Steady-state Behavior of Linear Systems

March 03, 2026

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#Control #LinearControlSystems We learned how to represent a linear system using differential equations and transfer functions here. We also learned about first-order and second-order linear systems and their transient characteristics here. In this note, we are going to learn about the steady-state behavior of a linear system. Perhaps the most important theorem in this regard is the final value theorem, which states that if a system is stable and has a final, constant value, this value can be found by $x_{ss} = \lim_{t \to \infty} x(t) = \lim_{s \to 0} s X(s)$ TODO: add a block diagram of the closed-loop system here. Consider a closed-loop system with $C (s)$ as the controller and $G (s)$ as the system. The relation between the reference and the output, also called complementary sensitivity is defined as $\frac{Y(s)}{R(s)} = \frac{C(s) G(s)}{1 + C(s)G(s)}$ and sensitivity is defined as $\frac{E(s)}{R(s)} = \frac{1}{1 + C(s)G(s)}$ To find the steady-state error of the system to different inputs, we define

position-error constant: $K_p = \lim_{s \to 0} C(s) G(s)$
velocity constant: $K_v = \lim_{s \to 0} s C(s) G(s)$
acceleration constant: $K_a = \lim_{s \to 0} s^2 C(s) G(s)$ Now let’s find the steady-state error to some basic inputs:
step input: We use the final value theorem. $\begin{align} e_{ss} = \lim_{s \to 0} s E(s) = \lim_{s \to 0} s \cdot \frac{1}{1 + C(s)G(s)} \cdot \underbrace{R(s)}_{\frac{1}{s}} = \lim_{s \to 0} \frac{1}{1 + C(s)G(s)} = \frac{1}{1 + \lim_{s \to 0} C(s)G(s)} = \frac{1}{1 + K_p} \end{align}$
ramp input: $\begin{align} e_{ss} = \lim_{s \to 0} s E(s) = \lim_{s \to 0} s \cdot \frac{1}{1 + C(s)G(s)} \cdot \underbrace{R(s)}_{\frac{1}{s^2}} = \lim_{s \to 0} \frac{1}{s + s C(s) G(s)} = \frac{1}{0 + \lim_{s \to 0} s C(s) G(s)} = \frac{1}{K_v} \end{align}$
parabola input: $\begin{align} e_{ss} = \lim_{s \to 0} s E(s) = \lim_{s \to 0} s \cdot \frac{1}{1 + C(s)G(s)} \cdot \underbrace{R(s)}_{\frac{1}{s^3}} = \lim_{s \to 0} \frac{1}{s^2 + s^2 C(s) G(s)} = \frac{1}{0 + \lim_{s \to 0} s^2 C(s) G(s)} = \frac{1}{K_a} \end{align}$ Systems that have a finite $K_p$ are called type 0 and have a non-zero steady-state error to a step input. A system with a finite $K_v$ is called type 1 and has a non-zero steady-state error to a ramp input. A system with a finite $K_a$ is called type 2 and has a non-zero steady-state error to a parabola input.

TODO: Add the Ruth-Hurwitz table

To read more about linear control systems, you can take a look at First and Second-order Systems or Root Locus.