Control System Background

True (for a stable system with normal negative feedback). Negative feedback reduces the time constant of the system, so the output reaches its final value sooner.

Take a first-order plant $G(s) = \dfrac{K}{\tau s + 1}$.

Open loop: for a unit step input

$$c(t) = K\left(1 - e^{-t/\tau}\right), \qquad \text{time constant} = \tau$$

Closed loop (unity feedback):

$$\frac{C(s)}{R(s)} = \frac{G}{1+G} = \frac{K}{\tau s + 1 + K} = \frac{\dfrac{K}{1+K}}{\dfrac{\tau}{1+K}s + 1}$$ $$c(t) = \frac{K}{1+K}\left(1 - e^{-t(1+K)/\tau}\right), \qquad \tau_{cl} = \frac{\tau}{1+K}$$

The closed-loop time constant is smaller by the factor $(1+K)$, so the response is $(1+K)$ times faster. The price is a lower gain, $K/(1+K)$ instead of $K$, which is made up by adding an amplifier in the forward path.

Example: $\tau = 1$ s, $K = 9$. Open loop: $\tau = 1$ s, settling time $4\tau = 4$ s. Closed loop: $\tau_{cl} = 1/10 = 0.1$ s, settling time $= 0.4$ s, i.e. ten times faster.

For second-order systems the same idea holds: feedback raises the natural frequency $\omega_n$, so rise time and peak time fall (but too much gain can increase overshoot).

A control system is an arrangement of physical components connected so that it commands, directs or regulates itself or another system to get a desired output. A closed-loop control system measures the actual output, compares it with the desired value and uses the difference (error) to drive the system.

                       disturbance d
                            |
 r   +----+  e   +------+ u +---+---+   c
---->| Σ  |----->|Contr-|-->| Plant |----+---->
     +----+      | oller|   +-------+    |
      - ^        +------+                |
        |  b    +-----------------+      |
        +-------| Feedback/sensor |<-----+
                +-----------------+

Function of each block

1. Reference input / set point $r$: the desired value of the output (e.g. 25 °C in a room). A reference-input element may convert it into a signal of the same type as the feedback.
2. Summing point (error detector, comparator): subtracts the feedback signal from the reference, $e = r - b$. Example: a potentiometer pair, differential amplifier.
3. Controller: processes the error and produces the control signal $u$ (proportional, PI, PID action). It decides how strongly and how fast to correct.
4. Actuator / final control element: converts the low-power control signal into a high-power action on the plant, e.g. motor, valve, servo, heater contactor. (Often shown inside the controller or plant block.)
5. Plant (process): the system being controlled, e.g. a furnace, a DC motor, a water tank. Its output is the controlled variable $c$.
6. Disturbance $d$: an unwanted input (load change, ambient temperature, noise) that affects the output.
7. Feedback element / sensor: measures the output and converts it to a signal $b$ comparable with $r$ (thermocouple, tachogenerator, potentiometer).

Working

If the output drifts from the set point, the error becomes non-zero, the controller and actuator act to reduce it, and the loop keeps repeating until $c \approx r$. For a negative-feedback system

$$\frac{C(s)}{R(s)} = \frac{G(s)}{1+G(s)H(s)}$$

Example: room temperature control. Thermostat sets $r$; the thermostat sensor measures room temperature $b$; the comparator finds the error; the controller switches the heater/compressor (actuator); the room is the plant.

Control system

A control system is a combination of components arranged so that a desired output is obtained by regulating, commanding or directing the system's input, e.g. temperature control of an oven, speed control of a motor.

Why closed loop dominates modern complex systems

Modern systems (aircraft, robots, power grids, process plants) must be accurate and reliable in spite of changing loads, ageing parts and noise. Open-loop systems cannot correct themselves; closed-loop systems can, because they measure the output and act on the error.

1. Accuracy despite parameter changes: the sensitivity of the closed-loop gain to plant variation is

$$S_G^T = \frac{1}{1+GH} \ll 1 \quad (\text{open loop: } S_G^T = 1)$$

2. Disturbance rejection: a disturbance at the plant gives output $\dfrac{G_2}{1+G_1G_2H}D$ instead of $G_2D$, i.e. it is reduced by $(1+G_1G_2H)$.
3. Faster response: a plant $K/(\tau s+1)$ gets time constant $\tau/(1+K)$ in closed loop.
4. Smaller steady-state error and the ability to track a changing reference (e.g. a missile tracking a target).
5. Stabilizing unstable plants: an inverted pendulum, a fighter aircraft or a magnetic levitation system is unstable open loop and can only work with feedback.
6. Automation: it removes the human operator from the loop, which suits large, fast or remote systems.

The disadvantages (higher cost, need for sensors, possible instability if gain is too high) are small compared with these gains, and cheap sensors and microcontrollers have made feedback easy to add. Hence closed-loop control is the normal choice in modern complex systems.

A permanent mismatch between reference and output is a steady-state error $e_{ss}$. It can be removed by using a closed-loop system with integral action (PI or PID controller), i.e. by raising the type number of the loop.

Justification

For unity feedback, $E(s) = \dfrac{R(s)}{1+G(s)}$ and

$$e_{ss} = \lim_{s\to0} sE(s)$$

For a unit step, $e_{ss} = \dfrac{1}{1+K_p}$ with $K_p = \lim_{s\to0}G(s)$.

