Hadi Ravanbakhsh

Hadi Ravanbakhsh, Sriram Sankaranarayanan

We present a technique for learning control Lyapunov-like functions, which are used in turn to synthesize controllers for nonlinear dynamical systems that can stabilize the system, or satisfy specifications such as remaining inside a safe set, or eventually reaching a target set while remaining inside a safe set. The learning framework uses a demonstrator that implements a black-box, untrusted strategy presumed to solve the problem of interest, a learner that poses finitely many queries to the demonstrator to infer a candidate function, and a verifier that checks whether the current candidate is a valid control Lyapunov function. The overall learning framework is iterative, eliminating a set of candidates on each iteration using the counterexamples discovered by the verifier and the demonstrations over these counterexamples. We prove its convergence using ellipsoidal approximation techniques from convex optimization. We also implement this scheme using nonlinear MPC controllers to serve as demonstrators for a set of state and trajectory stabilization problems for nonlinear dynamical systems. We show how the verifier can be constructed efficiently using convex relaxations of the verification problem for polynomial systems to semi-definite programming (SDP) problem instances. Our approach is able to synthesize relatively simple polynomial control Lyapunov functions, and in that process replace the MPC using a guaranteed and computationally less expensive controller.

Hadi Ravanbakhsh, Sriram Sankaranarayanan

We present a technique for learning control Lyapunov (potential) functions, which are used in turn to synthesize controllers for nonlinear dynamical systems. The learning framework uses a demonstrator that implements a black-box, untrusted strategy presumed to solve the problem of interest, a learner that poses finitely many queries to the demonstrator to infer a candidate function and a verifier that checks whether the current candidate is a valid control Lyapunov function. The overall learning framework is iterative, eliminating a set of candidates on each iteration using the counterexamples discovered by the verifier and the demonstrations over these counterexamples. We prove its convergence using ellipsoidal approximation techniques from convex optimization. We also implement this scheme using nonlinear MPC controllers to serve as demonstrators for a set of state and trajectory stabilization problems for nonlinear dynamical systems. Our approach is able to synthesize relatively simple polynomial control Lyapunov functions, and in that process replace the MPC using a guaranteed and computationally less expensive controller.

Hadi Ravanbakhsh, Sriram Sankaranarayanan

In this article, we consider the problem of synthesizing switching controllers for temporal properties through the composition of simple primitive reach-while-stay (RWS) properties. Reach-while-stay properties specify that the system states starting from an initial set I, must reach a goal (target) set G in finite time, while remaining inside a safe set S. Our approach synthesizes switched controllers that select between finitely many modes to satisfy the given RWS specification. To do so, we consider control certificates, which are Lyapunov-like functions that represent control strategies to achieve the desired specification. However, for RWS problems, a control Lyapunov-like function is often hard to synthesize in a simple polynomial form. Therefore, we combine control barrier and Lyapunov functions with an additional compatibility condition between them. Using this approach, the controller synthesis problem reduces to one of solving quantified nonlinear constrained problems that are handled using a combination of SMT solvers. The synthesis of controllers is demonstrated through a set of interesting numerical examples drawn from the related work, and compared with the state-of-the-art tool SCOTS. Our evaluation suggests that our approach is computationally feasible, and adds to the growing body of formal approaches to controller synthesis.

Hadi Ravanbakhsh, Sriram Sankaranarayanan, Sanjit A. Seshia

We consider the problem of learning structured, closed-loop policies (feedback laws) from demonstrations in order to control under-actuated robotic systems, so that formal behavioral specifications such as reaching a target set of states are satisfied. Our approach uses a ``counterexample-guided'' iterative loop that involves the interaction between a policy learner, a demonstrator and a verifier. The learner is responsible for querying the demonstrator in order to obtain the training data to guide the construction of a policy candidate. This candidate is analyzed by the verifier and either accepted as correct, or rejected with a counterexample. In the latter case, the counterexample is used to update the training data and further refine the policy. The approach is instantiated using receding horizon model-predictive controllers (MPCs) as demonstrators. Rather than using regression to fit a policy to the demonstrator actions, we extend the MPC formulation with the gradient of the cost-to-go function evaluated at sample states in order to constrain the set of policies compatible with the behavior of the demonstrator. We demonstrate the successful application of the resulting policy learning schemes on two case studies and we show how simple, formally-verified policies can be inferred starting from a complex and unverified nonlinear MPC implementations. As a further benefit, the policies are many orders of magnitude faster to implement when compared to the original MPCs.

