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Art. No. 39466 ISBN 978-91-44-11688-4 Edition 1:1 ÂŠ The Authors and Studentlitteratur 2016 www.studentlitteratur.se Studentlitteratur AB, Lund

Cover design by: Jens Martin, Signalera

Printed by Interak, Poland 2016

Contents 1 Systems and Models 1.1 To Describe Reality with Models . 1.2 Systems and Experiments . . . . . 1.3 What is a Model? . . . . . . . . . . 1.4 Models and Simulation . . . . . . . 1.5 How to Build Models . . . . . . . . 1.6 How to Verify Models . . . . . . . 1.7 What are Models Used for? . . . . 1.8 Modeling as a Scientific Discipline

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2 Examples of Models 2.1 Introduction . . . . . . . . . . . . 2.2 An Ecological System . . . . . . 2.3 A Flow System . . . . . . . . . . 2.4 A Simple Electrical Circuit . . . 2.5 Models of Human Speech . . . . 2.6 A Connected Mechanical System 2.7 Model Characteristics . . . . . . 2.8 Comments . . . . . . . . . . . . .

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3 Types of Models 3.1 Models . . . . . . . . . . . . . . . . . . 3.2 Black Boxes and Simple Experiments . 3.3 Variables and Signals . . . . . . . . . . 3.4 State-space Models . . . . . . . . . . . 3.5 Stationary Solutions and Linearization 3.6 Connecting State-space Models . . . .

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CONTENTS

3.7

Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4 Principles of Physical Modeling 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . 4.2 The Phases of Modeling . . . . . . . . . . . . . . 4.3 What are the Requirements for a Modeling Tool? 4.4 Dimensionless Variables and Scaling . . . . . . . 4.5 Simplified Models . . . . . . . . . . . . . . . . . . 4.6 Comments . . . . . . . . . . . . . . . . . . . . . . 4.7 Appendix . . . . . . . . . . . . . . . . . . . . . . 5 Some Basic Relationships in Physics 5.1 Introduction . . . . . . . . . . . . . . 5.2 Electrical Circuits . . . . . . . . . . 5.3 Mechanical Translation . . . . . . . 5.4 Mechanical Rotation . . . . . . . . . 5.5 Flow Systems . . . . . . . . . . . . . 5.6 Thermal Systems . . . . . . . . . . . 5.7 Domain Independent Modeling . . . 5.8 Connecting Different Domains . . . . 5.9 Comments . . . . . . . . . . . . . . .

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65 65 65 69 72 83 90 90

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93 93 93 98 102 105 109 112 113 115

6 Bond Graphs 6.1 Efforts and Flows . . . . . . . . . . . . . . . . 6.2 Junctions . . . . . . . . . . . . . . . . . . . . 6.3 Simple Bond Graphs . . . . . . . . . . . . . . 6.4 Transformers and Gyrators . . . . . . . . . . 6.5 Systems with Mixed Physical Variables . . . . 6.6 Simplifications in Bond Graphs . . . . . . . . 6.7 From Bond Graphs to Equations â€“ Causality 6.8 Causality and Equations . . . . . . . . . . . . 6.9 Controlled Elements . . . . . . . . . . . . . . 6.10 Systematic Bond Graph Modeling . . . . . . 6.11 Further Remarks . . . . . . . . . . . . . . . . 6.12 Comments . . . . . . . . . . . . . . . . . . . .

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117 117 120 124 126 127 130 133 141 146 151 154 156

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CONTENTS

7 DAE Models 7.1 Introduction . . . . . . . . . . . . . . . 7.2 DAE Models . . . . . . . . . . . . . . 7.3 Linear DAE Systems . . . . . . . . . . 7.4 Nonlinear DAE models . . . . . . . . . 7.5 DAE Models and Model Simplification 7.6 Comments . . . . . . . . . . . . . . . . 7.7 Appendix . . . . . . . . . . . . . . . .

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8 Object-oriented Modeling 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 8.2 Model Libraries and Graphical Interfaces . . . . . . 8.3 Object-oriented Modeling â€“ Simscape and Modelica 8.4 Model Libraries and Model Extensions . . . . . . . . 8.5 Aggregated Models in Modelica . . . . . . . . . . . . 8.6 Equation-based Models . . . . . . . . . . . . . . . . 8.7 Comments . . . . . . . . . . . . . . . . . . . . . . . . 9 Models with Disturbances 9.1 Discrete Time Models . . . . . . . . . . . . . . . . . 9.2 Disturbances in Dynamic Models . . . . . . . . . . . 9.3 Description of Signals in the Time Domain . . . . . 9.4 Description of Signals in the Frequency Domain . . . 9.5 Linking Continuous Time and Discrete Time Models 9.6 Comments . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . .

