An effective projection-based nonlinear adaptive control strategy for heavy vehicle suspension with hysteretic leaf spring

Abstract

A new nonlinear adaptive control strategy for electrohydraulic active suspensions with the hysteretic leaf spring is proposed to improve suspension performances of heavy vehicle. The nonlinear hysteresis property of leaf spring, described and experimentally validated with the Bouc–Wen model, is transformed into linear system by Takagi–Sugeno (T–S) fuzzy approach. Based on the derived T–S fuzzy model, a robust ${H}\infty$ dynamic feedback control with adaptive gain is employed to generate the desired target forces for hydraulic actuators. A new projection-based adaptive control (PAC) law is further proposed for actuators to track the target forces under parametric uncertainties. The PAC law is derived based on the global asymptotic stability conditions of Lyapunov function for the obtained T–S fuzzy model with parametric uncertainties under inputs from both outputs of robust ${H}\infty$ controller and errors of force tracking. The benefits of vehicle systems with PAC active suspension are compared to both sliding mode control (SMC) active suspension and passive suspension. The obtained results indicate that the PAC method can be able to cope with uncertainties and has better robustness than SMC method. Furthermore, the obtained results also indicate that the proposed nonlinear adaptive control strategy effectively improves the suspension performances of heavy vehicle with the hysteretic leaf spring.

Appendix

Lemma 1

[20]. For any vectors ${\mathbf{x}},{\mathbf{y}}\in R^{n}$, and two matrices ${\mathbf{U}}$ and ${\mathbf{S}}$ with appropriate dimensions, if the appropriate dimension ${\varvec{\Delta }}$ satisfies ${\varvec{\Delta }}^{{\mathbf{T}}}{\varvec{\Delta }}\le {\mathbf{I}}$, then for any scalar $\sigma _{0} >0$, we have

$$\begin{aligned} 2{\mathbf{x}}^{\mathrm{T}}{\mathbf{U}{\varvec{\Delta }} \mathbf{Sy}}\le \sigma _{0}^{-1} {\mathbf{x}}^{\mathrm{T}}{\mathbf{UU}}^{\mathrm{T}}{\mathbf{x}}+\sigma _{0} {\mathbf{y}}^{\mathrm{T}}{\mathbf{S}}^{\mathrm{T}}{\mathbf{Sy}} \end{aligned}$$

(47)

Proof of part A: Considering the tracking error ${\mathbf{e}}$ on the closed-loop system (15), the candidate Lyapunov function is defined as,

$$\begin{aligned} V_{2} (t)= & {} \overbrace{{\varvec{\upxi }}_{1}^{\mathrm{T}} {\tilde{\mathbf{R}}{\varvec{\upxi }}}_{1} +Tr[({\mathbf{K}}_{h} -{\mathbf{K}}_{o} )^{\mathrm{T}}{\varvec{\Theta }}({\mathbf{K}}_{h} -{\mathbf{K}}_{o} )]}^{V_{1} (t)}\nonumber \\&+1/2{\mathbf{e}}^{\mathrm{T}}{\mathbf{e}} \end{aligned}$$

(48)

where ${\varvec{\upxi }}_{1} =[{\mathbf{x}}^{\mathrm{T}}\quad {\mathbf{y}}^{\mathrm{T}}\quad ({\mathbf{K}}_{h} -{\mathbf{K}}_{o} )^{\mathrm{T}}]^{\mathrm{T}}$, ${\tilde{{\mathbf{R}}}}=\mathrm{diag}[{\mathbf{R}}\quad {\mathbf{0}}]$.

Computing the time derivative of $V_{1} (t)$ yields

$$\begin{aligned} {\dot{V}}_{2} (t)= & {} {\varvec{\upxi }}_{1}^{\mathrm{T}} \left[ {\begin{array}{cc} \mathrm{sys}({\mathbf{PA}}+{\mathbf{PB}}_{2} {\mathbf{K}}_{o} {\mathbf{C}})&{}*\\ {\mathbf{B}}_{2}^{{\mathbf{T}}} {\mathbf{P}}-{\varvec{\Theta } \mathbf{NH}}\hbox {C}&{}-\mathrm{sys}({\varvec{\Theta } \mathbf{M}}) \\ \end{array}} \right] {\varvec{\upxi }}_{1} \nonumber \\&+2{\varvec{\upxi }}_{1}^{\mathrm{T}} {\tilde{{\mathbf{R}}}\tilde{\mathbf{w}}}+2{\mathbf{e}}^{{\mathbf{T}}}{{\mathbf{B}}}_{2}^{\mathrm{T}} {\mathbf{Rx}}+{\mathbf{e}}^{{\mathbf{T}}}{\dot{\mathbf{e}}} \end{aligned}$$

