Time-, fuel-, and energy-optimal control of nonlinear norm-invariant systems

Author

Athans, M. ; Falb, P.L. ; Lacoss, R.T.

Author_Institution

Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

Volume

Issue

fYear

1963

fDate

7/1/1963 12:00:00 AM

Firstpage

196

Lastpage

202

Abstract

Nonlinear systems of the form $\\dot{X}(t)=g[x(t);t]+u(t)$ , where $x(t), u(t)$ , and $g[x(t); t]$ are $n$ vectors, are examined in this paper. It is shown that if $\\parallel x(t) \\parallel = \\sqrt{{x_{1}}^{2}(t) + \\cdots + {x_{n}}^{2}(t)}$ is constant along trajectories of the homogeneous system $\\dot{X}(t)=g[x(t); t]$ and if the control $u(t)$ is constrained to lie within a sphere of radius $M$ , i.e., $\\parallel u(t) \\parallel \\leq M$ , for all $t$ , then the control $u^{\\ast}(t)= - Mx(t) / \\parallel x(t) \\parallel$ drives any initial state $\\xi$ to 0 in minimum time and with minimum fuel, where the consumed fuel is measured by $\\int_{0}^{T}\\parallel u(t) \\parallel dt$ . Moreover, for a given response time $T$ , the control $\\tilde{u}(t) = -\\parallel\\xi\\parallel x(t)/T \\parallel x(t) \\parallel$ drives $\\xi$ to 0 and minimizes the energy measured by ${1 \\over 2}\\int_{0}^{T}\\parallel u(t) \\parallel^{2}dt$ . The theory is applied to the problem of reducing the angular velocities of a tumbling asymmetrical space body to zero.

Keywords

Fuel-optimal control; Minimum-energy control; Nonlinear systems; Time-optimal control; Angular velocity; Control systems; Delay; Energy measurement; Fuels; Linear systems; Nonlinear control systems; Nonlinear systems; Optimal control; Time measurement;

fLanguage

English

Journal_Title

Automatic Control, IEEE Transactions on

Publisher

ieee

ISSN

0018-9286

Type

jour

DOI

10.1109/TAC.1963.1105581

Filename

1105581

Link To Document

https://search.isc.ac/dl/search/defaultta.aspx?DTC=49&DC=865299