Solutions

This page contains a list of the exercises in the text, including supplemental problems. It is mainly intended for use by people who are generating solutions to the exercises. Instructors can also use this page to see what problems have solutions available.

Note:: if you are looking for solutions to the exercises, you won't find them here. Solutions to exercises are available to course instructors by contacting Princeton University Press.

Guide to entries

Each entry in the table below contains the following information (the two-letter code in parenthesis is used in the table entry macros):

• Exercise number (ex): the number of the exercise in the solutions manual. Items marked with a * appear in the printed version of the text.
• Directory (di): the name of the subdirectory where the exercise appears. This does not appear in the table directly, but is used to create the link to the exercise.
• Title (ti): a short descriptive title for the exercise.
• Filename (fi): the base filename used for the exercise. Clicking on the link will take you to the LaTeX source code for the exercise (requires login).
• Solution status (st): current status of the solutions. The following abbreviations are used:
  • C (complete) - complete solution is available
  • A (available) - solution is available, but not yet verified and/or edited
  • P (partial) - partial solution is available (some parts, missing figures/code, etc)
  • I (in progress) - solution is being written, but is not yet available
  • U (unavailable) or blank - no solution is currently available
  • X (not planned) - no solution is planned
• Author (au): initials for the author of the solutions
• Comment (co): additional comments.

How to sign up to write a solution for an exercise:

1. Find an exercise whose solution status is either unavailable (U) or partial (P)
2. Edit the table and add your name to the 're' entry in the table
3. Add the page to your watch list so that you can see when there is a change in status
4. Wait until you are assigned a set of problems to work on (status will be changed to 'I', with your initials listed in the author column.

Chapter 1 - Introduction

Num	Title	Filename	Stat	Auth	Comments
1.1*	Vestibulo-occular reflex	eyemotion	C	KJA
1.2*	Everyday examples of feedback systems	fbkexamps	C	Misc
1.3*	Human balance system	balance-human	C	CL
1.4*	Exploring the performance of a cruise controller	cruise-redesign	C	CL
1.5*	Integral action	integral	C	KJA
1.6*	Popular press articles	nytexamps	C	RMM
1.7	Milk supply chain	milksupply	P	KJA	Figure missing
1.8	Exploring the performance of a cruise controller (supplemental)	cruise-mlintro	A	CL	Cleaned up preliminary solution with model from 3.1.
1.9	MATLAB ball and beam	ballbeam-mlintro	P	Misc	Preliminary solution available; could be cleaned up
1.10	Web search: voltage clamp	voltageclamp			Give short intro plus web links
1.11	Web search: haptics and force feedback	haptics			Give short intro plus web links
1.12	Anti-rollover control	rollover			Give short summary of regulations

Chapter 2 - System Modeling

Num	Title	Filename	Stat	Auth	Comments
2.1*	Chain of integrators form	chainofint	C	SH
2.2*	Inverted pendulum	balance-bal2inv	C	RMM
2.3*	Disrete-time dynamics	discrete-solutions	C	SH
2.4*	Keynesian economics	discrete-keynes	C	SH
2.5*	Least squares system identification	sysid-leastsq	C	SH
2.6*	Normalized oscillator dynamics	oscillator	C	SH
2.7*	Electric generator	powergrid	A	RMM
2.8*	Admission control for a queue	queue-admcontrol	A	KJA
2.9*	Biological switch	biocircuits-switchmod	A	RMM
2.10*	Motor drive	dcmotor-modeling	C	SH
2.11	Uncertainty lemon	lemon	P
2.12	Frequency response	freqresp
2.13	Eye and head motion	occular-blockdiag	P
2.14	Second-order system identification	sysid-logdec
2.15	Insect vision	flyvision	P
2.16	State space model for balance systems	balance-statespace
2.17		fingerflame	P
2.18	Consensus and invariants	compsys-invariants	P
2.19	Electric motor	electricmotor	P
2.20		exothermic
2.21		caffeine	P
2.22		ballbeam-modeling	P
2.23		pvtol2invpend
2.24		flyflight
2.25		pendulum	P

Chapter 3 - Examples

Num	Title	Filename	Stat	Auth	Comments
3.1*	Cruise control	cruise-hillsim	A	CL	check figures
3.2*	Bicycle dynamics	bicycle-statespace	C	RMM
3.3*	Bicycle steering	bicycle-steering	C	RMM
3.4*	Operational amplifier circuit	opamp-twopole	A	SH
3.5*	Operational amplifier oscillator	opamp-osc	A	SH
3.6*	Congestion control using RED~\cite{Low+02:infocom}	congctrl-red	P		solution consists of reference to research paper
3.7*	Atomic force microscope with piezo tube	afm-preload	A	SH
3.8*	Drug administration	drugadmin-alcohol	C
3.9*	Population dynamics	logistic	A	SH
3.10*	Fisheries management	fishery-modeling	A
3.11	Cruise control with powertrain dynamics	powertrain	P

