Linear Quadratic Gaussian
Also known as: LQG, LQR with Kalman Filter
The Linear Quadratic Gaussian (LQG) controller combines the Linear Quadratic Regulator (LQR) with a Kalman Filter to handle stochastic systems with measurement noise and process noise. Developed by Kalman and later formalized by Athans and others, LQG is the natural stochastic extension of LQR and remains the gold standard for optimal linear control under noise, with applications spanning spacecraft, aircraft autopilot, and industrial process control.
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When to use it
Use LQG when you have a linear system subject to Gaussian process and measurement noise, when full state measurement is unavailable or expensive, and when you need optimal control under uncertainty. LQG is ideal for applications like spacecraft attitude control, inertial navigation, and industrial systems with uncertain disturbances. Avoid LQG if the system is strongly nonlinear (use extended or unscented Kalman filter variants), if noise is non-Gaussian, or if measurements arrive at irregular intervals.
Strengths & limitations
- Provably optimal for linear systems with Gaussian noise; minimizes the expected quadratic cost.
- Separation principle: Kalman Filter and LQR can be designed independently, simplifying analysis and debugging.
- Provides both optimal state estimation and optimal feedback control in a unified framework.
- Robustness properties inherited from both the Kalman Filter (noise rejection) and LQR (stability margins).
- Real-time implementation feasible; both Kalman filter update and LQR gain application are O(n^3) and O(n^2) respectively.
- Assumes Gaussian noise; real disturbances are often non-Gaussian with outliers or multi-modal distributions.
- Requires accurate knowledge of process and measurement noise covariances (Q, R); misspecified noise statistics degrade performance.
- Linear assumption; nonlinear systems require approximations (EKF, UKF) that lose optimality guarantees.
- Assumes linear measurements; nonlinear measurement relationships require extended or unscented variants.
- Cannot handle hard constraints; soft quadratic penalties only.
Frequently asked
What is the separation principle and why is it so important?
The separation principle states that for LQG, the optimal state estimator (Kalman Filter) and the optimal controller (LQR) can be designed independently, and combining them is optimal. This means you can first design a Kalman Filter ignoring control, then design an LQR controller ignoring noise, and apply both together without loss of optimality. This simplifies design and enables modular system development.
How do I estimate the process noise covariance Q and measurement noise covariance R?
The most principled approach is to estimate Q and R from data using maximum likelihood or covariance matching methods. Alternatively, start with rough estimates based on sensor specifications and disturbance magnitudes, then tune empirically by monitoring filter residuals (they should be white and uncorrelated). If residuals are correlated, increase Q or R; if the filter diverges, reduce them.
What happens if my system is nonlinear?
For mildly nonlinear systems, linearize around the nominal trajectory and use the Extended Kalman Filter (EKF) combined with LQR designed for the linearized system. For strongly nonlinear systems, consider the Unscented Kalman Filter (UKF) or particle filters, though these lose the optimality guarantee. Model Predictive Control with nonlinear models is another alternative.
Can I use LQG with time-varying systems?
Yes, the Kalman Filter and LQR both have time-varying variants. The time-varying Kalman Filter solves a Riccati equation backward in time from a terminal cost. The time-varying LQR solves a Riccati equation backward in time as well. Both are more computationally expensive than the steady-state versions but enable handling of trajectory tracking and slowly time-varying dynamics.
Sources
- Kalman, R. E. (1960). A new approach to linear filtering and prediction problems. Journal of Basic Engineering, 82(1), 35-45. DOI: 10.1115/1.3662552 ↗
- Athans, M. (1971). The role and use of the stochastic linear-quadratic-gaussian problem in control system design. IEEE Transactions on Automatic Control, 16(6), 529-552. DOI: 10.1109/TAC.1971.1099818 ↗
- Kwakernaak, H., & Sivan, R. (1972). Linear Optimal Control Systems. Wiley-Interscience. link ↗
How to cite this page
ScholarGate. (2026, June 3). Linear Quadratic Gaussian. ScholarGate. https://scholargate.app/en/control-theory/linear-quadratic-gaussian
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Extended Kalman FilterControl Theory↔ compare
- Kalman FilterBayesian↔ compare
- Linear Quadratic RegulatorControl Theory↔ compare