everyone.
I'm interested in control theory and I'm studying the topic of "discrete-time algebraic Riccati Inequality (DARE)". However, I have one question regarding the matrix inequality on this topic.
From Wikipedia, the discrete-time algebraic Riccati Inequality can be obtained as:
\begin{align} A^T P A - A^T P B (R+B^T P B)^{-1} B^T P A + Q - P > 0. \end{align} where $P>0, R > 0, Q > 0$ and $A, B$ are systems matrices with appropriate dimension. If this inequality is satisfied, then the controller can successfully stabilize the closed-loop system. I wonder why the above matrix inequality term should be positive definite. Is this concept different from the Lyapunov stability criterion where the energy of the function is decreasing as time flows?
Does anyone give me a clue for understanding this theory?
Much appreciated.
The positive definiteness of $Q$ and $R$ terms comes from their role in the linear-quadratic-Gaussian control problem. In particular, our goal is to minimize $\sum_{t=1}^\infty (\|x_t\|_1^2 + \|u_t\|_2^2)$ for some norms $\|\cdot\|_1, \|\cdot\|_2$. Instead of just minimizing the norms $\|\cdot\|_i$ (with $i = 1,2$) to simply be the Euclidean norm $\|x\| = \sqrt{x^Tx}$, we would like to be able to use a weighted norm so that we can tune the relative importance of state "size" vs. control-input "size", as well as the relative importance of components within the state or control-input vectors. With that flexibility in mind, it is helpful to be able to define our norms so that $$ \|x\|_1^2 = \|Lx\|^2 = x^T(\overbrace{L^TL}^Q) x, \quad \|u\|_2^2 = \|Mu\|^2 = u^T(\overbrace{M^TM}^R) u $$ for some invertible matrices $L,M$. Thus, $Q = L^TL$ and $R = M^TM$ are correspondingly positive definite.
Similarly, the matrix $P$ is also associated with a cost function: for an initial state $x$, define $$ V(x) = \min_u \sum_{t=1}^\infty (\|x_t\|_1^2 + \|u_t\|_2^2) \quad \text{s.t.} \quad x_0 = x, $$ where the minimum is taken over all possible input streams $u_1,u_2,u_3,\dots$. It turns out that $V(x)$ is necessarily a quadratic (squared-norm) function of $x$, which is to say that it can be expressed in the form $V(x) = x^TPx$ for some positive definite matrix $P$.
With $P,Q,R$ defined in this fashion, it follows that $P,Q,R$ satisfy the discrete-time algebraic Ricatti equation,
$$ A^T P A - A^T P B (R+B^T P B)^{-1} B^T P A + Q - P = 0. $$
From there, it turns out that this equation has a solution if and only if the (DARI) inequality
$$ A^T X A - A^T X B (R+B^T X B)^{-1} B^T X A + Q - X \succeq 0 $$
has at least solution $X$ satisfying the condition that $R + B^TXB \succ 0$. For any such solution $X$, we have $X \preceq P$, which is to say that the solution to the DARE is maximal among the solutions to the DARI.
This paper seems to be the origin of the CARI/DARI; it seems that the authors do not give an interpretation of $X$ in terms of the original LQG problem. The utility of framing the problem in terms of $X$ seems to be that, in some contexts, it is easier to show that the DARI is feasible than it is to show that the DARE has a solution or actually solve the DARE.