Ensuring safe driving and real-time execution is a crucial requirement in the motion planning process for autonomous vehicles. Hence, there is a compelling demand for advanced motion planning algorithms that exhibit effective management of inequality constraints and exceptional computational performance. This paper investigates a diffeomorphism-transformed iterative linear quadratic regulator (DTiLQR) algorithm for addressing constrained motion planning problems in autonomous vehicles with nonlinear dynamics and multiple inequality constraints. With regard to the state and input constraints, a novel state-and-input diffeomorphism is proposed to transform the constrained state/input space into an unconstrained one. Subsequently, these inequality constraints are systematically incorporated into the vehicle dynamics, thereby leading to the newly constructed system in this context. Then, we reformulate and incorporate the obstacle avoidance constraint into the objective function using state diffeomorphism and logarithmic barrier function. With this, the original optimization problem is converted to the unconstrained counterpart, adhering only to the constructed system dynamics. In this sense, featuring a streamlined single-loop architecture (which is essentially different from the dual-loop algorithmic design of existing constrained iLQR algorithms), DTiLQR is used to solve the optimization problem effectively while maintaining motion performance and constraint satisfaction for the resulting optimal trajectory. Ultimately, case studies across various driving situations showcase the effectiveness and exceptional computational efficiency of the proposed DTiLQR algorithm.

This work investigates a game-theoretic optimization approach towards robust control of uncertain dynamical systems with state and input constraints. The uncertainty involved is possibly rapidly time-varying but bounded within a prescribed fuzzy set. For this, the associated fuzzy dynamical system is appropriately established and constructed based on fuzzy set theory. To cope with the bounded state and input constraints, a novel state-and-input diffeomorphism technique is proposed, where a transformed system is formulated such that the prescribed inequality constraints are innovatively merged into the stabilization and trajectory tracking problems. Furthermore, a diffeomorphism-based robust control (DBRC) strategy is developed to ensure the uniform boundedness (UB) and uniform ultimate boundedness (UUB) of the transformed system. Under this proposed control architecture, the constraint satisfaction of the original system is thus always analytically ensured based on the rigorous properties of the diffeomorphism technique. The resulting control parameter optimization problem then has to take into account the multiple considerations (and compromise) amongst the factors of the steady-state performance; the finite convergence time; and the control effort. For this, a two-player Nash game is formulated and solved in an effective manner. The Nash equilibrium is obtained and the existence of the solution is also proved theoretically. With this methodology, and with the resulting attainment of the desired Nash equilibrium, the attendant outcome of superior system performance is achieved. Finally, numerical simulations on a steer-by-wire (SBW) system demonstrate the effectiveness of the proposed approach.

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 develops a diffeomorphism-based robust bounded control (DRBC) for the permanent magnet linear synchronous motor system subject to inequality constraints (i.e., bounded input and position constraints) and uncertainties. The uncertainties, including parameter uncertainties and external disturbances, are potentially nonlinear and fast time varying. The bound of the uncertainty is described by a fuzzy set. To overcome the bounded input constraint, a robust bounded control is proposed based on a novel input diffeomorphism scheme, which is in a deterministic form and not if–then rule based. Furthermore, to overcome the bounded position constraint, a transformed system is formulated by a state diffeomorphism scheme, which transforms the bounded state-constrained system to an unconstrained one. Thus, the output of the controlled system can be restricted to a prescribed range. The DRBC guarantees both the uniform boundedness and the uniform ultimate boundedness of the transformed system. A fuzzy performance index, which combines the steady-state performance (the average fuzzy performance) and control effort, is then established based on the fuzzy description of uncertainty. As a result, the design parameter optimization can be solved by minimizing the performance index. Experimental results demonstrate that the DRBC is of superior tracking performance and robustness without violating the prescribed inequality constraints.

While designing control for a class of underactuated mechanical systems (UMSs), the uncertainty and the prescribed nonholonomic tracking trajectories should be taken into consideration. Uncertainty considered in this article is time varying and bounded, and the bound of uncertainty is described using the fuzzy set theory, namely, fuzzy UMSs. An analytical dynamics-based view is taken in which the prescribed tracking trajectories are viewed as servo constraints which can be linear, nonlinear, holonomic, and nonholonomic. Using this view, a novel closed form solution of adaptive robust control is found with leakage type adaptive law to guarantee deterministic system performance, including uniform boundedness and uniform ultimate boundedness. In order to find the optimal dual gain parameters of the designed control, a two-player cooperative game is proposed for which the Pareto optimality can always be guaranteed. The effectiveness of the proposed control is shown through numerical simulation of a two-wheeled inverted pendulum vehicle.