Hadi Ravanbakhsh, Sriram Sankaranarayanan

In this article, the problem of synthesizing switching controllers is considered through the synthesis of a "control certificate". Control certificates include control barrier and Lyapunov functions, which represent control strategies, and allow for automatic controller synthesis. Our approach encodes the controller synthesis problem as quantified nonlinear constraints. We extend an approach called Counterexample Guided Inductive Synthesis (CEGIS), originally proposed for program synthesis problems, to solve the resulting constraints. The CEGIS procedure involves the use of satisfiability-modulo theory (SMT) solvers to automate the problem of synthesizing control certificates. In this paper, we examine generalizations of CEGIS to attempt a richer class of specifications, including reach-while-stay with obstacles and control under disturbances. We demonstrate the ability of our approach to handle systems with nonpolynomial dynamics as well. The abilities of our general framework are demonstrated through a set of interesting examples. Our evaluation suggests that our approach is computationally feasible, and adds to the growing body of formal approaches to controller synthesis.

Hadi Ravanbakhsh

The focus of this thesis is developing a framework for designing correct-by-construction controllers using control certificates. We use nonlinear dynamical systems to model the physical environment (plants). The goal is to synthesize controllers for these plants while guaranteeing formal correctness w.r.t. given specifications. We consider different fundamental specifications including stability, safety, and reach-while-stay. Stability specification states that the execution traces of the system remain close to an equilibrium state and approach it asymptotically. Safety specification requires the execution traces to stay in a safe region. Finally, for reach-while-stay specification, safety is needed until a target set is reached. The design task consists of two phases. In the first phase, the control design problem is reduced to the question of finding a control certificate. More precisely, the goal of the first phase is to define a class of control certificates with a specific structure. This definition should guarantee the following: "Having a control certificate, one can systematically design a controller and prove its correctness at the same time." The goal in the second phase is to find such a control certificate. We define a potential control certificate space (hypothesis space) using parameterized functions. Next, we provide an inductive search framework to find proper parameters, which yield a control certificate. Finally, we evaluate our framework. We show that discovering control certificates is practically feasible and demonstrate the effectiveness of the automatically designed controllers through simulations and real physical systems experiments.

Hadi Ravanbakhsh, Forrest Laine, Sanjit A. Seshia

In autonomous systems, a motion planner generates reference trajectories which are tracked by a low-level controller. For safe operation, the motion planner should account for inevitable controller tracking error when generating avoidance trajectories. In this article we present a method for generating provably safe tracking error bounds, while reducing over-conservatism that exists in existing methods. We achieve this goal by restricting possible behaviors for the motion planner. We provide an algebraic method based on sum-of-squares programming to define restrictions on the motion planner and find small bounds on the tracking error. We demonstrate our method on two case studies and show how we can integrate the method into already developed motion planning techniques. Results suggest that our method can provide acceptable tracking error wherein previous work were not applicable.

Shromona Ghosh, Yash Vardhan Pant, Hadi Ravanbakhsh et al.

Recent advances in learning-based perception systems have led to drastic improvements in the performance of robotic systems like autonomous vehicles and surgical robots. These perception systems, however, are hard to analyze and errors in them can propagate to cause catastrophic failures. In this paper, we consider the problem of synthesizing safe and robust controllers for robotic systems which rely on complex perception modules for feedback. We propose a counterexample-guided synthesis framework that iteratively builds simple surrogate models of the complex perception module and enables us to find safe control policies. The framework uses a falsifier to find counterexamples, or traces of the systems that violate a safety property, to extract information that enables efficient modeling of the perception modules and errors in it. These models are then used to synthesize controllers that are robust to errors in perception. If the resulting policy is not safe, we gather new counterexamples. By repeating the process, we eventually find a controller which can keep the system safe even when there is a perception failure. We demonstrate our framework on two scenarios in simulation, namely lane keeping and automatic braking, and show that it generates controllers that are safe, as well as a simpler model of a deep neural network-based perception system that can provide meaningful insight into operations of the perception system.

