A Novel Unknown-Input Estimator for Disturbance Estimation and Compensation

Transcription

1 A Novel Unknown-Input Estimator for Disturbance Estimation an Compensation Difan ang Lei Chen Eric Hu School of Mechanical Engineering he University of Aelaie Aelaie South Australia 5005 Australia Abstract A novel unknown-input estimator (UIE) for estimating an rejecting isturbances is propose in this paper Effective treatment of unknown isturbing inputs is of vital importance to maintaining satisfactory performance of control systems Avance methos that estimate these isturbances an cancel them accoringly outperform traitional approaches However limitations remain incluing requirements for some knowlege on unknown inputs erivatives of measure outputs inversion of plant ynamics constraine state observer esign parameter optimisation (global optimum not guarantee) or complicate structures he propose UIE is exempte from the aforementione limitations It consists of an estimation gain matrix a state observer an a low-pass-filter-characterise subsystem Comparison via simulation is rawn between the new UIE an a benchmark isturbance observer on a multi-input multi-output system he propose UIE is shown to be more effective in estimating an compensating unknown inputs 1 Introuction In reality a control system is always subject to unknown inputs in the form of internal uncertainties an exogenous loas that cannot be measure or inconvenient to measure If these isturbances are not properly treate poor performance of the control system may result especially in applications where high-precision set-point tracking or regulation of controlle variables (eg position spee or torque etc) is require [Schuhmann et al 01; Erenturk 013] o attenuate the negative impact from unknown inputs various methos can be use A common practice is to maximise the inherite robustness of a controller by careful parameter esign an tuning [Åström an Hägglun 006] Further robustness improvement can be achieve via avance techniques such as nonlinear control [Chen et al 009] H robust control [Dietz an Scherer 010] an aaptive intelligent control [Mohammazaheri an Chen 010a; Mohammazaheri an Chen 010b] A more attractive approach istinguishe from the aforementione comes with the concept of unknown-input estimation which brings isturbance rejection to a higher level as emonstrate by various inustrial applications [She et al 011; Mitsantisuk et al 01; Erenturk 013; Chanar an alole 014] his technique provies the controller with better tolerance to uncertainties by constructing counteractive control efforts to cancel the effects from unknown inputs accoring to the estimations With an extene state observer a class of partially unknown inputs moelle in the form of constant-coefficient ifferential equations are estimate together with the states of a linear time-invariant plant [Johnson 1971; Ohishi et al 1987] he well-known internal moel principle works in a similar way but the plant states are not estimate [Francis an Wonham 1975; Hara et al 1988] A prerequisite for all these methos is that characteristics of the exogenous inputs such as the frequency of each sinusoial component must be known Although such information can be obtaine via fiel or experimental ata there is possibility that the ata collecte may not cover all situations In this case isturbances cannot be sufficiently well moelle an unsatisfactory estimation performance may result Moreover it is sometimes ifficult or inconvenient to collect relevant ata for isturbance moelling When only the orer of the exogenous inputs is known an these isturbances perturb the system through channels other than those of the control inputs a generalise extene state observer can be use [Li et al 01; Gobole et al 013] For nonlinear systems nonlinear isturbance observers are also available [Yang et al 01; Wei an Chen 01] However higher-orer observers are require when ealing with complex isturbing exogenous inputs an this increases the observer imension as a result Instea of using a moel base on a priori assumptions or empirical knowlege estimating completely unknown inputs is possible by combining an unknown-input-ecouple observer (UIDO) with an aitional isturbance-estimation function [Hou an Müller 199] he former is use to observe plant states while the latter is for approximating the unknown inputs Without ecoupling unknown inputs an orinary state observer [Luenberger 1964] can still work

