## 1 Introduction  
In areas with high safety requirements such as spacecraft, the fault-tolerant control (FTC) has always attracted the attention [1; 2] of many scholars and engineers. In [1], a novel discontinuous adaptive failure compensation controller with system chattering rejection based on a common approximate function was designed for a rigid satellite with actuator failure. In [2], an ADP-based tracking FTC with an adaptive fault compensation was given for a spacecraft attitude system with additive faults. In the FTC community, a common and very important method is the sliding-mode control (SMC) that was widely applied in industrial and academic fields due to the extremely simple design and strong anti-disturbance ability [3]. The equivalent control of SMC can cope with matched uncertainties inherently, but for mismatched uncertainties, including faults or failures, it is an open problem [4; 5]. The SMC with a new sliding surface including the information of mismatched uncertainties through a disturbance observer was designed in [4]. The invariant manifold-based SS was applied to transform the multiple mismatched disturbances into matched ones in [5]. It has always been very interesting to find more and better ways to solve this open problem. The ADP approaches have developed a relatively complete paradigm for the optimal control [6]. An online adaptive algorithm was proposed in [7], which involves the simultaneous updating of both critic and actor NN. A classical thinking [8] of transformation from robust control into optimal control was used in RADP methods [9] for adaptive control problems subject to matched or mismatched uncertainties. Specifically, in [10], ADP was introduced to integral sliding-mode control (ISMC) for systems with mismatched disturbance, but the SS design was not considered. Hence, combining SMC with ADP to cope with mismatched uncertainties induces our attentions. The idea of using other control methods to design SS is an optional combination solution, which is an effective way to deal with nonlinear FTC problems. In [11], the classical control-based SS design combined two control algorithms. When the system states are in sliding mode, the classical control is conducted. The method [12] also adopted the above idea and used a pseudo control input in the SS design process. In [13], a novel fast terminal sliding mode control without the reaching phase was designed, which makes the derivative of an augmented state satisfy an optimal cost function. In [14], the optimal SS similar to [15] was given for a linear system. The robustness of SS to mismatched uncertainties were not considered in these two papers [13; 14]. In references [16; 17; 18; 19], optimal SSs were developed based on the first subsystem of the normal form [19]. In [16], an optimal sliding manifold was designed based on kinematics without uncertainties and the angular velocity was seen as the control input in this design process. In [17], optimal design of reaching phase and sliding phase of the SMC was given. In [18], the successive approximation approach was used to solve a novel optimal sliding mode that was designed by the same method. In [19], after an optimal equivalent control was first solved, a novel SS was derived from the above equivalent control. Recently, the ADP algorithm has also been introduced into the integral SMC design [10; 20]. In [10], ADP was used to solve the optimal guaranteed cost control problem of sliding mode dynamics. In [20], ADP was employed to design the nominal control, which aimed at stabilizing sliding mode dynamics. However, in these two papers [10; 20], ADP was adopted to stabilize sliding mode dynamics rather than to design SS directly. In this paper, we intend to explore a unifying design of sliding-mode FTC with the RADP-based SS for mismatched uncertain nonlinear cascade systems with actuator faults.