Backstepping control lyapunov function. , 2018b, Vaidyanathan et al.

Backstepping control lyapunov function The distinctive feature of this approach lies in its well-defined step-by-step design procedure, and the system stability can be rigorously guaranteed by a Jun 5, 2023 · The backstepping constraint control method, based on barrier Lyapunov functions (BLFs), is designed not only to track the desired trajectory but also to guarantee no violation of the position and Jul 5, 2024 · An asymmetric time-varying integral barrier Lyapunov function (ATIBLF) based integral reinforcement learning (IRL) control algorithm with an actor–critic structure is first proposed. The main idea of Backstepping technique is to make the complex system similar to cascaded first order subsystems, stable according to Lyapunov theory. In the presence of bounded disturbance torques, the attitude errors remain bounded within the constraint boundary. , 2015; Shukla et al. V (x). It is noticed that the switching parameters are different for each subsystem, leading to a nontrivial task to construct a common Lyapunov function for all subsystems, as pointed out in Ma and Zhao (2010), Wu (2009) and Zhao, Zheng et al. For the trajectory tracking of the AUV, the non-linear controller based on Lyapunov can be designed by backstepping control method. Backstepping allows more design flexibility in choosing the intermediate control functions. May 28, 2021 · In this paper, we present a backstepping adaptive hybrid force/position control based on Barrier Lyapunov Function for a robotic manipulator to prevent constraint violation of applied force and position simultaneously. e May 1, 2024 · To improve the robustness of the system, the authors in Wang et al. Xiao, C. But plenty of challenges await us still. Consider a nonlinear control-afne system: x_ = f(x)+ g (x)u; (1) with state x 2 R n, input u 2 R m, and functions f : R n! R n and g : R n! R nm assumed to be locally Lipschitz The outcomes of the work show that the closed-loop control system is exponentially stable. The backstepping control method is a recursive design procedure that links the choice of a control Lyapunov function with the design of a feedback controller and guarantees global asymptotic stability of strict feedback systems. Let us consider the following system: x˙ = f(x,u), x∈Rn, u∈R, f(0,0) = 0, Task: Find a feedback control law u= α(x) such that the equilibrium x= 0 of the closed-loop system x˙ = f(x,α(x)) Jan 1, 2019 · In this study, backstepping control integrated with Lyapunov-based model predictive control (BS-MPC) is proposed for nonlinear systems in a strict-feedback form. The longitudinal dynamic for the AHV-VGI is transformed into strict feedback form. Moreover, the performance metrics of various Lyapunov function candidates, namely quadratic Lyapunov function (QLF), symmetric BLF and asymmetric BLF, are compared for the Jan 1, 2020 · Based on barrier Lyapunov functions, a reinforcement learning control method is proposed for air-breathing hypersonic vehicles with variable geometry inlet (AHV-VGI) subject to external disturbances and diversified uncertainties. The virtual input of the first step is designed by solving the finite-horizon optimal control problem (FHOCP), and the real input is designed by the backstepping method. A smooth time varying feedback u = k(x, t) is usually con- sidered. Alternatively, VSA can use non-model-based control techniques such as PD control, but it does not show strong Mar 1, 2023 · Simulation results for an electromechanical system have demonstrated the effectiveness of the proposed adaptive control method compared with the TFABC and BC schemes. 2020. Oct 11, 2024 · The backstepping technique is one of the most powerful tools in nonlinear control, in which the control laws are designed recursively by constructing a series of control Lyapunov functions. The nonstrict-feedback nonlinear systems considered in this paper are divided into two strict-feedback nonlinear subsystems. In view of the normal backstepping method Apr 1, 2009 · The control design is based on backstepping, with a Barrier Lyapunov Function candidate employed in the first step, and Quadratic Lyapunov Function candidates in the remaining steps. Therefore, this method has been widely used, for systems such as, aircraft, mobile robots, and manipulators, because their complicated, but cascaded dynamic structures Oct 1, 2020 · To address this trouble, sliding-mode control (SMC) and terminal SMC are utilized to control chaos and unknown disturbances [17], [18]. In this section we revisit Barrier Functions, Control Bar-rier Functions and Lyapunov backstepping as a precursor to introducing Control Barrier