Integrable Hamiltonian systems defined on the Lie groups SO(3) and SU(2): an application to the attitude control of a spacecraft

This paper considers left-invariant control systems defined on the Lie groups SU(2) and SO(3). Such systems have a number of applications in both classical and quantum control problems. The purpose of this paper is two-fold. Firstly, the optimal control problem for a system varying on these Lie Groups, with cost that is quadratic in control is lifted to their Hamiltonian vector fields through the Maximum principle of optimal control and explicitly solved. Secondly, the control systems are integrated down to the level of the group to give the solutions for the optimal paths corresponding to the optimal controls. In addition it is shown here that integrating these equations on the Lie algebra su(2) gives simpler solutions than when these are integrated on the Lie algebra so(3).

Effects of reduced height (Rht) and photoperiod insensitivity (Ppd) alleles on yield of wheat in contrasting production systems

Near isogenic lines (NILs) varying for reduced height (Rht) and photoperiod insensitivity (Ppd-D1) alleles in a cv. Mercia background (rht (tall), Rht-B1b, Rht-D1b, Rht-B1c, Rht8c+Ppd-D1a, Rht-D1c, Rht12) were compared for interception of photosynthetically active radiation (PAR), radiation use efficiency (RUE), above-ground biomass (AGB), harvest index (HI), height, weed prevalence, lodging and grain yield, at one field site but within contrasting (‘organic’ v ‘conventional’) rotational and agronomic contexts, in each of three years. In the final year, further NILs (rht (tall), Rht-B1b, Rht-D1b, Rht-B1c, Rht-B1b+Rht-D1b, Rht-D1b+Rht-B1c) in Maris Huntsman and Maris Widgeon backgrounds were added together with 64 lines of a doubled haploid (DH) population [Savannah (Rht-D1b) × Renesansa (Rht-8c+Ppd-D1a)]. There were highly significant genotype × system interactions for grain yield, mostly because differences were greater in the conventional system than in the organic system. Quadratic fits of NIL grain yield against height were appropriate for both systems when all NILs and years were included. Extreme dwarfing was associated with reduced PAR, RUE, AGB, HI, and increased weed prevalence. Intermediate dwarfing was often associated with improved HI in the conventional system, but not in the organic system. Heights in excess of the optimum for yield were associated particularly with reduced HI and, in the conventional system, lodging. There was no statistical evidence that optimum height for grain yield varied with system although fits peaked at 85cm and 96cm in the conventional and organic systems, respectively. Amongst the DH lines, the marker for Ppd-D1a was associated with earlier flowering, and just in the conventional system also with reduced PAR, AGB and grain yield. The marker for Rht-D1b was associated with reduced height, and again just in the conventional system, with increased HI and grain yield. The marker for Rht8c reduced height, and in the conventional system only, increased HI. When using the System × DH line means as observations grain yield was associated with height and early vegetative growth in the organic system, but not in the conventional system. In the conventional system, PAR interception after anthesis correlated with yield. Savannah was the highest yielding line in the conventional system, producing significantly more grain than several lines that out yielded it in the organic system.

Integrable quadratic Hamiltonians on the Euclidean group of motions

In this paper, we discuss the problem of globally computing sub-Riemannian curves on the Euclidean group of motions SE(3). In particular, we derive a global result for special sub-Riemannian curves whose Hamiltonian satisfies a particular condition. In this paper, sub-Riemannian curves are defined in the context of a constrained optimal control problem. The maximum principle is then applied to this problem to yield an appropriate left-invariant quadratic Hamiltonian. A number of integrable quadratic Hamiltonians are identified. We then proceed to derive convenient expressions for sub-Riemannian curves in SE(3) that correspond to particular extremal curves. These equations are then used to compute sub-Riemannian curves that could potentially be used for motion planning of underwater vehicles.

Optimal control of a class of discrete–continuous non-linear systems — decomposition and hierarchical structure

A technique is derived for solving a non-linear optimal control problem by iterating on a sequence of simplified problems in linear quadratic form. The technique is designed to achieve the correct solution of the original non-linear optimal control problem in spite of these simplifications. A mixed approach with a discrete performance index and continuous state variable system description is used as the basis of the design, and it is shown how the global problem can be decomposed into local sub-system problems and a co-ordinator within a hierarchical framework. An analysis of the optimality and convergence properties of the algorithm is presented and the effectiveness of the technique is demonstrated using a simulation example with a non-separable performance index.

Dynamic integrated system optimization and parameter estimation for discrete time optimal control of nonlinear systems

An algorithm for solving nonlinear discrete time optimal control problems with model-reality differences is presented. The technique uses Dynamic Integrated System Optimization and Parameter Estimation (DISOPE), which achieves the correct optimal solution in spite of deficiencies in the mathematical model employed in the optimization procedure. A version of the algorithm with a linear-quadratic model-based problem, implemented in the C+ + programming language, is developed and applied to illustrative simulation examples. An analysis of the optimality and convergence properties of the algorithm is also presented.

A new approach for the solution of the optimal set-point tracking problem for nonlinear systems

In this paper, a discrete time dynamic integrated system optimisation and parameter estimation algorithm is applied to the solution of the nonlinear tracking optimal control problem. A version of the algorithm with a linear-quadratic model-based problem is developed and implemented in software. The algorithm implemented is tested with simulation examples.

Optimal control of nonlinear differential algebraic equation systems

A novel iterative procedure is described for solving nonlinear optimal control problems subject to differential algebraic equations. The procedure iterates on an integrated modified linear quadratic model based problem with parameter updating in such a manner that the correct solution of the original non-linear problem is achieved. The resulting algorithm has a particular advantage in that the solution is achieved without the need to solve the differential algebraic equations . Convergence aspects are discussed and a simulation example is described which illustrates the performance of the technique. 1. Introduction When modelling industrial processes often the resulting equations consist of coupled differential and algebraic equations (DAEs). In many situations these equations are nonlinear and cannot readily be directly reduced to ordinary differential equations.