Safe Control Synthesis Using Environmentally Robust Control Barrier Functions

Control Barrier Functions (CBFs) have emerged as a powerful means for guaranteeing control system safety in the form of set invariance [1]. CBFs are often used with Control Lyapunov Functions (CLFs) to simultaneously ensure stability and safety of the system along with state and input constraints [2]. This approach has been successfully implemented in numerous applications such as mobile robots [3], robotic manipulators [4], robotic swarms [5], aerial vehicles, racing drones [6], and spacecraft docking [7]. CBFs have also been extensively used in the realm of the autonomous vehicles to generate a safe control input in problems such as cruise control, on-ramp merging [1, 8], signal free intersections [9], and the lane change [10], to name a few.

One of the challenges in designing a control input using CBFs is that controllers rely on models or measurements that are assumed to be perfect and free of uncertainty. However, models or measurements are usually uncertain or imperfect and this can result in an unsafe behaviour if not accounted for properly. This problem has been mainly addressed through the development of robust CBFs. However, there exist different perspectives in considering robustness in safety-critical control using CBFs. Generally, the proposed approaches in the literature consider either input disturbances [11, 12, 13, 14, 15, 16, 17, 18, 19], dynamic model mismatch [20, 21], or measurement errors [22, 23, 24, 25, 26]. While the aforementioned works study robustness of CBF-based controllers, they do not take into account that the environment is dynamically changing and only consider a quasi-static environment. A good example of “environment” is the “other agents” in a multi-agent system with which the ego-agent is interacting, e.g., in the cruise control problem [1] the lead vehicle can be considered as the environment.

There exist limited works that consider the effect of a dynamic environment in the design of a safe control input via CBFs [27, 28, 29, 30, 31]. In [27, 28], the effect of a dynamic environment is considered through the notion of time-varying CBFs in which the time-derivative of the CBF is also added to the CBF constraint. Even though the authors in [27, 28, 29] have taken a dynamic environment into consideration, they do not study robustness and assume that perfect information on the environment is available. In [30], safe navigation in unknown environments is studied, where on-board range sensing is utilized to construct CBFs online. In the recent work [31], the notion of Environmental Control Barrier Functions (ECBFs) has been introduced and robust ECBFs against errors in the environment, in particular a time-delay, are investigated. In this paper, inspired by [31], we study the notion of Environmentally Robust Control Barrier Function (ER-CBFs). Similar to [31], worst-case error-based ER-CBFs are considered, however, unlike [31] ($i$) the Lipschitz constant is not used to define the worst-case error in an ER-CBF constraint and ($ii$) it is not assumed that the dynamics of the environment are known. Moreover, the original quadratic program (QP) for the control synthesis is converted to a second-order cone program (SOCP). Finally, we propose our main robust control scheme, by exploiting the closed-form solution of the original QP.

In summary, the contributions of this paper are as follows. An ER-CBF is introduced based on errors in the nominal CBF resulting from errors in a dynamically changing environment. This ER-CBF contains the norm control input (which is the optimization decision variable) and leads to the formulation of a control synthesis problem as a SOCP. Secondly, the nominally safe input obtained from a nominal CBF is considered as the desired input for the SOCP and a new ER-CBF is introduced which does not depend on the norm of the control input, hence the control synthesis problem leads to solving a QP. Then, the explicit solution to this QP is obtained. Finally, the effectiveness of the presented results is demonstrated in an adaptive cruise control example.

The rest of the paper is organized as follows. Preliminaries on CBFs and CLFs are reviewed in Section \Romannum2. The problem formulation is discussed in Section \Romannum3. Main results are presented in Section \Romannum4. An adaptive cruise control example is visited in Section \Romannum5. Numerical simulations on the cruise control example are presented in Section \Romannum6. Finally, concluding remarks are provided in the last section.

Consider a nonlinear control affine system

where $x\in\mathcal{X}\subset\mathbb{R}^{n}$, $u\in\mathbb{R}^{m}$ are state and control input, respectively, and $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ and $g:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n\times m}$ are locally Lipschitz continuous functions. To establish exponential stability of system (1) without having to define an explicit feedback controller, the notion of a Control Lyapunov Function (CLF) is introduced.

Given a stabilizing CLF $V(x)$ as in Definition 3, any Lipschitz continuous controller $u(t)\in\mathbb{R}^{m}$ that satisfies (3), $\forall t\geq 0$ exponentially stabilizes system (1) to the origin.

