# Control Systems

## Introduction

### System Configuration

**Open Loop** systems do not monitor or correct the output for disturbances; however, they are simpler and less expensive than closed-loop systems.

**Closed Loop** systems monitor the output and compare it to the input. If an error is detected, the system corrects the output and hence corrects the effects of disturbances.

### Analysis and Design Objectives

#### Transient Response

That part of the response curve due to the system and the way the system acquires or dissipates energy. In stable systems it is the part of the response plot prior to the steady-state response.

#### Steady-State Response

Also known as **Forced Response**, for linear systems, that part of the total response function due to the input. It is typically of the same form as the input and its derivatives.

#### Stability

That characteristic of a system defined by a natural response that decays to zero as time approaches infinity.

## Modeling in the Frequency Domain

## Time Response

## Reduction of Multiple Systems

### Block Diagrams

The block diagram of a linear, time-invariant system consists of four elements: *signals*, systems, summing junctions , and *pickoff points* . These elements can be assembled into three basic forms: *cascade*, *parallel* , and *feedback*.

#### Cascade Form

#### Parallel Form

From the above diagram, it is clear that each branch will be

#### Feedback Form

#### Moving Blocks

From these diagrams, the logic of moving each process can be validated when summing each signal and multiplying by each process, then doing the same for its equivalent system.

## Stability

### Routh-Hurwitz Criteria

### Routh-Hurwitz Special Cases

#### Zero Only in the First Column

Assume a value for the variable

Alternately, the reverse coefficients of the denominator can be used in analysis, as seen below:

#### Entire Row Is Zero

If an entire row is zero, return to the row above, and form the equation represented by those coefficients. For example, from the system below, at row

In the case of a row of zeroes formed by a row of even polynomials, for example

From the example above, the row of zeroes can be seen to occur in the

The findings for the even polynomials are then combined with the findings for the rest of the system:

## Steady State Errors

\textit Steady State Error is defined as the difference between the input and output as t

Most steady state errors

In the first case

From these systems we see that

### Steady State Error for Unity Feedback Systems

Steady-state error can be calculated from transfer function

#### Steady State Error in Terms of

### Static Error Constants and System Type

The steady-state error for unit step inputs is

The steady-state error for ramp inputs of unit velocity is

The steady-state error for parabolic inputs of unit acceleration is

The terms in the denominator are known as **static error** constants , representing position, velocity and acceleration, respectively.

Systems can also be defined by **system type**. This defines the number of pure integrations in the forward path, assuming a unity feedback system. Increasing the system type decreases the steady-state error as long as the system is stable.

This will also be apparent by the structure of the system. An

### Steady-State Error Specifications

The steady-state error is inversely proportional to the static error constant - the larger the constant, the smaller the steady-state error. Increasing gain increases the static error constant, thus, increasing the gain decreases the steady-state error if the system is stable.

### Steady State Disturbances

However,

### Forming Equivalent Unity from Non-Unity Systems

## Root Locus Techniques

The following sections apply to \textbf Negative Feedback Closed Loop systems.

### Sketching the Root Locus

#### Number of Branches

The number of branches in a root locus equal the number of poles.

#### Symmetry

The root locus is symmetrical about the real axis.

#### Real Axis Segments

On the real axis, for

#### Starting and Ending Points

The root locus begins at the finite and infinite poles of

#### Behavior at Infinity

When finding

### Refining the Sketch

#### Breakaway/Break-in Point

At the breakaway or break-in point, the branches of the root locus form an angle of 180°/

For all points on the root locus,

Or, conversely,

Where

-Axis Crossings

The crossing of the

The

#### Angles of Departure and Arrival

If we assume a point on the root locus **pole** , the sum of angles drawn from all finite poles and zeros to this point is an odd multiple of 180**pole** that is **pole**.

*or*

If we assume a point on the root locus **zero** , the sum of angles drawn from all finite poles and zeros to this point is an odd multiple of 180**zero** that is **zero** that is near the point. Thus, the only unknown angle in the sum is the angle drawn from the **zero** that is **zero**.

*or*

Note that finding the angle at these points is calculating the length from the poles or zeroes to this point.

### Plotting and Calibrating the Root Locus

When locating points on the root locus and finding their specified gain, for example as it crosses the radial line representing 20% overshoot, such as the below graph where

Evaluating the graph at points along the line, and summing the angles from poles and zeros, it can be determined if a point is on the root locus if the angles are a multiple of 180

From this value

### Generalized Root Locus

If finding the root locus of a system concerning a single parameter instead of gain

### Positive Feedback Systems

**Number of Branches**No change**Symmetry**No change**Real Axis Segments**On the real axis, the root locus for positive-feedback systems exists to the left of an even number of real-axis, finite open-loop poles and/or finite open-loop zeros.**Starting and Ending Points**The root locus for positive-feedback systems begins at the finite and infinite poles ofand ends at the finite and infinite zeros of . **Behavior at Infinity**The root locus approaches straight lines as asymptotes as the locus approaches infinity. Further, the equations of the asymptotes for positive-feedback systems are given by the real-axis intercept,, and angle, , as follows: