Sampled-Data Stabilization of Nonlinear Delay Systems with a Compact Absorbing Set and State Measurement

Abstract

We present a methodology for the global sampled-data stabilization of systems with a compact absorbing set and input/measurement delays. The methodology is based on a numerical prediction scheme, which is combined with a projection of the state measurement on an appropriate sphere. The stabilization is robust to perturbations of the sampling schedule and is robust with respect to measurement noise. The obtained results are novel even for the delay-free case.

Appendix: Proof of Lemma 3.3

Let $\sigma > 0$, $T_{H} > 0$ be sufficiently small constants so that Lemma 3.1 holds. Let $N \ge N^{ * }$ be an integer, where $N^{ * }$ is the integer constant in Lemma 3.2. Let a partition of $\Re_{ + }$ $\left\{ {\tau_{i} } \right\}_{i = 0}^{\infty }$ with $\mathop {\sup }\limits_{i \ge 0} \left( {\tau_{i + 1} - \tau_{i} } \right) \le T_{s}$, $\xi \in B(\Re_{ + } ;\Re^{k} )$ and consider a solution of (1.1), (1.2), (2.8), (2.9), (2.10) defined for all $t \ge 0$ and satisfying $x(t) \in S_{1}$ for $t \ge T$ for some $T \ge 0$. Since $\left\{ {\tau_{i} } \right\}_{i = 0}^{\infty }$ is a partition of $\Re_{ + }$, there exists an integer $l \ge 0$ such that $\tau_{l} \ge r + T$. It follows that $x(t) \in S_{1}$ for $t \ge \tau_{l} - r$.

Definition (3.1) and inequality (3.2) in conjunction with (2.9) imply that $z(\tau_{i} ) \in S_{2}$ for all integers $i \ge 0$. Therefore, Lemma 2.1 in conjunction with (2.8) implies that $z(t) \in S_{2}$ for all $t \in [\tau_{i} ,\tau_{i + 1} )$. Define:

$$ L_{X} : = \sup \left\{ {\left| {f(x,u) - f(z,u)} \right|/\left| {x - z} \right|:\,x,z \in S_{2} \,,\,x \ne z\,,\,u \in U} \right\} $$

(A.1)

Equations (1.1), (2.8), the fact that $x(t + \tau ) \in S_{1}$ for all $t \ge \tau_{i}$ with $i \ge l$, in conjunction with definition (A.1) imply that the following inequality holds for all $t \in [\tau_{i} ,\tau_{i + 1} )$ and for all integers $i \ge l$:

$$ \left| {z(t) - x(t + \tau )} \right| \le \left| {z(\tau_{i} ) - x(\tau_{i} + \tau )} \right| + L_{X} \int_{{\tau_{i} }}^{t} {\left| {z(s) - x(s + \tau )} \right|ds} $$

(A.2)

A direct application of the Gronwall–Bellman inequality to (A.2) gives the following inequality for all $t \in [\tau_{i} ,\tau_{i + 1} )$ and for all integers $i \ge l$:

$$ \left| {z(t) - x(t + \tau )} \right| \le \, \exp \left( {L_{X} T_{s} } \right)\left| {z(\tau_{i} ) - x(\tau_{i} + \tau )} \right| $$

(A.3)

Let $\tilde{x}(t)$ denote the solution of (1.1) with initial condition $\tilde{x}(\tau_{i} - r) = Q\left( {x(\tau_{i} - r) + \xi (\tau_{i} )} \right)$ corresponding to the same input $u$. Lemma 3.2 implies the existence of a constant $M_{4} > 0$ such that:

$$ \left| {z(\tau_{i} ) - \tilde{x}(\tau_{i} + \tau )} \right| \le M_{4} \left( {\,\left| {Q\left( {x(\tau_{i} - r) + \xi (\tau_{i} )} \right)} \right| + \left\| {{\mathop{u}\limits^{\smile}}_{{\tau_{i} }} } \right\|\,} \right)/N $$

(A.4)

By virtue of the fact that $x(t) \in S_{1}$ for all $t \ge \tau_{i} - r$ with $i \ge l$, in conjunction with definition (A.1), Eq. (1.1), inequality (2.5) and Lemma 2.1 (which implies that $\tilde{x}(t) \in S_{2}$ for all $t \ge \tau_{i} - r$ with $i \ge l$), we obtain the following inequality for all $t \ge \tau_{i} - r$ with $i \ge l$:

$$ \left| {\tilde{x}(t) - x(t)} \right| \le \left| {\xi (\tau_{i} )} \right| + L_{X} \int_{{\tau_{i} - r}}^{t} {\left| {\tilde{x}(s) - x(s)} \right|ds} $$

