An unmanned ground swarm (UGS) system consists of multiple vehicles exhibiting intelligent and coordinated behavior through mutual cooperation and information exchange. The system is expected to simultaneously achieve four key objectives: swarm tracking, compact formation, collision avoidance, and obstacle evasion. However, these objectives are inherently conflicting—for instance, maintaining compact formation and ac curate tracking may increase the risk of inter-agent collisions or obstacle encounters. To resolve these conflicts, we propose a novel integrated control framework that unifies the four objectives into a consistent set of constraints. This framework systematically ad dresses the performance trade-offs through a robust and adaptive control scheme capable of handling dynamic agent behaviors and system uncertainties. Robustness is ensured without prior knowledge of the uncertainties, while an adaptive mechanism further reduces the control effort required. Moreover, fuzzy-set theoretic formulation is employed to quantify uncertainty bound, enabling the expression of performance indices that link control parameters to system behavior. These indices are then used in a Pareto-game framework to achieve parameter optimality. The effectiveness of the proposed control design—characterized by its robustness, adaptability, and optimality—is validated through simulations of a UGS team scenario.

To enhance the efficiency and safety of transportation, low-altitude economy has garnered increasing attention. In this study, we consider air–ground systems composed of both uncrewed ground vehicles and uncrewed aerial vehicles. A novel integrated framework for modeling, control, and optimization is proposed, which creatively unifies kinematics, dynamics, and control into a structured setting for approximate constraint following. The framework consists of three key phases. First, in the modeling phase, desired system behaviors, such as collision avoidance, compact formation, and trajectory tracking, are formulated into a unified performance metric. This metric is then treated as a constraint, and the system's dynamics for approximate constraint following are derived using the Udwadia–Kalaba approach. Second, in the control phase, an adaptive robust control scheme is developed to compensate for system uncertainties. Notably, the method does not require prior knowledge of the uncertainty bounds, while still guaranteeing consistent system performance in uncertain environments. Third, in the optimization phase, the fuzzy nature of the uncertainty bounds is considered. The uncertainty bound is modeled as a fuzzy set characterized by a membership function, which is incorporated into a fuzzy-theoretic performance index. The Pareto-optimal game theory is then employed to determine the optimal control parameters. The proposed framework enables the air–ground logistics system to follow constraints with reduced error, as validated through simulation results.

This article proposes a Nash game-optimized adaptive robust control framework for bipedal parallel wheel-legged robots. Specifically, the framework targets a balance between tracking accuracy and control effort under underactuation, constraint coupling, and substantial uncertainties. To address these challenges, a constraint following adaptive robust controller is adopted, while gain selection is posed as a two-player Nash game between performance and cost objectives. Consequently, the resulting equilibrium yields controller gains without heuristic tuning. Furthermore, Lyapunov analysis establishes uniform ultimate boundedness of the closed-loop trajectories under bounded disturbances and model errors. Finally, numerical simulations verify that the proposed approach achieves a superior balance, concurrently enhancing tracking accuracy while significantly reducing control effort compared to conventional methods.

Human-machine systems (HMSs) are dedicated to integrating intelligent human decisions with machine operations to achieve synergistic operational functionality. We focus on constraint-following control within the HMS, considering potential uncertainties, environmental disturbances, and limited operational space. A hierarchical hybrid control scheme is proposed, consisting of a preemption algorithm and a human decision algorithm. Specifically, the preemption algorithm relies on online state feedback from mechanical system signals, such as position and velocity, which can be implemented in hardware or software; the human decision algorithm takes inputs from electrophysiological signals or language commands. In this development, a Lagrangian density function is constructed that integrates optimal decision making with a uniformly bounded threshold. The intelligent decision-making problem in the HMS is creatively analyzed and mathematically solved leveraging variational calculus, resulting in the analytical expression of the optimal membership function associated with human decisions. Furthermore, a series of numerical simulation experiments are conducted using a bionic upper-limb prosthetic system as an example, and the comparison results demonstrate the superiority and effectiveness of the proposed method.

Trajectory tracking in bimanual humanoid robots, whose closed-chain kinematic structures inherently amplify the effects of modeling errors, external disturbances, and time-varying parameters, is a challenging task. To address this, we reformulate the dual-arm tracking task as a servo constraint-following problem and derive the system dynamics under approximate constraints using the Udwadia-Kalaba method. The humanoid system is modeled as a constrained mechanical structure subjected to fast-varying, bounded uncertainties with unknown limits. On this basis, we propose a robust control framework that guarantees both uniform boundedness (UB) and uniform ultimate boundedness (UUB) of the tracking error, ensuring stability and performance even under severe parametric and dynamic uncertainties. To reconcile the trade-off between transient dynamics and steady-state accuracy—essential for service-oriented tasks such as door opening or coffee pouring—we integrate a Nash equilibrium-based optimization mechanism into the controller design. By formulating a two-player non-cooperative game over the controller's key tuning parameters, we analytically derive the existence, uniqueness, and closed-form solutions of the game, achieving an optimal balance between competing objectives. Comprehensive simulations on a reduced-order bimanual humanoid model validate the proposed approach, demonstrating superior tracking accuracy, disturbance rejection, and energy efficiency compared to benchmark methods. The proposed strategy offers a theoretically grounded and practically implementable solution for robust, constraint-compliant humanoid manipulation.

This paper addresses the performance regulation of Unmanned Ground Vehicle (UGV) swarm systems, where multiple vehicles travel together to complete a task. The system must satisfy two key performance requirements: maintaining vehicle coalition and avoiding collisions. These requirements are inherently conflicting, as staying close together increases the risk of collisions. Additionally, the system must contend with modeling uncertainties and disturbances. We propose a novel differential homeomorphic approach, introducing a$\beta _{i}$-performance measure that harmoniously combines these conflicting requirements. Our approach ensures that vehicles remain close to each other without collisions, even under uncertainty. To achieve this, we develop an adaptive robust control scheme that guarantees desirable$\beta _{i}$-performance. Furthermore, we optimize the control design parameters using a cooperative game-theoretic framework, establishing the existence, uniqueness, and closed-form solutions of Pareto optimality. Consequently, our approach meets four critical performance criteria for UGV swarm systems: coalition, collision avoidance, robustness, and optimality.

