Continuous-time Lyapunov stability analysis and systematic parametrization of robust adaptive sliding mode controller for systems with matched and unmatched dynamics

Abstract

This paper presents the development of a continuous-time robust adaptive sliding mode controller using the model reference adaptive control philosophy. The stability analysis of the controller considering a system subjected to matched and unmatched dynamics is provided using the Lyapunov stability criterion. This control strategy can be applied to plants that are partially modeled, systems with uncertain parameters, or unmodeled dynamics. The stability analysis elucidates the controller constraints and proves that the tracking error tends to a small residual value, even in the presence of unmodeled dynamics (matched or unmatched). In addition, a systematic controller parametrization procedure based on the sine–cosine algorithm is presented to automate this task. Simulation results of the robust adaptive continuous-time sliding mode controller applied to an unstable non-minimum-phase system are presented. A comparison of this controller with a robust model reference adaptive controller is also presented, where the benefits of the adaptive sliding mode controller stand out, obtaining a superior performance that reduces relevantly the error metrics of 56.28%, 28.57%, and 14.79% for mean absolute error, mean squared error, and root mean squared error, respectively. Furthermore, a processor-in-the-loop experiment considering a complex real-world engineering problem is also provided to corroborate the controller performance and discuss its robustness, demonstrating its feasibility.

Appendix A - Lemma 2

This appendix presents Lemma 2, which contains useful results for the stability analysis of the continuous-time RMRAC-SM.

Lemma 2

The value of $2\sigma \varvec{\phi } \varvec{\theta }$ is greater or, at maximum, equal to zero.

Proof

Let

$$\begin{aligned} {\left\| {{\varvec{\theta } ^*}} \right\| ^2}{} & {} = {\varvec{\theta } ^*}^T{\varvec{\theta } ^*} = {(\;\varvec{\theta } - \varvec{\phi } )^T}(\varvec{\theta } - \varvec{\phi } )\; \nonumber \\ {}{} & {} = \;{\varvec{\theta } ^T}\varvec{\theta } - 2{\varvec{\phi } ^T}\varvec{\theta } + {\varvec{\phi } ^T}\varvec{\phi }, \end{aligned}$$

(73)

or yet,

$$\begin{aligned} 2{\varvec{\phi } ^T}\varvec{\theta } = \;\;{\left\| \varvec{\theta } \right\| ^2}\; + \;{\left\| \varvec{\phi } \right\| ^2}\;\; - \;{\left\| {{\varvec{\theta } ^*}} \right\| ^2}. \end{aligned}$$

(74)

From (74), it is written as the following inequality, $2{\varvec{\phi } ^T}\varvec{\theta } \ge {\left\| \varvec{\theta } \right\| ^2} - {\left\| {{\varvec{\theta } ^*}} \right\| ^2}$, which can be expressed as

$$\begin{aligned} 2\sigma {\varvec{\phi } ^T}\varvec{\theta } \ge \sigma \left( {{{\left\| \varvec{\theta } \right\| }^{2}} - {{\left\| {{\varvec{\theta } {^*}}} \right\| }^{2}}} \right) . \end{aligned}$$

(75)

From (36) and (75), it can be concluded

$$\begin{aligned} \left\{ \begin{matrix} 2\sigma {{\bar{\varvec{\phi }} }^T}\bar{\varvec{\theta }}> 0&{}\mathrm{{ if }} &{}\left\| {\bar{\varvec{\theta }} } \right\| \ge 2{M_0}\\ 2\sigma {{\bar{\varvec{\phi }} }^T}\bar{\varvec{\theta }} > 0&{}\mathrm{{ if }}&{}{M_0}\mathrm{{< }}\left\| {\bar{\varvec{\theta }} } \right\| < 2{M_0}.\\ 2\sigma {{\bar{\varvec{\phi }} }^T}\bar{\varvec{\theta }} = 0&{}\mathrm{{ if }}&{}\left\| {\bar{\varvec{\theta }} } \right\| \le {M_0} \end{matrix} \right. \end{aligned}$$

(76)

Therefore, $2\sigma {\varvec{\phi } ^T}\varvec{\theta } \ge 0$.

$\square$