Adaptive quantized fuzzy control of stochastic nonlinear systems with actuator dead-zone
Introduction
In last years, networked control systems(NCSs), due to their such important merits as flexibility, low cost, and ease of maintenance, have aroused wide concern (see, e.g., [1]). What features the networked control systems is the transmitting messages from the controller to the plant through the limited communication cables. To avoid a network congestion and make the systems work properly within the given bandwidth, the control signal should be quantized before transmission. A key goal of a quantized control scheme is to ensure the performance of the control system at a low communication rate. The last years have witnessed many quantized control results, see, e.g., [2], [3], [4], [5], [6], [7], [8], [9], [10]. Among them, the stabilization of linear quantized systems was considered in [2], [3], [4], [5], the stabilization of nonlinear quantized systems was studied in [6], [7], [8], H∞ control of networked linear systems was investigated in [9], [10]. However, the models of the plants in [1], [2], [3], [4], [5], [6], [7], [8], [9], [10] are required to be completely known.
Due to the impact of random disturbance, the practically networked control systems are uncertain [11], [12], [13], [14], [15], [16]. By using robust control approaches, the uncertain systems with quantized input were investigated in [17], [18], [19], [20], [21], [22], [23]. By applying online learning ability of adaptive controller, some adaptive control approaches to uncertain systems with input quantization were developed in [24], [25], [26]. In [24] and [25], the quantized control strategies were set up for linear systems. In [26], by constructing a novel hysteretic quantizer, an adaptive quantized controller was established for nonlinear ones. However, the stability conditions of the systems in [24], [25], [26] are not easy to obtain ahead of time, because the stability conditions are determined by the value of control signal. To relax the stability conditions in [24], [25], [26], a new inequality concerning controller design parameters and system parameters was established in [27]. Although some progress has been made in [27], the restrictive assumptions of nonlinear functions of system and quantizer parameters are required in control design. If such restrictive assumptions are not satisfied, the control approach in [27] will become infeasible. More recently, to remove the rigorous assumptions, two novel adaptive fuzzy controllers were established for nonlinear system in [28], [29]. However, the above literatures on quantized work don’t take into account the negative effect of actuator nonlinearity. Moreover, the plants in [24], [25], [26], [27], [28], [29] are strict-feedback nonlinear systems, and the analytical approaches are unsuitable for non-strict feedback nonlinear systems. As mentioned in [30], [31], [32], [33], non-strict feedback nonlinear systems have achieved widespread application but for such systems the control is more difficult.
As an important non-smooth nonlinearity, a dead zone is common in many practical actuators, which may cause control failure or system instability. To cope with this dead-zone nonlinearity, two main ways are often used to design controllers. One is to create a smooth inverse of dead-zone to compensate the impact, as shown in [34], [35], [36], [37], [38], [39], [40]. The other is to construct a simplified dead zone model to facilitate the control design, as was done in [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57]. It must be emphasized, however, that the aforementioned results are based on the implicit assumption that the control command transmitted between the controller and the physical actuator doesn’t exhibit a quantized phenomena. As far as we know, the tracking control issue of nonlinear quantized systems with actuator dead zone has not been explored in spite of its potentialities in networked control systems, which encourages the current research. So this manuscript investigates the quantized control of stochastic nonlinear systems in non-strict feedback form with actuator dead zone. The goal of this manuscript is to provide an ordinary frame for the study of nonlinear quantized systems with actuator nonlinearity. The following are the main contributions of this manuscript.
- (1)
It is the first time that both the quantized effect and the unknown actuator nonlinearity are taken into consideration in stochastic nonlinear systems. In addition, the structure of the plants is in non-strict feedback form, and the control for this kind of stochastic quantized systems is more difficult than a strict-feedback form. Therefore, in contrast with the past works, the controlled system is more general and the control design is more challenging.
- (2)
Theoretically and technically, to cope with the difficulty resulting from the unknown actuator dead zone and the quantization effect of the control signal, a novel connection between the control signal and the system input is established. By applying this connection, an adaptive fuzzy control approach could be used to look into the quantized system. Furthermore, by adjusting the design parameters of the adaptive controller, the quantization error can be minimized and the desired tracking performance is guaranteed.
Section snippets
Stochastic stability
Consider the stochastic nonlinear system as follows:
where denotes the state variable, and satisfy ; w denotes an independent dimensional standard Brownian motion defined on the complete probability space.
Definition 1 According to [58], the differential operator of V(x) ∈ C2 can be defined as the following form:
where Tr is the trace of a matrix. Definition 2 The solution x(t), t ≥ 0 of the system (1) is called semi-globally uniformly[58]
[58]
Adaptive fuzzy tracking control design
In the subsequent controller design, Φi(Xi) will be adopted to approximate unknown nonlinear functions. Let and be an estimation of θi, then an estimation error is expressed as . Furthermore, the following coordinate transformation is adopted: where where ki > 0 and ai > 0 are design constants. with represents the basis vector function of the
Simulation example
Example Consider a non-strict feedback stochastic nonlinear quantized system with actuator nonlinearity as follows:
where Γ(Q(u)) denotes the input of the system defined in (7), which is subject to quantization effect and actuator dead zone, Q(u) is defined in (8). In simulation, the quantized parameters and the dead zone parameters are chosen as . The reference signal is
Conclusion
This paper investigates a quantized control issue of stochastic non-strict feedback nonlinear systems with actuator nonlinearity. By utilizing the proposed connection between the control signal and the system input, the stochastic nonlinear quantized control problem is transformed into the conventional nonlinear control of stochastic system with unknown control gain and bounded perturbation. Subsequently, an adaptive fuzzy compensation scheme is established to cancel the effects of unknown
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under Grant 61573108, Grant 61503223, and Grant U1501251, in part by the Ministry of Education of New Century Excellent Talent under Grant NCET-12-0637, in part by the Natural Science Foundation of Guangdong Province under Grant 2016A030313715 and the Science Fund for Distinguished Young Scholars under Grant S20120011437, in part by the Doctoral Fund of Ministry of Education of China under Grant 20124420130001,
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2020, ISA TransactionsCitation Excerpt :In high precision control, the dead-zone parameter with quantization nonlinearity may cause poor tracking performance. The quantization and dead-zone nonlinearity based control design has been addressed in [22,46]. These studies have been applied to a particular class of large scale and stochastic nonlinear systems.