Function backstepping. Aimed at stabilizing the attitude of a quadrotor, in this paper, we establish the dynamics of the quadrotor. , 2020; Huang et al Aug 17, 2022 · This paper investigates the improvement for the backstepping tracking control for nonlinear systems with output constraints and unknown functions. In our future research work, we will focus on designing of adaptive command filtered backstepping control using barrier Lyapunov functions with full state constraints. The ATIBLF items are appropriately arranged in every step of the optimized backstepping control design to ensure that the dynamic full-state constraints are . If θ is known, we would apply the static integrator backstepping to design a virtual control law α 1 = −c 1z 1 −φTθ (2. 2. Jan 10, 2022 · Zhang J, Wang S, Zhou P, et al. Especially, this paper considers the constraint of rotor displacement to avoid the damage and safety accidents of AMB systems due to excessive rotor displacement deviation. For example, the intermediate control function can be chosen to avoid the cancellation of “good” nonlinearity – the term 3 Nov 1, 2023 · As an extensively used nonlinear control method, the traditional backstepping control (T-BC) is advantageous in control using multi-Lyapunov functions (Sun et al. , then there is a quadratic Lyapunov function that proves it (we’ll prove this later) Basic Lyapunov theory 12–20 Jan 1, 2020 · (11) 2. Keywords: Attitude control, Backstepping, Barrier Lyapunov Function, Constraints, Lyapunov stability Oct 30, 2020 · Considering disturbance uncertainties and turning safety of air cushion vehicles (ACV), a novel trajectory tracking control scheme is proposed based on an exponential convergence disturbance observer, command filter and barrier Lyapunov function (BLF). Nguyen, K. Connect an integrator to input u 2 {\displaystyle u_{2}} so that the augmented system has input u 3 {\displaystyle u_{3}} and output states x . More specifically, the study aims to achieve the following specific research objectives: (a) To modify the backstepping design procedure by adding more control parameters that enhance the solvability of the optimization problem in (d). , 2020 , Zhang and Yang, 2020 ) are focused on the state-feedback case. Jul 15, 2021 · To construct the contraction constraints in (8), (11), it is necessary to find the state feedback controller and the corresponding Lyapunov function. A barrier Lyapunov function (BLF) in backstepping design procedure is employed and corresponding backstepping controller is presented to prevent output constraints violation. International Journal of Electrical Power & Energy Systems 121: 105991. , 1991), let the time derivative of Lyapunov function be VË™ = −ETKblfE, (8) where, [Kblf ] ∈ n×n is a positive definite diagonal matrix expressed as 5th International Conference on Advances in Control and Optimization of Dynamical Systems February 18-22 Aug 13, 2018 · At last, by using the common Lyapunov function approach in combination with the decoupled backstepping method, a common approximation-free adaptive controller is constructed to guarantee that all the signals in the closed-loop system are semiglobally uniformly bounded. MRAC for Linear Time Invariant Systems 3. 0165 www. 26) whose derivative is rendered negative definite Dec 20, 2021 · The backstepping control proposes a systematic scheme to design a control input law to trace a wanted reference signal by picking a decent Lyapunov function nominee. 5 Tracking Control Lyapunov Function In this paper, we consider a trajectory tracking problem. , 2018; Vaidyanathan and Azar Sep 1, 2017 · Backstepping is one of the most popular nonlinear controller and differentiable control Lyapunov function (CLF) design techniques. Backstepping is used for output stabilization or tracking using feedback similarly to feedback linearization. Jun 8, 2023 · The control Lyapunov function candidate employed in the paper is the asymmetric Barrier Lyapunov Function (ABLF), which aids in constraining the output from designed limit. Its basic 1 is called virtual control and will be determined in later discussion. , 2023 , Wang et al. IET Control Theory & Applications Research Article Full state constraints control of switched complex networks based on time-varying barrier Lyapunov functions ISSN 1751-8644 Received on 12th February 2020 Revised 30th May 2020 Accepted on 28th July 2020 E-First on 17th September 2020 doi: 10. Control Lyapunov Function 3. Next, force/position control Oct 1, 2023 · Compared with traditional backstepping control schemes, the value of virtual stabilizing functions is needless in our scheme. Control Lyapunov Function (CLF) We are interested in an extension of the Lyapunov function concept, called a control Lyapunov function (CLF). ) And how do we choose the best control law u= (x)? 