Design of Robust Model Reference Adaptive Controller for a Wider Class of Nonlinear Systems

Abstract

In this paper, a novel control approach, the state-dependent robust model reference adaptive controller, is presented for a broad class of nonlinear systems with unknown parameters and matched uncertainties. At first, the well-known model reference adaptive control (MRAC) is designed for the system by considering the pseudo-linearized description, making it possible to deal with a broad category of nonlinear systems with unknown parameters without linearization. Also, state-dependent matrices are used to calculate time-varying gains, which extends the domain of the controller. In many cases, the fast adaptation of parameters using the MRAC technique may result in undesired oscillations in the control law and system states. Then, the robust controller is designed to increase the robustness of the system against these oscillations. The error feedback signal is included in the system state equations as a control input to decrease the amplitude of oscillations by solving a series of arisen state-dependent linear matrix inequalities. The main contribution of this paper, rather than some of the other proposed robust MRAC controllers, is that it covers a broad class of nonlinear systems. Finally, to demonstrate the effectiveness of the proposed technique in terms of convergence rate, accuracy, and robustness in the reference trajectory tracking, it is applied to Genesio chaotic system having unknown parameters.

Appendix

More details about the proof of stability of the Lyapunov function in Theorem 1 and the procedure of obtaining (21) from (20) is as following:

$$\begin{aligned} {\dot{V}}\left( e\left( t \right) ,{\tilde{\theta }}\left( t \right) \right)&={{e}^{T}}\left( t \right) \left( A_{m}^{T}P+P{{A}_{m}} \right) e\left( t \right) \nonumber \\&\quad -2{{e}^{T}}\left( t \right) PB\left( x\left( t \right) \right) {{{{\tilde{\theta }}}}^{T}}\left( t \right) \varphi \left( x\left( t \right) \right) \nonumber \\&\quad +2trace\left( {{{\dot{{\tilde{\theta }}}}}^{T}}\left( t \right) {{\varSigma }^{-1}}{\tilde{\theta }}\left( t \right) \right) \end{aligned}$$

(41)

according to proposition 5 the derivative of $\theta$ is equal to zero, so we have:

$$\begin{aligned} {\tilde{\theta }}\left( t \right)&={\hat{\theta }}\left( t \right) -\theta \rightarrow \dot{{\tilde{\theta }}}\left( t \right) =\dot{{\hat{\theta }}}\left( t \right) \rightarrow \dot{{\tilde{\theta }}}\left( t \right) \nonumber \\&=\varSigma \varphi \left( x\left( t \right) \right) {{e}^{T}}\left( t \right) PB\left( x\left( t \right) \right) \end{aligned}$$

(42)

also by considering (43) and (44) corresponding (13) and (16)

$$\begin{aligned}&\dot{{\hat{\theta }}}\left( t \right) =\varSigma \varphi \left( x\left( t \right) \right) {{e}^{T}}\left( t \right) PB\left( x\left( t \right) \right) \end{aligned}$$

(43)

$$\begin{aligned}&A_{m}^{T}P+P{{A}_{m}}=-Q \end{aligned}$$

(44)

and the fact that $trace\left( x{{y}^{T}} \right) ={{x}^{T}}y$, we would have

$$\begin{aligned} \begin{aligned}&2trace\left( {{{\dot{{\tilde{\theta }}}}}^{T}}\left( t \right) {{\varSigma }^{-1}}{\tilde{\theta }}\left( t \right) \right) \\&\quad =2trace\left( {{{\dot{{\hat{\theta }}}}}^{T}}\left( t \right) {{\varSigma }^{-1}}{\tilde{\theta }}\left( t \right) \right) \\&\quad = 2trace\left( {{\left[ \varSigma \varphi \left( x\left( t \right) \right) {{e}^{T}}\left( t \right) PB\left( x\left( t \right) \right) \right] }^{T}}{{\varSigma }^{-1}}{\tilde{\theta }}\left( t \right) \right) \\&\quad = 2trace\left( {{B}^{T}}\left( x\left( t \right) \right) {{P}^{T}}e\left( t \right) {{\varphi }^{T}}\left( x\left( t \right) \right) {{\varSigma }^{T}}{{\varSigma }^{-1}}{\tilde{\theta }}\left( t \right) \right) \\&\quad = 2trace\left( {{B}^{T}}\left( x\left( t \right) \right) {{P}^{T}}e\left( t \right) {{\varphi }^{T}}\left( x\left( t \right) \right) {\tilde{\theta }}\left( t \right) \right) \\&\qquad \xrightarrow [{{\varphi }^{T}}\left( x\left( t \right) \right) {\tilde{\theta }}\left( t \right) ={{y}^{T}}]{{{B}^{T}}\left( x\left( t \right) \right) {{P}^{T}}e\left( t \right) =x} \\&\quad =2{{e}^{T}}\left( t \right) PB\left( x\left( t \right) \right) {{{{\tilde{\theta }}}}^{T}}\left( t \right) \varphi \left( x\left( t \right) \right) \\ \end{aligned} \end{aligned}$$

(45)

Finally

$$\begin{aligned} \begin{aligned}&\dot{V}\left( e\left( t \right) ,{\tilde{\theta }}\left( t \right) \right) \\&\quad =-{{e}^{T}}\left( t \right) Qe\left( t \right) -2{{e}^{T}}\left( t \right) PB\left( x\left( t \right) \right) {{{{\tilde{\theta }}}}^{T}}\left( t \right) \varphi \left( x\left( t \right) \right) \\&\qquad +2{{e}^{T}}\left( t \right) PB\left( x\left( t \right) \right) {{{{\tilde{\theta }}}}^{T}}\left( t \right) \varphi \left( x\left( t \right) \right) \\&\quad =-{{e}^{T}}\left( t \right) Qe\left( t \right) \prec 0\\ \end{aligned} \end{aligned}$$

(46)

Also, the same procedure could be followed for the derivative of Lyapunov function (30) in Theorem 2.