Glossary - System Analysis and Control

Keystone concepts¶

A handful of ideas recur throughout the book, each time with increasing depth. These are the “keystone” concepts worth keeping in view as you progress. The table below catalogs them and links to the sections where they are developed, roughly in order of increasing sophistication.

Table 1:Keystone concepts and where they are developed

Concept	Where it is developed
Stability	transfer functions (informal) → inverse Laplace (pole locations) → stability criteria (algebraic tests) → root locus (vary gain) → stability margins and delay (frequency domain)
Feedback / closed-loop	introduction (intuition) → op amps (negative vs positive feedback) → feedback & P-control (closed-loop transfer function) → PI/PID and beyond
Robustness	introduction → linearization → high-gain control → integral control → stability margins → delay / design / lead-lag
Poles and performance	inverse Laplace → first-order → second-order & performance → higher-order / dominant poles → zeros
Transfer function & canonical form	transfer functions → impedance → first-order → second-order
DC gain	transfer functions (definition) → first-/second-order canonical forms → final value theorem
Superposition / LTI representations	linear systems → Laplace transforms (zero-state/zero-input) → impulse response & convolution → higher-order (PFE) → frequency response (Fourier)
Steady-state error	P-control → system type & PI control → PID → sensitivity
Zeros	the effect of zeros → introduced by PID and root locus design → Bode plots
Pole-zero cancellation	PI control → PID control → root locus design
Resonance	frequency response (named) → Bode plots (read off) → first/second-order Bode (derived)
High-gain trade-off	feedback & P-control → PI control → lead/lag

Several physical examples also recur as keystones, each revisited as new tools are introduced:

Spring-mass-damper

linear mechanical systems, transfer functions, second-order systems, zeros.

Pendulum / inverted pendulum

rotational systems, linear systems, linearization, second-order systems, PID control, root locus design.

DC motor

DC motors, electromechanical systems, first-order & case studies, second-order, higher-order.

Cruise control

linearization, Laplace transforms, transfer functions, first-order systems.

Terminology¶

System representation¶

Transfer function: A mathematical representation of a linear time-invariant (LTI) system. It is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input, assuming zero initial conditions. The transfer function is a function of the complex variable $s$ . If the LTI system is finite-dimensional (representable by a standard ODE), its transfer function can be expressed as a rational function (ratio of polynomials).
Characteristic polynomial: The denominator of the transfer function of a system. It is a polynomial in $s$ whose roots are the poles of the system.
Pole: A value of $s$ for which the transfer function $G(s)$ becomes infinite. Poles are the roots of the denominator of the transfer function.
Zero: A value of $s$ for which the transfer function $G(s)$ becomes zero. Zeros are the roots of the numerator of the transfer function.
Integrator: A system with a transfer function of the form $G(s)=\frac{1}{s}$ , which corresponds to the operation of integration in the time domain. We also say that a system “has an integrator” if its transfer function has a pole at the origin (i.e., $s=0$ ), or a “has a double integrator” if it has two poles at the origin, etc.
Order (of a system): The order of a system is the degree of its characteristic polynomial, which is the same as the number of poles of the system (counting multiplicity). For example a spring-mass-damper system like $m\ddot{y}+c\dot{y}+ky=u$ has a second-order characteristic polynomial $ms^2+cs+k$ , so it is a second-order system. This can refer to either the open-loop transfer function or the closed-loop transfer function, depending on context.

System analysis¶

DC gain

The steady-state value of the system in response to a step input. It can be found by evaluating the transfer function at $s=0$ , i.e., $G(0)$ .

Dirac delta function

A mathematical function $\delta(t)$ that is zero everywhere except at $t=0$ , where it is infinitely large in such a way that its integral over the entire real line is equal to 1. It is often used as a test input to model an instantaneous impulse input to a system. Also called the “unit impulse function”.

Heaviside step function

A mathematical function $H(t)$ that is zero for $t<0$ and one for $t \geq 0$ . It is often used as a test input to model a sudden change in constant input to a system. Also called the “unit step function”.

