Design of Unknown Inputs Observer for a Chemical Process: Analysis of Existence and Observability Based on Phenomenological Model

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1 Design of Unknown Inputs Observer for a Cemial Proess: Analysis of Existene and Observability Based on Penomenologial Model Mario Giraldo Esuela de Meatrónia aultad de Minas Universidad Naional de Colombia Grupo de Automátia Universidad Naional - GAUNAL Medellín Colombia magiraldovas@unal.edu.o Hétor Botero Esuela de Meatrónia aultad de Minas Universidad Naional de Colombia Grupo de Investigaión en Proesos Dinámios - Kalman Medellín Colombia E Mail: abotero@unal.edu.o Abstrat is paper explains te existene and observability onditions for te design of an Unknown Inputs Observer (UIO for a emial proess. irst a penomenologial interpretation for te existene of te observer is given from te transfer funtion of te linearized system. And seondly te existene and observability onditions of te observer are verified algebraially evaluating te rank of a defined observability matrix. Bot ways of verifiation are ompared. inally an UIO is implemented and simulation results are presented. Keywords-unknown inputs observer; observabilit; stable observer; ausal systems. I. INRODUCION State observers are widely used wen a variable or set of variables of interest need to be observed or estimated beause measurements are unavailable due to diffiulties or impossibilities witin te proess ig measurement osts among oter reasons []. A partiular problem of state estimation using state observers arises wen one or more inputs of te system are unknown. is kind of problem an be found e.g. in emial reators were is diffiult measuring te onentration anges of substanes in te input flows. e proposed solution for tis problem is te Unknown Inputs Observer (UIO wi allows te estimation of te states vetor of a linear or nonlinear system measuring some of te inputs and available outputs [34]. e performane of UIO for linear time-invariant systems (LI and nonlinear systems ave been proved wit good results and te existene and observability onditions of te observer ave been well studied and several metods ave been proposed [567]. or linear time-invariant systems te existene of te UIO and te observability ondition an be verified by means of observability matrix rank onditions similar to te ones proposed by Kalman for systems wit known inputs [8]. However tere is a lak of a pysial interpretation of te rank onditions in te observability matrix of a dynami system. Some previous works to explain te relationsip between te system penomenology and te existene of an observer ave been presented [9] and oter give a penomenologial interpretation of te existene of an UIO and Unknown Inputs Observability for LI systems based on an eletromeanial system [0]. It was also stated tat te proedure ould be easily extended to oter dynami systems models. In te same way tis paper explains te existene and observability onditions for te design of an UIO for a emial proess wit pysial sense. Wit te obtained results and its omparison wit te previous work it an be proposed te analysis metod as a general one. e paper is organized in four setions. In setion are establised te algebrai onditions for an UIO to be an asymptotially stable observer. Next te design proedure for te observer is explained. e setion finises wit te existene and observability onditions of te observer by means of an algebrai evaluation of rank onditions of given interest matries. In setion 3 a nonlinear system is proposed to be observed. e system is linearized in order to give a pysial interpretation for te existene and observability onditions from te model penomenology. inally in setion 4 an UIO is implemented for te dynami system desribed in setion 3 and simulation results are presented. II. UNKNOWN INPUS OBSERVER DESIGN e problem of designing an unknown inputs observer onsiders a linear time-invariant system desribed by x = Ax + Bu + Dv ( y = Cx (

