Lyapunov Minimax Approach Research Articles

Robust Constraint-Following Control for Uncertain Mechanical Systems With Generalized Udwadia-Kalaba Approach

In this letter, we propose a robust constraint-following control approach for uncertain mechanical systems under both equality and inequality constraints. Particularly, both the global and local inequality constraints are systematically incorporated into the Udwadia-Kalaba (U-K) equation leveraging the diffeomorphism technique, wherein a novel smooth approximation of the local inequality constraints is proposed to address the non-differentiability resulting from its spatiotemporal dependence nature. Based on this development, the generalized U-K equation is mathematically established. With this, we develop a robust constraint-following control strategy to ensure satisfying system performance in the presence of uncertainties and various constraints. Moreover, by Lyapunov minimax approach, the proposed control strategy guarantees both uniform boundedness (UB) and uniform ultimate boundedness (UUB) of the system. Finally, numerical simulations on the lateral motion control of an autonomous vehicle demonstrate the effectiveness of the proposed approach.

Cooperative Game Approach to Robust Control Design for Fuzzy Dynamical Systems.

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.

Stackelberg Game Theory-Based Optimization of High-Order Robust Control for Fuzzy Dynamical Systems

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.

Controlling tractor‐semitrailer vehicles in automated highway systems: Adaptive robust and Lyapunov minimax approach

AbstractIn this paper, we design a coordinated steering and braking control scheme to ensure the tractor‐semitrailer vehicle system is practically stable. The tractor's steering input is designed to realize lane tracking and the semitrailer's differential braking torque is designed to improve the stability of the semitrailer. This control methodology is developed in two steps. Firstly, the expected steering input and the expected differential braking force are derived by introducing a set of given tracking constraints and considering the possible initial condition derivation from the constraints based on the Udwadia‐Kalaba approach. Secondly, we develop an adaptive robust control law to tackle the parameter uncertainty, which may be (possibly) time‐varying, and design the required braking torques, which are the actual inputs, to generate the desired braking forces. Furthermore, this control methodology can deal with nonlinear mechanical systems with both holonomic constraints and nonholonomic constraints. Numerical simulations demonstrate that the control algorithm could guarantee the vehicle dynamics are practically stable and achieve lane following maneuvering.

Optimal Design of Robust Control for Fuzzy Mechanical Systems: Performance-Based Leakage and Confidence-Index Measure

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.

Controlling the differential mobile robot with system uncertainty: Constraint-following and the adaptive robust method

We aim to control the differential mobile robot system to follow a class of pre-specified constraints sufficiently closely in the presence of system uncertainties. The mass and the moment of inertia of the differential mobile robot are considered as the uncertain parameters, which are (possibly) fast time-varying. In the first step, based on Udwadia and Kalaba’s approach, an adaptive robust control scheme is proposed to deal with the system uncertainties. The adaptive law is of leakage type, which can adjust itself based on the tracking error. Using this adaptive robust control scheme, we can obtain the desired armature currents of the two direct current (DC) motors, which are not the real control inputs. In the second step, taking the motor dynamics into account, a Lyapunov Minimax approach for the required actual control inputs (i.e., the input voltages of the two DC motors) is proposed to generate the desired armature currents. This two-step control methodology guarantees uniform boundedness and uniform ultimate boundedness, and renders the system to follow a class of pre-specified constraints approximately.

Stackelberg-Theoretic Approach for Performance Improvement in Fuzzy Systems.

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.

Robust Control Design of Uncertain Mechanical Systems: A Fuzzy Approach

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.

A Novel Optimal Robust Control Design of Fuzzy Mechanical Systems

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.

Optimal Robust Control for Rigid Serial Manipulators: A Fuzzy Approach

AbstractWe first investigate the fundamental properties of rigid serial manipulators as related to the control design. Then, a new optimal robust control is proposed for serial‐link mechanical manipulators with fuzzy uncertainty. Fuzzy set theory is used to describe the uncertainty. The desirable system performance is deterministic (assuring the bottom line) and also fuzzy (enhancing the cost consideration). The proposed control is deterministic and is not the usual if‐then rules‐based control. 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 cylindrical robot manipulator is chosen for demonstration.

Theory and Application of a Novel Optimal Robust Control: A Fuzzy Approach

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) and also fuzzy (enhancing the cost consideration). 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. 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 unique closed-form optimal gain and the cost are explicitly shown. The resulting control design is systematic and is able to guarantee the deterministic performance as well as minimizing the cost. In the end, an example is chosen for demonstration.

Robust uniformly ultimate boundedness control for uncertain switched linear systems

This paper presents uniformly ultimate boundedness (UUB) control design for switched linear systems with parametric uncertainties. Only the possible bound of the uncertainty is needed. Under arbitrary switching laws, a continuous state feedback control scheme is proposed in order to guarantee uniformly ultimate boundedness of every system response within an arbitrary small neighborhood of the zero state. The design techniques are based on common Lyapunov functions and Lyapunov minimax approach.

Control for tractor-semitrailer vehicle systems: a Lyapunov minimax approach

The problem of automated steering and independent braking control for a tractor-semitrailer vehicle system is addressed. The control is designed by combining the input/output linearization and Lyapunov minimax approach. The proposed design achieves, despite of the presence uncertainty, lane following and prevents the jack-knifing phenomenon of trailer yaw motion.