LQR Controller in Robotics

What is LQR Controller in Robotics?

The Linear Quadratic Regulator (LQR) is an optimal control technique that computes the best feedback gains to balance state error against control effort. Given a linear system and a quadratic cost function, LQR solves a Riccati equation and gives you the optimal full-state feedback matrix.

How It Works

You start with a linear state-space model dx/dt = Ax + Bu. You pick two weighting matrices: Q penalises state error and R penalises control effort. The solution to the algebraic Riccati equation gives a matrix P from which the optimal gain K = R⁻¹BᵀP. The controller is then u = −Kx — every state variable is fed back with the right weight to minimise the cost. Tuning Q and R is the practical art: bigger Q means tighter tracking, bigger R means lower control effort.

Real-World Example

Boston Dynamics Atlas and Spot use LQR-family controllers for whole-body balance. Quadcopter autopilots often use LQR for attitude stabilisation. Camera gimbal stabilisation (DJI, Insta360) uses LQR or LQR+MPC. ISRO uses LQR-derived controllers for satellite attitude control.

Why It Matters for Robotics

LQR is the gateway from PID to modern multivariable optimal control. Any robotics graduate programme and many senior roles ask about LQR design. Mastering LQR — its assumptions, limitations, and tuning — opens the door to MPC, robust control, and model-based RL.

Try It Yourself

Simulate an inverted pendulum in Python using SciPy. Compute the LQR gain using scipy.linalg.solve_continuous_are. Plot the pendulum stabilising with Q = diag(10,1,10,1), then try doubling each weight and watch how the response changes.

Quick Quiz

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Quick Quiz
3 questions
1.LQR finds the optimal feedback gain that minimises:
A.Time to charge a battery
B.A quadratic cost on state error and control effort
C.Camera brightness
D.Wi-Fi latency
2.LQR assumes the system is:
A.Highly nonlinear
B.Linear with Gaussian disturbances (for LQG)
C.Stochastic only
D.Quantum mechanical
3.Increasing the matrix Q in LQR generally:
A.Cools the motor
B.Penalises state error more — tighter tracking, higher control effort
C.Lowers battery use
D.Has no effect

Further Reading

• Model Predictive Control
• Pid Controller
• State Space Control
• Feedback Loop

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