## 1 Introduction  
Since [PERSON] and [PERSON] realized synchronization between two chaotic systems in 1990 [1, 2], chaos synchronization has been widely explored and studied due to its potential applications in vast areas of physics and engineering science [3, 4, 5, 6]. Synchronization is a process wherein two or more systems adjust a given property of their motion. Different types of synchronization phenomena have been numerically observed and experimentally verified in a variety of chaotic systems, such as complete synchronization [1, 7], phase synchronization [8, 9], anti-phase synchronization [10, 11], lag synchronization [12, 13], generalized synchronization [14, 15] and projective synchronization [16, 17]. Complete synchronization (CS) is characterized by a coincidence of states of two chaotic systems while evolving in time. CS only appears when two interacting systems are identical. However, it is difficult to construct two absolutely identical systems in the real world. Generalized synchronization (GS) refers to a functional relation of states of two different chaotic systems, which is an extension of CS. Projective synchronization (PS) is more complex than CS but simpler than GS. Additionally, PS is useful in extending binary digital to M-nary digital communication to achieve faster communication [18, 19, 20]. PS was first reported by [PERSON] et al. [16] in a class of systems with partial linearity in which the drive and response vectors evolve in a proportional scale. [PERSON] [21] showed that the scaling factor of PS in coupled partially linear systems is unpredictable and can be arbitrarily maneuvered by introducing a feedback control to master systems. The PS between two chaotic discrete dynamical systems was achieved by [PERSON] and [PERSON] [22] via linear state error feedback control. [PERSON] and [PERSON] [23] and [PERSON] and [PERSON] [24] extended the projective synchronization feature to general nonlinear systems, including non-partially linear chaotic systems, by applying controllers to response systems, which is called generalized projective synchronization (GPS) and was studied later [25, 26, 27]. CS and anti-synchronization are particular cases of GPS. As an extension of GPS, modified projective synchronization (MPS) was introduced by [PERSON] et al. [28] and [PERSON] et al. [29], where the responses of synchronized dynamical states can be synchronized up to a constant matrix. The combined projective synchronization (CPS) among different nonlinear systems has been reported by [PERSON] et al. [30] by means of the active backstepping design method. [PERSON] et al. [31; 32] considered two chaotic systems synchronized up to a scaling function factor, which is called function projective synchronization (FPS). The FPS between two novel chaotic systems has been investigated by [PERSON] et al. [33] and [PERSON] et al. [34] using the Lyapunov method of stability. [PERSON] et al. [35] and [PERSON] et al. [36] discussed modified function projective synchronization (MFPS), in which two chaotic systems can be synchronized up to a desired scaling function matrix. As another extension of MPS and FPS, in generalized function projective synchronization (GFPS), the scaling function matrix is more generalized than that of MFPS, as has been discussed by [PERSON] and [PERSON] [37], [PERSON] and [PERSON] [38], [PERSON] et al. [39] and [PERSON] [40]. Most of the previous works have focused on the constant scaling factor and mainly on chaotic systems with exactly known parameters. However, in many practical situations, GFPS needs to be investigated while some parameters are uncertain. It is easy to understand that GFPS may be greatly affected by these uncertainties. [PERSON] and [PERSON] [37] designed an adaptive controller to achieve GFPS between two different chaotic systems and gain some parameter update laws to estimate the unknown parameters of chaotic systems.