Showing papers in "Optimal Control Applications & Methods in 2015"

Optimization‐based autonomous racing of 1:43 scale RC cars

Abstract: Summary This paper describes autonomous racing of RC race cars based on mathematical optimization. Using a dynamical model of the vehicle, control inputs are computed by receding horizon based controllers, where the objective is to maximize progress on the track subject to the requirement of staying on the track and avoiding opponents. Two different control formulations are presented. The first controller employs a two-level structure, consisting of a path planner and a nonlinear model predictive controller (NMPC) for tracking. The second controller combines both tasks in one nonlinear optimization problem (NLP) following the ideas of contouring control. Linear time varying models obtained by linearization are used to build local approximations of the control NLPs in the form of convex quadratic programs (QPs) at each sampling time. The resulting QPs have a typical MPC structure and can be solved in the range of milliseconds by recent structure exploiting solvers, which is key to the real-time feasibility of the overall control scheme. Obstacle avoidance is incorporated by means of a high-level corridor planner based on dynamic programming, which generates convex constraints for the controllers according to the current position of opponents and the track layout. The control performance is investigated experimentally using 1:43 scale RC race cars, driven at speeds of more than 3 m/s and in operating regions with saturated rear tire forces (drifting). The algorithms run at 50 Hz sampling rate on embedded computing platforms, demonstrating the real-time feasibility and high performance of optimization-based approaches for autonomous racing. Copyright © 2014 John Wiley & Sons, Ltd.

A ph mesh refinement method for optimal control

Abstract: Summary A mesh refinement method is described for solving a continuous-time optimal control problem using collocation at Legendre–Gauss–Radau points. The method allows for changes in both the number of mesh intervals and the degree of the approximating polynomial within a mesh interval. First, a relative error estimate is derived based on the difference between the Lagrange polynomial approximation of the state and a Legendre–Gauss–Radau quadrature integration of the dynamics within a mesh interval. The derived relative error estimate is then used to decide if the degree of the approximating polynomial within a mesh should be increased or if the mesh interval should be divided into subintervals. The degree of the approximating polynomial within a mesh interval is increased if the polynomial degree estimated by the method remains below a maximum allowable degree. Otherwise, the mesh interval is divided into subintervals. The process of refining the mesh is repeated until a specified relative error tolerance is met. Three examples highlight various features of the method and show that the approach is more computationally efficient and produces significantly smaller mesh sizes for a given accuracy tolerance when compared with fixed-order methods. Copyright © 2014 John Wiley & Sons, Ltd.

Autogenerating microsecond solvers for nonlinear MPC: A tutorial using ACADO integrators

Abstract: Summary Nonlinear model predictive control (NMPC) allows one to explicitly treat nonlinear dynamics and constraints. To apply NMPC in real time on embedded hardware, online algorithms as well as efficient code implementations are crucial. A tutorial-style approach is adopted in this article to present such algorithmic ideas and to show how they can efficiently be implemented based on the ACADO Toolkit from MATLAB (MathWorks, Natick, MA, USA). Using its code generation tool, one can export tailored Runge–Kutta methods—explicit and implicit ones—with efficient propagation of their sensitivities. The article summarizes recent research results on autogenerated integrators for NMPC and shows how they allow to formulate and solve practically relevant problems in only a few tens of microseconds. Several common NMPC formulations can be treated by these methods, including those with stiff ordinary differential equations, fully implicit differential algebraic equations, linear input and output models, and continuous output independent of the integration grid. One of the new algorithmic contributions is an efficient implementation of infinite horizon closed-loop costing. As a guiding example, a full swing-up of an inverted pendulum is considered. Copyright © 2014 John Wiley & Sons, Ltd.

Optimal preview control: A review

Abstract: Summary The field of preview control has attracted many researchers for its applications in guidance of autonomous vehicles, robotics, and process control, as this field is well suited for use in design of systems that have reference signals known a priori. The paper presents the efforts of various researchers in the field of preview control. The literature available in this field, since 1966, is categorized based on formulation, method domain, solution approach, and objective. The preview control problem is formulated and solved using classical time-domain optimal control design tools for under water vehicle model. The key observation obtained from the discussions shows the enormous scope of work available in the field of preview control. Copyright © 2014 John Wiley & Sons, Ltd.