Tommaso Dreossi, Daniel J. Fremont, Shromona Ghosh et al.

We present VERIFAI, a software toolkit for the formal design and analysis of systems that include artificial intelligence (AI) and machine learning (ML) components. VERIFAI particularly seeks to address challenges with applying formal methods to perception and ML components, including those based on neural networks, and to model and analyze system behavior in the presence of environment uncertainty. We describe the initial version of VERIFAI which centers on simulation guided by formal models and specifications. Several use cases are illustrated with examples, including temporal-logic falsification, model-based systematic fuzz testing, parameter synthesis, counterexample analysis, and data set augmentation.

Hadi Ravanbakhsh, Sina Aghli, Christoffer Heckman et al.

We present an approach to path following using so-called control funnel functions. Synthesizing controllers to "robustly" follow a reference trajectory is a fundamental problem for autonomous vehicles. Robustness, in this context, requires our controllers to handle a specified amount of deviation from the desired trajectory. Our approach considers a timing law that describes how fast to move along a given reference trajectory and a control feedback law for reducing deviations from the reference. We synthesize both feedback laws using "control funnel functions" that jointly encode the control law as well as its correctness argument over a mathematical model of the vehicle dynamics. We adapt a previously described demonstration-based learning algorithm to synthesize a control funnel function as well as the associated feedback law. We implement this law on top of a 1/8th scale autonomous vehicle called the Parkour car. We compare the performance of our path following approach against a trajectory tracking approach by specifying trajectories of varying lengths and curvatures. Our experiments demonstrate the improved robustness obtained from the use of control funnel functions.

Hadi Ravanbakhsh, Sriram Sankaranarayanan

We introduce a counter-example guided inductive synthesis (CEGIS) framework for synthesizing continuous-time switching controllers that guarantee reach while stay (RWS) properties of the closed loop system. The solution is based on synthesizing specially defined class of control Lyapunov functions (CLFs) for switched systems, that yield switching controllers with a guaranteed minimum dwell time in each mode. Next, we use a CEGIS-based approach to iteratively solve the resulting quantified exists-forall constraints, and find a CLF. We introduce relaxations to guarantee termination, as well as heuristics to increase convergence speed. Finally, we evaluate our approach on a set of benchmarks ranging from two to six state variables. Our evaluation includes a preliminary comparison with related tools. The proposed approach shows the promise of nonlinear SMT solvers for the synthesis of provably correct switching control laws.

Hadi Ravanbakhsh, Sriram Sankaranarayanan

We investigate the problem of synthesizing switching controllers for stabilizing continuous-time plants. First, we introduce a class of control Lyapunov functions (CLFs) for switched systems along with a switching strategy that yields a closed loop system with a guaranteed minimum dwell time in each switching mode. However, the challenge lies in automatically synthesizing appropriate CLFs. Assuming a given fixed form for the CLF with unknown coefficients, we derive quantified nonlinear constraints whose feasible solutions (if any) correspond to CLFs for the original system. However, solving quantified nonlinear constraints pose a challenge to most LMI/BMI-based relaxations. Therefore, we investigate a general approach called Counter-Example Guided Inductive Synthesis (CEGIS), that has been widely used in the emerging area of automatic program synthesis. We show how a LMI-based relaxation can be formulated within the CEGIS framework for synthesizing CLFs. We also evaluate our approach on a number of interesting benchmarks, and compare the performance of the new approach with our previous work that uses off-the-shelf nonlinear constraint solvers instead of the LMI relaxation. The results shows synthesizing CLFs by using LMI solvers inside a CEGIS framework can be a computational feasible approach to synthesizing CLFs.