No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Lyapunov Stability Theory of Nonsmooth Systems
... where Ω V denotes the set of measure zero wherever ∇V is not defined [13,Def. 2.2]. ...
... [2, Remark 1.2] Following Assumption 2, H is positive definite.Define the concatenated state vector z : R ≥0 → R 3nN as z ≜ e ⊤ r . Using(11),(13), and (16), the closed-loop error system is expressed aṡ ...
... Based on the chain rule for differential inclusions in [13,Theorem 2.2], the derivative of t → V (ξ (t)) exists almost everywhere and is a solution toV (ξ) ...
This work presents a decentralized implementation of a Robust Integral of the Sign of the Error (RISE) controller for multi-agent target tracking problems with exponential convergence guarantees. Previous RISE-based approaches for multi-agent systems required 2-hop communication, limiting practical applicability. New insights from a Lyapunov-based design-analysis approach are used to eliminate the need for multi-hop communication required in previous literature, while yielding exponential target tracking. The new insights include the development of a new P-function which is developed which works in tandem with the inclusion of the interaction matrix in the Lyapunov function. Nonsmooth Lyapunov-based stability analysis methods are used to yield semi-global exponential convergence to the target agent state despite the presence of bounded disturbances with bounded derivatives. The resulting outcome is a controller that achieves exponential target tracking with only local information exchange between neighboring agents.
... The inclusion of internal dynamics significantly increases the complexity of the problem. To address this challenge, we decompose the system into two subsystems and introduce a novel non-differentiable piecewisecontinuous Lyapunov function that innovatively combines the Lyapunov functions of each subsystem whose stability conditions emerge from the determination of the one-sided directional derivative [18]. The control design achieves robust performance through the optimization of a guaranteed performance index, ensuring robust global stability. ...
... which reproduces (18). The proof is concluded. ...
... These conditions are ensured by Assumption (A3). In a practical setting, such conditions can be achieved through a proper sliding surface design, as illustrated in Section 7. Finally, it is also important to point out that (18) imposes thatv(ζ) < 0 only in a subset and not for all (ζ, x) ∈ R r × R 2n . ...
This paper presents a novel procedure for robust control design of linear time-invariant systems using a Multivariable Generalized Super-Twisting Algorithm (MGSTA). The proposed approach addresses robust stability and performance conditions, considering convex bounded parameter uncertainty in all matrices of the plant state-space realization and Lipschitz exogenous disturbances. The primary characteristic of the closed-loop system, sliding mode finite-time convergence, is thoroughly examined and evaluated. The design conditions, obtained through the proposal of a novel max-type non-differentiable piecewise-continuous Lyapunov function are formulated as Linear Matrix Inequalities (LMIs), which can be efficiently solved using existing computational tools. A fault-tolerant MGSTA control is designed for a mechanical system with three degrees of freedom, illustrating the efficacy of the proposed LMI approach.
... In what follows, the definitions of Filippov solution, generalized gradient and regular function are given according to [24,25,26]. ...
... In this subsection, a command force u i for each VTOL UAV will be synthesized. The main difficulties here are that the command force u i should comply with the nonsingular condition (25) and the desired position p r and its derivatives are not available in the command force u i and the subsequent applied torque τ i due to limited communication. ...
... whereâ z i = k γ tanh(γ z i ) and the property that | tanh(·)| < 1 have been applied. To this end, k η satisfying (33) is sufficient to guarantee that the developed command force u i in (30) for each UAV strictly satisfies the non-singular condition (25). Remark 3.2 By definingâ i = k γ tanh(γ i ) in the distributed estimator (8) and introducing the auxiliary dynamics (31), the developed command force u i for i ∈ V is equipped with a saturation property. ...
This paper investigates the coordinated trajectory tracking problem of multiple vertical takeooff and landing (VTOL) unmanned aerial vehicles (UAVs). The case of unidirectional information flow is considered and the objective is to drive all the follower VTOL UAVs to accurately track the trajectory of the leader. Firstly, a novel distributed estimator is developed for each VTOL UAV to obtain the leader's desired information asymptotically. With the outputs of the estimators, the solution to the coordinated trajectory tracking problem of multiple VTOL UAVs is transformed to individually solving the tracking problem of each VTOL UAV. Due to the under-actuated nature of the VTOL UAV, a hierarchical framework is introduced for each VTOL UAV such that a command force and an applied torque are exploited in sequence, then the position tracking to the estimated desired position and the attitude tracking to the command attitude are achieved. Moreover, an auxiliary system with proper parameters is implemented to guarantee the singularity-free command attitude extraction and to obviate the use of the unavailable desired information. The stability analysis and simulations effectively validate the achievement of the coordinated trajectory tracking of multiple VTOL UAVs with the proposed control approach.