Dyadic control is a new control frontier, which is to address two (often conflicting) objectives simultaneously. We consider rendering bothcompact formationandcollision avoidancein uncertain cooperative road transportation systems. These two tasks, however, present conflicting objectives, where excessive stress on formation tightness may lead to an increased risk of collisions. The tasks are creatively formulated as equality constraints and inequality constraints. Based on the generalized Udwadia-Kalaba (GUK) equation, two independent controllers are developed to handle these constraints, with orthogonality between the control components ensuring no mutual interference. The proposed method guarantees the uniform boundedness and uniform ultimate boundedness of the system, even in the presence of unknown uncertainties. The effectiveness of the control strategy is demonstrated through simulations of a four-vehicle fleet system.

This paper introduces an adaptive robust control approach tailored for underactuated mechanical systems encountering matched and mismatched uncertainty, employing a constraint-following methodology. The control strategy unfolds in two phases: initially, a nominal control scheme is devised neglecting uncertainty and deviations in initial conditions from constraints. Subsequently, uncertainty is categorized into matched and mismatched components, ensuring that mismatched uncertainties remain unobservable. Leveraging the structural characteristics of the uncertainty bound, a novel segmented adaptive law is proposed and seamlessly integrated into the adaptive robust control framework. By employing the Lyapunov minimax approach, the method ensures uniform boundedness and uniform ultimate boundedness simultaneously, thereby ensuring approximate adherence to constraints for underactuated mechanical systems facing both matched and mismatched uncertainties alongside initial condition deviations.

This study analyzes the role of human factors in semi-automomous vehicles. A novel human-machine system framework is proposed. In the human-machine systems, the human is in charge of decision-making, problem-solving and creativity, while the machine is responsible of strength, precision, computation and speed. Since the human behavior is inherently fuzzy, fuzzy set theory is legitimately employed to evaluate human influence. Further, a constraint-following control (CFC) method is proposed. The proposed control ensures the semi-autonomous vehicles global uniform boundedness (GUB) and global uniform ultimate boundedness (GUUB). Afterwards, the driver behavior is optimized to obtain better control effect. Two key performances are attained: robustness and optimality. Finally, the proposed control approach is validated through a simple example, showcasing its effectiveness.

A confined space is an area in an industrial facility that has limited access and allows only restricted movement due to physical constraints. Confined spaces often require special safety precautions and may be subject to specific regulations to ensure the well-being of workers. Flexible manufacturing cells typically work in confined spaces in order to increase efficiency and decrease cost. The operation can be further complicated if uncertainty is involved. We propose an adaptive robust controller for uncertain mechanical systems in a confined space to enhance system performance while ensuring system safety. The design procedure consists of five phases: constraint-following formulation, fuzzy uncertainty prescription, diffeomorphism transformation, adaptive robust control design, and Nash game-based optimization. The effectiveness of the control scheme is demonstrated by numerical simulation experiments for a humanoid robot arm.

Since the actual environment of mechanical systems is not ideal, there will be some unstable factors, namely uncertainty. Such uncertainty is changeable and bounded, but its boundary is usually uncertain. To describe the uncertain boundary, the fuzzy set theory is used in this paper, which is one of the innovations of this paper. On this basis, an adaptive robust control method based on two control parameters is proposed, which is for the fuzzy mechanical systems. It is proved by Lyapunov function that this control approach can ensure the global uniform boundedness (GUB) and global uniform ultimate boundedness (GUUB) performance of the controlled mechanical system. This is the first level of the control design. Afterwards, the second level is to ensure the control performance while reducing the control cost by optimizing two control parameters. In order to optimize two control parameters, the cooperative game theory is adopted, which is another one of the innovations of this paper. In the optimization process, two control parameters are regarded as two players, and the optimal control parameters are obtained by minimizing the cost function designed based on these two players. Finally, a serial robot model is adopted to verify the feasibility and superiority of the proposed control approach.

Path tracking for autonomous vehicles is one of the most critical tasks in intelligent transportation systems (ITS). The ITS performance, including efficiency, safety, flexibility, and resilience, are all based on it. The two central issues for a successful path tracking are resilience and smoothness. We endeavor to adopt a neural-activation based constraint-following approach to resolve these two issues concurrently. First, an adaptive robust constraint-following control scheme is proposed. The control tracks a desired trajectory with guaranteed performance even in the presence of uncertainty. Second, a neural-activation mechanism is proposed, which generates desired trajectory effectively based on traffic pattern with sufficiently smoothness. Third, the trajectory is embedded into the control scheme to ensure that the control conforms to any changing traffic pattern while in motion. As a result, the control can rapidly adapt to the changing traffic condition with smoothness and resilience.

Agile formation (AF) is a new frontier for intelligent swarm system formation. The AF pertains to perform various tasks in short phases of work and frequent reassessment and adaptation of plans. This greatly increases the applicability of swarm systems. There are however two major challenges for the control design: smooth task transitions and guaranteed collision avoidance. We adopt the constraint-following approach to address these. First, for agile formation, a plateau activation function is proposed to generate a sequence of consecutive and disjoint formations. For collision avoidance, a distance-gauge function is proposed. Second, by taking the objectives of agile formation control and collision avoidance as desirable constraints, the agile formation together with collision avoidance are both cast into a constraint following control problem. Third, to evaluate the constraint-following error, a performance measure$\beta $is introduced and then an agile formation control is designed to render the$\beta $-measure to be asymptotically convergent to zero. By this, the swarm system can follow the agile formation constraint and collision avoidance constraint. Therefore, agile formation and collision avoidance are both accomplished.

In order to improve the control performance of the Cyber-Physical Systems (CPSs), an integrated modelling-control-design trio framework is established in this paper. In the modelling part, CPS has two components, cyber component and physical component, so the system may be subject to (possibly fast) time-varying cyber interference and (possibly fast) time-varying physical uncertainty. A dynamic model encompassing these two phases of the CPS is established. In the control part, a novel control design is proposed based on the dynamic model. The problem of constraint-following for CPS operating under cyber interference and physical uncertainty is considered. In the design part, the choice of control parameters is investigated. This procedure consists of two stages. The first stage is to design a control scheme based on feasible control design parameters, so that it can guarantee the performance in the case of both cyber interference and physical uncertainty. The second stage is to seek the optimal design among the feasible control design parameters, which is resolved by an intelligent multi-agent game-theoretic approach. We invoke both Nash game and Stackelberg strategy to choose the optimal parameters. Interestingly, the optimal parameters obtained from different game settings are the same. This shows the conception of optimality we established spans in a broader context. The robustness and superiority of the system performance are demonstrated in the intelligent transportation system.

