Glossary - System Analysis and Control

System representation¶

transfer function: A mathematical representation of a linear time-invariant (LTI). 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 $s$ and is typically expressed as a ratio of polynomials in $s$ .
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¶

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 $u(t)$ . 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.

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
$G(s) = \frac{K}{\tau s + 1}$
(1)
second-order system: A system whose characteristic polynomial is second-order. The canonical form for a second-order system is
$G(s) = \frac{K\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}$
(2)
DC gain: The parameter $K$ in the canonical form of a system. It is also the steady-state value of the system in response to a step input.
time constant ( $\tau$ ): 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 ( $t_s$ ): 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 ( $\zeta$ ): 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 ( $\omega_n$ ): 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 ( $\omega_d$ ): 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 ( $\phi$ ): 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 $\theta = \arccos(\zeta)$ .
peak time ( $t_p$ ): For a second-order underdamped systems, it is the time it takes for the output to reach its maximum value in response to a step input.
percent overshoot ( $M_p$ ): For a second-order underdamped system, it 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, 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 where the controller is a combination of a proportional gain, an integrator, and a derivative term, i.e., $C(s) = k_p + \frac{k_i}{s} + k_d s$ .