Set-Membership identification of linear systems with input backlash

Transcription

1 Poceedings of the 006 Ameican Contol Confeence Minneapolis, Minnesota, USA, June 14-16, 006 ThA08.5 Set-Membeship identification of linea systems with input backlash V. Ceone, D. Reguto Abstact In this pape we pesent a two-stage pocedue fo deiving paametes bounds of linea systems with input backlash when the output measuement eos ae bounded. Fist, using steady-state input-output data, paametes of the nonlinea dynamic block ae tightly bounded. Then, given a suitable PRBS input sequence we evaluate tight bounds on the unmeasuable inne signal which, togethe with noisy output measuements ae employed fo bounding the paametes of the linea dynamic system. Index Tems Backlash, bounded uncetainty, output eos, eos-in-vaiable, paamete bounding, linea pogamming. I. INTRODUCTION Contol systems components, such as sensos and actuatos, often exhibits backlash which, indeed, is a typical chaacteistic of mechanical connections (see, e.g. [1]). Backlash can be classified as a had (i.e. non-diffeentiable) and dynamic nonlineaity. It is well known that this kind of nonlineaity may often cause delays, oscillations and inaccuacy which seveely limit the pefomance of contol systems (see, e.g. []). To cope with these limitations, eithe obust o adaptive contol techniques can be successfully employed (see, e.g., [3], and [4] espectively) which, on the othe hand, equie the chaacteization of the nonlinea dynamic block. Amazingly, thee ae only few contibutions in the liteatue on the identification of systems with backlash nonlineaity ([5],[6]). Theefoe, the identification of such systems is an open theoetical poblem of mao elevance to applications. The configuation we ae dealing with in this pape, shown in Fig. 1, closely esembles that of a Hammestein model which in tun consists of a static nonlinea pat N followed by a linea dynamic system. The identification of such a model elies solely on input-output measuements, while the inne signal x t is not assumed to be available. Identification of the Hammestein stuctue has attacted the attention of many authos, as can be seen in [7], [8]. It must be stessed that existing identification pocedues mostly equie that the nonlineaity be static and diffeantiable, usually a polynomial (see e.g., [9], [10], [11] and the efeences theein). On the side of linea systems with had input nonlineaities, Bai [5] consides the case of nonlineaities paameteized by one paamete. The poposed algoithm, based on the idea of sepaable least squaes, can be applied to seveal common static and nonstatic input nonlineaities. In identification, a common assumption is that the measuement eo η t is statistically descibed. A wothwhile al- The authos ae with the Dipatimento di Automatica e Infomatica, Politecnico di Toino, coso Duca degli Abuzzi 4, 1019 Toino, Italy; e- mail: vito.ceone@polito.it, diego.eguto@polito.it; Tel: ; Fax: tenative to the stochastic desciption of measuement eos is the bounded-eos chaacteization, whee uncetainties ae assumed to belong to a given set. In the bounding context, all paamete vectos belonging to the Feasible Paamete Set (FPS), i.e. paametes consistent with the measuements, the eo bounds and the assumed model stuctue, ae feasible solutions of the identification poblem. The inteested eade can find futhe details on this appoach in a numbe of suvey papes (see, e.g., [1], [13])and in the special issues edited by Noton [14], [15]. In this pape we pesent a scheme fo the identification of linea systems with input backlash. Moe pecisely, we addess the poblem of bounding the paametes of a stable single-input single-output SISO discete time linea system with unknown backlash at the input (see Fig. 1) when the output eo is consideed to be bounded. Note that the inne signal x(t) is