For the safe operation of system (1), a safe set $\mathcal{C}\subset\mathcal{X}\subset\mathbb{R}^{n}$ is defined as a superlevel set of a differentiable function $h:\mathbb{R}^{n}\times[0,+\infty)\rightarrow\mathbb{R}$, i.e.

To ensure that system (1) remains in the safety set $\mathcal{C}$, the notion of Control Barrier Function (CBF) can be defined as follows, in a similar manner as the concept of CLF:

Notice that CBFs used in this paper are time-varying functions. When $h(x,t)$ is time-invariant, it is denoted as $h(x)$ and the condition (5) is simplified [32] as follows:

In this study, a particular choice of extended class $\mathcal{K}_{\infty}$ function having the form of $\alpha(h(x,t))=\nu h(x,t)$ is considered where $\nu\geq 0$ is a CBF design parameter which controls the system behaviours near the boundary of $h(x,t)=0$. Safety of a set for a specific dynamical system is often stated in terms of the forward invariance property of that set with respect to the dynamical system:

The advantage of CLF and CBF formulations is that they allow the unification of control objectives (represented by CLFs) that are regulated to yield trajectories within desired sets (as enforced by CBFs). Given a CLF $V(x)$ and a CBF $h(x,t)$, they can be combined into a single controller by using a sequence of Quadratic Programs (QPs).

Most of the works to date on designing robust CBFs that account for uncertainty consider a quasi-static environment for which it is assumed that a perfect knowledge of the environment is available. Unlike previous studies, in this paper, similar to [31], we consider a robust safety-critical control problem with regard to existing uncertainty in the state of the dynamic environment. If $x_{s}\in\mathbb{R}^{p}$ represents the state of the dynamic environment, we redefine the safety set $\mathcal{C}$ and the time-varying CBF $h(x,x_{s})$ as

and $h:\mathbb{R}^{n}\times\mathbb{R}^{p}\rightarrow\mathbb{R}$. Consider $x_{s}=\hat{x}_{s}+e_{s}$ and $\dot{x}_{s}=\dot{\hat{x}}_{s}+\dot{e}_{s}$ where $\hat{x}_{s}$ denotes the measured state of the environment and $e_{s}$ corresponds to the uncertainty in the measured state. We assume that $e_{s}$ and $\dot{e}_{s}$ are bounded, i.e., $||e_{s}||<\mathcal{E}_{s}$ and $||\dot{e}_{s}||<\dot{\mathcal{E}}_{s}$. Our goal is to design a feedback controller $u(x,\hat{x}_{s})$ for a given system (1), such that the trajectories of the closed-loop system remain inside the safety set defined in (7). Note that since the dependence of $h$ on time comes only through $x_{s}(t)$, we wil henceforth write $h(x,x_{s})$ instead of $h(x,t)$ and drop $t$ from $x_{s}(t)$ in order to ease notation. Next, an Environmentally Robust CBF (ER-CBF) is defined.

Problem 1 seeks to find a robust safe control input based on a given desired control and error in the state of the environment, while Problem 2 only tries to robustify a given nominal safe control with respect to error in the surrounding environment. In the next section, we present our main results regarding the problems described in this section.

First, the CBF error is quantified due to estimating the state of the surrounding dynamical system, its gradient, and its time-derivative. Let

Since $e_{s}$ and $\dot{e}_{s}$ are bounded, it follows that $e_{h}$, $e_{\nabla h}$, and $e_{\frac{\partial h}{\partial t}}$ are bounded as well. In this study, it is intended to design a control input for (1) based on the worst-case bounds of $e_{h}$, $e_{\nabla h}$, and $e_{\frac{\partial h}{\partial t}}$ and the worst-case uncertainties for the above errors are considered as follows:

where it should be noted that $e_{h}^{*}$ and $e_{\frac{\partial h}{\partial t}}^{*}$ is obtained by minimizing, while $e_{\nabla h}^{*}$ is obtained by maximization. The reason for this is that (as shown in the next result) $e_{h}^{*}$ and $e_{\frac{\partial h}{\partial t}}^{*}$ appear as additive error terms in the residual term of the ER-CBF constraint, while $e_{\nabla h}^{*}$ appears as a multiplicative error term. From now on, in order to ease notation, the arguments $(x,x_{s},\hat{x}_{s})$ are omitted from $e_{h}^{*}$, $e_{\nabla h}^{*}$ and $e_{\frac{\partial h}{\partial t}}^{*}$. The next theorem provides a solution for Problem 1 and presents a sufficient condition in which an ER-CBF preserves system (1) forward invariance in the presence of bounded uncertainties in the state of the surrounding dynamical environment.

