L.G. Van Willigenburg

Efficient Differential Evolution algorithms for multimodal optimal control problems

Many methods for solving optimal control problems, whether direct or indirect, rely upon gradient information and therefore may converge to a local optimum. Global optimisation methods like Evolutionary algorithms, overcome this problem. In this work it is investigated how well novel and easy to understand Evolutionary algorithms, referred to as Differential Evolution (DE) algorithms, and claimed to be very efficient when they are applied to solve static optimisation problems, perform on solving multimodal optimal control problems. The results show that within the class of evolutionary methods, Differential Evolution algorithms are very robust, effective and highly efficient in solving the studied class of optimal control problems. Thus, they are able of mitigating the drawback of long computation times commonly associated with Evolutionary algorithms. Furthermore, in locating the global optimum these Evolutionary algorithms present some advantages over the Iterative Dynamic Programming (IDP) algorithm, which is an alternative global optimisation approach for solving optimal control problems. At present little knowledge is available to the selection of the algorithm parameters in the DE algorithm when they are applied to solve optimal control problems. Our study provides guidelines for this selection. In contrast to the IDP algorithm the DE algorithms have only a few algorithm parameters that are easily determined such that multimodal optimal control problems are solved effectively and efficiently.

The significance of crop co-states for receding horizon optimal control of greenhouse climate

While a tomato crop grows on the time-scale of weeks, the greenhouse climate changes on a time-scale of minutes. The economic optimal control problem of producing good quality crops against minimum input of resources is tackled by a two time-scale decomposition. First, the sub-problem associated to the slow crop evolution is solved off-line, leading to a seasonal pattern for the co-states of the amount of assimilates produced by photosynthesis, and the fruit and leaf weights. These co-states can be interpreted as the marginal prices of a unit of assimilate, leaf and fruit. Next, they are used in the goal function of an on-line receding horizon control (RHOC) of the greenhouse climate, thus balancing costs of heating and CO2-dosage against predicted benefits from harvesting, while profiting as much as possible from the available solar radiation. Simulations using the time-varying co-states are compared to experimental results obtained with fixed co-states. It appears that the on-line control is sensitive to the time evolution of the co-states, suggesting that it is advantageous to repeat the seasonal optimisation from time to time to adjust the co-states to the past weather and realised crop state. ? 2002 Elsevier Science Ltd. All rights reserved.

Optimal control of nitrate in lettuce by a hybrid approach: differential evolution and adjustable control weight gradient algorithms

Since high concentration levels of nitrate in lettuce and other leafy vegetables are undesirable, cultivation of lettuce according to specified governmental regulations is currently an important issue. Therefore, methods are sought in order to produce a lettuce crop that allow maximization of the profits of the grower while at the same time insuring the quality of the crops. Using a two-state dynamic lettuce model that predicts the amount of nitrate at harvest time, an optimal control problem with terminal constraints is formulated. The situation considered may be relevant in a plant factory where a fixed head weight should be reached in fixed time while minimizing light input. First, optimal trajectories of light, CO2 and temperature are calculated using the adjustable control weight (ACW) gradient method. Subsequently, novel, efficient and modified differential evolution (DE) algorithms are used to obtain an approximate solution to the same optimal control problem. While the gradient method yields a more accurate result, the optimum may be local. In order to exploit the salient characteristics of a DE algorithm as a global direct search method, a hybrid-combined approach is proposed. An approximate solution obtained with a DE algorithm is used to initialize the ACW gradient method. Although local minima did not seem to occur in this particular case, the results show the feasibility of this approach. # 2003 Published by Elsevier Science B.V.

Numerical Algorithms and Issues Concerning the Discrete-Time Optimal Projection Equations

The discrete-time optimal projection equations, which constitute necessary conditions for optimal reduced-order LQG compensation, are strengthened. For the class of minimal stabilizing compensators the strengthened discrete-time optimal projection equations are proved to be equivalent to first-order necessary optimality conditions for optimal reduced-order LQG compensation. The conventional discrete-time optimal projection equations are proved to be weaker. As a result solutions of the conventional discrete-time optimal projection equations may not correspond to optimal reduced-order compensators. Through numerical examples it is demonstrated that, in fact, many solutions exist that do not correspond to optimal reduced-order compensators. To compute optimal reduced-order compensators two new algorithms are proposed. One is a homotopy algorithm and one is based on iteration of the strengthened discrete-time optimal projection equations. The latter algorithm is a generalization of the algorithm that solves the two Riccati equations of full-order LQG control through iteration and therefore is highly efficient. Using different initializations of the iterative algorithm it is demonstrated that the reduced-order optimal LQG compensation problem, in general, may possess multiple extrema. Through two computer experiments it is demonstrated that the homotopy algorithm often, but not always, finds the global minimum.

Linear systems theory revisited

This paper investigates and clarifies how different definitions of reachability, observability, controllability, reconstructability and minimality that appear in the control literature, may be equivalent or different, depending on the type of linear system. The differences are caused by (1) whether or not the linear system has state dimensions that vary with time (2) bounds on the time axis of the linear system (3) whether or not the initial state is non-zero and (4) whether or not the system is time invariant. Also (5) time-reversibility of systems plays a role. Discrete-time linear strictly proper systems are considered. A recently published result is used to argue that all the results carry over to continuous time. Out of the investigation two types of definitions emerge. One type applies naturally to systems with constant dimensions while the other applies naturally to systems with variable dimensions. This paper reveals that time-varying (state) dimensions that are allowed to be zero are necessary to obtain equivalence between minimality and (weak) reachability together with observability at the systems level. Besides their theoretical significance the results of this paper are of practical importance for model reduction and control of time-varying discrete-time linear systems because they result in minimal realizations with smaller dimensions that are also computed more easily.