not supposed to be measuable. Results obtained in this pape can be staightly employed in the pocedue poposed by [3] whee they assume that bounds on the paametes of uncetain backlash ae available in ode to deive a sliding mode technique fo the stabilization of an intinsically nonlinea plant with an uncetain backlash in the actuato. To the autho s best knowledge, no contibution can be found in the liteatue which addess the above descibed identification poblem, except fo the authos wok [16]. The esults pesented hee, significantly impove pape [16], namely: (a) a moe geneal model of the backlash is consideed; (b) evaluation of the backlash paametes bounds and inne signal bounds does not ely any moe on gaphical inspection of the -dimensional paamete space, instead a couple of optimization esults ae given which povide tight bounds both on paametes and unmeasuable signal; (c) the simulated example has been evised accodingly. The pape is oganized as follow. Section II is devoted to the fomulation of the poblem. In Section III, paametes of the nonlinea block ae tightly bounded using input-output data collected fom the steady-state esponse of the system to a squae wave input. Then, in Section IV, though a dynamic expeiment, fo all u t belonging to a suitable Pseudo Random Binay Signal (PRBS) sequence {u t }, we compute tight bounds on the inne signal which, togethe with noisy output measuements ae used fo bounding the paametes of the linea pat. A simulated example is epoted in Section V. II. PROBLEM FORMULATION Conside the SISO discete-time linea system with input backlash depicted in Fig. 1, whee the nonlinea block tansfoms the input signal u t into the unmeasuable inne /06/$ IEEE 381

2 vaiable x t accoding to the following map (see, e.g., []) m l (u t + c l ) fo u t z l x t = m (u t c ) fo u t z (1) x t 1 fo z l <u t <z whee m l > 0, m > 0, c l > 0, c > 0 ae constant paametes chaacteizing the backlash and z l = x t 1 c l, z = x t 1 + c () m l m ae the u-axis values of intesections of the two m-slope paallel lines with the hoizontal inne segment containing x t 1. The linea dynamic pat is modeled by a discete-time system which tansfoms x t into the noise-fee output w t accoding to the linea diffeence equation A(q 1 )w t = B(q 1 )x t, (3) whee A( ) and B( ) ae polynomials in the backwad shift opeato q 1,(q 1 w t = w t 1 ), A(q 1 )=1+a 1 q a na q na, (4) B(q 1 )=b 0 + b 1 q b nb q nb. (5) In line with the wok done by a numbe of authos, in the contest of identification of block oiented systems, we assume that (i) the linea system is asymptotically stable (see, e.g., [17], [18], [19], [0], [1]); (ii) nb =0 b 0, that is, the steady-state gain is not zeo (see, e.g. [19], [0], [1]); (iii) the only a pioi infomation needed is an estimate of the pocess settling-time (see, e.g., []). Let y t be the noisecoupted output y t = w t + η t. (6) Measuements uncetainty is known to ange within given bounds Δη t, i.e., η t Δη t. (7) Unknown paamete vectos γ R 4 and θ R p ae defined, espectively, as γ T =[ γ 1 γ γ 3 γ 4 ]=[ m l c l m c ], (8) θ T =[ a 1... a na b 0 b 1...b nb ], (9) whee na + nb + 1 = p. It is easy to show that the paameteization of the stuctue of Fig. 1 is not unique. As a matte of fact, any paametes set b = α 1 b, = 1,,...,nb, and γ k = αγ k,k =1,, fo some nonzeo and finite constant α, povides the same input-output behaviou. To get a unique paameteization, in this wok we assume, without loss of geneality, that the steady-state gain of the linea pat be one, that is nb =0 g = b 1+ na i=1 a =1 (10) i In this pape we addess the poblem of deiving bounds on paametes γ and θ consistently with given measuements, eo bounds and the assumed model stuctue. u t N x t B(q 1 ) w t + y t A(q 1 ) Fig. 1. Single-input single-output discete-time linea system with input backlash N. ū i N η i w i + Fig.. Steady-state behaviou of the system unde consideation when g =1. III. ASSESSMENT OF TIGHT BOUNDS ON THE NONLINEAR STATIC BLOCK PARAMETERS Hee we exploit steady-state opeating conditions to bound the paametes of the backlash. The noisy output sequence is collected fom the steady-state esponse of the system to a set of squae wave inputs with diffeent amplitudes. Due to the fact that the backlash deadzone is unknown (its evaluation is the main pupose of this section) we suggest to choose the input amplitude in such a way that the output shows any nonzeo esponse. Fo each value of the input squae wave amplitude, only one steady-state value of the noisy output is consideed on the positive half-wave of the input and one steady-state value of the noisy output on the negative half-wave. Thus, given a set of squae wave inputs with M diffeent amplitudes, M steady-state values of the output ae taken into account. We only assume to have a ough idea of the settling time of the system unde consideation, in ode to know when steady-state conditions ae eached, so that steady-state data can be collected. Indeed, unde conditions (i), (ii) and (iii) stated in Section II, combining equations (1), (3), (6) and (10) at steady-state, we get the following input-output desciption involving only the paametes of the backlash: ȳ i = m (ū i c )+ η i, η t fo ū i x i 1 m +c i =1,...,M; (11) ȳ = m l (ū +c l )+ η, fo ū x 1 c l =1,...,M; m l (1) whee the tiplets {ū i, ȳ i, η i } and {ū, ȳ, η } ae collections of steady-state values of the known input signal, output obsevation and measuement eo taken duing the positive and the negative squae wave espectively. A block diagam desciption of the steady-state esponse is depicted in Fig. fo equation (11) only; equation (1) leads to a simila block diagam epesentation. Since the pais (m l,c l ) and (m,c ) affect the collected measuements, equations (11) and (1), sepaately, we can define the feasible paamete egion of the backlash as D γ = Dγ D l γ (13) 38

3 whee Dγ = {m,c R + :ȳ i = m (ū i c )+ η i, η i Δ η i ; i =1,...,M} (14) Dγ l = {m l,c l R + :ȳ = m l (ū + c l )+ η, η Δ η ; =1,...,M} (15) whee {Δ η i } and {Δ η } ae the sequences of bounds on measuements uncetainty. Fom definition (13) it can be seen that D γ is exactly descibed by the following constaints in the paamete space ȳ i m (ū i c ) Δ η i, ȳ i m (ū i c ) Δ η i, m > 0,c > 0, i =1,...,M (16) ȳ m l (ū + c l ) Δ η, ȳ m l (ū + c l ) Δ η, m l > 0,c l > 0, =1,...,M (17) Remak 1 Dγ l and Dγ ae -dimensional sets lying on the (m l,c l ) plane and the (m,c ) plane espectively, i.e. they ae disoint sets, which means that they can be handled sepaately. We also note that they have the same mathematical stuctue, which means that they enoy the same popeties. Thus, fom hee on the esults deived fo one of the two sets, say Dγ, will be also applicable to the othe set (Dγ). l Thoughout the pape it is assumed that Dγ (Dγ) l is a bounded set: to this end it suffices to collect at least two sets of measuements with diffeent inputs u. Below we pesent some possible desciptions of the feasible paamete set Dγ. Intoductoy definitions and peliminay esults ae fist given. A. Definitions and peliminay esults Definition 1 h + (u s ) and h (u s ) ae the constaints boundaies defining the FPS Dγ coesponding to the s-th sets of data: h + (u s ) =. {m R +,c R + : y s +Δη s = m (u s c )} h (u s ) =. {m R +,c R + : y s Δη s = m (u s c )} Definition Bounday of D γ. = H(D γ) Definition 3 The constaints boundaies h + (u s ) and h (u s ) ae said to be active if thei intesections with H(D γ) is not the empty set: h + (u s ) H(D γ) h + (u s ) is active. h (u s ) H(D γ) h (u s ) is active. Remak It is tivial to see that the constaints boundaies h + (u s ) and h (u s ) may eithe (a) intesect H(D γ) o (b) be extenal to H(D γ), hence be extenal