Constrained control of multi‐input systems with distinct input delays

Constrained Stabilization of Multi-Input Linear Systems with Distinct Input Delays

This paper addresses the design and analysis of a multivariable gradient-based stochastic extremum-seeking control method for multi-input systems with arbitrary input delays. The approach accommodates systems with distinct time delays across input channels. It achieves local exponential stability of the closed-loop system, guaranteeing convergence to a small neighborhood around the extremum point. By incorporating phase compensation for dither signals and a novel predictor- feedback mechanism with averaging-based estimates of the unknown gradient and Hessian, the proposed method overcomes traditional challenges associated with arbitrary, distinct input delays. Unlike previous work on deterministic multiparameter extremum-seeking with distinct input delays, this stability analysis is achieved without using backstepping transformations, simplifying the predictor design and enabling a more straightforward implementation. Specifically, the direct application of Artstein's reduction approach yields delay- and system-dimension-independent convergence rates, thus enhancing its practical applicability. A numerical example and a source-seeking application illustrate the robust performance and advantages of the proposed delay-compensated stochastic extremum-seeking method.

We develop a predictor-feedback control design for multi-input nonlinear systems with distinct input delays, of arbitrary length, in each individual input channel. Due to the fact that different input signals reach the plant at different time instants, the key design challenge, which we resolve, is the construction of the predictors of the plant's state over distinct prediction horizons such that the corresponding input delays are compensated. Global asymptotic stability of the closed-loop system is established by constructing a Lyapunov functional.

SummaryThis paper is concerned with the stabilization of linear systems with both state and distinct input delays. Nested predictor feedback controllers are designed to predict the future states such that the distinct input delays that can be arbitrarily large yet bounded are compensated completely. It is shown that the compensated closed‐loop system possesses the same characteristic equation as the closed‐loop system without distinct input delays. Both continuous‐time and discrete‐time time‐delay systems are studied in this paper. Moreover, the safe implementation problem for the continuous‐time nested predictor feedback controller is solved via adding input filters. Three numerical examples show the effectiveness of the proposed approaches.

Ensuring safety while retaining maximum performance is a basic requirement for automatic cyber-physical systems, especially for safety-critical applications. A quadratic programming optimization framework called MPC-CBF has recently been presented, which directly unifies model predictive control (MPC) with control barrier functions (CBFs) over the prediction time horizon. However, the conservative nature of CBFs can lead to feasibility problems in real applications. Based on the analysis of the role of the decay rate and the conservative accumulation phenomenon in standard CBF formulations, this paper proposes to directly optimize CBF constraints within the MPC framework. By regarding CBFs as a safety restriction level indicator and an optimizable constraint within the MPC framework, the trade-off between feasibility and safety can be adaptively optimized. The proposed Optimizable CBF (OCBF) model removes the hyper-parameters selection problem in standard CBFs and can adaptively adjust the safety restriction level and increase behavior diversity by adding the corresponding objects in the cost function in MPC. To eliminate the accumulation effects of actual values of the CBF constraints in previous time steps, this paper further proposes a General OCBF (GOCBF) formulation. Compared with existing formulations, the safety margin defined in our GOCBF has intuitive physical meanings and thus provides a more flexible and intuitive mechanism to compromise different objects in terms of ensuring safety while not undermining basic feasibility. Experimental results demonstrate that our algorithm provides a more flexible and intuitive mechanism to achieve this, thus improving feasibility and adding behavior diversity in the MPC-CBF framework.

Safety critical systems involve the tight coupling between potentially conflicting control objectives and safety constraints. As a means of creating a formal framework for controlling systems of this form, and with a view toward automotive applications, this paper develops a methodology that allows safety conditions -- expressed as control barrier functions -- to be unified with performance objectives -- expressed as control Lyapunov functions -- in the context of real-time optimization-based controllers. Safety conditions are specified in terms of forward invariance of a set, and are verified via two novel generalizations of barrier functions; in each case, the existence of a barrier function satisfying Lyapunov-like conditions implies forward invariance of the set, and the relationship between these two classes of barrier functions is characterized. In addition, each of these formulations yields a notion of control barrier function (CBF), providing inequality constraints in the control input that, when satisfied, again imply forward invariance of the set. Through these constructions, CBFs can naturally be unified with control Lyapunov functions (CLFs) in the context of a quadratic program (QP); this allows for the achievement of control objectives (represented by CLFs) subject to conditions on the admissible states of the system (represented by CBFs). The mediation of safety and performance through a QP is demonstrated on adaptive cruise control and lane keeping, two automotive control problems that present both safety and performance considerations coupled with actuator bounds.

This survey provides a brief overview on the control Lyapunov function (CLF) and control barrier function (CBF) for general nonlinear-affine control systems. The problem of control is formulated as an optimization problem where the optimal control policy is derived by solving a constrained quadratic programming (QP) problem. The CLF and CBF respectively characterize the stability objective and the safety objective for the nonlinear control systems. These objectives imply important properties including controllability, convergence, and robustness of control problems. Under this framework, optimal control corresponds to the minimal solution to a constrained QP problem. When uncertainties are explicitly considered, the setting of the CLF and CBF is proposed to study the input-to-state stability and input-to-state safety and to analyze the effect of disturbances. The recent theoretic progress and novel applications of CLF and CBF are systematically reviewed and discussed in this paper. Finally, we provide research directions that are significant for the advance of knowledge in this area.

