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 - Moving to Root Locus

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

Lecture 17 - 3/5 - Moving to Root Locus

• Root locus process