Cooperative distributed model predictive control for wind farms

Abstract: SUMMARY This paper focuses on cooperative distributed model predictive control (MPC) of wind farms, where the farms respond to active power control commands issued by the transmission system operator. A distributed MPC scheme is proposed, which aims at satisfying the requirements imposed by the grid code while minimizing the farm-wide mechanical structure fatigue. The distributed MPC control law is defined by a global finite-horizon optimal control problem, which is solved at every time step by distributed optimization. The computational approach is completely distributed, that is, every turbine evaluates its own globally optimal input by considering local measurements and communicating to neighboring turbines only. Two MPC versions are compared, in the first of which the farm-wide power output constraint is implemented as a hard constraint, whereas in the second, it is implemented as a soft constraint. As for distributed optimization methods, the alternating direction method of multipliers as well as a dual decomposition scheme based on fast gradient updates are compared. The performance of the proposed distributed MPC controller, as well as the performance of the distributed optimization methods used for its operation, are compared in the simulation on four exemplary scenarios. The results of the simulations imply that the use of cooperative distributed MPC in wind farms is viable both from a performance and from a computational viewpoint. Copyright © 2014 John Wiley & Sons, Ltd.

Costate approximation in optimal control using integral Gaussian quadrature orthogonal collocation methods

Abstract: Summary Two methods are presented for approximating the costate of optimal control problems in integral form using orthogonal collocation at Legendre–Gauss (LG) and Legendre–Gauss–Radau (LGR) points. It is shown that the derivative of the costate of the continuous-time optimal control problem is equal to the negative of the costate of the integral form of the continuous-time optimal control problem. Using this continuous-time relationship between the differential and integral costate, it is shown that the discrete approximations of the differential costate using LG and LGR collocation are related to the corresponding discrete approximations of the integral costate via integration matrices. The approach developed in this paper provides a way to approximate the costate of the original optimal control problem using the Lagrange multipliers of the integral form of the LG and LGR collocation methods. The methods are demonstrated on two examples where it is shown that both the differential and integral costate converge exponentially as a function of the number of LG or LGR points. Copyright © 2014 John Wiley & Sons, Ltd.

Optimal low‒thrust orbit transfers with eclipsing

Abstract: Summary By choosing the optimal steering history of a spacecraft, it is possible to maximize the mass delivered from a park orbit to a mission orbit. A low‒thrust orbit transfer that models coasts when passing through the Earth's shadow can be formulated as a large‒scale optimal control problem with many distinct phases. This paper presents a technique that constructs an initial guess using a receding horizon algorithm. A series of large‒scale multiphase optimal control problems are then solved to refine the phase structure of the problem. The final optimal solution incorporates high fidelity physics and mesh refinement techniques within a large sparse nonlinear programming approach.

Finite-time control and L1-gain analysis for positive switched systems

Abstract: Summary This paper is concerned with finite-time control and L1-gain analysis of positive switched systems. First, by using the multiple linear copositive Lyapunov functions approach, a sufficient condition for the finite-time stability of autonomous systems under consideration is established. Second, the finite-time boundedness with a weighted L1-gain performance of autonomous systems with uncertain disturbances is addressed. Furthermore, state-feedback controllers guaranteeing the finite-time stability and the finite-time boundedness of non-autonomous systems are constructed, respectively. Finally, an illustrative example is given to show the validity of the present design. Copyright © 2014 John Wiley & Sons, Ltd.