Accurate tracking control of autonomous cooperative transportation systems (CTS) remains challenging owing to the complexity of the mechanisms and the high requirements of coordination between carriers. In this paper, the trajectory tracking control of a CTS with a pair of autonomous vehicles serving as carriers is investigated. A novel constraint-oriented hierarchical modeling method is proposed to describe the dynamics of the system. By dividing the system dynamics into two portions: the lower-level individual modeling and the upper-level constraints abstraction, the modeling process is significantly simplified. Then an innovative constraint-following control law is designed to address the tracking control problem under the special system topology, based on the internal and external constraints designed in the modeling process. The asymptotic convergence of the tracking error is theoretically guaranteed. To reduce potential damage of the payload during transportation, a payload force optimization method is creatively proposed. It relies on the closed-form relationship between the control input and payload forces established by the constraint-oriented modeling. The normal and shear stress on the payload is successfully limited, without affecting the trajectory tracking performance. Simulation results show that the proposed control method and the payload force optimization strategy can help achieve accurate and safe autonomous cooperative transportation simultaneously.

In this paper, the dynamic modeling and collaborative control of a multi-vehicle cooperative transportation system for load carrying is explored. A hierarchical modeling and constraint-following control scheme is creatively proposed. In the dynamic modeling stage, the separate models of system components including the load and vehicle carriers are firstly established at the lower level. Then the internal and external constraints corresponding to the system topology and the transportation task are designed to integrate separate models at a higher level. In the system control stage, a distributed collaborative control law is proposed based on the closed-form constraint forces, with which the load can follow the external constraints actively and the carriers can maintain the internal constraints passively. In order to overcome the influence of time-varying multi-source uncertainties of the system on control effectiveness and stability, an adaptive robust control term is designed based on the Lyapunov min-max approach. Both uniform boundedness and uniform ultimate boundedness of the constraint-following error are guaranteed. Comprehensive validations show that our propose scheme can significantly reduce the modeling complexity despite the strongly coupled topology and nonlinearity of the system, as well as achieving more precise and robust trajectory following control compared with the baseline methods.

In this article, the trajectory generation and motion control of autonomous driving oversized cargo cooperative transportation systems (CTS) in static but bounded environment is investigated. Different from common vehicle systems, the challenges lie on the safety-guaranteed cooperation of independently controlled carriers with inherent connections brought by the rigid payload, which results in complex system dynamics and multiple time-variant uncertainties. A constraint-oriented “leader-follower” modeling and control framework is introduced, and a trajectory generation method based on the diffeomorphism is creatively proposed to generate closed-form collision-free trajectory for the payload in the bounded environment. To achieve safety-guaranteed trajectory following under uncertainties, a transformed adaptive robust control strategy (TARC) is designed through constraint relaxation, and the coordination of the carriers is realized. An implementation with comprehensive ablation studies demonstrates the effectiveness of our trajectory generation and tracking control framework. The collision-free trajectory set is efficiently generated, and the CTS can be kept strictly inside the safe corridor with high tracking accuracy, which is extremely hard for the baseline methods.

In this article, the uncertain system parameters and disturbances are considered and described by fuzzy sets theory. Then, a new form of robust control is designed for the fuzzy dynamical systems. The system is proved to be uniformly bounded and uniformly ultimately bounded according to Lyapunov approach. To seek a better system performance and lower control cost, an optimization problem with multi parameters is formulated. A two-level game structure is proposed to find the optimal solution. First, the generalized Stackelberg game theory is applied when there are one leader and two followers. Then, the Nash game theory is applied for the two followers. Numerical simulations are performed for verification.

A Stackelberg game-based design for an adaptive robust control for the fuzzy uncertain underactuated mechanical systems (UMSs) is proposed. The emphasis is on fuzzy-based uncertainty and inequality constraint. The uncertainty is time varying and bounded within a prescribed fuzzy set. For the inequality constraint, we creatively have it merge into constraint-following performance by a diffeomorphism technique. An adaptive robust control strategy is then proposed. Deterministic performance is guaranteed provided the control design parameters are within feasible regions. To further enhance the performance, we introduce a two-player Stackelberg game setting. The optimal choice of design parameters can be solved. The feasibility of this design is demonstrated on an autonomous wheeled mobile robot (AWMR), which is confined in a bounded space.

This article explores an adaptive robust control scheme for satellite formation-containment flying in a way of constraint following. For safe flight and uncertainty suppression, both collision avoidance and uncertainty suppression are addressed. First, for uncertainty suppression, (possibly fast) time-varying but bounded uncertainty is considered, and then an adaptive law is proposed to estimate the comprehensive uncertainty bounds online. Second, the problem of formation-containment control with collision avoidance is converted into another problem of constraint-following control by taking the objectives of collision avoidance, formation, and containment, respectively, as the collision-avoidance constraint, formation constraint, and containment constraint. Third, an η -measure is introduced to gauge the constraint-following error, and then an adaptive robust control is proposed to render the error to be uniformly bounded and uniformly ultimately bounded, regardless of the uncertainty. By this, the satellites can follow the above collision-avoidance constraint, formation constraint, and containment constraint approximately. As a result, satellite formation-containment control emphasis on collision avoidance and uncertainty suppression is achieved.

This article proposes an adaptive robust formation control scheme for the connected and autonomous vehicle (CAV) swarm system by utilizing swarm property, diffeomorphism transformation, and constraint following. The control design is processed by starting from a 2-D dynamics model with (possibly fast) time varying but bounded uncertainty. The uncertainty bounds are unknown. For compact formation, the CAV system is treated as an artificial swarm system, for which the ideal swarm performance is taken as a desired constraint. By this, formation control is converted into a problem of constraint following and then a performance measure β is defined as the control object to evaluate the constraint following error. For collision avoidance, a diffeomorphism transformation on space measure between two vehicles is creatively performed, by which the space measure is positive restricted. For uncertainty handling, an adaptive robust control scheme is proposed to render the β -measure to be uniformly bounded and uniformly ultimately bounded, that is, drive the controlled (CAV) swarm system to follow the desired constraint approximatively. As a result, the system can achieve the ideal swarm performance; thereout, compact formation is realized, regardless of the uncertainty. The main contribution of this article is exploring a 2-D formation control scheme for (CAV) swarm system under the consideration of collision avoidance and time-varying uncertainty.

