## 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 constant

If

#### Steady-State Error

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

If

### Frequency Analysis

#### Frequency Response

The transfer function in the frequency domain is obtained by substituting

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 (

### Frequency Analysis

#### Frequency Response

The transfer function in the frequency domain is obtained by substituting

#### 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 )

Where

### 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

### Lag Compensator

#### Transfer Function

Where

## 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.