A comparison of distributed MPC schemes on a hydro-power plant benchmark

Abstract: SUMMARY In this paper, we analyze and compare five distributed model predictive control (DMPC) schemes using a hydro-power plant benchmark. Besides being one of the most important sources of renewable power, hydro-power plants present very interesting control challenges. The operation of a hydro-power valley involves the coordination of several subsystems over a large geographical area in order to produce the demanded energy while satisfying constraints on water levels and flows. In particular, we test the different DMPC algorithms using a 24-h power tracking scenario in which the hydro-power plant is simulated with an accurate nonlinear model. In this way, it is possible to provide qualitative and quantitative comparisons between different DMPC schemes implemented on a common benchmark, which is a type of assessment rare in the literature. Copyright © 2014 John Wiley & Sons, Ltd.

Fractional linear control systems with Caputo derivative and their optimization

Abstract: SUMMARY In this paper, a fractional linear control system, containing Caputo derivative, with an integral performance index is studied. First, the existence and uniqueness of a solution to the mentioned control system is obtained. The main result is a theorem on the existence of optimal solutions to considered optimal control problems. Moreover, in order to find these solutions, the necessary optimality conditions (Pontryagin maximum principle) are derived. Our considerations consist of two parts: first, we consider starting a problem with zero initial condition and, next, with nonzero initial condition. All results are obtained by using results of such a type for equivalent fractional optimal control problem containing a Riemann–Liouville derivative. Copyright © 2014 John Wiley & Sons, Ltd.

Dominant speed factors of active set methods for fast MPC

Abstract: The paper presents a review of active set (AS) algorithms that have been deployed for implementation of fast model predictive control (MPC). The main purpose of the survey is to identify the dominant features of the algorithms that contribute to fast execution of online MPC and to study their influence on the speed. The simulation study is conducted on two benchmark examples where the algorithms are analyzed in the number of iterations and in the workload per iteration. The obtained results suggest directions for potential improvement in the speed of existing AS algorithms.

Linear-quadratic control problems with L1-control cost

Abstract: Summary We analyze a class of linear-quadratic optimal control problems with an additional L1-control cost depending on a parameter β. To deal with this nonsmooth problem, we use an augmentation approach known from linear programming in which the number of control variables is doubled. It is shown that if the optimal control for a given is bang-zero-bang and the switching function has a stable structure, the solutions are Lipschitz continuous functions of the parameter β. We also show that in this case the optimal controls for β * and a with | β − β * | sufficiently small coincide except on a set of measure . Finally, we use the augmentation approach to derive error estimates for Euler discretizations. Copyright © 2014 John Wiley & Sons, Ltd.

Optimal control of continuous systems with impulse controls

Abstract: Summary Impulse control problems, in which a continuously evolving state is modified by discrete control actions, have applications in epidemiology, medicine, and ecology. In this paper, we present a simple method for solving impulse control problems for systems of differential equations. In particular, we show how impulse control problems can be reformulated and solved as discrete optimal control problems. The method is illustrated with two examples. Published 2014. This article has been contributed to by US Government employees and their work is in the public domain in the USA.

Stability analysis of uncertain switched singular time‐delay systems with discrete and distributed delays

Abstract: Summary The issues of stability analysis of a class of continuous uncertain switched singular time-delay systems consisting of discrete and distributed delays is investigated in this paper Based on the LMIs, constructing the new extended Lyapunov-like Krasovskii functional, and the average dwell-time approach, delay-dependent sufficient conditions are derived to check the stability of such systems By solving the corresponding LMIs, the average dwell-time and switching signal condition are obtained This paper also highlights the relationship between the average dwell-time of the switched singular time-delay system, exponential rate of differential, and algebraic states Two numerical examples show the effectiveness of the proposed design method Copyright © 2013 John Wiley & Sons, Ltd

Plug-and-play distributed model predictive control with coupling attenuation†

Abstract: SUMMARY We consider the control of a large-scale system composed of state-coupled linear subsystems that can be added or removed offline. In this paper, we present plug-and-play (PnP) design methods based on model predictive control, meaning that (i) the design of a local controller requires information from parent subsystems only, (ii) the plugging in/out of a subsystem triggers at most the redesign of controllers associated to subsystems coupled to it, and (iii) plug-in/out operations are automatically denied if they compromise the stability of the overall system or constraint satisfaction. We advance previously proposed PnP decentralized control schemes by introducing a distributed control architecture that exploits communication between coupled subsystems. New controllers embody coupling attenuation terms that make PnP design applicable even when existing synthesis method are not. The main features of our approach are illustrated considering the PnP design of controllers for regulating the frequency of multiple generators in power networks. Copyright © 2014 John Wiley & Sons, Ltd.

