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The Algebraic Riccati Equation

Introducing the Algebraic Riccati Equation

Recall that, in the linear–quadratic regulator (LQR) problem, we consider the linear dynamics

\[ \dot x(t) = A x(t) + B u(t), \qquad x(0) = x_0. \]

We seek a state-feedback control law

\[ u(t) = -Kx(t) \]

that minimizes the cost

\[ J = \int_0^\infty \left( x(t)^\top Q\,x(t) + u(t)^\top R\,u(t) \right)\, dt, \]

where \(Q\) is positive semidefinite and \(R\) is positive definite.

The algebraic Riccati equation (ARE) is a matrix equation that plays a central role in solving the LQR problem. This equation is

\[ A^\top S + S A + Q - S B R^{-1} B^\top S = 0, \]

and the matrix variable to be solved for is \(S\). In general, the ARE does not admit a closed-form solution. However, it can be solved numerically for a positive semidefinite (and, under mild assumptions, positive definite) matrix \(S\). For example, in MATLAB, the command

\[ [\texttt{K}, \texttt{S}, \texttt{E}] = \texttt{lqr}(A,B,Q,R) \]

returns the ARE solution \(S\) as its second output.

Connection to the solution of the LQR problem

Once the ARE has been solved for \(S\), the optimal LQR feedback gain is recovered as

\[ K = R^{-1} B^\top S. \]

Moreover, the optimal value of the cost \(J\) (attained under the LQR controller) is

\[ J^\star = x_0^\top S x_0. \]

Thus, the solution \(S\) of the ARE determines both the optimal feedback gain and the optimal cost of the LQR problem.