Types of Controllers

Contents

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Key learnings:

Controller Definition: A controller is a device that adjusts a system’s output to match a set desired value, thus minimizing error and enhancing stability and accuracy.
Types of Controllers: Controllers are divided into continuous and discontinuous types, each suitable for different operational needs.
Integral Controller: The integral controller, or reset controller, modifies the output based on the integral of the error signal, effectively reducing the steady-state error but may impact system stability.
Combination Controllers: Controllers such as PI, PD, and PID combine elements of proportional, integral, and derivative control to optimize system response.
Fuzzy Logic Controllers: FLCs are effective in managing complex, nonlinear systems by using fuzzy sets and rules based on empirical knowledge, without needing precise mathematical models.

What is a Controller?

In control systems, a controller calculates a command for the plant from signals such as the setpoint, measured process variable or estimated state. Its design shapes tracking, disturbance rejection and robustness. Controllers are a central part of control engineering.

A controller must be selected and tuned for the plant, actuator, sensor and operating range. Common design objectives include:

Reduce steady-state error when the loop structure and controller action permit it.
Maintain closed-loop stability and adequate gain and phase margins.
Reject disturbances and reduce unwanted offsets.
Shape rise time, settling time and overshoot.
Limit how sensor noise reaches the actuator and output.
Increase response speed without exceeding actuator or robustness limits.

The industrial automotive overview covers automation equipment such as programmable logic controllers and SCADA systems. These platforms can implement the controller types described below.

Controllers can be grouped by how their output changes: continuous controllers and discontinuous or switching controllers.

In a discontinuous controller, the manipulated output switches between discrete values. Two-position, three-position and multi-position controllers are named for the number of available output states.

Discontinuous control suits switching final elements such as relays, contactors or valves with fixed positions. Hysteresis is often used to prevent rapid switching near a threshold.

A continuous controller can command any manipulated value within its output range. The manipulated variable is the controller output; the controlled variable is the measured process quantity.

In continuous controller theory, three basic feedback actions are:

Proportional controllers.
Integral controllers.
Derivative controllers.

These actions can be used alone or combined. The goal is to meet stated tracking, disturbance, actuator and robustness requirements rather than force zero error under every condition.

Proportional and integral controllers (PI Controller)
Proportional and derivative controllers (PD Controller)
Proportional integral derivative control (PID Controller)

The following sections explain each control action and its main trade-offs.

Proportional Controllers

A controller must be designed around the plant and performance requirements. For a proportional controller, check at least these two conditions:

The selected gain must keep the closed loop stable and sufficiently robust throughout the operating range.
The commanded output must stay within actuator limits and avoid unacceptable noise, overshoot or response speed.

A proportional controller produces an actuating signal proportional to the current error. If e(t) is setpoint minus measured output, the ideal control law is:

Introducing the proportional gain gives:

Here K_p is the proportional gain. Its units depend on the input and output signal units.

There is no universal requirement for K_p to exceed one. Signal scaling, plant dynamics, actuator limits and stability margins determine a suitable K_p. Increasing it raises loop gain and can improve speed or reduce error, but may also increase overshoot, noise sensitivity or instability.

Advantages of Proportional Controller

When tuned for the plant, proportional action can provide these benefits:

It can reduce steady-state error and improve response speed, although neither result guarantees better stability.
It is simple to implement and can increase closed-loop bandwidth within stability and actuator limits.

Disadvantages of Proportional Controller

Proportional-only control also has limits:

A type-zero plant generally retains a steady offset under a constant load or step command.
Excessive gain can increase overshoot, amplify measurement noise at the actuator or destabilise the loop.

The following proportional-controller example compares closed-loop stability, transient response and Steady State Error. Figure 1 shows the unity-feedback system.

proportional controller error amplifier block diagram — **Figure-1: A Feedback Control System with Proportional Controller**

The controller is the proportional gain K. The closed-loop characteristic equation is:

s³+3s²+2s+K=0

Applying the Routh-Hurwitz test gives the asymptotically stable range 0<K<6. At K=6, a pole pair lies on the imaginary axis. Values above 6 are unstable. K=0 is also a boundary case and removes proportional control action.

The Root locus in Figure 2 shows how the closed-loop poles move as K changes.

Root locus proportional controller time response — **Figure-2: Root locus of the system shown in Figure-1, Root Locus provides an idea that what should be the value of ‘K’**

A root locus is constructed from the open-loop transfer function G(s)H(s). Its branches trace the roots of the closed-loop characteristic equation. These roots are closed-loop poles, not zeros of that equation.

The root locus helps select K, but stability alone does not choose a design. The stable values K=0.2, K=1 and K=5.8 have different response speed, damping and robustness, as the following cases show.

For this plant, raising K reduces the step error but moves the dominant pole pair toward the imaginary axis as K approaches 6. The result is less damping and smaller stability margins. This trend is specific to the shown loop and should not be treated as a universal gain rule.

For a stable unity negative-feedback loop, the standard static error constants are:

$K_p =\lim_{s\rightarrow 0}KG(s)H(s)$ , Steady state error (e_ss)= $\frac{1}{1+K_p}$ for a unit-step input.

$K_v =\lim_{s\rightarrow 0}sKG(s)H(s)$ , Steady state error (e_ss)= $\frac{1}{K_v}$ for a unit-ramp input.

$K_a =\lim_{s\rightarrow 0}s^2KG(s)H(s)$ , Steady state error (e_ss)= $\frac{1}{K_a}$ for a unit-parabolic input.

If the loop remains stable and K multiplies the open-loop transfer function, increasing K scales finite static error constants and can reduce the associated error. Infinite or zero constants and stability limits require separate checks.

The following three cases use the exact characteristic roots and classical margins for the shown plant.

1. At K=0.2

The characteristic equation is s³+ 3s²+ 2s+0.2=0, with roots -2.088, -0.7909 and -0.1211. All three poles are real and negative, so the stable third-order response is non-oscillatory. The pole at -2.088 is a faster mode, but it still contributes and must not be discarded without an approximation check.

Figure 3 shows the unit-step response without oscillation. Calling this third-order response overdamped is informal; damping-ratio labels apply most directly to a second-order pole pair.

Time response over damped proportional controller — **Figure-3: Response has no oscillations, it is the response of the overdamped system**

The open-loop transfer function is $G(s)H(s)=\frac{0.2}{s(s+1)(s+2)}$ .

Its Gain Margin is 29.5 dB and its phase margin is 81.5°.

A non-oscillatory design can be desirable when overshoot is costly. Closed-loop poles do not need imaginary parts; acceptable locations follow from the response and robustness requirements.

For a standard second-order model, damping ratio above one is overdamped. No single damping ratio, including 0.8, is best for every plant or application.

2. At K=1

The characteristic equation is s³+ 3s²+ 2s+1=0, with roots -2.3247 and -0.3376 ±j0.5623. The real pole is a faster mode; the complex pair dominates the slower oscillatory response.

The complex-conjugate dominant poles have negative real parts, so this stable response is underdamped and oscillatory. Figure 4 shows its unit-step response.

Time response underdamped controller — **Figure-4: Response has oscillations, it is the response of an underdamped system**

The open-loop transfer function is $G(s)H(s)=\frac{1}{s(s+1)(s+2)}$ .

Its gain margin is 15.6 dB and its Phase Margin is 53.4°.

3. At K=5.8

K=5.8 remains inside the stable interval but lies close to the K=6 boundary. The roots show how close the dominant pair is to the imaginary axis.

The roots are -2.9816 and -0.0092±j1.39. The dominant pair has very small negative real parts, which produces slow decay and sustained-looking oscillation. Figure 5 shows the corresponding time response.