• Type-0 system, e.g. $G(s) = \dfrac{K}{\tau s+1}$: $K_p = K$, so $e_{ss} = \dfrac{1}{1+K} \ne 0$. The mismatch is permanent.
• Raising $K$ reduces the error (e.g. $K = 99$ gives 1%) but never makes it zero, and high gain can make higher-order systems oscillatory or unstable.
• Add an integrator (PI controller $K_p + K_i/s$): the loop becomes type 1, $K_p \to \infty$, so

$$e_{ss} = \frac{1}{1+\infty} = 0$$

The integrator keeps changing its output as long as any error exists, so the output is forced to follow the reference exactly.

Other measures

• If the system is open loop, convert it to closed loop with a sensor so that the error is measured and corrected.
• For ramp references, use a type-2 loop (double integration) or a lag compensator to raise $K_v$.
• Calibrate the sensor and feedback element: an error in $H$ appears directly in the output since $C/R \approx 1/H$.

In open loop the step response depends only on the plant; in closed loop the feedback changes the gain, speed, steady-state error and (for second-order systems) the damping.

First-order plant

Take a first-order plant $G(s) = \dfrac{K}{\tau s + 1}$.

Open loop: for a unit step input

$$c(t) = K\left(1 - e^{-t/\tau}\right), \qquad \text{time constant} = \tau$$

Closed loop (unity feedback):

$$\frac{C(s)}{R(s)} = \frac{G}{1+G} = \frac{K}{\tau s + 1 + K} = \frac{\dfrac{K}{1+K}}{\dfrac{\tau}{1+K}s + 1}$$ $$c(t) = \frac{K}{1+K}\left(1 - e^{-t(1+K)/\tau}\right), \qquad \tau_{cl} = \frac{\tau}{1+K}$$

So in closed loop: the time constant falls from $\tau$ to $\tau/(1+K)$ (faster response), and the steady-state gain falls from $K$ to $K/(1+K)$. The steady-state error for unit step is $e_{ss} = \dfrac{1}{1+K}$, which becomes small for large $K$.

Second-order case

For $G(s) = \dfrac{K}{s(s+a)}$, the open-loop step response grows without limit (a ramp after a transient), because of the pole at $s = 0$. With unity feedback:

$$\frac{C(s)}{R(s)} = \frac{K}{s^2 + as + K}, \quad \omega_n = \sqrt{K}, \quad \zeta = \frac{a}{2\sqrt{K}}$$

The output now settles at 1. Increasing $K$ raises $\omega_n$ (faster) and lowers $\zeta$ (more overshoot $M_p = e^{-\zeta\pi/\sqrt{1-\zeta^2}}$).

Summary

Example: $\tau = 1$ s, $K = 9$. Open loop: $\tau = 1$ s, settling time $4\tau = 4$ s. Closed loop: $\tau_{cl} = 1/10 = 0.1$ s, settling time $= 0.4$ s, i.e. ten times faster.

A closed-loop system is one in which the output is measured and fed back to be compared with the desired value; the difference (error) drives the system until the output equals the desired value.

Everyday example: automatic electric iron / water heater with thermostat

Take an electric water heater (geyser):

• Reference: temperature set on the thermostat knob, say 60 °C.
• Sensor (feedback): bimetal strip or thermostat bulb in the water.
• Comparator and controller: thermostat contacts compare actual and set temperature.
• Actuator: heating element switched ON/OFF.
• Plant: water in the tank; output: water temperature.
• Disturbance: cold water entering when hot water is used.

 set temp +   error  +-----------+   +-------+  water temp
 -------->(Σ)------->|Thermostat |-->|Heater |---+------>
           ^ -       |  contacts |   | +tank |   |
           |         +-----------+   +-------+   |
           +-------------[Bimetal sensor]<-------+

When water cools below 60 °C, the error makes the contacts close and heating starts; when 60 °C is reached the heater switches off. The temperature is held near the set point whatever the usage.

Importance of feedback

1. Accuracy: output follows the set point even if the heater ages or supply voltage changes (sensitivity reduced to $1/(1+GH)$).
2. Disturbance rejection: cold-water inflow is automatically corrected.
3. Faster response: time constant reduced from $\tau$ to $\tau/(1+K)$.
4. Safety and automation: no person has to watch the system; overheating is prevented.
5. Smaller steady-state error and ability to stabilize systems that would otherwise run away.

Feedback (a tachogenerator measuring speed) makes an armature-controlled DC motor run faster to respond, and keeps its speed nearly constant under load and parameter changes.

Open-loop model

Motor equations ($L_a$ neglected): $V_a = R_aI_a + K_b\omega$, $T = K_tI_a = J\dot\omega + B\omega + T_L$. Hence

$$\omega(s) = \frac{K_m}{\tau_m s+1}V_a(s) - \frac{K_L}{\tau_m s+1}T_L(s)$$ $$K_m = \frac{K_t}{R_aB + K_tK_b}, \quad K_L = \frac{R_a}{R_aB + K_tK_b}, \quad \tau_m = \frac{R_aJ}{R_aB + K_tK_b}$$

Steady-state speed drop on load: $\Delta\omega_{OL} = K_LT_L$. Any change in $R_a$, $K_t$ or $B$ changes the speed directly.