157 157 157 160 166 168 174 174

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177 . 177 . 178 . 179 . 184 . 188 . 191 . 192

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10 Correlation and Spectral Analysis 10.1 Correlation Analysis . . . . . . . . . . . . . . . . . . . 10.2 Fourier Analysis . . . . . . . . . . . . . . . . . . . . . 10.3 Estimation of Signal Spectra from Discrete Time Data 10.4 Estimating Transfer Functions Using Spectral Analysis 10.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . .

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221 221 224 228 237 241 242

11 Parameter Estimation in Linear Models 247 11.1 Grey-box Models . . . . . . . . . . . . . . . . . . . . . . 248 11.2 Linear Black-Box Models . . . . . . . . . . . . . . . . . 251

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CONTENTS

11.3 11.4 11.5 11.6

Fitting Parameterized Models Model Properties . . . . . . . Comments . . . . . . . . . . . Appendix . . . . . . . . . . .

to Data . . . . . . . . . . . . . . .

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259 267 278 279

12 Parameter Estimation in Nonlinear Models 12.1 The Basic Principle â€“ Minimize the Prediction Errors 12.2 Light-Grey-Box Nonlinear Models . . . . . . . . . . . 12.3 Dark-Grey-box Nonlinear Models . . . . . . . . . . . . 12.4 Neural Network as Nonlinear Black-box Models . . . . 12.5 Parameter Estimation Techniques . . . . . . . . . . . . 12.6 Comments . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 Appendix: Derivatives of the Prediction Function . . .

283 . 283 . 284 . 291 . 302 . 309 . 311 . 312

13 System Identification as a Tool for Modeling 13.1 Program Packages for Identification . . . . . 13.2 Design of Identification Experiments . . . . . 13.3 Post-treatment of Data . . . . . . . . . . . . 13.4 Choice of Model Structure . . . . . . . . . . . 13.5 Model Validation . . . . . . . . . . . . . . . . 13.6 An Example . . . . . . . . . . . . . . . . . . . 13.7 Comments . . . . . . . . . . . . . . . . . . . .

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349 . 349 . 349 . 359 . 360

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361 . 361 . 364 . 365 . 368

14 Simulation of Mathematical Models 14.1 Introduction . . . . . . . . . . . . . . 14.2 Simulating State-space Models . . . 14.3 Solution Methods for DAE Models . 14.4 Comments . . . . . . . . . . . . . . . 15 Model Validation and Model Use 15.1 Model Validation . . . . . . . . . 15.2 Domain of Validity of the Model 15.3 Remaining Critical of the Model 15.4 Use of Several Models . . . . . .

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315 316 318 325 331 339 342 346

A Linear Systems: Description and Properties 369 A.1 Continuous Time Systems . . . . . . . . . . . . . . . . . 369 A.2 Discrete Time Models . . . . . . . . . . . . . . . . . . . 371

CONTENTS

7

A.3 Links between Continuous and Discrete Time Models . . 373 A.4 Alias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 B Linearization 377 B.1 Continuous Time Models . . . . . . . . . . . . . . . . . 377 B.2 Discrete Time Models . . . . . . . . . . . . . . . . . . . 379 C Random Variables and Stochastic Processes 381 C.1 Probabilities and Random Variables . . . . . . . . . . . 381 C.2 Stochastic Processes . . . . . . . . . . . . . . . . . . . . 384 C.3 Two Basic Convergence Concepts . . . . . . . . . . . . . 385 D Signal Spectra 387 D.1 Time-continuous Deterministic Signal with Finite Energy387 D.2 Sampled Deterministic Signals with Finite Energy . . . . . . . . . . . . . . . . . . . . . . . . 388 D.3 Signals with Infinite Energy: The Power Spectrum . . . 389 D.4 Connections between the Continuous and the Sampled Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 D.5 Stochastic Processes . . . . . . . . . . . . . . . . . . . . 391

Chapter 3

Types of Models In this chapter, we shall characterize the different types of models that we will study further. We shall also describe component or block diagram models and show how the character of a block with (linear) dynamical systems can be found with simple experiments.