... It is noted that the right hand side of the closed-loop system (42) is discontinuous in (γ − γ). Thus, the solution of the closed-loop system (42) must be defined in the Filipov sense [3], [20]. Put the closed-loop system (42) in the following compact form:ẋ ...
... where [20]. It is known that, for any scalar x, the Filipov set of sgn(x) denoted by K[sgn](x) is as follows [20]: ...
... where [20]. It is known that, for any scalar x, the Filipov set of sgn(x) denoted by K[sgn](x) is as follows [20]: ...
In this paper, we study the problem of the distributed Nash equilibrium seeking of N-player games over jointly strongly connected switching networks. The action of each player is governed by a class of uncertain nonlinear systems. Our approach integrates the consensus algorithm, the distributed estimator over jointly strongly connected switching networks, and some adaptive control technique. Furthermore, we also consider the disturbance rejection problem for bounded disturbances with unknown bounds. A special case of our results gives the solution of the distributed Nash equilibrium seeking for high-order integrator systems.
... It is noted that the right hand side of the closed-loop system (30) is discontinuous in s i . Thus, the solution of the closedloop system (30) must be defined in the Filipov sense [3], [20]. Put the first three equations of the closed-loop system (30) to the following compact form: ...
... where K[f c ](x c , t) is the Filipov set of f c (x c , t) [3], [20]. It is known that, for any scalar x, the Filipov set of sgn(x) denoted by K[sgn](x) is as follows [20]: ...
... where K[f c ](x c , t) is the Filipov set of f c (x c , t) [3], [20]. It is known that, for any scalar x, the Filipov set of sgn(x) denoted by K[sgn](x) is as follows [20]: ...
In this paper, we first study the leader-following output synchronization problem for a class of uncertain nonlinear multi-agent systems over jointly connected switching networks. Our approach integrates the output-based adaptive distributed observer, the conventional adaptive control technique, and the output regulation theory. Compared with the existing results, our control law only relies on the output of the leader instead of the state of the leader and allows the followers and the leader to have different orders. Then, we further consider the rejection of a class of bounded disturbances with unknown bounds. Our problem includes the state consensus problem as a special case if the followers and the leader have the same order.
... where k 1 ∈ R n×n is a positive definite control gain matrix, sgn(·) is the sign function, andθ σ = θ (t u σ ) is the error bound at start of the GPS-denied interval derived in (29). Note that since the observer is discontinuous, solutions of (33) are interpreted in a generalized sense as Krasovskii solutions [25]. ...
... where k a > 0, k u > 0 and L 1 > 0 are user-defined gains, and the upper bound V u is selected such that V u < 1 2 η 2 . Proof: The generalized derivativeV (see [25,Equation 13]) of the candidate Lyapunov function in (6) along the flow of (36) and (38) is given bẏ ...
This paper presents a switched systems approach for extending the dwell-time of an autonomous agent during GPS-denied operation by leveraging memory regressor extension (MRE) techniques. To maintain accurate trajectory tracking despite unknown dynamics and environmental disturbances, the agent periodically acquires access to GPS, allowing it to correct accumulated state estimation errors. The motivation for this work arises from the limitations of existing switched system approaches, where increasing estimation errors during GPS-denied intervals and overly conservative dwell-time conditions restrict the operational efficiency of the agent. By leveraging MRE techniques during GPS-available intervals, the developed method refines the estimates of unknown system parameters, thereby enabling longer and more reliable operation in GPS-denied environments. A Lyapunov-based switched-system stability analysis establishes that improved parameter estimates obtained through concurrent learning allow extended operation in GPS-denied intervals without compromising closed-loop system stability. Simulation results validate the theoretical findings, demonstrating dwell-time extensions and enhanced trajectory tracking performance.
... it has the result that P(t) ≥ 0 ∀t ≥ t 0 from (34). (1), with gains selected according to (31) and (33), the controller (13) with adaptive laws (18) and (23) guarantees that all closed-loop signals remain bounded, and tracking errors asymptotically converge to zero as e e e 1 , ė e e 1 → 0, as t → ∞. ...
... Under Filippov's framework [30,31],V L (y y y, t) exists almost all time (a.a.t.) andV L (y y y, t) ∈ a.a.t.V L (y y y, t), whereV L (y y y, t) is defined aṡ ...