Chapter 4 - Dynamic Behavior

Num	Title	Filename	Stat	Auth	Comments
4.1*	Time-invariant systems	timeshift	A	SH
4.2*	Flow in a tank	tank-modeling	A	SH
4.3*	Cruise control	cruise-phaseplot	A	CL	Solution updated to match dynamics in text
4.4*	Lyapunov functions	lyap-secord	A	SH	V_2(x) has been modified
4.5*	Damped spring--mass system	lyap-oscillator	C
4.6*	Electric generator	powergrid	A
4.7*	Lyapunov equation	lyapqen	A
4.8*	Congestion control	congctrl-lyapstab	A
4.9*	Swinging up a pendulum	invpend-swingup	A
4.10*	Root locus diagram	rootlocus	A
4.11*	Discrete-time Lyapunov function	discrete-lyapunov	P		Sketch only
4.12*	Operational amplifier oscillator	opamp-oscnl	A	SH
4.13*	Self-activating genetic circuit	biocircuits-posfbk	A
4.14*	Diagonal systems	modal-form	P		Solution doesn't contain same parts as problem
4.15*	Furuta pendulum	furuta	A
4.16*	Routh-Hurwitz criterion	routh	A
4.17	Exponential stability	expstab
4.18	Population dynamics	popdyn-logistic	P
4.19	Predator prey	predprey-limit	P
4.20	Windup protection by conditional integration	windup-condit	P
4.21	Gain scheduling, stability	gainsched-stabiility
4.22	Spike generation in neurons	fitzhugh-nagumo	P
4.23	Consensus dynamics	compsys-consensus
4.24	Steering dynamics of a ship	tanker-dynamics	P
4.25	Pitchfork bifurcation	pitchfork
4.26	Stability examples	stabexmp	P
4.27	Lyapunov stability for cruise control	cruise-lyapunov	A	CL	Please add original author of solution to tex file
4.28	Lyapunov stability examples	lyapexmp	P

Chapter 5 - Linear Systems

Num	Title	Filename	Stat	Auth	Comments
5.1*	Response to the derivative of a signal	dersys	A	SH
5.2*	Impulse response and convolution	impulse-response	A	SH
5.3*	Pulse response for a compartment model	compartment-pulse	A	SH
5.4*	Matrix exponential for second-order system	matrixexp	P
5.5*	Lyapunov function for a linear system	lyap-linear	A	SH
5.6*	Nondiagonal Jordan form	jordan-nontrivial	A
5.7*	Rise time for a first-order system	risetime-firstord	A	SH
5.8*	Discrete-time systems	discrete-linsys	A
5.9*	Keynesian economics	discrete-keynes	A	SH
5.10*		linearization-scalar	A
5.11*	Transcriptional regulation	biocircuits-negfbk	A
5.12	Normalized coordinates for second-order system	transform-oscillator
5.13	Matrix exponential for Jordan form	matexp-jordan3x3
5.14	Solution of a second-order system	conveq-secord
5.15	Monotone step response	conveq-monotone
5.16		purecosine	P
5.17		linsys-diffeq	P
5.18		linexmp-coupled	P
5.19	State variables in compartment models	compartment-states
5.20		queue-linsys	P
5.21	Keynes model in continuous time	keynes-ode
5.22		cartpend	P
5.23		linsys-sinresp	P

Chapter 6 - State Feedback

Num	Title	Filename	Stat	Auth	Comments
6.1*	Double integrator	reachable-doubleint	A	SH
6.2*	Reachability from nonzero initial state	reachable-nonzeroic	A	SH
6.3*	Unreachable systems	unreachable	A	SH
6.4*	Integral feedback for rejecting constant disturbances	integral-noiserej	A
6.5*	Rear-steered bicycle	bicycle-rearsteer	A
6.6*	Characteristic polynomial for reachable canonical form	reachable-form	A	SH
6.7*	Reachability matrix for reachable canonical form	inverse-reachmat	A	SH
6.8*	Non-maintainable equilibria	pendcart-noref	A	SH
6.9*	Eigenvalue assignment for unreachable system	unreachable-assign	A	SH
6.10*	Cayley--Hamilton theorem	cayley-hamilton	A
6.11*	Motor drive	dcmotor-statefbk	A
6.12*	Whipple bicycle model	bicycle-whipple	A
6.13*	Atomic force microscope	afm-scalelqr	A
6.14*	Step response for a second order system	secord-zero	A
6.15*	Bryson's rule	brysons-rule	C
6.16		integral-refgain
6.17	Unreachable compartment model	compartment-nonreach.pst	P
6.18		refgain-direct
6.19		cruise-norefgain

