Stabilizing bit-rates in networked control systems with decentralized event-triggered communication

Abstract

In a decentralized event-triggered networked control system (NCS), an agent samples and transmits its local state information to the controller when some local event occurs. Such event-triggered NCSs were expected to be more efficient than traditional periodically sampled system in terms of communication channel usage. This paper studies the stability of decentralized event-triggered NCS in the presence of quantization and delays. We point out some potential issues in decentralized event-triggered design and propose an alternative decentralized event with a linear-affine threshold, which avoids infinitely fast data transmission. Conditions on quantizer and communication channel are derived, which, when satisfied, can guarantee stability of the resulting NCS. Based on these conditions, finite stabilizing bit-rates are provided.

Appendices

Appendix A: Proof of Lemma 1

Proof

Equation 2 implies $\|\dot x_i(t)\|=\|F_i\left(x(t), K(\hat x)\right)\|$. Since ||x(t)|| and $\|\hat x(t)\|$ are bounded by θ and $\hat \theta$, respectively, over $\left[r_i^k, f_i^{k+1}\right)$, we can apply Assumption 3 that F _i is locally Lipschitz and obtain

$$\begin{array}{rll} \|\dot x_i(t)\| &\le& a_i \|x(t)\| + b_i \|\hat x(t)\| \\ &\le& a_i\theta + b_i \hat \theta = \gamma_i, \end{array}$$

where the last inequality is obtained with the assumption ||x(t)|| ≤ θ and $\|\hat x(t)\| \le \hat \theta$. As a result,

$$ \frac{d}{dt} \|x_i(t)-x_i^k\|~ \le~ \|\dot x_i(t)\| ~\le~ \gamma_i $$

holds for any $t \in \left[r_i^k, f_i^{k+1}\right)$. Solving this inequality for any $t \in [r_i^{k+1}, f_i^{k+1})$, we obtain

$$\begin{array}{rll} \|x_i(t) -x_i^k \| &\le& \|x_i\left(r_i^{k+1}\right)-x_i^k\| + \gamma_i \left(t- r_i^{k+1}\right) \\ &\le& \rho_i\|x_i^k\|+ \omega_i+\gamma_i \Delta_i, \end{array}$$

where the second inequality comes from the triggering condition. Note that for any $t\in [r_i^k, r_i^{k+1}]$, the preceding inequality also holds. Consequently, we have

$$ \|x_i(t) -x_i^k \| \le \rho_i\|x_i^k\|+ \omega_i+\gamma_i \Delta_i, $$

for all $t\in [r_i^k, f_i^{k+1})$. Subtracting $\rho_i\|x_i(t) -x_i^k \|$ at both sides of the inequality above, we obtain

$$\begin{array}{rll} (1-\rho_i)\|x_i(t) -x_i^k \| &\le & \rho_i\|x_i^k\|+ \omega_i+\gamma_i \Delta_i - \rho_i\|x_i(t) -x_i^k \| \\ &\le& \rho_i\|x_i(t)\| + \omega_i+\gamma_i \Delta_i, \end{array}$$

which implies inequality (10). □

Appendix B: Proof of Lemma 2

Proof

Since ||x(t)|| ≤ θ and $\|\hat x(t)\| \le \hat \theta$ for any t ≥ 0, we apply Lemma 1 to obtain

$$ \|x_i(t) -x_i^k \| \le \frac{\rho_i}{1-\rho_i} \|x_i(t)\| + \frac{\omega_i+\gamma_i \Delta_i}{{1-\rho_i}} $$

for $\forall~t\in \left[f_i^k,f_i^{k+1}\right)$, $\forall~i \in {\mathcal{N}}$, and ∀ k ∈ ℤ⁺. Since $\hat x_i(t) = x_i^k$ for $\forall~t\in \left[f_i^k,f_i^{k+1}\right)$, we know

$$ \|x_i(t) - \hat x_i(t) \| \le \frac{\rho_i}{{1-\rho_i}} \|x_i(t)\| + \frac{\omega_i+\gamma_i \Delta_i}{{1-\rho_i}} $$

for ∀ t ≥ 0. Therefore,

$$\begin{array}{rll} \|\tilde x(t)\|= \|x(t)-\hat x(t)\| &\le& \left\|\left\langle \frac{\rho_i}{{1-\rho_i}} \|x_i(t)\| + \frac{\omega_i+\gamma_i \Delta_i}{{1-\rho_i}} \right\rangle \right\| \nonumber \\ &\le& \left\|\left\langle \frac{\rho_i}{{1-\rho_i}} \|x_i(t)\|\right\rangle\right\| + \left\|\left\langle \frac{\omega_i+\gamma_i \Delta_i}{{1-\rho_i}} \right\rangle \right\| \nonumber \\ &\le& p \|x(t)\| + d \end{array}$$

(37)

where $p := \max_{i\in {\mathcal{N}}}\frac{\rho_i}{{1-\rho_i}}$ and $d :=\left\|\left\langle \frac{\omega_i+\gamma_i \Delta_i}{{1-\rho_i}} \right\rangle \right\|$.

