3.1 Transient response analysis

Transient response overview

Transient response describes how a system behaves as it transitions from an initial state to a final steady state after receiving an input or disturbance. Every real control system takes some finite amount of time to reach its target, and the shape of that journey matters. Does the output overshoot? Does it oscillate? How long until it settles?

By quantifying these behaviors with measurable parameters, you can evaluate whether a system meets its design requirements and decide what needs to change if it doesn't.

First-order vs higher-order systems

A first-order system has a single energy storage element (a capacitor or an inductor, for example) and produces a smooth exponential response with no oscillations. Its transfer function has a first-degree polynomial in the denominator.

Higher-order systems (second-order and above) contain multiple energy storage elements and exhibit richer transient behavior, including oscillations and overshoots. The system's order equals the highest power of $s$ in the denominator of its transfer function. Second-order systems are especially important because they capture most of the interesting dynamics while remaining analytically tractable.

Time-domain specifications

Time-domain specifications turn the shape of a transient response into numbers you can compare against design requirements. The key ones are:

• Rise time ($t_r$)
• Settling time ($t_s$)
• Peak time ($t_p$)
• Percent overshoot ($\%OS$)
• Steady-state error ($e_{ss}$)

Each of these is defined for a step input, which is the standard test signal for transient analysis. The sections below cover each one in detail.

Rise time

Rise time ($t_r$) is the time it takes for the output to go from 10% to 90% of its final steady-state value after a step input is applied. A shorter rise time means the system reacts faster, which matters in applications like robotics or high-speed communications.

Rise time is governed by the system's natural frequency ($\omega_n$) and damping ratio ($\zeta$). In general, increasing $\omega_n$ decreases rise time, but this often comes at the cost of increased overshoot if $\zeta$ isn't adjusted accordingly.

Settling time

Settling time ($t_s$) is the time required for the output to stay within a specified band around the final value. The two most common bands are $\pm 2\%$ and $\pm 5\%$.

For a standard second-order system, a useful approximation is:

t_s \approx \frac{4}{\zeta \omega_n} \quad \text{(2% criterion)}

t_s \approx \frac{3}{\zeta \omega_n} \quad \text{(5% criterion)}

A shorter settling time means the system reaches steady state faster and spends less time oscillating. Both $\zeta$ and $\omega_n$ influence settling time directly.

Peak time

Peak time ($t_p$) is the time at which the output reaches its first (and largest) peak above the steady-state value. It only applies to underdamped systems, since overdamped and critically damped responses don't overshoot.

For a second-order underdamped system:

$t_p = \frac{\pi}{\omega_d}$

where $\omega_d = \omega_n\sqrt{1 - \zeta^2}$ is the damped natural frequency. A smaller $t_p$ means the system reaches its peak faster.

Percent overshoot

Percent overshoot ($\%OS$) measures how far the output exceeds the final steady-state value, expressed as a percentage:

$\%OS = 100 \cdot e^{-\pi\zeta / \sqrt{1 - \zeta^2}}$

This formula shows that percent overshoot depends only on the damping ratio $\zeta$. A higher $\zeta$ means less overshoot. For example, $\zeta = 0.5$ gives about 16.3% overshoot, while $\zeta = 0.7$ gives about 4.6%.

Excessive overshoot can stress mechanical components or push a process variable into an unsafe range, so many design specs place an upper limit on $\%OS$.

Steady-state error

Steady-state error ($e_{ss}$) is the difference between the desired output and the actual output once all transients have died out. A nonzero $e_{ss}$ means the system never perfectly tracks the reference input.

Common ways to reduce steady-state error:

• Increase the forward-path gain
• Add integral control action (which drives the accumulated error to zero)
• Increase the system type (the number of free integrators in the open-loop transfer function)

Note that steady-state error depends on both the system type and the type of input (step, ramp, parabolic). The final value theorem is the standard tool for computing it.

Transient response of first-order systems

First-order systems are the simplest dynamic systems and serve as building blocks for understanding more complex ones. Their transient behavior is fully characterized by a single parameter: the time constant.

The standard first-order transfer function is:

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

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

Time constant

The time constant ($\tau$) quantifies how fast a first-order system responds. Specifically, after a step input, the output reaches 63.2% of its final value at $t = \tau$.

Some useful benchmarks:

• At $t = 2\tau$, the output reaches about 86.5%
• At $t = 3\tau$, about 95%
• At $t = 4\tau$, about 98.2%
• At $t = 5\tau$, about 99.3%

This is why the settling time for a first-order system is often quoted as $t_s \approx 4\tau$ (2% criterion) or $t_s \approx 3\tau$ (5% criterion). A smaller $\tau$ means a faster system.

