MATLAB and Control System Performance Evaluation: Indicator Calculation and Result Analysis

# 1. Basics of Control System Performance Evaluation

In modern engineering practice, control systems are a key component in ensuring the reliable operation of processes, machinery, and systems. Performance evaluation of these systems is a crucial aspect, aimed at ensuring that they meet specific design requirements, including precision, response time, robustness, and stability. To achieve this goal, engineers typically need to use a range of performance metrics to conduct a comprehensive analysis of the control systems.

**2.1 The Importance of Performance Evaluation**

To quantify the performance of control systems, engineers have defined a series of quantitative performance indicators. These indicators include steady-state performance metrics such as steady-state error and steady-state accuracy, as well as dynamic performance metrics such as rise time, peak time, overshoot, and the number of oscillations. By calculating and analyzing these metrics, engineers can diagnose the current performance status of the system and take appropriate corrective measures to meet performance requirements.

**2.2 Steps in System Performance Evaluation**

Performance evaluation typically includes the following steps:

1. Set evaluation goals and performance indicators.
2. Collect necessary data and information.
3. Apply mathematical models and simulation tools for calculations.
4. Analyze the results and propose optimization suggestions.
5. Adjust or optimize the system.
6. Repeat steps 2 to 5 until performance requirements are met.

Performance evaluation is an iterative process that often requires repeated testing and adjustment. For complex control systems, this process is particularly important because it may involve multiple interrelated subsystems and variables. In this process, mathematical tools and software such as MATLAB play a core role, providing a platform for complex calculations and simulation experiments.

As an introductory overview of control system performance evaluation, the following chapters will delve into the application of MATLAB in this field and how to use it to conduct practical performance evaluations.

# 2. The Application of MATLAB in Control System Performance Evaluation

## 2.1 Introduction to MATLAB and Its Role in Control Systems

### 2.1.1 Basic Functions and Toolboxes of MATLAB

MATLAB (an abbreviation for Matrix Laboratory) is a high-performance numerical computing environment and fourth-generation programming language. It was developed by Professor Cleve Moler in the early 1980s at the University of New Mexico and Stanford University and is currently maintained and updated by MathWorks.

The core of MATLAB is matrix computation, which supports various data types such as integers, floating-point numbers, complex numbers, and characters. Its basic functions include matrix operations, mathematical calculations, drawing capabilities, and interaction with other programming languages. Another significant feature of MATLAB is its collection of toolboxes, which are developed by engineers or scientists in specialized fields. These toolboxes are sets of extended functions and applications designed to solve specific industry problems.

In the field of control systems, MATLAB provides the Control System Toolbox. This toolbox includes a wide range of functions for designing and analyzing control systems, such as transfer functions, state-space models, root locus analysis, and frequency response analysis.

% Sample code: Create a simple transfer function model
num = [2 5]; % Numerator polynomial coefficients
den = [1 3 2]; % Denominator polynomial coefficients
sys = tf(num, den); % Create a transfer function model

% Use functions from the Control System Toolbox to analyze system characteristics
step(sys); % Plot the system's step response
bode(sys); % Plot the system's Bode diagram

### 2.1.2 The Connection Between MATLAB and Control System Performance Evaluation

MATLAB's Control System Toolbox is closely related to control system performance evaluation. The functions and commands provided in the toolbox can be used to calculate system performance metrics in both the time domain and the frequency domain, such as steady-state error, rise time, peak time, overshoot, gain margin, and phase margin.

% Sample code: Use the stepinfo function to obtain performance metrics from step response
info = stepinfo(sys);
disp(info); % Display performance metrics

MATLAB allows users to perform rapid iterations when designing and analyzing control systems, thereby evaluating the impact of different designs on system performance. In addition, MATLAB supports automated scripts and GUI interfaces, making the performance evaluation process more efficient and intuitive.

## 2.2 Theoretical Foundations of Control System Performance Metrics

### 2.2.1 Steady-State Performance Metrics

Steady-state performance metrics mainly describe the behavior of a system as it approaches a stable state under the influence of input signals. For a control system, the most important steady-state performance metrics include steady-state error and steady-state gain.

Steady-state error refers to the difference between the output and the desired output of the system once it has stabilized. Generally, system design aims to reduce steady-state error to achieve more accurate control. Steady-state error can be calculated using the final value theorem or the area under the system's unit step response curve.

### 2.2.2 Dynamic Performance Metrics

Dynamic performance metrics are used to describe a system'***mon dynamic performance metrics include rise time, peak time, overshoot, and settling time.