to D γ. Definition 4 Edges of D γ. h + (u s ) =. h + (u s ) Dγ = { m,c Dγ : y s +Δη s = m (u s c )} h (u s ) =. h (u s ) Dγ = { m,c Dγ : y s Δη s = m (u s c )} Definition 5 Constaints intesections. The set of all the pais (m,c ) R whee intesections among the constaints occu is defined as Iγ = {(m,c ) R : {h + (u i ),h (u i )} {h + (u ),h (u )} ; i, =1,...,M; i } (18) Definition 6 Vetices of Dγ. The set of all the vetices of Dγ is defined as the set of all the intesection couples belonging to the feasible paamete set Dγ: V(Dγ)=I γ D γ (19) B. Exact desciption of D γ An exact desciption of D γ can be given in tems of edges, each one being descibed, fom a pactical point of view, as a subset of an active constaint lying between two vetices. An effective pocedue fo deiving active constaints, vetices and edges of D γ is epoted in [3]. C. Tight othotope desciption of Dγ Edges povide exact desciption of Dγ which, on the downside, could be not so easy to handle. A somewhat moe pactical desciption, although appoximate, can be obtained by the computation of the following othotope oute-bounding set Bγ tightly containing Dγ: B γ ={γ R : γ = γ c + δγ, δγ Δγ, =1, } (0) whee γ c = γmin γ min + γ max, Δγ = γmax γ min =minγ, γ Dγ γ max (1) =maxγ. () γ Dγ Since constaints (16) defining Dγ ae nonlinea in m and c, at least in pinciple the solution of the above optimization poblems () equies the use of nonconvex optimization techniques which, howeve, do not guaantee the finding of the global optimal solution. Poblems () can be solved thanks to the esult epoted below. Poposition 1 The global optimal solutions of poblems () occu on the vetices of D γ. Poof Fist (i) we notice that each level cuve of functionals () paallel lines to m -axis and c -axis espectively intesect the bounday of each constaint in (16) only once. Next, (ii) obective functions in () ae monotone which implies that the optimal solution lies on the bounday of D γ. Thanks to (i) the optimal value cannot lie on one edge between two vetices: if that was tue, it would mean that thee is a suboptimal value whee the functional intesect the edge twice: that would contadict (i). Then the global optimal solutions of poblems () can 383

4 only occu on the vetices of D γ. Remak 3 Given the set of vetices V(Dγ) computed via the pocedue epoted in [3], the evaluation of () is an easy task since it only equies the simple calculation of (a) the obective functions on a set of points 4M and (b) the maximum ove a set of eal values. IV. BOUNDING THE PARAMETERS OF THE LINEAR DYNAMIC MODEL In the second stage of ou pocedue we evaluate bounds on the paametes of the linea dynamic block. In this stage, we excite the system to be identified with a pseudo andom binay signal (PRBS) which takes the values ±u. We ecall that, thanks to its nice popeties, a PRBS input is successfully employed in linea system identification ([4], [5]). Although PRBS inputs ae not suitable fo nonlinea system identification in geneal ([5], [6]) since it may not adequately excite the unknown nonlineaity, in [7] it is shown that such a signal can be effectively used to decouple the linea and nonlinea pats in the identification of Hammestein model with static nonlineaity. In this pape we show that the use of a PRBS sequence is pofitable fo the identification of linea system with input backlash. The key idea undelying ou contibution is based on the following esult: Result 1 Unde a PRBS input whose levels ae ±u, u > c and u < c l, the output of a backlash descibed by (1) is still a PRBS with levels x = m (u c ), x = m l (u c l ). The poof of Result 1 is not epoted since it is a tivial one. Given the exact desciption of the feasible paamete set (FPS) Dγ, tight bounds on the magnitude x of the unmeasuable pseudo andom inne signal x t can be computed t though the following expessions