# Designing a Digital Speed Controller and Developing a Z-Domain Cascaded Control Plant: A Step-by-Step Approach

(10%) Develop Z-domain “cascaded” control plant (new system dynamic) with signal flow from ia*(k) => ia(k) => \omega (k) => \theta m(k) => \omega avg(k) . Show all functions with numerical values.
2. (25%)C(z) design based on continuous approximation: Design a digital speed controller with zero steady state error and 100Hz bandwidth (s = -2\pi \times 100Hz). You may apply the pole/zero cancellation in continuous-time.
Explain the design procedure and show the controller transfer function ( I*a(z)
\omega err(z) = ?) with both parameter and numerical values. Please pay attention on the non-minimum phase zero at Z = -3.1904.
1. Develop Z-domain “cascaded” control plant with signal flow from input to output:
• The system dynamics are described by the following signal flow: ia*(k) => ia(k) => ω(k) => θm(k) => ωavg(k).
• We need to express each function with numerical values.
2. Design a digital speed controller with zero steady-state error and 100Hz bandwidth based on continuous approximation:
• Design a controller transfer function that ensures zero steady-state error and a bandwidth of 100Hz (s = -2π × 100Hz).
• Apply pole/zero cancellation in continuous-time.
• Explain the design procedure and provide the controller transfer function with both parameter and numerical values. Pay attention to the non-minimum phase zero at Z = -3.1904.

Let’s address each part step by step:

### Part 1: Develop Z-domain “cascaded” control plant

• *ia(k)**: Input function.
• ia(k): Output of ia*(k).
• ω(k): Output of ia(k).
• θm(k): Output of ω(k).
• ωavg(k): Output of θm(k).

We need numerical values for each function. Let’s assume:

• ia*(k) = 10
• ia(k) = 8
• ω(k) = 6
• θm(k) = 4
• ωavg(k) = 2

### Part 2: Design a digital speed controller

• Objective: Design a controller transfer function with zero steady-state error and 100Hz bandwidth.
• Design Procedure: Apply pole/zero cancellation in continuous-time.
• Controller Transfer Function: Let’s denote it as 𝐶(𝑧).

Explanation of the design procedure:

• We aim to design a controller that ensures zero steady-state error, implying a proportional-integral (PI) or proportional-integral-derivative (PID) controller.
• To achieve a bandwidth of 100Hz, we need to ensure that the dominant poles of the controller transfer function are located at a frequency corresponding to 100Hz.
• By applying pole/zero cancellation in continuous-time, we can shape the controller transfer function to meet the desired specifications.

Let’s calculate the controller transfer function 𝐶(𝑧) with both parameter and numerical values, considering the non-minimum phase zero at Z = -3.1904.

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