Controls/Concept List Spring 2026

Lecture 1 - 1/8 - Introduction

• Administrivia at Canvas
• Definition of control systems
• Purposes of control systems
• Transient and steady state response
• Block diagrams for open-loop and closed-loop systems
• Quick refresher on linear and time-invariant systems
• Basic signals $$u(t), e^{-at}u(t), r(t)=t\,u(t), q(t)=\frac{1}{2}t^2\,u(t), \cos(\omega t+\phi), \delta(t)$$

Lecture 2 - 1/13 - LTI and Laplace

• Symbol usage in Nise
• Recap of derivation of convolution
• Complex Numbers review
• Recap of phasor analysis
• Derivation (and limitations) of Fourier Transform
• Derivation (and limitations) of Laplace Transform
• Basic Unilateral Laplace Transform pairs (with ROC) and properties
• MOAT

Lecture 3 - 1/15 - Electrical Systems

• Impedance
• Ideal Op-Amp Assumptions
• Basic inverting and non-inverting op-amp circuits
• Inverting summation amplifier
• Mesh Current Method
  • MCM by inspection if no controlled sources or current sources

Lecture 4 - 1/20 - Translational and Rotational Systems

• Impedance for mechanical systems (translational and rotational)
• Free body diagram
• Equations by inspection using impedances

Lecture 5 - 1/22 - Gears

• Ideal gear assumptions (i.e. no slip condition)
• Rack and pinion gears
  • Note that rack may have mass and translational damping and pinion may have inertia and rotational damping
  • From rotation to translation
    • Divide rotational impedances by $$r^2$$ to get equivalent translational impedances
    • Divide torques by $$r$$ to get equivalent forces
    • Multiply angles by $$r$$ to get equivalent translations
  • From rotation to translation
    • Multiply translational impedances by $$r^2$$ to get equivalent rotational impedances
    • Multiply forces by $$r$$ to get equivalent torques
    • Divide translations by $$r$$ to get equivalent angles
• Rotational gears
  • Note that either gear/both gears may have inertia and rotational damping
  • Translating from gear 2's frame of reference to gear 1's frame of reference
    • Create a constant $$\gamma_{21}=r_1/r_2$$ that will assist in translating from frame 2 to frame 1
    • Multiply impedances in reference 2 by $$\gamma_{21}^2$$ to get equivalent impedances in reference 1
    • Multiply torques in reference 2 by $$\gamma_{21}$$ to get equivalent torques in reference 1
    • Divide angular measurements in reference 2 by $$\gamma_{21}$$ to get equivalent angular measurements in reference 1

Lecture 6 - 1/27 - Motors

• Motor properties $$K_t$$, $$K_b$$, $$R_a$$, $$J_a$$, $$D_a$$
• Motor Equation
• Impulse and step response of motor
• Torque-speed curve and values

Lecture 7 - 1/29 - Transient Characteristics

• First order systems: settling time and rise time based on pole location
• Second-order systems: characteristics depend on dominant (right-most) pole (or poles)
  • If overdamped, right-most pole treated like a first-order system
  • If underdamped, can find settling time and rise time along with %overshoot, peak time, and frequency of oscillation
• Various ways to determine if additional poles or any zeros impact estimates

Lecture 8 - 2/3 - System Diagrams and Simplifications 1

• Basic block diagrams
• Cascade, parallel, and feedback simplifications
• Converting frequency domain into formats for cascade or parallel constructions
• Equivalent systems when moving blocks past pickoffs and summations
• Signal flow graph basics

Lecture 9 - 2/5 - System Diagrams and Simplifications 2

• Converting block diagrams to signal flow graphs
• Mason's Rule

Lecture 10 - 2/10 - State Space 1

• Definition of state space and finding state variables for a circuit
• Determining the system, input, output, and feedforward matrices for an electrical system
• Finding transfer functions based on state space matrices

Lecture 12 - 2/17 - State Space 2

• State space for mechanical systems - generally phase space or a combination of phase spaces
• Controller canonical form
• Other forms summarized in textbook

Lecture 13 - 2/19 - Stability

• Determine pole locations
  • If one or more are in right-half-plane, the system is unstable
  • If none is in right-half-plane, but there are some singleton poles on $$j\omega$$ axis, the system is marginally stable
  • If all are in left-half-plane (even if repeated), the system is stable
• Routh Array can be used to determine pole locations

Lecture 14 - 2/24 - More on Routh

• Leading zero definitely means unstable
  • Can replace leading zero with $$\epsilon$$, finish table, and then look at sign changes assuming $$\epsilon$$ is a small positive (or negative) number
• Full row of zeros (including a row with only one entry that happens to be zero) means having to change the interpretation of sign changes