Field programmable gate array based predictive control system for spacecraft rendezvous in elliptical orbits

Abstract: This work was supported by the Engineering and Physical Sciences Research Council Grant Number [EP/G030308/1] as well as industrial support from Xilinx, Mathworks, and the European Space Agency.

Nearly optimal simple explicit MPC controllers with stability and feasibility guarantees

Abstract: Summary We consider the problem of synthesizing simple explicit model predictive control feedback laws that provide closed-loop stability and recursive satisfaction of state and input constraints The approach is based on replacing a complex optimal feedback law by a simpler controller whose parameters are tuned, off-line, to minimize the reduction of the performance The tuning consists of two steps In the first step, we devise a simpler polyhedral partition by solving a parametric optimization problem In the second step, we then optimize parameters of local affine feedbacks by minimizing the integrated squared error between the original controller and its simpler counterpart We show that such a problem can be formulated as a convex optimization problem Moreover, we illustrate that conditions of closed-loop stability and recursive satisfaction of constraints can be included as a set of linear constraints Efficiency of the method is demonstrated on two examples Copyright © 2014 John Wiley & Sons, Ltd

Finite-horizon near optimal adaptive control of uncertain linear discrete-time systems

Abstract: SUMMARY In this paper, the finite-horizon near optimal adaptive regulation of linear discrete-time systems with unknown system dynamics is presented in a forward-in-time manner by using adaptive dynamic programming and Q-learning. An adaptive estimator (AE) is introduced to relax the requirement of system dynamics, and it is tuned by using Q-learning. The time-varying solution to the Bellman equation in adaptive dynamic programming is handled by utilizing a time-dependent basis function, while the terminal constraint is incorporated as part of the update law of the AE. The Kalman gain is obtained by using the AE parameters, while the control input is calculated by using AE and the system state vector. Next, to relax the need for state availability, an adaptive observer is proposed so that the linear quadratic regulator design uses the reconstructed states and outputs. For the time-invariant linear discrete-time systems, the closed-loop dynamics becomes non-autonomous and involved but verified by using standard Lyapunov and geometric sequence theory. Effectiveness of the proposed approach is verified by using simulation results. The proposed linear quadratic regulator design for the uncertain linear system requires an initial admissible control input and yields a forward-in-time and online solution without needing value and/or policy iterations. Copyright © 2014 John Wiley & Sons, Ltd.

An adaptive constraint tightening approach to linear model predictive control based on approximation algorithms for optimization

Abstract: Summary In this paper, we propose a model predictive control scheme for discrete-time linear invariant systems based on inexact numerical optimization algorithms. We assume that the solution of the associated quadratic program produced by some numerical algorithm is possibly neither optimal nor feasible, but the algorithm is able to provide estimates on primal suboptimality and primal feasibility violation. By adaptively tightening the complicating constraints, we can ensure the primal feasibility of the approximate solutions generated by the algorithm. We derive a control strategy that has the following properties: the constraints on the states and inputs are satisfied, asymptotic stability of the closed-loop system is guaranteed, and the number of iterations needed for a desired level of suboptimality can be determined. The proposed method is illustrated using a simulated longitudinal flight control problem. Copyright © 2014 John Wiley & Sons, Ltd.