2 n x R k u R m v R and p y R are te state were vetor te known inputs vetor te unknown inputs vetor and te output vetor of te system respetively. A B C and D are known onstant matries. Assuming tat p m and rank( D = m and rank( C = p an UIO for te system an be desribed as []: (3 z = Nz + Ly + Gu ˆx = z Ey (4 n were z R and N L G and E are unknown matries wi must be determined su tat te estimate state ˆx onverges asymptotially to x. A. Conditions for designing an asymptoti UIO e state observer desribed in (3 and (4 is proposed for te system desribed in ( and ( so te estimation error an be defined as [7] e= xˆ x= zx Ey (5 wit an observer error dynami ( ( (6 e= Ne+ NP+ LC PA x+ GPB updv wit P = In + EC (7 Establising onditions for ˆx to be an asymptoti estimation of x (6 an be redued to te omogeneous equation e = Ne (8 e onditions are establised as follows PD = 0 or ( I + EC D= 0 (9 n G = PB (0 NP + LC PA = 0 ( e equation (8 desribes an asymptotially stable error dynami if onditions (7 and (9 to ( are satisfied; additionally N must be Hurwitz. In order to use results obtained for lassial state observers witout unknown inputs in te definition of te observer dynamial equation as stated in [7] te equation ( an be written as were Substituting ( in (3 L is obtained as N = PA KC ( K = L+ NE (3 ( p L = K I + CE PAE (4 en te observer dynamial equation (3 beomes ( z = PA KC z + Ly + Gu (5 B. UIO alulation Having te onditions establised in te previous subsetion te proedure for alulation an UIO is straigtforward sine E P G and L an be obtained from (9 (7 (0 and (4 respetively. e problem of designing a state observer wit unknown inputs an be redued to find a matrix E satisfying (9 and a matrix K su tat ( PA KC is Hurwitz. is problem is equivalent to te standard observer design wit known inputs [6]. e eigenvalues of ( PA KC an be arbitrary loated. If te pair ( PA C is observable matrix K is osen suitably for te problem. If te pair ( PA C is not observable but detetable K must be found su tat te observer is asymptotially stable. e proedure to design te observer begins by alulating E matrix. rom (9 we ave ECD E exists if rank( CD = D (6 = m. One tis ondition is eked E an be obtained as follows E = D CD (7 ( If CD is not square its pseudoinverse matrix L ( CD = CD CD CD is alulated and a general solution of (6 an be written as L ( p L ( ( ( E = D CD + Y I CD CD (8 Having E P is alulated using (7. After te observability or detetability ondition of te pair ( PA C is verified matries N and K are alulated simultaneously wit (. G and L an be alulated straigtforward wit (0 and (4 respetively. C. Existene and observability onditions in UIO design eorem gives te neessary and suffiient onditions for te existene and observability onditions of a stable observer as (5: eorem

3 or te system desribed by ( and ( te observer (5 exists if and only if Condition (Existene ondition rank( CD = rank( D = m Condition (Condition to establis an arbitrary error dynami sp PA rank = n s Re( s 0 C e teorem demonstration an be found in [7]. III. PHYSICAL INERPREAION OR HE OBSERVABILIY WIH UNKNOWN INPUS In te previous setion an algebrai proedure as been desribed to verify te existene of an observer for a dynami system. But it is intended to establis wat are te pysial onditions or te system dynamis beaviour tat allow te designing and implementation of te UIO. In tis sense a emial proess as been analyzed like example. In tis ase a matematial model of a rossflow tubular eat exanger is presented. e diagram of te proess is sown in igure. e system is omprised by a refrigerant fluid (: ool and a work fluid (: ot. e temperature of te first is lower tan te seond. i o igure. Proess diagram: Crossflow tubular eat exanger. A energy balane produes te following model d Cp U a LMD = ( + dt Cv V Cv V o i o o i o ( ρ d Cp U a LMD = ( dt Cv V Cv V ( ρ (9 (0 were is te volumetri fluid flow Cp is te onstant pressure speifi eat Cv is te onstant volume speifi eat V is te volume U is te global eat transfer oeffiient a is te transferene area ρ is te fluid density and LMD is te differene of temperature of te fluids along te transferene i o area a. e sub indies i and o stand for old water ot water input and output respetively. e temperature differene anges point to point as te fluids flow troug te tubes ausing te eat exange dynami to ange point to point. is makes neessary to define te temperature differene as a Logaritmi Mean emperature Differene (LMD []. LMD = A. Model linearization ( ( i o o i ln i o o i ( e linearization of (9 and (0 is neessary in order to obtain te system transfer funtions and te state spae representation in te form of ( and ( from wi te UIO is designed and existene onditions are evaluated. e linearization of te system is done around an operation point by aylor series expansion. A detailed proedure an be found in [3]. e state spae model obtained in te linearization of (9 and (0 is x = αx+ αx + u ( x = α3x+ α4x + u (3 wit Cp U a l α = Cv V α = ( ρ Cv V ( ρ Cv V α = Cp U a l α4 = Cv V ( ( = i o ρ Cv V = i o ρ Cv V and 3 ( ρ Cv V ( ρ Cv V ( ( l = ln ( ( i o o i i o i o ln o i o i o i ( were x is te output old water temperature x is te output ot water temperature u is te volumetri old water flow and u is te volumetri ot water flow. ese variables are