Closed-loop scheme

 ωr  +    e  +-----+ Va +-------+   ω
 --->(Σ)---->| KA  |--->| Motor |---+--->
      ^ -    +-----+    +-------+   |
      |      +-------+              |
      +------| Kg    |<-------------+
             | tacho |
             +-------+

With amplifier gain $K_A$ and tacho constant $K_g$, loop gain $K = K_AK_mK_g$:

$$\frac{\omega(s)}{\omega_r(s)} = \frac{K_AK_m}{\tau_m s + 1 + K}, \qquad \frac{\omega(s)}{T_L(s)} = \frac{-K_L}{\tau_m s + 1 + K}$$

Comparison

1. Faster response: time constant is reduced by $(1+K)$.
2. Better speed regulation: the load-torque disturbance causes a speed drop $(1+K)$ times smaller.
3. Less sensitivity to changes in $R_a$ (heating), flux or friction.
4. Cost: lower overall gain (made up by $K_A$), a tachogenerator is needed, and with armature inductance included very high $K$ can cause oscillation.

Example: with $K = 19$, a 100 rpm open-loop drop on load becomes only 5 rpm.

The usual traffic-light system of Kathmandu is an open-loop control system, because the lights change according to a fixed time sequence set in a timer. The controller does not measure the actual traffic (queue length, waiting vehicles) and does not change its action according to it.

 timer     +------------+  lights  +---------+ vehicle flow
 setting ->| Timer/     |--------->| Traffic |------------>
 (fixed)   | controller |          | (plant) |
           +------------+          +---------+

Why it is open loop

• Input: preset timing (e.g. 60 s green, 5 s amber).
• Output: flow of vehicles / queue length.
• There is no sensor feeding the traffic condition back to the controller, so the output has no effect on the control action. Even when a road is empty, it still gets its fixed green time.

(When a traffic police officer watches the queues and changes the signal manually, the system becomes a manual closed loop, the officer acting as sensor and controller.)

How to make it closed loop

Install sensors such as inductive loop detectors, cameras with image processing or radar at each approach. The measured queue length or waiting time is fed back to an adaptive controller, which compares it with the desired value and lengthens or shortens the green time.

 desired  +   +----------+  +-------+  +-------+ queue
 queue/   -->(Σ)->|Adaptive |->|Lights |->|Traffic|--+-->
 delay       ^ -  |controller| +-------+  +-------+  |
             |    +----------+                       |
             +----[Loop detector / camera]<----------+

This reduces waiting time and congestion and adapts automatically to peak hours, festivals or accidents.

Internal disturbances are changes inside the system itself: variation of plant parameters (ageing, temperature, wear, supply changes) and internal noise. Feedback reduces their effect because the output is measured and the error is corrected.

1. Parameter variation

Sensitivity to parameter changes. Sensitivity of the overall gain $T$ to a change in $G$ is

$$S_G^T = \frac{\partial T/T}{\partial G/G} = \frac{\partial T}{\partial G}\cdot\frac{G}{T}$$

• Open loop: $T = G$, so $S_G^T = 1$ (a 10% change in $G$ gives a 10% change in output).
• Closed loop: $T = \dfrac{G}{1+GH}$, so $\dfrac{\partial T}{\partial G} = \dfrac{1}{(1+GH)^2}$ and

$$S_G^T = \frac{1}{(1+GH)^2}\cdot\frac{G(1+GH)}{G} = \frac{1}{1+GH}$$

Since $|1+GH| \gg 1$, the effect of a change in $G$ is reduced by the factor $(1+GH)$. (With $GH = 99$, a 10% change in $G$ changes the output by only 0.1%.)

2. Disturbance generated inside the plant

Disturbance rejection. Let the disturbance $D(s)$ enter between two forward blocks $G_1$ (controller) and $G_2$ (plant).

• Open loop: $C_D(s) = G_2(s)D(s)$.
• Closed loop (feedback $H$), with $R = 0$ (superposition):

$$C_D(s) = \frac{G_2(s)}{1 + G_1(s)G_2(s)H(s)}\,D(s)$$

The disturbance output is divided by $(1+G_1G_2H)$. If $|G_1G_2H| \gg 1$, $C_D \approx \dfrac{D}{G_1H}$, which is very small for large $G_1$.

Example

A DC motor's armature resistance rises as it heats. In open loop the speed falls in proportion. With tacho feedback and loop gain $K = 49$, the change in speed is only $1/(1+49) = 2\%$ of the open-loop change.

Hence the closed-loop system is far less affected by internal disturbances, as long as the loop gain is high and the sensor itself is accurate (sensitivity to $H$ is $S_H^T = -GH/(1+GH) \approx -1$).

A control system is an interconnection of components forming a system configuration that will provide a desired system response. It regulates its output (temperature, speed, voltage, position, flow) to match a desired value, usually using feedback.

Significance in engineering

1. Automation: machines and processes run without continuous human attention (washing machine, lifts, CNC machines, industrial plants).
2. Accuracy and quality: products and processes are held at precise values (paper thickness, chemical concentration, voltage regulation).
3. Disturbance rejection: loads and environmental changes are corrected automatically.
4. Speed of response: feedback makes systems respond faster ($\tau \to \tau/(1+K)$).
5. Safety: protection and automatic shutdown (boiler pressure control, aircraft autopilot).
6. Stabilizing unstable systems: rockets, inverted pendulums and fighter aircraft cannot work without control.
7. Energy saving: motors, HVAC and lighting run only as needed.
8. It is interdisciplinary: electrical, mechanical, chemical, civil, biomedical and computer engineering all use the same tools (transfer functions, stability, frequency response).

Example: frequency control of a hydropower generator

In a hydro plant in Nepal, the generator must stay at 50 Hz while consumer load keeps changing.