- Rise time refers to the time required for the system output to reach the final steady-state value for the first time;
- Peak time is the time required for the system output to reach its maximum value;
- Overshoot is the maximum percentage by which the system output exceeds the final steady-state value;
- Settling time is the time required for the system output to enter the final steady-state value error band and remain within that band.

The calculation of dynamic performance metrics typically requires analysis of the system's time-domain response.

## 2.3 Implementation of Control System Performance Evaluation in MATLAB

### 2.3.1 Functions for Linear System Performance Evaluation

In MATLAB, linear system performance evaluation functions can be directly applied to transfer functions or state-space models. For example, the `step` function is used to obtain the system's step response, and the `stepinfo` function can calculate and return dynamic performance metrics related to the step response.

% Sample code: Obtain and analyze system dynamic performance metrics
sys = tf(1, [1 2 1]); % Create a standard second-order system model
figure(1); % Create a new graphic window
step(sys); % Plot the step response
title('System Step Response'); % Graphic title
grid on; % Display grid

% Use the stepinfo function to obtain performance metrics
info = stepinfo(sys);
disp(info); % Display performance metrics

### 2.3.2 Methods for Nonlinear System Performance Evaluation

Nonlinear systems, due to their complexity, cannot be directly evaluated using methods for linear systems. MATLAB provides various methods to handle nonlinear systems, including using the Simulink simulation environment, using the `nlreg` function and `fmincon` function in the control system toolbox, etc.

When using Simulink, the control system model can be simulated as a nonlinear dynamic system, and the behavior of the system can be observed through simulation runs. `nlreg` and `fmincon` are optimization functions that can be used to find parameters for optimizing the performance of nonlinear systems.

When handling nonlinear systems in MATLAB, it is usually necessary to numerically solve the system equations and analyze characteristics such as the system's phase trajectory and attraction domain.

% Sample code: Use fmincon for parameter optimization of a nonlinear system
% Define the objective function and constraints of the nonlinear system
fun = @(x) (x(1)^2 + x(2)^2 - 1)^2 + (x(1)^2 - x(2))^2;
nonlcon = @(x) deal([], [x(1)^2 + x(2)^2 - 1]); 

% Set initial values and bounds for optimization parameters
x0 = [0.5; 0.5];
lb = [-1; -1];
ub = [1; 1];

% Call the fmincon function for optimization
options = optimoptions('fmincon', 'Display', 'iter', 'Algorithm', 'sqp');
[x_opt, fval] = fmincon(fun, x0, [], [], [], [], lb, ub, nonlcon, options);

% Output optimization results
disp(x_opt);
disp(fval);

Nonlinear system performance evaluation and optimization is a more complex process that requires expertise to ensure the accuracy of the simulation model and to rationally interpret simulation results.

# 3. MATLAB Performance Evaluation Metric Calculation

## 3.1 Calculation of Steady-State Performance Metrics in MATLAB

### 3.1.1 MATLAB Analysis of Steady-State Error

Steady-state error refers to the difference between the system output and the desired output after the input signal has stabilized. In MATLAB, a system's transfer function can be defined, and specific commands can be used to calculate the steady-state error. For example, for a unit step input, the steady-state error can be calculated using the following commands:

num = [1]; % Numerator coefficients
den = [1 3 2]; % Denominator coefficients
sys = tf(num, den); % Create a transfer function model
step(sys); % Perform unit step response analysis
es = stepinfo(sys, 'steadyStateValue'); % Calculate steady-state error

In the above code, `num` and `den` represent the numerator and denominator polynomial coefficients of the system, respectively. The `tf` function is used to create the transfer function model `sys`, and the `step` function is then used to analyze its unit step response. The `stepinfo` function is used to obtain detailed information about the response analysis, where the `'steadyStateValue'` option is used to obtain the steady-state error value.

Steady-state error is crucial for control system design because it directly relates to whether the system can ultimately reach the desired operating state. Through MATLAB, engineers can quickly analyze the steady-state error for different system designs and adjust controller parameters to meet design requirements.

### 3.1.2 MATLAB Methods for Stability Determination

System stability refers to whether the system can return to or maintain its equilibrium state after being disturbed. MATLAB provides various methods to determine a system's stability, such as the characteristic root method and the Nyquist method. Below is an example code using MATLAB's `rlocus` function for root locus analysis:

num = [1]; % Numerator coefficients
den = [1 3 2]; % Denominator coefficients
sys = tf(num, den); % Create a transfer function model
rlocus(sys); % Plot the root locus diagram

The `rlocus` function generates a root locus diagram that shows the trajectory of the system's poles as the control gain changes. By analyzing this diagram, the stability of the system under different gains can be determined. On the root locus diagra