x min = min m (u c ), fo u c x max = max m (u c ), m,c Dγ fo u c (3) m,c Dγ Computation of bounds in equation (3) equies, at least in pinciple, the solution of nonconvex optimization poblems with vaiables and 4M constaints. Howeve, the computational effots can be damatically educed thanks to the esults epoted below, whee we exploit the following definition: Definition 7 x-level cuve of the obective function to be optimized: g (u,x) =. {m R +,c R + : x = m (u c )} (4) Poposition The global optimal solutions of poblems (3) occu on the vetices of Dγ. Poof The poof of Poposition follows the same lines as the poof of Poposition 1. Fist (i) we notice that the each x-level cuve g (u,x) of functional (4) intesect each constaint bounday in (16) only once. Next, (ii) the obective function (4) is a monotone function which implies that the optimal solution lies on the bounday of D γ. Thanks to (i) the optimal value cannot lie on an edge between two vetices: if that was tue, it would mean that thee is a suboptimal value whee the functional intesect the edge twice: that would contadict (i). Then the global optimal solutions of poblems (3) can only occu on the vetices of D γ. Hee, same comments epoted in Remak 3 apply. Now, if we define the quantities x c t = x min + x max, Δx t = x max x min (5) the following elation can be established between the unknown inne signal x t and the cental value x c t: x c t = x t + δx t (6) δx t Δx t. (7) We can now fomulate the identification of the linea model in tems of the noisy output sequence {y t } and the uncetain inne sequence {x c t} as shown in Fig. 3. Such a fomulation is commonly efeed to as an eos-in-vaiables poblem (EIV), i.e. a paamete estimation poblem in a linea-inpaamete model whee the output and some o all the explanatoy vaiables ae uncetain. As a matte of fact, x t B(q 1 ) w t + y t A(q 1 ) δx t + + x c t Fig. 3. Eos-in-vaiables set-up fo bounding the paametes of the linea system. combining equations (3), (4), (5), (6), (6) we get na nb y t = (y t i η t i )a i + (x c t δx t )b +η t. (8) i=1 =0 The definition of the feasible paamete egion fo the linea system is: D θ = {θ R p :A(q 1 )[y t η t ]=B(q 1 )[x c t δx t ]; g =1; η t Δη t ; δx t Δx t ; t =1,...,N}. (9) whee g = 1 takes account of condition (10) on the steady-state gain. Fom equation (8) it can be seen that consecutive egessions ae elated deteministically by uncetain output samples and uncetain input samples; that occuence qualifies the poblem as a dynamic EIV. It is efeed to as a static EIV poblem when the uncetain vaiables appeaing in successive egessions ae supposed to vay independently. The elations between successive η t 384

5 egessions in the dynamic EIV case give ise to possibly nonlinea exact paamete bounds, which could be not easily and exactly computed [8]. On the othe end, in the static EIV case exact paamete bounds ae piecewise linea and, although geneally non convex, the feasible paamete egion is the union of at most p convex sets: each being the intesection of the FPS with a single othant of the p-dimensional paamete space (a detailed discussion on the geometical and topological stuctue of the feasible paamete egion fo static EIV poblems can be found in [9]). Thus, as shown in [8], the FPS of static EIV can be moe conveniently handled than the FPS of dynamic EIV. That motivates the use, in this pape, of esults fom the static EIV [9]; since in model (8) the uncetain vaiables appeaing in successive egessions ae deteministically elated, only oute appoximations of the exact feasible paamete egion will be obtained. Thus, in this wok, a polytopic oute appoximation D θ of the exact FPS D θ, i.e. D θ D θ, will be pesented, togethe with an othotope-oute bounding set B θ of D θ, which povides paamete uncetainties intevals. When we apply esults fom [9] to ou poblem we