Switched state-feedback controllers with multi-estimators for MIMO systems

Transcription

1 Proceedings of the th WEA Int Conf on COMPUTATIONAL INTELLIGENCE MAN-MACHINE YTEM AND CYBERNETIC Venice Ital November witched state-feedback controllers with mlti-estimators for MIMO sstems LIBOR TŮMA BOŘIVOJ HANUŠ AND JAROLAV HLAVA Department of Control Engineering Technical Universit of Liberec Faclt of Mechatronics 6 7 Liberec Hálkova 6 CZECH REPUBLIC Abstract: witched controllers can often otperform traditional adaptive control The paper starts with a brief overview of the development of the field of switched controllers After that it describes an original control architectre based on controller and estimator switching This architectre ses state-feedback control with switched controllers to cope with controlled sstem dnamics changes If conventional observer were sed to estimate the nmeasrable state variables the control performance wold deteriorate becase the estimation reslts cold be corrpted b distrbances acting at varios points of the controlled sstem To overcome this problem the proposed architectre incldes also a bank of switched estimators Each of these estimators is tned for a specific inpt point of a distrbance witching logic is sed to select the most sitable estimator that ields the most reliable estimate in an particlar control sitation A considerable deal of attention is devoted to etensive testing of the proposed architectre sing compter simlation The testing reslts are promising ewords: witched sstems Hbrid sstems witched controller witched-estimators Introdction In man practical sitations the controller design is complicated b considerable modelling ncertainties sstem parameters changes non-linearities and large nmeasrable distrbances There are some classical soltions to this kind of problems The aim of the robst control design methodolog is to design controller that achieves a specified level of performance nder an ncertaint from the assmed ncertaint set However if the ncertaint set is too large the controller becomes ver conservative with nacceptabl slggish nominal responses Another classical soltion is adaptive control In this approach the controller is selected on the basis of the crrent estimate of the ncertain process and this selection is done over a continosl parameteried famil of candidate controllers Adaptive control can be sed even if ncertaint is large and robst control design tools are inapplicable However this approach also has some well known inherent limitations In particlar if nknown parameters enter the process model in a complicated wa it ma be difficlt to constrct a continosl parametried famil of candidate controllers Moreover on-line identification of process model over a continm ma also be a difficlt problem Ths the design of adaptive control algorithms ma be fairl complicated and the final sccess will often depend on trial and error In this paper we will focs on an alternative approach that seeks to overcome some of the above mentioned problems Its main distingishing featre is that controller selection is done sing sitable switching logic not b continos tning witching control Unlike adaptive control which can be considered a classical field of the control theor the se of switched controllers is a relativel new idea Althogh its origins can be traced back to the eighties (the pioneering work is b Mårtensson [6] another important earl paper is []) most papers on switched controllers have been pblished dring the last decade Becase of the hbrid (mied continos/logical) strctre of the control sstem switched controllers can be stdied within a more general framework of hbrid control The algorithms proposed in the earl papers sed a ver simple switching mechanism The were based on a seqential or pre-roted search among a set of candidate controllers This blind search of an acceptable controller is time consming and as a reslt of it the control performance (in particlar the transient behavior) of sch controllers is poor Considerabl better reslts can be achieved with switching algorithms that evalate online the potential performance of each candidate controller and se this to direct their search These algorithms can frther be divided into two main categories:

