Modeling error or external disturbances can severely degrade the performance of Model Predictive Control (MPC) in real-world scenarios. Robust MPC (RMPC) addresses this limitation by optimizing over feedback policies but at the expense of increased computational complexity. Tube MPC is an approximate solution strategy in which a robust controller, designed offline, keeps the system in an invariant tube around a desired nominal trajectory, generated online. Naturally, this decomposition is suboptimal, especially for systems with changing objectives or operating conditions. In addition, many tube MPC approaches are unable to capture state-dependent uncertainty due to the complexity of calculating invariant tubes, resulting in overly-conservative approximations. This work presents the Dynamic Tube MPC (DTMPC) framework for nonlinear systems where both the tube geometry and open-loop trajectory are optimized simultaneously. By using boundary layer sliding control, the tube geometry can be expressed as a simple relation between control parameters and uncertainty bound; enabling the tube geometry dynamics to be added to the nominal MPC optimization with minimal increase in computational complexity. In addition, DTMPC is able to leverage state-dependent uncertainty to reduce conservativeness and improve optimization feasibility. DTMPC is demonstrated to robustly perform obstacle avoidance and modify the tube geometry in response to obstacle proximity.
A number of works have been published on the stability, feasibility, and performance of linear tube MPC @cite_10 @cite_13 @cite_8 . While this is an effective strategy to achieve robustness, decoupling the nominal MPC problem and controller design is suboptimal. Rakovi @cite_17 showed that the region of attraction can be enlarged by parameterizing the problem with the open-loop trajectory and tube size. The authors presented the homothetic tube MPC (HTMPC) algorithm that treated the state and control tubes as homothetic copies of a fixed cross-section shape, enabling the problem to be parameterized by the tube’s centers (i.e., open-loop trajectory) and a cross-section scaling factor. The work was extended to tubes with varying shapes, known as elastic tube MPC (ETMPC), but at the expense of computational complexity @cite_12 . Both HTMPC and ETMPC possess strong theoretical properties and have the potential to significantly improve performance but a nonlinear extension has yet to be developed.
@cite_8 @cite_10 @cite_13 @cite_12 @cite_17
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In this paper, we present a novel constraint tightening approach for nonlinear robust model predictive control (MPC). This approach uses a simple constructive constraint tightening based on growing tubes. Contrary to other approaches, we require no complex offline computations to obtain a stabilizing control law. Instead, we consider the notion of incremental stabilizability and design tubes based on an estimate of the achievable exponential decay rate. In addition, we show how this tightening can be used as an ad-hoc modification to improve the robustness of MPC without terminal constraints. We study the system theoretic properties of the resulting closed-loop system, including bounds on the region of attraction and the minimal robust positively invariant (RPI) set. Within an MPC framework without terminal constraints, the proposed constraint tightening leads to a nonlinear robust controller without complex design procedures, which makes it appealing for practical applications. This paper introduces elastic tube model predictive control (MPC) synthesis. The proposed framework is a natural generalization of the rigid and homothetic tube MPC design methods. The cross-sections of the employed state and control tubes are allowed to change more elastically, while the local component of the tubes control policy is permitted to take a more general form. The related stabilizing terminal conditions are also adequately generalized in order to take advantage of more flexible tubes and tubes control policy parameterizations. These novel features result in an improved tube MPC at the cost of a manageable increase in computational complexity. This paper addresses the attraction domain of model-based predictive control (MPC) for nonlinear systems with control input and state constraints. Based on a stability condition of nonlinear MPC, a method to determine the terminal weighting term in the performance index and the terminal stabilising control law to maximise the domain of attraction of the nonlinear MPC is proposed. The problem of maximisation of the attraction region is recast as a well-defined optimisation problem. By an LMI based optimisation approach, the terminal weighting item and fictitious terminal stabilising control law axe optimised to enlarge the attraction domain and hence the feasibility domain of the nonlinear MPC method. The proposed method is illustrated by a numerical example and favourably compared with existing results. This paper focuses on the design of a tube-based Model Predictive Control law for the control of constrained mobile robots in off-road conditions with longitudinal slip while ensuring robustness and stability. A time-varying trajectory tracking error model is used, where uncertainties are assumed to be bounded and additive. The robust tube-based MPC is compared with other motion control techniques through simulation and physical experiments. These tests show the satisfactory behavior of the presented control strategy. We consider a comparison of active set vs. interior point strategies for the solution of receding time horizon problems in nonlinear model predictive control (NMPC). For this study we consider a control algorithm where we form quadratic programs (QPs) in each time horizon by linearizing the model. We also ignore second order information on the model and constraints. This approach can be viewed as a direct nonlinear extension of MPC with linear models and is easily tailored to include stabilizing constraints. Using this framework we consider the application of three active set strategies as well as interior point methods applied to both the NMPC and the QP subproblem. The