The authors propose a new tracking control design strategy for uncertain non-linear systems which are convertible to Semi-Strict Feedback Form (SSFF). The system in SSFF is first converted into new variables via existing adaptive backstepping control techniques. The control law is obtained by combining adaptive backstepping procedure and higher order integral sliding mode. The component of control law designed via backstepping is continuous which shows robustness against parametric uncertainties where as the discontinuous control component provides robustness against unmodeled dynamics and external disturbances. Since, this strategy relies on an integral manifold of the adaptively developed variables, therefore, the reaching phase is eliminated in this approach, which is an advantage in term of robustness. Furthermore, the parameters update law correctly provides the estimation of parameters which is again results in enhanced robustness of the strategy. The stability of proposed method is analysed theoretically and validated through a numerical example.
The design process of nonlinear feedback control system is quite complex and challenging. Major concern is to design a controller that guaranties some specific behaviour of the system and is robust against unmodeled dynamics, parametric uncertainties, and external disturbances. There are no specific methods or set of analysis and design tools that can fit universally for wide range of situations and applications. However several tools are available which are generally applied to control nonlinear feedback system to achieve desired regulation or output tracking.
Sliding mode control (SMC) reported by [
A systematic design procedure that resolves the issue of relative degree is the backstepping algorithm [
Sira-Ramirez et al. [
Fascinated by the strategy of DISMC, the authors have combined the adaptive backstepping and Higher Order Integral Sliding Mode Control (HOISMC) (see, e.g., [
Consider a class of nonlinear systems which can be represented/transformed into the following SSFF [
It is required that the output
It is assumed that the
In this study, a generalised recursive approach analogous to that of [
Consider a new variable which represents the error between the actual output of the system and the desired output in the following form:
Now, a virtual control input for the stabilization of the first step is designed by considering the Lyapunov function as follows:
The time derivative of the new variable defined in (
Now, define a third new variable of the form
The variable
The time derivative of
The time derivative of the
Finally, define a variable
Note that,
In this study, which is the main contribution, a control design for the nonlinear systems convertible into SSFF is considered. The design is carried out by making use of first order and second order adaptive backstepping integral sliding mode control which are discussed in detail in forthcoming subsections.
The control design for the systems which can be converted in new variables
The development of the continuous control component is presented in the form of the following proposition.
Consider that the nonlinear system with new variables
Consider a Lyapunov candidate function of the form
Note that the update law mentioned in (
In [
Consider the transformed system with the state vector
We consider the Lyapunov function of the form
Therefore, the above expression reduces to the following form:
Thus,
Note that the parameter update law which appears before (
This subsection is dedicated to the study of second order adaptive integral sliding mode control design. Since higher order integral sliding modes (see, e.g., [
On the other hand, the real twisting controller needs the time derivative of the sliding manifold which will be recovered in this work via the uniform robust exact differentiator [
Note that the parameter update law in this case bears the first component of (
This section is dedicated to verify the aforementioned claims. Therefore, consider a second order nonlinear system reported in [
The above example is simulated with the control law and parameter update laws defined above. The tracking of the
Output tracking of desired signal
The estimated parameter.
Integral manifold
The applied control input.
The AHOISMC results of tracking are somewhat similar to that reported in Figure
The
The manifold
Output disturbance added.
From the aforementioned discussions of figures, it is verified that our new proposed techniques provide robustness from the very beginning via the integral manifold approach and the robustness against parametric variations is provided via adaptive backstepping. It is therefore claimed that the above development outshines the existing adaptive sliding mode techniques.
The backstepping technique which is famous for its robust nature against parameter variation is utilized in combination with integral higher order sliding mode control strategy in order to enhance the robustness of the system from the very beginning of the process with considerable attenuation in chattering. The approach is relying on an adaptively developed new system. The integral manifold is designed in the new state variables. In other words, the manifold is adaptive in nature because of the adaptively developed states variable. The control law is designed via higher order integral sliding mode with adaptation which is capable of providing robustness against uncertainties caused by external/internal disturbances. The stability analysis is elaborated in terms of a proposition and a theorem. A numerical simulation result has verified the design approach.
The authors declare that there is no conflict of interests regarding the publication of this paper.