Set-Membership Identification of Wiener models with noninvertible nonlinearity

Transcription

1 Set-Membership Identification of Wiener models with noninvertible nonlinearity V. Cerone, Dipartimento di Automatica e Informatica Politecnico di Torino (Italy) vito.cerone@polito.it, diego.regruto@polito.it Scuola di Dottorato SIDRA "A. Ruberti" 2007 Identificazione di sistemi nonlineari Bertinoro, 9-11 Luglio 2007

2 Wiener models Wiener models consist of a linear dynamic system followed by a static nonlinearity u t w t y t L x t N η t + + where: N : static (i.e. memoryless) nonlinearity L: linear subsystem x t : inner signal not measurable DAUIN - Politecnico di Torino 1

3 Identification of Wiener models Motivations: Main attractive feature: ability to embed process structure knowledge (e.g. nonlinearity in the measurement equipment). Applications: nonlinear filtering, acoustic echo cancellation, identification of biological systems, modelling of electrical stimulated muscles, modeling of distillation columns... many more DAUIN - Politecnico di Torino 2

4 Problem formulation u t w t y t L x t N η t + + w t = γ k x k t A(q 1 )=1+a 1 q a na q na B(q 1 )=b 0 + b 1 q b nb q nb q 1 x t = x t 1 y t = w t + η t DAUIN - Politecnico di Torino 3

5 Problem formulation Aim: compute bounds on the parameters γ T =[γ 1 γ 2... γ n ], θ T =[a 1.. a na b 0.. b nb ]. Prior assumption on the system: 1. stability; 2. n, na and nb are known; 3. the steady-state gain is not zero; 4. a rough upper bound on the settling time of the system is known; Prior assumption on the measurement uncertainty: 1. {η t } is UBB: {η t } η t ; 2. η t is known; DAUIN - Politecnico di Torino 4

6 Proposed solution: preliminary Three-stage solution: First stage: computation of bounds on the nonlinear block parameters γ. Second stage: computation of bounds on the inner (unmeasurable) signal x t. Third stage: computation of bounds on the linear block parameters. DAUIN - Politecnico di Torino 5

7 Proposed solution: preliminary Remark 1: The parameterization is not unique nb j=0 assume g dc = b j 1+ na i=1 a =1 i Remark 2: Stimulate the system with a set of step inputs with different amplitudes Steady-state operating conditions: N η s ū s w s ȳ s + + DAUIN - Politecnico di Torino 6

8 1st Stage: assessment of tight bounds on γ Get M n steady-state measurements: ȳ s = γ k ū k s + η s, s =1,...,M The feasible parameter set of the nonlinear block, is defined as: D γ = {γ R n :ȳ s = γ k ū k s + η s, η s η s ; s =1,...,M} D γ is a convex polytope: (a) Provides tight bounds on parameters γ (b) Can be computed through standard algorithm from the SM literature DAUIN - Politecnico di Torino 7

9 1st Stage: orthotope-outer bounding set B γ containing D γ The shape of D γ may result quite complex for large M and n Compute a tight orthotope outer-bound: B γ = {γ R n : γ j = γj c + δγ j, δγ j γ j /2,j =1,...,n} γ c j = γmin j + γ max j 2 γ j = γ max j γ min j γ min j =min γ D γ γ j γ max j =max γ D γ γ j Computational aspects: B γ is obtained solving 2n LP problems with n variables and 2M constraints. DAUIN - Politecnico di Torino 8

10 2nd Stage: bounds on the inner signal x t Simplified case: exactly known γ Task: for given {w t } compute {x t }, solving: w t = Σ k γ k x k t x 2 0 (1) x w INVERTIBLE 1 w 3 NON INVERTIBLE (2) x 3 x 3 (3) x 1 w t γ k x k t = p t(x t,w t )=0 Problem: polynomials are noninvertible Key idea: design {u t } to drive {x t } Λ such that: { pt (x t,w t )=0 x t Λ (1) has a unique solution for each t w t = Σ k γ k x k t w 2 INVERTIBLE x t x 1 (1) x 2 (1) w 2 NON INVERTIBLE x 2 (2) w 1 INVERTIBLE (4) (2) x 2 x x 2 (3) x t DAUIN - Politecnico di Torino 9