4 deviation variables ten representing anges of te system dynamis in te neigbourood of an operation point. B. Evaluation of te existene of te UIO from te pysial system model e transfer funtions of te system are obtained from ( and (3 depending on wi is te state measured te input measured te state to be estimated and te unknown input. Ea ombination is sown in te ases presented in able. able. Status of te variables of te system. Case Case Case 3 Case 4 Unknown input u u u u Known input u u u u Measured state x x x x state x x x x e transfer funtions for ea ase are: Case : s α X ( s = X ( s U ( s α α Case : α X ( s = X ( s + U ( s 3 sα4 sα4 Case 3: α X ( s = X ( s + U ( s sα sα Case 4: s α X ( s = X ( s U ( s 4 α3 α3 Cases and 4 sow transfer funtions tat are not proper te order of te numerator is greater tan te order of te denominator. It indiates tat te number of derivatives in te input is greater tan te number of derivatives in te output. It means tat it is neessary to know future values of te inputs to predit te urrent values of te outputs. Systems wi orrespond to tis desription are nonausal systems. On te oter and Cases and 3 sow transfer funtions tat are proper and terefore are ausal systems. In ase it is neessary to estimate x (ot water output temperature wit x (old water output temperature and u (old water volumetri flow known but u (ot water volumetri flow unknown. is is pysially impossible beause a ange in te ot water volumetri flow as a diret effet on te ot water output temperature and ten te eat exange ours to produe a ange in te old water temperature. erefore tere is no diret ausal relationsip between te ange in te unknown input and te ange in te measured variable. In ase 3 it is neessary to estimate x (old water output temperature wit x (ot water output temperature and u (old water volumetri flow known but u (ot water volumetri flow unknown. is is pysially possible beause a ange in te ot water volumetri flow as a diret effet on te ot water output temperature and ten te eat exange ours to produe a ange in te old water temperature. erefore tere is a diret ausal relationsip between te ange in te unknown input and te ange in te measured variable. Next an analysis based on ondition of eorem is performed. C. Evaluation of te existene and observability onditions using eorem In order to ek te results from te analysis performed in te previous subsetion te algebrai alulus proposed in eorem are arried out. A matrix is defined and te state vetor x remains te same for all ases as follows: Case : A α α x x o = α3 α = = 4 x o B = [ 0] C = [ 0] and D [ 0 ] Wi leads to [ ] CD = 0 rank( CD = 0 =. (4 Sine ondition from teorem is not satisfied a state observer annot be designed. Case 3: B = [ 0] C = [ 0 ] and D [ 0 ] Wi leads to [ ] CD = rank( CD = And te rank ondition matrix is: s + α α =. Sine ondition of teorem is fulfilled and te rank ondition matrix an be evaluated an observer an be designed. s α e error dynami an be arbitrarily adjusted if. is means tat te observability ondition of te system is lost for s = α. If ondition is not satisfied ausal relationsips

5 between te estimated states and te known inputs and outputs an be found altoug tose relationsips are restrited. e dynami error matrix N is alulated for ase 3: N= α α 0 k k α k If = te eigenvalues of te dynami error matrix indiate tat te error dynami annot be arbitrarily seleted sine λ = α is a fixed value inerent from te system dynamis. But k an be arbitrarily osen. en te UIO an be designed but te dynami error response is limited by te system dynamis. In summary sine ases and 3 represent ausal systems an UIO for su systems an be designed. Cases and 4 are nonausal systems. IV. SIMULAION: UIO IN HE REAL HEA EXCHANGER ollowing te proedure introdued in setion an UIO is designed for a eat exanger wi uses ot water as work fluid and old water as ooling medium. e parameters of te model of te eat exanger desribed by equations (9 to ( and its linearization ( and (3 are sown in able and able 3. e values