Outline 2(42) Sysid Course VT An Overview. Data from Gripen 4(42) An Introductory Example 2,530 3(42)

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1 Outline 2(42) Sysid Course T An Overview. Automatic Control, SY, Linköpings Universitet An Umbrella Contribution for the aterial in the Course The classic, conventional System dentification Setup The dentification Loop odel and odel Structures dentification ethods odel alidation xyz means: details are given on page xyz in the textbook Ljung: System dentification. Theory for the User, 2nd Edition, Prentice-Hall An ntroductory Example 2,530 3(42) ata from Gripen 4(42) Pitch rate, Canard, Elevator, Leading Edge Flap How do the control surface angles affect the pitch rate? Aerodynamical derivatives? How to use the information in flight data?

2 Aircraft ynamics: From input 1 5(42) Using all inputs 6(42) y(t) pitch rate at time t. u 1 (t) canard angle at time t. T = 1/60. Try y(t) = +b 1 u 1 (t T) + b 2 u 1 (t 2T) + b 3 u 1 (t 3T) + b 4 u 1 (t 4T) u 1 canard angle; u 2 Elevator angle; u 3 Leading edge flap; y(t)= a 1 y(t T) a 2 y(t 2T) a 3 y(t 3T) a 4 y(t 4T) +b 1 1 u 1(t T) b 4 1 u 1(t 4T) +b 1 2 u 2(t T) b 3 1 u 3(t T) b 3 4 u 3(t 4T) ashed line: Actual Pitch rate. Solid line: 10 step ahead predicted pitch rate, based on the fourth order model from canard angle only. System dentification: ssues 14 7(42) System dentification: State-of-the-Art Setup 8(42) Select a class of candidate models Select a member in this class using the observed data Evaluate the quality of the obtained model esign the experiment so that the model will be good. A Typical Problem Given Observed nput-output ata: Find a escription of the System that Generated the ata [Simulator or Predictor. Linear System: mpulse response or Bode plot]. Basic Approach Find a suitable odel Structure, Estimate its parameters, and compute the response of the resulting model Techniques Estimate the parameters by L techniques/pe (prediction error methods). Find the model structure by AC, BC or Cross alidation

3 The S Flow 15 9(42) The S Flow; odel Structures 10(42) o, try new o, try new (ˆ ) : The Experiment : The easured ata : The odel Set : The dentification ethod : The alidation Procedure o, try new o, try new (ˆ ) odels: General Aspects 79,161 11(42) Linear odels 79 12(42) A model is a mathematical expression that describes the connections between measured inputs and outputs, and possibly related noise sequences. They can come in many different forms The models are labeled with a parameter vector θ A common framework is to describe the model as a predictor of the next output, based on observations of past input-output data. Observed input output (u, y) data up to time t: Z t odel described by predictor: (θ) : ŷ(t θ) = g(t, θ, Z t 1 ). General escription y(t) = G(q, θ)u(t) + H(q, θ)e(t), q : shift op. e : white noise G(q, θ)u(t) = k=1 g k u(t k), H(q, θ)e(t) = 1 + k=1 h k e(t k) y(t) = G(q, θ)u(t) + v(t); Spectrum Φ v (ω) = λ H(e iω ) 2 Predictor ŷ(t θ) = G(q, θ)u(t) + [ H 1 (q, θ)][y(t) G(q, θ)u(t)]

4 Common Black-Box Parameterizations 81 13(42) BJ (Box-Jenkins) AR: G(q, θ) = B(q) F(q) ; C(q) H(q, θ) = (q) B(q) = b 1 q 1 + b 2 q b nb q nb F(q) = 1 + f 1 q f nf q nf θ = [b 1, b 2,..., f nf ] y(t) = B(q) A(q) u(t) + 1 e(t) or A(q) A(q)y(t) = B(q)u(t) + e(t) or y(t) + a 1 y(t 1) a na y(t na) = b 1 u(t 1) b nb u(t nb) Special Feature of AR-models 14(42) The AR odel is a Linear Regression The predictor for AR can be written ŷ(t θ) = ϕ T (t)θ ϕ(t) = [ y(t 1)... y(t na) u(t 1)... u(t nb) ] T θ = [ ] T a 1... a na b 1... b nb Common Black and Grey Parameterizations 97 15(42) Continuous Time (CT) odels 93 16(42) State-Space with Possibly Physically Parameterized atrices x(t + 1) = A(θ)x(t) + B(θ)u(t) + K(θ)e(t) y(t) = C(θ)x(t) + e(t) Corresponds to G(q, θ) = C(θ)(q A(θ)) 1 B(θ). H(q, θ) = C(θ)(q A(θ)) 1 K(θ) + Physical odel with unknown parameters ẋ(t) = F(θ)x(t) + G(θ)u(t) + w(t) y(t) = C(θ)x(t) + (θ)u(t) + v(t) Sample it (with correct nput ntersample Behaviour): x(t + 1) = A(θ)x(t) + B(θ)u(t) + K(θ)e(t) y(t) = C(θ)x(t) + e(t) ow apply the discrete time formalism to this model, which is parameterized in terms of the CT parameters θ