Research on adaptive control for time-varying parameters (TVPs) in nonlinear systems has gained considerable interest. The trajectory tracking problem for nonlinear Euler-Lagrange (EL) systems is investigated in the paper under TVPs and external disturbance, and we propose a continuous projection adaptive controller to improve tracking efficiency and reduce control amplitude. Firstly, two filtered tracking errors are designed to obtain faster convergence for tracking errors. Then, the adaptive update law is designed with continuous projection regarding the TVPs, especially for non-linearly-parameterized systems. Subsequently, the controller is designed by constructing a continuous auxiliary term to realize trajectory tracking. System stability is rigorously analyzed by utilizing Lyapunov technique and LaSalle-Yoshizawa corollary extension, and theoretical analysis demonstrates that all signals of the closed-loop error system remain bounded and tracking errors converge to zero. Comparative tests are performed to evaluate the effectiveness of the proposed strategy.
... where f i are the vector fields defined as the right-hand sides of the closedloop system (40),(48). Now, let P A i + A T i P = −Q i for some Q i = Q T i > 0, i ∈ M and let V : IR (n+1) ( M +M )−1 → IR be the Lipschitz, regular [18] and positive definite candidate Lyapunov function given by: ...
... The derivative of the Lyapunov function along Filippov closed-loop solutions is negative, hence the origin of the closed-loop system is globally stable in the Filippov sense [18]. Now following the approach in [21], from (57) for any closed-loop trajectory we have sup ...
This note presents an extension to the adaptive control strategy presented in [1] able to counter eventual instability due to disturbances at the input of an otherwise stable closed-loop system. These disturbances are due to the presence of affine terms in the plant and reference model. The existence of a common Lyapunov function is used to prove global convergence of the error system, even in the presence of sliding solutions, as well as boundedness of all the adaptive gains.
... The notion and computational properties of the generalized gradient are throughly studied in nonsmooth analysis [9]. In particular, tools for establishing stability and convergence properties of nonsmooth dynamical systems are presented in [3,15,27]. Finally, we refer to [7,17] for guidelines on how to design dynamical systems for optimization purposes, and to [4] for gradient descent flows in distributed computation in settings with fixed-communication topologies. ...
... The following result is a generalization of LaSalle principle for differential equations of the form (2.5) with nonsmooth Lyapunov functions. The formulation is taken from [3], and slightly generalizes the one presented in [27]. ...
This paper discusses dynamical systems for disk-covering and sphere-packing problems. We present facility location functions from geometric optimization and characterize their differentiable properties. We design and analyze a collection of distributed control laws that are related to nonsmooth gradient systems. The resulting dynamical systems promise to be of use in coordination problems for networked robots; in this setting the distributed control laws correspond to local interactions between the robots. The technical approach relies on concepts from computational geometry, nonsmooth analysis, and the dynamical system approach to algorithms.
... Due to the discontinuity of the control law (26) at f E = f * E , the solution of the system (28) is understood in the Filippov sense [35]. ...
... where ∈ represents the differential inclusion, and K[·] is the Filipov set-valued map defined in [35] and has the property that ...
This paper studies the elevation angle-based formation tracking control of multi-agent systems. Unlike bearing-based formation control, which uses bearing vectors, formation control using elevation angles eliminates the need for a global coordinate frame. This is beneficial because the coordinate frames of individual agents are often not aligned. Previous studies on elevation angle-based formation control have predominantly focused on leaderless formations. By incorporating a leader-follower structure, follower agents can effectively track and move with leaders, allowing the formation to move together and maneuver through obstacles. We consider three types of agents: single-integrator, double-integrator, and nonholonomic robots. For agents with a single integrator model, we first propose a control law for stationary leaders case and then modify the proposed control law to address moving leaders. Finite-time convergence to the desired formation is guaranteed. We then consider agents with double-integrator dynamics and nonholonomic robots, showing asymptotic convergence to the desired formation. Simulation results are provided to validate the proposed control laws.
... Lemma 2 (Chain rule [29][30][31]) Let ...
... Substituting Eqs. (18), (30), and (28) into Eq. (24) yieldṡ ...
This paper presents a novel and constructive solution for the velocity estimation of a large class of mechanical systems. By employing the technical results in non-smooth analysis, a specific eigenvalue analysis problem that can be systematically solved through structured computational procedures is formulated, and the resulting globally exponentially convergent observer can therefore be generated automatically for any appropriate model input with arbitrarily complex configurations. The proposed method applies directly to non-holonomic systems with Pfaffian constraints, and its structural simplification can be further achieved for systems with uniformly bounded inertia matrix. The established results provide a major breakthrough on both the dimension and practical realizability of the prevailing Immersion & Invariance velocity observer designs, and its effectiveness is verified with numerical simulations of several representative examples.
... Proposition 2.17. [3,24] (LaSalle's invariance principle for differential inclusions) Consider (4), and let f : R m → R be a locally Lipschitz and regular function. Let S ⊆ R m be compact and strongly invariant for (4), and assume that maxL F [F ] f (y) ≤ 0 for each y ∈ S. ...
