The control algorithm hierarchy of motion control for over-actuated mechanical systems with a redundant set of effectors and actuators commonly includes three levels. First, a high-level motion control algorithm commands a vector of virtual control efforts (i.e. forces and moments) in order to meet the overall motion control objectives. Second, a control allocation algorithm coordinates the different effectors such that they together produce the desired virtual control efforts, if possible. Third, low-level control algorithms may be used to control each individual effector via its actuators. Control allocation offers the advantage of a modular design where the high-level motion control algorithm can be designed without detailed knowledge about the effectors and actuators. Important issues such as input saturation and rate constraints, actuator and effector fault tolerance, and meeting secondary objectives such as power efficiency and tear-and-wear minimization are handled within the control allocation algorithm. The objective of the present paper is to survey control allocation algorithms, motivated by the rapidly growing range of applications that have expanded from the aerospace and maritime industries, where control allocation has its roots, to automotive, mechatronics, and other industries. The survey classifies the different algorithms according to two main classes based on the use of linear or nonlinear models, respectively. The presence of physical constraints (e.g input saturation and rate constraints), operational constraints and secondary objectives makes optimization-based design a powerful approach. The simplest formulations allow explicit solutions to be computed using numerical linear algebra in combination with some logic and engineering solutions, while the more challenging formulations with nonlinear models or complex constraints and objectives call for iterative numerical optimization procedures. Experiences using the different methods in aerospace, maritime, automotive and other application areas are discussed. The paper ends with some perspectives on new applications and theoretical challenges.

We present an algorithm for generating a binary search tree that allows efficient evaluation of piecewise affine (PWA) functions defined on a polyhedral partitioning. This is useful for PWA control approaches, such as explicit model predictive control (MPC), as it allows the controller to be implemented online with small computational effort. The computation time is logarithmic in the number of regions in the PWA function.

Abstract-This paper describes a concept for a collision avoidance system for ships, based on model predictive control. A finite set of alternative control behaviors are generated by varying two parameters: offsets to the guidance course angle commanded to the autopilot, and changes to the propulsion command ranging from nominal speed to full reverse. Using simulated predictions of the trajectories of the obstacles and ship, the compliance with COLREGS and collision hazards associated with each of the alternative control behaviors are evaluated on a finite prediction horizon, and the optimal control behavior is selected. Robustness to sensing error, predicted obstacle behavior, and environmental conditions can be ensured by evaluating multiple scenarios for each control behavior. The method is conceptually and computationally simple and yet quite versatile as it can account for the dynamics of the ship, the dynamics of the steering and propulsion system, forces due to wind and ocean current, and any number of obstacles. Simulations show that the method is effective and can manage complex scenarios with multiple dynamic obstacles and uncertainty associated with sensors and predictions.

Explicit solutions to constrained linear MPC problems can be obtained by solving multi-parametric quadratic programs (mp-QP) where the parameters are the components of the state vector. We study the properties of the polyhedral partition of the state-space induced by the multiparametric piecewise linear solution and propose a new mp-QP solver. Compared to existing algorithms, our approach adopts a different exploration strategy for subdividing the parameter space, avoiding unnecessary partitioning and QP problem solving, with a significant improvement of efficiency.

Abstract-Dynamic Takagi-Sugeno fuzzy models are not always easy to interpret, in particular when they are identified from experimental data. Ideally, it is desirable that a dynamic Takagi-Sugeno fuzzy model should give accurate global nonlinear prediction and at the same time that its local models are close approximations to the local linearizations of the nonlinear dynamic system. The latter is important in many applications where the constituent local models are used individually and aids validation and interpretation of the model considerably. This defines a multi-objective identification problem, namely, the construction of a dynamic model that is a good approximation of both local and global dynamics of the underlying system. While these objectives are often conflicting, it is shown that there exists a close relationship between dynamic Takagi-Sugeno fuzzy models and dynamic linearization when using affine local model structures, which suggests that a solution to the multi-objective identification problem exists. However, it is also shown that the affine local model structure is a highly sensitive parameterization when applied in transient operating regimes, i.e., far away from equilibrium. The reason is essentially that the constant term in the affine local model tends to dominate over the linear term during transients. In addition, it is inherently more difficult to design informative experiments in transient regions compared to near-equilibrium regions. Due to the multi-objective nature of the identification problem studied here, special considerations must be made during model structure selection, experiment design, and identification in order to meet both objectives. Some guidelines for experiment design are suggested and some robust nonlinear identification algorithms are studied. These include constrained and regularized identification and locally weighted identification. Their usefulness in the present context is illustrated by examples.

An algorithm for the construction of an explicit piecewise linear state feedback approximation to nonlinear constrained receding horizon control is given. It allows such controllers to be implemented via an efficient binary tree search, avoiding real-time optimization. This is of significant benefit in applications that requires low real-time computational complexity or software complexity. The method has a priori guarantee of asymptotic stability with region of attraction being a close inner approximation to the stabilizable set. This is achieved by ensuring that the approximation error does not exceed the stability margin.

A wheel slip controller is developed and experimentally tested in a car equipped with electromechanical brake actuators and a brake-by-wire ABS system. A gain scheduling approach is taken, where the vehicle speed is viewed as a slowly time-varying parameter and the model is linearized about the nominal wheel slip. Gain matrices for the different operating conditions are designed using an LQR approach. The stability and robustness of the controller are demonstrated via Lyapunov theory, frequency analysis and experiments using a test vehicle.

Abstract-We present two results on attitude estimation using vector and rate gyro measurements. The first result concerns an observer previously presented by Hamel, Mahony, and Pflimlin, with proven stability results when (i) the reference vectors are stationary; or (ii) the gyro measurements are unbiased. We prove semiglobal stability without either of these assumptions when a parameter projection is added, and convergence from all initial attitudes when using a resetting strategy. The second result is an algorithm for estimation of bias in the body-fixed vector measurements, which is analyzed in combination with the attitude and gyro bias observer.