Control Systems Lab - SC4070 System Identification and Linearization

Transcription

1 Control Systems Lab - SC4070 System Identification and Linearization Dr. Manuel Mazo Jr. Delft Center for Systems and Control (TU Delft) m.mazo@tudelft.nl Tel.: TU Delft, February 13, 2015 (slides modified from the original drafted by Robert Babuška) M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 1 / 46

2 Outline 1 References and overview of models 2 Parameter estimation in nonlinear models 3 Linear system identification in time domain 4 Experiment design and the identification procedure 5 Linearization of nonlinear models M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 2 / 46

3 Outline 1 References and overview of models 2 Parameter estimation in nonlinear models 3 Linear system identification in time domain 4 Experiment design and the identification procedure 5 Linearization of nonlinear models M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 3 / 46

4 References on parameter and model identification L. Ljung, System Identification: Theory for the User, 2nd edition, Prentice-Hall, 1999 L. Ljung, System Identification, Chapter 58 of the Control Engineering Handbook, pp , CRC Press, 1996 K.J. Aström & B. Wittenmark, Computer-Controlled Systems, Prentice-Hall, 1997, Chapter 13 Lecture notes of the courses Filtering & identification (SC4040) or System identification (SC4110) Manual of MATLAB System Identification Toolbox M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 4 / 46

5 General overview of models 1 Physical models in state-space form: d x(t) = f (x(t), u(t), θ) dt y(t) = h(x(t), u(t), θ) 2 Black-box models in frequency: 1st order & 2nd order linear models (in continuous-time): G 1(s) = 1 τs + 1 ω 2 G 2(s) = s 2 + 2ζω N s + ωn 2 linear models: ARX, ARMAX, OE, BJ (in discrete-time): y(k) = G(q, θ)u(k) + H(q, θ)e(k) nonlinear models (neural networks, fuzzy models,... ) M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 5 / 46

6 Outline 1 References and overview of models 2 Parameter estimation in nonlinear models 3 Linear system identification in time domain 4 Experiment design and the identification procedure 5 Linearization of nonlinear models M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 6 / 46

7 Physical models in state-space form Model: d x(t) = f (x(t), u(t), θ) dt y(t) = h(x(t), u(t), θ) What is the optimal value of θ? We want to estimate it Given a model, and a parameter θ construct output ŷ(t k θ) at t = t k = k h Given the measured signal y(t k ), define the prediction error: ε(k, θ) = y(t k ) ŷ(t k θ) Define the performance index: J(θ) = 1 N g(ε(k, θ)) with N: # data points N k=1 Compute the optimal value: ˆθN = arg min θ J(θ) M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 7 / 46

8 Physical models in state-space form Usually: g(ε) = ε 2 : 1 ˆθ N = arg min θ N N ε(k, θ) 2 k=1 In general: numerical, nonlinear optimization algorithms see course Optimization in Systems and Control (SC4090) Matlab optimization toolbox (lsqnonlin) (also used in Matlab NCD toolbox (nonlinear control design)) M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 8 / 46

9 Nonlinear least squares M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 9 / 46

10 In Matlab: lsqnonlin Syntax: x=lsqnonlin( fun,x0,lb,ub,options) with Denoting by x = θ, F k (x) = ε(k, θ) fun : m-file function returning F (x) and its Jacobian F (optional) [F,J]=fun(x) F=... % Compute Jacobian if required. if (nargout > 1) J=... end; x0 : initial starting point lb, ub : lower and upper bounds for x (optional) options : options structure (optional) M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 10 / 46

11 Most important options LargeScale : use a large-scale algorithm ( on ) or medium-scale algorithm ( off ). Display : controls display of (intermediate) values. Possible values: off, iter, and final Jacobian : indicates whether Jacobian is defined by user MaxIter : maximum number of iterations allowed TolFun, TolX : termination tolerance on the function value and on x LevenbergMarquardt : Choose Levenberg-Marquardt over Gauss-Newton algorithm (default: on ) M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 11 / 46

12 Outline 1 References and overview of models 2 Parameter estimation in nonlinear models 3 Linear system identification in time domain 4 Experiment design and the identification procedure 5 Linearization of nonlinear models M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 12 / 46