11 2nd Stage: bounds on the inner signal x t Definition 1 The set W R is an Output Invertibility Interval if for w t W each p t (x t,w t,γ) Π t shows either only one real root (n odd) or two real roots when (n even). Definition 2 The set X R is a Feasible Inner-signal Interval if the set of output values O = {w t R : w t = N (x t,γ), N (x t,γ) V t,x t X} is an Output Invertibility Interval DAUIN - Politecnico di Torino 10

12 Set-Membership Identification of Wiener models with noninvertible nonlinearity 2nd Stage: bounds on the inner signal xt Case n odd Case n even wt = Σk γk xt w = Σ γ xkt k 0.8 k k 0.6 x_ 0 t 0.4 _ x w _ 1 _ w 0.2 _ w 2 0 _ x x _ xt DAUIN - Politecnico di Torino xt 11

13 2nd Stage: bounds on the inner signal x t Proposition 1 (Output Invertibility Intervals) N (x t,γ) with γ D γ, shows the following Output Invertibility Intervals (case n odd): W =]w, + [ and W =],w[ where w =max x t Υ t max γ D γ Υ t = x t R: d dx t γ k x k t, w = min x t Υ t min γ D γ γ k x k t γ k x k t =0, for some γ D γ DAUIN - Politecnico di Torino 12

14 2nd Stage: bounds on the inner signal x t Proposition 2 (Feasible Inner-signal Intervals) The Wiener system with uncertain N (x t,γ), shows the following Feasible Inner-signal Intervals (case n odd and γ n > 0): X = ]x, + [ and X = ], x[ where x =max x t R : w x =min x t R : w γ k x k t =0, for some γ D γ γ k x k t =0, for some γ D γ. (2) DAUIN - Politecnico di Torino 13

15 2nd Stage: Input Sequence Design u t = u DC + u td such that x t = x DC + x td Feasible Inner-signal Interval g dc =1 u DC = x DC Proposition 3 (Input sequence design) For a given u DC x, each sample of the sequence {x t } belongs to X if: {u td } u DC x h up For given u DC x, each sample of the sequence {x t } belongs to X if: {u td } u DC x h up where: h 1 h up ; h: impulse response the linear block; h 1 : l 1 norm of the linear block; : l norm of a sequence. DAUIN - Politecnico di Torino 14

16 2nd Stage: Input Sequence Design Proposition 4 (Input signal design) All the samples of {w t } belong to W if the samples of {y t } satisfy the following inequalities (case n odd): where y t > y t or y t <y t, when n is odd (3) y = w + η t, y = w η t No bound h up is a priori known Tune amplitude of {u td } until {y t } satisfies (3) DAUIN - Politecnico di Torino 15

17 2nd Stage: bounds on {x t } 4 Proposition 5 (Inner-signal bounds) Given: N(x t,γ) with γ D γ {u t } which drives {x t } into X the measured output sequence {y t } Each sample x t of {x t } is bounded as follows: w t = Σ k γ k x k t x_ y t + η y t η _ x x t min x t max x min t x t x max t (4) 2 where (case γ n > 0 and X = X): 3 x min t x max t =min =max { { x t R : y t η t x t X : y t + η t γ k x k t =0, for some γ D γ } γ k x k t =0, for some γ D γ x t } DAUIN - Politecnico di Torino 16

18 3rd stage: bounds on θ Given the designed input sequence {u t }, the uncertain inner sequence {x t } we define: x c t = xmin t δx t x t + x max t 2 δx t u t B(q 1 ) x t + + x c t A(q 1 ) x t = xmax t x min t 2 Output Error problem with UBB errors (special case of EIV problem) (V. Cerone - Feasible parameter set of linear models with bounded errors in all variables, Automatica 93) DAUIN - Politecnico di Torino 17