(A.5)

A direct application of the Gronwall–Bellman inequality to (A.2) gives the following inequality for all integers $i \ge l$:

$$ \left| {\tilde{x}(\tau_{i} + \tau ) - x(\tau_{i} + \tau )} \right| \le \, \exp \left( {L_{X} (r + \tau )} \right)\left| {\xi (\tau_{i} )} \right| $$

(A.6)

Using (A.4), (A.6), (2.5) and the triangle inequality, we obtain the following inequality for all integers $i \ge l$:

$$ \begin{array} {l} \left| {z(\tau_{i} ) - x(\tau_{i} + \tau )} \right|N \le M_{4} \left( {\,\left| {Q\left( {x(\tau_{i} - r) + \xi (\tau_{i} )} \right)} \right| + \left\| {{\mathop{u}\limits^{\smile}}_{{\tau_{i} }} } \right\|\,} \right) + N\,\exp \left( {L_{X} (r + \tau )} \right)\left| {\xi (\tau_{i} )} \right| \\ \quad \le M_{4} \left( {\,\left| {x(\tau_{i} - r)} \right| + \left\| {{\mathop{u}\limits^{\smile}}_{{\tau_{i} }} } \right\|\,} \right) + \left( {N\,\exp \left( {L_{X} (r + \tau )} \right) + M_{4} } \right)\left| {\xi (\tau_{i} )} \right| \end{array} $$

(A.7)

Using (A.3), (A.7) we get for $t \in [\tau_{i} ,\tau_{i + 1} )$ and for all integers $i \ge l$:

$$ \begin{array} {l} N\left| {z(t) - x(t + \tau )} \right| \le \, \exp \left( {L_{X} T_{s} } \right)M_{4} \left( {\,\left| {x(\tau_{i} - r)} \right| + \left\| {{\mathop{u}\limits^{\smile}}_{{\tau_{i} }} } \right\|\,} \right) \\ \quad + \exp \left( {L_{X} T_{s} } \right)\left( {N\,\exp \left( {L_{X} (r + \tau )} \right) + M_{4} } \right)\left| {\xi (\tau_{i} )} \right| \end{array} $$

(A.8)

Define:

$$ K: = \sup \left\{ {\left| {k(x) - k(z)} \right|/\left| {x - z} \right|\,:\,x,z \in S_{2} \,,\,x \ne z\,} \right\} $$

(A.9)

Definition (A.9) combined with (A.8), the fact that $\mathop {\sup }\limits_{i \ge 0} \left( {\tau_{i + 1} - \tau_{i} } \right) \le T_{s}$ and the fact $x(t + \tau ) \in S_{1}$ for all $t \ge \tau_{i}$ with $i \ge l$, directly implies the following estimate for all $t \in [\tau_{i} ,\tau_{i + 1} )$ and for all integers $i \ge l$:

$$ \begin{aligned} & \left| {k(z(t)) - k(x(t + \tau ))} \right|\exp \left( {\sigma \,t} \right)N \le K\exp \left( {L_{X} T_{s} } \right)\left( {N\exp \left( {L_{X} (r + \tau )} \right) + M_{4} } \right) \\ & \quad \exp \left( {\sigma \,t} \right)\mathop {\sup }\limits_{{\tau _{l} \le s \le t}} \left( {\left| {\xi (s)} \right|} \right) + KM_{4} \exp \left( {L_{X} T_{s} } \right)\,\exp \left( {\sigma \,(T_{s} + r + \tau )} \right) \\ & \mathop {\quad \sup }\limits_{{\tau _{l} - r - \tau \le s \le t}} \left( {\left| {x(s + \tau )} \right|\exp (\sigma \,s)} \right) + KM_{4} \exp \left( {L_{X} T_{s} } \right)\, \\ & \quad \exp \left( {\sigma \,(T_{s} + r + \tau )} \right)\mathop {\sup }\limits_{{\tau _{i} - r - \tau \le s < t}} \left( {\left| {u(s)} \right|\exp (\sigma \,s)} \right) \\ \end{aligned} $$

(A.10)

Since the above inequality is independent of the integer $i \ge l$, it follows that inequality (A.10) holds for $t \ge \tau_{l}$. Define for all $t \ge 0$:

$$ q(t) = \left[ {t/T_{H} } \right]T_{H} $$

(A.11)

