Block diagram in z-domain

In summary, the conversation discusses the use of z-transform and block diagrams in determining the transfer function and input-output relationships in a system. It highlights the process of converting from s-domain to z-domain and the use of H(z) to obtain the output Y(z). The conversation also addresses a question about multiplying H(z) and U(z) when dealing with cascaded transfer functions. The conversation concludes by suggesting further study of the properties of Z transform and comparing it to the S transform to identify key differences.
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Hi this is a general question about z-transform and block diagram:

Suppose Y(z) is the output, U(z) is the input, and H(z) is the transfer function, then:

Y(z) = H(z) U(z).

Suppose we start with an open-loop transfer function G(s) under unity feedback, then to go from s to z, say we assume a zero-order-hold component in the open loop path. and obtain G(z) of that path. Then, we compute H(z) by H(z) = G(z) /( 1+G(z) ). Now, we transform the input from U(s) to U(z). We multiply H(z) by U(z) to get Y(z). Once we get Y(z), we could transform it back to Y(s) or to y(t).

Is this correct so far?

My question is, suppose you have two transfer functions in cascade A(s) and B(s), such that the combined transfer function is AB(s), we know that AB(z) does not equal to A(z)B(z), why could I multiply H(z) and U(z) to get Y(z)? Why am I allowed to multiply there?

Is there something wrong in what I said?

- Thanks
 
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1. What is a block diagram in the z-domain?

A block diagram in the z-domain is a graphical representation of a system or process using blocks to represent the different components or stages of the system. It is used in control systems engineering to visualize the flow of signals and information through the system.

2. How is a block diagram in the z-domain different from a block diagram in the time domain?

A block diagram in the z-domain is a representation of a system in the frequency domain, while a block diagram in the time domain represents the system in the time domain. This means that the blocks in a z-domain block diagram represent the transfer functions of the system, while the blocks in a time domain block diagram represent the physical components of the system.

3. What is the purpose of using a block diagram in the z-domain?

The main purpose of using a block diagram in the z-domain is to simplify the analysis and design of control systems. It allows engineers to easily identify the input-output relationships of the system and make changes to the system without having to go through complex mathematical calculations.

4. How do I draw a block diagram in the z-domain?

To draw a block diagram in the z-domain, you first need to identify the different components or stages of the system. Then, you can use the transfer functions of each component to represent them as blocks in the diagram. Finally, you connect the blocks using arrows to indicate the flow of signals through the system.

5. Are there any limitations to using a block diagram in the z-domain?

Yes, there are some limitations to using a block diagram in the z-domain. It is only applicable to linear time-invariant systems, meaning the system's behavior does not change with time and is not affected by external disturbances. Additionally, it assumes that all signals are in the discrete-time domain and that the system is operating in a stable state.

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