(49)

in which ${\tilde{{\mathbf{w}}}}=\left[ {{\bar{{\mathbf{w}}}}^{\mathrm{T}}\quad {\bar{{\mathbf{w}}}}^{\mathrm{T}}} \right] ^{\mathrm{T}}$. Substituting the adaptive control laws (37) and (38) into $2{\mathbf{e}}^{{\mathbf{T}}}{{\mathbf{B}}}_{2}^{\mathrm{T}} {\mathbf{Rx}}+{\mathbf{e}}^{{\mathbf{T}}}{\dot{\mathbf{e}}}$, the following transformation can be obtained

$$\begin{aligned} 2{\mathbf{e}}^{{\mathbf{T}}}{{\mathbf{B}}}_{2}^{\mathrm{T}} {\mathbf{Rx}}+{\mathbf{e}}^{{\mathbf{T}}}{\dot{\mathbf{e}}}\le -\kappa _{\mathrm{1}} {\mathbf{e}}^{\mathrm{T}}{\mathbf{e}}+\kappa _{\mathrm{2}} \end{aligned}$$

(50)

Then, substituting Eqs. (50) into (49), we have

$$\begin{aligned}&{\dot{V}}_{2} (t)\le {\varvec{\upxi }}_{1}^{\mathrm{T}} \left[ \begin{array}{cc} \mathrm{sys}({\mathbf{PA}}+{\mathbf{PB}}_{2} {\mathbf{K}}_{o} {\mathbf{C}})&{}*\\ {\mathbf{B}}_{2}^{{\mathbf{T}}} {\mathbf{P}}-{\varvec{\Theta } \mathbf{NH}}\hbox {C}&{}-\mathrm{sys}({\varvec{\Theta } \mathbf{M}}) \\ \end{array} \right] {\varvec{\upxi }}_{1} \nonumber \\&\quad +2{\varvec{\upxi }}_{1}^{\mathrm{T}} {\tilde{\mathbf{R}}\tilde{\mathbf{w}}}-\kappa _{\mathrm{1}} {\mathbf{e}}^{\mathrm{T}}{\mathbf{e}}+\kappa _{\mathrm{2}} \end{aligned}$$

(51)

Based on the inequality (19) and by using the Schur lemma, the following inequation holds

$$\begin{aligned}&\left[ {\begin{array}{cc} \mathrm{sys}({\mathbf{PA}}+{\mathbf{PB}}_{2} {\mathbf{K}}_{o} {\mathbf{C}})&{}*\\ {\mathbf{B}}_{2}^{{\mathbf{T}}} {\mathbf{P}}-{\varvec{\Theta } \mathbf{NH}}\hbox {C}&{}-\mathrm{sys}({\varvec{\Theta } \mathbf{M}}) \\ \end{array}} \right] +\gamma ^{-2}{\tilde{{\mathbf{R}}}\tilde{\mathbf{R}}} \nonumber \\&\quad +\left[ {\begin{array}{c} ({\mathbf{C}}_{1}^{\mathrm{T}} +{\mathbf{C}}^{\mathrm{T}}{\mathbf{K}}_{o}^{\mathrm{T}} {\mathbf{D}}_{12}^{\mathrm{T}} ) \\ {\mathbf{D}}_{12}^{\mathrm{T}} \\ \end{array}} \right] \left[ {\begin{array}{c} ({\mathbf{C}}_{1}^{\mathrm{T}} +{\mathbf{C}}^{\mathrm{T}}{\mathbf{K}}_{o}^{\mathrm{T}} {\mathbf{D}}_{12}^{\mathrm{T}} ) \\ {\mathbf{D}}_{12}^{\mathrm{T}} \\ \end{array}} \right] ^{\mathrm{T}}<0\nonumber \\ \end{aligned}$$

(52)