Combining (A.10) with definition (A.10) and definition (2.10), we obtain the following estimate for all $t \ge jT_{H}$, where $j$ is any integer with $jT_{H} \ge \tau_{l}$:

$$ \begin{array} {l} \mathop {\sup }\limits_{{jT_{H} \le s \le t}} \left( {\left| {u(s) - k(x(q(s) + \tau ))} \right|\exp (\sigma \,s)} \right)N \\ \quad \le KM_{4} \exp \left( {L_{X} T_{s} } \right)\,\exp \left( {\sigma \,(T_{H} + T_{s} + r + \tau )} \right)\mathop {\sup }\limits_{{\tau_{l} - r - \tau \le s \le t}} \left( {\left| {x(s + \tau )} \right|\exp (\sigma \,s)} \right) \\ \quad + KM_{4} \exp \left( {L_{X} T_{s} } \right)\,\exp \left( {\sigma \,(T_{H} + T_{s} + r + \tau )} \right)\mathop {\sup }\limits_{{\tau_{i} - r - \tau \le s < t}} \left( {\left| {u(s)} \right|\exp (\sigma \,s)} \right) \\ \quad + K\,\exp \left( {L_{X} T_{s} } \right)\left( {N\,\exp \left( {L_{X} (r + \tau )} \right) + M_{4} } \right)\exp \left( {\sigma \,t} \right)\mathop {\sup }\limits_{{\tau_{l} \le s \le t}} \left( {\left| {\xi (s)} \right|} \right) \\ \end{array} $$

(A.12)

Using (A.11), (A.12), the triangle inequality, (A.9) and the fact $x(t + \tau ) \in S_{1}$ for all $t \ge \tau_{i}$, we obtain for $t \ge jT_{H}$, where $j$ is any integer with $jT_{H} \ge \tau_{l}$:

$$ \begin{array} {l} \mathop {\sup }\limits_{{jT_{H} \le s \le t}} \left( {\left| {u(s) - k(x(q(s) + \tau ))} \right|\exp (\sigma \,s)} \right)N \\ \quad \le KM_{4} \exp \left( {L_{X} T_{s} } \right)\exp \left( {\sigma \,(T_{H} + T_{s} + r + \tau )} \right)\mathop {\sup }\limits_{{\tau_{i} - r - \tau \le s < jT_{H} }} \left( {\left| {u(s)} \right|\exp (\sigma \,s)} \right) \\ \quad + KM_{4} \exp \left( {L_{X} T_{s} } \right)\,\exp \left( {\sigma \,(T_{H} + T_{s} + r + \tau )} \right)\mathop {\sup }\limits_{{jT_{H} \le s < t}} \left( {\left| {u(s) - k(x(q(s) + \tau ))} \right|\exp (\sigma \,s)} \right) \hfill \\ \quad + KM_{4} \exp \left( {L_{X} T_{s} } \right)\,\exp \left( {\sigma \,(T_{H} + T_{s} + r + \tau )} \right)\left( {1 + K\,\exp \left( {\sigma \,T_{H} } \right)} \right)\mathop {\sup }\limits_{{\tau_{l} - r - \tau \le s \le t}} \left( {\left| {x(s + \tau )} \right|\exp (\sigma \,s)} \right) \hfill \\ \quad + K\,\exp \left( {L_{X} T_{s} } \right)\left( {N\,\exp \left( {L_{X} (r + \tau )} \right) + M_{4} } \right)\exp \left( {\sigma \,t} \right)\mathop {\sup }\limits_{{\tau_{l} \le s \le t}} \left( {\left| {\xi (s)} \right|} \right) \hfill \\ \end{array} $$

Selecting $N > K\,\exp \left( {L_{X} T_{s} } \right)M_{4} \,\exp \left( {\sigma \,(T_{H} + T_{s} + r + \tau )} \right)$, we obtain from the above estimate for all $t \ge jT_{H}$, where j is any integer with $jT_{H} \ge \tau_{l}$:

$$ \begin{array} {l} \left( {N - \tilde{K}} \right)\mathop {\sup }\limits_{{jT_{H} \le s \le t}} \left( {\left| {u(s) - k(x(q(s) + \tau ))} \right|\exp (\sigma \,s)} \right) \le \tilde{K}\mathop {\sup }\limits_{{\tau_{i} - r - \tau \le s < jT_{H} }} \left( {\left| {u(s)} \right|\exp (\sigma \,s)} \right) \\ \quad + \tilde{K}\left( {1 + K\,\exp \left( {\sigma \,T_{H} } \right)} \right)\mathop {\sup }\limits_{{\tau_{l} - r - \tau \le s \le t}} \left( {\left| {x(s + \tau )} \right|\exp (\sigma \,s)} \right) \\ \quad + K\,\exp \left( {L_{X} T_{s} } \right)\left( {N\,\exp \left( {L_{X} (r + \tau )} \right) + M_{4} } \right)\exp \left( {\sigma \,t} \right)\mathop {\sup }\limits_{{\tau_{l} \le s \le t}} \left( {\left| {\xi (s)} \right|} \right) \\ \end{array} $$

