Examples are laws such as “the temperature is higher than 25◦” and “the car started moving”. Examples are laws such as "the speed of the vehicle is 25 miles per hour" or "the rate of change of velocity is equal to -9.8 meters per second squared".
Our Planet. Our Knowledge. Our Destiny
Without a doubt, our knowledge of science, technology and social sciences has played a key role in shaping the modern world. In the history of human knowledge, this is a relatively new development, currently considered as separate disciplines.
Observe. Understand. Innovate
- Cyber-Physical Systems and Hybrid Systems
- Examples
- Computational vs. Physical Systems
- Biological and Intelligent Systems
In house we have cleaning robots, smart lighting systems and smart heating, ventilation and air conditioning systems (or HVAC). Existing systems such as these are representative of the areas where we can expect significant innovation and development in the future.
Developing New Products
Is the Field of Cyber-Physical Systems New?
Furthermore, many important phenomena at the quantum level cannot simply be viewed as continuous or discrete systems. In the context of the interdisciplinary field of systems engineering, it is seen as a measure of these combined properties.
What You Will Learn from This Book, and How
A Writing Tip
If we wanted to abbreviate "The field of operating systems is new", we would replace the term with OS. In contrast, if we say "Linux and BSD are both operating systems", we would replace the term with operating systems.
Chapter Highlights
Study Problems
Lab: Warm Up Exercises
Then make small changes to the model to test your understanding of how it works. Going through all three sequences is a really good way to brush up on the math concepts that will be used in the next few chapters.
Project
In terms of specific technical details, you can assume gravity g = 10 m/s2 and drag coefficient k = 0.01 m/s2. The target can appear at any point between positions −6 and 6. You will need to read the game model for full details.
To Probe Further
In this chapter we discuss principles for modeling physical systems; differential equations, with a focus on ordinary differential equations (ODEs); systems of equations; vector calculus;.
Reconnecting with the Physical World
Conservation Laws
Elements in Mechanical Systems
In the case of the example shown in Figure 2.2, the normal force will point upwards. In the first case there is no mass attached; in the second there is a single mass attached;.
Working in 2D and 3D
Elements in Electrical Systems
Capacitor This is an element where the voltage across is proportional to the integral of the current passing through the element. Inductor This is an element where the rate of change in current, that is, I = dI(t)/dt, is proportional to the voltage across it.
The Absence or Presence of Time in a Model
Arithmetic Equations, and Linear and Non-linear Systems of
Sometimes it is possible to use a similar method as above to solve such equations. As a result, it is often necessary to resort to iterative approximation methods to solve such equations, which can be computationally more expensive (even if done by a computer).
Where Different Numbers Come from
Time-Dependent and Differential Equations
Other types of differential equations exist outside of ODEs, such as partial differential equations (PDEs) and integral differential equations (IDEs), but these will not be covered in this book either. We will always use as the variable for time, and we differentiate with respect to time.
Remarks on the Basic Machinery for Solving Differential
Proof sketch: Assume that the solution is a polynomial of finite order, and the highest nominal value has a power.
Chapter Highlights
Study Problems
Assume that there is a gravitational force pulling the mass down and that the normal length to the spring is 0. a) Assume the usual spring law and a coeff. Consider the impact of a table tennis ball with a flat floor. a) Suppose the ping-pong ball is a point mass with position p= (x, y, z).
Lab: Spring Bouncing and Object Creation
For example, we can extract parts of a model that are related to a ball from the Main model, which is a separate model that represents the entire world we are simulating, into a separate model. Now, we can easily create a world in which there are two different balls starting at different heights once the world we are simulating starts.
Project: Mascot and Ping Pong Game
The distribution also comes with standard models for different phases of the project (called tournaments in the implementation). By changing the standard models, you will both develop the design of the robot player and control how different aspects of the design are visualized in simulations.
To Probe Further
Starting from a continuous environment, we introduce discrete events and look at the issues that arise in making such a transition. We analyze issues of zero-crossing and determinism; mode switching and its effect on derivatives; discrete transitions; and the behavior of Zeno.
