Principles of System Identification

List of Figures

1.1 (SEE COLOR INSERT) Identification is the task of using input-output data to

build a model: a mathematical abstraction of the process...............................................2

1.2 Applications of models in process systems engineering. ...............................................4

1.3 Temperature response of a reactor to changes in coolant flow rate is used in iden-

tification of the reactor system........................................................................................4

1.4 Plot of RH and Temperature ...........................................................................................6

1.5 Two approaches to modeling: first-principles vs. experimental (empirical). .................8

1.6 (SEE COLOR INSERT) The system being identified consists of the true process

and additional elements.................................................................................................13

1.7 (SEE COLOR INSERT) A generic iterative procedure for System Identification. ...20

1.8 A standard sampled-data system. .................................................................................20

1.9 System Identification involves application of concepts from four broad fields in

engineering, mathematics and statistics........................................................................26

2.1 The best fit and the error in the parameter estimates depend on the SNR. ..................34

2.2 Training data and order determination in Example 2.4. ...............................................36

2.3 Cross-validation of polynomial models in Example 2.4. .............................................36

2.4 Schematic of the liquid level system discussed in Section 2.4. ....................................38

2.5 Time-trends and spectra of flow and level measurements in the identification of

the liquid level system. .................................................................................................39

2.6 Impulse response estimates of the liquid level system. ................................................42

2.7 Step response estimates of the liquid level system. .....................................................43

2.8 Comparing one-step ahead predictions (deviations from steady-state) of the iden-

tified models on the training data. ................................................................................46

2.9 Correlation between residuals and lagged inputs for the output-error (bottom) and

equation-error (top) models. ........................................................................................47

2.10 Correlation analysis of residuals obtained from the best equation-error model. .........48

2.11 Auto-correlation function of the residuals from the output-error model. ....................49

2.12 Infinite-step ahead predictions from the output-error model on the test data set..........49

2.13 Plot of (log) Hankel singular values and correlation functions from the state-space

model. ...........................................................................................................................51

3.1 The inputs and outputs of a model are in general different from the physical inputs

and outputs of a process. ..............................................................................................57

3.2 (SEE COLOR INSERT) Types of models. ................................................................58

3.3 (SEE COLOR INSERT) Four broad categories of empirical models. ......................59

3.4 (SEE COLOR INSERT) Composite model from identification. ...............................66

4.1 Impulse response of an LTI system. .............................................................................69

4.2 Schematic depicting the derivation of a convolution operation. ..................................69

4.3 Impulse responses of different systems. .......................................................................70

4.4 Typical step response of a first-order LTI system. .......................................................74

4.5 Bode plots for the first-order system in Example 4.5. .................................................78

4.6 Bode plot for the pure delay system of Example 4.6 with D = 2. ...............................78

4.7 Two different ways of realizing a state-space representation. ......................................89

4.8 Snapshot of the input-output data. .............................................................................102

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