We consider mechanical systems with uncertainty. The uncertainty may be time varying. The bound of the uncertainty is described by its fuzzy characteristics. To design a feasible control, we start with a robust phase, which renders a control scheme that guarantees the system performance regardless of the actual value of the uncertainty. This robust phase is then followed by an optimal phase. There are design parameters in the control, which can be fine-tuned. We proposed multiple performance objectives. The goal of the choice of the control design parameters is to minimize the performance objectives. However, since these objectives are nonconciliating (meaning one's minimum is not the other one's minimum), we invoke the Stackelberg strategy for the optimal parameters. The game strategy mimics two players: one is the leader and one is the follower. Through the interplay between the two players, we show how to select the design parameters. The design procedure in both robust and optimal phases is demonstrated by a coupled inverted pendulum system.

The inequality constraints, system nonlinearities, parameter uncertainties, and external disturbances are always unavoidable in practical mechanical systems. This article proposes an adaptive robust control (ARC) algorithm from the view of servo constraint following to tackle the control problem of mechanical systems subject to the above factors. For the inequality constraints, a creative diffeomorphism which could convert the two-sided bounded state variables to the unbounded ones is explored, which could render the transformed nonlinear system free from inequality constraints. For the system uncertainties, a leakage-type ARC algorithm is developed, which could render the system the practical stability. The permanent magnet linear motor (PMLM) system is utilized as a typical application to verify the proposed state transformation and ARC approach. Numerical simulations show that the displacement of the PMLM system could well track the desired trajectory without violating the given bound line.

In this article, we propose a high-order robust control for fuzzy dynamical systems. The time varying but bounded uncertainty in this system is described by the fuzzy set theory. The control is deterministic and is not based on IF-THEN fuzzy rules. By the Lyapunov approach, we prove that the control is able to guarantee uniform boundedness and uniform ultimate boundedness. In addition, the tunable parameters in the high-order control are regarded as two players in a cooperative game. Two cost functions are also proposed based on the two players. These two cost functions are related to system performance and control cost. Then, the optimal design problem is solved by finding the Pareto-optimality parameters. Numerical simulations are performed for verification.

In this article, we apply a high-order control to a dynamical system with uncertainty. There are two characteristics. First, the uncertain part, which is time-varying but bounded, is described in a fuzzy aspect. Specifically, the uncertainty lies within a fuzzy set and the bound is regarded as a fuzzy number. Second, the systems are uniformly bounded and uniformly ultimately bounded with a deterministic controller based on the Lyapunov theory. To obtain better system performance and lower control input, we apply the noncooperative game theory to optimize the parameters by establishing a Nash game. Then, the D-operation is proposed for the uncertainty related to fuzzy numbers. Finally, we perform the numerical simulations of the steering-by-wire system for verification.

This article focuses on a robust control scheme for pointing control of the marching tank gun. Both matched and mismatched uncertainties, which may be nonlinear (possibly fast) time varying but bounded, are considered. First, the pointing control system is constructed as a coupled, nonlinear, and uncertain dynamical system with two interconnected (horizontal and vertical) subsystems. Second, for the horizontal pointing control, robust control is proposed to render the horizontal subsystem to be practically stable. Third, for the vertical pointing control, an uncertainty bound-based state transformation is constructed in a similar way of backstepping to convert the original mismatched system (i.e., the vertical subsystem) to be locally matched and then robust control is proposed to render the transformed system to be practically stable. Finally, it is proved that when the transformed system is rendered to be practically stable, the original system renders the same performance; therefore, vertical pointing control is achieved. This work should be among the first ever endeavor to cast all the coupling, nonlinearity, and (both matched and mismatched) uncertainty into the pointing control framework of the marching tank gun.

This article proposes an optimal indirect approach of constraint-following control for fuzzy mechanical systems. The system contains (possibly fast) time-varying uncertainty that lies in a fuzzy set. It aims at an optimal controller for the system to render bounded constraint-following error such that it can stay within a predetermined bound at all time and be sufficiently small eventually. First, for deterministic performance, the original system is transformed into a constructed system. A deterministic (not the usual IF-THEN rules-based) robust control is then designed for the constructed system to render it to be uniformly bounded and uniformly ultimately bounded, regardless of the uncertainty. Second, for optimal performance, a performance index, including the average fuzzy system performance and control effort, is proposed based on the fuzzy information. An optimal design problem associated with the control gain is then formulated and solved by minimizing the performance index. Finally, it is proved when the constructed system renders uniform boundedness and uniform ultimate boundedness, the original system achieves the desired performance of bounded constraint following.

There is uncertainty in the system, and we consider that uncertainty is (possibly fast) time varying, but with definite bound. Fuzzy set theory is used to describe the inexact boundary and then the problem of robust control of uncertain dynamical systems is studied. Based on two adjustable design parameters, a robust control method for general mechanical systems is proposed. The control is deterministic, not the conventional IF-THEN rule based. By using the Lyapunov minimax approach, it is proved that the proposed control can guarantee system performance to be uniformly bounded and uniformly ultimately bounded. In order to find the optimal solution in the prescribed range, a two-player cooperative game is used. To reduce costs while ensuring control performance, two performance indices are developed, each of which is controlled by an adjustable parameter (i.e., player). Both necessary and sufficient conditions for Pareto-optimality are established. Using these conditions, the Pareto-optimal solution can be obtained. The effectiveness of the control design is demonstrated by the simulation of the two-body pendulum.

