20220620 uniformly completely observable (UCO)

http://www.dem.ist.utl.pt/poliveira/Invest/IFACWC_17.pdf

Consider the LTV system given by
$$\{\begin{array}{l}\dot{\mathbf{x}}(t)=\mathbf{A}(t)\mathbf{x}(t)+\mathbf{B}(t)\mathbf{u}(t)\\ \mathbf{y}(t)=\mathbf{C}(t)\mathbf{x}(t)\end{array}$$
where $\mathbf{x}(t)\in {\mathbb{R}}^n,\mathbf{u}(t)\in {\mathbb{R}}^m$, and $\mathbf{y}(t)\in {\mathbb{R}}^p$ are the system state, input, and output, respectively, and $\mathbf{A}(t)\in {\mathbb{R}}^{n\times n}$, $\mathbf{B}(t)\in {\mathbb{R}}^{n\times m}$, and $\mathbf{C}(t)\in {\mathbb{R}}^{p\times n}$ are bounded continuous functions of time.

The following definitions introduce the concepts of observability Gramian and uniform complete observability for systems with bounded realizations Kalman (1960), Silverman and Anderson (1968), Jazwinski (1970), Silverman and Meadows (1967).

Definition 1. (Observability Gramian). The observability Gramian associated with the pair $(\mathbf{A}(t),\mathbf{C}(t))$ on $\left[t_0,t_f\right]$ is defined as
$${\mathcal{W}}_o\left(t_0,t_f\right):={\int }_{t_0}^{t_f}{\varphi }^T\left(t,t_0\right){\mathbf{C}}^T(t)\mathbf{C}(t)\mathbf{\varphi }\left(t,t_0\right)dt$$
where $\varphi \left(t,t_0\right)$ is the state transition matrix associated with $\mathbf{A}(t)$ from $t_0$ to $t$.

Definition 2. (Uniform complete observability). The pair $(\mathbf{A}(t),\mathbf{C}(t))$ is uniformly completely observable (UCO) if there exist positive constants $\alpha >0$ and $\delta >0$ such that, for all $t\ge t_0$,
$${\mathcal{W}}_o(t,t+\delta )⪰\alpha \mathbf{I}$$

Likewise, the dual definitions for control are as follows. Definition 3. (Controllability Gramian). The controllability Gramian associated with the pair $(\mathbf{A}(t),\mathbf{C}(t))$ on $\left[t_0,t_f\right]$ is defined as
$${\mathcal{W}}_c\left(t_0,t_f\right):={\int }_{t_0}^{t_f}\mathbf{\varphi }\left(t,t_0\right)\mathbf{B}(t){\mathbf{B}}^T(t){\varphi }^T\left(t,t_0\right)dt$$
where $\varphi \left(t,t_0\right)$ is the state transition matrix associated with $\mathbf{A}(t)$ from $t_0$ to $t$.

Definition 4. (Uniform complete controllability). The pair $(\mathbf{A}(t),\mathbf{B}(t))$ is uniformly completely controllable (UCC) if there exist positive constants $\alpha >0$ and $\delta >0$ such that, for all $t\ge t_0$,
$${\mathcal{W}}_c(t,t+\delta )⪰\alpha \mathbf{I}$$

Finally, a vector norm inequality is introduced next. Let $\mathbf{x}\in {\mathbb{R}}^n$. Then,
$$\Vert \mathbf{x}{\Vert }_{\infty }\le \Vert \mathbf{x}\Vert \le \sqrt{n}\Vert \mathbf{x}{\Vert }_{\infty },$$
where $\Vert \mathbf{x}\Vert$ and $\Vert \mathbf{x}{\Vert }_{\infty }$ are the Euclidean and infinity norms of $\mathbf{x}$, respectively.