Introduction
However, we can avoid the need to specify both separately because the equation can be further simplified. We can simply consider a situation where the inter-candidate ratio is high (such as 100); then we have all the information we need for a model of a ball that can undergo simple elastic bounce when it hits the floor.
Hybrid Automata
In particular, we would like to explicitly specify something about the temperature of the room; we may also want to specify the rate at which the room receives heat from the outside environment, and the rate at which the air conditioning system removes this heat when it is in the cooling state. In each of the two states, there is a dynamic equation that describes the rate of change of temperature.
Reset Maps
Zero-Crossing
Zeno Behavior
Modeling Elastic Collision
Let the velocity of the two balls be u1andu2 before the collision and v1 and v2 after the collision, respectively. Often energy loss is specified in terms of coefficient of restitution, the ratio of relative velocities before and after impact.
Chapter Highlights
Avoid Common Mistakes
Study Problems
Consider Figure 3.3, which shows “before the collision” (top) and “after the collision”. below) velocities for the collision of two masses (represented as cars). The other object is static before the collision, and both objects are fixed after the collision. a) Calculate the coefficient of restitution in this collision.
Lab: Discrete Bouncing
This creates a cycle of small calculation steps where the ball starts with a positive speed below the ground but is subjected to gravity, at the end of the cycle it has lost a small negative speed and height, and impact condition is detected and the speed is again set to positive half, and the process repeats, but the ball still loses altitude. Also compare your model to this one, and note the points you found required extra care to be able to model instantaneous bounces when you tried to build it yourself.
Project: Speed-Based Player for Ping Pong Robot
To Probe Further
Finally, we discuss the effect of implementing the controller on a digital computer and the effects of using finite representations of value (N bits to represent the value) and time (the ability to sample or activate only when the clock is ticking).
Introduction
For that purpose, we would like to have a regulator that is parameterized by our target for the plant's output. Of course, this also means that the problem can only be solved if the plant's behavior is itself an invertible function.
Feedback Control
In general, however, we would like to devise a controller that causes the plant to produce different output values depending on the value of another input that we supply. It is instructive here to note the following: If the plant's behavior can be seen as a mathematical function, then this problem can be solved by a regulator that is the inverse of the plant's function.
Proportional Feedback Control
Starting from this equation we can derive an expression for the ez terms as follows: first we note that the equation implies that. We can now consider what happens to the relation in (4.1) as we change the value of G.
Operational Amplifiers
In fact, the general behavior of the system is problematic for values of H between −1/Gand 0. For example, we may wish to determine the output of the system when z(t) = Z−t. In this case the equation becomes .
Multi-Dimensional Error and Proportional/Integral/Differential
Just using differentiation would give you a controller that tries to make the car have the same acceleration as the controller input appears to have. When we applied simple proportional feedback to a dynamic system that had an integrative effect, the key underlying differential equation had the form = −y.
Chapter Highlights
Study Problems
Assume that the gain for this operational amplifier is G. a) Write down the equation governing the relationship between VinandVout, using the gain G. b). Use the equation you wrote above to derive an equation for the ratio Vout/Vin in terms of G. c) Calculate the smallest value for the gain G that will ensure that Vout is always within 10% of Vin, that is, Vout is always between 90% and 110% of Vin. d) Insert a gain H between Vout and the negative (−) input of the op-amp.
Lab: Exploring Control
This type of controller will bring the value of x relatively quickly to get reasonably close to the target. Here the acceleration is used to drive the system and the value of the variable x is negatively driven to the target value of zero.
Project: Acceleration-Based Player for Ping Pong Robot
To Probe Further
It addresses the physical aspects of computing implementation, for both analog and digital computers, and covers quantization (and quantization levels), discretization (and sampling), the Nyquist–Shannon theorem, embedded hardware and software, and real-time systems and constraints.
Introduction
Our first impression of such a system might be that it looks like a vast under-utilization of electronics: instead of evaluating the continuum of possible values that an electronic circuit produces and seeing it as a single value, we we are only concerned with two levels (or, more precisely, two arrays of levels) representing a single binary digit. This design choice to work with "two levels" also means that we only compute with "quantized" values; ie, values that have been explicitly converted into a final, discrete quantity or representation.