... Lyapunov theory provides a powerful framework for analyzing the stability of systems without explicitly solving the governing differential equations (Shevitz and Paden, 1994). It allows us to draw conclusions about the behavior of trajectories, such as their convergence or divergence from an equilibrium point, without needing to determine the exact path of the trajectory. ...
... This concept was first proposed by Lyapunov in his famous works [9] and [10] for systems of ordinary differential equations, and thereafter gained numerous applications. Since that time, the concept of Lyapunov stability has been attracting much attention from researchers (see, in particular, [11]- [15]). ...
The paper is devoted to developing Lyapunov's methods for analyzing the stability of an equilibrium of a dynamical system in the space of probability measures that is defined by a nonlocal continuity equation. Sufficient stability conditions are obtained based on the basis of an analysis of the behaviour of a nonsmooth Lyapunov function in a neighbourhood of the equilibrium and the investigation of a certain quadratic form defined on the tangent space of the space of probability measures. The general results are illustrated by the study of the stability of an equilibrium for a gradient flow in the space of probability measures and the Gibbs measure for a system of connected simple pendulums. Bibliography: 28 titles.
... Next, we turn to verify the monotonic evolution of V (⃗ p) along the system trajectories given by the notion of Lie derivative in the nonsmooth setting, which requires maxLV (⃗ p) < 0, ∀⃗ p ̸ = ⃗ p ⋆ , withLV (⃗ p) being the set-valued Lie derivative of V regarding [∂J(⃗ x * (⃗ p)) − ⃗ p] in (8) at ⃗ p defined by [23], [28] LV (⃗ p) := a ∈ R : ∃v ∈ ∂J(⃗ x * (⃗ p)) − ⃗ p such that ...
Incentive-based coordination mechanisms for distributed energy consumption have shown promise in aligning individual user objectives with social welfare, especially under privacy constraints. Our prior work proposed a two-timescale adaptive pricing framework, where users respond to prices by minimizing their local cost, and the system operator iteratively updates the prices based on aggregate user responses. A key assumption was that the system cost need to smoothly depend on the aggregate of the user demands. In this paper, we relax this assumption by considering the more realistic model of where the cost are determined by solving a DCOPF problem with constraints. We present a generalization of the pricing update rule that leverages the generalized gradients of the system cost function, which may be nonsmooth due to the structure of DCOPF. We prove that the resulting dynamic system converges to a unique equilibrium, which solves the social welfare optimization problem. Our theoretical results provide guarantees on convergence and stability using tools from nonsmooth analysis and Lyapunov theory. Numerical simulations on networked energy systems illustrate the effectiveness and robustness of the proposed scheme.
... (36) Proof: Since the proposed observer and the error dynamics in (22) are discontinuous, non-smooth analysis must be used to prove the finite-time convergence of (22) [17], [13]. ...
We propose a distributed control strategy to allow the control of a multi-agent system requiring k-hop interactions based on the design of distributed state and input observers. In particular, we design for each agent a finite time convergent state and input observer that exploits only the communication with the 1-hop neighbors to reconstruct the information regarding those agents at a 2-hop distance or more. We then demonstrate that if the k-hop based control strategy is set-Input to State Stable with respect to the set describing the goal, then the observer information can be adopted to achieve the team objective with stability guarantees.
... The proof relies on the Lyapunov stability theory, which is the heart of the dynamic system analysis [54]. Lyapunov stability theory can be categorized into (a) the indirect method, which analyzes the convergence through the system state equation, and (b) the direct method, which explicitly describes the behavior of the system's trajectories and its convergence by making use of the Lyapunov function. ...
Hyperspectral images are typically composed of hundreds of narrow and contiguous spectral bands, each containing information regarding the material composition of the imaged scene. However, these images can be affected by various sources of noise, distortions, or data loss, which can significantly degrade their quality and usefulness. This paper introduces a convergent guaranteed algorithm, LRS-PnP-DIP(1-Lip), which successfully addresses the instability issue of DHP that has been reported before. The proposed algorithm extends the successful joint low-rank and sparse model to further exploit the underlying data structures beyond the conventional and sometimes restrictive unions of subspace models. A stability analysis guarantees the convergence of the proposed algorithm under mild assumptions, which is crucial for its application in real-world scenarios. Extensive experiments demonstrate that the proposed solution consistently delivers visually and quantitatively superior inpainting results, establishing state-of-the-art performance.
... The proof relies on the Lyapunov stability theory, which is the heart of the dynamic system analysis [54]. Lyapunov stability theory can be categorized into (a) the indirect method, which analyzes the convergence through the system state equation, and (b)the direct method, which explicitly describes the behavior of the system's trajectories and its convergence by making use of the Lyapunov function. ...