2 block block Proceedings of the th WEA Int Conf on COMPUTATIONAL INTELLIGENCE MAN-MACHINE YTEM AND CYBERNETIC Venice Ital November switching algorithms based on process estimation and algorithms based on a direct performance evalation of each candidate controller The first categor of algorithms can be represented eg b references [7] [8] and [] Its principle can briefl be eplained in the following wa The switching algorithm (spervisor) continosl compares the behavior of the process with the behavior a several admissible process models in order to determine which model is more likel to describe the actal process The decision is based on the estimation errors achieved with respective models The model with smallest estimation error is regarded as an estimate of the actal process and the spervisor places in the loop the candidate controller that is the most adeqate one for this estimated model Ths the strctre of an estimator-based spervisor consists of a mltiestimator responsible for evalating which admissible model best matches the process and a decision logic that selects which candidate controller shold be sed Algorithms based on direct performance evalation can be characteried b the fact that the spervisor attempts to assess directl the potential performance of ever candidate controller withot tring to estimate the model of the process To achieve this the spervisor comptes certain performance signals that provide a measre of how well each of the candidate controllers C n wold perform in a conceptal eperiment in which the actal control signal wold be generated b C n as a response to the measred process otpt This tpe of spervision is inspired b the idea of controller falsification introdced in [] The strctre of this kind of spervisor incldes a performance monitor that generates the performance signals and a decision logic that selects which candidate controller shold be sed This approach is the relativel less sed one An eample of a paper sing this approach is [9] In the seqel we will basicall follow the approach based on process estimation However the standard algorithm that makes the selection among different controllers in response to the controlled sstem dnamics changes will be agmented with another featre If the controller ses state feedback it is necessar not onl to estimate which of the process models is the most appropriate one bt it is also necessar to estimate the nmeasrable state variables If conventional observer were sed to estimate these nmeasrable state variables the control performance wold deteriorate becase the estimation reslts can be corrpted b distrbances acting at varios points of the controlled sstem (distrbance at sstem inpt or otpt distrbance acting at some other point of the controlled sstem) To overcome this problem the proposed architectre incldes also a bank of switched estimators Each of these estimators is tned for a specific inpt point of a distrbance witching logic is etended so that it cold select the most sitable estimator that ields the most reliable estimate in an particlar control sitation Control sstem architectre The basic strctre of the control sstem is shown in Fig A ver simple set of candidate controllers is sed for the sake of eplanation It incldes onl two controllers for two nominal models of the controlled sstem However etension to a larger set is straightforward A bank of estimators tned for different kinds of distrbances is associated with each controller The following distrbances are considered: d IN distrbance acting at sstem inpt d OUT distrbance at sstem otpt d ARX d ARMAX general distrbances acting anwhere in the sstem described b ARX or ARMAX model The discrete integrators (/(- - )) are added to the basic state feedback strctre in order to prevent stead state errors The state feedback controllers are tned for corresponding nominal models according to qadratic performance criterion (- - ) (- ) T T J Q + Q () d IN d ARX d ARMAX d OUT Controlled sstem (- ) - d - M Estimators Controller d Estimators Controller E M - E E E - w Fig The strctre of the control sstem with switched controllers and switched incremental estimators

3 Proceedings of the th WEA Int Conf on COMPUTATIONAL INTELLIGENCE MAN-MACHINE YTEM AND CYBERNETIC Venice Ital November Controlled sstem is described b A A + B + B d () where is vector of measrable state variables is vector of not measrable state variables d is a distrbance (d d IN d ARX d ARMAX d OUT otherwise d ) Estimators are described b where E E A A + E B + + E d ~ () are the state variables of the E estimator corresponding to the ^ time increments of the controlled sstem state variables d ~ - the inpt distrbance which is estimated b means of the estimator ( d ~ M ε s M B for d ~ d IN d ARMAX otherwise d ~ ) The block with the incremental estimators of the tpe E ARMAX and which also model the controlled sstem in the nominal regimes is connected to the otpt of the controlled sstem and forms inpt to the controller The estimators are tned for the tpes of the epected distrbances d s inpts The information abot the step distrbance inpt is estimated b means of the matri M (see Fig ) from the estimators error ε s and it is transferred frther into the estimator b means of the corrective feedback (some frther reslts regarding the strctre of these estimators were pblished b the athors in [] and []) The maimm nmber of distrbances entering the controlled sstem simltaneosl in one grop is eqal to the nmber of the measrable state variables The nmber of sch grops is nlimited The estimator error is also sed to estimate which nominal model tned in the block of the estimators is closest to the crrent behavior of the controlled sstem If discrete controller is sed the estimator ields the estimate of all nmeasrable state variables Ever estimator of ever nominal block is connected permanentl to the measrable otpt of the controlled sstem and its error ε s is evalated The estimator operates either in its nominal fnctions (eg E ARMAX ) or it is switched over in the fnction It is switched into the control operation if it is evalated as the most sitable one (after the evalation of its error ε s ) The strateg of the estimators switching aims at reaching the smallest vale of the estimators errors ε s as fast as possible The switching algorithm is realied b means of the logic fnctions Testing eperiments The following controlled sstem was sed for testing s+ s+ + s+ s+ () where is integer coefficient that parametries nominal models Now onl two nominal regimes are considered for simplicit ( ) The corresponding discrete time transfer fnctions with the sampling interval T s are for and for () (6) The discrete integrators are combined with the description of the controlled sstem A B + (7) A where is vector of controlled state variables is vector of not controlled state variables The description (7) is agmented with state variables of the integrators (see Figa) I A B + A B (8) A B / 6 a) / 6 6 continos case- t/( ) / discrete case 6 c) d) Fig tep response of the discrete integrator b)