Step response

The step response of a first-order system with DC gain $K$ and time constant $\tau$ is:

$y(t) = K\left(1 - e^{-t/\tau}\right), \quad t \geq 0$

This is a smooth exponential curve that starts at zero and asymptotically approaches $K$. There are no oscillations and no overshoot. The initial slope of the curve equals $K/\tau$, so a smaller time constant produces a steeper initial rise.

Impulse response

The impulse response of the same first-order system is:

$y(t) = \frac{K}{\tau} e^{-t/\tau}, \quad t \geq 0$

The output starts at $K/\tau$ and decays exponentially toward zero. Again, $\tau$ controls the decay rate. The impulse response is the derivative of the step response, which is a general relationship that holds for any LTI system.

Transient response of second-order systems

Second-order systems are where transient analysis gets interesting. Two energy storage elements interact, producing behavior that ranges from sluggish exponential decay to sustained oscillation, depending on the system parameters.

The standard second-order transfer function is:

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

The two parameters that control everything are the natural frequency $\omega_n$ and the damping ratio $\zeta$.

Natural frequency and damping ratio

Natural frequency ($\omega_n$) is the frequency at which the system would oscillate if there were no damping at all ($\zeta = 0$). It sets the overall speed of the response: higher $\omega_n$ means faster dynamics.

Damping ratio ($\zeta$) describes how much energy dissipation is present. It determines the character of the response:

• $\zeta = 0$: undamped (pure oscillation)
• $0 < \zeta < 1$: underdamped (oscillatory, decaying)
• $\zeta = 1$: critically damped (fastest non-oscillatory)
• $\zeta > 1$: overdamped (slow, no oscillation)

The poles of the standard second-order system are:

$s = -\zeta\omega_n \pm \omega_n\sqrt{\zeta^2 - 1}$

When $\zeta < 1$, the square root term becomes imaginary, producing a complex conjugate pole pair. When $\zeta \geq 1$, both poles are real.

Underdamped response

When $0 < \zeta < 1$, the system is underdamped. The step response oscillates around the final value with gradually decreasing amplitude. The output overshoots the target, comes back, undershoots slightly, and so on until it settles.

The step response for an underdamped second-order system is:

$y(t) = 1 - \frac{e^{-\zeta\omega_n t}}{\sqrt{1-\zeta^2}} \sin\left(\omega_d t + \phi\right)$

where $\omega_d = \omega_n\sqrt{1-\zeta^2}$ and $\phi = \cos^{-1}(\zeta)$.

A lower damping ratio produces more pronounced oscillations, higher overshoot, and a longer settling time. Most practical control systems operate in the underdamped region with $\zeta$ between about 0.4 and 0.8, balancing speed against overshoot.

Critically damped response

When $\zeta = 1$, the system is critically damped. It returns to steady state as fast as possible without any oscillation or overshoot. Both poles are real and equal at $s = -\omega_n$.

The step response is:

$y(t) = 1 - (1 + \omega_n t)e^{-\omega_n t}$

Critically damped designs are common in applications like door closers, instrument needle movements, and positioning systems where overshoot is unacceptable but speed still matters.

Overdamped response

When $\zeta > 1$, the system is overdamped. It has two distinct real poles, and the response creeps toward the final value without oscillating or overshooting. The tradeoff is that it's slower than both the critically damped and underdamped cases.

The response is the sum of two decaying exponentials, each associated with one of the two real poles. The slower pole (closer to the imaginary axis) dominates the long-term behavior.

Overdamped designs are appropriate when overshoot must be strictly avoided, such as in temperature control or large industrial processes where oscillation could cause damage.

Effect of poles on transient response

The location of poles in the $s$-plane gives you a visual map of transient behavior:

• Real part ($\sigma = -\zeta\omega_n$): Controls the decay rate of the envelope. Poles farther left mean faster settling.
• Imaginary part ($\omega_d$): Controls the oscillation frequency. Poles farther from the real axis mean faster oscillation.
• Distance from origin ($\omega_n$): Controls the overall speed of response.
• Angle from negative real axis ($\cos^{-1}\zeta$): Directly encodes the damping ratio.

Poles in the left half-plane (negative real part) produce stable, decaying responses. Poles in the right half-plane produce responses that grow without bound. Poles exactly on the imaginary axis produce sustained oscillations.

