Stability analysis of dynamic nonlinear interval type-2 TSK fuzzy control systems based on describing function

Abstract

This paper focuses on the limit cycles prediction problem to discuss the stability analysis of dynamic nonlinear interval type-2 Takagi–Sugeno–Kang fuzzy control systems (NIT2 TSK FCSs) with adjustable parameters. First, in order to alleviate computational burden, a simple architecture of NIT2 TSK FCS using two embedded nonlinear type-1 TSK fuzzy control systems (NT1 TSK FCSs) is proposed. Then, describing function (DF) of NIT2 TSK FCS is obtained based on the DFs of embedded NT1 TSK FCSs. Subsequently, integrating the stability equation and parameter plane approaches provides a solution to identify the limit cycle and the asymptotically stable regions. Moreover, particle swarm optimization technique is applied to minimize the limit cycle region. Furthermore, for robust design, a gain-phase margin tester is utilized to specify the minimum gain margin ($\hbox {GM}_{\mathrm{min}}$) and phase margin ($\hbox {PM}_{\mathrm{min}}$) when limit cycles can arise. Finally, two simulation examples are considered to validate the advantages of the presented method.

Appendix

1.1 A. Proof of Lemma 1:

$\forall t_1\in R$, let $x_1\equiv A\hbox {sin}\omega t_1$ and ${{\dot{x}}}_1\equiv \omega A\hbox {cos}\omega t_1$.

For LNT1 TSK FCS: As shown in Fig. 5, in the case

$\Phi _{k} \le x_{1} <\Phi _{k+1}$ and $\Psi _{l} \le {\dot{x}}_{1} <\Psi _{l+1}$, we have

$$\begin{aligned} {\underline{u}}(t_{1} )&=\frac{1}{{\underline{\zeta }}} \mathrm{((}{\underline{M}}_{k} (x_{1} ){\underline{Z}}_{k,l} (x_{1} ,{\dot{x}}_{1} )\mathrm{)}\Pi _{1}^{x_{1} } \mathrm{+(}{\underline{M}}_{k} (x_{1} ){\underline{Z}}_{k,l} (x_{1} ,{\dot{x}}_{1})\\&\quad +\,{\underline{M}}_{k+1} (x_{1} ){\underline{Z}}_{k+1,l} (x_{1} ,{\dot{x}}_{1} )\mathrm{)}\Pi _{2}^{x_{1} } \\&\quad +\,\mathrm{(}{\underline{M}}_{k+1} (x_{1} ){\underline{Z}}_{k+1,l} (x_{1} ,{\dot{x}}_{1} )\mathrm{)}\Pi _{3}^{x_{1} } \mathrm{),\; \; \; } \end{aligned}$$

where

$$\begin{aligned} {\underline{Z}}_{k,l} (x_{1} ,{\dot{x}}_{1} )&={\underline{N}}_{l} ({\dot{x}}_{1} ){\underline{u}}_{k,l} (x_{1} ,{\dot{x}}_{1} )\Pi _{1}^{{\dot{x}}_{1} } +({\underline{N}}_{l} ({\dot{x}}_{1} ){\underline{u}}_{k,l} (x_{1} ,{\dot{x}}_{1} )\\&\quad +\,{\underline{N}}_{l+1}({\dot{x}}_{1} ){\underline{u}}_{k,l+1} (x_{1} ,{\dot{x}}_{1} ))\Pi _{2}^{{\dot{x}}_{1} } \\&\quad +\,{\underline{N}}_{l+1} ({\dot{x}}_{1} ){\underline{u}}_{k,l+1} (x_{1} ,{\dot{x}}_{1} )\Pi _{3}^{{\dot{x}}_{1} }.\\ {\underline{Z}}_{k+1,l} ({\dot{x}}_{1} )=&{\underline{N}}_{l} ({\dot{x}}_{1} ){\underline{u}}_{k+1,l} (x_{1} ,{\dot{x}}_{1} )\Pi _{1}^{{\dot{x}}_{1} }\\&\quad +\,({\underline{N}}_{l} ({\dot{x}}_{1} ){\underline{u}}_{k+1,l} (x_{1} ,{\dot{x}}_{1} )\\&\quad +\,{\underline{N}}_{l+1} ({\dot{x}}_{1} ){\underline{u}}_{k+1,l+1} (x_{1} ,{\dot{x}}_{1} ))\Pi _{2}^{{\dot{x}}_{1}} \\&\quad +\,{\underline{N}}_{l+1} ({\dot{x}}_{1} ){\underline{u}}_{k+1,l+1} (x_{1} ,{\dot{x}}_{1} )\Pi _{3}^{{\dot{x}}_{1} }. \end{aligned}$$