Impulse response

The output of a system when the input is a Dirac delta function $\delta(t)$ . Intuitively, it represents the system’s response to an instantaneous “kick”.

Step response

The output of a system when the input is a unit step function $H(t)$ . It represents the system’s response to a sudden change from zero to one.

Steady-state value

The value that the output of a system approaches as time goes to infinity in response to a given input.

Transient response

The part of the system’s response that occurs before the system reaches its steady-state value. It captures the dynamics of how the system transitions from its initial state to its steady-state.

Final value theorem

(FVT) A mathematical result that allows us to compute the steady-state value of a system’s response to a given input using the Laplace transform. It states that if $Y(s)$ is the Laplace transform of the output $y(t)$ , and $U(s)$ is the Laplace transform of the input $u(t)$ , then assuming $s Y(s)$ is stable, we have:

y_{ss} = \lim_{t \to \infty} y(t) = \lim_{s \to 0} s Y(s)

(1)

A common use of the FVT is to compute the steady-state value of the output $y(t)$ of a system with transfer function $G(s)$ in response to an input $u(t)$ . In this case, we can write:

y_{ss} = \lim_{t \to \infty} y(t) = \lim_{s \to 0} s\, G(s) U(s)

(2)

In the case where $u(t)$ is a step input, we have $U(s) = \frac{1}{s}$ , so the formula simplifies to: $y_{ss} = G(0)$ , which is the DC gain of the system.

Partial fraction expansion

(PFE) A mathematical technique used to decompose a rational function (such as a transfer function) into a sum of simpler fractions. This is useful because it allows us to easily compute the inverse Laplace transform of the function, or to see which poles contribute to the transient response of the system.

Residue

The coefficient of a term in the partial fraction expansion of a transfer function. For example, if we have a term of the form $\frac{r}{s-p}$ in the partial fraction expansion, then $r$ is the residue associated with the pole at $s=p$ .

First and second-order systems¶

Canonical form

A standard way of writing a transfer function that makes it easy to identify key parameters. We saw canonical forms for first-order and second-order systems.

First-order system

A system whose characteristic polynomial is first-order. The canonical form for a first-order system is

G(s) = \frac{K}{\tau s + 1}

(3)

where $K$ is the DC gain and $\tau$ is the time constant.

Second-order system

A system whose characteristic polynomial is second-order. The canonical form for a second-order system is

G(s) = \frac{K\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}

(4)

where $K$ is the DC gain, $\omega_n$ is the natural frequency, and $\zeta$ is the damping ratio.

Time constant

For a first-order system, the time constant $\tau$ is the time it takes for the system’s step response to reach approximately 63.2% of its final value. It is a measure of how quickly the system responds to changes in input.

Settling time

The settling time $t_s$ is the time it takes for the output of a system to enter and remain within 2% of its steady-state value. For a first-order systems, $t_s \approx 4\tau$ . For a second-order underdamped system, $t_s \approx \frac{4}{\zeta\omega_n}$ .

Damping ratio

The parameter $\zeta$ in the canonical form of a second-order system. It is a dimensionless measure of how oscillatory the system’s response is. A system is underdamped if $\zeta<1$ , critically damped if $\zeta=1$ , and overdamped if $\zeta>1$ .

Undamped system

A second-order system with a damping ratio $\zeta=0$ . Undamped systems exhibit sustained oscillations in their step response.

Underdamped system

A second-order system with a damping ratio $\zeta<1$ . Underdamped systems exhibit oscillatory behavior in their step response.

Critically damped system

A second-order system with a damping ratio $\zeta=1$ . Critically damped systems do not exhibit oscillations and have a repeated real pole.

Overdamped system

A second-order system with a damping ratio $\zeta>1$ . Overdamped systems do not exhibit oscillations in their step response. They have two real poles, so they can be thought of as the sum of two first-order systems.

Natural frequency

The parameter $\omega_n$ in the canonical form of a second-order system. It represents the frequency at which the system would oscillate if it were undamped (i.e., if $\zeta=0$ ).