13 ARX and ARMAX models ARMAX and ARX models e ARX: q-notation: q 1 x(k) = x(k 1) u C + B 1/A y ARMAX ARMAX y(k) + a 1y(k 1) +... a ny(k n) = u(k) + b 1u(k 1) + b mu(k m) + e(k) y(k) = B(q) A(q) u(k) + 1 A(q) e(k) ARMAX: y(k) + a 1y(k 1) +... a ny(k n) = u(k) + b 1u(k 1) b mu(k m)+ e(k) + c 1e(k 1) c l e(k l) y(k) = B(q) C(q) u(k) + A(q) A(q) e(k) u + B 1/A e 39 y ARX ARMAX and ARX models ARXeq C B + 1/A e u y ARX B + 1/A ARXeq M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 13 / 46 u e y ARMAX ARMAX

14 Output-Error and Box-Jenkins C/D models BJ u B/F + y BJeq Output-Error: y(k) = B(q) u(k) + e(k) F (q) u e OE y B/F + OEeq Box-Jenkins and Output-Error mod 38 e Box-Jenkins: y(k) = B(q) C(q) u(k) + F (q) D(q) e(k) u B/F C/D + y BJ BJeq u B/F + e y OE OEeq 38 M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 14 / 46

15 ARX, ARMAX, OE, BJ Models Simulate parameterized model (parameters: a i, b i, c i,...) Obtain real data from experiments Optimize over parameters M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 15 / 46

16 Identification of 1st-order models G 1(s) = Ke Ls τs + 1 Note: L is a delay in the action of the input. Step response: ( y(t) K(1 exp( (t L)/τ)) ) M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 16 / 46

17 2nd-order models ẍ+2ω N ζẋ+ωnx 2 = K ssωnu, 2 y = q Transfer function: K ssω 2 N G 2(s) = s 2 + 2ζω N s + ωn 2 In this experiment u = 0. ω = ω N 1 ζ 2 M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 17 / 46

18 2nd-order models K ssω 2 N G 2(s) = s 2 + 2ζω N s + ωn 2 In this experiment u = step(t). πζ 1 ζ M p = e 2 compute ζ T s 4 ζω N compute ω N M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 18 / 46

19 Summary of prediction error methods Alternative: state-space (subspace) identification M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 19 / 46

20 Outline 1 References and overview of models 2 Parameter estimation in nonlinear models 3 Linear system identification in time domain 4 Experiment design and the identification procedure 5 Linearization of nonlinear models M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 20 / 46

21 Physical versus black-box models Physical models + general (nonlinear models) + based on physical considerations - nonlinear optimization local minima, time-consuming black-box models + often very effective + intuitive behavior (in particular for 1st & 2nd order models) + identification is fast + many control design methods for linear models - black-box relation with physical system? - linear not general, only valid locally M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 21 / 46

22 The identification procedure M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 22 / 46

23 Experiment design M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 23 / 46

24 Experiment design Experiment design comprises a.o.: experiment duration: N data points selection of frequency band of interest selection of sampling frequency 1 Hz h use of anti-aliasing filters M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 24 / 46

25 Choice of input signal Input signal must excite system, i.e., contain enough frequencies Telegraph signal random shift between two levels M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 25 / 46

26 Choice of input signal For linear identification: do not use full range of input signals! relatively small amplitude around an operating point For nonlinear identification: two levels (H, L) may not be sufficient use multiple levels or smoothly varying signals (multi-sine)! Take care that the output signal amplitude is significantly larger than the noise amplitude signal-to-noise ratio should be large enough Check: measure output under zero input to set output noise level M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 26 / 46

27 Choice of input signal It is a good idea to first generate the input sequence off-line, and to examine its properties before applying it to the system first try it with the simulation model! Binary signals (telegraph, PRBN) often suitable to identify linear systems For nonlinear models multi-level telegraph signal Choose the frequency range of input signal so most of its energy is situated in frequency bands of interest, i.e., let input signal contain pulses that occasionally allow step response to more or less settle, and also clearly excite interesting fast modes in the system M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 27 / 46

28 Post-treatment of data M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 28 / 46

29 Post-treatment of data Subtract mean (for identification around operating point) Removing drift: detrend(y) If a filter is used for identification use it in the control design! M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 29 / 46