4 Proceedings of the th WEA Int Conf on COMPUTATIONAL INTELLIGENCE MAN-MACHINE YTEM AND CYBERNETIC Venice Ital November a) + ( ) c) + ( ) d) ( ) b) T s and T s The proposed hbrid control strctre was tested with the discretied model (() and (6)) of the real continos sstem () Parameter of the model can be changed The otpt controlled variables of the model are connected permanentl to the estimators ( ) witch are tned for the step distrbances (d ARX d IN d OUT ) Control sstem switches over the estimator fnctions ( ) and connects the estimator to the controller and the controller to the disrete integrator The control performance was tested with different vales of and different step distrbance inpts in the controlled sstem The responses are compared with the control responses of a controlled sstem with all state variables measrable The controllers of both control sstems are eqall tned according to the performance criterion () and to the parameter of the controlled sstem Ever single test is carried ot for one nominal regime of the controlled sstem ( or ) and for one-step distrbance (d IN d OUT and w) in the time s The test incldes the responses of on the sstem with measrable state variables and the responses on the sstem with estimators the differences - and and the changes of the nominal regime and of the tpe of the estimators dring the process The testing reslts are in Fig to Fig Fig distrbance d IN Fig distrbance d IN E - E E Fig distrbance d IN E Fig6 distrbance d IN E E

5 Proceedings of the th WEA Int Conf on COMPUTATIONAL INTELLIGENCE MAN-MACHINE YTEM AND CYBERNETIC Venice Ital November Fig7 distrbance 6 8 d OUT E Fig distrbance d OUT E E Fig8 distrbance d OUT E E Fig9 distrbance d OUT - E Fig distrbance Fig distrbance w E - - w E

6 Proceedings of the th WEA Int Conf on COMPUTATIONAL INTELLIGENCE MAN-MACHINE YTEM AND CYBERNETIC Venice Ital November Fig distrbance Fig distrbance Conclsion w - - E E - w - - E E - A novel hbrid control architectre based on controller and estimator switching was described and tested sing the compter simlation with a model of linear mltivariable forth order sstem with two nominal models The control sstem consists of two blocks with the estimators tned for the epected distrbances (d IN d OUT d ARX ) each block with one of the nominal transfer fnctions The vale of the distrbances is evalated from the estimator errors in the first sampling instant (sing matri M) The tpe of the distrbance and the nominal model of the controlled sstem are estimated in the second sampling instant b means of the comparison of the estimator otpt and of the response of the controlled sstem The reference vale w changes are controlled withot error b means of the estimator of an tpe The error vales from the following sampling instants can be accmlated for frther evalation The sampling intervals of the estimators and of the controllers ma be different The testing reslts are promising and the confirm the epectations Frther testing of this control architectre will be done sing the laborator plant for eperiments with hbrid control that was described b the athors in [] Acknowledgement: This research has been spported b the Cech cience Fondation within project //8 References [] F M and Barmish B R Adaptive stabiliation of linear sstems via switching control IEEE Transactions on Atomatic Control Vol 986 pp 97- [] Hanš B and Tůma L Estimators for hbrid mltidimensional control sstem with variable strctre Proceedings of the th International Conference on Process Control Štrbské pleso lovak Repblic Ma [] Hanš B and Tůma L Hbrid Control cheme With Discrete Estimator for Efficient Distrbance Rejection Preprints of the 6th IFAC World Congress Prage Cech Repblic Jl [] Hlava J and Šlc B An Eperimental Approach to Verification of Control Algorithms for Hbrid Continos-Discrete stems WEA Transactions on stems Vol No 6 pp 6-6 [] Liberon D witching in stems and Control Birkhäser [6] Mårtensson B The order of an stabiliing reglator is sfficient a priori information for adaptive stabiliation stems & Control Letters Vol 6 No 98 pp 87 9 [7] Morse A pervisor control of families of linear set-point controllers part : eact matching IEEE Transactions on Atomatic Control Vol No 996 pp - [8] A Morse pervisor control of families of linear set-point controllers - part : robstness IEEE Transactions on Atomatic Control Vol No 997 pp - [9] Mosca E Capecchi F and Casavola A Designing predictors for MIMO switching spervisor control International Jornal of Adaptive Control and ignal Processing pecial Vol No pp 6 86 [] afonov MG and Tsao T-C The nfalsified control concept and learning Proceedings of the rd Conference on Decision and Control pp 89 8 Dec99