Sliding Mode Control with State Derivative Output Feedback in Reciprocal State Space Form

Abstract

This paper investigates the novel sliding mode control design with state derivative output feedback in nontraditional reciprocal state space (RSS) form. The concepts and the need of RSS form are comprehensively reviewed and explained. Novel switching function and approaching condition based on the derivative of sliding surface are introduced. In addition, a sufficient condition for finding the upper bound of system uncertainty to guarantee the stability in sliding surface is developed for robustness analysis. A compact sliding mode controller utilizing only state derivative related output feedback is proposed for systems with system uncertainty, matched input uncertainty, and matched external disturbance. Simulation results for a circuit system successfully verify the validities of the proposed algorithms. Our derivation is basically parallel to that for systems in standard state space form. Therefore, those who understand the concepts of sliding mode control can easily apply our method to handle more control problems without being involved in complex mathematics.

Model-Following Designs Using Direct State Derivative Measurement Feedback in Novel Reciprocal State Space Form

“…To provide supplementary design algorithms for systems which cannot be modeled in standard state space form and to utilize direct state derivative measurement feedback in control designs in similar ways as state measurement feedback control designs in standard state space form, the following novel reciprocal state space (RSS) form was proposed [14] [15]. When the matrix F in (3) is non-singular (full rank) and only state derivatives are directly measured and fed back, the system can be expressed in reciprocal state space (RSS) form as follows.…”

“…It also showed that applying state derivative feedback in RSS form can easily carry out some popular control design techniques such as pole placement, eigenstructure assignment, and linear quadratic regulator (LQR) [14] [17]. Note that some singular systems with impulse mode can also be expressed in RSS form to easily carry out state derivative measurement feedback control designs [15] [17].…”

“…Consequently, a globally asymptotically stable system can be expressed in both state space form and RSS form. From system constraint in (15), we can conclude that if closed loop system is asymptotically stable, c A in (15) is invertible and both x and x  can be expressed by each other. We can also conclude that that both x and x  must vanish with time when closed loop system is properly controlled and asymptotically stable.…”

“…Bear in mind that the closed loop poles are the reciprocals of matrix c A in (15). We can observe that the (24)-(26) used to solve for the optimal state derivative measurement feedback gain K in (27) are identical to those of a standard state space system with state output feedback to minimize the traditional performance index as follows [18].…”

Control Design for System with Lipschitz Nonlinearity of State Derivative Variables in Reciprocal State Space Form

“…It is obvious that singular system cannot be expressed in state space form to carry out control design. To provide supplementary design algorithms of state derivative feedback in state space form and handle some of singular systems, a direct state derivative feedback control scheme was developed in "Reciprocal state space" (RSS) form [9] [11][12][13][14][15] by the author of this paper. If F in (2) is invertible, we can have the following system in RSS form and carry out control design with state derivative feedback.…”