## What are multiple subsystems?

The multiple subsystems can be explained as the interrelationship of different types of subsystems. Generally, there are two modes to describe multiple subsystems, signal flow graphs and block diagrams. The block diagrams are helpful in design purposes as well as frequency domain analysis, whereas the signal flow graphs are utilized in indicating the correlation among signals.

In the context of a particular system, if the number of users increases, the number of subsystems would also increase because a single subsystem is not sufficient for multiple users. Hence, these subsystems are reduced to a single function using various techniques.

## Advantages of multiple subsytem

The advantages of dividing the users into multiple subsystems are given in the following points.

### Improve interactive subsystem startup time

The startup time for the single subsystem can be kept shorter by dividing the specific work into multiple subsystems.

### Additional options for performance tuning

The setup of subsystems, including small routing entries, can happen by using multiple subsystems.

### Improved manageability of work

Whenever each subsystem performs a specific work, we can keep track. There are various examples of multiple subsystems like server jobs, a remote command to a different subsystem, etc. The isolation of different groups of jobs can also be done with the help of multiple subsystems.

### Improved error tolerance in interactive subsystems

Whenever a network failure occurs, then a device's recovery process can be managed by multiple subsystems.

### Reduced downtime impact for users

There are some examples for this; let us consider one of them as suppose that every afternoon the system is brought to the restricted state for backup purposes, the users can be taken offline by ending the subsystems one at a time.

### Improved scalability and availability

On having the single subsystem, the work can be done for a few users, doing this the system will be less busy and more efficient with the work that needs to be handled by that subsystem.

## How are multiple subsystems represented?

The representation of multiple subsystems can perform in block diagrams and signal flow graphs. The block diagram concept helps in frequency domain analysis, whereas signal-flow graphs help in state-space analysis. For the block diagram reduction, block diagram algebra helps, whereas Mason's rule would help in the reduction of signal-flow graphs.

## Block diagrams

A subsystem is represented as a block with an input, an output, and a transfer function. Many systems are composed of multiple subsystems. When multiple subsystems are interconnected, a few more schematic elements must be added to the block diagram. These new elements are: summing junctions and pickoff points. All parts of a block diagram for a linear, time-invariant system are shown in the figure. The characteristic of the summing junction shown in figure (c) is that the output signal $C\left(s\right)$, is the algebraic sum of the input signals ${R}_{1}\left(s\right)$, ${R}_{2}\left(s\right)$, and ${R}_{3}\left(s\right)$. A pick-off point, as shown in figure (d), distributes the input signal $R\left(s\right)$ undiminished, to several output points.

## Cascade form

The representation of the cascade subsystem is given below. Intermediate signal values show the output of each subsystem.

Each signal is derived from the product of the input times the transfer function. The equivalent transfer function, ${G}_{e}\left(s\right)$, shown in figure (b), is the output Laplace transform divided by the input Laplace transform from the figure (a), or ${G}_{e}\left(s\right)={G}_{3}\left(s\right)\xb7{G}_{2}\left(s\right)\xb7{G}_{1}\left(s\right)$*.*

## Parallel form

In the parallel form, the equivalent transfer function ${G}_{e}\left(s\right)$ is the algebraic sum of the subsystems’ transfer functions,

${G}_{e}\left(s\right)={G}_{1}\left(s\right)\pm {G}_{2}\left(s\right)\pm {G}_{3}\left(s\right)$

## Feedback Form

The feedback system is the basis for our study of control system engineering. Let us derive the transfer function that represents the system.

The typical feedback system is shown in figure (a). A simplified model is shown in figure (b). Directing our attention to simplified model,* $E\left(s\right)=R\left(s\right)\pm C\left(s\right)H\left(s\right)$*. But since *$C\left(s\right)=E\left(s\right)\xb7G\left(s\right)$*, substituting *E(s)* in the second equation to the first equation and solving for transfer function *${G}_{e}\left(s\right)=C\left(s\right)/R\left(s\right)$*, we obtain the closed loop transfer function shown in figure (c),

${G}_{e}\left(s\right)=\frac{G\left(s\right)}{1\pm G\left(s\right)H\left(s\right)}$

The product $G\left(s\right)\xb7H\left(s\right)$ is called the open-loop. So far, we have explored three different configurations for multiple subsystems. Since these three forms are combined into complex arrangements in physical systems, recognizing these topologies is a prerequisite to obtaining the equivalent transfer function. Now, we will reduce complex systems.

## Moving blocks to create familiar forms

Moving block to generate familiar forms represents the establishment of a basic block that moves. The movement of the block towards the left or right and pickoff points are shown in the following figure.

**Figure: **Block diagram algebra for summing junctions - equivalent forms for moving a block

- to the left past a summing junction;
- to the right past a summing junction.

**Figure:** Block diagram algebra for pickoff points - equivalent forms for moving a block

- to the left past a pickoff point
- to the right past a pickup point

**Example: Reduce the block diagram shown in the figure to a single transfer function.**

**Solution:** First, the three summing junctions can be collapsed into a single summing junction as shown in (a). Second, recognize that three feedback functions, ${H}_{1}\left(s\right),{H}_{2}\left(s\right)$ and ${H}_{3}\left(s\right)$ are connected in parallel. The equivalent function is ${H}_{1}\left(s\right)-{H}_{2}\left(s\right)+{H}_{3}\left(s\right)$. Also, recognize that ${G}_{2}\left(s\right)$ and ${G}_{3}\left(s\right)$ are connected in cascade. Thus, the equivalent transfer function is the product ${G}_{3}\left(s\right)\xb7{G}_{2}\left(s\right)$ as shown in *(b)*. Finally, the feedback system is reduced and multiplied by* ${G}_{1}\left(s\right)$* to yield the equivalent transfer function shown in *(c).*

## Signal-flow graphs

The signal-flow graphs refer to as an alternative to block diagrams. A block diagram consists of blocks, signals, and pickoff points, whereas a signal-flow graph consists only of branches representing the system and node's signals. These elements are shown in Figures (a) and (b), respectively.