Transient response underdamped controller — **Figure-5: Response has oscillations, it is the response of the underdamped system (Response in Figure-4 is also belongs to the underdamped system)**

The open-loop transfer function is $G(s)H(s)=\frac{5.8}{s(s+1)(s+2)}$ .

Its Gain Margin is 0.294 dB and its phase margin is 0.919°.

Both classical margins are close to zero, consistent with the dominant poles lying near the imaginary axis. Small gain, delay or model changes can therefore cross the stability boundary.

Integral Controllers

In ideal integral controllers, the actuating signal is proportional to the accumulated error. Integral action continues changing the command while a persistent error remains.

The ideal integral relationship is:

Introducing the integral gain gives:

Here Ki is the integral gain. Integral action is also called reset action because it can drive a persistent error toward zero in a stable loop with sufficient actuator authority.

Advantages of Integral Controller

Integral action can remove constant offset after a constant disturbance or step command when the closed loop is stable and the actuator is not saturated. The result depends on the plant, feedback structure and system type.

Disadvantages of Integral Controller

Integral action adds phase lag and can reduce stability margins if tuned too aggressively. During actuator saturation, accumulated error can cause integrator windup, so practical designs often use output limits and anti-windup logic.

Derivative Controllers

Derivative controllers respond to the rate of change of error. Pure derivative action is rarely used alone because it has no DC action and is sensitive to high-frequency measurement noise.

Derivative action has zero gain for constant error, so it does not remove steady-state offset.
Ideal differentiation amplifies high-frequency noise and can drive the actuator sharply; practical designs use a filtered derivative term.

For ideal derivative action, the actuating signal is proportional to the derivative of the error.

The ideal derivative relationship is:

Introducing the derivative gain gives:

Here K_d is the derivative gain. Derivative action is also called rate action. A practical first-order filter limits its gain at high frequency.

Advantages of Derivative Controller

When properly filtered and tuned, derivative action can add damping, reduce overshoot and improve settling. Its effect depends on the plant and on whether it acts on error or measurement.

Proportional and Integral Controller

A proportional-integral controller adds proportional action on current error to integral action on accumulated error.

The PI relationship can be written in time or transfer-function form.

Before introducing the two gain constants, the control action has this form:

With proportional and integral gains, it becomes:

Here K_i is the integral gain and k_p is the proportional gain.

PI control combines the immediate response of proportional action with the offset removal of integral action. Tuning must still balance speed, overshoot, stability margins, actuator limits and windup.

An ideal PI transfer function adds one pole at the origin and one zero at the negative ratio of integral gain to proportional gain. Positive gains place that zero in the left half-plane.

The integrator can reduce steady-state error but also adds phase lag. A tuned PI loop can remain stable and robust; the result depends on both gains and the plant.

Figure 6 shows the PI loop. Figure 7 shows its unit-step response for K=5.8 and K_i=0.2. The proportional-only design already had very small margins, and this added integral action makes the illustrated closed loop unstable.

This example does not mean every integrator destabilises a loop. Retuning the proportional and integral gains can produce a stable PI design with adequate margins.

Integral Controller time response — **Figure-6: The closed loop control system with PI Controller**

Integral controller response — **Figure-7: The response of the system shown in Figure-6, with K=5.8, Ki=0.2**

Proportional and Derivative Controller

A proportional-derivative controller adds proportional action on current error to derivative action on the error rate. Practical implementations normally filter the derivative term.

Before introducing the two gain constants, the ideal control action has this form:

With proportional and derivative gains, it becomes:

Here K_d is the derivative gain and K_p is the proportional gain.
PD control can improve damping and transient response, but it does not add integral action to remove constant offset and it requires noise-aware filtering.

A PD term adds a controller zero in the ideal transfer-function model. A zero can reshape the root locus and phase response, but it does not automatically improve stability.

Controller poles and zeros must be selected from the plant model and design requirements. Their effect depends on location, gain, unmodelled dynamics and the complete closed-loop characteristic equation.