We now consider $\dot V$ at time t. By Eq. 5, the time derivative of V at time t satisfies

$$ \dot V \le -\beta(x(t))\left(\phi(\|x(t)\|) - \psi\left(\|x(t)\|, \|\tilde x(t)\|\right)\right). $$

Since ||x(t)|| and $\|\hat x(t)\|$ are both bounded for any t ≥ 0 , we can apply Assumption 2 to the preceding inequality and obtain

$$\begin{array}{rll} \dot V &\le & -\beta(x(t))\left(L\|x(t)\| - B\|\tilde x(t)\| \right) \\ &\le & -\beta(x(t))\left(L\|x(t)\| - B p \|x(t)\| - Bd \right) \end{array}$$

(38)

where the second inequality is obtained using inequality (37). Therefore there must exist T ≥ 0 such that

$$ \|x(t)\| \le \alpha_1^{-1} \circ \alpha_2\left(\frac{Bd}{L-Bp}\right) $$

holds for ∀ t ≥ T, as shown in Khalil (2002, p. 169). □

Appendix C: Proof of Theorem 1

Proof

First, we prove that V(x(t)) ≤ V(x ₀) holds for all t > 0 by contradiction. To simplify the notations, we use V(t) to denote V(x(t)) if it is clear in context. Suppose that there is a time instant $\bar t > 0$, such that $V(\bar t) > V(x_0)$. Note that $\dot V(t)|_{t=0} < 0$. Therefore, there must exist a time instant $t^* \in (0,\bar t)$ and a small positive constant ϵ such that

$$ V(t) < V(t^*) =V(x_0),~~ \forall~t\in [0,t^*) $$

(39)

$$ \dot V(t) > 0,\;\;\forall~t\in (t^*-\epsilon,t^*]. $$

(40)

These inequalities imply

$$ \|x(t)\| \le \theta, $$

(41)

$$ \|\hat x(t)\| \le \hat \theta $$

(42)

for ∀ t ∈ [0, t ^*]. Following a similar analysis as in the proof of Lemma 2, we obtain

$$ \dot V(t) \le -\beta(x(t))\left(L\|x(t)\| - B p \|x(t)\| -Bd\right) $$

(43)

for all t ∈ [0,t ^*], where $p := \max_{i\in {\mathcal{N}}}\frac{\rho_i}{{1-\rho_i}}$ and $d :=\left\|\left\langle \frac{\omega_i+\gamma_i \Delta_i}{{1-\rho_i}} \right\rangle \right\|$. By inequality (40), we know $\dot V(t) > 0$ for ∀ t ∈ (t ^* − ϵ, t ^*] . Combining this with the preceding inequality yields

$$ 0 < \dot V(t) \le -\beta(x(t))\left(L\|x(t)\| - B p \|x(t)\| -Bd \right) $$

for any t ∈ (t ^* − ϵ, t ^*], which implies

$$\|x(t)\| < \frac{Bd}{L-Bp} $$

(44)

for any t ∈ (t ^* − ϵ, t ^*], since β is a positive definite function. Note that by Eq. 11, $\gamma_i = a_i \theta + b_i \hat \theta$. Therefore, inequality (44) yields

$$\begin{array}{rll} \|x(t)\| &<& \frac{B \left\|\left\langle \frac{\omega_i+ (a_i\theta +b_i\hat \theta) \Delta_i}{{1-\rho_i}} \right\rangle \right\| }{L-Bp}\\ &\le& \alpha_2^{-1}\circ V(x_0) \end{array}$$

for any t ∈ (t ^* − ϵ, t ^*], where the second inequality is obtained using inequality (17). As a result, we have $V(t^*) \le \alpha_2(\|x(t^*)\|) < V(x_0)$, which contradicts inequality (39). Therefore, we can conclude that V(t) ≤ V(x ₀) always holds.