Hyperspectral images are typically composed of hundreds of narrow and contiguous spectral bands, each containing information regarding the material composition of the imaged scene. However, these images can be affected by various sources of noise, distortions, or data loss, which can significantly degrade their quality and usefulness. This paper introduces a convergent guaranteed algorithm, LRS-PnP-DIP(1-Lip), which successfully addresses the instability issue of DHP that has been reported before. The proposed algorithm extends the successful joint low-rank and sparse model to further exploit the underlying data structures beyond the conventional and sometimes restrictive unions of subspace models. A stability analysis guarantees the convergence of the proposed algorithm under mild assumptions , which is crucial for its application in real-world scenarios. Extensive experiments demonstrate that the proposed solution consistently delivers visually and quantitatively superior inpainting results, establishing state-of-the-art performance.
... The following two lemmas are useful for proving the stability of discontinuous systems using nonsmooth Lyapunov functions. The next result can be found in [20], [26], [27], [30]. ...
This paper studies decentralized formation control of multiple vehicles when each vehicle can only measure the local bearings of their neighbors by using bearing-only sensors. Since the inter-vehicle distance cannot be measured, the target formation involves no distance constraints. More specifically, the target formation considered in this paper is an angle-constrained circular formation, where each vehicle has exactly two neighbors and the angle at each vehicle subtended by its two neighbors is pre-specified. To stabilize the target formation, we propose a discontinuous control law that only requires the sign information of the angle errors. Due to the discontinuity of the proposed control law, the stability of the closed-loop system is analyzed by employing a locally Lipschitz Lyapunov function and nonsmooth analysis tools. We prove that the target formation is locally finite-time stable with collision avoidance guaranteed. The evolution of the vehicle positions in the plane is also characterized.
... is not a solution, which is evident from Equation ( (Shevitz, D. & Paden, B. (1994)). Therefore, the proposition is proven. ...
The single wheel, gyroscopically stabilized robot - Gyrover, is a dynamically stable but statically unstable, underactuated system. In this paper, based on the dynamic model of the robot, we investigate two classes of nonholonomic constraints associated with the system. Then, based on the backstepping technology, we propose a control law for balance control of Gyrover. Next, through transferring the systems states from Cartesian coordinate to polar coordinate, control laws for point-to-point control and line tracking in Cartesian space are provided.
... Lemma 1 (Nonsmooth Barbalat' s Lemma [36] ) Assume that for all ≥ 0 , there exists a compact set Ω such that the Filippov solution of = ( , ) always remains within it. If is an empty set, there is a regular function of the change over time : Ω → R with ≤ 0 ∀ ∈ is ordinary. ...
This paper uses a bioinspired neurodynamic (BIN) approach to investigate the formation control problem of leader-follower nonholonomic multiagent systems. In scenarios where not all followers can receive the leader's state, a distributed adaptive estimator is presented to estimate the leader's state. The distributed formation controller, designed using the backstepping technique, utilizes the estimated leader states and neighboring formation tracking error. To address the issue of impractical velocity jumps, a BIN-based approach is integrated into the backstepping controller. Furthermore, considering the practical applications of nonholonomic multiagent systems, a backstepping controller with a saturation velocity constraint is proposed. Rigorous proofs are provided. Finally, the effectiveness of the presented formation control law is illustrated through numerical simulations.
... where the constants β > η and k ≥ 0. PROOF. Since u(t) is discontinuous, the solution of the system is understood in Filippov sense [15]. Consider ...
This paper presents novel algorithms for the Fermat-Weber Location Problem, guiding an autonomous agent to the point that minimizes the weighted sum of Euclidean distances to some beacons using only bearing measurements. The existing results address only the simple scenario where the beacons are stationary and the agent is modeled by a single integrator. In this paper, we propose a number of bearing-only algorithms that let the agent, which can be modeled as either a single-integrator or a double-integrator, follow the Fermat-Weber point of a group of stationary or moving beacons. The theoretical results are rigorously proven using Lyapunov theory and supported with simulation examples.
... This concept was first proposed in the famous works of A. M. Lyapunov [9], [10] for systems of ordinary differential equations, after which it received numerous applications. Since then, the concept of Lyapunov stability attracts a great attention (see, in particular, [11]- [15]). ...
The paper is concerned with the development of Lyapunov methods for the analysis of equilibrium stability in a dynamical system on the space of probability measures driven by a non-local continuity equation. We derive sufficient conditions of stability of an equilibrium distribution relying on an analysis of a non-smooth Lyapunov function. For the linear dynamics we reduce the stability analysis to a study of a quadratic form on a tangent space to the space of probability measures. These results are illustrated by the studies of the stability of the equilibrium measure for gradient flow in the space of probability measures and Gibbs measure for a system of coupled mathematical pendulums.
