Modular‐based adaptive control of uncertain nonlinear time‐varying systems with piecewise jumping parameters

Abstract

Nonlinear time-varying systems exist widely in practice. Therefore, it is of great theoretical importance and practical value to investigate the problem of controlling such systems. However, the results available in developing adaptive control to address such a problem are still limited. Especially a majority of them are restricted to be slowly time-varying linear systems. This paper presents a modular-based adaptive control scheme for parametric strict feedback nonlinear time-varying systems. The parameters considered include both continuous and piecewise time-varying parameters, and they are not necessarily restricted to be slowly time-varying or infrequent jumping. The technique of adaptive backstepping with nonlinear damping is employed in the control design module, while the parameter projection algorithm is performed on the parameter estimation module. It is proved that the uniform boundedness of all closed-loop system signals can be guaranteed with the proposed control scheme. The performance of the tracking error in the mean square sense with respect to the parameter variation rate is also established. Furthermore, perfect asymptotically tracking can be achieved when the varying rates of unknown parameters are in the L 2 space.A few adaptive nonlinear control design tools described in [7] and [8] have been applied to LTV systems [9] [10,11] and nonlinear time-varying systems [12,13]. Although these existing results achieve stability and improve performance, they are mainly dependent upon integrator backstepping method with tuning function and Kreisselmeier/Marino&Tomei (K/MT) filters; on the other hand, they also require the time variations of the parameters to be either sufficiently slow or infrequent jumps. Namely, the rate of parameter variations is constrained by a sufficiently small upper bound.In this paper, on the basis of our recent preliminary work in [14], we further address the issue of controlling uncertain nonlinear time-varying systems by utilizing the modular-based adaptive control scheme in which the controller module and the parameter identifier module are designed separately, which is different from the tuning function approach adopted in [9], [15], [16,17], and [18] and the output feedback approach in [19] and [20]. More importantly, the usual restrictive parameter assumption of sufficiently slow time variations and infrequent jumps is removed. This implies that the variation rates of unknown continuous time-varying parameters can be bounded by arbitrary constants while piecewise time-varying parameters are allowed to jump with any finite frequencies and they are not limited to be piecewise constant like in [14]. Actually, there is no result available to control nonlinear systems with discontinuous piecewise time-varying parameters. It is shown that the proposed scheme can ensure the uniform boundedness of all the closed-loop system signals. A relationship between the tracking error and the rates of parameter variations is also established. It is shown that the slower the time variation...

Robust adaptive output feedback control for uncertain nonlinear systems with quantized input

“…307–313 of may be applied to address the issue. For the case of time‐varying control gains, we need to consider their time derivatives, which can be handled by following the approaches proposed for time‐varying systems, see and for examples.…”

Linear backstepping output feedback control for uncertain linear systems

“…In this part, a Lyapunov-based identifier to estimate the unknown parameter vector  is designed. The update law of O  is chosen as follows: [9] and [11], we now derive a similar Proj … ¹ º to [12][13][14]. Choose a smooth convex…”

Adaptive control of feedforward nonlinear systems with uncertain time‐varying parameters and an unknown time‐varying delay in the input

“…Regarding uncertain time‐varying parameters, there are several existing results 17‐21 . In Reference 18, a robust regulation scheme is proposed for uncertain systems with a known constant delay in the states and unknown time‐varying parameters.…”

“…So, the method of Reference 20 is not applicable when there are time‐varying parameters in both states and input. In Reference 21, they propose a state feedback controller for nonlinear systems having uncertain time‐varying parameters occurring with the input and nonlinearity in which parameters are not necessarily slowly varying. However, since time‐varying rates of parameters associated with the input and nonlinearity are bounded, their control method is not applicable to nonlinear systems having arbitrarily fast‐varying parameters.…”

“…Although in Reference 22, the authors propose a memoryless controller to regulate nonlinear systems with an unknown time‐varying delay in the input, their result cannot treat nonlinear systems with uncertain time‐varying parameters. Specifically, our considered problems have the following generalized features: (i) the delay is time‐varying and unknown where there is no information on the upper bound of delay over; 1,4,6 (ii) there exists uncertain time‐varying parameters associated with main diagonal states and input where these uncertain parameters are not necessarily small over 3,12,14,20 ; (iii) there is no restriction on the time‐varying rates of delay and parameters over; 14,17‐21 (iv) nonlinearities are extended by containing unknown growth rate over 5‐7,14 . Overall, all the aforementioned existing results 1‐7,12,14,17‐22 are not applicable to our considered system.…”