An adaptive control using fuzzy basis function expansions for a class of nonlinear systems (Q1389066)

This paper present a robust model reference adaptive sliding mode tracking control scheme for a class of nonlinear systems in the form of \(x^{(n)} +f({\mathbf x}) =bu\) or \[ \dot{\mathbf x} =A{\mathbf x} +{\mathbf b} u+{\mathbf f}, \] \[ \text{ where }A= \left[\begin{matrix} \mathbf{0}_{(n-1) \times 1} & I_{n-1}\\ 0 & \text\textbf{0}_{1\times (n-1)} \end{matrix} \right]_{n\times n}, \] \[ {\mathbf b}= [0,\dots, 0,b]^\tau,\;{\mathbf f} =[0, \dots, 0,-f]^\tau \text{ and } {\mathbf x} =[x(t), \dot x(t), \dots, x^{(n-1)} (t)]^\tau \] using the fuzzy basis function expansion \(\varphi ({\mathbf x})= \sum^M_{ j=1} \theta_j p_j({\mathbf x})\), where \(\theta_j\) is the \(j\)th linear combination coefficient, the fuzzy basis function and the membership function of \(x_i\) in the fuzzy set \(A^j_i\) are respectively as follows \[ p_j({\mathbf x})= {\prod^n_{i=1} \mu_{A_i^j} (x_i)\over \sum^M_{j=1} \prod^n_{i=1} \mu_{A_i^j} (x_i)},\;\mu_{A^j_i} (x_i)= \exp\left[ -\left({x_i-c_{A_{ij}} \over \sigma_{ A_{ij}}} \right)^2 \right],\;j=1,2, \dots,M. \] The desired reference model with input \(r(t)\) is given by \[ \dot{\mathbf x}_m= A_m{\mathbf x}_m +{\mathbf b}_mr(t) \qquad ({\mathbf x}_m =[x_m,\dot x_m, \dots, x_m^{(n-1)}]^\tau) \] and thus the dynamics of the output tracking error vector \({\mathbf e}= {\mathbf x} -{\mathbf x}_m\) is \(\dot{\mathbf e} =A{\mathbf e} +(A-A_m) {\mathbf x}_m +{\mathbf f} -{\mathbf b}_m r+ {\mathbf b}u\). The main conclusions in this paper are: When taking the sliding mode variable \(s={\mathbf c}^\tau {\mathbf e}\), \({\mathbf c} =[c_1,c_2, \dots, c_n]^\tau\), \(c_n>0\), and the control gain \(b>b_1>0\), the output tracking error asymptotically converges to zero, if the control input is designed such that in the case of \(| f({\mathbf x}) |<f_0\) and \(f_0>0\) \[ u=-{\text{sign} (s)\over c_nb_1} \biggl[| {\mathbf c}^\tau A{\mathbf e} |+\bigl| {\mathbf c}^\tau (A-A_m){\mathbf x}_m \bigr| +c_nf_0+ |{\mathbf c}^\tau {\mathbf b} r|\biggr] \] or in the case of unknown \(1/b_1\) and \(f_0/b_1\) to be respectively learned by two fuzzy basis function expansion networks \(\widehat\theta^\tau_1 \varphi ({\mathbf x})\) and \(\widehat \theta_2^\tau \psi ({\mathbf x})\) in the Lyapunov sense (and the output of the fuzzy networks are then used as the parameters of the controller to adaptively compensate for the effects of system uncertainties) if the input is chosen as \[ u=- \widehat\theta^\tau_1 \varphi({\mathbf x}) {\text{sign} (s)\over c_n} \biggl[|{\mathbf c}^\tau A{\mathbf e} |+\bigl |{\mathbf c}^\tau (A-A_m) {\mathbf x}_m \bigr| +| {\mathbf c}^\tau {\mathbf b}_m r|- \widehat\theta_2^\tau \psi({\mathbf x}) \text{sign} (s)\biggr] \] where the weight vectors are adjusted by using the following adaptive mechanisms: \[ {\overset {.} {\widehat\theta}}_1 =\eta_1\biggl[| {\mathbf c}^\tau A{\mathbf e}| +\bigl| {\mathbf c}^\tau (A-A_m) {\mathbf x}_m \bigr|+ | {\mathbf c}^\tau {\mathbf b}_m r|\biggr] | s| \varphi ({\mathbf x}),\;{\overset {.} {\widehat\theta}}_2= \eta_2c_n | s|\psi ({\mathbf x}). \] So, using this scheme, strong robustness with respect to unknown system dynamics and nonlinearities can be obtained. Simulation results demonstrate the effectiveness, simplicity and practicality of the proposed control scheme.