 50 Hz +    +---------+  +-------+ +--------+ +-----+ f
 ---->(Σ)-->|Governor |->|Servo/ |>|Turbine |>|Gen. |-+->
      ^ -   |         |  | gate  | |        | |     | |
      |     +---------+  +-------+ +--------+ +-----+ |
      +-----------[Speed / frequency sensor]<---------+

When load increases, speed and frequency fall. The sensor detects this, the governor computes the error, and the servomotor opens the wicket gate or needle to admit more water. Turbine power rises and frequency returns to 50 Hz. Without this control system, frequency and voltage would swing with every load change, damaging equipment of consumers. This shows how control systems give stable, accurate and safe operation of engineering systems.

Two important advantages of a closed-loop (feedback) system over an open-loop system are (1) reduced sensitivity to parameter variations and (2) faster response. (Better disturbance rejection is a third.)

Advantage 1: Reduced sensitivity to parameter changes

Sensitivity to parameter changes. Sensitivity of the overall gain $T$ to a change in $G$ is

$$S_G^T = \frac{\partial T/T}{\partial G/G} = \frac{\partial T}{\partial G}\cdot\frac{G}{T}$$

• Open loop: $T = G$, so $S_G^T = 1$ (a 10% change in $G$ gives a 10% change in output).
• Closed loop: $T = \dfrac{G}{1+GH}$, so $\dfrac{\partial T}{\partial G} = \dfrac{1}{(1+GH)^2}$ and

$$S_G^T = \frac{1}{(1+GH)^2}\cdot\frac{G(1+GH)}{G} = \frac{1}{1+GH}$$

Since $|1+GH| \gg 1$, the effect of a change in $G$ is reduced by the factor $(1+GH)$. (With $GH = 99$, a 10% change in $G$ changes the output by only 0.1%.)

Advantage 2: Faster response (smaller time constant)

Take a first-order plant $G(s) = \dfrac{K}{\tau s + 1}$.

Open loop: for a unit step input

$$c(t) = K\left(1 - e^{-t/\tau}\right), \qquad \text{time constant} = \tau$$

Closed loop (unity feedback):

$$\frac{C(s)}{R(s)} = \frac{G}{1+G} = \frac{K}{\tau s + 1 + K} = \frac{\dfrac{K}{1+K}}{\dfrac{\tau}{1+K}s + 1}$$ $$c(t) = \frac{K}{1+K}\left(1 - e^{-t(1+K)/\tau}\right), \qquad \tau_{cl} = \frac{\tau}{1+K}$$

The time constant becomes $\tau/(1+K)$, so the system responds $(1+K)$ times faster. The loss of gain ($K \to K/(1+K)$) is easily recovered with a preamplifier.

Example: $\tau = 1$ s, $K = 9$. Open loop: $\tau = 1$ s, settling time $4\tau = 4$ s. Closed loop: $\tau_{cl} = 1/10 = 0.1$ s, settling time $= 0.4$ s, i.e. ten times faster.

True. An air-conditioning system with a thermostat is a closed-loop control system, because the room temperature (output) is measured and compared with the set temperature, and the difference decides whether the compressor runs.

 set   +  e  +----------+  +----------+  +------+ temp
 ----->(Σ)-->|Thermostat|->|Compressor|->| Room |-+->
        ^ -  +----------+  +----------+  +------+ |
        |                                         |
        +--------[Temperature sensor]<------------+

• Reference: set temperature, e.g. 24 °C.
• Sensor: thermistor in the indoor unit measures room temperature.
• Controller: compares and switches or varies (inverter AC) the compressor speed.
• Plant: the room; disturbances: sunlight, people, opening doors.

If the room becomes warmer than 24 °C, the error is positive and cooling increases; when the set value is reached, the compressor slows or stops. The output itself affects the control action, which is the defining feature of a closed loop.

Note: an old cooler or fan that runs at a fixed setting without a thermostat would be open loop.

The traffic-light system in Kathmandu is open loop because its control action (which light is on and for how long) does not depend on the output (the actual traffic flow or queue length). The lights change according to a timer that is set in advance.

 timer     +------------+  lights  +---------+ vehicle flow
 setting ->| Timer/     |--------->| Traffic |------------>
 (fixed)   | controller |          | (plant) |
           +------------+          +---------+

Reasons

1. No feedback sensor: there are no vehicle detectors (loops, cameras) feeding traffic data to the controller.
2. Fixed time sequence: green, amber and red times are preset; an empty road gets the same green time as a crowded one.
3. No comparison with a desired value: the controller has no error signal; it cannot know whether the queue is growing.
4. Disturbances are not corrected: a sudden rush, accident or VIP movement is not handled automatically. In practice traffic police switch to manual control, which itself shows the automatic system has no feedback.

To make it closed loop, sensors would measure queue length or waiting time, and an adaptive controller would adjust the green time accordingly.

Negative feedback reduces the overall gain and increases the speed of response, both by the factor $(1+GH)$.

Effect on gain

$$\text{Open loop: } T = G, \qquad \text{Closed loop: } T = \frac{G}{1+GH}$$

The gain falls by $(1+GH)$. Example: $G = 100$, $H = 0.1$ gives closed-loop gain $100/11 = 9.09$. For large $GH$, $T \approx 1/H$, so the gain depends mainly on the (accurate) feedback element. The lost gain is recovered with an amplifier.

Effect on speed of response

Take a first-order plant $G(s) = \dfrac{K}{\tau s + 1}$.