As proved in Kim et al. (2000), for $t_2\equiv t_1-({\pi }/{\omega })$, $x_2=-x_1$ and ${{\dot{x}}}_2=-{{\dot{x}}}_1$. Therefore,

$$\begin{aligned} {\underline{u}}(t_{2} )&=\frac{1}{{\underline{\zeta }}} \mathrm{((}{\underline{M}}_{-k} (x_{2} ){\underline{Z}}_{-k,-l} (x_{2} ,{\dot{x}}_{2} )\mathrm{)}\Pi _{1}^{x_{2} }\\&\quad +\,\mathrm{(}{\underline{M}}_{-k} (x_{2} ){\underline{Z}}_{-k,-l} (x_{2} ,{\dot{x}}_{2} )\\&\quad +\,{\underline{M}}_{-k-1} (x_{2}){\underline{Z}}_{-k-1,-l} (x_{2} ,{\dot{x}}_{2} )\mathrm{)}\Pi _{2}^{x_{2} } \mathrm{+(}{\underline{M}}_{-k-1} (x_{2} )\\&\quad {\underline{Z}}_{-k-1,-l} (x_{2} ,{\dot{x}}_{2} )\mathrm{)}\Pi _{3}^{x_{2} } \mathrm{)}=-{\underline{u}}(t_{1} ). \end{aligned}$$

Therefore, ${\underline{u}}(t-\left( \frac{\pi }{\omega } \right) )=-{\underline{u}}(t).$ As shown in Fig. 4, in the case $\Phi _{k} \le x<\Phi _{k+1}$ and $\Psi _{l} \le {\dot{x}}<\Psi _{l+1}$, we have

$$\begin{aligned} {\bar{u}}(t)&=\frac{1}{{\bar{\zeta }}} (({\bar{M}}_{k-1} (x){\bar{Z}}_{k-1,l} (x,{\dot{x}}))\Pi _{1}^{x} +({\bar{M}}_{k} (x){\bar{Z}}_{k,l} (x,{\dot{x}})\\&\quad +\,{\bar{M}}_{k+1} (x){\bar{Z}}_{k+1,l} (x,{\dot{x}}))\Pi _{2}^{x} \\&\quad +\,({\underline{M}}_{k+2} (x){\bar{Z}}_{k+2,l} (x,{\dot{x}}))\Pi _{3}^{x} ), \end{aligned}$$

where

$$\begin{aligned} {\bar{Z}}_{k-1,l} (x,{\dot{x}})&={\bar{N}}_{l-1} {\bar{u}}_{k-1,l-1} (x,{\dot{x}})\Pi _{1}^{{\dot{x}}}\\&\quad +\,({\bar{N}}_{l} {\bar{u}}_{k-1,l} (x,{\dot{x}})+{\bar{N}}_{l+1} {\bar{u}}_{k-1,l+1} (x,{\dot{x}}))\Pi _{2}^{{\dot{x}}} \\&\quad +\,{\bar{N}}_{l+2} {\bar{u}}_{k-1,l+2} (x,{\dot{x}})\Pi _{3}^{{\dot{x}}} .\\ {\bar{Z}}_{k,l} (x,{\dot{x}})=&{\bar{N}}_{l-1} {\bar{u}}_{k,l-1} (x,{\dot{x}})\Pi _{1}^{{\dot{x}}} +({\bar{N}}_{l} {\bar{u}}_{k,l} (x,{\dot{x}})\\&\quad +\,{\bar{N}}_{l+1} {\bar{u}}_{k,l+1} (x,{\dot{x}}))\Pi _{2}^{{\dot{x}}} +{\bar{N}}_{l+2} {\bar{u}}_{k,l+2} (x,{\dot{x}})\Pi _{3}^{{\dot{x}}}.\\ {\bar{Z}}_{k+1,l} (x,{\dot{x}})=&{\bar{N}}_{l-1} {\bar{u}}_{k+1,l-1} (x,{\dot{x}})\Pi _{1}^{{\dot{x}}} +({\bar{N}}_{l} {\bar{u}}_{k+1,l} (x,{\dot{x}})\\&\quad +\,{\bar{N}}_{l+1} {\bar{u}}_{k+1,l+1} (x,{\dot{x}}))\Pi _{2}^{{\dot{x}}} \\&\quad +\,{\bar{N}}_{l+2} {\bar{u}}_{k+1,l+2} (x,{\dot{x}})\Pi _{3}^{{\dot{x}}} .\\ {\bar{Z}}_{k+2,l} (x,{\dot{x}})=&{\bar{N}}_{l-1} {\bar{u}}_{k+2,l-1} (x,{\dot{x}})\Pi _{1}^{{\dot{x}}} +({\bar{N}}_{l} {\bar{u}}_{k+2,l} (x,{\dot{x}})\\&\quad +\,{\bar{N}}_{l+1} {\bar{u}}_{k+2,l+1} (x,{\dot{x}}))\Pi _{2}^{{\dot{x}}} \\&\quad +\,{\bar{N}}_{l+2} {\bar{u}}_{k+2,l+2} (x,{\dot{x}})\Pi _{3}^{{\dot{x}}} . \end{aligned}$$

The proof of ${\overline{u}}\left( t\right) =-{\overline{u}}\left( t-({\pi }/{\omega })\right)$ is similar to the proof of ${\underline{u}}\left( t\right) =-{\underline{u}}\left( t-({\pi }/{\omega })\right)$.

