Homework

Base neutralization process

In this homework, you will design a controller for a base neutralization system that operates in two stages, as illustrated in the figure below.

Neutralization process

The bulk of the neutralization takes place in the first tank, where a large amount of acid (u_1) is mixed with the basic solution (d) to reduce the pH (y_1) to approximately 10. In the second tank, the pH (y_2) is further adjusted to a target value of around 7 by adding a smaller amount of acid (u_2).

The system can be modeled using the following signal flow diagram

Signal flow diagram

Where the transfer functions are given as

G_1(s) = \frac{1.5}{100s+1}, \quad G_2(s) = \frac{1.0}{50s+1}.

Your goal is to design a controller which ensures y_2 \approx r with a closed-loop bandwidth \omega_b = 1.0 rad/s, while also keeping u_2 small at low frequencies. The controller \bm{K} should be of the form \begin{bmatrix} u_1 \\ u_2 \end{bmatrix} = \bm{K}(s) \begin{bmatrix} -y_1 \\ r - y_2 \end{bmatrix}.

You should use the H_\infty signal-based design method to achieve this goal, that is, design the controller by solving \operatorname*{minimize}_{\bm{K} \text{ stabilizing}} \lVert \operatorname{LFT}(\bm{P}, \bm{K}) \rVert_{\infty}, where \bm{P} is the generalized plant, which includes the dynamics of the base neutralization process along with weighting functions used to express performance objectives. It maps the exogenous inputs (e.g., reference r, disturbance d) and the control inputs (u_1, u_2) to the performance outputs (such as tracking error and control effort) and to the measured signals used by the controller.

The operator \operatorname{LFT}(\bm{P}, \bm{K}) denotes the lower linear fractional transformation, which represents the closed-loop transfer function from the exogenous inputs to the performance outputs when the controller \bm{K} is connected to the generalized plant \bm{P} via feedback.

Tasks

You should follow these steps to complete the homework

  1. Create the generalized plant \bm{P} using the transfer functions G_1(s) and G_2(s), along with your chosen exogenous inputs, performance outputs, and weighting filters. The weighting filters should be designed to ensure that the closed-loop system meets the performance objectives, such as reference tracking and control effort requirements.

  2. Use the hinfsyn_lmi function from JuliaSimControl to solve the H_\infty optimization problem. This function will compute the optimal controller \bm{K} and the appropriate \gamma upper bound on the H_\infty norm of the LFT. We suggest using γrel = 1.05 as the relative tolerance for the optimization.

  3. Prepare a concise report (hw.pdf) and include the controller design code (hw.jl). Compress the report and code into a zip file named hw.zip and submit it to the BRUTE system.

JuliaSim

JuliaSimControl is not a package that is available in the Julia package registry because it is not a free software. It is a commercial package that is part of the JuliaSim suite, which is a collection of modeling and simulation tools for Julia. However, to our luck, it is for non-commercial academic teaching and research purposes.

Installing JuliaSimControl is more involved than your run of the mill Julia package, but not by much.

First, you need to register yourself on the JuliaHub website.

After that, follow the instruction on Using JuliaSim Packages Locally via the JuliaHub Registry to add the JuliaHub registry to your Julia installation.

Once you have done that, you can install JuliaSimControl in the standard way using the Julia package manager.

Report instructions

Your report should contain the following

  1. A description of the generalized plant \bm{P}, including your choice of exogenous inputs, performance outputs, and weighting functions. Briefly explain the reasoning behind your design choices.

  2. Closed-loop performance analysis, namely show

    • the time response of the output y_2 and control input u_2 to a step change in the reference r.
    • the frequency response for a transfer function from the reference r to the output y_2, and from the reference r to the control input u_2.

  3. Brief comment on whether the bandwidth and low-frequency control objectives were met, based on your plots.

Template

using JuliaSimControl
using LinearAlgebra, Plots

s = tf("s")

G₁ = 1.5/(100s+1) |> ss
G₂ = 1/(50s+1) |> ss

## TODO: Form the generalized plant P

## TODO: Find the controller K using hinfsyn_lmi

## TODO: Simulate the closed-loop system
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