First Order Control System: What is it? (Rise & Settling Time Formula)

Contents

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Key learnings:

First Order Control System Definition: A first order control system uses a simple type of differential equation to relate inputs and outputs, focusing on the first derivative of time only.
Transfer Function: The transfer function for such systems typically takes the form K/(Ts+1), where K is DC Gain and T is the time constant, explaining how input affects output.
Rise Time: Rise time in these systems is calculated by the time it takes for the output to rise from 10% to 90% of its final value, a crucial indicator of response speed.
Settling Time: Settling time is the duration the system takes to stabilize within a small percentage (usually 2% to 5%) of its final output value, indicating efficiency.
System Analysis Tools: Tools like MATLAB are essential for analyzing the poles and zeros of the transfer function, providing qualitative data that can predict system behavior.

What is a First Order Control System?

A first-order control system has one independent dynamic mode in its reduced input-output model. Its governing differential equation contains a first derivative, and its transfer function has a first-degree denominator after common factors are cancelled. A transfer function assumes zero initial conditions.

As an example, let us look at the block diagram of the control system shown below.

First Order Control System Block Diagram — (a) Block Diagram of a First-order Control System; (b) Simplified Block Diagram

The standard transfer function for this control system is:

$\begin{align*} \frac{C(s)}{R(s)} = K \frac{1}{Ts+1} \end{align*}$

Where:

K denotes the DC gain. For a stable system, K equals the steady output divided by a constant input.
T is the time constant, in seconds. A smaller positive T gives a faster stable response.

The differential-equation order is its highest derivative order. In a proper rational transfer function, the system order is normally the degree of the reduced denominator polynomial in $s$ .

The denominator Ts+1 contains $s$ only to the first power ( $s^1 = s$ ), so the model has one pole at s = -1/T. For T > 0, that pole lies in the left half-plane. Hence the block diagram above represents a stable first-order control system.

In a theoretical alternate example, let us say that the transfer function was equal to:

$\begin{align*} \frac{C(s)}{R(s)} = K \frac{1}{Ts^2+1} \end{align*}$

Here $s$ reaches the second power ( $s^2$ ), so the transfer function represents a second-order control system if no common factor cancels. The expression Ts^2+1 has imaginary-axis poles for positive T, so it is not a standard damped second-order example.

Many thermal, electrical and mechanical processes can be approximated by a first-order model over a stated operating range. A higher-order system may use a first-order approximation when one stable pole dominates the response and the neglected modes are sufficiently faster.

Dynamic order and feedback topology are separate properties. A control system may be open loop or closed loop, and either topology can have first-order or higher-order dynamics.

An open-loop controller does not use measured output feedback to correct the command. Its response still depends on the plant dynamics and disturbances.

A closed-loop system measures output and compares it with the reference. The difference is the error; its limiting value, when that limit exists, is the steady state error. The controller uses this feedback to change the plant input.

The output produced by a unit-step input is the unit-step response. It shows the transient approach to the final value and supports measurements such as time constant, rise time and settling time. First- and second-order models are common approximations, not the only possible system orders.

A first-order differential equation has no derivative above the first; a second-order equation includes a second derivative. Transfer-function order follows the reduced denominator degree.

A first-order dynamic model has one state or one independent energy-storage mode in a minimal realisation. It need not contain a pure integrator: the standard stable lag has one pole at a negative real value.

System equations can be solved in time using differential equations or in the s domain using the Laplace Transform. One common decomposition writes the response as forced and natural parts.

The natural response solves the homogeneous equation and depends on system poles and initial conditions. A particular response is driven by the input; it may contain transient terms and should not always be treated as identical to the steady-state response.

The following analysis uses the control engineering concepts of poles, zeros and transfer function. Poles set the natural modes; zeros shape the input-output response.

First Order Control System Transfer Function

A transfer function is the Laplace-domain output-to-input ratio for a linear time-invariant system under zero initial conditions. It describes one selected input-output pair.

