← Dynamic Systems Engineering Foundations · Semester 1, Week 5
Mathematical Models: The Language of Dynamic Systems
Theme
Observation and reasoning are not enough for engineering. Engineers require a precise language capable of describing systems, predicting future behavior, and testing competing models. Mathematics provides this language. Unlike a traditional mathematics course, the purpose of this week is not to memorize formulas but to understand why mathematical models are indispensable for describing dynamic systems.
Learning Objectives
By the end of this week, students should be able to:
- Explain why mathematics is a modeling language.
- Represent systems using variables.
- Distinguish constants from variables.
- Recognize relationships between variables.
- Understand functions as mappings between system states.
- Appreciate rates of change without requiring advanced calculus.
- Build simple predictive mathematical models.
Core Vocabulary
Variable · Constant · Parameter · Function · Relationship · Domain · Range · Rate of Change · Probability · Optimization · Approximation · Prediction · Model · Simulation
Fundamental Principles
Connecting to Previous Weeks
Week 1 asked: "What do we know?"
Week 2 asked: "What is the system?"
Week 3 asked: "What causes change?"
Week 4 asked: "What cannot be directly observed?"
This week asks: "How can we represent everything we have learned in a precise language?"
Anchor Example — Thermostat
Variables: Current Temperature · Desired Temperature · Outside Temperature · Heater Output
State Variable: Room Temperature
Relationships: When room temperature falls below the desired temperature, the controller increases heating.
Discussion: How changing one variable influences the others.
Anchor Example — Human Learning
Variables: Study Time · Practice Problems · Sleep · Stress · Knowledge Estimate · Assessment Score
Task: Construct a simple conceptual model showing how multiple variables influence learning outcomes — learning depends on interacting variables rather than a single cause.
Anchor Example — AI Assistant
Variables: Prompt Quality · Retrieved Information · Conversation Context · Reasoning Depth · Response Quality
Discussion: Why improving one variable alone may not maximize overall performance.
Anchor Example — Traffic
Variables: Vehicle Density · Average Speed · Traffic Flow · Road Capacity
Task: Examine how increasing density initially increases throughput but eventually reduces traffic flow.
Anchor Example — Small Business
Variables: Customers · Revenue · Operating Costs · Inventory · Marketing · Employee Productivity
Task: Identify which variables are directly measurable and which require estimation.
AI Laboratory
Choose one anchor system. Use AI to identify independent variables, dependent variables, hidden variables, and possible mathematical relationships. Ask AI to justify every proposed relationship using observations from previous weeks. Reject any relationship that lacks evidence.
Reflection Questions
- Which variables matter most?
- Which variables can be ignored?
- What assumptions are built into my model?
- How would I know if my model is incorrect?
Assignment
Construct a variable map for one anchor system. Include: system purpose, state variables, independent variables, dependent variables, estimated variables, unknown variables, and relationships between variables. Clearly identify which relationships are supported by evidence and which remain hypotheses.
End-of-Week Competency
Students can translate observations, causal reasoning, and state estimation into structured mathematical models that support prediction, simulation, and validation. This establishes the formal language used throughout Dynamic Systems Engineering.