SYSTEM IDENTIFICATION The System Identification Problem is to

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

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)} If we assume the system is linear we can write: - using standard z-transform notation 281

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

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

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

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

giving the difference equation: identification problem determine n , na, nb, nc (structure) estimate (parameters) 286

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

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). 288

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

MATLAB statements and results: (ARX n , na = 2, nb = 2) 290

ARX model: 291

MATLAB Demo 292

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

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. 294

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) 295

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

Auto-tuning parameters K, Ti, Td + _ PID controller Process 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. 297

MRAS Model Reference Adaptive Systems model ym regulator parameters ideal output adjustment mechanism uc regulator u process y actual output 298

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

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

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

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 302

Example - SIMULINK Simulation of MRAS 303

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MATLAB Demo 305

INTRODUCTION TO THE KALMAN FILTER State Estimation Problem w(t) u(t) v(t) y(t) x(t) SYSTEM 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: 306

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). 307

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

Filter equation : L(t) is a time dependent matrix gain. 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: R E Kalman 309

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: 310

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. 311

Linear Quadratic Estimator Design Using MATLAB 312

Example: produces: 313

giving the filter equations: where l 1 = 0. 5562, l 2 = 0. 1547 314

w(t) u(t) = 0 v(t) x 1 x 2 + -1 y(t) + SYSTEM -1 _ + + + l 1 FILTER l 2 315

SIMULINK SIMULATION 316

Comparison of actual (solid) and measured (dash) states x 1 317

Comparison of actual (solid) and measured (dash) states x 2 318

Measurement signal y(t) 319

MATLAB Demo 320