A system represents by a line with an arrow showing the direction of signal flow through the system. Adjacent to the line, we write the transfer function. A signal is a node with the signal’s name written adjacent to the node.

Figure (c) shows the interconnection of the systems and the signals.

Each signal is the sum of* *signals flowing into it. For example, the signal V(s),

$V\left(s\right)={R}_{1}\left(s\right){G}_{1}\left(s\right)-{R}_{2}\left(s\right){G}_{2}\left(s\right)+{R}_{3}\left(s\right){G}_{3}\left(s\right)\phantom{\rule{0ex}{0ex}}For{C}_{2}\left(s\right),\phantom{\rule{0ex}{0ex}}{C}_{2}\left(s\right)=V\left(s\right){G}_{5}\left(s\right)\phantom{\rule{0ex}{0ex}}={R}_{1}\left(s\right){G}_{1}\left(s\right){G}_{5}\left(s\right)-{R}_{2}\left(s\right){G}_{2}\left(s\right){G}_{5}\left(s\right)+{R}_{3}\left(s\right){G}_{3}\left(s\right){G}_{5}\left(s\right)\phantom{\rule{0ex}{0ex}}{C}_{3}\left(s\right)=-V\left(s\right){G}_{6}\left(s\right)\phantom{\rule{0ex}{0ex}}=-{R}_{1}\left(s\right){G}_{1}\left(s\right){G}_{6}\left(s\right)+{R}_{2}\left(s\right){G}_{2}\left(s\right){G}_{6}\left(s\right)-{R}_{3}\left(s\right){G}_{3}\left(s\right){G}_{6}\left(s\right)$

**Example:** Convert the block diagram shown in the figure to a signal flow diagram.

**Solution:** Begin by drawing the nodes as figure (a). Next, interconnect the nodes as figure (b). Notice that the negative signs at the summing junction of the block diagram are represented by the negative transfer function of the signal flow diagram. Finally, if desired, simplify the signal-flow graph to the one shown in figure (c) by eliminating signals that have a single flow in and a flow out, such as ${V}_{2}\left(s\right),{V}_{7}\left(s\right),\mathrm{and}{V}_{8}\left(s\right)$.

## The loop gains

The concept of branch gains found by traversing a path that starts at a node and ends at the same node, following the direction of the signal flow, without passing through any other node more than once, is refers as loop gains. For examples of loop gains, see the following figure.

There are four loop gains:

$1.{G}_{2}\left(s\right){H}_{1}\left(s\right)\phantom{\rule{0ex}{0ex}}2.{G}_{4}\left(s\right){H}_{2}\left(s\right)\phantom{\rule{0ex}{0ex}}3.{G}_{4}\left(s\right){G}_{5}\left(s\right){H}_{3}\left(s\right)\phantom{\rule{0ex}{0ex}}4.{G}_{4}\left(s\right){G}_{6}\left(s\right){H}_{3}\left(s\right)$

## Common Mistakes

Students often get confused with the difference in the open loop control system and the closed loop control system. Always keep in mind that in case of open-loop control systems, the desired output does not depend on the controlled act whereas, in closed-loop, the required output mainly depends on the controlled act.

## Context and Applications

This topic is significant in the professional exams for both graduate and postgraduate courses, especially for

- Bachelor of Technology in Mechanical Engineering

- Master of Technology in Mechanical Engineering

## Related Concepts

- Closed-loop transfer function
- State variables
- State space

## Practice Problems

**Q1.** Which system is also known as an automatic control system?

a) open-loop control system

b) closed-loop control system

c) either 1 or 2

d) neither 1 nor 2

**Correct option: (b)**

**Explanation:** The closed-loop system is also expressed as an automatic control system because it can be controlled automatically until the required response is obtained.

**Q2.** The advantage of an open-loop system is/are?

a) simple and economical

b) accurate

c) reliable

d) all of the above

**Correct option: (a) **

**Explanation:**

The below points are some of the benefits of the open-loop control system.

- Simple and easy to design
- Economical as compared to other systems
- Simple maintenance

**Q3.** Which of the following are the disadvantages of a closed-loop control system?

a) reduces the overall gain

b) complex and costly

c) oscillatory response

d) all of the above

**Correct option: (d)**

**Explanation:**

The following points are some disadvantages of a closed-loop control system:

- The overall gain is lessened
- Complex
- The control system is expensive
- Oscillatory response

**Q4.** The output of the system affects the input quantity, then the system is a?

a) open-loop control system

b) closed-loop control system

c) either 1 or 2

d) none of the above

**Correct option: (b)**

**Explanation:** In comparing closed-loop and open-loop systems, the output returns to input only in a closed-loop system. Therefore, the input quantity does get affected by the output.

**Q5.** By using which of the following elements, mechanical translational systems are obtained?

a) mass element

b) spring element

c) dash-pot

d) all of the above

**Correct option: (d)**

**Explanation:**

The following are the three basic elements included in translating mechanical system.

- Mass element
- Spring element
- dash pot

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