2 properly uner the situation that the unknown inputs are cancelle by corresponing control inputs Similarly in this case an estimation function is still neee besies the state observer to preict unknown inputs Among these stuies some methos require the erivatives of measure outputs [Liu an Peng 00; Chanar an alole 014] which make the estimation sensitive to noises or moelling errors; Although some approaches o not ifferentiate the measure outputs [Corless an u 1998; Liu an Peng 000; Xiong an Saif 003; She et al 008; Liu et al 013] some other limitations are impose Specifically some rank an bouneness conitions apply to unknown inputs in the work of Corless an u [1998] Liu an Peng [000] an Xiong an Saif [003]; he computation becomes another problem using the metho of Liu an Peng [000] when the number of unknown inputs increases; he complexity of parameter selection rises ue to the less restrictive conitions on unknown inputs [Xiong an Saif 003; Kim an Rew 013]; he state observer gain is constraine by the estimation scheme for unknown inputs an in the mean time limite esign freeom is given to the unknown-input estimation [She et al 008; Liu et al 013] As a counterpart of esign in the state space the isturbance observer (DOB) synthesise in the frequency omain relies on a customise low-pass filter an the inversion of plant ynamics [Ohnishi 1987; Umeno an Hori 1991; Umeno et al 1993; Kempf an Kobayashi 1999; Choi et al 003] he custom low-pass filter esign which influences the stability of the plant can be intricate when an optimal set of filter coefficients are esire through an optimization proceure he plant inversion also causes some ifficulties in employing DOBs in multi-input multi-output (MIMO) systems since an inverse cannot always be foun he generalisation of the DOB to MIMO cases bypasses the plant inversion an aopts parameterise controller esign through an optimisation process [Shahruz 009; Du et al 010] Nevertheless the global optimum is not guarantee in these methos he work in this paper therefore aims to evelop an alternative approach for unknown-input estimation exempte from the aforementione limitations without sacrificing its performance Instea of iscussing unknown-input estimation as a subsystem or an a-on of an existing regulator or tracking controller the stuy presente in this paper solely focuses on the new unknown-input estimation scheme an treats it as a stanalone mechanism he outcomes of this stuy not only contribute to the theory framework of unknown-input estimation but also eliver potentials of proviing existing controllers in literature with enhance tolerance to isturbances for further improvement on regulation or trajectory tracking performance o avoi confusion an for unity the scheme for unknown-input estimation is terme as unknown-input estimator (UIE) in this paper Subsequent sections are arrange as follows A novel UIE is erive an analyse in Section ; Simulations are presente in Section 3 in the form of comparisons between the propose UIE an a benchmark DOB on an MIMO system Conclusions are rawn in Section 4 Unknown-Input Estimator (UIE) 1 Problem Statement an Assumptions he general plant moel to be consiere is given by x Ax Bu B (1) y C x nx ny nu where x y an u are vectors of system states measure outputs an control inputs with n n x n y an n u elements respectively; xn A x n represents system internal ynamics; xn B u an nxn B enote the istribution of control an unknown inputs respectively; C escribes output nyn x n ynamics; contains n unknown inputs that the system is subjecte to incluing uncertain nonlinear time-varying an state-epenent terms he following two assumptions are mae: Assumption 1: A B is controllable Assumption : C A is observable he first assumption is not irectly associate with the UIE to be iscusse but mae to confine the plant of interest to the controllable system category in inustry he secon assumption is necessary for state estimation which is one of the essential components of the UIE propose in this paper Note that B can be either known or unknown with no rank conition require an there is no restriction on the number of unknown inputs his means there can be more unknown inputs than the control inputs an the measure outputs When both B an n are unknown it is ifficult or even impossible to precisely estimate the components of Given the fact that the estimation of unknown inputs in most servo systems are for isturbance rejection to maintain prescribe performance it is thus possible to assume the existence of equivalent unknown inputs entering the plant through the control input channels B his is a practical technique commonly aopte in DOBs an it has been shown by She et al [008] that such equivalent quantities exist for the unknown inputs in a system as in Eq (1) uner Assumptions 1 an Denote the equivalent unknown inputs by e then Eq (1) can be expresse as x Ax Bu e () y C x Now we can use Eq () as an equivalent of Eq (1) an the control problem at this point is to estimate an fee back ( ) to cancel e t UIE Structure In orer to cancel the effects from unknown inputs or equivalently e the control input u is efine as u ˆ (3) e

