Minimization of Integral-square-error for Non-Linear Control Systems of Third and Higher Order

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Abstract
The paper deals principally with a control system which has a plant consisting of three integrators with a saturable control input. The command signal input is a step plus a ramp plus a parabola. The switching surface which minimizes integral-square-error is found, partly algebraically and partly numerically, by methods which start from Pontryagin's maximum principle. All optimum trajectories have an infinite number of switches before the origin is reached, except for two trajectories which have no switches. Some optimum trajectories have the property that the ratio of any two successive switching intervals is constant. All other optimum trajectories (apart from the two exceptional cases) converge rapidly towards these constant ratio trajectories. Thus, when finding optimum trajectories by backwards numerical computation from near the origin of the Hamiltonian system of equations, it is necessary to adjust the ‘initial’ values to be, not only small, but also close to a constant ratio trajectory. The above behaviour is similar to that for an analogous linear system. Approximate analysis of the non-linear Hamiltonian system by means of describing function techniqnes also indicates this behaviour; in fact, the constant ratio trajectories were only discovered as a result of preliminary linear and quasi-linear studies. It is conjectured that similar behaviour occurs for other plants and other performance criteria.

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