From Code to Motion: Simulating and Deploying Robots

🔧 What is Linear Control?

Linear Control refers to a class of control systems where the relationship between the system's inputs and outputs is governed by linear differential equations. These systems follow the principle of superposition: the response to a sum of inputs is equal to the sum of responses to each input.

📐 Mathematical Form

A linear system is typically represented in state-space form:

    𝑥̇(t) = A·x(t) + B·u(t)
    y(t) = C·x(t) + D·u(t)
  

• x(t) is the state vector
• u(t) is the control input
• y(t) is the output
• A, B, C, D are constant matrices

⚙️ Common Linear Controllers

• P Controller: Proportional control
• PI Controller: Adds integral action for zero steady-state error
• PID Controller: Adds derivative action for improved damping
• LQR (Linear Quadratic Regulator): Optimal control method for linear systems
• State Feedback Control: Uses full state for control, e.g., u = -Kx

✅ Advantages

• Simpler to analyze and design
• Well-established tools (Bode plot, Root Locus, etc.)
• Efficient computational implementation

🚫 Limitations

• Only accurate for systems that are naturally linear or approximated as linear
• Fails for highly nonlinear behaviors (e.g., saturation, friction, complex kinematics)

🤖 In Robotics

Linear control is often used when:

• The robot operates near an equilibrium point
• A nonlinear system is linearized around a setpoint
• Controllers like PID or LQR are sufficient

For example, balancing a two-wheeled robot or stabilizing a drone can use linear control after linearizing the dynamics.

📌 Summary

• Linear control uses linear equations to model and stabilize systems
• It's the foundation of classical and modern control theory
• Widely used in robotics for regulation, stabilization, and trajectory tracking