# Applications and Examples

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AM05 has a set of examples and exercises that form a thread through the book and attempt to describe the key concepts in terms of applications that are familiar to different audiences.

## Major Examples

Major examples appear in most chapters in the book to illustrate key concepts. All major examples are introducded in Chapter 2 - System Modeling or Chapter 3 - Examples and appear in later chapters as worked examples and/or exercises. Each major example has a page on the wiki that provides a model for that example to allow numerical exercises.

### Cruise Control

Description: Longitudinal control of vehicle dynamics. Simplest versions include only the mass of the car and damping (linear), but more complex versions include engine dynamics and saturation. This example is first introduced in Chapter 1 and then used through the text to illustrate various techniques.

Materials:

Textbook Usage:

 Ch Body Homework Wiki 1 Use as example of uncertainty management Work out solutions for linear case and demonstrate robustness Explore nonlinear model in simulation 2 Defined in detail, as separate section Implement in simulation and explore robustness + performance improvements Simulation model available 3 Not used Phase plot Numerical phase plot, using SIMULINK (too hard?)

### Balance Systems

Description: This application includes a class of systems in which motion at the base of a mechanical system is used to stabilize (balance) a mass above the base. When it is initially introduced, the Segway transportation system and a rocket are used as examples. A simplified version of the problem, motivated through the example of a rocket, allows a direct acceleration at the base of a pendulum.

 Ch Body Homework Wiki 1 Not used Not used Not used 2 Balance systems described and equations for two main cases (with and without base position) are given TBD TBD: need to put code for examples + MATLAB model

### Queing Systems

Description: This class of examples covers situations in computing and information systems in which data is queued for later usage. Specific applications include congestion control on the internet and queue processing for databases (I think). The basic concept here is the management of the flow of information on the queue.

 Ch Body Homework Wiki 1 Not used Not used Not used 2 Defined in a separate section; currently very short - needs to be expaned None yet Nothing yet 3 Not used Not used Not used

### Active Filters

Description: This example covers active filters made using op amps. The goal in these examples is two-fold: to connect to electrical engineering and physics students and to provide some practical examples for students in all disciplines.

 Ch Body Homework Wiki 1 Not used Not used Not used 2 Defined in a separate section Nothing yet Nothing yet 3 Not used Not used Not used

### Pharmacokinetics

 Ch Body Homework Wiki 1 Not used Not used Not used 2 Defined in a separate section; currently very short Nothing yet Nothing yet 3 Not used Not used Not used

### Population Dynamics

Description: This class of systems includes various situations in which one wants to analyze and control population dynamics. The primary example is the predator-prey system, but other variations are possible. We may put in some "one child" analysis, but I am worried that this might be a bit political.

 Ch Body Homework Wiki 1 Not used Not used Not used 2 Discrete time version described in details Nothing yet Several FAQs; need to put a copy of the model 3 Introduce continuous time version of the model; use as major example of phase plots TBD Nothing yet; need to put cts version of model online

## Canonical Examples

Canonical examples are not attached to any specific application but cover common classes of systems and/or systems with meaningful structure.

### Second Order Systems

Description: Second order systems are used in several places in the text, especially the frequency domain portions of the book, to illustrate control techniques. A number of special cases are identified in Chapter 2 and used through the remainder of the text as the simplest examples. They are motivated starting in Chapter 7 as describing the dominant behavior of a system.

Special Cases:

• Double integrator - two poles at the origin
• Damped oscillator - stable, complex pair of eigenvalues
• Inverted pendulum - unstable, complex pair of eigenvalues
 Ch Body Homework Wiki 1 Not used Not used Not used 2 Not used Not used Not used 3 Not used Not used Not used 4 Introduced in their own section and the various special cases defined Show that some of the example systems from Ch 2 are equivalent to the second order special cases, via change of variables Not used

### Discrete Time Systems

Description: Discrete time systems are introduced in Chapter 2 and used in advanced sections, examples and exercises through the remainder of the text (but mainly in the state space portions of the book). In particular, these are used in exercises to demonstrate how the concepts from state space control can be easily extended to discrete time systems.

 Ch Body Homework Wiki 1 Not used Not used Not used 2 Defined in a separate section, including examples (predator-prey, queing) Conversion of continuous time model to discrete time model via approximate differentiation Not used

### Reachable Canonical Form

 Ch Body Homework Wiki 1 Not used Not used Not used 2 Introduced briefly in converting nth order system into state space form Nothing yet Nothing yet 3 Not used Not used Not used