5 on-linear odels (42) The S Flow: Estimation: 18(42) (θ) : ŷ(t θ) = g(t, θ, Z t 1 ). g non-linear in Z. ore in the L lecture notes Physical Grey-Box odels Perform physical modeling (e.g. in OELCA) and denote unknown physical parameters by θ. Collect the model equations as ẋ(t) = f (x(t), u(t), θ) y(t) = h(x(t), u(t), θ) (ˆ ) LAR models Block Oriented odels, Wiener-Hammerstein and similar Local Linear odels, Linear Parameter arying odels. Linear model G(z, p) parameterized by a measured regime variable p o, try new o, try new Criteria: 1. Pragmatically (42) f a model, ŷ(t θ), essentially is a predictor of the next output, is is natural to evaluate its quality by assessing how well it predicts: Form the Prediction error and measure its size: ε(t, θ) = y(t) ŷ(t θ), l(ε(t, θ)) (ote: Linear system ε = 1 H (y Gu)) Typically l(x) = x2. How has it performed historically? Criteria: 2. Statistically (42) aximum Likelihood (L): Which model makes the recorded observations most likely? (pdf: probability density function) max p(y θ) p : the pdf of the outputs for a given parameter value f the innovations e of the measured data have the pdf f (x), then (θ) = l(ε(t, θ)) t=1 log p(y θ) = (θ) = l(ε(t, θ)) t=1 l(x) = log f (x) Which model in the structure performed best? ˆθ = arg min θ (θ) So, what minimizes the pragmatic fit is also the L estimate! Gaussian pdf l(x) x 2

6 Estimation: ariants 21(42) ore in the Special ssues Lecture notes Regularization: To curb the flexibility of the model structure and to provide better numerics in the minimization we can postulate a regularized criterion: 221 odel Estimate Properties ( ˆθ ) 22(42) W (θ) = (θ) + λ(θ θ ) T R((θ θ ) Bayesian iew: The parameter vector θ is a random vector with a certain prior pdf, and we seek the posterior pdf, given the observations. Suppose the prior is θ (θ, R 1 /λ) 213 o, try new o, try new (ˆ ) Sub-space methods 208 -techniques 224 (The E algorithm, Errors-in-ariable problems) odel Estimate Properties, 223, (42) As the number of data,, tends to infinity ˆθ θ arg min θ El(ε(t, θ)) the best possible predictor in f contains a true description of the system Cov ˆθ = λ [Eψ(t)ψT (t)] 1 [ψ(t) = dθ d ŷ(t θ), λ : noise level] is the Cramér-Rao lower bound for any (unbiased) estimator. E: Expectation. These are very nice optimal properties: The model structure is large enough: The L/PE estimated model is (asymptotically) the best possible one. Has smallest possible variance (Cramér- Rao) The model structure is not large enough: The L/PE estimate converges to the best possible approximation of the system. The estimate has smallest possible asymptotic bias. Proof: Formal Calculations 24(42) We illustrate the main ideas for a simple curve-fitting problem: System: y(t) = g 0 (x t ) + e(t), e white noise. odel: g(x t, θ) Convergence: (LL= Law of Large umbers) (θ) = 1 (g 0 (x t ) + e(t) g(x t, θ)) 2 = 1 (g 0 (x t ) g(x t, θ)) e 2 (t) + 2 (g 0 (x t ) g(x t, θ))e(t) 1 LL: (g 0 (x t ) g(x t, θ))e(t) 0 so (θ) E(g 0 (x t ) g(x t, θ)) 2 (uniformly in θ!) = lim 1 (g 0 (x t ) g(x t, θ)) 2 as