3.1

Models

Mathematical Models With a mathematical model we mean a description of the system where the relationships between the modelâ€™s variables and signals are expressed in mathematical terms. In the examples in the previous chapter we saw that modeling naturally leads to differential (or difference) equations. We saw also in Section 2.5 that there is not much difference in modeling a physical system and a physical signal. Our mathematical models of signals and dynamical systems will thus essentially consist of a collection of differential equations. In Section 3.4 and Chapter 7 we will treat the formal, mathematical aspects of such descriptions: state-space models, DAE systems, etc.

Component Models In addition to these mathematical and exact descriptions, it is very useful to have models that illustrate the connections between a systemâ€™s signals in more qualitative and pictorial way. Figure 2.8 is an 35

(c) The Authors and Studentlitteratur

36

Chapter 3

Types of Models

example of a more complex system that is built up by simpler components: a component model. A component model of a system is a logical and/or physical decomposition of the functions of a system, and it points to the interaction between the various components. These interactions are illustrated by links between the components (blocks). A given system can be decomposed in components in several different ways. It is most productive to deal with such large models in an object oriented software environment. We return to that issue in Chapter 8.

Bond Graphs Bond graphs can be seen as an intermediary concept between a mathematical model and a component model. Bond graphs as a graphical way of describing models will be discussed in Chapter 6.

Block Diagram Models A block diagram of a system is a logical decomposition of the functions of the system and shows how the different parts (blocks) influence each other. This interaction is illustrated by arrows between the blocks. A given system can usually be represented by several different block diagram models, depending on how detailed we want to make them. Example 3.1: Block Diagram for the Water Tank Consider the tank in Figure 2.4. The outflow q depends on the inflow u, which we can illustrate by a simple block diagram, according to Figure 3.1. We can also make a more detailed description, which in addition contains the level h in the tank. The level h depends on the inflow u and the outflow q (Figure 3.2). The outflow q in turn depends on the level h (and the outflow area a), according to Bernoulliâ€™s law (see Figure 3.3). The left picture in Figure 3.3 is preferable if the outflow area is fixed and cannot be influenced. If the outflow area can be varied, for example, by placing a valve in the outflow, the right picture is however more natural. From the subsystems in Figures 3.2 and 3.3, we now have a block diagram for the tank according to Figure 3.4. (c) The Authors and Studentlitteratur

3.1 Models

37

u

q

Tank

Figure 3.1: Block diagram for the tank in Section 2.3. u q

h

Tank

Figure 3.2: Block diagram describing how the level in the tank depends on the inflow and the outflow.

Note the difference between the schematic picture in Figure 2.4 and the block diagram in Figure 3.4. Schematic pictures are often used for simple illustration of the function of a system. These are, however, based on the physical construction of the system, whereas the block diagram is based on the logical description. The flows in Figure 3.4 are information flows and not water as in Figure 2.4. Block diagrams are very useful when structuring a system, especially for larger and more complex ones. As models for the system they can be compared to the verbal models we discussed in the introductory chapter. They also constitute a very important starting point for the mathematical model building. When, for example, we built the model (2.10), (2.11) for the tank system, we started from the basic

a

q

h (a)

q

h (b)

Figure 3.3: (a) Outflow as a function of the level, and (b) as a function of the level and the outflow area. (c) The Authors and Studentlitteratur

38

Chapter 3

Types of Models

a

u q

h

q

Figure 3.4: Block diagram for the tank system. equations (2.9) and (2.7)-(2.8), which correspond to the two blocks in Figure 3.4. In Chapter 4 we will discuss in more detail the use of block diagrams in model building. Block diagrams are also used as models in sciences where quantitative models usually cannot be constructed, as in, for example, ecology, sociology (sociograms), and so on.

3.2

Black Boxes and Simple Experiments

A black box is a model or sub-model (a block in a block diagram or a component in a component model) which only deals with the connections between inputs and outputs, without considering the underlying physical phenomena. The description of the black box model could be given as a collection of mathematical equations, or more graphically in terms of a curve or a table. It could e.g., be a step response or an impulse response from input to output. For a linear system the frequency response (like a Bode plot) is often illustrative (see, e.g., Appendix A). In this section we shall discuss how to obtain transient (step- or impulse-) responses and frequency responses from simple experiments.