first two active set methods (QPOPT and and QKWIK) are general purpose solvers that have been incorporated into SQP algorithms previously, while the third is a Schur complement approach that can easily exploit the sparse structure of the KKT matrix in MPC. Abstract A robust model predictive control (RMPC) approach for linear systems with bounded state-dependent uncertainties is proposed. Such uncertainties can arise from unmodeled non-linearities or external disturbances, for example. By explicitly considering the state dependency of the uncertainty sets in the RMPC approach, it is shown how closed-loop performance can be improved over existing approaches that consider worst-case uncertainty. Being able to handle state-dependent uncertainties is particularly relevant in learning-based MPC where the system model is learned from data and confidence in the model typically varies over the state space. The efficacy of the proposed approach for learning-based RMPC is illustrated with a numerical example, where uncertainty sets are obtained from data using Gaussian Process regression. The robust model predictive control for constrained linear discrete time systems is solved through the development of a homothetic tube model predictive control synthesis method. The method employs several novel features including a more general parameterization of the state and control tubes based on homothety and invariance, a more flexible form of the terminal constraint set and a relaxation of the controlled dynamics of the sets that define the state and control tubes. Under natural assumptions, the proposed method is computationally efficient and it induces strong system theoretic properties. Abstract Visual tracking algorithms based on structured output support vector machine (SOSVM) have demonstrated excellent performance. However, sampling methods and optimization strategies of SOSVM undesirably increase the computational overloads, which hinder real-time application of these algorithms. Moreover, due to the lack of high-dimensional features and dense training samples, SOSVM-based algorithms are unstable to deal with various challenging scenarios, such as occlusions and scale variations. Recently, visual tracking algorithms based on discriminative correlation filters (DCF), especially the combination of DCF and features from deep convolutional neural networks (CNN), have been successfully applied to visual tracking, and attains surprisingly good performance on recent benchmarks. The success is mainly attributed to two aspects: the circular correlation properties of DCF and the powerful representation capabilities of CNN features. Nevertheless, compared with SOSVM, DCF-based algorithms are restricted to simple ridge regression which has a weaker discriminative ability. In this paper, a novel circular and structural operator tracker (CSOT) is proposed for high performance visual tracking, it not only possesses the powerful discriminative capability of SOSVM but also efficiently inherits the superior computational efficiency of DCF. Based on the proposed circular and structural operators, a set of primal confidence score maps can be obtained by circular correlating feature maps with their corresponding structural correlation filters. Furthermore, an implicit interpolation is applied to convert the multi-resolution feature maps to the continuous domain and make all primal confidence score maps have the same spatial resolution. Then, we exploit an efficient ensemble post-processor based on relative entropy, which can coalesce primal confidence score maps and create an optimal confidence score map for more accurate localization. The target is localized on the peak of the optimal confidence score map. Besides, we introduce a collaborative optimization strategy to update circular and structural operators by iteratively training structural correlation filters, which significantly reduces computational complexity and improves robustness. Experimental results demonstrate that our approach achieves state-of-the-art performance in mean AUC scores of 71.5 and 69.4 on the OTB2013 and OTB2015 benchmarks respectively, and obtains a third-best expected average overlap (EAO) score of 29.8 on the VOT2017 benchmark. Controller design faces a trade-off between robustness and performance, and the reliability of linear controllers has caused many practitioners to focus on the former. However, there is renewed interest in improving system performance to deal with growing energy constraints. This paper describes a learning-based model predictive control (LBMPC) scheme that provides deterministic guarantees on robustness, while statistical identification tools are used to identify richer models of the system in order to improve performance; the benefits of this framework are that it handles state and input constraints, optimizes system performance with respect to a cost function, and can be designed to use a wide variety of parametric or nonparametric statistical tools. The main insight of LBMPC is that safety and performance can be decoupled under reasonable conditions in an optimization framework by maintaining two models of the system. The first is an approximate model with bounds on its uncertainty, and the second model is updated by statistical methods. LBMPC improves performance by choosing inputs that minimize a cost subject to the learned dynamics, and it ensures safety and robustness by checking whether these same inputs keep the approximate model stable when it is subject to uncertainty. Furthermore, we show that if the system is sufficiently excited, then the LBMPC control action probabilistically converges to that of an MPC computed using the true dynamics. More than 15 years after model predictive control (MPC) appeared in industry as an effective means to deal with multivariable constrained control problems, a theoretical basis for this technique has started to emerge. The issues of feasibility of the on-line optimization, stability and performance are largely understood for systems described by linear models. Much progress has been made on these issues for non-linear systems but for practical applications many questions remain, including the reliability and