Since the hypotheses of Lemma 1 are satisfied, we know that x(t) ∈ Λ, where Λ is defined in Eq. 13. Consequently, ||x(t)|| ≤ θ and $\|\hat x(t)\| \le \hat \theta$ hold for all t ≥ 0. Applying Lemma 2, we know that the overall NCS is UUB. Meanwhile, since ||x(t)|| and $\hat x(t)$ are bounded, we know that the growth rate of $\|x_i(t)-x_i^k\|$ is bounded. Therefore, with a linear-affine threshold, we conclude that the inter-event time intervals are bounded by a positive constant from below. □

Appendix D: Proof of Theorem 2

Proof

By inequalities (19) and (23), we know

$$\begin{array}{rll} \sigma \alpha_1^{-1} \circ V(x_0) &\le& \alpha_1^{-1}\circ \alpha_2 \left(\sigma \alpha_1^{-1} \circ V(x_0) \right) \\ &\le& \mu \kappa \left( \sigma \alpha_1^{-1} \circ V(x_0) \right) \\&<& \kappa \left( \sigma \alpha_1^{-1} \circ V(x_0) \right) < \alpha_2^{-1}\circ~V(x_0), \end{array}$$

which implies that inequality (17) in Theorem 1 is satisfied with $\omega_i^0 = \tau_i \alpha_1^{-1} \circ V(x_0)$. Then by Theorem 1, we know x(t) ∈ Λ for all t ≥ 0, which means

$$\begin{array}{rll} \|x(t)\| &\le& \alpha_1^{-1}\circ V(x_0),\\ \|\hat x(t)\| &\le& \left\|\left\langle \frac{\tilde \delta_i \tau_i + 1}{1-\tilde \delta_i \rho_i}\right\rangle\right\| \alpha_1^{-1}\circ V(x_0). \end{array}$$

Since the time between the switches of $\omega_i^s$ is long enough, we can apply Lemma 2 to show that there exists t ₁ > 0 such that every $\omega_i^0$ has been switched to $\omega_i^1$ and the inequality

$$\begin{array}{rll} \|x(t)\| &\le & \alpha_1^{-1} \circ \alpha_2\left( \sigma \alpha_1^{-1} \circ V(x_0) \right) \\ &\le& \mu \kappa\left( \sigma \alpha_1^{-1} \circ V(x_0) \right) \nonumber \\ &\le & \mu \alpha_2^{-1}\circ V(x_0) \le \mu \alpha_1^{-1}\circ V(x_0) \end{array}$$

holds for any t ≥ t ₁, where the third inequality is obtained using inequality (23). Consequently,

$$ \|\hat x(t)\|\le \mu \left\|\left\langle \frac{\tilde \delta_i \tau_i + 1}{1-\tilde \delta_i \rho_i}\right\rangle\right\| \alpha_1^{-1}\circ V(x_0) $$

holds for all t ≥ t ₁. Then we can re-apply Lemma 2 to get the new ultimate bound on ||x(t)||, i.e. there exists t ₂ > t ₁ such that every $\omega_i^1$ has been switched to $\omega_i^2$ and

$$\begin{array}{rll} \|x(t)\| &\le& \mu^2\alpha_1^{-1}\circ V(x_0),\\ \|\hat x(t)\| &\le& \mu^2\left\|\left\langle \frac{\tilde \delta_i \tau_i + 1}{1-\tilde \delta_i \rho_i}\right\rangle\right\| \alpha_1^{-1}\circ V(x_0). \end{array}$$

holds for all t ≥ t ₂. Keeping this procedure, we know that there exists t _k  > 0, such that

$$ \|x(t)\| \le \mu^s \alpha_1^{-1} \circ V(x_0) $$

holds for all t ≥ t _s . Since μ ∈ (0, 1), as k → ∞, the preceding inequality implies x(t) → 0, which implies asymptotic stability of the NCS.

We now show the lower bound on the inter-event time intervals. Note that during the time interval when the event is $\|x_i(r_i^k) -x_i^k\| \le \rho_i\|x_i^k\| +\omega_i^s$, the bound on the state is $\|x(t)\| \le \mu^s \alpha_1^{-1} \circ V(x_0)$. Therefore, we can follow a similar analysis in Lemma 1 and obtain

$$\begin{array}{lll} &&\|x_i\left(r_i^{k+1}\right) -x_i^k \| \\ &&\quad \le \|x_i\left(r_i^k\right) - x_i^k\| + \mu^s\underbrace{\left( a_i+b_i\left\|\left\langle \frac{\tilde \delta_i \tau_i + 1}{1-\tilde \delta_i \rho_i}\right\rangle\right\| \right)\alpha_1^{-1}\circ V(x_0)}_{\varrho_i} \left(r_i^{k+1}- r_i^k\right) \\ && \quad\le \tilde \delta_i \left(\rho_i\|x_i^k\| + \omega^s_i\right) +\mu^s \varrho_i T_i^k. \end{array}$$

