Efficient Convex Optimization for Linear MPC

This paper presents a continuous time solution to the problem of the design of a relatively optimal control, precisely, a dynamic control which is optimal with respect to a given initial condition and it is stabilizing for any other initial state. This technique provides a drastic reduction of the complexity of the controller and successfully applies to systems in which (constrained) optimality is necessary for some "nominal operation" only. The technique is combined with a pole assignment procedure. It is shown that once the closed-loop poles have been fixed and an optimal trajectory originating from the nominal initial state compatible with these poles is computed, a stabilizing compensator which drives the system along this trajectory can be derived in closed form. There is no restriction about the optimality criterion and the constraints. The optimization is carried out over a finite-dimensional parameterization of the trajectories. However, although the complexity of the compensator is not affected by the choice of cost and constraints, (the compensator dimension is fixed as a design data) it is not in general easy to optimize in the proposed space since, in general, the constraints are infinite dimensional so approximations are necessary. In the case of quadratic optimization with simultaneous pole assignment, an efficient solution based on convex quadratic programming is proposed

Simulating (elastically) deformable models that can collide with each other and with the environment remains a challenging task. The resulting contact problems can be elegantly approached using Lagrange multipliers to represent the unknown magnitude of the response forces. Typical methods construct and solve a Linear Complementarity Problem (LCP) to obtain the response forces. This requires the inverse of the generalized mass matrix, which is generally hard to obtain for deformable-body problems. In this article, we tackle such contact problems by directly solving the Mixed Linear Complementarity Problem (MLCP) and omitting the construction of an LCP matrix. Since a convex quadratic program with linear constraints is equivalent to an MLCP, we propose to use a Conjugate Residual (CR) solver as the backbone of our collision response system. By dynamically updating the set of active constraints, the MLCP with inequality constraints can be solved efficiently. We also propose a simple yet efficient preconditioner that ensures faster convergence. Finally, our approach is faster than existing methods (at the same accuracy), and it allows accurate treatment of friction.

Model predictive control (MPC) has become a practical approach for implementing active safety control strategies in autonomous vehicles (AVs). This paper introduces a Fast Iterative MPC (FI-MPC) framework with rule-based iteration convergence criteria to tackle the significant challenges of control precision and solving time discrepancies between linear and nonlinear MPC algorithms within tracking control strategies for AVs. Differing from linear and nonlinear MPC structures, the core concept of FI-MPC involves incorporating the unmodeled nonlinear dynamics of the vehicle or tires into a linear framework to achieve optimal iterative control solutions. The goal is to mitigate the mismatch between linear and high-fidelity nonlinear dynamics models, thus achieving real-time high-precision tracking control of nonlinear vehicle dynamics under challenging conditions. In addition, rule-based convergence criteria are designed to enhance the iteration mechanism of FI-MPC, further conserving the computational resources of the onboard controller by reducing unnecessary iteration steps. To validate the effectiveness of the FI-MPC algorithm, this study investigates the trajectory tracking problem for a four-wheel-drive AV for a weave test scenario. Co-simulation and hardware-in-the-loop testing results at various speeds confirm that FI-MPC enables high-precision control of nonlinear dynamics while ensuring excellent real-time performance.

This paper presents a geomagnetic-inertial fusion navigation method that addresses heading ambiguity caused by gradient vanishing in mapless navigation scenarios. The key innovation lies in reformulating the fusion navigation problem in underwater environment without prior geomagnetic maps as a convex quadratic programming (CQP) optimization approach, fundamentally transforming the traditional Kalman filter-based sensor fusion into a constrained model predictive control (MPC) scheme. This CQP formulation enables the direct integration of target geomagnetic information while ensuring computational efficiency and global optimality. Crucially, we introduce a dynamic weighting mechanism, driven by an anomaly detector, which allows the system to intelligently select reliable magnetic data and revert to inertial navigation in distorted fields. This ensures high robustness, computational efficiency, and real-time performance. The proposed method facilitates robust multi-destination navigation without requiring precise initial parameter calibration and effectively mitigates cumulative drift during long-range navigation. Comprehensive validations through Monte Carlo simulations and BlueROV2-based experiments demonstrates substantial improvements: 81.9% and 84.8% improvements in circular error probable 50% (CEP<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">50</sub>) and CEP90 accuracy compared to traditional geomagnetic navigation, and 56.1% and 57.4% improvements compared to standalone inertial navigation, respectively.