3 Now we construct ˆ e ( t ) in the following form: ˆ ˆ e ev + K y yˆ (4) where ˆ n ev is an aaptive auxiliary variable for nn y estimating the equivalent unknown inputs K is an estimation gain an y ˆ is an estimate of the plant outputs he auxiliary variable ˆ ev is obtaine an upate in real time via a subsystem Af Bf C f with ˆ e ( t ) as its input: ( ) ( ) ˆ x f t Af x f t Bfe (5) ˆ ev C f x f nf n f nf n n n f where Af Bf an C f are the internal input an output ynamics of the subsystem respectively with n f enoting the number of the subsystem states Note that Eq (5) only gives a general input-output form of the subsystem In aition x f o not have any irect physical implication in this general form an n f epens on the etaile structure of the subsystem It is because the esign of ( Af Bf C f ) is flexible an varies among ifferent applications Proper selection of the structure an parameters for the subsystem is aresse in etail in Sections 3 an 4 for ease of iscussion in consieration of its irect impact on system close-loop characteristics In orer to obtain the estimate plant outputs y ˆ a state observer that takes e into account is neee an constructe as: ˆ( ) ˆ( ) [ ( ) ˆ x t Ax t B u t e ] L[ y yˆ ] (6) yˆ Cxˆ nxn y where L is the estimation gain for the state observer an x ˆ is the estimate of plant states In summary the subsystem Af Bf C f the gain K an the state observer are the three major components to esign for the propose UIE 3 Close-loop Analysis Since we are able to use system () as an equivalent of Eq (1) the mechanism of estimating an suppressing unknown inputs can be reveale by the relation between the equivalent unknown inputs e an the system outputs y in Eq () From Eqs (4) (5) an (6) we have in frequency omain: 1 yˆ C ( s I A LC ) Ly G y (7) nx ˆ 1 C ( si A ) B ˆ G ˆ (8) ev f nf f f e e ˆ 1 e ( s ) Inu G ( s ) K y ( s ) yˆ ( s ) (9) Gy y yˆ an thus y Iny P Gy Iny Gyy 1 P e (10) G e where P is the transfer function matrix of the MIMO plant from u to y ; G ( ) G ( ) G ( ) an yy s s yy y s G are the transfer function matrices of corresponing input/output pairs; I nx I nf I nu an I ny are ientity matrices with a imension of nx nx nf nf nu nu an ny ny respectively Note that Eq (10) is in an MIMO form For ease of iscussion the i row an j column element in G an th th G is enote by G ( s ) an G ( s ) respectively As shown in Eq (10) for a given P the effects from unknown inputs can be effectively suppresse by minimising G ( i 1 ny ; j 1 nu ) an this can be achieve by a proper esign of G y In other wors if a G y exists so that G is sufficiently small then estimates of the equivalent unknown inputs can be obtaine with aequate accuracy Feeing these estimates back into the system can effectively counteract the actual unknown inputs alleviating isturbances Accoring to Eq (9) this requires appropriate esign of G an K Specifically for any positive scalars an 3 1 with a properly etermine K 0 assume if ii 1 G ; ( i 1 n ) (11) n u ii j0 1 an G G ; ( i j 1 n ) (1) then from Eqs (9) an (10) G 3; ( i 1 ny ; j 1 nu ) an as a result y 0 in Eq (10) for t o avoi confusion efine y as the error between the isturbe an unisturbe outputs hat is y 0 in Eq (10) means y P ( )[ ( ) ˆ n s e s e ] 0 which leas to ˆ e ( s ) e ( s ) Accoring to Eq (5) it is also true that ˆ ev e at steay state With a properly esigne G satisfying Eqs (11) an (1) the spee of estimation can be further fine tune via ajustment of K as inicate by Eq (9) 4 UIE Design Selection of State Observer Gain L Accoring to Eqs (7) an (10) as well as the state estimation principle there is limite contribution from G yy in minimising G for a given G y herefore the state observer gain L can be esigne normally as the one for an orinary observer When a system is subject to unknown inputs that are non-gaussian an vary ynamically an orinary state observer assuming isturbances as white noises can have consierable estimation error However with the presence of the propose UIE these non-gaussian an time-varying unknown inputs can be mostly cancelle an the Kalman filter technique can be employe to compute L Remark 1: he ynamics of state estimation is given by e x ALC ex B (13) where ( ) ˆ e t e As shown in Eq (13) state estimation of aequate accuracy is ensure by the presence u u