of te parameters were obtained by experimentation and parametri identifiation of a eat exanger in te bioproesses laboratory at Universidad Naional de Colombia Medellín site. able. Heat exanger parameters. Parameter Value Units U W/m K a 0.04 m Hot water parameters values ρ Kg/ m 3 Cp 480 J/KgK Cv 480 J/KgK V m 3 Cold water parameters ρ Kg/ m 3 Cp 479 J/KgK Cv 479 J/KgK V m 3 able 3. Steady state values. Variable Value Units m 3 /s m 3 /s i K i 33.5 K o K o 35.5 K e UIO was designed for a eat exanger in wi u = v= (ase 3 of able te measured variable or state was o x in te linearized system and te estimated variable or state was o x in te linearized system. e performane of te observer is tested introduing anges or perturbations tat ange te operation onditions of te system. able 4 reports te perturbations introdued to te system. able 4. Perturbations introdued to te system. Variable ime (s Magnitude (m 3 /s u v Simulation results are sown in following figures. In igure te estimated estate ( o onverges but its error dynami is restrited by te onstant α =. e time of response of te system dynamis are about five onstant times. It is expeted tat te observer reaes and follows te referene for at a time of seonds. o emperature (K Cold water temperature Referene ime (s igure. Cold water temperature dynami. igure 3 is a zoom of te initial 6 seonds of simulation and it sows tat te observer reaes te referene in te stipulated time. emperature (K Cold water temperature ime (s Referene igure 3. Cold water temperature dynami. Initial transient zoom.

6 e error dynami for te measured state an be osen arbitrarily in igure 4 k = so it is expeted tat te observer reaes te referene at 5 seonds and in igure 5 k = 0 so te time response is 0.5 seonds. emperature (K emperature (K Hot water temperature Referene ime (s igure 4. Hot water temperature dynami ( k = Hot water temperature Referene ime (s igure 5. Hot water temperature dynami ( k = 0. V. CONCLUSIONS e existene and observability onditions for te design of an UIO for a emial proess are given and tey were explained wit pysial sense. e suggested analysis metod to give a pysial interpretation of te existene and observability onditions of a system proved to work wit good results. Bot algebrai and penomenologial analysis pointed te same regarding te existene of an observer. is work and previous works support te generalization of te analysis. [] Moreno J. Unknown inputs observers for SISO nonlinear systems. Proeedings of te 39 t IEEE onferene on Deision and Control. Sydney Australia 000. [3] Radke A. and Gao Z. A survey of state and disturbane observers for pratitioners Proeedings of te 006 Amerian Control Conferene pp Minneapolis Minnesota USA June [4] Raff. Laner. and Allgower. A finite time unknown input observer for linear systems 4t Mediterranean Conferene on Control and automation June 006. [5] Basile G. and Marro G. On te observability of linear time-invariant systems wit unknown inputs Journal of optimization teory and appliations vol. 3 N 6. pp [6] Aboky C. and Vivalda J-C. Observability for linear systems wit unknown inputs Proeedings of te 39 t IEEE onferene on Deision and Control. Sydney Autralia pp [7] Daroua M. Zasadzinsky M. and Xu S.J. ull order observers for linear systems wit unknown inputs. IEEE transations on automati ontrol volume [8] Kalman R. (96. On te general teory of ontrol systems. Proeedings of te firts IAC ongress volume. [9] Hautus M.L.J. and Sontag E.D. (980. An approa to detetability and observers. Letures in Applied Matematis. Vol [0] Botero H. Gómez L. and Giraldo M. Diseño de observadores de estado on entradas desonoidas: Análisis de la existenia y la observabilidad desde la fenomenología del modelo. XIV Congreso Latinoameriano de Automátia. Santiago de Cile 00. [] Yang. and Wilde R. (988. Observers for linear systems wit unknown inputs. IEEE transations on automati ontrol. Vol 33. N 7 July 988. [] Kern D. Proess eat transfer. MGraw Hill Méxio [3] Marín H. Muñoz D. and Murillo J. Evaluaión y análisis de ontrolabilidad para la operaión unitaria interambio alório omo una aproximaión a la integraión Diseño-Control. rabajo de grado. Ingeniería químia. aultad de Minas. Universidad Naional de Colombia REERENCES [] Oliveira J. Santos J.N. Selegim Jr. P. Inverse measurement metod for deteting bubbles in a fluidized bed reator toward te development of an intelligent temperature sensor Powder enology vol. 69 pp