Open-Loop and Closed-Loop Control

A control system is an interconnection of components arranged to command, direct, or regulate a plant so that its output follows a desired reference. In its simplest form,

$$\boxed{\text{input command}\longrightarrow\text{control system} \longrightarrow\text{controlled output}}.$$

Examples include motor-speed control, room-temperature control, antenna positioning, tank-level control, aircraft autopilots, and regulated power supplies.

Basic Terminology

Term	Meaning
Reference input	Desired value of the output; also called the command input.
Controlled output	Quantity that the system must regulate.
Plant or process	Physical system being controlled.
Controller	Element that generates the control action.
Actuator	Device that applies control energy to the plant.
Sensor or feedback element	Measures the output and produces a feedback signal.
Error signal	Difference between the reference and feedback signals.
Disturbance	Unwanted input that affects the output.

Basic control-system terms.

For the standard feedback representation,

$$\boxed{e(t)=r(t)-b(t)},$$

where $r(t)$ is the reference, $b(t)$ is the feedback signal, and $e(t)$ is the error. The controller acts on this error to reduce the difference between the desired and actual outputs.

Open-Loop Control

An open-loop control system has control action that is independent of the output. There is no feedback path and therefore no automatic correction of output error. If $G_c(s)$ is the controller transfer function and $G_p(s)$ is the plant transfer function, then

Caution

Key Formula — Open-loop input–output relation

$$\frac{C(s)}{R(s)}=G_c(s)G_p(s).$$

System	Why it is open loop
Electric toaster	Heating time is preset; toast colour is not measured.
Timer-based washing machine	The cycle runs for a fixed time regardless of cleanliness.
Fixed-time traffic signal	Switching follows a schedule rather than measured traffic density.
Open-loop stepper motor	Pulses command position, but actual shaft position is not measured.

Typical open-loop systems.

Note

Key Point — Advantages of open-loop control

Open-loop systems are simple, inexpensive, easy to maintain, need no sensor, and can respond quickly when their calibration remains accurate. They are not subject to instability created by a feedback loop.

Their limitations are the absence of automatic error correction, high sensitivity to disturbances and parameter variations, reduced accuracy, frequent recalibration, and inability to compensate automatically for load changes.

Closed-Loop Control

A closed-loop control system measures the output, compares it with the reference, and uses the resulting error to generate corrective action.

Canonical single-loop feedback system.

For the negative-feedback loop in the figure,

$$\begin{aligned} E(s)&=R(s)-B(s), & B(s)&=H(s)C(s), & C(s)&=G(s)E(s). \end{aligned}$$

Substitution gives

$$\begin{aligned} C(s)&=G(s)\bigl[R(s)-H(s)C(s)\bigr],\\ C(s)\bigl[1+G(s)H(s)\bigr]&=G(s)R(s). \end{aligned}$$

Hence,

Caution

Key Formula — Closed-loop transfer functions

$$\boxed{\frac{C(s)}{R(s)}=\frac{G(s)}{1+G(s)H(s)}} \qquad\text{(negative feedback)}.$$

For unity feedback, $H(s)=1$ and $C/R=G/(1+G)$. For positive feedback,

$$\boxed{\frac{C(s)}{R(s)}=\frac{G(s)}{1-G(s)H(s)}}.$$

The corresponding characteristic equations are $1+G(s)H(s)=0$ and $1-G(s)H(s)=0$, respectively.

Closed-loop control improves accuracy, corrects errors automatically, reduces the effects of disturbances and parameter variation, increases useful bandwidth, and often improves linearity. These benefits require extra sensors and hardware; poor loop design can cause instability, and noisy feedback can inject measurement noise into the control action.

Open-Loop versus Closed-Loop Systems

Feature	Open loop	Closed loop
Feedback	Absent	Present
Control action	Independent of output	Depends on output
Error correction	Not automatic	Automatic
Accuracy	Lower	Higher
Disturbance sensitivity	High	Low when designed correctly
Parameter sensitivity	High	Reduced by loop gain
Stability	No feedback-induced instability	May become unstable
Complexity and cost	Low	Higher
Calibration	Often required	Less frequent
Examples	Toaster, timer-based washer	Thermostat, speed control, autopilot

Comparison of open-loop and closed-loop control.

Tip

Exam Focus

Use open-loop control when the input–output relation is accurately known and disturbances are negligible. Use closed-loop control when accuracy, disturbance rejection, tracking, and robustness are important.

Feedback Concepts

Feedback is the process of returning a measured portion of the output to the input for comparison with the reference. The comparison produces the error signal that drives the controller.

Type	Error signal	Principal effect
Negative feedback	$E(s)=R(s)-H(s)C(s)$	Opposes the error, usually improving accuracy and robustness when the loop has adequate stability margins.
Positive feedback	$E(s)=R(s)+H(s)C(s)$	Reinforces the error, increasing effective gain and tending toward oscillation or instability.