Brief Minimal and non-minimal optimal fixed-order compensators for time-varying discrete-time systems

Using the minimality property of finite-horizon time-varying compensators, established in this paper, and the Moore-Penrose pseudo-inverse instead of the standard inverse, strengthened discrete-time optimal projection equations (SDOPE) and associated boundary conditions are derived for finite-horizon fixed-order LQG compensation. They constitute a two-point boundary value problem explicit in the LQG problem parameters which is equivalent to first-order necessary optimality conditions and which is suitable for numerical solution. The minimality property implies that minimal compensators have time-varying dimensions and that the finite-horizon optimal full-order compensator is not minimal. The use of the Moore-Penrose pseudo-inverse is further exploited to reveal that the optimal projection approach can be generalised, but only to partially include non-minimal compensators. Furthermore, the structure of the space of optimal compensators with arbitrary dimensions is revealed to a large extent. Max-min compensator dimensions are introduced and their significance in solving numerically the two-point boundary value problem is explained. The numerical solution is presented in a recently published companion paper, which relies on the results of this paper.

Centimetre-precision guidance of moving implements in the open field: a simulation based on GPS measurements

High quality and sustainable agricultural production in the open field will be supported by centimetre-precision guidance of agricultural implements. In this paper, a complete guidance system is proposed and simulated using real sensor data obtained from a real time kinematic (RTK) differential global positioning system (DGPS). This type of the satellite navigation system became available commercially in 1997 and is claimed to reach cm accuracy. It uses real time kinematics to improve the accuracy of a position fix by phase comparison of the carrier signal. The simulation primarily aims to investigate the accuracy of the guidance system that can be obtained with RTK DGPS. Using real DGPS measurement errors the simulated system showed a tracking error less than 0.005 m when the PID-controller had settled.

Computation and implementation of digital time-optimal feedback controllers for an industrial X-Y robot subjected to path, torque, and velocity constraints

For an industrial X-Y robot, in which the links are subjected to torque and velocity constraints, the time-optimal control prob lem is solved where the robot motion is constrained to follow an arbitrary path. A numerical procedure to compute the solu tion is presented and demonstrated. The solution consists of a continuous-time state trajectory and open-loop control. Because the X-Y robot is controlled by a digital computer, a recently developed numerical procedure to compute optimal tracking digital controllers is applied to the solution to arrive at an implementable digital time-optimal feedback controller that accounts for modeling errors as well. Experimental results ob tained after implementation of the digital time-optimal feedback controllers are presented for two paths. The robot dynamics include both viscous and Coulomb friction. The extension of the solution and numerical procedures to general rigid ma nipulators, including both viscous and Coulomb friction, is straightforward.For an industrial X-Y robot, in which the links are subjected to torque and velocity constraints, the time-optimal control prob lem is solved where the robot motion is constrained to follow an arbitrary path. A numerical procedure to compute the solu tion is presented and demonstrated. The solution consists of a continuous-time state trajectory and open-loop control. Because the X-Y robot is controlled by a digital computer, a recently developed numerical procedure to compute optimal tracking digital controllers is applied to the solution to arrive at an implementable digital time-optimal feedback controller that accounts for modeling errors as well. Experimental results ob tained after implementation of the digital time-optimal feedback controllers are presented for two paths. The robot dynamics include both viscous and Coulomb friction. The extension of the solution and numerical procedures to general rigid ma nipulators, including both viscous and Coulomb friction, is straightforward.

Temporal Linear System Structure

Piecewise constant rank systems and the differential Kalman decomposition are introduced in this note. Together these enable the detection of temporal uncontrollability/unreconstructability of linear continuous-time systems. These temporal properties are not detected by any of the four conventional Kalman decompositions. Moreover piecewise constant rank systems admit the state dimension to be time-variable. As demonstrated in this note linear continuous-time systems with variable state dimensions enable the well rounded realization theory suggested already by Kalman. The differential Kalman decomposition introduced in this note is associated with differential controllability and differential reconstructability. The system structure obtained from the differential Kalman decomposition may be interpreted as the temporal linear system structure. This note reveals that the difference between controllability and reachability as well as reconstructability and observability is entirely due to changes of the temporal linear system structure. Also, this note reveals how the differential Kalman decomposition relates to the conventional ones.

UDU Factored Discrete-Time Lyapunov Recursions Solve Optimal Reduced-Order LQG Problems

A new algorithm is presented to solve both the finitehorizon time-varying and infinite-horizon timeinvariant discrete-time optimal reduced-order linear quadratic Gaussian (LQG) problem. In both cases the first order necessary optimality conditions can be represented by two non-linearly coupled discrete-time Lyapunov equations, which run forward and backward in discrete time. The algorithm iterates these two equations forward and backward in discrete time, respectively, until they converge. In the finite-horizon time varying case the iterations start from boundary conditions and the forward and backward in time recursions are repeated until they converge. The discrete-time recursions are suitable for UDU factorisation. It is illustrated how UDU factorisation may increase both the numerical efficiency and accuracy of the recursions. By means of several numerical examples and the benchmark problem proposed by the European Journal of Control, the results obtained with the new algorithm are compared to results obtained with algorithms that iterate the strengthened discrete-time optimal projection equations forward and backward in time. The convergence properties are illustrated to be comparable. Especially when the reduced compensator dimensions are significantly smaller than those of the controlled system, the algorithm presented in this paper is more efficient.