(A.13)

where $\tilde{K}: = K\,\exp \left( {L_{X} T_{s} } \right)M_{4} \,\exp \left( {\sigma \,(T_{H} + T_{s} + r + \tau )} \right)$. Using (A.13) and (3.3), we get the above estimate for all $t \ge jT_{H}$, where $j$ is any integer with $jT_{H} \ge \tau_{l}$:

$$ \begin{array} {l} \left( {N - \tilde{K}} \right)\mathop {\sup }\limits_{{jT_{H} \le s \le t}} \left( {\left| {x(s + \tau )} \right|\exp (\sigma \,s)} \right) \le M_{2} \left( {N - \tilde{K}} \right)\exp \left( {\sigma \,jT_{H} } \right)\left| {x(jT_{H} + \tau )} \right| \\ \quad + M_{3} \tilde{K}\mathop {\sup }\limits_{{\tau_{i} - r - \tau \le s < jT_{H} }} \left( {\left| {u(s)} \right|\exp (\sigma \,s)} \right) + M_{3} \tilde{K}\left( {1 + K\,\exp \left( {\sigma \,T_{H} } \right)} \right)\mathop {\sup }\limits_{{\tau_{l} - r - \tau \le s \le t}} \left( {\left| {x(s + \tau )} \right|\exp (\sigma \,s)} \right) \\ \quad + M_{3} K\,\exp \left( {L_{X} T_{s} } \right)\left( {N\,\exp \left( {L_{X} (r + \tau )} \right) + M_{4} } \right)\exp \left( {\sigma \,t} \right)\mathop {\sup }\limits_{{\tau_{l} \le s \le t}} \left( {\left| {\xi (s)} \right|} \right) \\ \end{array} $$

(A.14)

Using (A.13) and selecting $N > \tilde{K} + \tilde{K}M_{3} \left( {1 + K\,\exp \left( {\sigma \,T_{H} } \right)} \right)$, we get the above estimate for all $t \ge jT_{H}$, where $j$ is any integer with $jT_{H} \ge \tau_{l}$:

$$ \begin{array} {l} \left( {N - \tilde{K} - \tilde{K}M_{3} \left( {1 + K\,\exp \left( {\sigma \,T_{H} } \right)} \right)} \right)\mathop {\sup }\limits_{{jT_{H} \le s \le t}} \left( {\left| {x(s + \tau )} \right|\exp (\sigma \,s)} \right) \le M_{3} \tilde{K}\mathop {\sup }\limits_{{ - \tau \le s < jT_{H} }} \left( {\left| {u(s)} \right|\exp (\sigma \,s)} \right) \\ \quad M_{2} \left( {N - \tilde{K}} \right) + \tilde{K}M_{3} \left( {1 + K\,\exp \left( {\sigma \,T_{H} } \right)} \right)\mathop {\sup }\limits_{{ - \tau \le s \le jT_{H} }} \left( {\left| {x(s + \tau )} \right|\exp (\sigma \,s)} \right) \\ \quad + M_{3} K\,\exp \left( {L_{X} T_{s} } \right)\left( {N\,\exp \left( {L_{X} (r + \tau )} \right) + M_{4} } \right)\exp \left( {\sigma \,t} \right)\mathop {\sup }\limits_{r \le s \le t} \left( {\left| {\xi (s)} \right|} \right) \\ \end{array} $$

(A.15)

Inequality (A.15) implies the existence of ${\Lambda}_{1} \ge 1$ (constant independent of $j$) for which the following inequality holds for $t \ge 0$ and for all integers $j$ with $jT_{H} \ge \tau_{l}$:

$$ \begin{array} {l} \mathop {\sup }\limits_{ - r - \tau \le s \le t} \left( {\left| {x(s + \tau )} \right|\exp (\sigma \,s)} \right) \le {\Lambda}_{1} \mathop {\sup }\limits_{{ - r - \tau \le s \le jT_{H} }} \left( {\left| {x(s + \tau )} \right|\exp (\sigma \,s)} \right) \\ + {\Lambda}_{1} \mathop {\sup }\limits_{{ - r - \tau \le s < jT_{H} }} \left( {\left| {u(s)} \right|\exp (\sigma \,s)} \right) + {\Lambda}_{1} \exp \left( {\sigma \,t} \right)\mathop {\sup }\limits_{0 \le s \le t} \left( {\left| {\xi (s)} \right|} \right) \\ \end{array} $$

(A.16)

