A control system is an arrangement of physical components connected or related in such a manner as to command, direct or regulate itself or another system.

## Open And Closed-Loop Systems

An open-loop system is one in which the control action is independent of the output.

A closed-loop system is one in which the control action is dependent upon the output.

In the above definitions the control action represents the quantity responsible for activating the system to produce a desired output.

## Comparison

Open Loop System | Closed Loop System |
---|---|

Input command is the sole factor responsible for providing the control action. | The control action is provided by the difference between the input command and the corresponding output. |

Its ability to perform is determined by its calibration. As the calibration deteriorates, so too does its performance. | It faithfully reproduces the input owing to feedback. Generally accuracy is high. |

Generally easier build. | Generally complicated and costly. |

Generally not troubled with problems of instability. | Due to feedback action, the system has tendency to oscillate. |

Presence of non-linearity causes malfunctioning. | It usually performs accurately even in the presence of non-linearities. |

## Feedback

Feedback is that characteristic of closed-loop control systems which distinguishes them from open-loop systems. It permits the output to be compared with the input to the system so that the appropriate control action may be formed as some function of the output and input.

### Characteristics Of Feedback

- Increased accuracy.
- Reduced sensitivity of the ratio of output to input to variations in system characteristics.
- Reduced effects of non-linearities and distortion.
- Increased bandwidth.
- Tendency towards oscillation or instability.

**Servo Mechanism** is a power-amplifying feedback control system in which the controlled variable is mechanical position, or a time derivative of position such as velocity or acceleration.

**Regulator Or Regulating System **is a feedback control system in which the reference input or command is constant for long periods of time, often for the entire time interval during which the system is operational.

## Analogue Systems

**Force-Current Analogy**

Mechanical Transnational System | Mechanical Rational System | Electrical System |
---|---|---|

Force (F) | Torque (T) | Current (I) |

Mass (M) | Moment of inertia (M.I) | Capacitance (C) |

Viscous frictional coefficient (f) | Viscous friction coefficient (f) | Reciprocal of resistance (1/R) |

Spring stiffness (K) | Torsional spring stiffness (K) | Reciprocal of inductance (1/L) |

Displacement (x) | Angular Displacement (Q) | Magnetic flux linkage (∅) |

Velocity (x) | Angular velocity (Θ) | Voltage (E) |

**Force-Voltage Analogy**

MECHANICAL TRANSNATIONAL SYSTEM | MECHANICAL RATIONAL SYSTEM | ELECTRICAL SYSTEM |
---|---|---|

Force (F) | Torque (T) | Voltage (E) |

Mass (M) | Moment of inertia (M.I) | Inductance (L) |

Viscous frictional coefficient (f) | Viscous friction coefficient (f) | Resistance (E) |

Spring stiffness (K) | Torsional spring stiffness (K) | Reciprocal of capacitance (1/C) |

Displacement (x) | Angular Displacement (Q) | Charge (q) |

Velocity (x) | Angular velocity (Θ) | Current (i) |

**Thermal, Liquid level, Pneumatic and Electrical Systems**

Electrical System | Thermal System | Liquid Level System | Pneumatic System |
---|---|---|---|

Charge (Coulomb) | Heat flows (Joules) | Liquid flow (cu m) | Air flow (cu m) |

Current (Amperes) | Heat flow rate (Joules/min) | Liquid flow rate (cu m)/min | Air flow rate (cu m/min) |

Voltage (Volts) | Temperature (ºC) | Head (meters) | Pressure (N/m×m) |

Resistance (Ohms) | Resistance (ºC/(Joules/min)) | Resistance (M/((cu m)/min) | Resistance ((N/m×m)/(cu m/min)) |

Capacitance (Farads) | Capacitance (Joules/ºC) | Capacitance ((cu m)/m) | Capacitance ((cu m)/(N/m×m)) |

## Transfer Function

**Open loop control system** is a system in which output has no effect on the control action. The output is not feedback for comparison with the input.

