Closed-form Control Law Research Articles

A New Adaptive Explicit Nonlinear Model Predictive Control Design for a Nonlinear MIMO System: An Application to Twin Rotor MIMO System

This paper proposed an adaptive explicit nonlinear model predictive control (AENMPC) technique using multiple estimation models with a convex combination framework [18] for a class of nonlinear MIMO systems. Here, the explicit solution for the control signal is obtained from an optimal performance index which can be formulated without online optimization. In this work, a closed-form control law is developed by approximating the tracking error in the receding horizon by its Taylor series expansion. The control performance of any model-based control technologies explicitly depends on the quality of the unknown system parameters hence an adaptive parameter estimator is used to estimate the system parameter online [16,17]. To ensure the boundedness of the estimated parameter within a predefined compact region, a projection based adaptive law is used [43]. Using an aerodynamic laboratory set-up, known as the twin-rotor multi-input multi-output system (TRMS), the effectiveness of the proposed control algorithm has been verified. The complete state information of the system to the proposed adaptive controller is given from an extended Kalman filter based state observer. The performance of the proposed adaptive control algorithm has been verified successfully in simulations as well as real-time experimental setup of the TRMS model and compared with an existing control approach.

Autonomous Vehicle Convoy Formation Control with Size/Shape Switching for Automated Highways

Today’s semi-autonomous vehicles are gradually moving towards full autonomy. This transition requires developing effective control algorithms for handing complex autonomous tasks. Driving as a group of vehicles, referred to as a convoy, on automated highways is a highly important and challenging task that autonomous driving systems must deal with. This paper considers the control problem of a vehicle convoy modeled with linear dynamics. The convoy formation requirement is presented in terms of a quadratic performance index to minimize. The convoy formation control is formulated as a receding horizon linear-quadratic (LQ) optimal control problem. The receding horizon control law is innovatively defined via the solution to the algebraic Riccati equation. The solution matrix and therefore the receding horizon control law are obtained in the closed-form. A control architecture consisting of four algorithms is proposed to handle formation size/shape switching. The closed-form control law is at the core of these algorithms. Simulation results are provided to justify the models, solutions, and proposed algorithms.

Perimeter Surveillance Based on Set-Invariance

A solution to the perimeter surveillance problem for one intruder and multiple surveillance robots based on set-invariance is presented. The surveillance robots, constrained to move on the perimeter of a polygonal region, intercept the intruders as they cross the perimeter. The proposed closed-form control laws only depend on the maximum speed of the robots and their distances to the endpoints of the line segments that make the sides of the polygon. The presented results allow for groups of robots with members of different characteristics, such as size and maximum speed, to defend polygonal regions. Simulations are used to show the application and effectiveness of the theoretical results.

Modeling and trajectory tracking control of a two-wheeled mobile robot: Gibbs–Appell and prediction-based approaches

SUMMARYThis study deals with the problem of trajectory tracking of wheeled mobile robots (WMR's) under non-holonomic constraints and in the presence of model uncertainties. To solve this problem, the kinematic and dynamic models of a WMR are first derived by applying the recursive Gibbs–Appell method. Then, new kinematics- and dynamics-based multivariable controllers are analytically developed by using the predictive control approach. The control laws are optimally derived by minimizing a pointwise quadratic cost function for the predicted tracking errors of the WMR. The main feature of the obtained closed-form control laws is that online optimization is not needed for their implementation. The prediction time, as a free parameter in the control laws, makes it possible to achieve a compromise between tracking accuracy and implementable control inputs. Finally, the performance of the proposed controller is compared with that of a sliding mode controller, reported in the literature, through simulations of some trajectory tracking maneuvers.

Explicit Reference Governor for Constrained Nonlinear Systems

This technical note introduces a novel control law that dynamically modifies the reference of a pre-compensated nonlinear system to ensure the satisfaction of constraints. This is done by translating the state space constraints into an upper bound on the value of the Lyapunov function and by manipulating the velocity of the applied reference to enforce this bound. The main advantage of this approach is that it systematically provides a closed-form control law able to enforce constraints. The theory is introduced for general nonlinear systems subject to constraints and is then specialized to the large and highly relevant class of nonlinear systems admitting a Lyapunov function lower-bounded by a quadratic form and subject to linear constraints. Numerical simulations show the effectiveness of the proposed method.

Indirect Robust Control of Spacecraft via Optimal Control Solution

A new method is proposed for robust control of spacecraft in proximity orbit servicing operations from an optimal control formulation. It can be shown that the robust control of the nonlinear uncertain system is equivalent to the optimal controlled nominal system with a modified cost functional including uncertainty bounds such that both stabilization and optimal performance can be achieved. The -D technique is utilized to design a closed-form control law, which facilitates the onboard implementation. The pursuer spacecraft is controlled to autonomously approach a target at a specified safe relative position and align its docking port towards the one on the target while keeping their attitude synchronized. Flexible motion, parameter uncertainties, and external disturbance are considered in the control design to enhance the robustness.