19 2nd Stage: Computational aspects Computation of Υ t : find the roots of the uncertain polynomial: p t (x t,γ)= d dx t γ k x k t = n kγ k x k 1 t Proposed approach: gridding on x t one LP problem for each x t. Computation of w and w: solve the following two nonlinear programming problems: w =max x t Υ t max γ D γ γ k x k t and w = min x t Υ t min γ D γ γ k x k t Proposed approach: gridding on the set Υ t one LP problem for each x t. DAUIN - Politecnico di Torino 18

20 2nd Stage: Computational aspects Computation of x and x: solve the following problems (n odd, γ n > 0): and x =max{x t R : p t (x t,γ,w) =0, for some γ D γ } (5) x =min{x t R : p t (x t,γ,w)=0, for some γ D γ } (6) Computation of x max t and x min t : solve the following problems: x max t =max{x t X : p t (x t,γ,y t + η t )=0, for some γ D γ } (7) x min t =min{x t R : p t (x t,γ,y t η t )=0, for some γ D γ } (8) DAUIN - Politecnico di Torino 19

21 Algorithm 1 (Computation of x) 1. Set α = α 0 and ɛ = prescribed tolerance. 2. Compute r =max{x t R : pt nom (x t,γ c, w) =0}. 3. Set x m = r. 4. Set x M = x m + α. 5. If γ D γ : p t (x M,γ, w) =0 then x m = x M ; else If x M x m <ɛ then x = x M ; return x ; stop algorithm. else α = α/2; end if end if. 8. Repeat from 4. 2nd Stage: Computational aspects Main properties: Remark: Algorithm 1 can be also used to compute x, x max t 1. Algorithm 1 is convergent. 2. Algorithm 1 provides an upper bound x of x with x x ɛ. 3. Step 5 is a LP problem. and x min t DAUIN - Politecnico di Torino 20

22 Parameters of the simulated system: γ 1 =-5, γ 2 =-4, γ 3 =1; A(q 1 )=(1 1.1q q 2 ); B(q 1 )=(0.1q q 2 ) Signal to noise ratio: SNR = 10 log Example: / N N w 2 t η 2 t t=1 t=1 Measurement output errors: Bounded absolute output errors have been considered: η t η t ; η t belongs to the uniform distribution U[ η t, + η t ] η s η s ; η s belongs to the uniform distribution U[ η s, + η s ]. η t = η s = η DAUIN - Politecnico di Torino 21

23 Nonlinear block parameter: central estimates and parameters uncertainty bounds Number of steady-state samples: M =50 SNR γ j True γj c γ j (db) Value 58.2 γ e-3 γ e-4 γ e γ e-2 γ e-3 γ e γ e-1 γ e-2 γ e-3 DAUIN - Politecnico di Torino 22

24 Inner signal estimate true signal (blue), central estimate (red) and bounds (black) 8 inner signal x samples DAUIN - Politecnico di Torino 23

25 Linear block parameters: central estimates and parameters uncertainty bounds Number of transient samples: N = 100 SNR θ j True θj c θ j (db) Value 58.2 θ e-3 θ e-3 θ e-4 θ e θ e-2 θ e-2 θ e-3 θ e θ e-1 θ e-1 θ e-2 θ e-2 Number of transient samples: N = 1000 SNR θ j True θj c θ j (db) Value 58.2 θ e-3 θ e-3 θ e-4 θ e θ e-2 θ e-2 θ e-3 θ e θ e-1 θ e-1 θ e-2 θ e-2 DAUIN - Politecnico di Torino 24

26 Conclusions and references The proposed three-stage parameter bounding procedure provides: tight bounds on the parameters of the nonlinear block using steady-state inputoutput data; overbounds on the parameters of the linear part, through the computation of bounds on the unmeasurable inner signal x t ; The numerical example has showed the effectiveness of the proposed procedure. The approach is computationally tractable (in the above example the experiment with M =50and N = 1000 required about 3 minutes (AMD 3200 processor)). Reference paper: V. Cerone,, Parameter bounds evaluation of Wiener models with noninvertible polynomial nonlinearities, Automatica, vol. 42, pp , DAUIN - Politecnico di Torino 25