Therefore, with $\|x_i(r_i^{k+1}) -x_i^k \| = \rho_i\|x_i^k\| + \omega^s_i$, we have

$$ T_i^k \ge \frac{(1-\tilde \delta_i) \tau_i \alpha_1^{-1} \circ V(x_0)}{\varrho_i}. $$

□

Appendix E: Proof of Lemma 3

Proof

We prove the statement by considering two cases:

Case I: :

when $(1-\rho_i)\|x_i^k\| \ge \omega_i$. By the triggering condition, we have

$$ \|x_i\left(r_i^{k+1}\right)\| \ge \left(1-\rho_i\right)\|x_i^{k}\|-\omega_i $$

which implies

$$\begin{array}{rll} \rho_i\|x_i\left(r^{k+1}\right)\|+\omega_i &\ge & \rho_i\left(\left(1-\rho_i\right)\|x_i^{k}\|-\omega_i\right)+\omega_i \\ &=&(1-\rho_i)\left(\rho_i\|x_i^{k}\|+\omega_i\right) \end{array}$$

Using inequality (29) and the inequality above, we obtain

$$ \frac{\|e_{{\mathcal{Q}},i}^{k+1}\|}{\rho_i\|x_i(r_i^{k+1})\|+\omega_i} \le \frac{\|e_{{\mathcal{Q}},i}^{k+1}\|}{(1-\rho_i)(\rho_i\|{x}_i^{k}\|+\omega_i)} \le \frac{\pi \sqrt{n-1}}{p(1-\rho_i)}. $$

(45)

Case II: :

when $(1-\rho_i)\|x_i^k\| < \omega_i$. In this case the following inequalities

$$\begin{array}{lll} & &\omega_i-(1-\rho_i)\|{x}_i^{k}\|\le\|x_i\left(r_i^{k+1}\right)\|\le (1+\rho_i)\|{x}_i^{k}\|+\omega_i \\ &&\rho_i\|x_i\left(r_i^{k+1}\right)\|+\omega_i \ge \omega_i \end{array}$$

hold. Using inequality (29) and the preceding inequalities, we obtain

$$\begin{array}{rll} \frac{\|e_{{\mathcal{Q}},i}^{k+1}\|}{\rho_i\|x_i(r_i^{k+1})\|+\omega_i}&\le& \frac{\|e_{{\mathcal{Q}},i}^{k+1}\|}{\omega_i} \le \frac{\pi \sqrt{n-1}}{p}\cdot\frac{\rho_i\|{x}_i^{k}\|+\omega_i}{\omega_i}\nonumber\\ &\le& \frac{\pi\sqrt{n-1}}{p}\cdot\left(1+\frac{\rho_i\|{x}_i^{k}\|}{(1-\rho_i)\|{x}_i^{k}\|}\right)\nonumber\\ &=& \frac{\pi\sqrt{n-1}}{p(1-\rho_i)} \end{array}$$

(46)

The right hand side of inequalities (45) and (46) are the same and can be made arbitrarily small by increasing p. □

Appendix F: Proof of Corollary 1

Proof

Consider $\|x_i(t)-\hat x_i(t)\|$ over $t\in [r_i^k,r_i^{k+1})$. Note that

$$\begin{array}{rll} \frac{d}{dt} \|x_i(t)-x_i^k\| & \le & \|\dot x_i(t)\| \\ &\le& a_i \|x(t)\| + b_i \|\hat x(t)\| \\ &\le& a_i\theta + b_i \hat \theta = \gamma_i \end{array}$$

holds for any $t \in \left[r_i^k, r_i^{k+1}\right)$, which implies that for any $t \in [r_i^k, r_i^{k+1})$, the inequality

$$\begin{array}{rll} \|x_i(t) -x_i^k \| &\le& \|x_i(r_i^k) - x_i^k\| + \gamma_i \left(t- r_i^k\right) \\ &\le& \tilde \delta (\rho_i\|x_i^k\| + \omega_i) +\gamma_i T_i^k \end{array}$$

holds and therefore

$$ \|x_i(r_i^{k+1}) -x_i^k \| \le \tilde \delta (\rho_i\|x_i^k\| + \omega_i) +\gamma_i T_i^k. $$

Based on the triggering condition, $\|x_i(r_i^{k+1}) -x_i^k \| = \rho_i\|x_i^k\| + \omega_i$. Applying this equation to the preceding inequality yields the satisfaction of inequality (33). □