**Closed loop control system** is a system with one or more feedback paths to reduce system error.

**Transfer Function **is the ratio of Laplace transform of the output Y(s) to Laplace of input X(s).

G(s) = Y(s)/U(s) with initial conditions set to zero.

Step involved in obtaining the transfer function are:

- We write the differential equation of the system.
- Replace terms involving d/dt by s and ∫dt by 1/s.
- Eliminate all but the desired variable.

In case of closed loop system where H is the feedback function, the actuating signal E is dependent upon a portion of the output signal HC, as well as reference input R, such that

E = R-HC

(The feedback signal is opposite in sign from the reference input)

But C=G(R-HC)

Hence, C = (G/(1+HG))×R

Thus transfer function of a closed loop system, T is T=C/R = (G/(1+HG))

In general, closed loop transfer function

= Direct transfer function/1+loop transfer function

Also the transfer function of a linear time-invariant system is the ratio of the Laplace transform of the output variable to the Laplace transform of the input variable under the assumption that all initial conditions are zero.

## Stability

A system is stable if its impulse response approaches zero as time approaches infinity. Alternatively a system is stable if every bounded input produces a bounded output.

### System Stability

Stability of a system implies that small changes in the system input do not result in large changes in system output. Stability is a very important characteristic of the transient performance of the system. Almost all working systems are designed to be stable.

The stability of a system is judged by the roots of its characteristic equation:

- A system is said to be absolutely stable when the roots are negative real, or in the case of complex roots have negative real parts, the transient terms all decay with time.
- A system is said to be absolutely unstable when any one of the real roots is positive, or if any of the complex roots has a positive real parts, the transient terms associated with these roots increase with time.
- Marginally stable has some roots with real parts equal to zero, but none with positive real parts.

### Routh Stability Criterion

The routh stability criterion is a method for determining system stability that can be applied to an n^{th} order characteristic equation of the form.

a_{n}s^{n} + a_{n-1}s^{n-1} + ………….. + a_{1}s + a_{0} = 0

A routh table is prepared as defined below:

Where a_{n},a_{n-1}, ……… a_{0} are the coefficients of the characteristic equation and

The table is continued horizontally and vertically until only zeros are obtained.

After the array is completed, the following criterion is applied, the number of changes in sign for the terms in the first column equals the number of roots of the characteristic equation with positive real parts. Hence, by the routh criterion, for a system to be stable the array resulting from its characteristic equation must have a first column with terms of the same sign.

Shortcomings of Routh’s criterion:

- It assumes that characteristic equation is available in polynomial form, which is not necessarily always true.
- Although this criterion gives information about absolute stability, it conveys little or no information about how close the system may be to becoming unstable.
- The Routh array may show no change in sign in the first column but the ensuring dynamic response may be characterized by overshoots so excessive as to render the system useless for control purpose. Thus the system may be relatively unstable in spite of the fact that it is absolutely stable.
- Routh method does not provide the facility for selecting in a simple and direct fashion the parameters of a system component to stabilize the system when it is found to be absolutely unstable.

### Hurwitz Stability Criterion

The Hurwitz stability criterion is another method for determining whether or not all the roots of a characteristic equation have negative real parts. This criterion is applied through the use of determinants formed from the coefficients of characteristic equation. It is assumed that the first coefficient a_{n} is positive. The determinants Δi for i=1,2,3 ……. n-1 are formed as the principle minor determinants of the determinant.All the roots of the characteristic equation have negative real parts if and only if Δi > 0 for i=1,2,3…….. n.

## Other Methods Of Analysis

### Nyquist Criterion

It is frequency domain test which determines whether there are any roots of the characteristic polynomial in the right half of the S plane. It is graphical test based on conformal mapping and complex variable theory.

We know that for stability all roots of the characteristic equation [1+G(s)H(s)=0] must lie on left hand side for the S-plane. Nyquist stability criterion relates the number of zeros and poles of 1+G(s)H(s) that lie on R.H.S of the S-plane to the open-loop frequency response G(jω)H(jω).

Nyquist plot is obtained by drawing its mirror image, ω varies from ∞ to -∞ and number of encirclement’s of point -1+j0 is observed. Clockwise encirclement’s are taken as negative.