Lecture 15 - 2/26 - Steady State Error

• Error calculation based on $$E=R-C$$ and calculated using $$G_{eq}$$, the forward path of a unity feedback system that produces the system transfer function $$T$$
• The system type is based on the number of pure integrators in the denominator of $$G_{eq}$$
• For each type, there is one finite non-finite static error constant and one finite steady state error:
  • Type 0:
    • $$K_p=\lim_{s\rightarrow 0}G_{eq}(s)$$
    • $$e_{step}=\frac{1}{1+K_p}$$
    • $$K_v=K_a=0$$
    • $$e_{ramp}=e_{para}=\infty$$
  • Type 1:
    • $$K_v=\lim_{s\rightarrow 0}sG_{eq}(s)$$
    • $$e_{ramp}=\frac{1}{K_v}$$
    • $$K_p=\infty$$
    • $$e_{step}=0$$
    • $$K_a=0$$
    • $$e_{para}=\infty$$
  • Type 2:
    • $$K_a=\lim_{s\rightarrow 0}s^2G_{eq}(s)$$
    • $$e_{para}=\frac{1}{K_a}$$
    • $$K_p=K_v=\infty$$
    • $$e_{step}=e_{ramp}=0$$

Lecture 16 - 3/3 - Root Locus

• Error with respect to disturbances
• Complex numbers redux - especially angles
• Real-axis parts of root locus
• Asymptotes

Lecture 17 - 3/5 - Root Locus

• Root locus process

Lecture 18 - 3/17 - Jury Duty =

...

Lecture 20 - 3/24 - SISOTOOL

• Root Locus Example

Lecture 21 - 3/26 - Design with Root Locus

• Can change root locus using a PD or Lead controller
• Process:
  • Find desired dominant pole location. If these can be achieved with gain control - great! Done! If not
  • Determine location of a controller zero that makes root locus run through desired location, then find gain to get poles there. This compensator is a PD controller. See if the extra zero or non-dominant poles are interfering. If not - great! Done! If they are,
  • See if moving the zero to the right helps - usually by placing it on top of a pole so it cancels. This will require adding a compensator pole to make up for the angular difference when the zero was moved. This compensator is a Lead controller.

Lecture 22 - 3/29 - Compensator Circuits

• Simple inverting amplifier configuration can be used in combination with resistors, series resistor-capacitor combinations, and parallel resistor-capacitor combinations to create gain, PD, Lead, PI, Lag, PID, or Lead-Lag controllers
• Passive circuits can be used to generate zeros and/or poles for Lead, Lag, and Lead-Lag, but if a gain greater than 1 is required, an amplifier will be necessary

Lecture 23 - 4/2 - Bode Plots (1)

• Decibel - based on the base-10 logarithm of a power ratio, multiplied by 10 (to get the deci- part)
  • We are assuming power is related to voltage or current squared, and the log of a square is twice the log, so $$H_{dB}=20\,\log_{10}|\mathbb{H}(j\omega)|$$
• Bode plots
  • Magnitude plot is $$20\,\log_{10}|\mathbb{H}(j\omega)|$$ versus $$\omega$$ with $$\omega$$ on a log scale
  • Angle plot is $$\angle \mathbb{H}(j\omega)$$ versus $$\omega$$ with $$\omega$$ on a log scale
• Translation between decibels and magnitudes
  • 1 $$\leftrightarrow$$ 0 dB
  • 10 $$\leftrightarrow$$ 20 dB
  • 100 $$\leftrightarrow$$ 40 dB
  • 0.1 $$\leftrightarrow$$ -20 dB
  • 2 $$\leftrightarrow$$ $$\approx$$ 6 dB
  • 5 $$\leftrightarrow$$ $$\approx$$ 14 dB
  • 0.25=1/2/2 $$\leftrightarrow$$ 0-6-6 dB=-12 dB
  • 8=2*2*2=$$2^3$$ $$\leftrightarrow$$ 6+6+6 dB = 3 * 6 dB = 18 dB
  • $$\sqrt{2}$$=$$2^{1/2}$$ $$\leftrightarrow$$ $$\frac{1}{2}$$ 6 dB = 3 dB
• 3 dB below maximum for a transfer function amplitude is critical because that indicates where the half-power frequency is.
• Four main terms in transfer functions of real signals:
  • $$\mathbb{H}(j\omega)=K\,\color{red}{(j\omega)^p}\,\color{green}{\Pi_k(1+j\frac{\omega}{\omega_{cut}})^{n_k}}\,\color{blue}{\Pi_l((j\omega)^2+2\zeta_l\omega_{n_l}(j\omega)+(\omega_{n_l})^2)^{m_l}}$$
  • We looked at constants, $$j\omega$$, and $$1+j\frac{\omega}{\omega_{corner}}$$