Figure 8 shows the PD controller in the open loop G(s)H(s). In a practical filtered PD controller, the derivative filter also adds a high-frequency pole.

Proportional Derivative controller — **Figure-8: Closed-loop control system with PD Controller**

Figure 9 shows the unit-step response for K=5.8 and Td=0.5. Comparing it with Figure 5 shows the effect of derivative action for this plant and these gains.

Proportional derivative controller Time response — **Figure-9: Response of system shown in Figure-8, with K=5.8, Td=0.5**

The ideal PD transfer function is K+Tds, or Td(s+K/Td), so its zero is at -K/Td when Td is non-zero. The gain ratio determines that zero location.

A zero far beyond the loop bandwidth has little effect on the dominant response. A zero near the dominant dynamics has a stronger effect, which may help or harm the design. Check the complete root locus, frequency response, actuator demand and robustness rather than applying a fixed location rule.

As a design tendency, PD action can improve damping and transient response, while PI action can reduce steady-state offset. Neither result is automatic; both controllers must be tuned for the plant.

Proportional plus Integral plus Derivative Controller (PID Controller)

PID controllers are widely used to regulate process variables such as temperature, flow, pressure and speed. They act on current error, integrate error over time and respond to its rate of change.

PID Controller, Proportional integral derivative controller — **Figure-10: Closed loop control system with PID Controller**

Using the notation on this page, the ideal unfiltered PID transfer function is:

$Tds+K+\frac{Ki}{s}$ or $\frac{Tds^2+Ks+ Ki }{s}$

The ideal form has one pole at the origin. The parameters T_d, K and Ki determine up to two numerator zeros. A practical derivative filter adds another pole and limits high-frequency gain.

The gain choices can produce real or complex zeros in the ideal form. Extra parameters give more tuning freedom than P or PI control, but also create more trade-offs. Software can calculate candidate gains, yet the engineer must still validate stability, robustness, noise response, saturation and performance across the operating range.

Figure 11 shows the unit-step response for K=5.8, K_i=0.2 and T_d=0.5. Compare it with Figure 9 to see how integral action changes the PD result for this example.

Time response of PID Controller — **Figure-11: Response of system shown in Figure-10, with K=5.8, Td=0.5, Ki=0.2**

General Guidelines for Designing a PID Controller

A practical PID design process includes these general steps:

Define measurable requirements for tracking, disturbance rejection, noise, actuator use and stability margins.
Build or identify a plant model, then select proportional gain K with stability and robustness checks.
Add integral action when offset removal is required, then check saturation and add anti-windup protection where needed.
Add filtered derivative action when added damping or phase lead is useful. Do not impose a universal damping range; tune against the stated response requirements.
Use analysis or tuning software to inspect closed-loop poles, time responses, margins and sensitivity, then verify the result on the real operating range.
Test model uncertainty, disturbances, sensor noise, actuator limits and implementation timing before accepting the design.

Fuzzy Logic controllers

Fuzzy Logic controllers (FLCs) implement a fuzzy inference system. They can be useful when expert rules or data describe a nonlinear control relationship more conveniently than one fixed linear controller.

An FLC can be designed from expert knowledge or data without using one precise analytical plant model. This does not guarantee good control or disturbance rejection; the rule base, membership functions, input range and closed-loop behaviour still require validation.

An FLC uses fuzzy sets whose membership values range between zero and one. Overlapping membership functions allow a smooth transition between linguistic regions.

Fuzzy inference maps inputs to outputs through membership functions, logical operators, if-then rules, aggregation and defuzzification. Performance relative to PID or another controller depends on the plant, rules, tuning and test criteria.

Fuzzy-controller synthesis starts by defining input and output variables, their valid ranges, membership functions and a rule base. Expert knowledge, measured data or both can support these choices.

Common inputs include measured states, error, error rate and accumulated error. Select only signals that serve the control objective, then validate the inference surface and closed-loop response with representative and edge-case data.