Pointing tracking control of marching turret-barrel system is one of the important topics in exploration of intelligent ground combat platform. This paper focuses on an adaptive robust control scheme for pointing tracking of marching turret-barrel system driven by a motor and an electric cylinder. Three types of possibly fast time-varying but bounded uncertainty are considered: system modeling error, external disturbance and road excitation. The uncertainty bounds are not necessary to be known. First, the pointing tracking system is constructed as a coupled, nonlinear and uncertain dynamical system with two interconnected (horizontal and vertical) subsystems. Second, a tracking error$e$is defined as a gauge of control objective, and then the dynamical equation of the pointing tracking system is built in state-space form. Third, for uncertainty control, a comprehensive uncertainty bound$\alpha $is derived to measure the most conservative influence of the uncertainty, and then an adaptive law is proposed to evaluate it in real time. Finally, for pointing tracking control, an adaptive robust control is proposed to render the pointing tracking system to be practically stable; thereout, the objective of pointing tracking is achieved. This work should be among the first ever endeavours to cast all thecoupling,nonlinearityandbound-unknown uncertaintyinto the pointing tracking framework of marching turret-barrel system.

A novel class of high-order robust controls is presented for uncertain fuzzy systems in this article. The optimal tunable parameters are also obtained based on the Stackelberg game theory. First, a dynamical system structure is formulated with uncertainty. The uncertain portion in the system is bounded, nonlinear, and time-varying which is within prescribed fuzzy set. Thus, this is a fuzzy system. Then, the proof based on the Lyapunov minimax approach shows that the novel high-order robust controls are able to assure deterministic system performance, which is uniform boundedness and ultimate uniform boundedness. Furthermore, the optimal choice of the tunable parameters in the control is considered. We creatively apply the Stackelberg strategy to solving the optimization problem. Two parameters are regarded as leader and follower in a sequential game, respectively. Based on the Stackelberg game rules, we are able to design different cost functions for different parameters. The cost functions include both performance measures and control cost. Finally, the simulations of an electronic throttle system are presented for demonstration.

This article addresses a cooperative game-oriented optimal constraint-following problem for fuzzy mechanical systems. The state of the concerned system is affected by possibly (fast) time-varying uncertainty. The fuzzy set theory is adopted to describe such uncertainty. The task is to drive the system to obey a set of prescribed constraints optimally. Since the control objective may be changing along with the system uncertainty, a time-varying$\beta $-measure is defined to gauge the constraint-following error; based on which, an adaptive robust control scheme with two tunable parameters is then proposed to render it to be uniform boundedness and uniform ultimate boundedness. For the seeking of the optimal design parameters, two cost functions, each of which is dominated by one tunable parameter, are developed with the fuzzy information, and thereout a two-player cooperative game is formulated. Finally, the optimal design problem is successfully solved: with the existence, uniqueness, and analytical expression of the Pareto optimality.

Mechanical systems with (possibly fast) time-varying but bounded uncertainty are considered. The exact value of the boundary is unknown. All the designer can get is that the boundary values are in a (known) fuzzy set. On the basis, a robust control method is proposed, which can ensure the uniformly bounded and uniformly ultimately bounded of the controlled mechanical system. The control contains two flexibly selectable design parameters. We seek to choose the optimal parameters. For a superior performance, two fuzzy-set-based performance indices are proposed to reflect the transient performance as well as the steady-state performance. Each performance index can be influenced by two design parameters. However, influences are nonconciliatory. This poses a design dilemma: the increase of a parameter may harness one performance while inflict the other. Therefore, the “optimal” choice of the design parameters is not intuitively clear. To resolve this dilemma, the two-player Nash-based noncooperative game theory is adopted, which is a notable feature of this article. Once the problem is formulated, we show that there is always a Nash-equilibrium solution to the two-player problem. We also show how to find it. The approach is very general, which also can be extended to$n$-player Nash game for future research. The control is applied to a compressor powered by permanent magnet synchronous motor (PMSM) as a demonstration. The resulting performance shows that this Nash-based robust control design is both practical and effective.

This article addresses the control design problem for a fuzzy vehicular platoon system consisting of one leading vehicle and N following vehicles. Due to external disturbances, uncertainty exists in the platoon system and is usually nonlinear and possibly fast time varying. In order to deal with uncertainty, fuzzy theory is employed to describe the platoon system, thereby the so-called fuzzy vehicular platoon system. Based on the fuzzy property of uncertainty, a type of adaptive law is proposed. Since the original state of the system is one-side bounded when collision avoidance is taken into account, a state transformation is made, by which a new global state is obtained. Then, the swarm intelligence is embedded into the platoon system via a function, which mimics the swarm behavior. For the control design, we focus on adaptiveness and optimization. A switching-type adaptive robust control is proposed. The control is not IF-THEN rule based. We further explore the problem of control parameter optimization. A performance index consisting of transient control cost and average control cost is proposed. Aiming at minimizing the performance index, the optimal problem is formulated. The solution to the optimal problem (i.e., optimal control parameter) exists and is unique. The control with optimal parameter is called optimal control, which guarantees both deterministic performance (uniform boundedness, uniform ultimate boundedness, and collision avoidance) and fuzzy performance of the platoon system.

To enhance the lower limbs' rehabilitation training of stroke patients, in this article, a deterministic adaptive robust control with control gain parameters optimized by a novel cooperative game theory is proposed for the two-degree-of-freedom (DOF) lower limb exoskeleton robot system (LLERs) with uncertainties and external disturbances. On the one hand, the deterministic adaptive robust control put forward will guarantee the uniform boundedness and uniform ultimate boundedness of gait constraint deviation and parameter estimation error. On the other hand, for uncertainties and external disturbance (possibly fast time-varying), which will arise in the two-DOF LLERs inevitably, we suppose that these uncertainties and disturbances are bounded, and their bounds are lying within the fuzzy sets, which can be characterized by membership functions; with such descriptions and control performance analysis, a novel cooperative game with two players participated will be formulated to seek the Pareto optimal solutions for the control gains of deterministic adaptive robust control proposed, and furthermore, the existence of optimal solutions is also shown by invoking a numerical technique. Eventually, the simulation results presented have verified the effectiveness of our proposed methodology on improving rehabilitation training of lower limbs.

This article proposes a new robust control design framework for uncertain mechanical systems, which may be fully actuated or underactuated. The uncertainty is (possibly fast) time varying, which lies in prescribed fuzzy sets (hence fuzzy mechanical systems) and may be unbounded. The control goal is formulated as servo constraints (hence constraint-following control), which may be holonomic or nonholonomic. We introduce the possibility theory into the Lyapunov stability analysis (LSA), proposing possibility-based LSA (PBLSA), which allows a maximum failure possibility (generally small) prescribed by designers. It can be viewed as a generalization of the conventional LSA, and the resultant performance is interpreted in the context of possibility. By the PBLSA, a class of robust constraint-following controls that isnotIF–THEN heuristic rules based is proposed, which renders approximate constraint following for the system performance with a prescribed maximal failure possibility. Optimal design of a control parameter considering both system performance and control cost is investigated. The benefits of the proposed design framework are discussed and simulations on two applications are given for demonstrations.

