LqrEdit
Lqr, formally the Linear Quadratic Regulator, is a foundational method in control engineering for designing feedback controllers for linear dynamic systems. It computes a state-feedback gain that minimizes a quadratic cost function that balances deviation from desired behavior against the effort of control actions. The resulting control law is typically written as u = -K x, with x representing the state vector and K the gain matrix obtained from solving a matrix equation. Because of its clear objective and transparent tuning, Lqr has become a workhorse in industries ranging from aerospace and robotics to industrial automation and automotive systems.
In practice, Lqr is most effective when the system can be well approximated by a linear model, and when the cost structure reflects a pragmatic trade-off between performance and energy or actuator usage. It is common to implement Lqr in combination with state estimation, using a Kalman filter to reconstruct the full state from measurements. This pairing forms the LQG framework and extends the approach to scenarios with noise and partial observability. The mathematical backbone rests on the solution of a Riccati equation, yielding a gain with provable optimal properties under the model assumptions. For a continuous-time model, these ideas are tied to the algebraic Riccati equation, while discrete-time formulations rely on a discrete counterpart.
Theory and formulation
System models - Lqr operates on a state-space representation of linear dynamics. In continuous time, the evolution is written as ẋ = A x + B u, where A and B encode the plant dynamics and input channels. In discrete time, the model is x_{k+1} = A x_k + B u_k. The matrices A and B must capture the essential behavior of the plant for the controller to be effective. See state-space representation.
Quadratic cost - The performance index is a quadratic function of states and inputs, typically J = ∫ (x^T Q x + u^T R u) dt in continuous time or J = ∑ (x^T Q x + u^T R u) in discrete time. Here Q is a symmetric nonnegative semidefinite matrix penalizing deviations in the state, and R is a symmetric positive definite matrix penalizing control effort. The choices of Q and R encode the priorities of the control task, such as tighter regulation of certain states or a stronger emphasis on energy efficiency. See quadratic cost.
Optimal gain computation - The Lqr problem seeks the gain K that minimizes J subject to the system dynamics. In the continuous-time case, the optimal gain is K = R^{-1} B^T P, where P is the unique symmetric positive semidefinite solution to the algebraic Riccati equation A^T P + P A - P B R^{-1} B^T P + Q = 0. In discrete time, P satisfies P = A^T P A - A^T P B (R + B^T P B)^{-1} B^T P A + Q, with the same K = (R + B^T P B)^{-1} B^T P A. The standard existence results require Q ≥ 0 and R > 0, and a controllability condition of the pair (A, B) to ensure a stabilizing solution. See Riccati equation and algebraic Riccati equation.
Interpreting the solution - The K gain implements a stabilizing state-feedback law that minimizes the chosen trade-off. Closed-loop stability is guaranteed under the usual assumptions: the pair (A, B) is controllable and Q, R are chosen to reflect the desired performance, yielding a stable A − B K. The resulting objective of keeping the state small translates into a predictable, energy-conscious controller—an attractive feature for systems where reliability and efficiency are valued. See Lyapunov stability.
Extensions and practicalities - Integral action can be added to handle steady-state errors, leading to augmented state formulations. The basic Lqr framework also extends to time-varying or parameter-varying systems in a model-based way and to constrained settings through combinations with model predictive control (MPC) or robust design methods. In practice, Lqr is often used in tandem with state estimators; the combination, known as LQG (Linear Quadratic Gaussian), addresses measurement noise and partial observability via a Kalman filter for state reconstruction. See model predictive control and Kalman filter.
Tradeoffs and robustness - A central practical consideration is sensitivity to model accuracy. Lqr assumes a faithful linear model; mismatches in A, B, Q, or R can degrade performance or even destabilize the closed loop in edge cases. This has led practitioners to augment Lqr with robust control ideas (for example, H-infinity control) or to adopt adaptive or robust observers alongside Lqr. Proponents emphasize that the method provides a clear, provable optimum for the given model and cost, which translates into reliable, repeatable behavior in many well-characterized systems. Critics note that optimality under a fixed model does not guarantee resilience to large disturbances or unmodeled dynamics.
Discrete vs continuous formulations - The core ideas exist in both continuous-time and discrete-time frameworks, and the choice depends on how the plant is naturally modeled or controlled in practice. Digital implementations typically operate in a discrete-time setting, with the discrete Riccati equation guiding the gain computation. See linear time-invariant and discrete-time.
Connections to related methods - Lqr sits within the broader landscape of optimal control and is a building block for more comprehensive strategies. It relates closely to state-space representation modeling, the quadratic cost design philosophy, and, in extended form, to LQG and to alternatives in the robust or predictive traditions such as Model predictive control and H-infinity control. See also Riccati equation.
Applications
Aerospace and autonomous systems - Lqr has a long track record in aerospace for flight control and autopilots, where predictable performance and energy-aware control are paramount. The approach is well suited to systems with well-characterized dynamics and where safety and reliability are paramount. See aerospace.
Robotics and automation - In robotics, Lqr provides principled regulation of joint positions, velocities, and spacecraft-like dynamics in robotic arms and legged platforms. It is common to pair Lqr with state estimation to handle real-world sensing limitations. See robot and state-space representation.
Automotive and energy systems - Modern automotive control, including stability and drive-by-wire subsystems, often relies on Lqr principles to achieve smooth, energy-efficient behavior. In energy and process industries, Lqr-like formulations help regulate systems with clear performance and cost objectives, such as temperature, pressure, or speed control. See automotive and electric power.
Industry practice and policy implications - A key advantage of Lqr is its transparency: once Q and R are chosen, the resulting K is fixed and the closed-loop behavior follows a known mathematics. This predictability aligns well with engineering culture that prioritizes testable specifications, verifiable stability, and cost-effective implementation. It also fits into a broader ecosystem of model-based design and standards that reward efficiency, reliability, and clear return on investment.