2. Aug 7, 2024 · This research proposes a novel BLF-based backstepping controller for path tracking of Autonomous Vehicles (AVs) with unknown dynamics and unmeasurable states. Belta, “Control Barrier Functions for Systems with High Relative Degree”, 2021. However, two crucial issues are often encountered when we Nov 17, 1997 · Control of stochastic nonlinear systems where the noise vector field does not vanish at the origin is the topic of our future study. 2. The tracking control Lyapunov function (TCLF), defined as follows, has been developed to design a smooth time-varying controller for trajectory tracking . How do we nd a Lyapunov function from which we can derive a control law? (Such a function is called a control Laypunov function (CLF). 1) Nov 1, 2020 · By incorporating the lower bounds of virtual control coefficients into the barrier Lyapunov functions (BLFs), two adaptive backstepping control design schemes are proposed based on tuning function method, bound estimation approach and some smooth functions, which makes the controller powerful enough to compensate the unknown virtual control Feb 4, 2020 · This paper focuses on an adaptive cross-backstepping control for a class of nonstrict-feedback nonlinear systems affected by time-varying partial state constraints. The proposed framework includes: (1) forming geometric-dynamic model of the vehicle by combining the dynamics of the vehicle with the kinematics of the visual measurement system, (2) designing a fixed-time Extended-State Observer (ESO Jan 1, 2025 · Such constraints are often unavoidable in practice. An ETC mathematic model with Jun 1, 2025 · It addresses each state variable in the subsystem by recursively constructing Lyapunov function without requiring full state information upfront. Time-varying barrier Lyapunov function is introduced and utilized to make the rotor displacement in a • then there exists a Lyapunov function that proves it a sharper converse Lyapunov theorem is more specific about the form of the Lyapunov function example: if the linear system x˙ = Ax is G. In particular, we'll be investigating the backstepping controller. In this summary we'll be investigating how such controllers work. Let's examine a function. Meanwhile, the one-order subsystem is employed to eliminate the effect of system uncertainties based on the adaptive control scheme and the congelation of variables technique. So, it is more suitable for designing powerful and well-defined control function for strict-feedback systems. This method is recursively designed, considering some state variables as virtual control inputs to devise intermediate visual control laws. Before we're discussing systems, we start with some formal function de nitions. Unlike the traditional literature on fuzzy logic system or neural networks Jan 1, 2021 · In order to ensure stability, adaptation rules need to be derived from stability theorems such as Lyapunov Stability Theorem. Jan 1, 2021 · The backstepping control method is a recursive design procedure that links the choice of a control Lyapunov function with the design of a feedback controller and guarantees global asymptotic stability of strict feedback systems (Kokotovic, 1992; Kokotovic and Arcak, 2001; Krstić et al. 3. Apr 29, 2024 · Bypassing the conventional barrier Lyapunov function method, the constrained system is transformed into its unconstrained counterpart, thereby obviating the need for feasibility conditions. 1. (2020) propose an ET backstepping control strategy to address the issues of unknown control directions and sensor faults. Aug 1, 2024 · The goal of the paper is to provide a systematic method for control parameter tuning of a backstepping-based controller. , 2018; Hao et al. Taylor 10 High-Order Control Barrier Functions [3] Q. [4] W. S. Aug 14, 2019 · This paper presents the output constraint problem of permanent magnet synchronous motor servo system. This technique was detailed in the articles [15], [16]. Jan 1, 2022 · Backstepping is defined as a recursive Lyapunov-based scheme used for controlling the nonlinear systems. Lyapunov Equation 3. The added design flexibility can be explored to by-pass some of the artificial problems associated with the I/O linearization. Safety-critical systems in general, require safe performance even during the reinforcement learning (RL) period. It not only Jan 1, 2025 · This paper proposed an adaptive trajectory tracking control scheme with collision avoidance for a class of high-order nonlinear systems with unknown nonlinearities. Jun 5, 2023 · The backstepping constraint control method, based on barrier Lyapunov functions (BLFs), is designed not only to track the desired trajectory but also to guarantee no violation of the position and angle constraints. 