Control design and performance enhancement for uncertain mechanical systems are pursued in this article. Uncertainty in a physical system is often inevitable in practice, which is best characterized by its possible bound. Mechanical systems with uncertain nonlinearity are considered. Furthermore, even the knowledge of the coefficients in the bound is unknown, which can only be described by its fuzzy association to a set. In controlling the system, there is a hierarchical performance requirement. The first level is deterministic, including uniform boundedness and uniform ultimate boundedness. This is the part the system must meet regardless of the actual value of the uncertainty. The second level is optimality, in terms of minimizing a fuzzy-theoretic performance index. We propose a novel control design with a tunable design parameter. The control guarantees the first-level performance when the design parameter falls in a range. We, then, take the advantage of this range flexibility to address the second-level requirement. The optimal choice of the design parameter can be made by solving an optimization problem. This problem is completely solved. Both the analytic (i.e., closed form) expressions of the design parameter and the resulting minimum cost are given. As a result, we accomplish a two-level control design task.

In this article, we are dedicated to coping with the control design for underactuated mechanical system (UMS) with system uncertainties, which is also subject to a set of servo constraints. For the nominal system without any initial condition deviations, this article provides a general model-based servo control with the idea of second-order constraints, which can be derived from both holonomic and nonholonomic ones. For the uncertain system, with the uncertainties being separated into matched portions and mismatched portions, this article provides an adaptive robust control scheme to guarantee the UMS to attain the deterministic performance on the basis of nominal control. The leakage-type adaptive law is to size up the unknown uncertainty bounds. The cart-pole system and the rotational-translational actuator system are used to verify the availability of the proposed adaptive robust control.

This paper investigates the robust control for dynamical systems subject to uncertainty. The uncertainty is assumed to be (possibly fast) time varying and bounded. The bound is unknown but lies within a prescribed fuzzy set (hence the fuzzy dynamical system). We propose an approach for the robust control design which is implemented in two steps. First, a class of robust controls is proposed based on tunable parameters. The proposed controls are deterministic and are not conventionally IF-THEN rules based. By the Lyapunov minimax approach, we prove that the proposed controls are able to guarantee deterministic system performance, namely, uniform boundedness and ultimate uniform boundedness. Second, optima seeking from the proposed controls is considered to improve fuzzy system performance. We formulate the optima-seeking problem as a two-player (one leader and one follower) Stackelberg game by developing two cost functions, each of which is in charge of one tunable parameter (i.e., the player). Each cost function consists of an average fuzzy system performance index and the associated player's control effort. We show that the solution of the optimal design problem (i.e., the optima of the tunable parameters), which is called the Stackelberg strategy, always exists and how to obtain the backwards-induction outcome is provided. Simulation results on the walking control of a biped robot model are presented for demonstration.

A class of generalized nonlinear dynamic systems is first approximated by N fuzzy-based linear subsystems using the identification of input-output data or the linearizing system with respect to suitable operating points. To obtain null trajectory tracking error in fixed time, a fuzzy fixed-time control (FFTC) with nonlinear switching surface and switching gain is first designed. It can be said that the FFTC is based on a class of passive and distributive models with uncertainties. To compensate enormous uncertainties, a fuzzy fixed-time learning control (FFTLC) by learning two unknown coefficients for the upper bound of uncertainties in each subsystem is designed. As compared with radial basis function neural network, the computational complexity for the compensated uncertainties is much simple. It can be said that an FFTLC is based on a class of online active and distributive uncertain models. Due to the fixed-time control design, the transients often occur, particularly for the larger uncertainties or initial tracking error. Hence, the saturated input of nonlinear dynamic system is addressed and online compensated. Finally, the compared simulations and application to two-link robot manipulator confirm the effectiveness, robustness, and less computation as compared with previous studies.

This article proposes a novel robust control design for mechanical systems based on constraint following and multivariable optimization. The state of the concerned system is affected by (possibly fast) time-varying and bounded uncertainty. The objective is to drive the system to obey a set of prescribed constraints. A β-measure is defined to gauge the constraint-following error; based on which, a feedback robust control scheme, which invokes design parameters, is proposed. For the seeking of optimal design parameters, a multivariable constrained optimization problem is formulated. The problem is successfully solved: with the existence, uniqueness, and analytical expression (i.e., closed form) of the optimal design parameters demonstrated. With the optimal parameters, the proposed robust control can render dual performance: guaranteed and optimal. As the guaranteed performance, the β-measure is assured to be uniform boundedness and uniform ultimate boundedness. As the optimal performance, the performance index is globally minimized. This article is the first ever endeavour to cast both the constraint following and multivariable optimization into the control framework for uncertain mechanical systems.

The optimal design problem of adaptive robust control for fuzzy mechanical systems with uncertainty is investigated in this paper. The uncertainty that may be nonlinear and (possibly fast) time-varying is assumed to be bounded, and the knowledge of the bound only lies within a prescribed fuzzy set. Based on the Udwadia and Kalaba's approach, an adaptive robust controller, which is deterministic and is not the usual if-then rules-based is proposed to render the system to follow a class of prespecified constraints approximately. The adaptive law is of leakage type that can adjust the magnitude of the adaptive parameter based on the nonlinear performance-dependent gain. The resulting controlled system is uniformly bounded and uniformly ultimately bounded, which is proved via the Lyapunov minimax approach. Furthermore, we propose a novel concept: fuzzy confidence to measure the expectation value of a fuzzy number. Then, a fuzzy-based system performance index that includes the expectation value of the uniform ultimate boundedness (the average fuzzy performance) and the control cost is formulated. The optimal design problem associated with the control can then be solved by minimizing the performance index. As a result, the performance of the fuzzy mechanical system is both deterministically guaranteed and fuzzily optimized under this control.

