ECNG 3019: Advanced Control System Design

Assessment

Introduction

Unmanned Aquatic Vehicles (UAVs) have become increasingly prominent in recent years, owing to their unique maneuverability and capacity to operate in underwater environments that are too hazardous or inaccessible for human divers. Their applications span oceanographic research, environmental monitoring, search and rescue, and even defense, making them critical assets in both scientific and industrial domains. As these vehicles are tasked with navigating complex and dynamic underwater conditions, the robotics and control systems community has placed significant emphasis on modeling, analysis, and the development of effective control strategies for such platforms.

A fundamental prerequisite to designing controllers for UAVs lies in developing a clear understanding of their underlying dynamics. Careful analysis allows for the identification of system behaviors that influence controllability, stability and other relevant properties, thereby guiding the selection and design of appropriate control laws. Thus, this work aims to analyze a UAV, as a precursor to its control.

Mathematical Preliminaries

In this section, the system’s mathematical model is provided and detailed. The UAV is shown in Figure 1. Note the table of model parameters in Appendix A. The vehicle can move in all 6-Degrees of Freedom (DOFs), of which the linear motions are surge (x), sway (y) and heave (z) and the angular motions, roll (p), pitch (q) and yaw (r). The UAV’s motion is made capable by its three actuators, each located at its tail, i.e., the propeller, rudder fins (vertical) and stern fins (horizontal). The main actuator is the propeller, which generates a thrust, Xprop ∈ R which results in a surge motion. The propeller also results in a rolling motion due to the reaction torque, Kprop ∈ R. These quantities are given by:

where KT , KQ ∈ R are the thrust and torque coefficients, respectively, n ∈ R is the propeller’s angular velocity in revolutions per minute, ρ is the density of seawater and D is the propeller’s diameter.

Objectives

Before starting the assignment, be sure to read the ‘Important Guidelines’ box first. Using modern control theory, this coursework must achieve and document the following in a Springer journal format (template available on Myelearning), to be presented in a research seminar-styled presentation:

• Build out the system’s nonlinear mathematical model in MATLAB Simulink using an S-Function.
• Determine the class of system provided, stating all justifications/citations/etc., stating the significance of this in terms of the system’s behavior. Determine and discuss the equilibrium point/s of the system. How does the evolution of the state trajectories around this point/s change as the center of gravity moves in the negative x-direction (using diagram/math/etc)? Determine the stability of a chosen equilibrium point using Lyapunov analysis. Using your knowledge of the meaning behind Lyapunov theory (in particular the meaning behind, V < ˙ 0), discuss the possible design of a controller that allows tracking errors to converge within a prescribed time period (do not design a controller, just discuss the mathematical conditions/proofs/etc). Relate the system’s stability properties to the design of a control law using any necessary theory/theories. Note, use diagrams/graphs to aid in your explanations.
• Express the UAV as a block diagram and find the irreducible realization of the system. Discuss
• Solve for the general analytical expression of the system’s response to a rectangular (or square) pulse and square periodic wave input. Compare plots of the responses obtained from your analytical, Simulink (nonlinear system) and SimuNex results, giving reasons for all comparisons made.
• Project the system into a different coordinate basis. Justify your choice of basis, and interpret the result.
• Document your findings in a Springer double column journal format (max. 25 pages, excluding references). Despite the page limit, you can include appendices. These do not count towards the page limit. Results must include both Simulink and SimuNex simulations. The results for your SimuNex simulations must include both plots, and screenshots of the 3D environment.

Summary of Assessment Requirements

The given assessment focuses on analyzing, modeling, and understanding the control dynamics of Unmanned Aquatic Vehicles (UAVs) using modern control theory and simulation tools. The core objective is to demonstrate the student’s ability to model nonlinear systems, identify system behaviors, and assess stability through mathematical and simulation-based analysis. The final output must be presented in a Springer journal format and include both analytical and simulated results using MATLAB Simulink and SimuNex.

