Transformation of CLF to ISS-CLF for Nonlinear Systems with Disturbance and Construction of Nonlinear Robust Controller with Gain Performance

A new nonlinear control law for a class of nonlinear systems with disturbance is proposed. A control law is designed by transforming control Lyapunov function (CLF) to input-to-state stability control Lyapunov function (ISS-CLF). The transformed CLF satisfies a Hamilton-Jacobi-Isaacs (HJI) equation. The feedback system by the proposed control law has characteristics of gain. Finally, it is shown by a numerical example that the proposed control law makes a controller by feedback linearization robust against disturbance. 1. Introduction It is difficult to build strict mathematical models of actual systems and there may exist disturbance such as modelling errors and parameter variations of systems. It is one of the most important problems in control theory to construct a control law considering disturbance. If an input-to-state stability control Lyapunov function (ISS-CLF) or gain exists, then the stability of a nonlinear system with disturbance can be assured. If an ISS-CLF can be found, it is possible to construct a control law considering disturbance which depends on the state of a system. If disturbance is lower than a certain value, then a system may be asymptotically stable to the origin. But the construction way of an ISS-CLF is provided only for particular systems and a complicated procedure is required [1–3]. On the other hand, if the existence of gain can be shown, then it is assured that state and control input remain in set and a system is stable even if there exists disturbance belonging to set [4, 5]. It is possible to construct gain if a Hamilton-Jacobi-Isaacs (HJI) equation can be solved [6]. But the general solution to a HJI equation is not found and the solution to a HJI equation may not exist for a specified gain. Though the solution to a HJI equation by numerical calculation is also provided, it is complicated and there are some constraints [7]. It is also possible to construct the solution which satisfies a HJI equation by applying an inverse optimal method [8]. A control law provided by an inverse optimal method can optimize a certain meaningful objective function but an inverse optimal method can be applied only if ISS-CLF can be found. In addition, a certain meaningful objective function and gain may not be the desired ones. The main purpose of this paper is to propose a transformation method from CLF to ISS-CLF and to provide a lower bound condition of gain. A CLF is transformed to an ISS-CLF so that an ISS-CLF satisfies a HJI equation by using a proper transformation coefficient. At the same time, a lower bound condition of gain