1049/iet-cta. One constrained subsystems and another unconstrained subsystems. Keywords Autonomous Vehicle · Backstepping Control · Barrier Lyapunov Function · Fixed-time Observer, Path Tracking Apr 1, 2022 · We propose an alternative approach for addressing safety of higher-order systems through Control Barrier Function Backstepping. Feb 11, 2023 · Due to the complexity of modeling and the strong transmission coupling, the rich background of rigid actuator control has not been transferred to variable stiffness actuator (VSA). Backstepping is a nonlinear control design tool for underactuated systems. However, for asymptotic stabilization of systems defined on noncontractible manifolds, there exists no differentiable CLF; the semiconcave CLF design problem based on the backstepping has not been discussed. To address this issue, a Barrier Lyapunov Function-based safe RL (BLF-SRL) algorithm is proposed here for the May 1, 2018 · Robust adaptive control scheme: The scheme is designed based on common Lyapunov function approach. In contrast to the predominant focus on constant stepping coefficients in existing research Jan 1, 2018 · (7) To fulfil the requirement of Lyapunov stability theory (Slotine et al. In section III, key theoretical and methodological innovations to a class of systems are presented and discussed, i. An integral-multiplicative Barrier Lyapunov function was integrated into the backstepping procedure to remove the dynamics mismatching issue of the existing SUM-type BLF. (2020a) Novel prescribed performance-tangent barrier Lyapunov function for neural adaptive control of the chaotic PMSM system by backstepping. Feb 6, 2021 · This paper presents an adaptive constraint control approach for Electronic Throttle Control System (ETCS) with asymmetric throttle angle constraints. Jan 1, 2022 · In the first case, the attitude errors are asymptotically driven to zero while strictly satisfying the quaternion constraints. Introduction We consider the problem of global feedback stabilization of systems of the form i= f(x)+F(x)O+o(x)u, xER", uER, (1. (2015). The design of stepping manifolds in the backstepping control incorporates the Lyapunov function, posing a challenge in determining optimal stepping parameters for each manifold. Lyapunov Design via Lyapunov Equation 3. Drawing inspiration from the method of Lyapunov backstepping, we provide a constructive framework for synthesizing safety-critical controllers and CBFs for higher-order systems from a top-level dynamics safety backstepping lemma which recovers our earlier design. , 2020), but the design of T-BC has the following shortcomings. For safety reasons, the position and heading of DP ship are to be maintained in certain range. Moreover, a command filter is High-Order Control Barrier Functions Andrew J. ietdl. First, the task space is partitioned according to the constrained and unconstrained directions, and a new representation of dynamics is introduced. To overcome this obstacle, backstepping control (BC) has been reported [19]. To cope with this, you need a controller that adapts to these parameters: an adaptive controller. Jul 19, 2016 · Backstepping is a recursive non-linear control method where the feedback control law and the associated Lyapunov functions can be designed following a step-by-step procedure . , 1995; Vaidyanathan et al. Keywords: Nonlinear control; Adaptive stabilization; Adaptive control Lyapunov functions; Backstepping 1. That is, under feedback control law , the Lyapunov function decays to zero as the states return to the origin. A. Conference paper; First Online: 05 June 2025 pp 462–474 based adaptive control. Numerical examples May 3, 2023 · Backstepping is a mature and powerful Lyapunov-based design approach for a specific set of systems. From the control point of view, it can achieve stabilization and tracking goals [1, 2]. Another preparatory comment of potential interest to the reader with technical expertise in backstepping designs is that the Lyapunov function that we construct is not of the form V = ~ z2 but of the form V = ~ z4. Jul 12, 2022 · Guaranteed safety and performance under various circumstances remain technically critical and practically challenging for the wide deployment of autonomous vehicles. MRAC for Nonlinear Systems 4. The adaptive constraint control method, which is based on barrier Lyapunov function (BLF), is designed not only to track the desired throttle angle but also to guarantee no violation on the throttle angle constraints. In this method, PPF defines the