Open loop: for a unit step input

$$c(t) = K\left(1 - e^{-t/\tau}\right), \qquad \text{time constant} = \tau$$

Closed loop (unity feedback):

$$\frac{C(s)}{R(s)} = \frac{G}{1+G} = \frac{K}{\tau s + 1 + K} = \frac{\dfrac{K}{1+K}}{\dfrac{\tau}{1+K}s + 1}$$ $$c(t) = \frac{K}{1+K}\left(1 - e^{-t(1+K)/\tau}\right), \qquad \tau_{cl} = \frac{\tau}{1+K}$$

The time constant is reduced by $(1+K)$ and the bandwidth increases by the same factor, so the response is faster. Gain and speed are traded: gain $\times$ bandwidth stays constant ($K/\tau$).

Example: $\tau = 1$ s, $K = 9$. Open loop: $\tau = 1$ s, settling time $4\tau = 4$ s. Closed loop: $\tau_{cl} = 1/10 = 0.1$ s, settling time $= 0.4$ s, i.e. ten times faster.

Which system is more sensitive to disturbance

The open-loop system is more sensitive to disturbance.

Disturbance rejection. Let the disturbance $D(s)$ enter between two forward blocks $G_1$ (controller) and $G_2$ (plant).

• Open loop: $C_D(s) = G_2(s)D(s)$.
• Closed loop (feedback $H$), with $R = 0$ (superposition):

$$C_D(s) = \frac{G_2(s)}{1 + G_1(s)G_2(s)H(s)}\,D(s)$$

The disturbance output is divided by $(1+G_1G_2H)$. If $|G_1G_2H| \gg 1$, $C_D \approx \dfrac{D}{G_1H}$, which is very small for large $G_1$.

Ratio of disturbance outputs:

$$\frac{C_{D,\,cl}}{C_{D,\,ol}} = \frac{1}{1+G_1G_2H}$$

Since $|1+G_1G_2H| > 1$ (usually $\gg 1$), the closed-loop output is much less affected. Example: $G_1G_2H = 49$ means the disturbance effect is only 2% of that in open loop. The open loop has no way to sense the change in output, so the disturbance passes fully to the output.

Effect of gain on the response

Take unity feedback with $G(s) = \dfrac{K}{s(s+2)}$:

$$\frac{C(s)}{R(s)} = \frac{K}{s^2+2s+K}, \quad \omega_n = \sqrt{K}, \quad \zeta = \frac{1}{\sqrt{K}}$$

As gain $K$ increases:

1. Speed increases: $\omega_n$ rises, rise time and peak time fall.
2. Steady-state error falls: $e_{ss}$ for step ($1/(1+K_p)$) or ramp ($1/K_v$) decreases.
3. Damping falls: $\zeta$ decreases, so overshoot and oscillation increase.
4. Disturbance and sensitivity effects fall as $1/(1+GH)$.
5. Stability may be lost: for third- and higher-order systems, e.g. $\dfrac{K}{s(s+1)(s+2)}$, Routh's test shows instability for $K > 6$.

So the gain is chosen as a compromise between accuracy/speed and relative stability.

Closed-loop systems need sensors, comparators and more careful design, so they cost more and can become unstable. Still they are widely preferred because their performance advantages are much larger than these costs.

Justification

1. Accuracy: the output is forced to follow the reference; sensitivity to plant changes is $1/(1+GH)$ instead of 1.
2. Disturbance rejection: load changes and noise are corrected; their effect is reduced by $(1+GH)$.
3. Faster response: time constant $\tau \to \tau/(1+K)$.
4. Can use cheaper, less precise plant components: feedback compensates for their tolerance and ageing, which often saves overall cost.
5. Stabilizes unstable systems and allows full automation with less human labour.
6. Cheap electronics today: sensors and microcontrollers are inexpensive, so the extra cost is small.

Examples

In each case open-loop control would give large errors whenever load or environment changes, so the extra cost of feedback is justified.

Speed of response is measured by the time constant (first order) or natural frequency (second order); both improve as the loop gain $K$ rises.

First-order system

With unity feedback around $G(s) = \dfrac{K}{\tau s+1}$:

$$\frac{C(s)}{R(s)} = \frac{K}{\tau s+1+K} = \frac{K/(1+K)}{\dfrac{\tau}{1+K}s+1}$$ $$\tau_{cl} = \frac{\tau}{1+K}, \qquad t_s \approx 4\tau_{cl} = \frac{4\tau}{1+K}$$

As $K$ increases, $\tau_{cl}$ decreases, so the response reaches its final value faster. Example ($\tau = 1$ s): $K = 1$ gives $\tau_{cl} = 0.5$ s; $K = 9$ gives $0.1$ s.

Second-order system

For $G(s) = \dfrac{K}{s(s+a)}$: $\omega_n = \sqrt{K}$. Rise time and peak time $t_p = \pi/(\omega_n\sqrt{1-\zeta^2})$ decrease as $K$ rises (though overshoot increases).

Hence speed of response increases with gain.

i) Governor system of a hydropower station: closed-loop control

The governor keeps the turbine-generator speed (hence frequency, 50 Hz) constant while the electrical load changes. It measures the actual speed and corrects the water flow, so it is a closed-loop (feedback) system.

 Nref +   e   +---------+ u +--------+ q +-------+ N  
 ---->(Σ)---->|Governor |-->|Servo + |-->|Turbine|-+->
      ^ -     |(PID)    |   |gate    |   |+ Gen. | |
      |       +---------+   +--------+   +-------+ |
      |                            load ΔPL -->    |
      +------------[Speed sensor / PMG]<-----------+

When load rises, speed falls, $e$ becomes positive, the gate opens and the turbine power increases until speed returns to normal.

ii) Traffic light system of Kathmandu: open-loop control

The lights change according to preset timers; no measurement of traffic is fed back.

 preset   +-----------+ signal +---------+ vehicle flow
 timing ->| Timer /   |------->| Traffic |------------->
          | controller|        | junction|
          +-----------+        +---------+
                                   ^ disturbance:
                                   | traffic volume

Because the output (queue length) never affects the controller, the system is open loop. It can be made closed loop by adding vehicle detectors or cameras whose data adjust the green time.