... As the right-hand side of the above equation is discontinuous, the solution of e i is understood in Filippov sense [38]. Using the Lyapunov function V = 1 2 e i 2 , we havė ...
This paper studies matrix-scaled consensus algorithms for linear dynamical agents interacting over an undirected network. Under the proposed algorithms, the state vectors of all agents asymptotically agree up to some scaling matrix weights. First, the geometry of the matrix-scaled consensus space and the matrix-scaled Laplacian are studied. Second, matrix-scaled consensus algorithms for networks of single-integrators are proposed in two scenarios: with and without constant parametric uncertainties. Nonlinear and finite-time matrix-scaled consensus algorithms are also discussed. Finally, observer-based matrix-scaled consensus algorithms are designed for homogeneous and then extended for heterogeneous linear-time invariant agents. The validity of the proposed algorithms is established through rigorous mathematical analysis and demonstrated by numerical simulations.
... which is nonnegative on K and vanishes only on SOL(K, F ). To apply the nonsmooth Lyapunov techniques [25], we need to show that (9) is Lipschitz continuous w.r.t. its first and second arguments, respectively. ...
We leverage best response dynamics to solve monotone variational inequalities on compact and convex sets. Specialization of the method to variational inequalities in game theory recovers convergence results to Nash equilibria when agents select the best response to the current distribution of strategies. We apply the method to generalize population games with additional convex constraints. Furthermore, we explore the robustness of the method by introducing various types of time-varying disturbances.
... denotes the set-valued Filippov map of the system (17). Due to the presence of an intersection operation on the right side of (21), it indicates that whileV S2 (t) is not empty and there exists ξ ∈ SGN[(D ⊤ ⊗ I n )x] such that ξ ⊤ (D ⊤ ⊗ I n )f < 0 for allf ∈ K[f ], as a resultV S2 (t) moves into the negative halfplane of the real axis. ...
This paper presents a distributed continuous-time optimization framework aimed at overcoming the challenges posed by time-varying cost functions and constraints in multi-agent systems, particularly those subject to disturbances. By incorporating tools such as log-barrier penalty functions to address inequality constraints, an integral sliding mode control for disturbance mitigation is proposed. The algorithm ensures asymptotic tracking of the optimal solution, achieving a tracking error of zero. The convergence of the introduced algorithms is demonstrated through Lyapunov analysis and nonsmooth techniques. Furthermore, the framework's effectiveness is validated through numerical simulations considering two scenarios for the communication networks.
... Since (A1) has discontinuous right-hand-side, the solution of (A1) is understood in the Filippov's sense. 47 By using the formula xK[sign(x)] = |x|, we havė Thus, vec(s, − max ) is uniformly bounded, and 0 ≤ V(t) + ∫ t 0 ||s( )|| 2 d ≤ V(0). Thus, lim t→+∞ V(t) exists and is finite, and (t) is bounded. ...
In this paper, we proposed several disturbance observer‐based matrix‐weighted consensus algorithms. A new disturbance observer is firstly designed for linear systems with unknown matched or mismatched disturbances representable as the multiplication of a known time‐varying matrix with a unknown constant vector. Under some assumptions on the boundedness and persistent excitation of the regression matrix, the disturbances can be estimated at an exponential rate. Then, a suitable compensation input is provided to compensate the unknown disturbances. Second, disturbance‐observer based consensus algorithms are proposed for matrix‐weighted networks of single‐ and double‐integrators with matched or mismatched disturbances. We show that both matched and mismatched disturbances can be estimated and actively compensated, and the consensus system uniformly globally asymptotically converges to a fixed point in the kernel of the matrix‐weighted Laplacian. Depending on the network connectivity, the system can asymptotically achieve a consensus or a cluster configuration. The disturbance‐observer based consensus design is further extended for a network of higher‐order integrators subjected to disturbances. Finally, simulation results are provided to support the mathematical analysis.
... Subsequently, we revisit the Chain Rule, a mathematical principle that enables us to discriminate Lipschitz regular functions. Theorem 1 (Chain Rule [34]): Let x(·) be a Filippov solution in (1) and V (x(t)) be a Lipschitz regular function. Then V is absolutely continuous, and ...