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 investigates the trajectory tracking problem of underactuated mechanical systems (UMSs) with companion nonholonomic servo constraints and uncertainties. For such motion tasks, the existing approaches in the literature attempt unrealistically to furnish a reliable closed-form solution, rendering it difficult to have high-quality tracking performance with theoretical support. In addition, the uncertainties typically pose substantial difficulty in the controller synthesis. Here, by invoking the methodology of fuzzy sets, the uncertainties in the UMSs are elegantly represented; and with this, the formulation becomes such that a closer link between the uncertain dynamical model of the UMSs and the real world is established. The reference trajectories are regarded appropriately as servo constraints, and subsequently an adaptive robust controller is designed to accomplish the trajectory tracking task from a specific viewpoint of servo constraint tracking. As supported by rigorous proofs, the closed-form solution to the proposed controller is obtained with guaranteed Lyapunov stability. Leveraging on the closed-form solution, the global optimizer to the controller gain parameter can be determined, which is shown to exhibit several important properties including existence and uniqueness. Finally, a numerical example is presented to demonstrate the effectiveness of the designed method.

This article proposes a novel Stackelberg-game-oriented optimal control approach to address the bounded constraint-following control problem for uncertain mechanical systems. First, the uncertainties (possibly fast time-varying) in the system are assumed to be bounded with an unknown boundary, which lies in a specified fuzzy evidence number. In practical engineering, bounded system performance is always demanded, such as the inequality constraint. A diffeomorphism transformation approach is proposed to transform the constrained system into a restructured one satisfying the bounded constraint. Second, we propose an adaptive robust control oriented by the constraint-following control to render the restructured system to follow the specified constraints accurately with deterministic performance (guaranteeing uniform boundedness and uniform ultimate boundedness). The self-adjusting adaptive law (leakage-type) can compensate for the uncertainties and avoid overcompensation. Third, a Stackelberg-game-oriented optimization approach is proposed to obtain the optimal control parameters based on the fuzzy evidence theory. In the optimization approach, the two control parameters$\sigma$and$\varepsilon$are considered as two players with respective cost functions related to system performance and control cost. Furthermore, the optimization problem is solved by obtaining the Stackelberg strategy, which is proved to exist in analytic form. Ultimately, the permanent magnet synchronous linear motor system simulation is presented to show the design process and the excellent performance of the proposed optimal control scheme.

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.

For a specific class of under-actuated mechanical systems, non-holonomic servo constraints and model uncertainties are usually encountered. For such systems, this paper investigates the design of an adaptive robust controller with parameter optimization. A tighter link between the fuzzy set theory and the control of UMSs is bridged appropriately. Based on the UMSs with fuzzy information, an adaptive robust control method is then designed, and an analytical solution of the control input is determined, even if the servo constraints are non-holonomic. Furthermore, a concomitant parameter in the designed controller is analyzed, and a feasible controller admitting the optimal performance can be determined by minimizing a predefined performance index, such that the deterministic system performance can be ensured to be at a satisfying level. As supported by rigorous proofs, the existence and the uniqueness of the global solution to the optimization problem are presented. Finally, a numerical experiment is implemented to demonstrate the effectiveness of the proposed control design methodology.

This article addresses the problems of the adaptive robust control design and the parameters optimization for a class of underactuated mechanical systems. We describe the uncertainty by using the fuzzy set theory to form the fuzzy dynamical systems. A novel adaptive robust control with two tunable gain parameters is designed based on the formed system, and the nonholonomic servo constraints are also considered in the control design. By employing the principle of the Nash game theory, we establish an optimization method for the two gain parameters, and the Nash equilibrium orients the optimal values of the two gain parameters. Finally, the effectiveness of the designed optimal control is shown by a numerical simulation.

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.

To address the problem of the non-holonomic reference trajectory in trajectory tracking controller design for mechanical systems, we design a novel RBF-Udwadia controller in this paper. The framework of the Udwadia controller is employed to design the controller, which helps to deal with the non-holonomic reference trajectory. The radial basis function (RBF) neural network is employed to approximately model the uncertainty. The stability of the designed controller is analyzed by the Lyapunov method, and the effectiveness of the designed controller is verified by a numerical experiment.

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.

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.

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.

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.

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.