1.2 B. Proof of Lemma 2:

To obtain ${{\underline{a}}}_{r,k,l}$, $\ {{\underline{b}}}_{r,k,l}$, ${{\underline{c}}}_{r,k,l}$, and ${{\underline{d}}}_{r,k,l}$ and ${\; }\Phi _{k} +\tau _{e} \le x<\Phi _{k+1} -\tau _{e} ,{\; \; }\Psi _{l} +\tau _{{\dot{e}}} \le {\dot{x}}<\Psi _{l+1} -\tau _{{\dot{e}}}$.

$$\begin{aligned} {\underline{u}}(x,\dot{x})&= {1 {/} {{{\underline{\zeta }}}}}({{{\bar{M}}}_k}(x){{\underline{N}}_l}({\dot{x}}) {{\underline{u}}_{k,l}}(x,{\dot{x}})\\&\quad +\,{{\underline{M}}_k}(x){{\underline{N}}_{l + 1}}({\dot{x}}) {{\underline{u}}_{k,l + 1}}(x,{\dot{x}})\\&\quad +\, {{\underline{M}}_{k + 1}}(x){{\underline{N}}_l}({\dot{x}}){{\underline{u}}_{k + 1,l}}(x,{\dot{x}})\\&\quad + \,{{\underline{M}}_{k + 1}}(x){{\underline{N}}_{l + 1}}({\dot{x}}) {{\underline{u}}_{k + 1,l + 1}}(x,{\dot{x}}))\\&=1 {/} {{{\underline{\zeta }}} }(\left( {\frac{{- x}}{{\Delta {\Phi _k} - {\tau _e}}} + \frac{{{\Phi _{k + 1}} - {\tau _e}}}{{\Delta {\Phi _k} - {\tau _e}}}} \right) \\&\quad \left( {\frac{{- {\dot{x}}}}{{\Delta {\Psi _l} - {\tau _{{\dot{e}}}}}} + \frac{{{\Psi _{l + 1}} - {\tau _{{\dot{e}}}}}}{{\Delta {\Psi _l} - {\tau _{{\dot{e}}}}}}}\right) \\&\quad {{\underline{u}}_{k,l}}(x,{\dot{x}}) + \left( {\frac{{ - x}}{{\Delta {\Phi _k} - {\tau _e}}} + \frac{{{\Phi _{k + 1}} - {\tau _e}}}{{\Delta {\Phi _k} - {\tau _e}}}} \right) \\&\quad \left( {\frac{{{\dot{x}}}}{{\Delta {\Psi _l} - {\tau _{{\dot{e}}}}}} - \frac{{{\Psi _l} + {\tau _{{\dot{e}}}}}}{{\Delta {\Psi _l} - {\tau _{{\dot{e}}}}}}} \right) \\&\quad {{\underline{u}}_{k,l + 1}}(x,{\dot{x}})+ \left( {\frac{x}{{\Delta {\Phi _k} - {\tau _e}}} - \frac{{{\Phi _k} + {\tau _e}}}{{\Delta {\Phi _k} - {\tau _e}}}} \right) \\&\quad \left( {\frac{{ - {\dot{x}}}}{{\Delta {\Psi _l} - {\tau _{{\dot{e}}}}}} + \frac{{{\Psi _{l + 1}} - {\tau _{{\dot{e}}}}}}{{\Delta {\Psi _l} - {\tau _{{\dot{e}}}}}}} \right) \\&\quad {{\underline{u}}_{k + 1,l}}(x,{\dot{x}}) + \left( {\frac{x}{{\Delta {\Phi _k} - {\tau _e}}} - \frac{{{\Phi _k} + {\tau _e}}}{{\Delta {\Phi _k} - {\tau _e}}}} \right) \\&\quad \left( {\frac{{{\dot{x}}}}{{\Delta {\Psi _l} - {\tau _{{\dot{e}}}}}} - \frac{{{\Psi _l} + {\tau _{{\dot{e}}}}}}{{\Delta {\Psi _l} - {\tau _{{\dot{e}}}}}}} \right) \\ {{\underline{u}}_{k + 1,l + 1}}(x,{\dot{x}}))=&\frac{1}{{\Delta {\Phi _k} + {\tau _e}}}\frac{1}{{\Delta {\Psi _l} + {\tau _{{\dot{e}}}}}}\\&\quad ({{\underline{a}}_{1,k,l}}{x^2}{\dot{x}} + {\underline{a}_{2,k,l}}x{{\dot{x}}^2} + {{\underline{a}}_{3,k,l}}{x^2}{{\dot{x}}^3} + {{\underline{a}}_{4,k,l}}{x^3}{{\dot{x}}^2}) \\&\quad +\, \frac{1}{{\Delta {\Phi _k} + {\tau _e}}}({{\underline{b}}_{1,k,l}}{x^2} + {\underline{b}_{2,k,l}}x{\dot{x}} + {{\underline{b}}_{3,k,l}}{x^2}{{\dot{x}}^2}\\&\quad +\,{{\underline{b}}_{4,k,l}}{x^3}{\dot{x}})+ \frac{1}{{\Delta {\Psi _l} + {\tau _{{\dot{e}}}}}}({{\underline{c}}_{1,k,l}}x{\dot{x}} + {\underline{c}_{2,k,l}}{{\dot{x}}^2} \\&\quad +\,{{\underline{c}}_{3,k,l}}x{{\dot{x}}^3} + {{\underline{c}}_{4,k,l}}{x^2}{{\dot{x}}^2}) + ({\underline{d}_{1,k,l}}x{{\dot{x}}^2}\\&\quad +\, {{\underline{d}}_{2,k,l}}{x^2}{\dot{x}} + {{\underline{d}}_{3,k,l}}x + {{\underline{d}}_{4,k,l}}{\dot{x}}) \end{aligned}$$