Quantization
Digital circuits use analog electronic components, but unlike analog circuits, their primary function is to connect them and build circuits from them to transfer, process, and output only two voltage levels. Any value outside of these two levels is considered an indication that the system has not yet completed the transition between those two levels.
Discretization: How Fast Can Your Circuit Go?
In particular, it determines the maximum rate at which the circuit can sample (and process) external inputs. To support memory and repetition, digital computer circuits generally contain "loops" of wiring that feed several outputs as inputs to the circuit.
Detour: Boundedness of Digital Memory
As a result, determining the time it takes a circuit to produce the final, stable output can be challenging. This means that any system implemented using a digital computer can only observe a continuous signal at specific times and with a minimal gap between such samples.
Detour: From Hardware to Software—Storing Executable
The Effect of Quantization and Discretization on Stability
Abstract Modeling of Computational Effects
Note that the above equation is a very compact representation of the system we described above (proportional control). It can be seen as an idealized sensor that can read the exact position of the mass.
Modeling Quantization
For linear systems, we can more precisely quantify the effects of discretization and incorporate them accurately into the design of the entire system. To quantify the "amount of instability" in this example, we can simply look at the maximum height of the last wave at the end of the simulation.
Modeling Discretization
The most significant effect of quantization can be seen just by simulating the model and observing what happens to the variable x: with the addition of quantization, the system is no longer stable. Intuitively, we can view quantization as a kind of delay, where the system doesn't really see the value of the input until that input has changed enough since the last reading.
Detour: Discretization, Sampling Rates, and Loss
It is another matter to ensure that the computation performed on this signal does not introduce additional loss. The fact that no information is lost in the signal does not mean that performing a naive analog of the dense-time computation is the right thing to do to get corresponding behavior.
The Effects of Quantization and Discretization Easily
The first is that, in the context of building control systems, it just tells us that no information is lost in sampling. There are many cases where, based on different types of assumptions about the signal, limited sampling can lead to meaningful or complete information about the signal.
Chapter Highlights
Study Problems
Lab: Stability Exercises
Also write down a description of the effect you expect this quantization to have on the overall behavior of the system. In general, however, these can be directly proportional to the cost of computing resources required to perform the calculation.
Project: Quantization and Discretization
To Probe Further
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Introduction
Another key complication to consider is that mapping changes such as velocity or acceleration from Cartesian to Polar depend not only on the magnitude of the change itself, but also on the absolute position. A key analytical skill to keep in mind here is to look for patterns in the results of each differentiation step and replace each repeated variable with a single variable seen previously.
Coordinate Transformation
In particular, by being careful with independent and dependent variables and using the chain rule, we can compute closed representations of the mapping of changes (or derivatives) from one coordinate system to another. Now we can consider the conversion in the opposite direction, namely going from 2D Cartesian to 2D polar coordinates.
Chapter Highlights
Study Problems
Consider the simple two-link system shown in Figure 6.2. a) Write the formula for (x, y) in terms of the other variables in the diagram. We further assume that the end of the second wing is a point at (x, y) with respect to the (x0, y0) coordinate system.
Lab: Coordinate Transformations
There are generally many options for how to do this, but imagine we have to build for separate controllers, one for the angle of the arm and one for the extension of the arm. Once you've calculated the new controller signals, replace the zero on the lines that say "Fix me".
Project: Spherical-Actuation for Ping Pong Robot
As part of this activity you should reflect on how you derived the correct coordinate transformation. Since this is a project, it is also helpful to reflect on how you developed your player and what you learned from this experience.
To Probe Further
Consider whether you think it was helpful to ignore these effects initially or whether it would have been better to include them from the start.
Selected Topics
The Role of Game Theory in CPS Design
With the increasing pace of CPS innovation, the number of interacting systems is also increasing. Predicting the behavior of interacting autonomous systems can be challenging, and increasing the number of such systems only makes the situation more so.
