This is a note on Lyapunov and Riccati inequalities.
References:
V. Balakrishnan & L. Vandenberghe, ‘‘Semidefinite programming duality and linear time-invariant systems,’’ IEEE Trans. Autom. Control, 48(1):30–41, 2003.
J. C. Willems, ‘‘Least squares stationary optimal control and the algebraic Riccati equation,’’ IEEE Trans. Autom. Control, 16(6):621–634, 1971.
Let \(A \in \mathbb{C}^{m \times n}\). The spectral abscissa of \(A\) is defined by
\[ \alpha(A) := \max \{\mathrm{Re}[\lambda] : \text{\(\lambda\) is an eigenvalue of \(A\)}\}. \]
Theorem (strict Lyapunov inequality).
There exists a Hermitian matrix \(P \in \mathcal{H}^n\) such that \(P \succ 0\) and
\[ A^*P + PA \prec 0 \]
if and only if \(\alpha(A) < 0\).
Remark: The set of Hurwitz matrices is not convex, whereas Lyapunov inequalities can be solved by SDP. More restrictive classes can be convex, e.g., the set of square matrices with negative logarithmic norms.
Theorem (nonstrict Lyapunov inequality).
There exists a nonzero Hermitian matrix \(P \in \mathcal{H}^n\) such that \(P \succeq 0\) and
\[ A^*P + PA \preceq 0 \]
if and only if \(\alpha(A) \le 0\).
If a matrix \(A \in \mathbb{R}^{n \times n}\) is Hurwitz, then the Lyapunov equation
\[ A^\mathsf{T}P + PA + Q = 0, \quad Q = Q^\mathsf{T} \succ 0, \]
is feasible from the strict version of the above theorems. The unique solution is given by
\[ P = \int_0^\infty \mathrm{e}^{tA^\mathsf{T}} Q \mathrm{e}^{tA} \,\mathrm{d}t. \]
Let us define a quadratic form \(V \colon \mathbb{R}^n \to \mathbb{R}\) by
\[ V(x) = x^*Px. \]
This is a (strict) Lyapunov function for the linear ODE \(\dot{x}(t) = Ax(t)\).
Consider the control system described by
\[ \dot{x}(t) = Ax(t) + Bu(t). \]
In this case, there exists a matrix \(K \in \mathbb{R}^{n \times m}\) such that \(A + BK\) is Hurwitz if and only if there exist a symmetric matrix \(X \in \mathcal{S}^{n \times n}\) and a matrix \(U \in \mathbb{R}^{n \times m}\) such that \(X \succ 0\) and
\[ \begin{bmatrix} A & B \end{bmatrix} \begin{bmatrix} X \\ U \end{bmatrix} + \begin{bmatrix} X & U^\mathsf{T} \end{bmatrix} \begin{bmatrix} A^\mathsf{T} \\ B^\mathsf{T} \end{bmatrix} \prec 0. \]
The stabilizing gain is determined by \(K = UX^{-1}\). Moreover, the positive-definite matrix \(X\) satisfies
\[ X (A + BK)^\mathsf{T} + (A + BK) X \prec 0. \]
This is known as a dual Lyapunov inequality.
Let \(A \in \mathbb{C}^{m \times n}\), \(B \in \mathbb{C}^{n \times m}\), and \(M \in \mathcal{H}^{n + m}\). Let \(M\) be partitioned as
\[ M = \begin{bmatrix} M_{11} & M_{12} \\ M_{12}^* & M_{22} \end{bmatrix}. \]
Theorem (strict Riccati inequality).
Suppose that \(M_{22} \prec 0\).
There exists a Hermitian matrix \(P \in \mathcal{H}^n\) such that
\[ \begin{bmatrix} A^*P + PA & PB \\ B^*P & 0 \end{bmatrix} + M \prec 0 \]
if and only if
\[ (\mathrm{j} \omega I - A) x = Bu \implies \begin{bmatrix} x \\ u \end{bmatrix}^* M \begin{bmatrix} x \\ u \end{bmatrix} < 0 \]
for all \(\omega \in \mathbb{R}\) and all \((x,u) \in \mathbb{C}^{n + m} \setminus \{0\}\).
Remark: We need not consider \(\omega = \infty\), which means that \((\mathrm{j} \omega I - A) x = Bu\) holds with \(x = 0\) and \(u\) arbitrary. If \(x = 0\), then the condition is automatically satisfied because we have assumed that \(M_{22} \prec 0\). This assumption is necessary for feasibility of the Riccati inequality.