Transient Analysis The first step in modeling is to decide which quantities and variables are important to describe what happens in the system. We shall call this the structuring phase in Section 4.2. It is then also necessary to decide, or guess, how the variables affect each other, which time (c) The Authors and Studentlitteratur

3.2 Black Boxes and Simple Experiments

39

constants are important, and which relationships can approximately be described as static ones (compare Section 4.5). It is a rather demanding task for the modeler to answer these questions. Considerable knowledge and insights about the system will be required. Often simple experiments on the real system have to be carried out to support the work in this phase. A simple and common kind of experiment that shows how and in what time span various variables affect each other is called step-response analysis or transient analysis. In such experiments the inputs vary (typically one at the time) as a step: u(t) = u0 , t < t0 ; u(t) = u1 , t â‰Ľ t0 . The other measurable variables in the system are recorded during this time. We thus study the step response of the system, using terminology from Appendix A. An alternative would be to study the impulse response of the system by letting the input be a pulse of short duration. From such measurements, information of the following nature can be found: 1. The variables affected by the input in question. This makes it easier to draw block diagrams for the system and to decide which influences can be neglected. 2. The time constants of the system. This also allows us to decide which relationships in the model can be described as static (that is, they have significantly faster time constants than the time scale we are working with). 3. The characteristic (oscillatory, poorly damped, monotone, and the like) of the step responses, as well as the levels of static gains. Such information is useful when studying the behavior of the final model in simulation. Good agreement with the measured step responses should give a certain confidence in the model.

Example 3.2: Tank Dynamics Mixing tanks are common in process industry. Their purpose is to smooth variations in concentration in a liquid by letting it pass through a big tank, where it is mixed. At a paper mill (SkÂ¨arblacka, Sweden) three identical mixing vessels, coupled together as indicated in Figure 3.5, are used to smooth the concentration of pulp. The dynamics (c) The Authors and Studentlitteratur

40

Chapter 3

Types of Models

A

B Vessel 1

Vessel 2

Vessel 3

Figure 3.5: Three coupled mixing vessels. of this system were investigated by an impulse response experiment. The actual concentration of the pulp could not be manipulated since the experiment had to be done during normal operation. The problem was solved in the following way: A bucket of water with 2070 grams of radioactive lithium, with short half-life, was poured into the first mixing vessel at point A in Figure 3.5. The radioactivity was then measured at point B during 5 hours. This radioactivity will obviously be proportional to the concentration of lithium, after correction for the half-life and background radiation. Figure 3.6 shows the measurements. Even if the measurements are disturbed by a fair amount of noise, a clear picture of a typical time response is obtained. To correctly scale the impulse response we argue as follows: Let both the input and the output have the unit mg/liter. Then the coefficients of the impulse response will be dimensionless quantities. The total flow through the system was about 8300 liters/minute during the time of the experiment. The sudden addition of 2070 grams of lithium then corresponds to an impulse u(t) = u0 δ(t) with u0 = 2070 grams/minute = (2070/8300) ∗ 103 ≈ 250 milligrams per liter. Consequently we must divide the lithium concentration with this number to get the impulse response. This is shown in the lower plot of Figure 3.6. In this figure we also show the impulse response of the system G(s) =

1 sτ + 1

(c) The Authors and Studentlitteratur

3

for τ = 72 min

(3.1)

3.2 Black Boxes and Simple Experiments

Lithium concentration (mg/liter)

1

xx

0.8 0.6

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Figure 3.6: Upper plot: Lithium concentration at point B. Lower plot: Cross: the experimental impulse response. Solid line: a fitted impulse response. This gives a reasonable fit to the measurements and can be used as a model for further considerations with good approximation. Note that we have used a certain amount of physical insight in (3.1): The three cascaded mixers are identical, and therefore G(s) must be the product of three identical transfer functions. Moreover, the static gain must be 1. (Everything that is poured into the tanks will eventually leave them.) We may also note that the tanks each have a volume of 600 m3 . With a mean flow of 8300 liters/minute this gives, for a perfectly mixed tank, a theoretical time constant (mean residence time) of 600/8.3 â‰ˆ 72 minutes. This is in excellent agreement with the result of the transient analysis.