Distributed MPC for urban traffic networks: A simulation-based performance analysis†

Abstract: SUMMARY The operation of urban traffic networks with distributed model predictive control (DMPC) can be more flexible than centralized strategies because DMPC allows for graceful expansion of the control infrastructure, localized reconfiguration, and tolerance to faulty operation. Yet, computational performance is less efficient than with centralized model predictive control (MPC) because of the added complexity brought about by distribution schemes. To assess the trade-off between flexibility and performance, in this paper we assess the tolerance to failure and performance of DMPC in contrast with centralized control. The problem of concern is the signal setting of green time in a representative traffic network modeled in a commercial microscopic traffic simulator. A software tool is developed for implementing and simulating the DMPC framework in tandem with the simulator. Comparisons of DMPC with MPC and a baseline feedback control strategy that does not use constrained optimization show that DMPC can achieve performance gains with respect to the baseline case and enhance tolerance to failure. Computations for DMPC are less efficient than with centralized MPC; nevertheless, the time taken by DMPC is well below the required for field use. Although the true distributed deployment of DMPC requires special hardware, its implementation in a central cluster can be made without loss of operational flexibility. Copyright © 2014 John Wiley & Sons, Ltd.

Robust reliable guaranteed cost control for uncertain T‐S fuzzy neutral systems with interval time‐varying delay and linear fractional perturbations

Abstract: Summary The robust reliable guaranteed cost control for Takagi–Sugeno fuzzy systems with interval time-varying delay is considered in this paper. Some free weighting matrices and non-negative terms are provided to improve the conservativeness of our main results. An LMI optimization approach is applied to solve the problems of robust reliable guaranteed cost control and minimization of cost function. Copyright © 2014 John Wiley & Sons, Ltd.

Optimal energy management for grid‐connected storage systems

Abstract: Summary This paper proposes three near-optimal (to a desired degree) deterministic charge and discharge policies for the maximization of profit in a grid-connected storage system. The changing price of electricity is assumed to be known in advance. Three near-optimal algorithms are developed for the following three versions of this optimization problem: (1) The system has supercapacitor type storage, controlled in continuous time. (2) The system has supercapacitor or battery type storage, and it is controlled in discrete time (i.e., it must give constant power during each sampling period). A battery type storage model takes into account the diffusion of charges. (3) The system has battery type storage, controlled in continuous time. We give algorithms for the approximate solution of these problems using dynamic programming, and we compare the resulting optimal charge/discharge policies. We have proved that in case 1 a bang off bang type policy is optimal. This new result allows the use of more efficient optimal control algorithms in case 1. We discuss the advantages of using a battery model and give simulation and experimental results. Copyright © 2014 John Wiley & Sons, Ltd.

An optimal control approach to spacecraft rendezvous on elliptical orbit

Abstract: Summary In this paper, we consider a linear quadratic regulator control problem for spacecraft rendezvous in an elliptical orbit. A new spacecraft rendezvous model is established. On the basis of this model, a linear quadratic regulator control problem is formulated. A parametric Lyapunov differential equation approach is used to design a state feedback controller such that the resulting closed-loop system is asymptotically stable, and the performance index is minimized. By an appropriate choice of the value of a parameter, an approximate state feedback controller is obtained from a solution to the periodic Lyapunov differential equation, where the periodic Lyapunov differential equation is solved on the basis of a new numerical algorithm. The spacecraft rendezvous mission under the controller obtained will be accomplished successfully. Several illustrative examples are provided to show the effectiveness of the proposed control design method. Copyright © 2014 John Wiley & Sons, Ltd.

Linear quadratic regulator control of multi‐agent systems

Abstract: Summary This paper considers a collection of agents performing a shared task making use of relative information communicated over an information network. The designed suboptimal controllers are state feedback and static output feedback, which are guaranteed to provide a certain level of performance in terms of a linear quadratic regulator (LQR) cost. Because of the convexity of the LQR performance region, the suboptimal LQR control problem with state feedback is reduced to the solution of two inequalities, with the minimum and maximum eigenvalues of the Laplacian matrix as the coefficients. The advantage of the method is that the LQR control problem of network multi-agent systems can be converted into the LQR control of a set of single-agent systems, and the structure constraint on the feedback gain matrix can be eliminated. It can be shown that the size of the LQR control problem will not increase according to the number of the node in the fairly general framework. The method can be extended to the synthesis of the static output feedback, which is derived from the weighting matrices in LQR. Through some coordinate transformation and the augmentation of the output matrix, the LQR synthesis is provided on the basis of the output measurements of the adjacent agents. Numerical examples are presented to illustrate the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.

