I'm facing the following problem obtaining the solution of the Discrete Algebraic Riccati Inequality.
Notation and assumption:
$\succeq, \succ,\preceq,\prec$ refers to matrix definiteness
$R\succ0$, $P\succeq 0$ $\quad (A,B)$ controllable pair
Let's consider first the continuous-time case
$(1)\quad A^TP + P A + PBR^{-1}B^TP + Q \preceq 0$
This is a quadratic matrix inequality that can be easily recast as LMI through the Schur Complement Lemma having $R^{-1} \succ 0$
$(2)\quad\begin{bmatrix}\matrix{-(A^TP + PA +Q) & (PB) \\ (PB)^T & R}\end{bmatrix} \succeq 0$
Now, I'm trying doing the same in the discrete-time case.
Consider
$(3)\quad A^TPA – P – A^TPB\ (R+B^TPB)^{-1}\ B^TPA +Q \preceq 0$
which is a Nonlinear Matrix Inequality.
Although tempting the Schur Complement does not work here because the nonlinear term inner part $-(R+B^TPB)^{-1}\prec 0$ is negative definite.
Clearly I could use standard algorithms to find the solution of the associated Matrix Equation (replacing $\preceq$ by $=$ in Eq.(3) ). However I still wonder if the above problem Eq.(3) has a simple linear form using some 'LMI trick'.
EDIT:
Eq. (1) is incorrect as suggested by Johan. I got this formulation from a rather popular article see [1,sec 4.7]. Hence the derivation in Eq. (2) is wrong as well.
The correct one is
$(1\prime)\quad A^TP + P A – PBR^{-1}B^TP + Q \preceq 0$
obtained from the Lyapunov equation
$(4)\quad (A+BK)^T P + P(A+BK) \preceq -Q – K^T R K$
and replacing the LQ optimal controller $K=-R^{-1} B^TP$. (I'm not entirely sure if such a substitution makes really sense). However as far as I can see the quadratic matrix function $(1\prime)$ is for general $P\succ 0$ indefinite. Existence of the strict solution (equality $=$) is provided by the solution of the ARE.
[1] VanAntwerp, Jeremy G., and Richard D. Braatz. "A tutorial on linear and bilinear matrix inequalities." Journal of process control 10.4 (2000): 363-385.
Best Answer
Your question is incorrect to begin with, as the continuous-time ARE is
$\quad A^TP + P A - PBR^{-1}B^TP + Q = 0$
Hence, just as bad form.
The LMI formulations of LQ in both discrete-time and continuous-time would typically be done in both the Riccati matrix and the feedback matrix.
Find minimum-trace $P$ such that
$ (A-BK)^T P (A-BK) - P + K^TRK + Q \preceq 0$
Multiply with $P^{-1}$ from left and right, define $Y = P^{-1}$ and $F = KP^{-1}$, and apply suitable Schur complements to arrive at linear form.