Remark: Using the Schur complement, we can transform the Riccati inequality into
\[ A^*P + PA + M_{11} - (PB + M_{12}) M_{22}^{-1} (PB + M_{12})^* \prec 0, \quad M_{22} \prec 0. \]
Theorem (nonstrict Riccati inequality).
Suppose that \(M_{22} \preceq 0\) and that all uncontrollable modes of \((A,B)\) are nondefective and lie on the imaginary axis.
There exists a Hermitian matrix \(P \in \mathcal{H}^n\) such that
\[ \begin{bmatrix} A^*P + PA & PB \\ B^*P & 0 \end{bmatrix} + M \preceq 0 \]
if and only if
\[ (\mathrm{j} \omega I - A) x = Bu \implies \begin{bmatrix} x \\ u \end{bmatrix}^* M \begin{bmatrix} x \\ u \end{bmatrix} \le 0 \]
for all \(\omega \in \mathbb{R}\) and all \((x,u) \in \mathbb{C}^{n + m}\).
Consider the infinite-horizon LQR problem:
\[ \begin{aligned} \mathtt{minimize}& \quad \int_0^\infty [x(t)^\mathsf{T}Qx(t) + u(t)^\mathsf{T}Ru(t) + 2x(t)^\mathsf{T}Su(t)] \,\mathrm{d}t \\ \mathtt{subject\ to}& \quad \dot{x}(t) = Ax(t) + Bu(t), \quad x(0) = x_0, \quad \lim_{t \to \infty} x(t) = 0 \end{aligned} \]
Note that we do not impose any assumptions on \(Q\), \(R\), and \(S\) except that \(Q\) and \(R\) are symmetric. We define the optimal cost
\[ V^+(x_0) := \inf_{u \in \mathcal{L}^2_\mathrm{loc}[0,\infty)} \left\{ \int_0^\infty [x(t)^\mathsf{T}Qx(t) + u(t)^\mathsf{T}Ru(t) + 2x(t)^\mathsf{T}Su(t)] \,\mathrm{d}t : x(0) = x_0,\ \lim_{t \to \infty} x(t) = 0 \right\}. \]
Now, we assume that \((A,B)\) is controllable. This assumption implies that \(V^+(x_0) < \infty\) because we can take a \(u\) such that \((x,u) \in \mathcal{L}^2[0,\infty)\). Moreover, \(V^+(x_0) > - \infty\) if and only if there exists a symmetric matrix \(P \in \mathcal{S}^n\) such that
\[ \begin{bmatrix} A^\mathsf{T}P + PA & PB \\ B^\mathsf{T}P & 0 \end{bmatrix} + \begin{bmatrix} Q & S \\ S^\mathsf{T} & R \end{bmatrix} \succeq 0. \]
This is known as a reversed Riccati inequality.
Consider the LTI system described by
\[ \dot{x}(t) = Ax(t) + Bu(t), \quad y(t) = Cx(t) + Du(t). \]
Let the supply rate be given by
\[ w(u,y) = \begin{bmatrix} y \\ u \end{bmatrix}^\mathsf{T} \begin{bmatrix} Q & S \\ S^\mathsf{T} & R \end{bmatrix} \begin{bmatrix} y \\ u \end{bmatrix}. \]
Suppose that there exists a symmetric matrix \(P \in \mathcal{S}^n\) such that \(P \succeq 0\) and
\[ \begin{bmatrix} A^\mathsf{T}P + PA & PB \\ B^\mathsf{T}P & 0 \end{bmatrix} - \begin{bmatrix} C & D \\ 0 & I \end{bmatrix}^\mathsf{T} \begin{bmatrix} Q & S \\ S^\mathsf{T} & R \end{bmatrix} \begin{bmatrix} C & D \\ 0 & I \end{bmatrix} \preceq 0. \]
Then, the LTI system has the storage function \(W(x) = x^\mathsf{T}Px\) for which the dissipation inequality
\[ W(x(t_1)) \le W(x(t_0)) + \int_{t_0}^{t_1} w(u(\tau),y(\tau)) \,\mathrm{d}\tau \]
is satisfied for all \(t_0,t_1 \ge 0\) with \(t_0 \le t_1\) and all \((x,u,y) \in \mathfrak{B}\).