Chapter 7 - Output Feedback

Num	Title	Filename	Stat	Auth	Comments
7.1*	Coordinate transformations	coordxform	A	SH
7.2*		unobservable	A	SH
7.3*	Observable canonical form	obsform-transform	A	SH
7.4*	Bicycle dynamics	bicycle-observability	A
7.5*	Integral action	integral-action	A
7.6*	Vectored thrust aircraft	pvtol-ssctrl	A
7.7*	Uniqueness of observers	observer-assign	A
7.8*	Observers using differentiation	diff	A	SH
7.9*	Observer for Teorell's compartment model	compartment-teorellobs	A
7.10*	Observer design for motor drive	dcmotor-observer	A
7.11*	Feedforward design for motor drive	dcmotor-feedforward	A
7.12*	Whipple bicycle model	bicycle-whipple	A
7.13*	Discrete-time random walk	kalman-randomwalk	A
7.14*	Kalman decomposition	kalman-decomp-2x2	A	SH
7.15		secord-observer	P
7.16	Balance system	balance-observable
7.17	Balance system with biased measurement	balance-statectrl
7.18	Duality	observer-duality
7.19	Balance system	cartpend-observer
7.20	Trajectory generation	trajgen-scalar
7.21	Selection of eigenvalues	compartment-poleplace
7.22	Kalman filter for scalar ODE	kalman-scalar	P

Chapter 8 - Transfer Functions

Num	Title	Filename	Stat	Auth	Comments
8.1*	Steady state response to a sinusoid	sininput	A	SH
8.2*	Modal response for a first order system	modalinput	A	SH
8.3*	Inverted pendulum	invpend-xferfcn	A	SH
8.4*	Solutions corresponding to poles and zeros	pzsolns	A	SH
8.5*	Operational amplifier	opamp-pictrl	A	SH
8.6*	Transfer function for state space system	ssxferfcn	A	SH
8.7*	Kalman decomposition	kalmandecomp	A
8.8*	Transfer functions for control systems	ctrlxferfcns	A	SH
8.9*	Bode plot for a simple zero	bode-zero	A	SH
8.10*	Vectored thrust aircraft	pvtol-lateraltf	A	SH
8.11*	Common poles	missing	A
8.12*	Congestion control	congctrl-xferfcns	P
8.13*	Inverted pendulum with PD control	invpend-pzcancel	A	SH
8.14*	Vehicle suspension	vehicle-quartercar	A	SH
8.15*	Vibration absorber	vibdamper	A	SH
8.16		freqexamps	P
8.17		model-simplification
8.18		cruise-pictrl	P
8.19		dcmotor-xferfcn
8.20		xferfcn-match
8.21		heat-propogation	P
8.22		queue-xferfcn	P
8.23		lq-scalar

Chapter 9 - Frequency Domain Analysis

Num	Title	Filename	Stat	Auth	Comments
9.1*	Operational amplifier	opamp-unitygain	A
9.2*	Atomic force microscope	afm-tapping	A
9.3*	Heat conduction	heatcond-nyquist	P
9.4*	Vectored thrust aircraft	pvtol-looptf	A
9.5*	Vehicle steering	steering-margins	A
9.6*	Stability margins for second-order systems	secord-margins	A
9.7*	Congestion control in overload conditions	congctrl-overload	A
9.8*	Bode's formula	bode-weight	A
9.9*	Pad\'e approximation to a time delay	pade-approx	A	SH	'a' is not defined
9.10*	Inverse response	inverse-response	A
9.11*	Describing function analysis	descfcn-hystrelay	A	SH
9.12	Right half-plane pole	nyquist-rhp	P
9.13	Third-order system	nyquist-thirdorder
9.14		bodenyq	P
9.15	Congestion control using TCP/Reno	congctrl-margins
9.16		opamp-routh	P
9.17		cruise-nyquist-alt	P
9.18	Limitations	invpend-limits
9.19	Limitations	invpend-delay
9.20	Delay differential equations	delayde	P
9.21	Cruise control design	cruise-pictrl	P	CL	Model may need updating; check for duplication with similar exercise in previous chapter