Constraint removal accelerates model predictive control by detecting inactive constraints at the yet unknown optimal solution and removing them from the online optimization problem. We show in this paper that the number of removed constraints can be increased further by generalizing previously used inactivity criteria. The proposed generalization does not depend on information available at previous time steps, and consequently can also be applied at the initial state. In addition, we provide a detailed analysis of the computational complexity of the proposed variant and of existing constraint removal methods, applied to both active-set (AS) and interior-point (IP) solvers. Finally, we compare the different constraint removal variants in numerical experiments to corroborate the complexity analysis carried out, showing the greatest benefits of the proposed variant, especially with IP solvers.

An MPC Approach to Transient Control of Liquid-Propellant Rocket Engines

In this paper, we propose a measurement-based approach to the real-time economic dispatch (ED). The realtime ED is a widely used market scheduling problem seeking to economically balance electricity system supply and demand and provide locational marginal prices (LMPs) while respecting system reliability requirements. The ED is a convex optimization problem with a linear or quadratic objective, typically the minimization of generator costs or the maximization of social surplus. The constraints capture power balance and network flow capacity limits and are formulated using a linearized power flow model. Our approach utilizes power system sensitivities estimated from phasor measurement unit (PMU) measurements to reformulate the model-based power flow and network flow constraints. The resulting measurement-based real-time ED overcomes the vulnerabilities of the model-based real-time ED. The dispatch instructions and LMPs calculated with our measurement-based real-time ED accurately, and adaptively, reflect real-time system conditions. We illustrate the strengths of the proposed approach via several case studies.

We show that exact solutions to nonlinear MPC problems for a special class of bilinear systems can be obtained by solving a finite number of convex linear MPC problems. The underlying reformulation is realized through exact (input-state) linearization and supported by special biquadratic stage costs. While such stage costs are unusual, we show that they preserve beneficial properties such as positive definiteness and that they allow for meaningful tuning. We further show that the proposed reformulation reveals some interesting insights on the structureof the optimal control law and we illustrate our approach with a numerical example.

Regional MPC with active set updates

In this paper, the problem of control allocation - distribution of control power among redundant control effectors, under a set of constraints - for the inner loop of a re-entry vehicle guidance and control system is studied. Our control allocation scheme extends a previously developed model-predictive algorithm by providing asymptotic tracking of time-varying control input commands. The approach accounts for non-negligible dynamics of the actuators with hard constraints, setting it apart from most existing control allocation schemes, where a static relationship between control surface deflections (actuator outputs) and moments about a three-body axis (plant inputs) is assumed. The approach is readily extended to encompass a variety of linear actuator dynamics without the need for redesign of the overall control allocation scheme, allowing for increased effectiveness of the inner loop in terms of speed of maneuverability. Simulation results, with consideration given toward implementation, are provided for an experimental reusable launch vehicle, and are compared to those of static control allocation schemes.

We examine the problem of designing optimal trajectories to establish connectivity in a network of initially scattered dynamic agents, specifically minimizing the squared integral of the total control effort. The network edges are modeled by proximity relationships between endpoint agents, leading to a dynamic state-dependent network topology. We formulate an optimal control problem with specified initial states, linear dynamics, and a connectivity constraint on the final induced topology. Our approach utilizes the Hamiltonian and resultant Euler-Lagrange equations to restructure the optimal control formulation as a parameter optimization problem based on final agent states. We provide both a heuristic approach and an iterative semidefinite programming (SDP) relaxation to efficiently approximate a solution of the resulting combinatorial optimization problem. Simulation results for double integrator agent dynamics are first provided to demonstrate feasibility for both approaches, and the results are compared with those obtained from exhaustive global search and random sampling. Additional simulation is performed for a specific spacecraft formation problem requiring the design of a stable connected network between a collection of fractionated spacecraft modules to illustrate the practicability and indicate the range of applications of the proposed approaches.