efficiency of the on-line computation scheme. To deal with model uncertainty ‘rigorously’ an involved dynamic programming problem must be solved. The approximation techniques proposed for this purpose are largely at a conceptual stage. Among the broader research needs the following areas are identified: multivariable system identification, performance monitoring and diagnostics, non-linear state estimation, and batch system control. Many practical problems like control objective prioritization and symptom-aided diagnosis can be integrated systematically and effectively into the MPC framework by expanding the problem formulation to include integer variables yielding a mixed-integer quadratic or linear program. Efficient techniques for solving these problems are becoming available. In this paper the concept of Hybrid Parameter-Varying Model Predictive Control (HPV-MPC) is applied for autonomous vehicle steering. Parameter-varying in the MPC context means that a prediction model with non-constant, parameter-varying system matrices is employed. In the investigated scenarios, the displacement of a car on an icy road due to a side wind gust shall be mitigated, and a double lane-change maneuver shall be performed autonomously. In order to explore a possible reduction of online computations and the inherent degradation of control performance, the nonlinear model of the lateral dynamics is approximated in various ways. A comparison between controllers using prediction models varying from the full nonlinear model, as an indication for the maximal achievable performance, to a linear model was performed. Particular emphasis was put on the hybrid parameter-varying prediction model, to investigate their potential in terms of computational effort and control performance. This paper proposes a simple method for estimating dense and accurate optical flow field. It revitalizes an early idea of piecewise parametric flow model. A key innovation is that, we fit a flow field piecewise to a variety of parametric models, where the domain of each piece (i.e., each piece's shape, position and size) is determined adaptively, while at the same time maintaining a global inter-piece flow continuity constraint. We achieve this by a multi-model fitting scheme via energy minimization. Our energy takes into account both the piecewise constant model assumption and the flow field continuity constraint, enabling the proposed method to effectively handle both homogeneous motions and complex motions. The experiments on three public optical flow benchmarks (KITTI, MPI Sintel, and Middlebury) show the superiority of our method compared with the state of the art: it achieves top-tier performances on all the three benchmarks. The maximum a posteriori (MAP) configuration of binary variable models with sub-modular graph-structured energy functions can be found efficiently and exactly by graph cuts. Max-product belief propagation (MP) has been shown to be suboptimal on this class of energy functions by a canonical counterexample where MP converges to a suboptimal fixed point (Kulesza & Pereira, 2008). In this work, we show that under a particular scheduling and damping scheme, MP is equivalent to graph cuts, and thus optimal. We explain the apparent contradiction by showing that with proper scheduling and damping, MP always converges to an optimal fixed point. Thus, the canonical counterexample only shows the suboptimality of MP with a particular suboptimal choice of schedule and damping. With proper choices, MP is optimal. We focus on a replanning scenario for quadrotors where considering time efficiency, non-static initial state and dynamical feasibility is of great significance. We propose a real-time B-spline based kinodynamic (RBK) search algorithm, which transforms a position-only shortest path search (such as A * and Dijkstra) into an efficient kinodynamic search, by exploring the properties of B-spline parameterization. The RBK search is greedy and produces a dynamically feasible time-parameterized trajectory efficiently, which facilitates non-static initial state of the quadrotor. To cope with the limitation of the greedy search and the discretization induced by a grid structure, we adopt an elastic optimization (EO) approach as a post-optimization process, to refine the control point placement provided by the RBK search. The EO approach finds the optimal control point placement inside an expanded elastic tube which represents the free space, by solving a Quadratically Constrained Quadratic Programming (QCQP) problem. We design a receding horizon replanner based on the local control property of B-spline. A systematic comparison of our method against two state-of-the-art methods is provided. We integrate our replanning system with a monocular vision-based quadrotor and validate our performance onboard. This work addresses the problem of motion planning among obstacles for quadrotor platforms under external disturbances and with model uncertainty. A novel Nonlinear Model Predictive Control (NMPC) optimization technique is proposed which incorporates specified uncertainties into the planned trajectories. At the core of the procedure lies the propagation of model parameter uncertainty and initial state uncertainty as high-confidence ellipsoids in pose space. The quadrotor trajectories are then computed to avoid obstacles by a required safety margin, expressed as ellipsoid penetration while minimizing control effort and achieving a user-specified goal location. Combining this technique with online model identification results in robust obstacle avoidance behavior. Experiments in outdoor scenarios with virtual obstacles show that the quadrotor can avoid obstacles robustly, even under the influence of external disturbances. In this paper, we study resource allocation for a multicarrier-based cognitive radio (CR) network. More specifically, we investigate the problem of secondary users’ energy-efficiency (EE) maximization problem under secondary total power and primary interference constraints. First, assuming cooperation among the secondary base stations (BSs), a centralized approach is considered to solve the EE optimization problem for the CR network where the primary and