Taylor participated in the design of the project, theoretical analysis and writing of the article. Taylor participated in the design of the project, algorithm design and theoretical analysis, simulation code implementation, and writing of the article. Taylor participated in the conception of the project, algorithm design and theoretical analysis, and writing of the article.
Taylor participated in the design of the project, the design of the algorithm, and the writing of the article. Taylor participated in project design, algorithm design and theoretical analysis, implementation of simulation and experimental code, performance of experiments, and writing of the article. Taylor participated in the design of the project, the design of the algorithms, the implementation of the simulation and experimental code, the execution of the experiments, and the writing of the article.
Taylor participated in the conception of the project, theoretical analysis, simulation code implementation and writing of the article. Taylor participated in the conception of the project, algorithm design, simulation code implementation, and writing of the article.
LIST OF ILLUSTRATIONS
A phase portrait showing the evolution of the system state leaves the safe set (black ellipse) with the nominal model-based controller (CBF-QP) while remaining within the application of the learning-based controller (CBF-QP). ) controller. A phase portrait showing the evolution of the system state leaves the safe set (black bar) with the nominal model-based controller (CBF-QP) while remaining within the application of the learning-based controller (CBF-QP). . The barrier value meets the corresponding worst-case lower bound with and without using learning to calculate δ.
The barrier value satisfies the corresponding worst-case lower bound with and without learning used to calculate δ. The red dotted line is the trajectory of the system evolving under kCBF and remaining within the set C. Track distance D and value of CBF h using the nominal controller (4.59) and (CBF-QP) controller. on board the truck during a test drive.
Both true and measured trajectories are reliable demonstrating the robustness of the (MR-BS-SOCP) controller when compared to the (BS-QP) controller. The state trajectories for both drivers remain within the safe set for the length of the simulation.
LIST OF TABLES
NOMENCLATURE
INTRODUCTION
My second contribution is a characterization of the impact of residual learning errors on stability and safety, built through the objective of Input-to-State Stability (ISS) and Input-to-State Safety (ISSf) (Sections 3.4 and 3.6 ). My contribution is an extension of the work in [81] that uses discrete-time approximate models with respect to CLFs to synthesize optimization-based convex stabilizing controllers that significantly outperform continuous-time CLF-based controllers. None of this work has considered the use of discrete-time approximate models along the lines of work in [81], which often performs extremely well even at low sample rates [94].
In section 4.2 I present some of the limitations of ISSf and ISSf-CBFs stated in [58] regarding robustness and performance. In section 3.7 I will explore some of the challenges associated with the richness of data that arise when using the control-affine learning models in sections 3.3-3.6. In section 4.2 I will explore some limitations of the partial robustness concept of Input-to-State Safety (ISSf).
In Sections 4.3 and 4.4, the bounds of ISSf will be approximated by viewing the robustness parameter ϵ as a function of the value of C(x). Consequently, it will be assumed that given a measurement y, an estimate of the state x can be produced (although the state dynamics still uses x+e(x)).
BACKGROUND
Let xe ∈ E be the equilibrium point of the closed-loop system (2.2) and assume that the functions f,g and are continuously differentiable on E. Let xe ∈ E be the equilibrium point of the closed-loop system (2.2) and assume that the function k:E →Rm continuous on E, but only locally Lipschitz continuous on E \ {xe}. Let xe ∈ E be the unforced or forced equilibrium point of the open-loop system (2.1).
Let xe ∈ E be an unforced or forced equilibrium point of the open loop system (2.1), and let V : E → R≥0 be a local exponential CLF of the open loop system (2.1) and the equilibrium point xe with corresponding open set D ⊆ E. Let xe ∈ E be an unforced equilibrium point of the open loop system (2.1), and let V : E → R≥0 be a local exponential CLF of the open loop system (2.1) and equilibrium point xe. Let xe∈E be a forced equilibrium point of the open-loop system (2.1) for equilibrium input ue, and let V : E → R≥0 be a local exponential CLF of the open-loop system (2.1) and the equilibrium point xe .