Damped frequency

For a second-order underdamped system, the damped frequency $\omega_d$ is the frequency of oscillation in the step response. It is related to the natural frequency and damping ratio by $\omega_d = \omega_n \sqrt{1-\zeta^2}$ .

Pole angle

For a second-order underdamped system, the pole angle $\phi$ is the angle that the poles make with the negative real axis in the complex plane. It is related to the damping ratio by $\phi = \acos(\zeta)$ .

Peak time

For a second-order underdamped systems, the peak time $t_p$ is the time it takes for the output to reach its maximum value in response to a step input.

Percent overshoot

For a second-order underdamped system, the percent overshoot $M_p$ is the maximum percentage by which the output exceeds its steady-state value in response to a step input.

Feedback control¶

Plant: The system that we want to control. It is typically represented by a transfer function $G(s)$ .
Controller: The system that we design to achieve desired performance and stability. It is typically represented by a transfer function $C(s)$ . Used interchangeably with “compensator”.
Compensator: The system that we design to achieve desired performance and stability. It is typically represented by a transfer function $C(s)$ . Used interchangeably with “controller”.
Positive feedback: A feedback configuration where the feedback signal is added to the input signal. This tends to amplify the input and can lead to instability.
Negative feedback: A feedback configuration where the feedback signal is subtracted from the input signal. This tends to reduce the effect of disturbances and can improve stability.
Unity feedback: A feedback configuration where the output is fed back directly to the input without any modification. In this case, the closed-loop transfer function is $\frac{GC}{1+GC}$ , where $G$ is the plant and $C$ is the controller (assuming negative feedback).
Closed-loop transfer function: The transfer function from the reference input to the output in a feedback system. For a unity feedback system, it is given by $\frac{GC}{1+GC}$ . If we also include a sensor with transfer function $H$ and a pre-filter with transfer function $F$ , then the closed-loop transfer function is $\frac{GCF}{1+GCH}$ .
Loop gain: The product of the transfer functions of the plant and the controller in a feedback system. For a unity feedback system, the loop gain is $G(s)C(s)$ . If we also have a sensor in the loop, then the loop gain is $G(s)C(s)H(s)$ , where $H(s)$ is the transfer function of the sensor.
Relative degree: The relative degree of a transfer function is the difference between the degree of the denominator and the degree of the numerator. For example, the transfer function $G(s)=\frac{1}{s^2+3s+2}$ has a relative degree of 2, while $G(s)=\frac{s}{s^2+3s+2}$ has a relative degree of 1. You can also think of the relative degree as the number of poles minus the number of zeros of the transfer function.
Type (of a system): The type of a system is defined as the number of integrators in its loop gain. For example, in a unity feedback setup, if $G(s)C(s)=\frac{1}{s^2+3s+2}$ , it is type 0, while $G(s)C(s)=\frac{1}{s^2(s+1)}$ is type 2.
Proportional control: (P-control) A feedback control strategy where the controller is a proportional gain, i.e., $C(s) = k$ .
Integral control: (I-control) A feedback control strategy where the controller is an integrator multiplied by some proportional gain, i.e., $C(s) = \frac{k}{s}$ .
Proportional-integral control: (PI-control) A feedback control strategy where the controller is a combination of a proportional gain and an integrator, i.e., $C(s) = k_p + \frac{k_i}{s}$ .
Proportional-derivative control: (PD-control) A feedback control strategy where the controller is a combination of a proportional gain and a derivative term, i.e., $C(s) = k_p + k_d s$ .
Proportional-integral-derivative control: (PID-control) A feedback control strategy that combines P, I, and D control, i.e., $C(s) = k_p + \frac{k_i}{s} + k_d s$ .
Steady-state error: The difference between the reference input and the output of a system as time goes to infinity. It can be computed using the final value theorem as $e_{ss} = \lim_{s \to 0} s E(s)$ , where $E(s)$ is the Laplace transform of the error signal $e(t) = r(t) - y(t)$ . The usual way to compute this is to express $E(s)$ in terms of the reference input $R(s)$ . For a standard unity feedback system with plant $G(s)$ and compensator $C(s)$ , we have
$E(s) = \frac{R(s)}{1 + G(s)C(s)}$
(5)