This paper addresses the consensus control problem of heterogeneous multi-UAV systems against Byzantine attacks. A drone compromised by Byzantine attacks transmits erroneous values to its neighbors while applying wrong input signals for itself, which is aggressive and challenging to defend. Inspired by the concept of digital twin technology, we introduce a new hierarchical protocol equipped with a virtual twin layer (TL), which decouples the challenges into two defense schemes: one against Byzantine edge attacks on the TL and the other against Byzantine node attacks on the cyber-physical layer (CPL). In the TL, we provide a topology reconfiguration strategy that enhances the resilience of the communication network by judiciously adding a minimal number of key edges. We rigorously demonstrate that the control strategy attains asymptotic consensus within a finite timeframe, given that the topology on the TL adheres to a strongly
(2f+1)
-robustness criterion. Within the CPL, decentralized chattering-free controllers are proposed to ensure the resilient output consensus for the heterogeneous multi-UAV systems against Byzantine node attacks. Furthermore, the derived consensus controller exhibits an exponential convergence characteristic. The effectiveness and practicality of the obtained theoretical results are verified by a UAV swarm flight experiment.
Note to Practitioners
—Cooperative control of UAVs presents significant prospects for application and development, becoming a focal point of automatic control. By modeling UAV swarms as multi-agent systems, various complex distributed control methods have been conveniently proposed and implemented economically in practical systems. However, when certain agents are compromised and interfere with their neighbors, the whole network may become highly susceptible to failure. This paper specifically studies the resilient consensus control against the significant active internal threats, Byzantine attacks. The published results have primarily focused on cases where the leader UAV has no input signals. In practical applications, however, the leader often has a pre-established trajectory sent by the host and the followers are unaware of this input information. This significantly complicates the task of identifying Byzantine attackers. In this work, we introduce a new hierarchical protocol inspired by the concept of digital twin technology, which decouples the challenges into defense against Byzantine edge attacks on the TL and the defense against Byzantine node attacks on the cyber-physical layer. The experiment shows the feasibility and security of our control scheme, which provides valuable guidance for the practical applications of drones.
... Intermittent control strategies are crucial for systems where continuous control is impractical due to resource limitations or inherent system constraints. Key strategies include sampled-data control, impulse control, event-triggered control, hybrid systems, reset control, and model predictive control [19][20][21][22][23][24]. These approaches enable efficient and effective control across various applications by balancing control performance with practical limitations. ...
In this article, we delve into the exponential stability of uncertainty systems characterized by stochastic differential equations driven by G-Brownian motion, where coefficient uncertainty exists. To stabilize the system when it is unstable, we consider incorporating a delayed stochastic term. By employing linear matrix inequalities (LMI) and Lyapunov–Krasovskii functions, we derive a sufficient condition for stabilization. Our findings demonstrate that an unstable system can be stabilized with a control interval within (θ*,1). Some numerical examples are provided at the end to validate the correctness of our theoretical results.
To enhance tracking capability of the robust integral of the sign of the error controller (RISE) structure, an RISE controller with time-varying gains is proposed in this paper, which is incorporated with feedforward estimation from a fully connected neural network (FCNN). Specifically, for uncertain Euler–Lagrange dynamic systems, two error filters are designed to obtain faster convergence for tracking errors. Then, an adaptive update law is constructed for adjusting FCNN weights online, so as to approximate unknown nonlinear component in dynamic systems. To efficiently track trajectories, a self-tuning RISE feedback controller is designed to decrease high gains and avoid chattering phenomenon. System stability is rigorously analyzed by utilizing Lyapunov technique and LaSalle–Yoshizawa corollary extension, and asymptotic tracking convergence is proved. Finally, regarding robotic mechanisms, comparative tests are conducted to validate tracking performance of the proposed strategy.
Статья посвящена развитию методов Ляпунова для анализа устойчивости положения равновесия динамической системы в пространстве вероятностных мер, задаваемой нелокальным уравнением неразрывности. Получены достаточные условия устойчивости, опирающиеся как на анализ поведения негладкой функции Ляпунова в окрестности положения равновесия, так и на исследование квадратичной формы, заданной на касательном пространстве к пространству вероятностных мер. Общие результаты проиллюстрированы исследованием устойчивости положения равновесия для градиентного потока в пространстве вероятностных мер и меры Гиббса для системы связанных математических маятников. Библиография: 28 названий.
Chaotic jerk (CJ) system stabilization technology represents an innovative breakthrough with diverse applications in different fields. However, it is challenging to stabilize these systems due to their sensitivity to external inputs and initial conditions. Recent adaptive control methods for the CJ system’s stabilization offer fast convergence; these methods cancel nonlinearities and reduce closed-loop performance. In addition, using discontinuous functions in the control protocols accelerates state convergence to zero but results in chattering that hinders control performance. This article presents a new nonlinear adaptive control law that overcomes these limitations and significantly improves CJ system stabilization performance. The proposed adaptive chaos stabilization scheme uses continuous functions to stabilize the CJ system, setting this method apart from traditional adaptive control techniques. It eliminates chattering and achieves state variables’ robust, fast, and smooth convergence to the vicinity of zero. The proposed control strategy for suppressing the CJ system reduces stabilization time significantly, enhances energy efficiency, and assures an asymptotic stable closed loop. It incorporates parameter adaptation laws to estimate the bounds of uncertainties and external disturbances in the CJ system, leading to improved adaptation. The direct Lyapunov stability analysis proves the stability of the closed loop. Simulation results demonstrate that the proposed adaptive method offers superior performance in stabilizing the CJ system compared to existing adaptive schemes.