Games, Players, Strategies, Utilities, and Independent
Rationality, Independence and Strictly Dominant (or Dominated)
- The Independence Pattern
- The Cost of Lacking Communication and Trust Can Be
The most elementary utility pattern is independence, which is the situation where one player's utility for a strategy is better than the other's and is independent of the other player's choice. By hiding the other player's (irrelevant) utility, it makes clear to each player that strategy B is always positive.
Coordination, Intelligence, and Nash Equilibrium
- The Coordination Pattern
- Nash Equilibrium
- Determining the Nash Equilibrium
- Eliminating Strictly Dominated Strategies Preserves
So if both players reasoned rationally, considering the other player's utilities as options, the first player would choose (C) and the second would play (C). In addition, eliminating a choice for one game can reveal other dominated strategies to the other player (since they are also intelligent and can determine for themselves that the first player would never play that strategy).
Competitiveness, Privacy, Mixed Strategies
- Mixed Strategy Games
- Selecting a Mixed Strategy (or, Mixed Strategy Nash
To do this, we (and each player) will need to analyze the other player's expected payoff. Indeed, even in this game, player 2's optimal strategy will not be an even split.
Chapter Highlights
The uneven distribution of probabilities in this case can be explained as follows: if player 2 played an even split between (A) and (B), player 1 would eventually notice and start playing (A) more often because, on average, it gives a higher payoff than playing (B). In contrast, the probabilities we calculated for player 2 result in player 1's payoffs and give player 1 an incentive to maximize his expected payoff and reduce the chance that the other player can predict the next play.
Study Problems
To Probe Further
128 7 Game Theory Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits the use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give proper credit to the original author(s) and source, provide a link to the Creative Commons license, and indicate whether changes have been made. In this chapter, we turn to the second question, and in particular, to fundamental concerns of what constitutes communication, what are the hard limits of telecommunications, how such limits arise, and how we can model their effects.
Communication, Certainty, Uncertainty, and Belief
When we are certain of a value, we either believe it to be true or we believe it to be false. It is the notion of uncertainty, and consequently the notion of information, that is usually handled by methods of analysis based on sets and intervals, whether applied to numerical calculations or programs.
Messages: From Information to Representation
For example, the message "Tom and Jerry are here" contains information for a person who is unaware of their presence. They are here They are Tom & Jerry They are Tom & Jerry Tom & Jerry are here Tom & Jerry are here.
Belief, Knowledge, and Truth
- Broader Implications
Tom and Jerry are here φ Tom and Jerry are here Tom and Jerry are here Tom and Jerry are here Tom and Jerry are here Tom and Jerry are here φ They are here They are Tom & Jerry They are Tom & Jerry φ Tom & Jerry are here Tom & Jerry are here.
Carrier Signal, Medium, and Link
The following three modes generally require a medium, but it is worth noting that this is not always necessary. The last two are examples of more rare forms of communication, but it is helpful to be aware of such examples for both cooperative and non-cooperative situations.
Link Characteristics
- Latency
- Bandwidth
- Reliability
Assuming the recipient of the ice cream gets the information that the sender likes it, this is a like/neutral signal. So that said information can be represented by two ice cream/one ice cream/no ice cream.
Fundamental Limits from Physics
Limits Due to Component Dynamics
- Electrical Signal Transmission
- Variability in Component Parameters
- Light and Radio Transmission
The second equation models the way in which the voltage at the signal's target (represented by the capacitor) is a function of the current being transmitted and the elapsed time. The most important thing about running this simulation is that it takes some time for the voltage at the target, Q/R, which is simply Q in this case, to reach the value of the source.
Limits Due to Noise
In contrast to electrical signals, light and radio wave transmission can have an advantage in terms of maximum bandwidth and latency. Electromagnetics in free space can have frequencies in the THz, therefore bandwidth can in principle also approach these frequencies.
Limits Due to Energy Dissipation
A wide variety of methods are used to enable the recipient to verify the integrity of the message received. At the very least, it allows the recipient to ignore the message, but more often it allows to request a re-send.
Other Sources of Limitations
However, clock speeds must often match the timing needs of the most complex components in the system. In simple designs this may need to be matched or timed to the clock of the transmitting or receiving device.