4 of the UIE with estimation errors subjecte only to the resiual equivalent unknown inputs 0 as t Selection of G or ( Af Bf C f ) As iscusse in Section 3 the esign of G shoul satisfy the conitions as in Eqs (11) an (1) for effective unknown-input estimation an cancellation However it is known that G ( i 1 ny ; j 1 n ) cannot be minimise ieally throughout [ 0 ) For this reason an assumption is mae that the natural frequency of the unknown inputs to be suppresse are below c his hols true for a majority of servo systems as most isturbances are from low-frequency sources herefore it is viable to introuce G in the form of a low-past-filter-characterise subsystem so that max G ( j) is minimise for effective estimation an suppression of unknown inputs of frequency within ( 0 c ) instea Noises of higher frequency are filtere For simplicity ecoupling of multiple unknown inputs is preferre an thus G can be constructe irectly as a iagonal matrix: 11 ii G ( j) iag G ( j11 ) G ( jii ) (14) where ii ii with ii an i 1 nu In this paper a 1 st -orer filter is use for illustration purpose given its least phase lag an simplest structure: ii 1 1 G (15) 1 iis 1 s 1 cii th where cii is the cut-off frequency of the i iagonal element an ii is the corresponing time constant he state-space representation ( Af Bf C f ) can be obtaine through an appropriate transform from G Remark : It is worth emphasis that the esign of the subsystem G or ( Af Bf C f ) is not limite to the low-pass-filter-characterise structure iscusse herein but flexible an can vary for situations other than the inustrial servo systems commonly seen he most suitable configuration of the subsystem epens on the iniviual applications uner consieration an the esign can simply follow the guielines provie in this paper Selection of K From Eqs (9) an (10) K of a higher matrix norm contributes to smaller G ( j) for frequencies within 0 c However it is impractical to have infinitely large K which can amplify noises an reuce amping For a balance esign of K the linear-quaratic-regulator (LQR) computation can be performe on the reconstructe plant ynamics e x ( ALC ) ex Bev (16) ey Cex where e x xˆ x e y yˆ y an ev K C e x Accoring to uality this is to select that minimises the performance inex ex t Q ex ev R ev 0 K ( BK ) J ( ) t v (17) with symmetric positive-efinite weighting Q an R of respective size nx nx an nu nu As a result K = B K (18) where B is the Moore-Penrose pseuo inverse of B Remark 3: Unlike the estimator of She et al [008] some relative inepenence in the state observer esign is retaine in our approach although ynamics of the state observer are associate with the UIE he introuction of an aitional parameter K in our metho gives a thir egree of freeom in the UIE esign an hence the performance of the UIE is not merely epenent on the state observer gain an the low-pass-filter-characterise subsystem As a result less constraint is impose on the state observer esign his feature not only offers more flexibility in UIE tuning but also plays an important role when the propose UIE is integrate with an existing controller esigne in state space such as a stanar linear-quaratic-gaussian (LQG) tracking controller o be specific such flexibility allows inepenent an separate esign of the propose UIE as an a-on to an existing state-space controller for enhance robustness hese two components can share the same state observer without imposing aitional constraints on the state observer esign (he integration of the propose UIE into an existing controller is not the focus of the current stuy) 3 Simulations In this section the performance of the propose new UIE is evaluate via simulations For comparison a DOB specifically generalise for MIMO systems [Shahruz 009] is selecte as a benchmark his DOB takes the form of a conventional DOB with inherite reliability Moreover the H optimisation technique is applie in esign which gives aitional creits to such a DOB for its claime superior performance A mass-spring-amper system of two egrees of freeom as in Figure 3 with two control inputs u [ u1 u ] an two measure outputs q [ q1 q ] is consiere [Shahruz 009] Exogenous unknown inputs are not shown in the figure because no restrictions are impose on the number an istribution of these isturbances he corresponing equation of motion is given by Mq Rq Sq Bu B u (19) where the mass amping an stiffness matrices are M R an 3 1 S 1 1 v