7 Proof: Formal Calculations, ctn d 25(42) Asymptotics for linear models 263, (42) Asymptotic istribution (CLT= Central Limit Theorem) 0 = ( ˆθ ) = (θ ) + (θ )( ˆθ θ ) ( ˆθ θ ) = [ (θ )] 1 (θ ) (θ) = 2 (y(t) g(x t, θ))g (x t, θ) (θ ) = 2 e(t)ψ(t); ψ(t) = g (x t ; θ ) LL: (θ ) = 2 ψ(t)ψ T (t) + 2 e(t)g (x t, θ ) 2Eψψ T CLT: 1 e(t)ψ(t) (0, λeψψ T ) [Φ u, Φ v : Spectra of input and additive noise v = He. G 0 true transfer function.] π ˆθ θ = arg min θ π CovG(e iω, ˆθ ) n G(e iω, θ) G 0 (e iω ) 2 Φ u (ω) H(e iω, θ) 2 dω Φ v (ω) as n, Φ u (ω) n : model order ( ˆθ θ ) (0, λ[eψψ T ] 1 ) Experiment esign: 27(42) odel alidation: 28(42) (ˆ ) (ˆ ) o, try new o, try new o, try new o, try new See Lecture otes Experiment esign

8 odel alidation Techniques (42) Twist and turn the model(s) to check if they are good enough for the intended application. Essentially a subjective decision. Several basic techniques are available: 1. Simulation (prediction) cross validation Check how well the estimated model can reproduce new, validation data 2. Residual Analysis Are the "leftovers" unpredictable? 3. Comparing ifferent odels Which one performs best? 4. Bias-ariance Trade-off What is the suitable complexity (flexibility) of a model? Simulation Cross alidation (42) Collect a estimation data set Z e and a validation data set Z v. Estimate the model(s) using the estimation data and simulate it using the input signal in the validation data. Plot that simultated model output together with the measured validation output. m = arx(z1(1:150),[2 2 1]); compare(z1(151:end),m) The ultimate test. eeds no probabilistic justification. Plot simulated outputs and measured outputs for the fresh data set. Alternatively one may use predictions over longer periods. Typically the performance can be evaluated as sum of squared mismatches. ( FT ) odel alidation: Residual Analysis (42) Typical Plots 32(42) 1 Correlation function of residuals. Output # ε(t) = ε(t, ˆθ ) = y(t) ŷ(t ˆθ ) ε(t) should be independent of u(t τ), τ > 0 ε(t) should ideally be white noise. Compute the corresponding correlation functions Confidence intervals for the estimates can also be computed. This requires some care! lag 0.6 Cross corr. function between input 1 and residuals from output lag True delay: 4 guessed delay: 5

9 Typical Plots, ct d 33(42) odel alidation Techniques (42) 1 Correlation function of residuals. Output # lag 0.2 Cross corr. function between input 1 and residuals from output lag Guessed delay = true delay. Twist and turn the model(s) to check if they are good enough for the intended application. Essentially a subjective decision. Several basic techniques are available: 1. Simulation (prediction) cross validation Check how well the estimated model can reproduce new, validation data 2. Residual Analysis Are the "leftovers" unpredictable? 3. Comparing ifferent odels Which one performs best? 4. Bias-ariance Trade-off What is the suitable complexity (flexibility) of a model? Comparing ifferent odels (42) What to compare? simulation performance prediction performance compare(zv,th,k) Comparing models on fresh data sets (Simulation cross validation as before) Comparing models on second-hand data sets Comparing odels on Second-Hand ata (42) Problem: Larger models will always perform better on estimation data. BASC EA: Compensate for over-fit FT OEL ORER Add odel Complexity Penalty Akaike: AC, FPE Rissanen L Hypothesis test Check if decrease in fit is larger than expected

10 t (s) t (s) OUTPUT #1 PUT # AC, FPE, L (42) Comparing ifferent odels (42) AC for Gaussian case: min min θ [(1 + 2d ) 1 ε 2 (t, θ )], t=1 [ -log likelihood + d ] d = dim θ FPE similar, with (1 + 2d ) replaced by +d d. Aims at estimating the fit that would be obtained for a fresh data set. L (or BC) is like AC but with a 2d log penalty. What to compare? simulation performance prediction performance compare(zv,th,k) Comparing models on fresh data sets (Simulation cross validation as before) Comparing models on second-hand data sets Comparing many models simulaneously Bias ariance Trade Off (42) Any estimated model is incorrect. The errors have two sources: Bias: The model structure is not flexible enough to contain a correct description of the system. ariance: The disturbances on the measurements affect the model estimate, and cause variations when the experiment is repeated, even with the same input. ean Square Error (SE) = Bias 2 + ariance. When model flexibility,bias and ariance. To minimize SE is a good trade-off in flexibility. n state-of-the-art dentification, this flexibility trade-off is governed primarily by model order. Status of the State-of-the-Art Framework 40(42) Well established statistical theory Optimal asymptotic properties Efficient software any applications in very diverse areas. Some examples: Aircraft ynamics: Brain Activity (fr): Pulp Buffer essel: ;