Frequency Analysis A linear system is uniquely determined by its impulse response or its frequency response G(iĎ‰) (the Laplace transform of the impulse re(c) The Authors and Studentlitteratur

42

Chapter 3

Types of Models

sponse evaluated at s = iω). While transient analysis aims at direct estimates of the impulse response, there are several techniques to directly estimate the frequency response. We will first describe frequency analysis. If a linear system has the transfer function G(s) and the input is u(t) = u0 cos ωt,

(3.2)

then the output will be y(t) = y0 cos(ωt + ϕ)

(3.3)

after that possible transients have faded away (see Appendix A). Here y0 = |G(iω)| · u0 ϕ = arg G(iω)

(3.4) (3.5)

If the system is driven by the input (3.2) for a certain u0 and ω1 and we measure y0 and ϕ from the output signal, it is possible to determine the complex number G(iω1 ) using (3.4)–(3.5). By repeating this procedure for a number of different ω, we can get a good estimate of the function G(iω). This method is called frequency analysis. Sometimes it is possible to see or measure u0 , y0 , and ϕ directly from graphs of the input and output signals. (See for example Figure 3.8.) Most of the time, however, there will be noise and irregularities that make it difficult to determine ϕ directly. A suitable procedure is then to correlate the output with cos ωt and sin ωt in the way that is evident later from equation (10.11) in Section 10.2. This procedure is called frequency analysis with the correlation method. Example 3.3: Eye Dynamics It is well known that the pupil of an eye reacts to incoming light so that the light intensity at the retina is more or less independent of the outside brightness. We also know that this reaction is not immediate, but dynamic. It takes about a second or so before the eye adapts to new light conditions. Let us construct a model for how the pupil reacts to incoming light. Consider a system according to Figure 3.7. The dynamic properties of this system depend on how the nerve impulses (c) The Authors and Studentlitteratur

3.2 Black Boxes and Simple Experiments

Light

Pupil

Retina

43

Brain

Figure 3.7: Diagram for the adaptation of the pupil

that register the light at the retina are processed and sent to the pupil muscle. They also depend on the reaction time of the pupil muscle. It is not easy to write down reliable equations for this process. Instead we will use an experiment and build a model using frequency analysis. The experiments are described in the work carried out by L. Stark in 1959. A complication of the experiment in this case is that there is always feedback from the output to the input in the system in Figure 3.7. The area of the pupil will affect the light intensity at the retina; this indeed is the purpose of the reaction. To disable the feedback during the experiment, a light ray with a very small area was used and aimed at the center of the pupil. The intensity was then varied as a sinusoid, and the area of the pupil was measured. These measurements were made using a wide infrared light beam, also aimed at the pupil. From the reflected intensity it was possible to compute the area of the pupil. Figure 3.8 shows corresponding inputs and outputs at the frequency ω = 3.75 rad/s. The output is not exactly sinusoidal, which shows that the system is not exactly linear and/or that errors affect the measurements. It can still be described as a sinusoid with reasonable approximation, and it is not difficult to determine the gain |G(iω)| and the phase delay arg G(iω) from the figure. By repeating the experiment for a number of frequencies and graphing log |G(iω)| and arg G(iω) as functions of log ω, the points in the diagram in Figure 3.9 were obtained. These points could then be adjusted to transfer function curves for linear systems. In Figure 3.9 (c) The Authors and Studentlitteratur

44

Chapter 3

Types of Models

Light (millilumen)

0.4 0.3 0.2 0.1 0 0

1

2

3

4

5 Time (s)

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7

8

9

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25 20 15 10

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8

10 Time (s)

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Figure 3.8: Inputs and outputs from pupil experiment. Amplitude

10 0

10 -1

*

*

*

* *

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10 0 0

-200 -400 -600 10 0

*

*

* *

*

10 1 Frequency (rad/s)

10 2

Phase *

*

*

* *

*

10 1 Frequency (rad/s)

*

*

* *

10 2

Figure 3.9: Experimentally determined transfer function for the pupil system.

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3.2 Black Boxes and Simple Experiments

45

0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

Figure 3.10: Step-response for the system (3.6). they have been adjusted to the transfer function G(s) = e−0.28s

0.19 (1 + 0.09s)3

(3.6)

The factor e−0.28s means that there is a pure time delay of 0.28 s before anything at all happens [the Laplace transform of u(t − T ) is 0.19 e−sT U (s)]. The factor (1+0.09s) 3 is a third-order system that describes how the output reacts after the time delay. The step-response of the model (3.6) is shown in Figure 3.10. This model can be physiologically interpreted as a time delay corresponding to the time to transmit and process the information in nerves and synapses, while the third-order system corresponds to the dynamics of the muscle.

(c) The Authors and Studentlitteratur

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