To obtain the safe input $u$ with respect to a desired control input $u_{\text{des}}$, the following optimization problem is required to be solved at each time step to determine a value $u$ kept constant over this time step:

Since the optimization variable $u$ appears in $\|f(x)+g(x)u\|$ above, this problem is no longer a QP. By writing $\frac{1}{2}\|u-u_{\text{des}}\|^{2}=\frac{1}{2}(u-u_{\text{des}})^{T}(u-u_{\text{des}})=\frac{1}{2}\|u\|^{2}+u_{\text{des}}^{T}u+\frac{1}{2}\|u_{\text{des}}\|^{2}$ and removing the term $\frac{1}{2}\|u_{\text{des}}\|^{2}$ from the objective function as it is a constant, a slack variable $q$ can be used to restate the optimization problem as follows:

with the decision variables $u$ and $q\geq 0$. This problem is a second-order cone program (SOCP). The second constraint above can be written as a rotated second-order cone condition [33], which leads to the following SOCP:

The difficulty with SOCPs is that, unlike QPs, SOCPs tend to be infeasible very easily [34] and, therefore, sometimes another slack variable is added to the constraint (13) to ensure the feasibility of the SOCP [22], which can deteriorate the safe input. Another problem with SOCPs, as reported in [20], is that the computation time to solve them is higher than that of QPs, which can affect the real-time control synthesis as well. In [16], it is assumed that the control input $u$ is not overly restrictive, thus permitting sufficient control authority to preserve safety in the presence of uncertainty.

To circumvent solving a SOCP for ER-CBFs at each time step, we propose a novel ER-CBF which is based on computing the control input using the nominal CBF and modifying it whenever the ER-CBF constraint in (13) is violated. In this way, the control input produced by the nominal CBF is considered as the desired input and it is modified minimally so as to robustly guarantee safety. First, we denote the control input $u$ in the optimization problem with nominal CBFs and ER-CBFs as $u_{\text{nom}}$ and $u_{\text{rob}}$, respectively. To begin with, we present the closed-form solution for the nominal safe input framed as the following QP (solved at each time step):

For this QP, the closed-form solution can be obtained using the following Lemma.

Our method to synthesize the control input using ER-CBFs is inspired by Lemma 1, where we intend to use $u_{\text{nom}}^{\ast}$ as the desired input and find an extra control effort such that the resulting input ensures robustness in the presence of environmental errors. First, let

Then, in order to design an environmentally safe controller with ER-CBFs, one needs to consider the following SOCP:

Since $u^{*}_{\text{nom}}$ which is the solution of (CBF-QP), will be taken as the desired input for (ER-CBF-SOCP), from now on, the desired input for (ER-CBF-SOCP) is denoted by $u^{(s)}_{\text{des}}$ to distinguish it with $u_{\text{des}}$ in (CBF-QP). Let us define $u_{\delta}\triangleq u_{\text{rob}}^{*}-u_{\text{nom}}^{*}$ as the amount of change in the control input $u_{\text{nom}}^{*}$ such that robustness to the environmental uncertainties is guaranteed. Next, based on Lemma 1, an upper bound on $u_{\delta}$ is obtained. Note that $u_{s}$ in Lemma 1 for the case where $u\in\mathbb{R}$ reduces to

Theorem 3 finds an upper bound on the size of the necessary modification in the nominal safe input to meet the robustness condition. Using the results of Theorem 3, a new constraint is established for the ER-CBF which is no longer dependent on the norm of the control input, hence the control design problem can be formulated as a QP rather than a SOCP.

Notice that the residual term in Theorem 4, unlike the one in Theorem 2, does not depend on the control input, which is the decision variable in the optimization-based control design. In fact, in the residual term of Theorem 4, we have $u_{\text{nom}}^{*}$ and $\bar{u}_{\delta}$ which are not decision variables and hence they are available before solving the optimization problem. Hence, one can utilize a QP to compute the control input instead of a SOCP. Thus, to design an environmentally safe controller with ER-CBFs, the following QP is considered:

In the sequel, the closed-form solution of (ER-CBF-QP) tackling Problem 2 is presented.