Then the controller kCLF : D → Rm is continuous on D, locally Lipschitz continuous on D\ {xe}, gives xe an equilibrium point of the closed-loop system (2.2), and satisfies kCLF(x)∈KCLF(x) for all x ∈D. Letxe ∈E be an unforced or forced equilibrium point of the open-loop system (2.1) with equilibrium input. More generally, (2.25) and the fact that ∂Φ∂x(xe) is full rank implies that xeis an unforced (forced) equilibrium point of the open-loop system (2.1) if and only.
Then the controller is kISS: D → Rm continuous on D, locally Lipschitz continuous on D\{xe}, represents the equilibrium point of the closed-loop system (2.2), and satisfies kISS(x)∈KISS(x) for all x∈ D. Adding and subtracting the nominal model open-loop system (3.1) from the real open-loop system (2.1) yields the following description of the true. However, an error in the estimator cW˙ deteriorates the local exponential stability of the real closed-loop system (2.2) with respect to the equilibrium point xe.
In this section, I will present an analysis of the effects of residual learning error on the local exponential stability of the true closed-loop system (2.2) using a learning-informed (CLF-QP) controller. Let xe ∈ E be an equilibrium point of the nominal model open-loop system (3.1) and the true open-loop system (2.1) with known equilibrium inputs ube and ue, respectively. This controller corresponds to the (CLF-QP) controller that uses the nominal model open-loop system (3.1) for the time derivative of the local exponential CLFV.
However, the error in the estimator Sb˙ degrades the safety of the true closed-loop system (2.2) with respect to the set C. In this section, I will present an analysis of the effects of the residual learning error on the safety of the true closed-loop system (2.2) using a self-learning controller (CBF -QP). The feasibility of the (DR-CCF-SOCP) controller at a given state is determined by the model error structure setU(x) defined in (3.166).
Value of the certificate function C(x) (top) and fut u (bottom) over time for stability (Top pair) and security (Bottom pair) experiments.
DISTURBANCE-ROBUST SAFETY-CRITICAL CONTROL
In Section 4.2 I will discuss some of the limitations of ISSf and ISSf-CBFs as presented in Sections 2.7 and 2.9. In particular, this work observed that if the CBF property was satisfied on the set E, including outside the set C, then asymptotic stability of the set C could be established under a CBF-based controller. Importantly, this work established a paradigm for safety-critical control design that focuses on checking the extension of the setC that is kept forward invariant, rather than trying to explicitly enforce the forward invariance of the set C itself.
This is achieved by relaxing the stability conditions on an ISSf-CBF deep inside C, and requiring that stability only apply near the boundary of C. In parallel with the development of ISSf, other notions of robust CBFs were formulated that ensure the forward invariance of group C in the presence of perturbations. Instead of considering disturbances simply as piecewise continuous signals, it is often interesting to attribute more structure through the form of a statement on the probability distribution of the disturbance value, leading to a stochastic disturbance signal.
This contrasts much of the work studying perturbations by ISSf and other robustness approaches mentioned above, which typically assume that the perturbation can (and will) always take the worst value when establishing safety guarantees. Because it is still possible for a stochastic disturbance signal to take these worst-case values (perhaps with low probability), it is important that the resulting safety guarantees are designed to not only take this situation into account, but use to make of the fact that a stochastic disturbance often does not take the worst values. Underlying the sample-and-hold input to the system is the continuous-time evolution of the system dynamics (with the input held constant).
A challenge with this approach is that a sudden failure of the closed-loop CBF constraint does not necessarily mean that the system is unsafe, but rather that the change in CBF value is greater than desired. Much of this work focuses on bounding the probability that the system state leaves the cluster. My work in [235], discussed in Section 4.5, also develops the work in [59] in conjunction with the work in [69] to develop online optimization-based CBF controllers for discrete-time systems that provide limited security guarantees in time.
In addition, this work uses an ISSf-like approach by providing probabilistic security guarantees based on increments of the set C instead of just C. This approach focuses on the continuous evolution of the entire distribution of CBF values that can be captured by solving the convection partial differential equation and diffusion. Alternatively, the tools developed in this work were revisited in [64] to propose a method for achieving long-term security guarantees using the prior invariance of sets of security probability levels.
