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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 5.18: Bifurcation analysis of the predator–prey system Figure 5.19: Stability plots for a bicycle moving at constant velocity Figure 5.1: Response of a damped oscillator Figure 5.21: Simulation of noise cancellation 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 6.10: Response of a compartment model to a constant drug infusion Figure 6.12: Active band-pass filter Figure 6.13: AFM frequency response 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 7.6: State feedback control of a steering system Figure 8.13: Vehicle steering using gain scheduling See the software page for more information on how to run these scripts. Additional Information