Chapter 10 - PID Control

Num	Title	Filename	Stat	Auth	Comments
10.1*	Ideal PID controllers	pid-ideal	A	SH
10.2*	PI for second order process	secord-pid	A	SH
10.3*	Integral control for two pole system	integral-twopole	A
10.4*	Ziegler-Nichols tuning	zntuning-delint	A
10.5*	Vehicle steering	steering-pi	A
10.6*	Congestion control	congctrl-pituning	A
10.7*	Motor drive	dcmotor-pdctrl	A
10.8*	Motor drive with first order filter	derfilter	A
10.9*	Tuning rules	zntuning-examples	A
10.10*	Windup and anti-windup	antiwindup-intpi	A	SH
10.11*	Windup protection by conditional integration	antiwindup-condint	A
10.12		pidmethods	P
10.13		pidexmps	P
10.14	Compartment model	compartment-monotone
10.15		plasma-oven	P
10.16		pirlocus	P
10.17		pvtol-pid	P
10.18	Ball and beam	ballbeam-pid
10.19		balance-pid	P

Chapter 11 - Frequency Domain Synthesis

Num	Title	Filename	Stat	Auth	Comments
11.1*	Gang of Four transfer functions	gangoffour	A
11.2*	Gang of Four for system with RHP pole, proportional control	rhppole-cancel	A		Missing figures
11.3*	Gang of Four for H infinity structure	gangoffour-hinfstruc	A
11.4*	Feedforward compensator for spring-mass system	springmass-feedforward	A
11.5*	Sensitivity of feedback and feedforward	sensitivity-gyr	A
11.6*	Equivalence of controllers with two degrees of freedom	twodof-equiv	A
11.7*	Disturbance attenuation	sensitivity-atten	A
11.8*	Disturbance reduction through feedback	sensitivity-twopole	A
11.9*	High frequency measurement noise approximation	noiseapprox	A
11.10*	Attenuation of low-frequency sinusoidal disturbances	sinusoid-atten	A
11.11*	Lead compensation	lead-phasecalc	A
11.12*	RHP pole/zero pair	rhppzpair	A
11.13*	Sensitivity bounds for RHP pole/zero pair	sensitivity-pzdel	A
11.14*	Phase margin formulas	sensitivity-pm	A
11.15*	Stabilization of an inverted pendulum with visual feedback	balance-visual	A
11.16*	Rear-steered bicycle	bicycle-rearsteer	A
11.17*		bode-compsens	A
11.18	Rise-time/bandwidth product	risetime-bandwidth	P
11.19		laplace-smalltime	P
11.20		znlimits	P
11.21	Pole in the right half-plane and time delay	rhppole-delay	P
11.22	Cancellation of unstable process zero	steering-reverse
11.23	Limitations on achievable phase lag	rhppole-limitations	P
11.24		pvtol-gangoffour
11.25	Performance characteristics of second-order systems	secord-perf	P
11.26		secord-rhpz	P
11.27	Integral of complementary sensitivity	bode-compsens-alt	P
11.28	Bode's integral	bode-compsens-norolloff	P
11.29	Lead compensator for an insect flight control system	flylead	P
11.30	Control of a magnetic levitation system	maglev	P
11.31		pvtol-gangof4
11.32	Relationship between gain and sensitivity crossover frequencies	sensitivity-margins	P	KJA

Chapter 12 - Robust Performance

Num	Title	Filename	Stat	Auth	Comments
12.1*	Uncertainty between two first-order systems	uncertainty-onepole	A
12.2*	Uncertainty between two second-order systems	uncertainty-twopole	A
12.3*	Difference in sensitivity functions	sensitivity-difference	A
12.4*	The Riemann sphere	vinnecombe-sphere	A
12.5*	Stability margins	sensitivity-maxpm	A
12.6		idealbode-margins	A
12.7*	Fractional loop transfer function	idealbode-fracsys	A
12.8*	Smith predictor	smith-predictor	A
12.9*	Ideal delay compensator	delay-compensation	A
12.10*	Vehicle steering	steering-nyquist	A
12.11*	Design with dominant pole specification	design-fourthord	A
12.12*	AFM nanopositioning system	afm-robustpid	A
12.13*	$H_\infty$ control	hinf-singval	A
12.14*	Vinnicombe metric computation	hinf-vinnecombe	A		Solution consists of a citation
12.15*	Robustness of an observer-based design	robust-statespace	A		Solution is very long
12.16*	Robustness using the Nyquist criterion	robust-nyquist	P		Missing figures
12.17	Robustness inequalities	robustness-conditions
12.18		vinnecombe-nonunity
12.19		vinnecombe-onepole
12.20	Disk drive tracking	missing