According to Lemma 1, we have $2{\varvec{\upxi }}_{1}^{\mathrm{T}} {{\tilde{\mathbf{R}}}{\tilde{\mathbf{w}}}}\le \varepsilon ^{-1}{\varvec{\upxi }}_{1}^{\mathrm{T}} {\tilde{\mathbf{R}}{\tilde{\mathbf{R}}}\upxi }_{1} +\varepsilon {\tilde{{\mathbf{w}}}}^{\mathrm{T}}{\tilde{{\mathbf{w}}}}$ in which $\varepsilon$ is a positive number. Thus, the following transformation can be expressed as

$$\begin{aligned} {\dot{V}}_{2} (t)< & {} -(\gamma ^{-2}-\varepsilon ^{-1}){\varvec{\upxi }}_{1}^{\mathrm{T}} {\tilde{\mathbf{R}}\tilde{\mathbf{R}}\upxi }_{1} \nonumber \\&+\varepsilon {\tilde{{\mathbf{w}}}}^{\mathrm{T}}{\tilde{{\mathbf{w}}}}-\kappa _{\mathrm{1}} {\mathbf{e}}^{\mathrm{T}}{\mathbf{e}}+\kappa _{\mathrm{2}} \end{aligned}$$

(53)

The positive number $\varepsilon$ is chosen to satisfy $\gamma ^{-2}-\varepsilon ^{-1}>0$ and define $\varepsilon _{0} =\min \left\{ {\lambda _{\min } ({\tilde{{\mathbf{R}}}})(\gamma ^{-2}-\varepsilon ^{-1}),2\kappa _{1} } \right\}$ in which $\lambda _{\min } ({\tilde{{\mathbf{R}}}})$ is the minimal nonzero eigenvalue of matrix ${\tilde{{\mathbf{R}}}}$. Then, we can obtain

$$\begin{aligned} {\dot{V}}_{2} (t)<-\varepsilon _{0} V_{2} (t)+\varepsilon _{1} \end{aligned}$$

(54)

where $\varepsilon _{1} =2\varepsilon ^{-1}\max \left\| {{\bar{{\mathbf{w}}}}} \right\| _{2} +\kappa _{\mathrm{2}} +\varepsilon _{0} \max (Tr[({\mathbf{K}}_{h} -{\mathbf{K}}_{o} )^{\mathrm{T}}\mathrm{\varvec{\Theta }}({\mathbf{K}}_{h} -{\mathbf{K}}_{o} )])$, with ${\tilde{{\mathbf{w}}}}^{\mathrm{T}}{\tilde{{\mathbf{w}}}}\le 2\max \left\| {{\bar{{\mathbf{w}}}}} \right\| _{2}$ and $Tr[({\mathbf{K}}_{h} -{\mathbf{K}}_{o} )^{\mathrm{T}}\mathrm{\varvec{\Theta }}({\mathbf{K}}_{h} -{\mathbf{K}}_{o} )]\le \max (Tr[({\mathbf{K}}_{h} -{\mathbf{K}}_{o} )^{\mathrm{T}}\mathrm{\varvec{\Theta }}({\mathbf{K}}_{h} -{\mathbf{K}}_{o} )])$. Thus, we can note that the Lyapunov function $V_{2} (t)$ is bounded by

$$\begin{aligned} V_{2} (t)<V_{2} (0)e^{-\varepsilon _{0} t}-\frac{\varepsilon _{1} }{\varepsilon _{0} }(1-e^{-\varepsilon _{0} t}) \end{aligned}$$

(55)

which demonstrates that all the signals ${\mathbf{x}}(t)$ and ${\mathbf{e}}(t)$ of the closed-loop system (2) are bounded.

Proof of part B: For the stability instruction in the presence of parameter uncertainties, we consider the augmented Lyapunov function as $V(t)=V_{2} (t)+1/2{\tilde{{\varvec{\uptheta }} }}^{\mathrm{T}}{\varvec{\Gamma }}^{-1}{\tilde{{{\varvec{\uptheta }} }}}$, and the time derivative of V(t) is written as

$$\begin{aligned} {\dot{V}}(t)= & {} {\varvec{\upxi }}_{1}^{\mathrm{T}} \left[ {\begin{array}{cc} \mathrm{sys}({\mathbf{PA}}+{\mathbf{PB}}_{2} {\mathbf{K}}_{o} {\mathbf{C}})&{}*\\ {\mathbf{B}}_{2}^{{\mathbf{T}}} {\mathbf{P}}-{\varvec{\Theta } \mathbf{NH}}\hbox {C}&{}-\mathrm{sys}({\varvec{\Theta } \mathbf{M}}) \\ \end{array}} \right] {\varvec{\upxi }}_{1} \nonumber \\&+2{\mathbf{e}}^{{\mathbf{T}}}{{\mathbf{B}}}_{2}^{\mathrm{T}} {\mathbf{Rx}}+{\mathbf{e}}^{\mathrm{T}}{\dot{\mathbf{e}}}+{\tilde{{\varvec{\uptheta }}}}^{\mathrm{T}}{ \varvec{\Gamma }}^{-1}{\dot{\tilde{\varvec{\uptheta }}}} \end{aligned}$$