30 Choice of model class and structure M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 30 / 46

31 Choice of model class and structure To answer selection issues prior knowledge is required: from crude first-principles models from simple pre-identification experiments (step response, etc.) time constants which relationships can be described as static? characteristics (oscillatory, poorly damped, monotone,... ) Also useful for model validation M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 31 / 46

32 Choice of linear model class/structure Try Simple Things First ARX ARMAX OE BJ simple to determine (prediction error is linear in parameters) high order necessary to model noise, unrealistic nonlinear optimization realistic for input noise (when B = C) nonlinear optimization realistic for (white) sensor noise nonlinear optimization most general, most difficult to compute M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 32 / 46

33 Validation M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 33 / 46

34 Model errors E{e} = E{n 2 } + E{(y u ŷ) 2 } E{(y u ŷ) 2 } = E{[ŷ E{ŷ}] 2 } + [y E{ŷ}] Variance error: E{[ŷ E{ŷ}] 2 } due to noise influence can be reduced by using longer measurement sequences: with λ : noise variance N : # data samples θ 0 : true parameters E [ (ˆθ N θ 0 ) (ˆθ N θ 0 ) T ] λ N M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 34 / 46

35 Model errors E{e} = E{n 2 } + E{(y u ŷ) 2 } E{(y u ŷ) 2 } = E{[ŷ E{ŷ}] 2 } + [y E{ŷ}] Bias error: y E{ŷ} due to deficiencies in model structure (present even if noise-free or N ) M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 35 / 46

36 Over-fitting Example: Consider {y 0, y 1, y 2, y 3} generated from the true system y k = ay k 1 + e k. Then we can exactly fit model y k = θy k 1 + γy k 2 to the data. M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 36 / 46

37 Over-fitting Cures to avoid problem of over-fitting: 1 Cross-validation: e.g., θ = arg min J [0, 2 3 t obs ] (θ) while model quality is given by J [ 2 3 t obs,t obs ] (θ) 2 Penalize the number of parameters (e.g. Akaike): J N (θ) = (1 + 2d N N ) ε 2 (k, θ) k=1 with d the total number of model parameters M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 37 / 46

38 Useful Matlab commands Model estimation: arx, armax, oe, bj Filter design: butter Filtering: filter d2c poles, gain (compare with white-box model) step, bode validation, check with prior knowledge / experiments M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 38 / 46

39 Outline 1 References and overview of models 2 Parameter estimation in nonlinear models 3 Linear system identification in time domain 4 Experiment design and the identification procedure 5 Linearization of nonlinear models M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 39 / 46

40 Linearization of nonlinear models ẋ nl = f (x nl, u nl ) y nl = g(x nl, u nl ) Choose x 0, y 0 or u 0 such that 0 = f (x 0, u 0) and y 0 = g(x 0, u 0) (equilibrium). Linearize the above model: ẋ nl f (x nl x 0) + f (u nl u 0) xnl =x 0 unl =u 0 x nl u nl y nl y 0 g (x nl x 0) + g (u nl u 0) xnl =x 0 unl =u 0 x nl u nl M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 40 / 46

41 Linearization of nonlinear models ẋ nl f (x nl x 0) + f (u nl u 0) xnl =x 0 unl =u 0 x nl u nl y nl y 0 g (x nl x 0) + g (u nl u 0) xf nl=x 0 unl =u 0 x nl u nl d δx dt = Aδx + Bδu δy = Cδx + Dδu with δx = x nl x 0, δu = u nl u 0, δy = y nl y 0. M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 41 / 46

42 Comparison of models through simulation u Nonlinear system y Linearized model y δ u 0 y 0 δu = u nl u 0 y nl = δy + y 0 M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 42 / 46

43 Alternatively... u 0 y 0 u Nonlinear system y Linearized model δ u nl = δu + u 0 δy = y nl y 0 M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 43 / 46

44 Control by local linear controller u u 0 Nonlinear system Linearized model x init =0 y 0 y Linear controller x init =0 M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 44 / 46

45 Two-degree-of-freedom control y 0 u 0 y 0 Reference Feedforward controller Nonlinear system δy δ Feedback controller Linear controllers must use signals δu, δy, δx! M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 45 / 46

46 The End Thanks for your attention! Questions? M. Mazo Jr. (DCSC/TUD) System Identification and Linearization 46 / 46