Chapter Highlights
Study Problems
Assume that the carrier frequency is 100 Hz. b) Assume that the target knows the exact transmission frequency, but not the phase. Model the target and explain the mechanism for determining the phase for the carrier signal.
To Probe Further
Essentially, the signal and carrier are multiplied to generate the transmitted signal. a) Model a source and signal generation for such a communication, assuming that the transmission channel is perfect. The terms sensing and activation are used to refer to obtaining information about the world and to influencing physical objects, respectively.
Everyday Input and Output
In cyber-physical systems, an interesting aspect of exploring sensing and actuation is that it provides us with a natural opportunity to learn more about how computational components work today, and in particular, which can be directly realized using circuits of based on semiconductors, and which they require other intermediate steps to realize. For many microprocessors, all that would be required would be to connect an LED followed (in series) with a small resistor to the wire carrying the digital signal we wish to observe.
Symmetry: LEDs and Photo-Voltaic Cells
- Diodes
- The Photo-Voltaic Effect
- Transistors and Amplifiers
In abundance of light and with suitably configured circuits, such cells can also be used to harvest electrical energy from this light. For example, a small change in the voltage (or current) supplied by the middle terminal can have a significant effect on the current that can flow across the other two terminals.
Analog-to-Digital Conversion (ADC)
Depending on the application, we may choose to build circuits that realize other rounding operations, such as those that calculate an upper bound or the nearest integer value. Even if we have more bits or if we have a smaller range of input values, we can let each integer represent a fraction.
Digital-to-Analog Conversion (DAC)
Sensing Temperature
Sensing Position
A wide range of techniques can be used to measure position remotely, i.e. without physical connection. Depending on the environment, one or more cameras may be used to provide positional information.
Actuating Mechanical Systems
It is a simple and reliable way to measure relative position, and can be used in both linear and angular configurations. In outdoor environments, systems such as the Global Positioning System (GPS) or LIDAR can be used, depending on the requirements of the situation.
Chapter Highlights
Study Problems
To Probe Further
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution, and reproduction in any medium or form. , as long as you give proper credit to the original author(s) and source, provide a link to the Creative Commons license, and indicate if changes have been made.
Background
The Acumen Environment and Graphical User Interface
Basic Structure of An Acumen Model
Model Parameters and the “Initially” and “Always” Sections
Model Instantiation
Expressions
- Variable Names
- Literals
- Vector and Vector Generators
- Matrices
- Summations
In addition, they can be generated by specifying a start value, step size and end value. Theif statement can be omitted when there is no filtering, that is, when its condition is always true.
Formulae
- Continuous Formulae
- If Formulae
- Match Formulae
- Discrete Formulae
- Foreach Formulae
- Collections of Formulae
It can be seen as a generalization of an if formula, enabling different formulas under several different cases depending on the value of a particular expression that we are matching on. A discrete formula has a left side that must be the next value of either a variable or the derivative of a variable, and a right side that can be any expression.
How a Model Is Simulated: Order of Evaluation
For each model instance, each parent's struct actions are executed first, and then all children's struct actions are executed.
Visualization Using the _3D Panel
- Colors
- Transparency
- Coordinate System
- Text
- Box
- Cylinders
- Cone
- Spheres
- OBJ Mesh Objects
- Default Values
- Composites
- Shapes, Their Parameters, and Their Default Values
- Animation = Dynamic _3D Values
- Manual Control of the View of the _3D Scene
- In-model Control of the View of the _3D Scene
- Camera View
Note that unlike the case of the color illustration above, the triples here are coordinates in three-dimensional space and not color intensities. Rotations are specified as a triple of angles (in radians) about the center of the object and are applied in the order described in Figure A.6.
Built-In Functions
Function Declarations
Operator Precedence
Simulator Settings
Command Line Parameters
A good practice is to use a test? for the parameter names (as shown in the example above) to make sure they were passed by the command line call, then use the default values if they weren't. An added advantage of doing this is that the models can be easily run in both command line and interactive mode.
Print to Standard Output (stdout) or Console
BNF of Acumen
