 # 280 SYSTEM IDENTIFICATION The System Identification Problem is to estimate a model of a system based on input-output data. Basic Configuration continuous.

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280 SYSTEM IDENTIFICATION The System Identification Problem is to estimate a model of a system based on input-output data. Basic Configuration continuous System disturbance (not observed) v(t)v(t) y(t)y(t) u(t)u(t) output (observed) input (observed) discrete System { v ( k )} { y ( k )} { u ( k )}

281 We observe an input number sequence ( a sampled signal ) { u ( k )} = { u (0), u (1),..., u ( k ),..., u ( N )} and an output sequence { y ( k )} = { y (0), y (1),..., y ( k ),..., y ( N )} using standard z-transform notation If we assume the system is linear we can write:-

282 G( z ) U( z ) Y( z ) V( z ) + The disturbance v ( k ) is often considered as generated by filtered white noise :- G( z ) U( z ) Y( z ) V( z ) + H( z ) (z)(z) white noise filter disturbance output input process giving the description:

283 Parametric Models ARX model ( autoregressive with exogenous variables ) where G( z ) U( z ) Y( z ) V( z ) + H( z ) (z)(z)

284 giving the difference equation : and represents an extra delay of n  sampling instants. identification problem determine n , n a, n b ( structure ) estimate ( parameters )

285 ARMAX model (autoregressive moving average with exogenous variables) where G( z ) U( z ) Y( z ) V( z ) + H( z ) (z)(z)

286 giving the difference equation : identification problem determine n , n a, n b, n c ( structure ) estimate ( parameters )

287 General Prediction Error Approach Process Predictor with adjustable parameters  + - Algorithm for minimising some function of e ( t,  ) u(t)u(t) y(t)y(t) e ( t,  )  Predictor based on a parametric model Algorithm often based on a least squares method.

288 Consistency A desirable property of an estimate is that it converges to the true parameter value as the number of observations N increases towards infinity. This property is called consistency Consistency is exhibited by ARMAX model identification methods but not by ARX approaches (the parameter values exhibit bias ).

289 Example of MATLAB Identification Toolbox Session Input and Output Data of Dryer Model

290 MATLAB statements and results: (ARX n  , n a = 2, n b = 2)

291 ARX model :

292 MATLAB Demo

293 ADAPTIVE CONTROL PERFORMANCE ASSESSMENT & UPDATING MECHANISM REGULATORPROCESS parameters slowly varying ref + _ outputs (fast varying) disturbances fast varying regulator parameters K J Astrom

294 Adaptive control is a special type of nonlinear control in which the states of the process can be separated into two categories:- (i) slowly varying states (viewed as parameters (ii) fast varying states (compensated by standard feedback) In adaptive control it is assumed that there is feedback from the system performance which adjusts the regulator parameters to compensate for the slowly varying process parameters.

295 Adaptive Control Problem An adaptive controller will contain :- characterization of desired closed-loop performance (reference model or design specifications) control law with adjustable parameters design procedure parameters updating based on measurements implementation of the control law (discrete or continuous)

296 Overview of Some Adaptive Control Schemes Gain Scheduling regulator process output y control signal u command signal gain schedule regulator parameters operating conditions The regulator parameters are adjusted to suit different operating conditions. Gain scheduling is an open-loop compensation.

297 Auto-tuning PID controller Process + _ parameters K, T i, T d PID controllers are traditionally tuned using simple experiments and empirical rules. Automatic methods can be applied to tune these controllers. (i) experimental phase using test signals; then:- (ii) use of standard rules to compute PID parameters.

298 Model Reference Adaptive Systems MRAS regulator process actual output y u ucuc model ideal output ymym adjustment mechanism  regulator parameters

299 The parameters of the regulator are adjusted such that the error e = y - y m becomes small. The key problem is to determine an appropriate adjustment mechanism and a suitable control law. where  determines the adaptation rate. This rule changes the parameters in the direction of the negative gradient of e 2 MIT rule adjustment mechanism

300 Combining the MIT rule with the control law: and computing the sensitivity derivatives produces the scheme: process  u multiplier ucuc y _ +   e filter integrator model + _ ymym multiplier Note : steady-state will be achieved when the input to the integrator becomes zero. That is when y = y m

301 Self Tuning Regulators STR regulator process actual output y u ucuc regulator parameters  design process parameters estimation

302 The process parameters are updated and the regulator parameters are obtained from the solution of a design problem. The adaptive regulator consists of two loops :- (i) inner loop consisting of the process and a linear feedback regulator (ii) outer loop composed of a parameter estimator ( recursive ) and a design calculation. (To obtain good estimates it is usually necessary to introduce perturbation signals) Two problems :- (i) underlying design problem (ii) real time parameter estimation problem

303 Example - SIMULINK Simulation of MRAS

304

305 MATLAB Demo

306 INTRODUCTION TO THE KALMAN FILTER State Estimation Problem Vectors w(t) and v(t) are noise terms, representing unmeasured system disturbances and measurement errors respectively. They are assumed to be independent, white, Gaussian, and to have zero mean. In mathematical terms:- w(t)w(t) v(t)v(t) u(t)u(t)x(t) y(t)y(t) SYSTEM

307 where Q and R are symmetric and non negative definite covariance matrices. (E is the expectation operator) Only u(t) and y(t) are assessable. The state estimation problem is to estimate the states x(t) from a knowledge of u(t) and y(t). (and assuming we know A, B, G, C, D, Q, and R).

308 Construction of the Kalman-Bucy Filter SYSTEM u(t)u(t) y(t)y(t) x(t)x(t) Filter equation :- CB A D L(t) u(t)u(t) y(t)y(t) + _ + FILTER +

309 The estimation problem is now to find L(t) such that the error between the real states x(t) and the estimated states is minimized. This can be formulated as: Filter equation :- L(t) is a time dependent matrix gain. R E Kalman

310 Duality Between the Optimum State Estimation Problem and the Optimum Regulator Problem It can be shown that the optimum state estimation problem: subject to: is the dual of the optimum regulator problem: subject to:

311 Thus L(t) can be obtained by solving the matrix Ricatti equation: Furthermore for large measurement times L(t) converges to: a constant matrix gain.

312 Linear Quadratic Estimator Design Using MATLAB

313 Example: produces:

314 giving the filter equations: where l 1 = 0.5562, l 2 = 0.1547

315 + u(t) = 0 w(t)w(t)v(t)v(t) + y(t)y(t) x1x1 x2x2 SYSTEM ++ _ + l1l1 l2l2 FILTER

317 Comparison of actual (solid) and measured (dash) states x1x1

318 Comparison of actual (solid) and measured (dash) states x2x2

319 Measurement signal y(t)

320 MATLAB Demo

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