An efficient cost reduction procedure for bounded-control LQR problems
- PDF / 1,073,543 Bytes
- 22 Pages / 439.37 x 666.142 pts Page_size
- 27 Downloads / 216 Views
An efficient cost reduction procedure for bounded-control LQR problems Vicente Costanza1 · Pablo S. Rivadeneira1,2 · John A. Gómez Múnera1
Received: 8 April 2016 / Revised: 19 June 2016 / Accepted: 15 October 2016 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2016
Abstract A novel approach has been developed for approximating the solution to the constrained LQR problem, based on updating the final state and costate of a related regular problem, and on slightly shifting the switching times (the instants when the control meets the constraints). The main result is the expression of a suboptimal control in feedback form using the solution of some compatible Riccati equation. The gradient method is applied to reduce the cost via explicit algebraic formula for its partial derivatives with respect to the hidden final state/costate of the related regular problem and to the switching times. The numerical method is termed efficient because it does not involve integrations of states or cost trajectories, and reduces to its minimum the dimension of the unknown parameters at the final condition. All the relevant objects are calculated from a few auxiliary matrices, which are computed only once. The scheme is here applied to two case studies whose optimal solutions are known. The first example is a two-dimensional model of the ‘cheapest stop of a train’ problem. The second one refers to the temperature control of a metallic strip leaving a multi-stand rolling mill, a problem with a high-dimensional state. Keywords Optimal control · Restricted controls · LQR problem · Gradient methods Mathematics Subject Classification Primary 93C05; Secondary 49N10
Communicated by Antonio Leitao.
B
Vicente Costanza [email protected]
1
Instituto de Desarrollo Tecnológico para la Industria Química (INTEC, UNL-CONICET), Güemes 3450, 3000 Santa Fe, Argentina
2
Universidad Nacional de Colombia - Sede Medellín, Facultad de Minas, Grupo “GITA”, Cra. 80#65-223, Medellín, Colombia
123
V. Costanza et al.
1 Introduction The linear-quadratic regulator (LQR) problem is probably the most studied and used in the optimal control literature. Concurrently, the Hamiltonian formalism has also been at the core of the development of modern optimal control (Agrachev and Sachkov 2004; Athans and Falb 2006; Costanza and Rivadeneira 2014b; Pontryagin et al. 1964; Troutman 1996). In this paper, a Hamiltonian approach to the bounded control LQR problem will be pursued. When the n-dimensional finite-horizon problem for unbounded controls is regular, it leads to a set of 2n linear ordinary differential equations (ODEs) with two-point boundary-value conditions, known as the Hamilton Canonical Equations (HCEs). There are well-known methods (see for instance Costanza and Neuman 2009; Sontag 1998) to transform the boundary-value problem into an initial-value one. In the infinite-horizon, bilinear-quadratic regulator, and also in the change of set-point servo problems, there are also some attempts to find the missing initial condi
Data Loading...