The position control and vibration suppression problems of the flexible solar array drive system containing uncertainty are considered in this paper. The uncertainty in system may be due to unknown parameters and external disturbance. The uncertainty bound can be described via a fuzzy set. In addition, there exists the flexible vibration in the system. A new optimal robust control approach with input shaping is proposed for the flexible solar array drive system. By designing the position command trajectory, the input shaper is proposed to suppress the flexible vibration. To enhance the position control performance, the optimal robust control is proposed by fuzzy description of the uncertainty bound. Neither the system nor the control is fuzzyif–thenrule based. The global solution to this optimal design problem is demonstrated to be always existent and unique. The resulting control is able to guarantee the uniform boundedness and uniform ultimate boundedness of the system in the presence of uncertainty, while minimizing a fuzzy-based performance index associated with both the fuzzy performance and the control cost. In addition, the flexible vibration can be effectively suppressed. The novelty of this research is a systematic control approach by blending input shaping technology, control theory, fuzzy set theory, and optimization theory into an integrated framework, for solving the position control and vibration suppression problems of flexible solar array drive system with mismatched conditions.

The robust control design problem for uncertain dynamical systems is considered in this study. The uncertainty is time varying (possibly fast) and bounded, and the bound lies within a prescribed fuzzy set (hence the fuzzy dynamical system). We design the robust control in two steps. First, we propose a class of robust controls based on tunable parameters, which is in deterministic form and not conventionally IF-THEN fuzzy rule based. It is shown that these controls are able to guarantee deterministic system performance, namely uniform boundedness and ultimate uniform boundedness. Second, we seek the optima of tunable parameters in the control by formulating a two-player Nash game, which is based on two performance indexes (i.e., the cost functions). It is shown that the Nash equilibrium (i.e., the optima of tunable parameters) always exists. The procedure of obtaining the Nash equilibrium is provided. Under the proposed control, the system performance is both deterministically guaranteed and fuzzily optimized from the Nash game perspective. The effectiveness of the control design is illustrated by the simulation control of a unicycle robot.

We consider an artificial swarm mechanical system consisting of multiple agents. The agents are composed of mechanical components. The ideal kinematic performance includes mutual attractions and repulsions. This kinematic performance is embedded into the dynamics by being treated as a constraint. The Udwadia-Kalaba theory is then used to generate the required servo constraint force to assure the constraint is met for the nominal system. The system also includes uncertainty. The uncertainty in the swarm mechanical system is time-varying, whose value falls within a prescribed fuzzy set. For the robust control design, a creative β-measure-based approach is introduced. The robust control guarantees uniform boundedness and uniform ultimate boundedness regardless of the actual value of the uncertainty. For the optimal choice of a control design parameter, a fuzzy-theoretic performance index is introduced. The resulting optimization problem is proven to be tractable, with the global solution to be existent and unique. Furthermore, the analytic expression of this solution is obtained. As a result, the optimal design problem is completely solved. To further demonstrate its effectiveness, we compare the performances of the swarm mechanical system under the robust control and linear-quadratic regulator control through simulation results with an illustrative example.

This study investigates the robust control for uncertain dynamical systems. The uncertainty is (possibly fast) time-varying but bounded. We adopt the fuzzy set theory to describe the uncertainty in the system (hence, called the fuzzy dynamical system). A class of robust controls is proposed based on tunable parameters. The controls are deterministic and are not conventional IF-THEN fuzzy rules based (such as Mamdani type). The proposed controls are able to guarantee deterministic system performance, namely uniform boundedness and uniform ultimate boundedness. In the phase of searching for the optima from the pool of admissible control design parameters, we formulate this as a two-player cooperative game by developing two performance indexes (i.e., the cost functions), each of which is dominated by one tunable parameter (i.e., the player). By the cooperative game theory, we are able to obtain the Pareto-optimality (i.e., the optima of tunable parameters). Simulation results on an electric vehicle motion control problem are presented for demonstration.

In this paper, we propose a fuzzy approach for optimal robust control design of an automotive electronic throttle (ET) system. Compared with the conventional ET control systems, we establish the fuzzy dynamical model of the ET system with parameter uncertainties, nonlinearities, and external disturbances, which may be nonlinear, (possibly fast) time varying. These uncertainties are assumed to be bounded, and the knowledge of the bound only lies within a prescribed fuzzy set. A robust control that is deterministic and is not the usual if-then rules-based control is presented to guarantee the controlled system to achieve the deterministic performance: uniform boundedness and uniform ultimate boundedness. Furthermore, a fuzzy-based system performance index including average fuzzy system performance and control cost is proposed based on the fuzzy information. The optimal design problem associated with the control can then be solved by minimizing the fuzzy-based performance index. With this optimal robust control, the performance of the fuzzy ET system is both deterministically guaranteed and fuzzily optimized.

The control design problem for a class of constrained uncertain systems is considered in this paper. The uncertainty in the system, including unknown system parameters and external disturbance, is nonlinear and time-varying. The bound of the uncertainty is described via a fuzzy set. The states of the system are constrained to be bounded. A one-to-one state transformation is proposed to convert the bounded state constrained system into the unconstrained system. A new robust control scheme is then proposed for the transformed system, which is in deterministic form and not fuzzy if-then rule based. By fuzzy description of the uncertainty bound, the optimal design of the control gain is proposed, which minimizes a fuzzy performance index associated with both the fuzzy system performance and the control effort. The analytic solution to the optimization problem is demonstrated to always exist and be unique. The resulting control can guarantee uniform boundedness and uniform ultimate boundedness of the uncertain system, while minimizing the fuzzy-based performance index. In addition, the state constraint can be always guaranteed.

A fuzzy-based optimal approach to robust control design is proposed for interconnected uncertain systems with mismatching conditions, which were previously unavailable. The interconnected system contains uncertainty, which may include initial conditions, unknown system parameters and input disturbance. The uncertainty bound lies within a prescribed fuzzy set. The system does not satisfy the matching condition. The robust control design in this paper consists of a control scheme design and control gain optimization. A new robust control scheme is first proposed, whose structure is deterministic and not if-then fuzzy rule based. The control gain design problem is then formulated as constrained optimization by fuzzy description of the uncertainty bound, which minimizes the fuzzy system performance and the control effort. It is shown that the global solution to this optimization problem always exists and is unique. The closed-form solution and closed-form minimum cost are presented. The resulting control is able to render the system performance in twofold. First, it guarantees uniform boundedness and uniform ultimate boundedness regardless of the actual value of uncertainty. Second, it minimizes a fuzzy-based performance index. The novelty of this research is a new and carefully orchestrated effort in blending several creative methods and tools; including simultaneous state transformation and control design, dual deterministic and fuzzy features of the performance index, and raised control order; into an integrated framework, resulting in a tractable design problem.