Linear time-invariant (LTI) system

A system is linear if it obeys the principle of superposition (additivity and homogeneity): if $r_1 \to c_1$ and $r_2 \to c_2$, then $a r_1 + b r_2 \to a c_1 + b c_2$. It is time-invariant if its parameters do not change with time, so a delayed input gives an equally delayed output: $r(t-T) \to c(t-T)$.

An LTI system is described by a linear differential equation with constant coefficients, e.g.

$$a_2\ddot c + a_1\dot c + a_0c = b_0r$$

and can be represented by a transfer function $G(s) = C(s)/R(s)$. Example: an RLC circuit with fixed R, L, C.

"Modern complex systems are more pronounced with closed-loop control"

Modern systems (power grids, aircraft, robots, process plants) have many interacting parts, changing loads and strict accuracy needs. Closed-loop control suits them because:

1. Low sensitivity: $S_G^T = \dfrac{1}{1+GH}$, so ageing and parameter changes hardly affect the output.
2. Disturbance rejection: load and noise effects are cut by $(1+GH)$.
3. Faster response: time constant $\tau/(1+K)$.
4. Stabilization: inherently unstable systems (fighter aircraft, maglev, rockets) can be operated.
5. Automation: cheap sensors and digital controllers make feedback easy and allow operation without humans.

So the statement is justified; open loop is used only where the system is simple and well known (washing-machine timer, traffic timer).

A closed-loop system measures the output and corrects any error, so it both rejects disturbances and tracks the command input better than an open-loop system.

Disturbance rejection

Disturbance rejection. Let the disturbance $D(s)$ enter between two forward blocks $G_1$ (controller) and $G_2$ (plant).

• Open loop: $C_D(s) = G_2(s)D(s)$.
• Closed loop (feedback $H$), with $R = 0$ (superposition):

$$C_D(s) = \frac{G_2(s)}{1 + G_1(s)G_2(s)H(s)}\,D(s)$$

The disturbance output is divided by $(1+G_1G_2H)$. If $|G_1G_2H| \gg 1$, $C_D \approx \dfrac{D}{G_1H}$, which is very small for large $G_1$.

In open loop the whole disturbance $G_2D$ reaches the output, because nothing senses it.

Command input tracking

Tracking error for unity feedback: $E(s) = \dfrac{R(s)}{1+G(s)}$, and $e_{ss} = \lim_{s\to0}sE(s)$.

• Step input: $e_{ss} = \dfrac{1}{1+K_p}$; with an integrator in $G$ (type 1), $e_{ss} = 0$.
• In open loop, the output equals the reference only if the plant gain is exactly calibrated; any change in the plant gives an error, since sensitivity is 1. In closed loop the sensitivity is $1/(1+GH)$.

Also the closed-loop time constant $\tau/(1+K)$ is smaller, so the output follows a changing reference more quickly.

Example: in a motor speed system with loop gain 99, a load disturbance or 10% change in motor gain changes speed by only about 0.1% in closed loop, compared with 10% in open loop.

A general (closed-loop) control system is shown below with its main blocks and signals.

                                disturbance d
                                     |
 v  +-------+ r  +    e  +-------+ u  v  +-------+  c
 -->|Ref.   |--->(Σ)---->|Contr- |--->+->| Plant |--+-->
    |input  |     ^ -    |oller +|       |       |  |
    |element|     |      |actua- |       +-------+  |
    +-------+     |  b   |tor    |                  |
                  |      +-------+                  |
                  +------[Feedback element H]<------+

Blocks

• Reference input element: converts the command $v$ (e.g. knob setting) into reference signal $r$ of the same kind as the feedback.
• Summing point (error detector): forms $e = r - b$.
• Controller (with actuator): converts error into manipulated variable $u$ (P, PI, PID action; motor, valve, amplifier).
• Plant / process: the system to be controlled.
• Feedback element (sensor): measures $c$ and gives $b = Hc$.

Variables

The relation is $\dfrac{C(s)}{R(s)} = \dfrac{G(s)}{1+G(s)H(s)}$.

The statement is justified (with the condition that the gain increased is the one in the loop before the point where the disturbance enters, and the system stays stable).

Let the disturbance $D(s)$ enter between $G_1(s)$ (controller) and $G_2(s)$ (plant), with feedback $H(s)$.

 R  +      +----+ +  +----+     C
 -->(Σ)--->| G1 |-->(Σ)->| G2 |--+-->
     ^ -   +----+    ^ D +----+  |
     |       +----+              |
     +-------| H  |<-------------+
             +----+

With $R = 0$ (superposition), the output due to the disturbance is

$$C_D(s) = \frac{G_2(s)}{1+G_1(s)G_2(s)H(s)}\,D(s)$$

For large loop gain, $|G_1G_2H| \gg 1$:

$$C_D(s) \approx \frac{G_2}{G_1G_2H}D(s) = \frac{D(s)}{G_1(s)H(s)}$$

So increasing $G_1$ and/or $H$ makes $C_D$ smaller; the disturbance is suppressed. In open loop, $C_D = G_2D$, with no reduction at all.