This article studies the problem of stabilizing a leader-follower formation specified by a set of bearing constraints while disturbed by some unknown uniformly bounded disturbances. A set of leaders are positioned at their desired positions while each follower is modeled by a single integrator with an additive time-varying disturbance. Adaptive variable-structure control laws using displacements or only bearing vectors are applied to stabilize the desired formation. Thanks to the adaptive mechanisms, the proposed control laws require neither information on the bearing Laplacian nor the directions and upper bounds of the disturbances. It is further proved that when the leaders are moving with the same bounded uniformly continuous velocity, the moving target formation can be achieved under the proposed control laws. Simulation results are provided to support the stability analysis.
The enhanced model reference adaptive control (EMRAC) algorithm is an effective full‐state adaptive solution to control plants affected by nonlinear unmodelled dynamics and persistent disturbances. The EMRAC strategy improves the tracking performance by equipping the MRAC algorithm with adaptive switching and adaptive integral control actions. However, the need for the plant state prevents the applicability of the EMRAC algorithm to engineering control problems where only the plant output is measurable. To cover this gap and extend the range of plants controllable by EMRAC solutions, this article presents an output‐based EMRAC algorithm leveraging the closed‐loop (CL) MRAC formulation. The robustness of the closed‐loop control system is analytically analysed, not only with respect to plant parameter uncertainties and square measurable disturbances, but also to ℒ∞ unmodelled terms and disturbances. The ultimate boundedness of the closed‐loop control system is assessed with respect to ℒ∞ unknown nonlinear terms and disturbances, by using Lyapunov theory for Filippov systems, as the adaptive switching control action makes the closed‐loop system discontinuous. To assess the effectiveness of the CL‐EMRAC strategy to impose reference trajectories despite the unmodelled plant dynamics and persistent bounded disturbances, the problem of vehicles' direct yaw moment control is used as an engineering case study. The closed‐loop tracking performance is also quantitatively evaluated through a set of key performance indicators and compared to those provided by four benchmark controllers, that is, two LQ‐based strategies and two MRAC‐based control solutions. The CL‐EMRAC and benchmark controllers are implemented and tested in a co‐simulation environment based on a high‐fidelity IPG CarMaker vehicle model.
This chapter proposes a novel robust control strategy for three-phase power converters operated under unbalanced grid conditions. A consolidated control objective is obtained in the stationary frame, which can be flexibly adjusted according to the degree of oscillation in the active and reactive powers and the balance of the three-phase current. Based on the dynamics of the converter and control objective, a control scheme in a cascaded framework is presented, in which an AO is applied to estimate the positive and negative sequences of the grid voltage. In the current tracking loop, a STA current controller coupled with a STD is implemented to track the current references, featuring rapid dynamics and improved robustness. Additionally, in the voltage regulation loop, an effective composite controller is developed to regulate the dc-link voltage, where a STO is used to estimate the load disturbance, thereby improving the performance of the converter. The experimental results are provided to confirm the effectiveness and superiority of the proposed control strategy.
The value function of Mayer’s problem arising in optimal control is investigated, and lower semicontinuous solutions of the associated Hamilton–Jacobi–Bellman equation are defined in three (equivalent) ways. Under quite weak assumptions about the control system, the value function is the unique solution. Moreover, it is stable with respect to perturbations of the control system and the cost. It coincides with the viscosity solution whenever it is continuous.
This paper develops a calculus for computing Filippov's differential inclusion which simplifies the analysis of dynamical systems described by differential equations with a discontinuous right-hand side. In particular, when a slightly generalized Lyapunov theory is used, the rigorous stability analysis of variable structure systems is routine. As an example, a variable structure control law for rigid-link robot manipulators is described and its stability is proved.
The design of variable-structure control (VSC) systems for a class
of multivariable, nonlinear, time-varying systems is presented. Using
the Utkin-Drazenovic method of equivalent control and generalized
Lyapunov stability concepts, the VSC design is described in a unified
manner. Complications that arise due to multiple inputs are examined,
and several approaches useful in overcoming them are developed. Recent
developments are investigated, as is the kinship of VSC and the
deterministic approach to the control of uncertain systems. All points
are illustrated by numerical examples. The recent literature on VSC
applications is surveyed