Intelligent transportation has become an essential field of cyber-physical systems. Among various intelligent transportation technologies, the automated highway system (AHS) has its unique advantage of being able to coordinate a platoon of vehicles as a whole unit. The major challenge of building a robust AHS is the nonlinear and (potentially) fast time-varying uncertainty induced by parameter variations and external disturbances. Finally, reflected as the spacing between neighboring vehicles, such uncertainties can be a serious concern for maintaining safety. This paper addresses the problem by proposing a mathematical transformation scheme to bound the spacing error and build a distributed control algorithm on such a basis. The propose algorithm achieves a spacing error satisfying both uniformly boundedness and uniformly ultimate boundedness. Our decentralized algorithm is communication efficient in the sense that it only requires the state information of the preceding car and the acceleration feedback and does not need to communicate with all other cars.

We consider an uncertain mechanical system. The uncertainty includes the initial condition and system parameter. The uncertain parameter in the system is (possibly fast) time varying. The only known information about the uncertainty is that it lies in a fuzzy set. The mechanical system is to follow a set of constraints, which may include many engineering applications, even in the presence of uncertainty. For this purpose, we propose a β-measure for constraint obedience, which reflects how much this constraint is obeyed. Based on a very general Lyapunov function, a control scheme is proposed to render a twofold performance: guaranteed and optimal. In the guaranteed phase, the β-measure is assured to be uniformly bounded and uniformly ultimately bounded, regardless of the actual value of the uncertainty. In the optimal phase, a fuzzy-theoretic-based performance, by which both the “average” β-measure and control effort are considered, is minimized. As a result, the control serves the practical engineering purposes: The mechanical system is guaranteed to follow the desired task with the minimum cost. This paper is part of a unique endeavor to cast both the descriptions of the uncertainty and desired performance index into a fuzzy framework.

We first investigate the fundamental properties of the mechanical system as related to the control design. Then a new robust control is proposed for mechanical systems with fuzzy uncertainty. Fuzzy set theory is used to describe the uncertainty in the mechanical system. The desirable system performance is deterministic. The proposed control is deterministic and is not the usual if-then rules-based. The resulting controlled system is uniformly bounded and uniformly ultimately bounded proved via the Lyapunov minimax approach. The resulting control design is systematic and is able to render the deterministic performance. A mechanical system is chosen for demonstration.

The control problem for a mechanical system to follow a constraint is formulated. The uncertainty is considered in both the system dynamics and constraint. It is bounded, and the knowledge of the bound only lies within a fuzzy set. We propose an adaptive robust control for the mechanical system for approximate constraint-following. Both the mechanical system model and control scheme are deterministic and not if-then heuristic rules-based. An optimal design problem using the fuzzy description of the uncertainty is proposed. We prove that the global solution to the resulting optimization problem always exists and is unique. Under the control, the performance of the mechanical system is both deterministically guaranteed and fuzzily optimized.

Motion control for uncertain mechanical systems is considered. Emphasis is on constraint following, while the constraint may be redundant. The uncertainty in the system, which is (possibly) fast time varying, is bounded. The bound is unknown but is assumed to be within a prescribed fuzzy set. An adaptive robust control is proposed based on the fuzzy property of the uncertainty. Both the mechanical system and the control are deterministic and, hence, not if-then fuzzy rule based. The performance of the resulting controlled system is twofold. First, some deterministic performance is guaranteed, regardless of the value of the uncertainty. Second, the minimization of a fuzzy-based performance index is guaranteed based on an optimal choice of a control design parameter. The effectiveness of the design procedure is illustrated by the simulation control of a digging excavator.

We first investigate the fundamental properties of the mechanical system related to the control design. Then, a new optimal robust control is proposed for mechanical systems with fuzzy uncertainty. Fuzzy set theory is used to describe the uncertainty in the mechanical system. The desirable system performance is deterministic (assuring the bottom line) as well as fuzzy (enhancing the cost consideration). The proposed control is deterministic and is not the usual if-then rule based. The resulting controlled system is uniformly bounded and uniformly ultimately bounded proved via the Lyapunov minimax approach. A performance index (the combined cost, which includes average fuzzy system performance and control effort) is proposed based on the fuzzy information. The optimal design problem associated with the control can then be solved by minimizing the performance index. The resulting control design is systematic and is able to guarantee the deterministic performance, as well as minimizing the cost. In the end, a mechanical system is chosen for demonstration.

We propose a new approach for the control design of fuzzy dynamical systems. The system may contain uncertainty, which includes unknown parameter and input disturbance. The uncertainty lies within a prescribed fuzzy set. The control structure is deterministic, and, hence, not if-then rule-based. The desired controlled system performance includes uniform boundedness and uniform ultimate boundedness. In addition, we propose a quadratic cost index, which reflects the fuzzy system performance. We then formulate a control parameter design problem as a constrained optimization problem. It is proven that the global solution to this problem always exists and is unique. The closed-form solution and the closed-form minimum cost are derived.

The hybrid (position and force) control problem of a robot manipulator has been cast into the framework of control of dynamical systems whose mathematic model contains uncertainties. The uncertainties involved can be due to imperfect modeling, friction, payload change, and external disturbances. Based solely on the bound of these uncertainties, controllers can be constructed. A two-joint SCARA-type robot is discussed as an illustrative example.>

The problem of hybrid (position and force) control of robot manipulators is studied as a control of a dynamical system whose mathematical model contains uncertainties. It is shown that robust control which renders the system globally practically stable can be designed. A simplification of the robust control can be made provided the uncertainty is cone-bounded. This occurs when the modeling error in the Coriolis and centrifugal forces is negligible. Further elaboration on the quadratically bounded uncertainty (which occurs as substantial modeling error in Coriolis and centrifugal forces arises) is investigated. The simplified robust control is shown to be applicable provided the control design parameters are chosen in a prescribed way. A control for a two-joint SCARA type robot is provided as an illustrative example. Excellent system performance is observed.>