Output due to reference: $C_R = \dfrac{G_1G_2}{1+G_1G_2H}R \approx \dfrac{R}{H}$, which is almost independent of the gains, so tracking is not harmed.

Limits

• Raising $G_2$ alone does not help much: the result tends to $D/(G_1H)$, which does not depend on $G_2$.
• Raising $H$ also lowers the closed-loop gain ($\approx 1/H$), so it must be chosen with the required output level in mind.
• Very high gain can reduce damping or cause instability in higher-order systems, and it amplifies sensor noise.

A control system is an arrangement of physical components connected so as to command, direct or regulate itself or another system to give a desired output. In a closed-loop system the output is measured and fed back, and the error between the desired and actual output drives the system.

                       disturbance d
                            |
 r   +----+  e   +------+ u +---+---+   c
---->| Σ  |----->|Contr-|-->| Plant |----+---->
     +----+      | oller|   +-------+    |
      - ^        +------+                |
        |  b    +-----------------+      |
        +-------| Feedback/sensor |<-----+
                +-----------------+

Function of each block

1. Reference input $r$ (set point): desired value of the output.
2. Error detector / summing point: compares $r$ with feedback $b$ and gives error $e = r - b$.
3. Controller: acts on the error (P, PI, PID) and produces control signal $u$.
4. Actuator: power element (motor, valve, amplifier) that applies $u$ to the plant.
5. Plant / process: the system being controlled; gives the controlled output $c$.
6. Disturbance $d$: unwanted input acting on the plant (load change, noise).
7. Feedback element / sensor: measures $c$ and converts it to $b$, comparable with $r$.

The closed-loop transfer function is $\dfrac{C(s)}{R(s)} = \dfrac{G(s)}{1+G(s)H(s)}$.

Example: in a motor speed control, $r$ = speed set by a potentiometer, controller = amplifier, actuator and plant = DC motor and load, sensor = tachogenerator.

Advantages of closed loop over open loop

1. More accurate: output follows the reference; steady-state error is small (zero with integral action).
2. Less sensitive to parameter changes: $S_G^T = \dfrac{1}{1+GH}$ compared with 1 for open loop, so ageing, temperature and supply variations hardly affect the output.
3. Disturbance rejection: a disturbance at the plant is reduced by $(1+GH)$.
4. Faster response, larger bandwidth: a plant $K/(\tau s+1)$ gets time constant $\tau/(1+K)$.
5. Can stabilize unstable plants (e.g. inverted pendulum, aircraft).
6. Automatic operation: no human operator is needed to watch and correct the output.
7. Reduces non-linear effects and distortion (used in feedback amplifiers).

(Its drawbacks are higher cost, more components, lower overall gain and the possibility of instability if the gain is too high.)

Feedback changes the dynamic (transient) response of a system: it changes the time constant, bandwidth, damping and stability.

1. Time constant and speed

Take a first-order plant $G(s) = \dfrac{K}{\tau s + 1}$.

Open loop: for a unit step input

$$c(t) = K\left(1 - e^{-t/\tau}\right), \qquad \text{time constant} = \tau$$

Closed loop (unity feedback):

$$\frac{C(s)}{R(s)} = \frac{G}{1+G} = \frac{K}{\tau s + 1 + K} = \frac{\dfrac{K}{1+K}}{\dfrac{\tau}{1+K}s + 1}$$ $$c(t) = \frac{K}{1+K}\left(1 - e^{-t(1+K)/\tau}\right), \qquad \tau_{cl} = \frac{\tau}{1+K}$$

Feedback reduces the time constant by $(1+K)$, so the response is faster and the bandwidth increases from $1/\tau$ to $(1+K)/\tau$.

2. Damping and overshoot (second order)

With $G(s) = \dfrac{K}{s(\tau s+1)}$ and unity feedback:

$$\frac{C(s)}{R(s)} = \frac{K/\tau}{s^2 + s/\tau + K/\tau}, \quad \omega_n = \sqrt{\frac{K}{\tau}}, \quad \zeta = \frac{1}{2\sqrt{K\tau}}$$

Raising the feedback loop gain $K$ increases $\omega_n$ (faster rise) but reduces $\zeta$, so overshoot $M_p = e^{-\pi\zeta/\sqrt{1-\zeta^2}}$ increases. Adding rate (tachometer) feedback $K_ts$ changes damping to

$$\zeta' = \frac{1 + KK_t}{2\sqrt{K\tau}}$$

which reduces overshoot without changing $\omega_n$.

3. Stability

Feedback moves the poles. It can stabilize an unstable plant (e.g. $1/(s-1)$ with gain $K > 1$ gives the stable pole $s = 1 - K$) or destabilize a stable one when gain is too high (e.g. $K/[s(s+1)(s+2)]$ is unstable for $K > 6$).

4. Reduced effect of disturbances and parameter changes on the transient

Both are reduced by $(1+GH)$, so the transient shape stays nearly the same even when plant parameters drift.

A control system is judged by three basic requirements: how fast it responds, how accurately it follows the input, and whether it is stable.

(i) Speed

Speed is the quickness of the transient response. It is measured by:

• time constant $\tau$ (first order), settling time $t_s \approx 4\tau$;
• rise time $t_r$, peak time $t_p = \dfrac{\pi}{\omega_n\sqrt{1-\zeta^2}}$, settling time $t_s = \dfrac{4}{\zeta\omega_n}$ (2% band);
• bandwidth in frequency response (larger bandwidth = faster).

