Greedy Synthesis of Event- and Self-Triggered Controls
with Control Lyapunov-Barrier Function

The sets of real numbers, real vectors of length $n$, and real matrices of size $n\times m$ are denoted by $\mathbb{R}$, $\mathbb{R}^{n}$, and $\mathbb{R}^{n\times m}$, respectively. The sets of nonnegative numbers and nonnegative integers are denoted by $\mathbb{R}_{\geq 0}$ and $\mathbb{N}$, respectively.

$L_{f}V(x)$ denotes the Lie derivative of $V(x)$ along the vector field $f(x)$, i.e. $L_{f}V(x)=\frac{\partial V(x)}{\partial x}f(x)$.

A continuous function $\gamma:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}$ is said to belong to class $\mathcal{K}$ if it is strictly increasing and $\gamma(0)=0$. A continuous function $\alpha:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}$ is said to belong to class $\mathcal{K}_{\infty}$ if it belongs to class $\mathcal{K}$ and $\alpha(r)\rightarrow\infty$ as $r\rightarrow\infty$.

This paper deals with a nonlinear affine system in the form of

where $x\in\mathcal{D}\subset\mathbb{R}^{n}$ and $u\in\mathbb{R}^{m}$ denote the state and the control input of the system, respectively, and $\mathcal{C}\subseteq\mathcal{D}$ is a safe set that is defined later. It is assumed that $\mathcal{D}$ is bounded, and the functions $f(x)$ and $g(x)$ are Lipschitz.

The existence of a CLF guarantees asymptotic stabilization of the nonlinear control system (1) with any Lipschitz continuous feedback controller $u(x)$ that satisfies (2) for all $x\in\mathcal{D}$ [25, 16].

Let define the safe set $\mathcal{C}$ as the superlevel set of a continuously differentiable function $h:\mathcal{D}\subset\mathbb{R}^{n}\rightarrow\mathbb{R}$

The existence of a CLF guarantees that the control system is safe [15].

Motivated by the results on CLF and CBF, it is of interest to obtain a controller that satisfies

so that it is a safe stabilizing controller.

To design such a controller, a CLF-CBF QP was constructed in [26, 15]:

where

a positive definite matrix $Q\in\mathbb{R}^{m+1\times m+1}$ and $c\in\mathbb{R}^{m+1}$ are weights and $\delta$ is a relaxation variable that ensures the solvability of the QP. If $\delta$ is forced to be nonpositive, then the existence of a feasible controller will guarantee the monotonic decrease of the Lyapunov function.

The primary idea behind event-triggered control is to update the control input only when necessary to achieve a specified performance condition, thereby reducing the frequency of updates. In line with the standard structure of an event-triggered controller, we consider the control inputs that are maintained constant between successive event times, i.e.,

where $u_{k}$ is the control input computed at time $t_{k}$, which is the time instance when the control input is re-computed and the actuator signals are updated. The time instance $t_{k}$ is determined by

Here, we introduce a greedy approach for computing the control input $u_{k}$ at trigger time $t_{k}$. To implement it into an event-triggered control, we are interested in a control law that maximizes the inter-execution time. For this purpose, we seek the control input that brings the state away from the boundaries of the constraints (5), (6). This is achieved by maximizing the slack variables $\rho_{1}$ and $\rho_{2}$ in the new constraints:

where $b_{\text{clf}}(x)$ and $b_{\text{cbf}}(x)$ are defined in (8).

Based on the constraints (11), we propose to modify the CLF-CBF QP in (7) as follows:

where a positive semidefinite matrix $Q\in\mathbb{R}^{m+2\times m+2}$ and $c\in\mathbb{R}^{m+2}$ are weights, $\rho_{1}$ and $\rho_{2}$ are slack variables, and $\varepsilon_{\text{cbf}}>0$ is a constant (design parameter) that forces the states to be away from the boundary. This is still QP and the solvability of the QP is still ensured as long as $\rho_{1}$ is not constrained. Let $[u^{*}(x),\rho_{1}^{*}(x),\rho_{2}^{*}(x)]^{\top}=\mathbf{v}^{*}(x)$.

Typically, a small norm control input $u$ is desired to minimize the control effort while large $\rho_{1}$ and $\rho_{2}$ are desired to maximize the inter-execution time. Hence, a possible choice for $Q$ and $q$ is

where $w_{1}\in\mathbb{R}^{m\times m}$ is positive definite, and $w_{2},w_{3}\geq 0$.

With (12), the proposed controller implements the control input $u_{k}$ defined by

where $x_{k}=x(t_{k})$.

For the states to remain in the safe region, the safety constraint (6) should be always satisfied. However, the satisfaction of the stability constraint (5) cannot be guaranteed together with the satisfaction of (6) in general. This is the same for the proposed controller. Yet, we do our best to minimize the time in which (5) is violated.

Define

where

Note that the time derivative of the Lyapunov function is $\dot{V}(x)=-p_{k}(x)$, thus $p_{k}(x)\geq 0$ is desired for the stability, while $q_{k}(x)\geq 0$ is desired for the safety.

