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Additional Exercises The following exercises cover some of the topics introduced in this chapter. Exercises marked with a * appear in the printed text. Exercise: Exploring the dynamics of a rolling mill Exercise: Popular articles about control Frequently Asked Questions Question: Can a control system include a human operator as a component? Question: How are stability, performance and robustness different? Question: How can I go from a continuous time linear ODE to a discrete time representation? Question: How can we tell from the phase plots if the system is oscillating? Question: How do you know when your model is sufficiently complex? Question: In the predator prey example, where is the fox birth rate term? Question: What is "closed form"? Question: What is a state? How does one determine what is a state and what is not? Question: What is a stochastic system? Question: What is the advantage of having a model? Question: What is the definition of a system? Question: Why does the effective service rate f(x) go to zero when x = 0 in the example on queuing systems? Question: Why isn't there a term for the rabbit death rate besides being killed by the foxes? Errata

Python Code The following Python scripts are available for producing figures that appear in this chapter. Figure 1.11: A feedback system for controlling the velocity of a vehicle, Figure 1.18: Air–fuel controller based on selectors, Figure 2.11: Response to a step change in the reference signal for a system with a PI controller having two degrees of freedom, Figure 2.12: Responses of a static nonlinear system, Figure 2.14: Responses of the systems with integral feedback, Figure 2.19: System with positive feedback and saturation, Figure 2.8: Step responses for a first-order, closed loop system with proportional and PI control, Figure 2.9: Responses to a unit step change in the reference signal for different values of the design parameters, Figure 3.11: Simulation of the forced spring–mass system with different simulation time constants, Figure 3.12: Frequency response computed by measuring the response of individual sinusoids, Figure 3.22: Queuing dynamics, Figure 3.24: Consensus protocols for sensor networks, Figure 3.26: The repressilator genetic regulatory network, Figure 3.28: Response of a neuron to a current input, Figure 3.2: Illustration of a state model, Figure 3.4: Input/output response of a linear system, Figure 3.8: Discrete-time simulation of the predator–prey model, Figure 4.12: Internet congestion control, Figure 4.13: Internet congestion control for N identical sources across a single link, Figure 4.20: Simulation of the predator-prey system, Figure 4.2: Torque curves for typical car engine, Figure 4.3: Car with cruise control encountering a sloping road, Figure 5.10: Phase portraits for a congestion control protocol running with N = 60 identical source computers, Figure 5.11: Comparison between phase portraits for a nonlinear system and its linearization, Figure 5.12: Stability analysis for a tanker, Figure 5.13: Solution curves for a stable limit cycle, Figure 5.15: Stability of a genetic switch, Figure 5.16: Dynamics of a genetic switch, Figure 5.17: Stabilized inverted pendulum, Figure 5.1: Response of a damped oscillator, Figure 5.3: Phase portraits, Figure 5.4: Equilibrium points for an inverted pendulum, Figure 5.5: Phase portrait and time domain simulation for a system with a limit cycle, Figure 5.6: Illustration of Lyapunov’s concept of a stable solution, Figure 5.7: Phase portrait and time domain simulation for a system with a single stable equilibrium point, Figure 5.8: Phase portrait and time domain simulation for a system with a single asymptotically stable equilibrium point, Figure 5.9: Phase portrait and time domain simulation for a system with a single unstable equilibrium point, Figure 6.14: Linear versus nonlinear response for a vehicle with PI cruise control, Figure 6.1: Superposition of homogeneous and particular solutions, Figure 6.5: Modes for a second-order system with real eigenvalues, Figure 8.13: Vehicle steering using gain scheduling See the software page for more information on how to run these scripts. Additional Information