(ii) Accuracy

Accuracy is how closely the output follows the input in steady state, measured by steady-state error

$$e_{ss} = \lim_{s\to0} sE(s)$$

with error constants $K_p$, $K_v$, $K_a$: step $e_{ss} = 1/(1+K_p)$, ramp $1/K_v$, parabola $1/K_a$. Higher type number or gain gives better accuracy.

(iii) Stability

A system is stable if every bounded input gives a bounded output (BIBO); for an LTI system all closed-loop poles must lie in the left half of the s-plane. Tests: Routh–Hurwitz, root locus, Bode, Nyquist. Relative stability (how far from instability) is measured by damping ratio, peak overshoot, gain margin and phase margin.

These three conflict: higher gain improves speed and accuracy but reduces damping and can cause instability, so design is a compromise.

A feedback system senses the effect of the disturbance on the output and produces a correcting control action, so the disturbance is largely cancelled.

Disturbance rejection. Let the disturbance $D(s)$ enter between two forward blocks $G_1$ (controller) and $G_2$ (plant).

• Open loop: $C_D(s) = G_2(s)D(s)$.
• Closed loop (feedback $H$), with $R = 0$ (superposition):

$$C_D(s) = \frac{G_2(s)}{1 + G_1(s)G_2(s)H(s)}\,D(s)$$

The disturbance output is divided by $(1+G_1G_2H)$. If $|G_1G_2H| \gg 1$, $C_D \approx \dfrac{D}{G_1H}$, which is very small for large $G_1$.

Example: $G_1G_2H = 99$ means only 1% of the disturbance effect remains at the output, compared with 100% in open loop. In a motor speed system, a sudden load torque lowers the speed; the tachometer detects this, the error rises, and the amplifier increases armature voltage to restore the speed.

True (for negative feedback in a stable system). Feedback reduces the time constant and increases the bandwidth.

Take a first-order plant $G(s) = \dfrac{K}{\tau s + 1}$.

Open loop: for a unit step input

$$c(t) = K\left(1 - e^{-t/\tau}\right), \qquad \text{time constant} = \tau$$

Closed loop (unity feedback):

$$\frac{C(s)}{R(s)} = \frac{G}{1+G} = \frac{K}{\tau s + 1 + K} = \frac{\dfrac{K}{1+K}}{\dfrac{\tau}{1+K}s + 1}$$ $$c(t) = \frac{K}{1+K}\left(1 - e^{-t(1+K)/\tau}\right), \qquad \tau_{cl} = \frac{\tau}{1+K}$$

The time constant falls from $\tau$ to $\tau/(1+K)$, so the response is $(1+K)$ times faster, at the cost of reduced gain $K/(1+K)$.

Example: $\tau = 1$ s, $K = 9$. Open loop: $\tau = 1$ s, settling time $4\tau = 4$ s. Closed loop: $\tau_{cl} = 1/10 = 0.1$ s, settling time $= 0.4$ s, i.e. ten times faster.

True, provided the gain is increased in the forward path before the point where the disturbance enters (and stability is kept).

Let the disturbance $D(s)$ enter between $G_1(s)$ and $G_2(s)$, with feedback $H(s)$. With $R = 0$:

$$C_D(s) = \frac{G_2}{1+G_1G_2H}\,D(s) \approx \frac{D(s)}{G_1H} \quad (|G_1G_2H| \gg 1)$$

Increasing the forward gain $G_1$ increases the denominator, so the output caused by the disturbance falls. Meanwhile the output due to the reference, $C_R = \dfrac{G_1G_2}{1+G_1G_2H}R \approx R/H$, is almost unchanged.

Example: raising $G_1G_2H$ from 9 to 99 cuts the disturbance effect from $1/10$ to $1/100$ of the open-loop value.

Note: increasing only $G_2$ (after the disturbance) does not help, since $C_D \to D/(G_1H)$; and very high gain may reduce damping or cause instability.

Open-loop control system

A system in which the control action does not depend on the output. The output is neither measured nor fed back.

 r   +------------+  u   +-------+   c
---->| Controller |----->| Plant |------>
     +------------+      +-------+

Examples: electric toaster with timer, washing machine on timer, fixed-time traffic lights. They are simple and cheap but cannot correct errors caused by disturbances or parameter changes.

Closed-loop control system

A system in which the output is measured and fed back to the input, and the control action depends on the error between the desired and actual output. Examples: room air conditioner with thermostat, motor speed control with tachometer, voltage regulator.

                       disturbance d
                            |
 r   +----+  e   +------+ u +---+---+   c
---->| Σ  |----->|Contr-|-->| Plant |----+---->
     +----+      | oller|   +-------+    |
      - ^        +------+                |
        |  b    +-----------------+      |
        +-------| Feedback/sensor |<-----+
                +-----------------+

Role of each block

1. Reference input $r$ (set point): desired value of the output.
2. Error detector / summing point: compares $r$ with feedback $b$ and gives error $e = r - b$.
3. Controller: acts on the error (P, PI, PID) and produces control signal $u$.
4. Actuator: power element (motor, valve, amplifier) that applies $u$ to the plant.
5. Plant / process: the system being controlled; gives the controlled output $c$.
6. Disturbance $d$: unwanted input acting on the plant (load change, noise).
7. Feedback element / sensor: measures $c$ and converts it to $b$, comparable with $r$.

The overall transfer function is $\dfrac{C(s)}{R(s)} = \dfrac{G(s)}{1+G(s)H(s)}$, where $G$ is the forward path (controller, actuator, plant) and $H$ the feedback element.

Comparison