Discrete Quadratic Linear System Optimal Control Problem with MATLAB Implementation

This document provides the complete MATLAB source code for solving optimal control problems in discrete quadratic linear systems. To facilitate better understanding of the program implementation, let's first define the core mathematical concepts involved. Discrete systems are dynamic systems whose behavior is characterized at distinct time instants, typically modeled using difference equations. Quadratic systems employ quadratic functions to describe relationships between system variables, while linear systems maintain proportionality between inputs and outputs through linear transformations. The MATLAB code presented here implements optimal control solutions using quadratic programming algorithms, specifically designed for discrete-time linear systems with quadratic performance indices. The program structure includes: - System matrix initialization (A, B for state-space representation) - Quadratic cost function formulation (Q, R matrices for state and control weighting) - Discrete-time Riccati equation solver for optimal gain calculation - State feedback control law implementation This engineering problem is crucial for optimizing system performance under constraints, commonly applied in robotics, aerospace control, and industrial automation. The code utilizes MATLAB's built-in matrix operations and control system toolbox functions for efficient computation. To execute the program, users should: 1. Define system matrices (A, B) representing discrete state-space model 2. Specify weighting matrices (Q, R) for cost function optimization 3. Set initial conditions and simulation parameters 4. Run the main function to compute optimal control sequence The algorithm automatically handles constraint violations and provides convergence monitoring for the iterative solution process. Users can easily modify cost function parameters to study different optimization scenarios while maintaining system stability through eigenvalue analysis.