This paper presents an original adaptive sliding mode control strategy for a class of nonlinear systems on the basis of uncertainty and disturbance estimator. The nonlinear systems can be with parametric uncertainties as well as unmatched uncertainties and external disturbances. The novel adaptive sliding mode control has several advantages over traditional sliding mode control method. Firstly, discontinuous sign function does not exist in the proposed adaptive sliding mode controller, and it is not replaced by saturation function or similar approximation functions as well. Therefore, chattering is avoided in essence, and the chattering avoidance is not at the cost of reducing the robustness of the closed-loop systems. Secondly, the uncertainties do not need to satisfy matching condition and the bounds of uncertainties are not required to be unknown. Thirdly, it is proved that the closed-loop systems have robustness to parameter uncertainties as well as unmatched model uncertainties and external disturbances. The robust stability is analyzed from a second-order linear time invariant system to a nonlinear system gradually. Simulation on a pendulum system with motor dynamics verifies the effectiveness of the proposed method.
Sliding mode control (SMC) is one of distinguished control methods because of its strong external disturbance rejection and parameter variations insensibility performance when matching condition holds. Since 1950s, SMC has attracted many attentions both in theory study and application area; see [
As known to all, one of the main obstacles for application of SMC is chattering, and when discontinuous term exists in control signal, chattering cannot avoid essentially. Many researches have proposed lots of methods to reduce or eliminate chattering. Reference [
In addition to chattering, some other disadvantages of SMC include that the bounds of uncertainty and external disturbances are required to be known usually and SMC merely guarantees complete robustness to uncertainties and external disturbances which satisfy matching condition. Hence, some researchers endeavor to improve SMC from these aspects. Reference [
So far, there is little adaptive SMC methods that can complete the following goals at the same time: (i) to avoid chattering in essence, (ii) to have strong robustness to parameter uncertainties as well as unmatched model uncertainties and external disturbances, and (iii) to avoid knowing the bounds of uncertainties. Besides, on one hand, a common way to eliminate chattering is replacing sign function
The purpose of this paper is to eliminate chattering fundamentally and to deal with parameter uncertainties as well as unmatched model uncertainties and external disturbances without requiring to know the bound of uncertainties.
In order to realize the objective, a novel uncertainty and disturbance estimator based adaptive sliding mode control (UDE-based ASMC) method is presented to avoid chattering in essence. According to I&I adaptive control strategy, parameter uncertainties can be handled well, and a controller component which acts as an equivalent control is obtained. While applying uncertainty and disturbance estimator (UDE), another controller component which is used to deal with model uncertainties and external disturbances, is constructed. This controller component is continuous, sign function
The remainder of this paper is organized as follows. In Section
First of all, consider the following second-order linear time invariant (LTI) system:
Define a sliding surface,
On one hand, one can see from (
On the other hand, the control objective is to realize that the equilibrium
Hence, let
Thus, to construct
According to I&I adaptive control approach, it is suitable to suppose a parameter estimation law as
Since
Due to that fact that (
Now, one can choose parameter estimation law as
Thus,
Let the required control be expressed as
Because of (
Differentiating (
Then
Substituting (
Here, we select
Then
With (
Because
According to Lyapunov stability theory, the derivative of the Lyapunov function should be negative in order to guarantee the stability of systems. Therefore, it is expected that the term
Rewrite (
Clearly,
Suppose
Now, the estimate variable
Because
Up to now, adaptive sliding mode control law for second-order LTI system is obtained:
For second-order LTI systems (
According to the above derivation process, the results can be directly obtained from Barbashin-Krasovskii theorem [
The above result is based on the premise that (
When the low-pass filter (
The smooth function
Consider the following nonlinear system, which is with parametric uncertainties as well as unmatched model uncertainties and external disturbances,
In the following, when there is no confusion, we will use
For nonlinear systems (
There exists a full information bounded control law
Designable parameters
The change rate of
Based on the UDE method, the way to obtain control component
Equation (
Rewrite (
Suppose
Now, the estimate variable
Because
Substituting (
According to estimation error (
Based on (
Select Lyapunov function as
Then, due to (
Assumptions
In LTI systems (
Similar to that of Remarks
From (
In order to verify the validity of the proposed UDE-based ASMC method, a pendulum with motor dynamics given in [
According to the form of (
Select
Then
According to (
In Figures
Comparison about states
Comparison about sliding mode
Figures
To make comparison, simulation results with the method proposed by [
From the comparison, one can find that, under the presented UDE-based ASMC, the states converge faster than that of [
Comparison between the two methods.
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|
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Method in this paper | Peak value | 1 | −1.0301 | 10.2015 | 20.9345 |
Converge time | 2 | 3 | 2.5 | 3 | |
Method in Kwan (1995) [ |
Peak value | 1 | −0.4604 | 10.8797 | 81.243 |
Converge time | 7 | 6 | 8 |
|
In this paper, a novel uncertainty and disturbance estimator-based adaptive sliding mode control (UDE-based ASMC) method is presented for nonlinear systems, which are with parameter uncertainties as well as unmatched model uncertainties and external disturbances. Parameter estimation law is obtained according to immersion and invariance (I&I) adaptive control approach, and uncertainty and disturbance estimator is employed to realize chattering elimination in essence. UDE-based ASMC can guarantee closed-loop systems that have strong robustness without requiring to know the bounds of uncertainties. At the end of this paper, UDE-based ASMC is applied to a pendulum with motor dynamics, simulation results illustrate the approving performance, including chattering elimination, control peak decreasing, fast convergence, and strong robustness to parameter uncertainties and external uncertainties.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the Natural Science Foundation of China (51275249) and the Talent Introduction Foundation of Engineering College Nanjing Agricultural University (Rcqd11-06).