Root locus¶

Pole-zero plot: A graphical representation of the poles and zeros of a transfer function in the complex plane. Poles are represented by “x” marks, while zeros are represented by “o” marks.
Excess poles: The number of poles of a transfer function that exceed the number of zeros. For example, if a transfer function has 5 poles and 3 zeros, it has 2 excess poles.
Characteristic equation: The equation obtained by setting the denominator of the closed-loop transfer function to zero. For a unity feedback system with loop gain $G(s)C(s)$ , the characteristic equation is $1 + G(s)C(s) = 0$ .
Root locus: A graphical representation of all possible closed-loop pole locations as we vary the proportional gain $k$ in a feedback system. If the plant is $G(s)$ , this corresponds to the solutions of the characteristic equation $1 + kG(s) = 0$ as $k$ varies from 0 to $\infty$ .
Branch: A continuous curve in the complex plane that represents the trajectory of a closed-loop pole as the gain $k$ is varied. Each branch starts at an open-loop pole (when $k=0$ ) and ends at an open-loop zero (as $k \to \infty$ ), or goes to infinity if there are excess poles. No branch can ever cross itself.
Real locus: The portion of the root locus that lies on the real axis of the complex plane. The real locus can be determined by the rule that a point on the real axis is part of the root locus if the total number of poles and zeros to the right of that point is odd.
Asymptote: A straight line that the root locus branches approach as $k \to \infty$ when there are more poles than zeros. The number of asymptotes is equal to the difference between the number of poles and zeros, and their configuration is always the same. E.g., when we have one excess pole, the asymptote is always at 180°, and when we have three excess poles, they are always at 60°, 180°, and 300° degrees.
Centroid: The point where the asymptotes intersect. This always occurs on the real axis and depends on the pole and zero locations.
Breakaway point: A point on real axis where two branches of the locus meet and then break away from each other at right angles and become complex conjugates.
Break-in point: A point on the real axis where two complex conjugate poles meet from opposite sides and then break in at right angles along the real axis.
Collision point: A generalized breakaway/break-in point where some number of branches meet and then break away in different directions. This can happen on the real axis or elsewhere in the complex plane.
Negative locus: The root locus corresponding to a proportional feedback with a negative gain. This amounts to using the characteristic equation $1 + kG(s) = 0$ with $k < 0$ .

Frequency response¶

Frequency response: The steady-state response of a system to sinusoidal inputs at different frequencies. It is typically represented by the magnitude function $M(\omega)$ and phase function $\phi(\omega)$ of the system’s output given a sinusoidal input with amplitude 1 and frequency $\omega$ . The magnitude and phase are the length and angle, respectively, of the complex number $G(j\omega)$ , where $G(s)$ is the transfer function of the system.
Bode plot: A graphical representation of a system’s frequency response, consisting of two plots: the magnitude plot (in decibels) and the phase plot (in degrees), both plotted against frequency on a logarithmic scale, with the frequency axis shared between the two plots.
Decade: A factor of 10 in frequency. For example, the frequencies 0.5 rad/s, 5 rad/s, and 50 rad/s are each one decade apart.
Octave: A factor of 2 in frequency. For example, the frequencies 3 rad/s, 6 rad/s, and 12 rad/s are each one octave apart.
Decibel (dB): A logarithmic unit used to express the magnitude of a system’s frequency response. The magnitude in decibels is defined as $M_\text{dB} = 20 \log_{10} M$ , where $M$ is the magnitude of the frequency response. See this table for reference.
Corner frequency: A feature of a Bode magnitude plot where the slope has a characteristic “elbow” and transitions from one slope to another.
High-frequency roll-off: The behavior of a system’s frequency response at high frequencies, where the magnitude typically decreases (rolls off) at a certain rate (e.g., -20 dB/decade for a first-order system).
Low-pass filter: Another name for a system that has a relatively flat magnitude response at low frequencies, a corner frequency, and a high-frequency roll-off. It “lets the low frequencies pass” while attenuating the high frequencies. The simplest example is a first-order system, $G(s)= \frac{K}{\tau s + 1}$ , which has a corner frequency at $\omega_c = \frac{1}{\tau}$ and a high-frequency roll-off of -20 dB/decade.
Resonance peak: A feature of a Bode magnitude plot where the magnitude reaches a local maximum at a certain frequency, often due to the presence of highly underdamped second-order dynamics.
Resonance frequency: The frequency $\omega_r$ at which a resonance peak occurs in the Bode magnitude plot.
Resonance peak magnitude: The magnitude $M(\omega_r)$ of the frequency response at the resonance frequency.