desired bounds, while BLF enforces these constraints through its unique Oct 1, 2023 · In the first approach, different robust controllers such as terminal sliding mode, adaptive sliding mode, integral sliding mode, dynamic surface control, finite-time control, fixed-time control, and model predictive control are designed to provide fault tolerance for different marine crafts (Baldini et al. Sreenath, “Exponential Control Barrier Functions for Enforcing High Relative-Degree Safety-Critical Constraints”, 2016. Nevertheless, the chattering in SMC is a severe barrier to its extension. The disturbance observer with exponential convergence is used to estimate the uncertain environment disturbances. Throughout the development over three decades, innovative theories and practices have extended backstepping to stabilization and tracking problems for nonlinear systems with growing complexity. org Aug 17, 2022 · In this manuscript, the rotor displacement control problem of active magnetic bearings system is considered and resolved. Backstepping controller, proposed in the beginning of 1990’s (Zhou and Wen, 2008) is one of the post popular Lyapunov based recursive control methods to provide stable behavior for nonlinear systems (Zhou and Wen, 2008; Sundarapandian and Ahmad, 2021; Krstic and Jun 5, 2023 · The backstepping constraint control method, based on barrier Lyapunov functions (BLFs), is designed not only to track the desired trajectory but also to guarantee no violation of the position and angle constraints. However, the aforementioned results ( Li et al. Jul 28, 2017 · The dynamics of a quadrotor has the characteristics of underactuation, strong coupling, and nonlinearities, which makes the controller design difficult. As the order of the controlled system increases, the T-BC will create the problem of “differential explosion”. To address this problem, prescribed performance functions (PPF) can be utilized alongside the barrier Lyapunov function (BLF) technique to ensure desired performance [[30], [31], [32]]. A specially designed reinforcement learning (RL) algorithm, featuring an observer-critic-actor architecture, is deployed in an adaptive optimal control the integrator backstepping, the dynamic surface control, command filtered backstepping, finite-time control, fuzzy logic system, neural network, Nussbaum function, barrier Lyapunov function, and hyperbolic tangent function. , 2018b, Vaidyanathan et al. Since backstepping design has been well studied and mature, we omit the details of the procedure. Feb 3, 2019 · This paper presents a backstepping controller using barrier Lyapunov function (BLF) for dynamic positioning (DP) system. Contents 1. 5 Sontag’s formula Let’s suppose that we have a control problem of the form Jan 1, 2023 · Much research has been done on adaptive nonlinear control by backstepping, with and without mechanical sensors [7], [12], [13], [14]. Although the ordinary backstepping control can ensure the stability of the system, it cannot guarantee the safety of the system or meet certain special conditions. Based on such a characteristic, that backstepping technique has been proposed originally in Kanellakopoulos, Kokotovic, and Morse (1991) to address the tracking control of nonlinear strict-feedback systems. Oct 8, 2024 · This paper presents a hybrid control approach, termed Model Predictive Backstepping Control, for steering angle manipulation. The attractions of the backstepping-like approach are the recursive design processes and modularized However, finding a suitable control Lyapunov function is a challenging problem for nonlinear control systems. Definition 3. This paper proposes to introduce the barrier Lyapunov function (BLF) in the servo system backstepping controller. 25) with the control Lyapunov function V = 1 2 z2 1 + 1 2 z2 2 (2. org Aug 1, 2023 · A main characteristic of the strict-feedback system is that the nonlinear functions appear in systems in an unmatched manner. Introduction 2. Therefore, most model-based control techniques developed for VSA require feedback linearization first. The process begins by recursively stabilizing the subsystems forming virtual control Keywords: Control Lyapunov Function, Lyapunov Design, Model Reference Adaptive Control, Adaptive Control, Backstepping Design. 4 days ago · Output Constrained Control Based Barrier Lyapunov Functions for Lateral Dynamics of Autonomous Vehicles. In addition, it can provide a faster response and result in more accurate path tracking compared to previous BLF-based control systems. Based on the simplified dynamics model, we design a backstepping based controller using Barrier Lyapunov Functions. vntass ukcqfkv eiq onjk mmi vyg yil bysuuz pxy xbtiv