Negative and positive feedback.

Note

Key Point — Effects of negative feedback

With a properly designed loop, negative feedback improves accuracy and linearity, reduces steady-state error, parameter sensitivity, and disturbance effects, and usually increases bandwidth. These benefits are obtained at the cost of reduced overall gain. Negative feedback does not guarantee stability: frequency-dependent phase lag can make the returned signal effectively positive and destabilize the loop.

For example, let a first-order plant $G(s)=K/(1+s\tau)$ have constant negative feedback $H$. Its closed-loop transfer function can be written as

Caution

Key Formula — First-order bandwidth effect

$$T(s)=\frac{K/(1+KH)}{1+s\tau/(1+KH)}, \qquad \tau_c=\frac{\tau}{1+KH}.$$

Thus, for $K>0$, $H>0$, and a stable closed loop, the pole bandwidth increases by the factor $1+KH$ while the DC gain decreases to $K/(1+KH)$.

The product of the forward and feedback paths is the loop gain:

Caution

Key Formula — Loop gain

$$\boxed{L(s)=G(s)H(s)}.$$

It governs accuracy, sensitivity, disturbance rejection, and stability margins. Since

$$T(s)=\frac{G(s)}{1+G(s)H(s)} =\frac{1}{H(s)}\frac{L(s)}{1+L(s)},$$

then $T(s)\approx 1/H(s)$ where $|L(s)|\gg 1$, provided the loop remains well away from the critical point $L=-1$. The closed-loop behavior there is set mainly by the feedback element rather than by forward-path gain variations.

Classification of Control Systems

Control systems are classified independently along several axes; a given system may, for example, be nonlinear, time varying, discrete time, and MIMO at once.

Linear and Nonlinear Systems

A linear system obeys homogeneity and additivity, and hence superposition:

$$ax_1(t)+bx_2(t)\ \longrightarrow\ ay_1(t)+by_2(t).$$

Ideal RLC networks, small-signal amplifier models, and linearized motor models are common examples. A nonlinear system does not satisfy superposition; saturation, dead zone, backlash, relay action, diode characteristics, and magnetic hysteresis are typical nonlinearities.

Time-Invariant and Time-Varying Systems

A time-invariant system has parameters that do not change with time, such as a fixed RLC network. A time-varying system has one or more time-dependent parameters; the decreasing mass of a missile as fuel burns is a standard example.

Continuous-Time and Discrete-Time Systems

Type	Signal nature	Example
Continuous time	Signals are defined for every value of time.	Analog motor-speed control.
Discrete time	Signals are represented at distinct sampling instants.	Digital control using a microcontroller.

Classification by the time variable.

SISO and MIMO Systems

Type	Meaning	Example
SISO	Single input, single output.	Heater temperature control.
MIMO	Multiple inputs, multiple outputs.	Aircraft flight-control system.

Classification by numbers of inputs and outputs.

Regulator and Tracking Systems

A regulator holds its output at a prescribed value despite disturbances; a voltage regulator is the standard example. A tracking system makes its output follow a changing command, as when a radar antenna follows a target.

Deterministic and Stochastic Systems

A deterministic model has prescribed inputs and parameters with no random quantities. A stochastic model represents random inputs, parameters, or noise statistically.

Lumped and Distributed Parameter Systems

In a lumped-parameter model, state variables vary only with time and are described by ordinary differential equations. In a distributed-parameter model, variables also depend on spatial position and are generally described by partial differential equations.

Servomechanisms

A servomechanism is a closed-loop control system whose controlled output is a mechanical position, velocity, or acceleration. Feedback makes that mechanical output follow the command accurately.

Closed-loop servo position-control system with position feedback.

In the position-control loop of the figure, the error detector compares commanded and measured positions. The amplifier raises the error signal to a useful power level, the servomotor produces mechanical motion, and the sensor closes the loop.

Component	Function
Error detector	Compares desired and actual position.
Servo amplifier	Amplifies and conditions the error signal.
Servomotor	Converts the electrical control signal into mechanical motion.
Load	Mechanical system that must be positioned or driven.
Feedback sensor	Measures position or speed and produces the feedback signal.

Elements of a basic servo system.

Applications include radar-antenna positioning, CNC machine tools, robotic arms, aircraft control surfaces, disk-drive head positioning, and automatic steering systems.

Feature	DC servomotor	AC servomotor
Supply	DC	AC
Control	Armature or field control	Usually control-phase voltage
Response	Fast and straightforward to control	Smooth and reliable
Maintenance	Brushes require maintenance	Many types are brushless and need less maintenance
Applications	Robotics and position control	Instrument servos and low-power control

Comparison of DC and AC servomotors.

Tip

Exam Focus — Servo identification

Every servomechanism is a closed-loop tracking system, but not every closed-loop system is a servomechanism. The word servo specifically indicates controlled mechanical position, velocity, or acceleration.