From Code to Motion: Simulating and Deploying Robots

🔍 What is an Observer in Control Systems?

An observer is a computational model used to estimate internal state variables of a dynamic system when not all of them can be directly measured. It uses system inputs and outputs to reconstruct unmeasured states in real time.

🚦 Why Do We Need Observers?

• Not all states (like velocity, position, angles) can be measured directly
• Sensors may be noisy, expensive, or unavailable
• Many control laws (e.g., LQR, backstepping) need full state feedback
• Observers bridge the gap by estimating hidden variables

🧠 Basic Concept

Given a system:

      𝑥̇ = A·x + B·u
      y = C·x
    

You may only measure y. The observer estimates x using:

      𝑥̂̇ = A·𝑥̂ + B·u + L(y - 𝑦̂)
      𝑦̂ = C·𝑥̂
    

Where L is the observer gain matrix that determines how fast and accurately the observer converges.

🧰 Types of Observers

• Luenberger Observer: Basic linear observer; fast and effective for simple systems
• Kalman Filter: Optimal estimator in presence of Gaussian noise
• Extended Kalman Filter (EKF): Handles nonlinear dynamics via linearization
• Sliding Mode Observer: Robust against modeling uncertainties
• High-Gain Observer: Rapid convergence but sensitive to measurement noise
• Nonlinear Observers: Customized for complex systems like robots and AUVs

🎯 Example: Luenberger Observer

For a linear system:

      𝑥̇ = A·x + B·u
      y = C·x
    

The observer is designed as:

      𝑥̂̇ = A·𝑥̂ + B·u + L(y - C·𝑥̂)
    

The term L(y - C·𝑥̂) corrects the estimate based on the error between real output and predicted output.

🛠️ Application Areas

• Robotics (state estimation for mobile robots, AUVs)
• Aerospace (attitude estimation, navigation)
• Automotive (observer-based fault detection, vehicle control)
• Process Control (temperature, flow, pressure estimation)

⚖️ Observer vs Sensor