(56)

Substituting the inequation (52) into Eq. (56), we can obtain

$$\begin{aligned} {\dot{V}}(t)< & {} -\gamma ^{-2}{\varvec{\upxi }}_{1}^{\mathrm{T}} {\tilde{{\mathbf{R}}}\tilde{\mathbf{R}}\upxi }_{1} +{\mathbf{e}}^{{\mathbf{T}}}(2{{\mathbf{B}}}_{2}^{\mathrm{T}} {\mathbf{Rx}}+{\dot{\mathbf{e}}})\nonumber \\&+\,{\tilde{{\varvec{\uptheta }}}}^{\mathrm{T}}{\varvec{\Gamma }}^{-1}{\dot{\tilde{\varvec{\uptheta }}}} \end{aligned}$$

(57)

Combining Eqs. (37) and (38) with the inequation (57) results in

$$\begin{aligned}&{\dot{V}}(t)<-\gamma ^{-2}{\varvec{\upxi }}_{1}^{\mathrm{T}} {\tilde{{\mathbf{R}}}\tilde{\mathbf{R}}\upxi }_{1} -\kappa _{\mathrm{1}} {\mathbf{e}}^{\mathrm{T}}{\mathbf{e}} \nonumber \\&\quad -\,\overbrace{{\mathbf{e}}^{\mathrm{T}}{\tilde{{\varvec{\Pi }}}}(-{\mathbf{F}}-A_{s}^{\mathrm{2}} {\mathbf{C}}_{\mathrm{3}} ({\mathbf{Ax}}+{\mathbf{B}}_{\mathrm{2}} {\mathbf{F}})+{\varvec{\upsigma }})}^{{\tilde{\varvec{\uptheta }}{} \mathbf{E}{\varvec{\Lambda }} }}\nonumber \\&\quad +\,{\tilde{{\varvec{\uptheta }}}}^{\mathrm{T}}{\varvec{\Gamma }}^{-1}{\dot{\tilde{\varvec{\uptheta }}}} \end{aligned}$$

(58)

Due to the adaptation function ${\varvec{\upchi }}$ satisfying ${\varvec{\upchi }}={\mathbf{E}{\varvec{\Lambda }}}$, we have

$$\begin{aligned} {\mathbf{e}}^{\mathrm{T}}{\tilde{{\varvec{\Pi }}}}(-{\mathbf{F}}-A_{s}^{\mathrm{2}} {\mathbf{C}}_{\mathrm{3}} ({\mathbf{Ax}}+{\mathbf{B}}_{\mathrm{2}} {\mathbf{F}})+{\varvec{\upsigma }})={\tilde{\varvec{\uptheta }}{} \mathbf{E}{\varvec{\Lambda }}} \end{aligned}$$

(59)

Then, substituting Eq. (59) into inequation (58), the transformation of inequation (58) can be obtained

$$\begin{aligned} {\dot{V}}(t)<-\gamma ^{-2}{\varvec{\upxi }}_{1}^{\mathrm{T}} {\tilde{{\mathbf{R}}}\tilde{\mathbf{R}}\upxi }_{1} -\kappa _{\mathrm{1}} {\mathbf{e}}^{\mathrm{T}}{\mathbf{e}}+{\tilde{{\varvec{\uptheta }}}}^{\mathrm{T}}({ \varvec{\Gamma }}^{-1}{\dot{\tilde{\varvec{\uptheta }}}}-{\mathbf{E}\varvec{\Lambda }})\nonumber \\ \end{aligned}$$

(60)

By the property P2 of Eq. (35), the following inequation holds

$$\begin{aligned} {\dot{V}}(t)\le -\gamma ^{-2}{\varvec{\upxi }}_{1}^{\mathrm{T}} {\tilde{\mathbf{R}}\tilde{\mathbf{R }}{\varvec{\upxi }} }_{1} -\kappa _{\mathrm{1}} {\mathbf{e}}^{\mathrm{T}}{\mathbf{e}}<0 \end{aligned}$$

(61)

which demonstrates that all the signals ${\mathbf{x}}(t)$ and ${\mathbf{e}}(t)$ of the closed-loop system (2) can asymptotically converge to zero with a finite time.