secondary users are using either orthogonal frequency-division multiplexing (OFDM) or filter bank based multicarrier (FBMC) modulations. We propose an alternating-based approach to solve the joint power-subcarrier allocation problem. More importantly, in the first place, subcarriers are allocated using a heuristic method for a given feasible power allocation. Then, we conservatively approximate the nonconvex power control problem and propose a joint successive convex approximation-Dinkelbach algorithm (SCADA) to efficiently obtain a solution to the nonconvex power control problem. The proposed algorithm is shown to converge to a solution that coincides with the stationary point of the original nonconvex power control subproblem. Moreover, we propose a dual decomposition-based decentralized version of the SCADA. Second, under the assumption of no cooperation among the secondary BSs, we propose a fully distributed power control algorithm from the perspective of game theory. The proposed algorithm is shown to converge to a Nash-equilibrium (NE) point. Moreover, we propose a sufficient condition that guarantees the uniqueness of the achieved NE. Extensive simulation analyses are further provided to highlight the advantages and demonstrate the efficiency of our proposed schemes. This paper proposes a novel two-stage optimization method for robust model predictive control (RMPC) with Gaussian disturbance and state estimation error. Since the disturbance is unbounded, it is impossible to achieve zero probability of violating constraints. Our goal is to optimize the expected value of an objective function while limiting the probability of violating any constraints over the planning horizon (joint chance constraint). Prior arts include ellipsoidal relaxation approach [1] and particle control [2], but the former yields very conservative result and the latter is computationally intensive. Our approach divide the optimization problem into two stages; the upper-stage that optimizes risk allocation, and the lower-stage that optimizes control sequence with tightened constraints. The lower-stage is a regular convex optimization, such as linear programming or quadratic programming. The upper-stage is also convex, but the objective function is not always differentiable. We developed a fast descent algorithm for the upper-stage called iterative risk allocation (IRA), which yield much smaller suboptimality than ellipsoidal relaxation method while achieving a substantial speedup compared to and particle control. This paper presents the design of a variable structure-like tracking controller for a mobile robot system. The controller provides robustness with regard to bounded disturbances in the kinematic model. Through the use of a dynamic oscillator and a Lyapunov-based stability analysis, we demonstrate that the position and orientation tracking errors exponentially converge to a neighborhood about zero that can be made arbitrarily small (i.e., the controller ensures that the tracking error is globally uniformly ultimately bounded (GUUB)). In addition, we illustrate how the proposed tracking controller can also be utilized to achieve GUUB regulation to an arbitrary desired setpoint. An extension is also provided that illustrates how a smooth, time-varying control law can be utilized to achieve setpoint regulation despite parametric uncertainty in the kinematic model. Simulation results are presented to demonstrate the performance of the proposed controllers. @S0022-0434 00!00504-9# In this paper, we present a convex optimization problem to generate Center of Mass (CoM) and momentum trajectories of a walking robot, such that the motion robustly satisfies the friction cone constraints on uneven terrain. We adopt the Contact Wrench Cone (CWC) criterion to measure a robot's dynamical stability, which generalizes the venerable Zero Moment Point (ZMP) criterion. Unlike the ZMP criterion, which is ideal for walking on flat ground with unbounded tangential friction forces, the CWC criterion incorporates non-coplanar contacts with friction cone constraints. We measure the robustness of the motion using the margin in the Contact Wrench Cone at each time instance, which quantifies the capability of the robot to instantaneously resist external force torque disturbance, without causing the foot to tip over or slide. For pre-specified footstep location and time, we formulate a convex optimization problem to search for robot linear and angular momenta that satisfy the CWC criterion. We aim to maximize the CWC margin to improve the robustness of the motion, and minimize the centroidal angular momentum (angular momentum about CoM) to make the motion natural. Instead of directly minimizing the non-convex centroidal angular momentum, we resort to minimizing a convex upper bound. We show that our CWC planner can generate motion similar to the result of the ZMP planner on flat ground with sufficient friction. Moreover, on an uneven terrain course with friction cone constraints, our CWC planner can still find feasible motion, while the outcome of the ZMP planner violates the friction limit. We present a novel, general-purpose Model-Predictive Control (MPC) algorithm that we call Control Particle Belief Propagation (C-PBP). C-PBP combines multimodal, gradient-free sampling and a Markov Random Field factorization to effectively perform simultaneous path finding and smoothing in high-dimensional spaces. We demonstrate the method in online synthesis of interactive and physically valid humanoid movements, including balancing, recovery from both small and extreme disturbances, reaching, balancing on a ball, juggling a ball, and fully steerable locomotion in an environment with obstacles. Such a large repertoire of movements has not been demonstrated before at interactive frame rates, especially considering that all our movement emerges from simple cost functions. Furthermore, we abstain from using any precomputation to train a control policy offline, reference data such as motion capture clips, or state machines that break the movements down into more manageable subtasks. Operating under these conditions enables rapid and convenient iteration when designing the cost functions.