# Control Systems/Block Diagrams

When designing or analyzing a system, often it is useful to model the system graphically. **Block Diagrams** are a useful and simple method for analyzing a system graphically. A "block" looks on paper exactly what it means:

## Systems in Series[edit | edit source]

When two or more systems are in series, they can be combined into a single representative system, with a transfer function that is the product of the individual systems.

If we have two systems, *f(t)* and *g(t)*, we can put them in series with one another so that the output of system *f(t)* is the input to system *g(t)*. Now, we can analyze them depending on whether we are using our classical or modern methods.

If we define the output of the first system as *h(t)*, we can define *h(t)* as:

Now, we can define the system output *y(t)* in terms of *h(t)* as:

We can expand *h(t)*:

But, since convolution is associative, we can re-write this as:

Our system can be simplified therefore as such:

### Series Transfer Functions[edit | edit source]

If two or more systems are in series with one another, the total transfer function of the series is the product of all the individual system transfer functions.

In the time-domain we know that:

But, in the frequency domain we know that convolution becomes multiplication, so we can re-write this as:

We can represent our system in the frequency domain as:

### Series State Space[edit | edit source]

If we have two systems in series (say system F and system G), where the output of F is the input to system G, we can write out the state-space equations for each individual system.

System 1:

System 2:

And we can write substitute these equations together form the complete response of system H, that has input u, and output y_{G}:

[Series state equation]

[Series output equation]

## Systems in Parallel[edit | edit source]

Blocks may not be placed in parallel without the use of an adder. Blocks connected by an adder as shown above have a total transfer function of:

Since the Laplace transform is linear, we can easily transfer this to the time domain by converting the multiplication to convolution:

## State Space Model[edit | edit source]

The state-space equations, with non-zero A, B, C, and D matrices conceptually model the following system:

In this image, the strange-looking block in the center is either an integrator or an ideal delay, and can be represented in the transfer domain as:

- or

Depending on the time characteristics of the system. If we only consider continuous-time systems, we can replace the funny block in the center with an integrator:

### In the Laplace Domain[edit | edit source]

The state space model of the above system, if *A*, *B*, *C*, and *D* are transfer functions *A(s)*, *B(s)*, *C(s)* and *D(s)* of the individual subsystems, and if *U(s)* and *Y(s)* represent a single input and output, can be written as follows:

We will explain how we got this result, and how we deal with feedforward and feedback loop structures in the next chapter.

## Adders and Multipliers[edit | edit source]

Some systems may have dedicated summation or multiplication devices, that automatically add or multiply the transfer functions of multiple systems together

## Simplifying Block Diagrams[edit | edit source]

Block diagrams can be systematically simplified. Note that this table is from Schaum's Outline: Feedback and Controls Systems by DiStefano et al