The definition of the norms $\left\| {x_{t} } \right\|$ and $\left\| {{\mathop{u}\limits^{\smile}}_{t} } \right\|$ give for all $t \ge 0$:

$$ \mathop {\sup }\limits_{0 \le s \le t} \left( {\left\| {x_{s} } \right\|\exp (\sigma s)} \right) \le \, \exp \left( {\sigma (r + \tau )} \right)\mathop {\sup }\limits_{ - r - \tau \le s \le t} \left( {\left| {x\left( {\tau + s} \right)} \right|\exp (\sigma s)} \right) $$

(A.17)

$$ \mathop {\sup }\limits_{0 \le s \le t} \left( {\left\| {{\mathop{u}\limits^{\smile}}_{s} } \right\|\exp (\sigma s)} \right) \le \, \exp \left( {\sigma (r + \tau )} \right)\mathop {\sup }\limits_{ - r - \tau \le s \le t} \left( {\left| {u(s)} \right|\exp (\sigma s)} \right) $$

(A.18)

When $t \le jT_{H}$ we have from (A.18) that $\mathop {\sup }\limits_{0 \le s \le t} \left( {\left\| {{\mathop{u}\limits^{\smile}}_{s} } \right\|\exp (\sigma s)} \right) \le \, \exp \left( {\sigma (r + \tau )} \right)\mathop {\sup }\limits_{{ - r - \tau \le s \le jT_{H} }} \left( {\left| {u(s)} \right|\exp (\sigma s)} \right)$. When $t \ge jT_{H}$, we obtain from (A.9), (A.11), (A.13), (A.16), the facts that $k(0) = 0$, $0 \le s - q(s) \le T_{H}$, $x(t) \in S_{1}$ for all $t \ge jT_{H}$, the existence of a constant ${\Lambda}_{2} \ge 1$ (independent of $j$) for which the following inequalities hold for $t \ge jT_{H}$:

$$ \begin{array} {l} \mathop {\sup }\limits_{0 \le s \le t} \left( {\left\| {{\mathop{u}\limits^{\smile}}_{s} } \right\|\exp (\sigma s)} \right) \le {\Lambda}_{2} \exp (\sigma \,t)\mathop {\sup }\limits_{0 \le s \le t} \left( {\left| {\xi \left( s \right)} \right|} \right) \\ \quad + {\Lambda}_{2} \mathop {\sup }\limits_{{ - r - \tau \le s \le jT_{H} }} \left( {\left| {u(s)} \right|\exp (\sigma s)} \right) + {\Lambda}_{2} \mathop {\sup }\limits_{{ - r - \tau \le s \le jT_{H} }} \left( {\left| {x(\tau + s)} \right|\exp (\sigma s)} \right)\\ \end{array} $$

Combining the two cases ($t \le jT_{H}$ and $t \ge jT_{H}$) and using (A.17), (A.18), we obtain the existence of a constant ${\Lambda}_{3} \ge 1$ (independent of $j$) for which the following inequality holds for all $t \ge 0$ and for all integers $j$ with $jT_{H} \ge \tau_{l}$:

$$ \begin{array} {l} \mathop {\sup }\limits_{0 \le s \le t} \left( {\left\| {{\mathop{u}\limits^{\smile}}_{s} } \right\|\exp (\sigma s)} \right) + \mathop {\sup }\limits_{0 \le s \le t} \left( {\left\| {x_{s} } \right\|\exp (\sigma s)} \right) \le {\Lambda}_{3} \mathop {\sup }\limits_{{ - r - \tau \le s \le jT_{H} }} \left( {\left| {u(s)} \right|\exp (\sigma s)} \right) \\ \quad + {\Lambda}_{3} \exp (\sigma \,t)\mathop {\sup }\limits_{0 \le s \le t} \left( {\left| {\xi (s)} \right|} \right) + {\Lambda}_{3} \mathop {\sup }\limits_{{ - r - \tau \le s \le jT_{H} }} \left( {\left| {x(\tau + s)} \right|\exp (\sigma s)} \right) \\ \end{array} $$

(A.19)

We notice that since $\mathop {\sup }\limits_{i \ge 0} \left( {\tau_{i + 1} - \tau_{i} } \right) \le T_{s}$, the smallest the integer $l \ge 0$ with $\tau_{l} \ge r + T$ also satisfies $\tau_{l} \le r + T + T_{s}$. Therefore, we conclude that inequality (A.19) holds for all $t \ge 0$ and for all integers $j$ with $jT_{H} \ge r + T + T_{s}$. Inequality (3.6) is a direct consequence of estimate (A.19). The proof is complete. $\triangleleft$