Control of time-delay systems represents one important class of control problems. A lot of work has been done in recent years. Much of the research work has been focused on the stability analysis and stabilization of time-delay systems based on the Lyapunov functional and linear matrix inequality (LMI) approaches. While the LMI approach does provide an e±cient tool, the results are mostly only su±cient and only numerical solutions are available. Our research aims to present analytical solutions by some algebraic approaches, which can provide an insightful understanding of control problem of time-delay systems. On the other hand, there has also been significant recent interest in networked multi-agent system (MAS) where unavoidably there exist delays in information acquisition as well as information exchange between agents in addition to possible input/state delays in each agent. This motivates the study of MASs with input and communication delays. The thesis is divided into two parts: Part I is focused on optimal control of single agent systems with multiple input delays; Part II is on distributed consensus control of MAS with communication and input delays. More specifically, in Part I, we shall study the linear quadratic regulation (LQR) and tracking for a single agent with linear dynamics. We first focus on LQR for linear discrete-time systems with multiple delays in a single input channel. By introducing a backward stochastic system, we establish a relationship between the LQR problem for the original system and optimal estimation for a backward stochastic system which can be solved by using projection approach. The explicit optimal controllers are developed for both finite time and infinite time horizon cases. A parallel result for linear continuous-time systems is also established, where the optimal feedback gain is given in terms of the solution of Riccati-type partial differential equations (PDEs).

A high-performance flight control approach for quadrotors using a modified active disturbance rejection technique

Fast Verification of Control Barrier Functions via Linear Programming

Despite advances made toward the automation of medical treatments, existing work has predominantly focused on the automation of individual medical treatments while overlooking interferences between them. In our prior work, we developed an automation architecture to administer interfering medical treatments with safety, which (i) monitors internal physiological state of a patient using an extended Kalman filter, (ii) mediates medical treatments to bound the estimated internal state within a prescribed safe regime based on control barrier functions, and (iii) treats the patient to a prespecified treatment target through a number of intermediate targets derived from reachability analysis. The goal of this work was to investigate the scalability of this automation architecture in more complex and challenging medical treatment scenarios with a number of conflicts. Using a critical care resuscitation scenario including fluid resuscitation and intravenous sedative-vasopressor infusion, we examined if our automation architecture can achieve treatment goals while ascertaining the safety of internal state in a large number of diverse in silico patients. The results suggested that (i) the extended Kalman filter could continuously monitor a patient's internal physiological state, (ii) the control barrier functions could mediate interfering medical treatments and protect patients against unsafe internal physiological state, and (iii) the reachability analysis could treat a patient as closely as possible to a treatment target while ensuring the safety of the patient's internal physiological state, all despite complex and entangled conflicts between them. Our automation architecture may provide a viable means to autonomously de-conflict interfering medical treatments.

Control barrier functions (CBFs) have become a popular tool to enforce safety of a control system. CBFs are commonly utilized in a quadratic program formulation (CBF-QP) as safety-critical constraints. A class <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathcal{K}$</tex> function in CBFs usually needs to be tuned manually in order to balance the trade-off between performance and safety for each environment. However, this process is often heuristic and can become intractable for high relative-degree systems. Moreover, it prevents the CBF-QP from generalizing to different envi-ronments in the real world. By embedding the optimization procedure of the exponential control barrier function based quadratic program (ECBF-QP) as a differentiable layer within a deep learning architecture, we propose a differentiable safety-critical control framework that enables generalization to new environments for high relative-degree systems with forward invariance guarantees. Finally, we validate the proposed control design with 2D double and quadruple integrator systems in various environments.

Designing safety-critical controllers for acceleration-controlled unicycle robots is challenging, as control inputs may not appear in the constraints of control Lyapunov functions (CLFs) and control barrier functions (CBFs), leading to invalid controllers. Existing methods often rely on state-feedback-based CLFs and high-order CBFs (HOCBFs), which are computationally expensive to construct and fail to maintain effectiveness in dynamic environments with fast-moving, nearby obstacles. To address these challenges, we propose constructing velocity obstacle (VO)-based CBFs (VOCBFs) in the velocity space to enhance dynamic collision avoidance capabilities, instead of relying on distance-based CBFs that require the introduction of HOCBFs. Additionally, by extending VOCBFs using variants of VO, we enable reactive collision avoidance between robots. We formulate a safety-critical controller for acceleration-controlled unicycle robots as a mixed-integer quadratic programming (MIQP), integrating state-feedback-based CLFs for navigation and VOCBFs for collision avoidance. To enhance the efficiency of solving the MIQP, we split the MIQP into multiple suboptimization problems and employ a decision network to reduce computational costs. Numerical simulations demonstrate that our approach effectively guides the robot to its target while avoiding collisions. Compared to HOCBFs, VOCBFs exhibit significantly improved dynamic obstacle avoidance performance, especially when obstacles are fast moving and close to the robot. Furthermore, we extend our method to distributed multirobot systems.

Ensuring UAV safety: A deep learning-based fault-tolerant approach with control barrier functions

Safe exploration in model-based reinforcement learning using control barrier functions