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This paper addresses the need of the novel reciprocal state space (RSS) framework to be the supplement of standard state space system in control designs. A model following design based on variable structure control design approach utilizing direct state derivative measurement for an RSS system with nonlinear dead-zone input is proposed. Simulations for both nonlinear dead-zone input and nonlinear sector input are carried out for an unstable RSS system to successfully verify the proposed algorithm. It shows that the design procedure is straightforward for systems with direct state derivative measurement if we express the system in RSS framework.

There are two major problems that people usually have to face when they carry out optimal control designs in state space system or descriptor system. One is no state derivative considered in cost functional, the other is that some systems cannot easily design optimal control in either state space system or generalized system frameworks. To overcome the problems, a novel state derivative space system (SDS system) framework is proposed. In this novel framework, novel differential Lagrange multiplier should be used to adjoin SDS system constraint to the cost functional that is function of state derivative to straightforwardly carry out optimal control design. It showed that some singular systems with impulse mode can easily carry out optimal control in SDS system. The purpose of this research is to develop novel and simple optimal control algorithms in SDS system so that wider ranges of problems can be solved without much of mathematics overhead.