CY3A2 System identification Modelling Elvis Impersonators Fresh evidence that pop stars are more popular dead than alive. The University of Missouri’s Jean Gaddy Wilson told a recent press conference in Dallas that, in 1977 when Elvis Died there where 48 professional Elvis impersonators. Today there are If that growth is projected, by the year 2012 one person in four on the face of the globe will be an Elvis Impersonator Royal Statistical Society News, 1996 Assume Elvis’s first hit was in 1955 and that the first impersonator started in that year. Assume that there is an exponential growth in elvis impersonators i.e. that the model is of the form EI = exp(b1*year +b0)

CY3A2 System identification y=Log(EI)=b1*year + b0 We can form a matrix of independent variables We can also form a vector of dependent output variables The Least squares fit to this is The prediction for 2012 is then If Jean Gaddy Wilson is right, either there will be a dramatic drop in world population or growth of Elvis Impersonators is more dramatic than an exponential model will allow.

CY3A2 System identification

Time series models (ARMAX) General form of the discrete time model used for system identification is the ARMAX model. Autoregressive, Moving Average, Exogeneous inputs. Autoregressive refers to the fact that the output is a linear combination of previous values of the output. Moving Average refers to the noise model. Exogeneous implies that there is an input to the system along with knowledge of its previous values. Thus the model is

CY3A2 System identification The picture is : Use a delay block

CY3A2 System identification

Variants are Autoregressive Moving Average (ARMA) - No access to knowledge of the input Autoregressive exogeneous (ARX) - Assume that only disturbance is white noise Finite Impulse response (FIR) - Output is a linear combination of only past input values. The output will drop to zero in finite time if the input becomes zero. Note on z transform We can use the z transform on the ARMAX model and its variants to specify the z domain transfer function as

CY3A2 System identification L.S. parameter calculation of ARMAX models We can put the general ARMAX model into a vector form for instant i as For all data values, taken over a range of data i=1, …n, form a vector y, and a matrix Φ values thus all the data can be collected together to form the following

CY3A2 System identification As you’ve guessed it, at time n the least squares solution to this is But now add a new input value u and a new output value y and we need to recalculate the entire thing. Recursive identification methods Would like a way of efficiently recalculating the model each time we have new data. Ideal form would be Thus if the model is correct at time n-1 and the new data at time n is indicative of the model then the correction factor would be zero.

CY3A2 System identification Advantages of recursive model estimation Gives an estimate of the model (all be it poor) from the first time step Can be computationally more efficient and less memory intensive, especially if we can avoid doing large matrix inverse calculations Can be made to adapt to a changing system, eg online system identification allows telephone systems to do echo cancellation on long distance lines. Can be used for fault detection, model estimates start to differ radically from a norm Forms the core of adaptive control strategies and adaptive signal processing Ideal for real-time implementations

CY3A2 System identification Example: Estimation of a constant (scalar) model: y i i This is the mean level of the signal, derived by LS method.

CY3A2 System identification If we introduce a subscript n to represent the fact that n data points are used in deriving the mean, such that The above equation is the least squares in recursive form.

CY3A2 System identification General form of Recursive algorithms where is a vector of model parameters estimate is the difference between the measured output and the estimated output at time n is the scaling - sometimes known as the Kalman Gain