Poles of a Transfer Function

The poles are values of the value of Laplace Transform variable s that make the reduced transfer-function denominator zero. They determine the system’s natural exponential modes.

The denominator is a polynomial; its roots are poles. Common numerator and denominator factors must be cancelled before interpreting the minimal input-output dynamics.

Zeros of a Transfer Function

The zeros are values of the Laplace variable s that make the reduced transfer-function numerator zero.

The numerator is a polynomial; its roots are zeros. A zero can change response shape without adding an independent natural mode.

First Order Control System

Consider a stable first-order control system with no finite zero. Its pole location sets the exponential response rate and the time needed to approach the final value.

For a unit step, R(s) = 1/s. Multiplying the input by the transfer function gives C(s). The example uses , where is the first-order transfer function.

The product C(s) has an input pole at s = 0 from the step and a system pole at s = -a. The transfer function itself has only the system pole. For a>0, this pole lies on the negative real axis and produces a decaying exponential.

MATLAB’s pzmap command can plot model poles and zeros, but the roots can also be found directly. Software should verify the reduced model and the stated stability assumptions.

Taking the inverse Laplace transform gives . For the stated zero initial condition, this contains the constant final term and a decaying natural term.

The step-input pole at the origin supplies the constant part of this response. The stable system pole at -a supplies the transient exponential. This pole-input interpretation applies to the stated simple-pole example.

For unit DC gain, a unit step, zero initial output and a>0, the time response of the first-order system is c(t) = 1-e^-at. The constant term is 1 and the transient term is -e^-at. Here a = 1/T; a non-unit gain K multiplies the complete response.

The same response follows from the time-domain differential equation or the inverse Laplace Transform. Either method should retain the input amplitude, DC gain and initial condition.

Poles and zeros provide qualitative information, but numerical response times require their values and a stated threshold. A stable first-order system approaches its final value asymptotically rather than reaching it at a finite instant.

The next sections define time constant, 10%-to-90% rise time and settling time for the canonical stable first-order step response.

Time Constant of a First Order Control System

The time constant T is the time for a zero-state step response to complete 1-e^-1, or about 63.2%, of its total change. Here T = 1/a. The rate a has units of reciprocal seconds.

The parameter a is the exponential decay rate. The natural term e-at has derivative -a at t = 0, and its time scale is 1/a. The time constant therefore summarises the transient speed of a stable first-order control system.

For stable real poles, moving the pole farther left increases a and shortens T. Controller design cannot move a pole arbitrarily: actuator limits, noise, unmodelled dynamics, robustness and stability constrain the achievable response.

Rise Time of a First Order Control System

The 10%-to-90% rise time is the interval for the response to move from 10% to 90% of its total change. Substituting these two output fractions into the canonical response gives the crossing times.

At 10% of the final change

At 90% of the final change

Subtracting the 10% crossing time from the 90% crossing time gives

Thus tr = ln(9)/a, approximately 2.2/a or 2.2T, for this stable first-order response.

Settling Time of a First Order Control System

Settling time is the first time after which the response remains within a stated band around its final value. Common bands are 2% and 5%, and the chosen band must accompany the reported value.

For fractional band δ, the exact result T_s = -ln(δ)/a evaluates to about 3.9/a for 2%, commonly rounded to 4/a, and about 3.0/a for 5%.

Time constant, rise time and settling time summarise the same exponential response. State the gain, initial condition and threshold before comparing values from equations, plots or software.

Conclusion of First Order Control Systems

For the stable, zero-free 1st order control system considered here:

The unit-step input contributes a pole at the origin to C(s), but that pole is not a pole of the transfer function.
The transfer-function pole generates the natural exponential mode.
A stable real pole at -a generates e-at. A larger positive a places the pole farther left and makes the transient decay faster.
Poles and zeros help predict response shape and stability; controller changes must also respect robustness and physical limits.