Dynamic surface control based on neural network for an air-breathing hypersonic vehicle

Abstract: Summary In this paper, an adaptive dynamic surface control approach is presented for the longitudinal motion of an air-breathing hypersonic vehicle. Fully tuned radial basis function neural network that regulates weights, width, and center of Gaussian function simultaneously is developed to estimate aerodynamic uncertainties and atmospheric disturbances. The nonlinear control law is subsequently designed by dynamic surface control approach for the vehicle model converted into strict block feedback form by input–output linearization method. Simulation results show that the velocity can be successfully tracked over a large range from Mach 11 to Mach 12 and an altitude range from 26 to 30 km. The presented approach has perfect ability of restraining unknown and time-varying nonlinear dynamics during flight. Copyright © 2014 John Wiley & Sons, Ltd.

A BMI approach to guaranteed cost control of discrete-time uncertain system with both state and input delays

Abstract: SUMMARY In this study, the guaranteed cost control of discrete time uncertain system with both state and input delays is considered. Sufficient conditions for the existence of a memoryless state feedback guaranteed cost control law are given in the bilinear matrix inequality form, which needs much less auxiliary matrix variables and storage space. Furthermore, the design of guaranteed cost controller is reformulated as an optimization problem with a linear objective function, bilinear, and linear matrix inequalities constraints. A nonlinear semi-definite optimization solver—PENLAB is used as a solution technique. A numerical example is given to demonstrate the effectiveness of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.

Fixed-structure H-2 controller design for polytopic systems via LMIs

Abstract: In this paper a new approach for fixed-structure H2 controller design in terms of solutions to a set of linear matrix inequalities are given. Both discrete- and continuous-time single-input single-output (SISO) time- invariant systems are considered. Then the results are extended to systems with polytopic uncertainty. The presented methods are based on an inner convex approximation of the non-convex set of fixed-structure H2 controllers. The designed procedures initialized either with a stable polynomial or with a stabilizing controller. An iterative procedure for robust controller design is given that converges to a suboptimal solution. The monotonic decreasing of the upper bound on the H2 norm is established theoretically for both nominal and robust controller design.

Analysis of a periodic optimal control problem connected to microalgae anaerobic digestion

Abstract: In this work, we study the coupling of a culture of microalgae limited by light and an anaerobic digester in a two-tank bioreactor. The model for the reactor combines a periodic day-night light for the culture of microalgae and a classical chemostat model for the digester. We first prove the existence and attraction of periodic solutions of this problem for a 1 day period. Then, we study the optimal control problem of optimizing the production of methane in the digester during a certain timeframe, the control on the system being the dilution rate (the input flow of microalgae in the digester). We apply Pontryagin's Maximum Principle in order to characterize optimal controls, including the computation of singular controls. We present numerical simulations by direct and indirect methods for different light models and compare the optimal 1-day periodic solution to the optimal strategy over larger timeframes. Finally, we also investigate the dependence of the optimal cost with respect to the volume ratio of the two tanks.

Synthesis of optimal control in a mathematical model of tumour–immune dynamics

Abstract: Summary An optimal control problem for a mathematical model of tumour–immune dynamics under the influence of chemotherapy is considered. The toxicity effect of the chemotherapeutic agent on both tumour and immunocompetent cells is taken into account. A standard linear pharmacokinetic equation for the chemotherapeutic agent is added to the system. The aim is to find an optimal strategy of treatment to minimize the tumour volume while keeping the immune response not lower than a fixed permissible level as far as possible. Sufficient conditions for the existence of not more than one switching and not more than two switchings without singular regimes are obtained. The surfaces in the extended phase space, on which the last switching appears, are constructed analytically.Copyright © 2013 John Wiley & Sons, Ltd.