Robust Control of PVTOL Aircraft with a Nonlinear Optimal Control Solution

In this paper, a new nonlinear control strategy is proposed for hovering control of planar vertical takeoff and landing (PVTOL) aircraft. The control of this typical nonlinear, nonminimum-phase system is treated as a robust control problem. The robust control design is then converted to an equivalent optimal control design so that both stabilization and performance can be guaranteed. The primary contribution of this work is to provide a closed-form solution to the resultant nonlinear optimal control problem by employing a recently emerging nonlinear control approach, called the θ−D technique. The θ−D technique is based on optimal control theory and derived from an approximate solution to the Hamilton-Jacobi-Bellman equation via a perturbation process. This formulation will not only enhance the system performance but also simplify the design process in that robustness and optimality are integrated in one unified optimal control framework. Furthermore, the closed-form control law is very suitable for onboard implementation. The effectiveness of the proposed method is demonstrated by simulation results and comparison studies with a similar linearization based optimal control formulation.

Feedback Linearizability and Explicit Integrator Forwarding Controllers for Classes of Feedforward Systems

We identify a class of feedforward nonlinear systems that are linearizable by a coordinate change. Then we develop explicit expressions for the Lyapunov-based integrator forwarding recursive procedure of Sepulchre, Jankovic, and Kokotovic, which has its roots in a coordinate transformation proposed by Mazenc and Praly. The explicit expressions that we develop allow us to also find closed-form control laws for several classes of systems that are not feedback linearizable, including some that are in the feedforward form and others that are in what we refer to as the "block-feedforward" form. Performance advantages of Lyapunov-based forwarding controllers over nested saturation controllers have been well illustrated in the literature on examples. The analytical expressions for the Lyapunov functions and the control laws allow us to give quantitative performance bounds.

Closed-form control laws for linear time-varying systems

Closed-form control laws are developed for continuous, linear, time-varying (LTV) systems based on approximate solutions to a receding-horizon control problem. These control laws can be derived in the first- or higher order closed forms. Once obtained, the control laws need no explicit gain-scheduling or online integrations to implement. The notion of practical stability is used, and practical or uniform asymptotic stability of the closed-loop system, depending on conditions imposed on the system, is established.

Approximate receding-horizon control laws for linear time-varying systems

Abstract Closed-form control laws are developed for linear, time-varying systems based on approximate solutions to a receding-horizon control problem. These control laws can be derived in the first- or higher-order forms. Once obtained, the control laws need no iterations or online integration to implement. Stability of the closed-loop system is guaranteed for sufficiently small values of a controller parameter.

Closed-Loop Planning of Multistage Production Under Resource Constraints

The objective of this paper is to address two important issues in multistage production planning: integration of production capacity constraints and responsiveness to demand and supply changes. A natural, although non classical way to tackle these problems is to build a closed-loop production policy based on a state representation of the manufacturing system. The positive invariance approach provides a methodology, a closed-form control law and efficient LP-based algorithms to solve the considered state feedback constrained control problem

Modelling and design of a reversible three-phase switching mode rectifier

A reversible three-phase switching mode rectifier consisting of a four-active-switch and without current sensor is proposed. To achieve controllable power factor, clean sinusoidal input current, adjustable DC output voltage, and bidirectional power flow capability, a closed-form pulse width modulation (PWM) duty cycle function is derived. The popularly used state space averaging technique is extended for modelling the reversible three-phase four-active-switch rectifier. The space vector representation technique is then used to simplify the modelling process without sacrificing accuracy and valid frequency range. Both steady-state and small signal analyses are made. Results show that the proposed closed-form control law for the rectifier can indeed achieve the desired property. Moreover, the rectifier also possesses a bidirectional power flow capability, which is useful for many applications. Finally, a prototype hardware circuit was constructed and experimental results are presented for demonstration. Guidelines for determining the LC parameters and the PI controller gains are described briefly in the text.

Deadbeat control of linear multivariable discrete-time systems

A simple closed-form control law for the deadbeat control of a class of linear multivariable discrete-time systems is presented.

Optimal inner-product angular momentum controllers

An optimization technique is presented to reduce to zero the angular velocities of a spinning object while minimizing a general performance criterion. Closed-form control laws are obtained for a controller structure with a single nonlinear transducer in the feedback loop that employs the square of the Euclidean distance to the origin as an error signal. Consideration is also given to the inverse problem of optimal control. Linearization and generalization of the nonlinear plant are discussed.