Nyquist stability criterion is as “For a closed loop system to be stable, the Nyquist plot of G(s)H(s) must encircle the point -1+j0 as many times as the number of poles G(s)H(s) that are in the right half of s-plane. Clockwise encirclement’s are taken as negative.

Equation generally used is N= P-Z

where N = Number of encirclement’s of point -1+j0

P = Number of poles of G(s)H(s) in right half s-plane.

Z = Number of zeros of 1+G(s)H(s) in right half s-plane.

For stability of closed loop, Z should be zero or N=P.

### Gain Margin

It is the factor by which the system gain can be increased to drive it to the verge of instability, and is reciprocal of the gain at the frequency at which phase angle becomes 180º.

### Phase Margin

It is the amount of additional phase lag at the gain cross over frequency to drive the system to the verge of instability or in other words it the distance from the GH(jω) curve to the critical point (-1+j0).

## Basic Control Components

Transducer, Potentiometer, Synchros, Tachometer, Accelerometer, Strain Gauge, Temperature, Pressure, Gyroscope, Power Actuators, DC Generators, DC Motors.

## Transient And Steady State Response

The transient response is the part of the time response which becomes zero as time becomes large, whereas the steady state response is that part which remains after the transient has died out. The standard test signals for the study of these responses are step input, Ramp input, impulse input and parabolic input.

### Transient Response Characteristic

**Natural frequency**ω**Damping ratio**τ**Delay time**t_{d}. It is the time for response to reach half of its final value.**Rise time**t_{r}. It is the time required by the step function response to reach from 10 to 90% or 0 to 100% of its final value. For undamped systems 0 to 100% is used and for over-damped systems 10 to 90% rise time is used.**Peak time**t_{p}. It is the time taken by the controlled system to reach the peak of the first overshoot.**Maximum overshoot**Mp. Maximum percent overshoot (MPO). MPO decides the relative stability of the system.

= (Maximum value of first overshoot – SS value × 100)/SS value.

**Setting Time**t_{s}. It is the time required by the system to reach and settle within a prescribed percentage of the final or steady value. Usually it is 2 to 5% that is 4 to 5 times constant (τ=1/ω).

## Frequency Response

### Frequency Response Analysis

Here we study the behavior of the system from the steady-state response of the system to a sinusoidal input. In this approach we do not determine the roots of the characteristic equation and the frequency response tests are in general simple and can be made accurately by use of readily available sinusoidal signal generators etc. These methods can be applied to systems which do not have rational functions and to non linear control systems.

## PID Controls

### Compensation Techniques

Unstable system have roots in the right half plane. To stabilize on unstable system, the compensator or controller must move these roots into the left half plane. We can move the roots by

- Placing a dynamic element in the forward transmission path.
- Placing a dynamic element in the feedback path.
- Feeding back all or some states.

Accordingly following compensation techniques are used

- Series or cascade compensation.
- Feedback compensation.
- Load compensation.

Increasing the gain increases the bandwidth and make the response faster and more accurate but usually decrease damping. The damping is improved by introducing a derivative signal and for substantial increase in accuracy, an integrator is used. Based on these concepts commercial controllers are available as:

PD = Proportional+Derivative → G(s)= KP + KD

PI = Proportional+Integral → G(s) = KP + KI/S

PID = Proportional+Integral+Derivative → G(s) = KP + KI/S + KDS

The adjustable parameters are the gains KP,KI and KD.

In **PD controller**,a free zero is introduced and design problem is to place the zero when it will do the most good and adjust the gain accordingly. This is accomplished by Bode or root locus methods.

In **PI controller**, the design problem is again to place the zero at the most suitable location while adjusting the gain. Bode and root locus methods will work.

In **PID controller**, a pole is placed at the origin and two zeros for adjustment of the dynamic response. These 2 zeros may be real or complex depending on the gains used but in either event both zeros will be in the left half plane. The design problem is to place the 2 zeros appropriately, adjust the loop gain and determine the required values of the 3 adjustable gains. Clearly this can be done with bode or root locus methods.