Control Systems Formulas

Representation of Dynamic Systems

Transfer Function

The transfer function of a linear time-invariant (LTI) system relates the output to the input in the Laplace domain.

Transfer Function ()

Where:

• : Transfer function.
• : Output in the Laplace domain.
• : Input in the Laplace domain.
• : Numerator (polynomial of the inputs).
• : Denominator (characteristic polynomial of the system).

Differential Equation and State-Space Model

Differential Equation

State-Space Representation

Where:

• : State vector.
• : System matrix (state).
• : Input matrix.
• : Output matrix.
• : Direct feedback matrix.

Stability of Systems

Routh-Hurwitz Stability Criterion

A system is stable if all the roots of the characteristic polynomial have negative real parts.

Stability Condition

For a system with the characteristic polynomial:

The Routh-Hurwitz criterion states that all coefficients of the first row of the Routh table must have the same sign.

Nyquist Stability Criterion

The Nyquist criterion uses the frequency response to determine the stability of a closed-loop system through the Nyquist plot.

Analysis of Time Response

The time response of a system describes how its output changes over time after a specific input.

a) Unit Step Response (Step Input)

For a second-order system with transfer function:

Where:

• : Natural frequency of the system.
• : Damping ratio.

Response Characteristics:

Delay Time ()

Rise Time ()

Approximately for an underdamped system:

Maximum Overshoot ()

Settling Time ()

For a 2% criterion:

First Order Systems

General Transfer Function

The transfer function of a first-order system has the following form:

Where:

• : Transfer function
• : Static gain of the system
• : Time constant
• : Variable in the Laplace domain

State-Space Equation

The state-space representation of a first-order system can be expressed as:

Where:

• : Derivative of the state variable
• : Input of the system
• : Output of the system

Time Response

Unit Step Response

The unit step response of a first-order system is:

Where:

• : Output at time
• : Static gain
• : Time constant
• : Time

Rise Time ()

The rise time for a first-order system, defined as the time it takes for the response to go from 10% to 90% of its final value, is:

Settling Time ()

The settling time, defined as the time it takes for the system to reach and stay within 2% of its final value, is:

Delay Time ()

The delay time, defined as the time it takes for the output to reach 50% of its final value, is:

Stability Analysis

Stability Conditions

A first-order system is stable if its time constantis positive:

If, the system is unstable.

Steady-State Error

The steady-state error for a unit step input is:

If, the steady-state error approaches zero.

Frequency Analysis

Frequency Response

The transfer function in the frequency domain is obtained by substitutinginto the transfer function:

Where:

• : Angular frequency

Magnitude

The magnitude of the frequency response is:

Phase

The phase of the frequency response is:

Second Order Systems

General Transfer Function

The transfer function of a second-order system is given by the following equation:

Where:

• : Undamped natural frequency
• : Damping ratio
• : Variable in the Laplace domain

System Parameters

Natural Frequency ()

The natural frequency refers to the frequency at which the system oscillates in the absence of damping:

Where:

• : Stiffness of the system
• : Mass of the system

Damping Ratio ()

The damping ratio determines the rate of decay of oscillations. It is given by:

Where:

• : Damping coefficient
• : Mass
• : Stiffness

Damped Frequency ()

The damped frequency is the frequency at which an underdamped system oscillates:

Time Response

Unit Step Response

The time response of a second-order system to a step input depends on the damping ratio ().

Underdamped System ()

The response is oscillatory and is described as:

Critically Damped System ()

The system does not oscillate and the response is:

Overdamped System ()

The system also does not oscillate and the response is:

Maximum Overshoot ()

The maximum overshoot is the maximum deviation above the final value in an underdamped system ():

Peak Time ()

The time at which the maximum overshoot occurs is:

Settling Time ()

The time it takes for the response to remain within a certain percentage of the final value (usually 2% or 5%) is:

Rise Time ()

The time it takes for the response to go from 0% to 100% of the final value for underdamped systems is approximately:

Stability Analysis

System Poles

The poles of the second-order system are the roots of the denominator of the transfer function:

Types of Poles:

• Underdamped (): Complex conjugate poles.
• Critically Damped (): Equal real poles.
• Overdamped (): Distinct real poles.

Stability Criteria

The system is stable if all poles have negative real parts (). If any pole has a positive real part, the system is unstable.

Frequency Analysis

Frequency Response

The transfer function in the frequency domain is obtained by substitutinginto the transfer function:

Magnitude

The magnitude of the frequency response is:

Phase

The phase of the frequency response is:

Peak Frequency ()

The frequency at which the maximum value of the magnitude occurs is:

Classical Controllers

The most common controllers are the proportional (P) controller, the proportional-integral (PI) controller, and the proportional-integral-derivative (PID) controller.

Proportional Controller (P)

Equation

Where:

• : Proportional gain.
• : Error between the input and output.

Proportional-Integral Controller (PI)

Equation

Where:

• : Integral gain.

Proportional-Integral-Derivative Controller (PID)

Equation

Where:

• : Derivative gain.

Frequency Domain Analysis

The frequency response describes the behavior of a system in response to sinusoidal inputs of different frequencies.

Gain and Phase

Gain ()

Phase ()

Bode Diagrams

The Bode diagrams show the magnitude and phase of the frequency response of a system.

Cut-off Frequency (where the magnitude drops to)

Whereis the time constant of the system.

Gain Margin and Phase Margin

Gain Margin

The amount of gain that can be increased before the system becomes unstable.

It is measured at the phase crossover.

Phase Margin

The angle that can be increased before the system becomes unstable.

It is measured at the gain crossover 0 dB.

Compensation

Compensators are designed to improve the performance of the system, either by increasing stability, improving response time, or adjusting the frequency response.

Lead Compensator

Transfer Function

Where, provides greater stability and higher phase margin.

Lag Compensator

Transfer Function

Where, improves gain margin and reduces steady-state error.

Steady-State Errors

Steady-state errors depend on the type of input and the type of system.

Static Error Coefficient

Error for Step Input ()

Error for Ramp Input ()

Error for Parabolic Input ()

Where:

• : Position error coefficient.
• : Velocity error coefficient.
• : Acceleration error coefficient.