Transient response design

Once you understand how pole locations map to transient behavior, design becomes a matter of placing poles where you need them. The goal is to meet specifications on rise time, settling time, overshoot, and steady-state error while maintaining stability.

Dominant pole concept

In higher-order systems, not all poles contribute equally to the transient response. The dominant poles are the ones closest to the imaginary axis, because they decay the slowest and therefore shape the response for the longest time.

If the non-dominant poles are at least 5 to 10 times farther from the imaginary axis than the dominant pair, you can approximate the higher-order system as a second-order system using just the dominant poles. This simplification makes hand calculations and initial design work much more manageable.

Pole placement

Pole placement is a design technique where you choose desired pole locations based on your transient response specs, then compute the controller gains needed to put the poles there.

The typical workflow:

1. Translate your specs ($\%OS$, $t_s$, $t_r$) into required $\zeta$ and $\omega_n$ values
2. Determine the desired pole locations: $s = -\zeta\omega_n \pm j\omega_n\sqrt{1-\zeta^2}$
3. Use state feedback (or output feedback) to compute the gain matrix that places the closed-loop poles at those locations
4. Verify the design with simulation, checking that non-dominant poles and zeros don't significantly alter the expected response

Pole placement via state feedback requires the system to be controllable. You can check this using the controllability matrix.

Transient response improvement techniques

Several compensation strategies can reshape the transient response:

• Lead compensation: Adds a zero-pole pair where the zero is closer to the origin than the pole. This increases phase margin, speeds up the response, and reduces settling time. Think of it as adding "anticipation" to the controller.
• Lag compensation: Adds a pole-zero pair where the pole is closer to the origin. This boosts low-frequency gain to reduce steady-state error without significantly affecting the transient speed. The tradeoff is a slight increase in settling time.
• PID tuning: Adjusting $K_p$, $K_i$, and $K_d$ (covered below) is the most common practical approach.
• Notch filters: Target and attenuate a specific resonant frequency that causes unwanted oscillations in the system.

Transient response in control systems

Transient response analysis isn't just an academic exercise. It directly determines whether a control system performs acceptably in practice. A system that overshoots too much, settles too slowly, or oscillates excessively will fail to meet its design objectives.

Effect of feedback on transient response

Negative feedback is the primary tool for shaping transient response. Its effects include:

• Reduced sensitivity to parameter variations and disturbances
• Increased bandwidth, which generally speeds up the response
• Modified pole locations, allowing you to trade off between speed, damping, and stability

The catch is that feedback can also degrade performance if designed poorly. Too much gain can push poles toward the right half-plane, causing instability. The root locus technique is a standard way to visualize how poles move as gain changes, helping you avoid this trap.

Transient response of PID controllers

PID controllers combine three control actions, each targeting a different aspect of transient performance:

• Proportional ($K_p$): Produces an output proportional to the current error. Increasing $K_p$ reduces rise time and steady-state error but increases overshoot. Too much $K_p$ can make the system oscillatory.
• Integral ($K_i$): Produces an output proportional to the accumulated error over time. It eliminates steady-state error entirely (for step inputs) but tends to increase overshoot and settling time. Excessive $K_i$ causes integral windup and sluggish, oscillatory behavior.
• Derivative ($K_d$): Produces an output proportional to the rate of change of error. It acts as a damper, reducing overshoot and improving settling time. However, it amplifies high-frequency noise, so it's often used with a low-pass filter in practice.

The PID transfer function is:

$C(s) = K_p + \frac{K_i}{s} + K_d s$

Tuning these three gains to meet transient specs simultaneously is one of the core practical skills in control engineering. Methods like Ziegler-Nichols provide starting-point tuning rules.

Transient response in state-space models

State-space models offer a more general framework for transient analysis, especially for multi-input multi-output (MIMO) systems. The standard form is:

$\dot{x} = Ax + Bu$ $y = Cx + Du$

The transient response is governed by the eigenvalues of the system matrix $A$, which are equivalent to the poles of the transfer function. Designing a state feedback controller $u = -Kx$ changes the closed-loop system matrix to $A - BK$, and you choose $K$ to place the eigenvalues where you want them.

Advanced techniques like LQR (Linear Quadratic Regulator) automate this process by finding the gain matrix $K$ that minimizes a cost function balancing state deviations against control effort. LQG (Linear Quadratic Gaussian) extends this to handle noisy measurements by combining LQR with a Kalman filter (observer). These methods produce transient responses that are optimal in a well-defined mathematical sense, though the resulting pole locations may not match simple second-order specs exactly.