We set the trigger condition in (10) as

1. 1.

   if $p_{k}(x_{k})\geq\varepsilon_{\text{clf}}$:

   $\displaystyle p_{k}(x)=0\text{ or }q_{k}(x)=0$

   (19)

2. 2.

   else:

   $\displaystyle q_{k}(x)=0\text{ or }t=t_{k}+\tau_{bd}$

   (20)

where $\varepsilon_{\text{clf}}>0$ and $\tau_{bd}>0$ are design parameters.

The first case is that, if the time-derivative of the Lyapunov function is sufficiently negative at the time of update $t_{k}$, then the next update time is when either the safety constraint (5) is satisfied with equality or the time-derivative of Lyapunov function becomes zero. This guarantees the satisfaction of both safety and stability between $t_{k}$ and $t_{k+1}$.

The second case is that, if the time-derivative of the Lyapunov function is close to zero or positive at the time of update $t_{k}$, then we compromise the controller design only focusing on the safety constraint. The next update time is when the safety constraint (5) is satisfied with equality or small time $\tau_{bd}$ passes, whichever occurs first. This limits the duration of time during which the stability constraint is violated to $\tau_{bd}$ with the same control input.

This subsection shows that the events cannot be triggered an infinite number of times in any finite time period with the proposed controller, i.e., the proposed controller is Zeno-free under mild conditions.

Define the inter-execution time

It is of interest to show the existence of a lower bound $\tau^{*}$ such that $\tau_{k}\geq\tau^{*}$ for all $k\in\mathbb{N}$.

If the problem is considered in a finite time horizon, the first assumption is sufficient.

Here, we see the parameter $\varepsilon_{\text{cbf}}>0$ in (12) can be used as a parameter to tune the inter-execution time.

As in Section III, let $t_{k}$ be the triggering time instance. Self-triggered control computes the control input $u_{k}$ as well as the next time instance $t_{k+1}$ at execution time $t_{k}$. Similar to the event-triggered control (9), the control input $u_{k}$ remains constant in between $t_{k}$ and $t_{k+1}$. Unlike the event-triggered condition, however, no sampling or computation is required between $t_{k}$ and $t_{k+1}$.

As for the event-triggered control, the controller implements the control input $u_{k}$ in (14) by solving (12).

The next execution time $t_{k+1}$ is computed based on the measured state $x(t_{k})=x_{k}$ at $t_{k}$:

where the map $\Gamma:\mathbb{R}^{n}\rightarrow\mathbb{R}_{\geq 0}$ determines the triggering time $t_{k+1}$ as a function of the state $x_{k}$ at the time $t_{k}$. Thus, the inter-execution time is given by $\Gamma(x)$, i.e., $\tau_{k}=\Gamma(x_{k})$.

In the following, we present approaches to how to design the map $\Gamma(x)$.

In Section III, the trigger condition (19)-(20) is designed to satisfy the safety constraint (6) and minimize the violation of the stability constraint (5). Here, again, we design the map $\Gamma$ that satisfies the safety constraint (6) all the time, while allowing the violation of the stability constraint:

Ideally, we would like to find $t$ that achieves an equality in (37). However, it is difficult in general for nonlinear systems, thus, we aim at finding a lower bound by revising the approach in Section III-D.

Again, we assume that Assumption III.1 is satisfied. Then, we may use the following map:

The difference between this map $\Sigma$ and the lower bound $\tau^{*}$ in the event-triggered control is only the upper bound on the norm of $f(x)+g(x)u$. Although a uniform $M$ may be used to design a constant $\Gamma=\tau^{*}$, such a law will shorten the inter-execution time because $M$ is likely to be much larger than $M_{k}$. However, one difficulty of implementing this approach is actually in the computation of $M_{k}$ in (47) where $M_{k}$ appears in both sides of the equation. To compute exact $M_{k}$, it is required to compute the evolution of (1) starting at $t_{k}$ by gradually increasing the time duration. In the actual implementation, this might not be a big problem because implementation is done digitally, which we discuss next.

Similarly to [20], we now consider the following discrete-time versions of $p_{k}(x)$ and $q_{k}(x)$ based on a sampling time $\Delta>0$, which are defined by

Let $\tau_{\min}$ and $\tau_{\max}$ be design parameters and let $N_{\min}=\lfloor\tau_{\min}/\Delta\rfloor$, $N_{\max}=\lfloor\tau_{\max}/\Delta\rfloor$.

Then, by choosing $\tau_{bd}=\Delta$, the map can be simplified as

where

This satisfies

and $n(x)\in[N_{\min},N_{\max}]$.

With a smaller choice of $\Delta$, the stability constraint is more strictly forced, i.e., while the stability constraint is violated, the control input is updated every sampling time. Such a choice of $\Delta$ may result in an increase in the number of execution. Of course, it is possible to use different values for $\tau_{bd}$ and $\Delta$ with a slight modification.

We may simply choose $\tau_{\min}=0$ and a sufficiently large $\tau_{\max}$. However, the value of $\tau_{\max}$ enforces the robustness of the implementation and limits the computational complexity [20].