5 Figure 1: A mass-spring-amper system of two egrees of freeom with two control inputs (image aapte from [Shahruz 009]) an the control input istribution matrix B is fixe as B while the istribution of unknown inputs B is not etermine herein as ifferent assumptions are to be mae in simulations that follow Let x [ q1 q q 1 q ] hen a state-space representation for Eq (19) is obtaine as in Eq (1) hroughout simulations the propose UIE takes the following esign: K L an Af B f 0 8 an C f for subsystem ( Af Bf C f ) equivalent to a two-channel low-pass filter of 1 st orer as in Eqs (14) an (15) with cut-off frequency c11 c 100 ra/s Note that the DOB of Shahruz [009] is synthesise in frequency omain with corresponing close-loop transfer function given by y [ P P K P ] e (0) an the parameter K is optimise as K11 K1 K K1 K where ( s s ) K11( s) s s ( s ) K1 s 198 1s ( s ) K1 s 198 1s an ( s s ) K s s he control effort provie by the DOB is the same as in Eq (3) but with ˆe obtaine via ˆ ( s ) K ( s )[ P ( s ) ( s ) P ( s ) u ( s )] e a s a where P ( ) enotes the actual ynamics of the plant In the following simulations two scenarios are consiere Case I: Exogenous unknown inputs are to be estimate uner the conition that B B his allows irect evaluation on the performance of unknown-input estimation by comparing the isturbances with corresponing control efforts accoring to Eq (3) he isturbances are simulate as: 1 sin( t) an a sin( t) a sin( t) 1 1 a1 t cos t where a cos t 5 0 5t 1 It is foun in simulation that both the UIE an DOB can effectively estimate the simulate unknown inputs with high accuracy an the associate estimation errors can be barely ientifie irectly from plots with an ˆe as in Figures an 3 he estimation performance can be better reveale by the corresponing estimation errors e ˆ e u As shown in Figures 4 an 5 both the UIE an DOB prouce very small estimation errors It is also notable that the UIE has much smaller estimation errors than the DOB Figure : Disturbance 1 an estimations Figure 3: Disturbance an estimations

6 Figure 4: Estimation error of isturbance 1 Figure 6: Open-loop system outputs uner isturbances Figure 7: Close-loop system outputs uner isturbances Figure 5: Estimation error of isturbance Case II: As mentione in Section it is expecte that exogenous unknown inputs can also be effectively cancelle uner the situation of B B an n n u In this case the performance of estimation an compensation can be evaluate by observing the system outputs hat is y 0 when t inicates satisfactory results Specifically 1 sin( t ) B an sin sin Without the UIE an DOB the outputs y of the system oscillate into large amplitue uner isturbances (see Figure 6) he oscillation can be effectively suppresse however by the UIE or DOB an notably the UIE outperforms the DOB as shown in Figures 7 an 8 It is worth emphasis that the H optimisation metho is performe for the DOB esign which can be verifie by the satisfactory isturbance suppression results of the DOB In comparison the much better isturbance compensation achieve by the UIE emonstrates a notable avantage of the propose UIE over the optimise DOB 4 Conclusion In this paper a novel UIE capable of effective estimation an compensation of completely unknown inputs for MIMO systems is propose he new UIE is exempte from the various limitations of existing unknown-input estimation Figure 8: Close-loop system outputs uner isturbances schemes while superior performance is maintaine Comparison through simulation is rawn between the propose UIE an an optimise benchmark DOB on an MIMO mass-spring-amper system an it is shown that the new UIE outperforms the DOB Further analysis on characteristics of the UIE an its integration into existing state-space servo controllers will be aresse in the future References [Åström an Hägglun 006] K J Åström an Hägglun Avance PID control Instrumentation Systems an Automation Society Research riangle Park NC 006 [Chanar an alole 014] S Chanar an S E alole Improving the performance of UDE-base controller using a new filter esign Nonlinear Dynamics 77(3): [Chen et al 009] L Chen F He an K Sammut Vibra-

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