. For trackng performance the error,ω e y r Sr ee Fgure - and let n d and we get that feedback effectve n term of mprovng performance a long a the relatve error e / r S reaonably mall, whch we may defne to be S. 77. We then get the followng defnton: Defnton - The cloed-loop bandwdth, ω B, the frequency where S jω frt croe db from below. Remark. Another nterpretaton to ay that control effectve f t gnfcantly change the output repone. For trackng performance, the output y Tr ee Fgure - and let n d, we may ay that control effectve a long a T reaonably large, whch we may defne to be larger than.77. Th lead to an alternatve defnton whch ha been tradtonally ued to defne the bandwdth of a control ytem. Defnton - The cloed-loop bandwdth, ω BT, the hghet frequency at whch T jω croe db from above. However, we would argue that th alternatve defnton, although beng cloer to how the term ued n ome other feld, le ueful for feedback control. Another mportant defnton for bandwdth a follow: 8

39 Chapter Lecture Note of Multvarable Control Defnton -5 The gan croover frequency, ω, defned a the frequency where L jω frt croe from above, alo ometme ued to defne cloed-loop bandwdth. c It ha the advantage of beng mple to compute and uually gve a value between ωb and ω BT. Specfcally, for ytem wth PM < 9 o B c BT mot practcal ytem we have ω ω ω -7 In concluon ω B whch defned n term of S and alo ω c n term of L are good ndcator of cloed-loop performance, whle ω BT n term of T may be mleadng n ome cae. Example -7 Comparon of ω B and ω BT a ndcator of performance. Followng an example where ω BT a poor ndcator of performance. z z L ; T ; z., τ τ τz z τ o For th ytem, both L and T have a RHP-zero at z. and we have M., PM 6., M.9, and M. We fnd that ω. 6 and ω. 5 are both le than z. a T B one hould expect becaue peed of repone lmted by the preence of RHP-zero, wherea ω BT / τ ten tme larger than z.. The cloed-loop repone to a unt tep change n the reference hown n Fgure -. The re tme. ec whch cloe to / ω 8.ecbut very dfferent from performance than ω BT. c / ωbt.ec llutratng that ω B a better ndcator of cloed-loop B Fgure - Step repone for ytem T.. 9

40 Chapter Lecture Note of Multvarable Control Fgure - Plot of S and T for ytem T.. The Bode plot of S and T are hown n Fgure -. We ee that T up to aboutω BT. However, n the frequency range from ω B to ω BT the phae of T not hown drop from about o to about o, o n practce trackng poor n th frequency range. For example, at frequency ω 8. 6 we have T. 9 and the repone to a nuodally varyng reference t nω t completely out of phae,.e. y t.9r t. r 8 We thu conclude that T by telf not a good ndcator of performance, we mut alo conder t phae. The reaon that we want T n order to have good performance, and t not uffcent that T. On the other hand, S by telf a reaonable ndcator of performance, t not neceary to conder t phae. The reaon for th that for good performance we want S cloe to and th wll be the cae f S rrepectve of the phae of S. - Proof the equaton -. Exerce - Derve the pre and pot-multplcaton matrce change Π to Π n example -. - Derve the pre and pot-multplcaton matrce change Π to Π n example -.

41 Chapter Lecture Note of Multvarable Control - Derve the LMFD of the ytem n example Conder followng ytem. x y u x x ] [ & a Fnd the SMM form of the ytem. b Fnd the pole and zero polynomal of the ytem. c Fnd the RMFD and LMFD of the ytem. -6 Conder followng tranfer matrx: d Fnd the SMM form of the ytem. e Fnd the pole and zero polynomal of the ytem. f Fnd the RMFD and LMFD of the ytem. -7 Conder followng tranfer matrx: g Fnd the SMM form of the ytem. h Fnd the pole and zero polynomal of the ytem.

42 Chapter Lecture Note of Multvarable Control Fnd the RMFD and LMFD of the ytem. -8 Conder g wth a unty negatve feedback. a Draw the tep repone of the ytem. b From the fgure derved n part a determne: re tme, ettlng tme, overhoot, decay rato, teady tate offet and exce varaton. c Fnd exce varaton and IAE by eq.-8 and -9 repectvely. -9 By ue of fgure -6 derve equaton -56 and In the example. of the man reference Skogetd,5 uppoe the acceptable varaton n room temperature are.5 o K, furthermore, the heat nput can vary between W and W, fnally, the expected varaton n outdoor temperature are - o K, and o K. Fnd the caled tranfer functon. - By ue of Fgure - derve equaton By ue of the man reference Skogetd,5 how by an example thatω B a better ndex for performance thanω. BT Reference Skogetad Sgurd and Potlethwate Ian. 5 Multvarable Feedback Control: England, John Wley & Son, Ltd. Macejowk J.M Multvarable Feedback Degn: Adon-Weley.