Key Pointers to be Covered in the Assessment:

• Develop a nonlinear mathematical model of the UAV in MATLAB Simulink using an S-Function.
• Identify and classify the system type, justifying the classification through mathematical and theoretical reasoning.
• Determine and analyze the equilibrium points of the system and study the effect of shifting the center of gravity.
• Conduct a stability analysis of the chosen equilibrium point using Lyapunov theory, including interpretation of V˙<0>
• Discuss the design conditions for a potential controller, without developing one, focusing on convergence properties.
• Construct and analyze a block diagram representation, including the irreducible realization of the system.
• Derive and compare analytical and simulated responses of the system to rectangular and square pulse inputs using MATLAB Simulink and SimuNex.
• Transform the system into a different coordinate basis and justify the choice mathematically.
• Present all findings in a structured research paper format with supporting figures, diagrams, and simulation results.

Academic Mentor’s Step-by-Step Guidance and Process

The Academic Mentor guided the student through a systematic approach to ensure conceptual clarity, practical application, and research-quality documentation.

Step 1: Understanding System Dynamics

The mentor began by helping the student understand the six Degrees of Freedom (DOFs) of UAV motion surge, sway, heave, roll, pitch, and yaw and how actuators (propeller, rudder, and stern fins) influence these motions. The student was guided to interpret the provided mathematical equations governing thrust and torque to establish a foundation for the Simulink model.

Step 2: Developing the Mathematical Model

The mentor assisted in translating the theoretical dynamics into a nonlinear Simulink model using an S-Function. The focus was on ensuring the accurate representation of actuator forces and environmental parameters like seawater density and propeller characteristics.

Step 3: System Classification and Equilibrium Analysis

Next, the mentor guided the student through determining the class of system (linear, nonlinear, time-variant, etc.) by analyzing state-space representations. The student then identified equilibrium points and explored their sensitivity to center of gravity shifts, visualizing how state trajectories evolve under such variations using MATLAB plots.

Step 4: Stability Analysis Using Lyapunov Theory

The mentor introduced the concept of Lyapunov stability and its significance in nonlinear control systems. Together, they formulated the Lyapunov function (V) and analyzed V˙\dot{V}V˙ to determine system stability. The mentor emphasized the theoretical implication of V˙<0>

Step 5: Controller Design Discussion

Although the task did not require implementing a controller, the mentor guided the student in discussing the mathematical conditions for tracking error convergence and how Lyapunov-based design principles could be applied in practice for UAV stability enhancement.

Step 6: Block Diagram and Realization

The student, under mentor guidance, expressed the UAV dynamics in block diagram form and reduced it to its irreducible realization. The mentor explained how this aids in understanding controllability and observability, which are crucial for future control design stages.

Step 7: Analytical and Simulation Comparisons

The mentor instructed the student to derive the analytical response for rectangular and square pulse inputs, then compare it with Simulink nonlinear model outputs and SimuNex 3D simulations. Detailed comparison of plots helped the student validate the mathematical accuracy and model fidelity.

Step 8: Coordinate Transformation and Interpretation

The mentor helped the student perform a coordinate basis transformation (e.g., from inertial to body-fixed frame), guiding the interpretation of results and the physical meaning of the transformed equations.

Step 9: Structuring the Springer Journal Report

Finally, the mentor provided guidance on documenting the work in the Springer double-column journal format, ensuring the report maintained academic tone, logical flow, and adherence to formatting standards. The student incorporated graphs, equations, system diagrams, and simulation screenshots in the appendices.

Outcome and Learning Objectives Achieved

By the completion of the project, the student successfully:

• Developed a comprehensive nonlinear model of the UAV system in MATLAB Simulink.
• Understood the mathematical foundations of system dynamics and stability.
• Applied Lyapunov theory to evaluate system stability and controller feasibility.
• Enhanced skills in MATLAB/Simulink modeling, simulation, and comparative analysis.
• Gained experience in research-style documentation and academic presentation.

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