A EROSPACE engineering applications greatly stimulated the development of optimal control theory during the 1950s and 1960s, where the objective was to drive the system states in such a way that some defined cost was minimized. This turned out to have very useful applications in the design of regulators (where some steady state is to be maintained) and in tracking control strategies (where some predetermined state trajectory is to be followed). Among such applications was the problem of optimal flight trajectories for aircraft and space vehicles. Linear optimal control theory in particular has been very well documented and widely applied, where the plant that is controlled is assumed linear and the feedback controller is constrained to be linear with respect to its input. However, the availability of powerful low-cost microprocessors has spurred great advantages in the theory and applications of nonlinear control. The competitive era of rapid technological change, particularly in aerospace exploration, now demands stringent accuracy and cost requirements in nonlinear control systems. This has motivated the rapid development of nonlinear control theory for application to challenging, complex, dynamical real-world problems, particularly those that bear major practical significance in aerospace, marine, and defense industries. Infinite-time horizon nonlinear optimal control (ITHNOC) presents a viable option for synthesizing stabilizing controllers for nonlinear systems by making a state-input tradeoff, where the objective is to minimize the cost given by a performance index. The original theory of nonlinear optimal control dates from the 1960s. Various theoretical and practical aspects of the problem have been addressed in the literature over the decades since. In particular, the continuous-time nonlinear deterministic optimal control problem associated with autonomous (time-invariant) nonlinear regulator systems that are affine (linear) in the controls has been studied by many authors. The long-established theory of optimal control offers quite mature and well-documented techniques for solving this control-affine nonlinear optimization problem, based on dynamic programming or calculus of variations, but their application is generally a very tedious task. Bellman’s dynamic programming approach reduces to solving a nonlinear first-order partial differential equation (PDE), expressed by the Hamilton–Jacobi–Bellman (HJB) equation. The solution to the HJB equation gives the optimal performance/cost value (or storage) function and determines an optimal control in feedback form under some smoothness assumptions. Alternatively, in the classical calculus of variations, optimal control problems can be characterized locally in terms of the Hamiltonian dynamics arising from Pontryagin’s minimum principle. These are the characteristic equations of the HJB PDE, which result in a nonlinear, constrained two-point boundary value problem (TPBVP) that, in general, can only be solved by successive approximation of the optimal control input using iterative numerical techniques for each set of initial conditions. Numerically, even though the nonlinear TPBVP is somewhat easier to solve than the HJBPDE, control signals can only be determined offline and are thus best suited for feedforward control of plants for which the state trajectories are known a priori. Therefore, contrary to the dynamic programming approach, the resultant control law is not generally in feedback form. Open-loop control, however, is sensitive to random disturbances and requires that the initial state be on the optimal trajectory. In contrast, nonlinear optimal feedback has inherent robustness properties (inherent in the sense that it is obtained by ignoring uncertainty and disturbances). The potential difficulty with the HJB approach is that no efficient algorithm is available to solve the PDE when it is nonlinear and the problem dimension is high, making it impossible to derive exact expressions for optimal controls for most nontrivial problems of interest. The optimal can only be computed in special cases, such as linear dynamics and quadratic cost, or very low-dimensional systems. In particular, if the plant is linear time invariant (LTI) and the (infinite-time) performance index is quadratic, then the corresponding HJB equation for this infamous linear-quadratic regulator (LQR) problem reduces to an algebraic Riccati equation (ARE). Contrary to the well-developed and widely applied theory and computational tools for theRiccati equation (for example, see [1]), theHJB equation is difficult, if not impossible, to solve for most practical applications. The exact solution for the optimal control policies is very complex

An effective genetic algorithm approach to the quadratic minimum spanning tree problem

We describe the formal algorithm and numerical applications of a novel convex quadratic programming (QP) strategy for performing the variational minimization that underlies natural resonance theory (NRT). The QP algorithm vastly improves the numerical efficiency, thoroughness, and accuracy of variational NRT description, which now allows uniform treatment of all reference structures at the high level of detail previously reserved only for leading "reference" structures, with little or no user guidance. We illustrate overall QPNRT search strategy, program I/O, and numerical results for a specific application to adenine, and we summarize more extended results for a data set of 338 species from throughout the organic, bioorganic, and inorganic domain. The improved QP-based implementation of NRT is a principal feature of the newly released NBO 7.0 program version. © 2019 Wiley Periodicals, Inc.

Visual servoing enables tracking and grasping of static and moving objects and locating regions of interest. Its implementation requires the coordination of low-level control tasks with high-level supervision and planning. The use of the PID-based manipulation control addresses only the dynamical response and requires the use of optimization-based techniques at the supervisory level. On the contrary, model predictive control (MPC) simplifies the design as it combines both tasks within a single algorithm. However, successful MPC implementation depends on the quality of the internal model utilized by the algorithm. Robots are intrinsically and significantly nonlinear control objects, so their models are inherently complex. This leads to computational problems. Standard solutions are to shorten the MPC horizons or to parallelize calculations. The approach suggested here returns to the problem roots, i.e. to the model. This study proposes simplified models that allow the use of the MPC. General Hammerstein–Wiener-like model is applied to an arm, which is further simplified into the linear dynamics used as the predictive supervisory control over the torque control. The head uses linear dynamics used by the supervisory MPC working with the base servomotors. It is shown that they are computationally efficient and sufficiently accurate. It is shown that the linear MPC controllers, which use the suggested simplified dynamic robot models, can successfully support visual servoing with accurate control. They are compared with standard PID-based structures and show their superiority. The proposed approach is successfully validated using Matlab and Gazebo robot simulator and is ultimately confirmed by experiments on a real robot.