where for $r=1,2,3,4$

$$\begin{aligned} {\underline{a}}_{r,k,l} =&\,m_{r,k,l} +m_{r,k,l+1} +m_{r,k+1,l} +m_{r,k+1,l+1},\\ {\underline{b}}_{r,k,l} ({\dot{x}})=&-\frac{\Psi _{l+1} -\tau _{{\dot{e}}} }{\Delta \Psi _{l} -\tau _{{\dot{e}}} } {\underline{m}}_{r,k,l}\\&+\,\frac{\Psi _{l} +\tau _{{\dot{e}}} }{\Delta \Psi _{l} -\tau _{{\dot{e}}} } {\underline{m}}_{r,k,l+1} +\frac{\Psi _{l+1} -\tau _{{\dot{e}}} }{\Delta \Psi _{l} -\tau _{{\dot{e}}} } {\underline{m}}_{r,k+1,l} \\&-\,\frac{\Psi _{l} +\tau _{{\dot{e}}} }{\Delta \Psi _{l} -\tau _{{\dot{e}}} } {\underline{m}}_{r,k+1,l+1} ),\\ {\underline{c}}_{r,k,l} ({\dot{x}})=&\,-\frac{\Phi _{k+1} -\tau _{e} }{\Delta \Phi _{k} -\tau _{e} } {\underline{m}}_{r,k,l} +\frac{\Phi _{k+1} -\tau _{e} }{\Delta \Phi _{k} -\tau _{e} } {\underline{m}}_{r,k,l+1}\\&-\, \frac{\Phi _{k} +\tau _{e} }{\Delta \Phi _{k} -\tau _{e} } {\underline{m}}_{r,k+1,l} -\frac{\Phi _{k} +\tau _{e} }{\Delta \Phi _{k} -\tau _{e} } {\underline{m}}_{r,k+1,l+1},\\ {\underline{d}}_{r,k,l} ({\dot{x}})=&\,(\Phi _{k+1} -\tau _{e} )(\Psi _{l+1} -\tau _{{\dot{e}}} ){\underline{m}}_{r,k,l} \\&-\,(\Phi _{k+1} -\tau _{e} )(\Psi _{l} +\tau _{{\dot{e}}} ) {\underline{m}}_{r,k,l+1} \\&-\,(\Phi _{k} +\tau _{e} )(\Psi _{l+1} -\tau _{{\dot{e}}}) {\underline{m}}_{r,k+1,l} \\&+\,(\Phi _{k} +\tau _{e} )(\Psi _{l} +\tau _{{\dot{e}}} ){\underline{m}}_{r,k+1,l+1}. \end{aligned}$$

The proof of ${{\underline{a}}}_{r,k,l}$,$\ {{\underline{b}}}_{r,k,l}$, ${{\underline{c}}}_{r,k,l}$, and ${{\underline{d}}}_{r,k,l}$ for other intervals and ${{\overline{a}}}_{r,k,l}$, ${{\overline{b}}}_{r,k,l}$, ${{\overline{c}}}_{r,k,l}$ and ${{\overline{d}}}_{r,k,l}$ is similar to this proof.