Stability and robustness margins¶

Sensitivity function: The sensitivity function $S(s)$ of a feedback system is defined as $S(s) = \frac{1}{1+G(s)C(s)}$ , where $G(s)$ is the plant and $C(s)$ is the controller. It quantifies how sensitive the closed-loop system is to changes in the plant or disturbances. A smaller sensitivity function (i.e., $|S(j\omega)| < 1$ ) indicates that the system is less sensitive to disturbances at that frequency. The sensitivity function is also the transfer function from reference to error.
Complementary sensitivity function: The complementary sensitivity function $T(s)$ of a feedback system is defined as $T(s) = \frac{G(s)C(s)}{1+G(s)C(s)}$ . It quantifies how sensitive the closed-loop system is to changes in the reference input. A smaller complementary sensitivity function (i.e., $|T(j\omega)| < 1$ ) indicates that the system is less sensitive to changes in the reference input at that frequency. The complementary sensitivity function is also the transfer function from reference to output.
Robustness: The ability of a control system to maintain stability and acceptable performance despite uncertainty, such as imperfect models, external disturbances, measurement noise, and time delays. Feedback control is valued largely because it improves robustness; this is a recurring theme throughout the course, made quantitative through stability margins.
Stability margin: A general term for how far a closed-loop system is from the boundary of instability, and hence a measure of its robustness. The gain margin, phase margin, and delay margin are specific stability margins that quantify tolerance to gain changes, phase lag, and time delay, respectively.
Gain crossover frequency: The frequency $\omega_{cg}$ at which the Bode magnitude plot is equal to 0 dB.
Phase crossover frequency: The frequency $\omega_{cp}$ at which the Bode phase plot is equal to -180°.
Gain margin: For an open-loop stable system, the gain margin $\GM$ is the distance below the 0 dB line of the magnitude plot at the phase crossover frequency. It indicates the amount of gain increase (in dB) that can be tolerated before the closed-loop system becomes unstable.
Phase margin: For an open-loop stable system, the phase margin $\PM$ is the distance above the -180° line of the phase plot at the gain crossover frequency. It indicates the amount of additional phase lag (in degrees) that can be tolerated before the closed-loop system becomes unstable.
Delay margin: The maximum amount of time delay that can be added to the system before it becomes unstable. It is related to the phase margin by the formula $\tau = \frac{\PM}{\omega_{cg}}$ .
Bandwidth: The frequency $\omega_b$ at which the magnitude of closed-loop Bode plot drops to -3 dB below its DC gain (zero-frequency value). It is a measure of how quickly the system can respond to changes in the reference input. A common approximation for the bandwidth is the gain crossover frequency $\omega_{cg}$ of the open-loop Bode plot.
Lead compensator: A type of compensator that adds a pole-zero pair to the system, where the zero is closer to the origin than the pole (zero has lower frequency). This can behave similarly to a PD controller, and is often used to increase the phase margin of a system.
Lag compensator: A type of compensator that adds a pole-zero pair to the system, where the pole is closer to the origin than the zero (pole has lower frequency). This can behave similarly to a PI controller, and is often used to increase the low-frequency gain of a system without significantly affecting the phase margin, or decreasing the high-frequency gain to reduce bandwidth and improve phase margin without affecting low-frequency gain.
Lead-lag compensator: A type of compensator that adds both a lead and a lag compensator, i.e., two pole-zero pairs, to the system. This can behave similarly to a PID controller.