Algebraic Riccati Equation (ARE)

The Algebraic Riccati Equation is a nonlinear matrix equation, typically expressed in its continuous-time form as:

AᵀP + PA - PBR⁻¹BᵀP + Q = 0

Where:

• P is the unknown symmetric, positive-definite solution matrix.
• A and B are the system matrices from the linear state-space model ẋ = Ax + Bu.
• Q (state cost) and R (input cost) are symmetric weighting matrices, with Q ≥ 0 and R > 0.

Its discrete-time counterpart is the Discrete-Time Algebraic Riccati Equation (DARE).

The primary application of the ARE is solving the Infinite-Horizon Linear Quadratic Regulator (LQR) problem. The solution matrix P directly yields the optimal state-feedback control law:

u*(t) = -K x(t) where K = R⁻¹BᵀP

This feedback law minimizes the quadratic cost function: J = ∫₀∞ (xᵀQx + uᵀRu) dt

The resulting controller guarantees asymptotic stability for stabilizable and detectable systems, making the ARE solution synonymous with optimal, stabilizing linear feedback.

Solving the ARE is intimately connected to analyzing a Hamiltonian matrix:

H = [ A, -BR⁻¹Bᵀ; -Q, -Aᵀ ]

The stable solution P corresponds to the stabilizing invariant subspace associated with the eigenvalues of H that lie in the open left-half complex plane. Numerical solvers (like the Schur vector method) compute this subspace to find P. This link ensures the optimal controller places the closed-loop system poles optimally for the given cost weights Q and R.

In Linear MPC, the ARE solution is critical for two principal design elements:

1. Terminal Cost: For stability, a common approach is to set the terminal cost in the MPC finite-horizon problem as xᵀ_N P x_N, where P is the ARE solution. This approximates the cost-to-go from the end of the horizon to infinity.
2. Terminal Constraint Set: The ARE solution helps define a terminal invariant set (often an ellipsoid) where the LQR controller is feasible and stabilizing. Enforcing that the final predicted state lies in this set guarantees closed-loop stability.

This bridges finite-horizon MPC optimization with infinite-horizon optimality and stability guarantees.

A unique, stabilizing solution to the ARE is not guaranteed for all systems. Key conditions include:

• Stabilizability: The pair (A, B) must be stabilizable. This ensures a control law exists that can stabilize the system.
• Detectability: The pair (A, Q) must be detectable. This ensures all unstable states are penalized by the cost function. If Q is positive definite, this condition is satisfied.

When these conditions hold, there exists a unique positive semidefinite solution P that stabilizes the closed-loop system (A - BK is Hurwitz).

Solving the ARE is a fundamental numerical task in control design. Standard methods include:

• Schur Vector Method: The most reliable. It uses the QZ algorithm or Schur decomposition on the Hamiltonian matrix to find its stable invariant subspace.
• Iterative Methods: The Kleinman algorithm (a Newton-type method) iteratively solves a sequence of Lyapunov equations for rapid convergence.
• Software Tools: Solvers are built into control design suites (MATLAB's care, dare), numerical libraries (SciPy), and MPC-specific toolkits (ACADO Toolkit).

Robust numerical solution is essential for implementing LQR and designing stable MPC laws.

The Linear Quadratic Regulator (LQR) is the foundational optimal feedback controller for linear time-invariant systems. Its design solves an infinite-horizon optimal control problem, minimizing a quadratic cost function that penalizes both state deviation and control effort.

• Key Relationship to ARE: The optimal LQR feedback gain matrix K is computed directly from the solution P of the Algebraic Riccati Equation: K = R⁻¹BᵀP, where R is the control penalty matrix and B is the input matrix.
• Role in MPC: The LQR solution often provides the terminal cost matrix in finite-horizon MPC formulations, a critical element for proving closed-loop stability.

An Optimal Control Problem (OCP) is the core mathematical formulation that MPC solves online. It defines the objective of finding a control input sequence that minimizes a cost function subject to system dynamics and constraints over a future horizon.

• Finite vs. Infinite Horizon: The ARE arises from solving the infinite-horizon, unconstrained version of the Linear Quadratic (LQ) OCP. This analytical solution provides a benchmark and theoretical foundation for the more complex, constrained finite-horizon problems solved in MPC.
• Components: A standard OCP is defined by: 1) A dynamic model (e.g., ẋ = Ax + Bu), 2) A cost function (e.g., J = ∫ (xᵀQx + uᵀRu) dt), and 3) A set of state and input constraints.

Terminal cost and terminal constraint are design techniques in finite-horizon MPC used to ensure closed-loop stability by approximating the properties of the infinite-horizon solution.

• Terminal Cost: Often chosen as a quadratic form V_f(x) = xᵀPx, where P is the solution to the ARE. This penalizes the state at the end of the prediction horizon, encouraging the controller to steer the system toward a region where the infinite-horizon optimal controller (LQR) would take over.
• Terminal Constraint: May require the final predicted state to lie within a terminal set, which is often designed to be positively invariant under the LQR feedback law derived from the ARE.

A Lyapunov function is a scalar, energy-like function used to prove the stability of an equilibrium point of a dynamical system. For a stable system, this function decreases over time.

• Connection to ARE: For the closed-loop LQR system (u = -Kx), the optimal cost-to-go V(x) = xᵀPx (where P solves the ARE) serves as a quadratic Lyapunov function. This provides a formal proof of global asymptotic stability for the unconstrained LQ problem.
• Role in MPC Stability Proofs: In MPC, the optimal value function of the solved OCP is often used as a candidate Lyapunov function. The terminal cost derived from the ARE solution is crucial in showing that this value function decreases from one time step to the next.

The Riccati Differential Equation (RDE) is the time-varying counterpart to the Algebraic Riccati Equation. It is a matrix differential equation that must be solved backward in time over a finite horizon.

• Context: The RDE arises in the solution of the finite-horizon Linear Quadratic (LQ) optimal control problem, where the terminal time is fixed.
• Steady-State Solution: As the horizon extends to infinity, the solution to the RDE converges to a constant matrix, which is the unique stabilizing solution of the Algebraic Riccati Equation (ARE). This illustrates how the finite-horizon LQ controller approaches the infinite-horizon LQR controller.

The Kalman Filter is the optimal state estimator for linear systems with Gaussian noise. A profound duality principle exists between optimal estimation (Kalman Filter) and optimal control (LQR).

• Dual Riccati Equation: The Kalman Filter gain is computed by solving an Algebraic Riccati Equation that is the dual of the control ARE. Where the control ARE uses system matrices (A, B) and cost weights (Q, R), the estimation ARE uses the transposed dynamics (Aᵀ, Cᵀ) and noise covariance matrices.
• Unified Framework: This duality means that solving the ARE is a fundamental skill for both optimal feedback control and optimal state estimation, two pillars of modern autonomous systems.