get the following desciption of the feasible paamete set D θ at the single time t (φ t Δφ t ) T θ y t +Δη t, (φ t +Δφ t ) T θ y t Δη t (30) [ ] θ = 1 (31) whee φ T t = [ y t 1... y t na x c t x c t 1...x c ] t nb (3) Δφ T t =[Δη t 1 sgn(a 1 )... Δη t na sgn(a na ) Δx t sgn(b 0 ) Δx t 1 sgn(b 1 )... Δx t nb sgn(b nb )] (33) Equation (31) takes account of condition (10) on the steadystate gain. The othotope-oute bounding set B θ is defined as B θ = {θ R p : θ = θ c + δθ, δθ Δθ, =1,...,p}, (34) whee θ c = θmin + θ max, Δθ = θmax θ min θ min =minθ, θ D θ θ max (35) =maxθ. (36) θ D θ Paamete vectos γ c and θ c ae Chebishev centes in the l nom of D γ and D θ espectively and ae commonly efeed to as cental estimates. V. A SIMULATED EXAMPLE In this section we illustate the poposed paamete bounding pocedue though a numeical example. The system consideed hee is chaacteized by a linea block with A(q 1 ) = (1 + q 1 0.1q ) and B(q 1 ) = (0.q 1 +1.q ) and a nonsymmetic backlash with m l = 0.4, m = 0.6, c l = 0.035, c = Thus, the tue paamete vectos ae γ = [m l c l m c ] T = [ ] T and θ = [a 1 a b 1 b ] T = [ ] T. We emphasize that the backlash paametes have been ealistically chosen: as a matte of fact we consideed the paametes of a eal wold pecision geabox which featues a gea atio equal to and a deadzone as low as ad ( 3 o ) and simulated a possible fictitious nonsymmetic backlash with gea atio m l =0.4, m = 0.6 and deadzone c l =0.035 ( o ), c =0.070 ( 4 o ). Bounded absolute output eos have been consideed when simulating the collection of both steady state data, {ū s, ȳ s }, and tansient sequence {u t,y t }. Hee we assumed η t Δη t and η s Δ η s whee η t and η s, ae andom sequences belonging to the unifom distibutions U[ Δη t, +Δη t ] and U[ Δ η s, +Δ η s ] espectively. Bounds on steady-state and tansient output measuement eos wee supposed to have the same value, i.e., Δη t = Δ η s. = Δη. Eight diffeent values of Δη wee chosen in such a way as to simulate the measuement eos of eight commecial absolute binay encode with a numbe of bits n bit vaying fom 8 to 15. Fo a given Δη, the length of steady-state and the tansient data ae M =50and N = [100, 1000] espectively. The steadystate input samples ū s ae equally spaced values fom 0.6 and 3, while the tansient input sequence {u t } is a PRBS which takes the values ±1. Results about the nonlinea and the linea block ae epoted in Figues 4, 5 and 6 espectively. Fo low noise level (n bit > 10 bits) and fo all N, the cental estimates of both the nonlinea static block and the linea model ae consistent with the tue paametes. Fo highe noise levels (n bit 10 bits), both γ c =[m c,c c ] and θ c give satisfactoy estimates of the tue paametes. As the numbe of obsevations inceases (fom N = 100 to N = 1000), paamete uncetainty bounds Δθ deceases, as expected. VI. CONCLUSION A two-stage paamete bounding pocedue fo linea systems with input backlash in pesence of bounded output eos has been outlined. Fist, using steady-state inputoutput data the two paametes of the backlash have been tightly bounded. Then, fo a given input tansient sequence we have computed bounds on the unmeasuable inne signal which, togethe with output noisy measuements have been used to ovebound the paametes of the linea pat. The numeical example showed the effectiveness of the poposed pocedue. ACKNOWLEDGMENT This eseach was patly suppoted by the italian Ministeo dell Istuzione, dell Univesità e della Riceca (MIUR), unde the plan Robustness and Optimization techniques fo high pefomance contol systems. REFERENCES [1] M. Nodin and P. O. Gutman, Contolling mechanical systems with backlash asuvey, Automatica, vol. 38, pp , 00. [] G. Tao and P. Kokotovic, Adaptive contol of systems with actuato and senso nonlineaities. New Yok, NY: Wiley,

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