Approximation-free output feedback control for hydraulic active suspensions with prescribed performance

Abstract

This paper investigates an approximation-free output feedback prescribed performance control for a half-vehicle active suspension systems to improve driving comfort. Different from prior results that ignore actuator dynamics, this paper factored hydraulic actuators into the controller design. To solve the nonlinearities of the hydraulic active suspension system, an approximation-free, backstepping-free control scheme is developed, where function approximators (e.g. neural networks and fuzzy systems) and the explosion of complexity in backstepping design can be avoided. In this sense, the heavy computational burden can be removed. Moreover, by using a high-gain observer (HGO) and a prescribed performance function, the proposed controller simply requires the system outputs to be available and can achieve prescribed transient and steady-state performance of system outputs. To stop the propagation of peak phenomena caused by the HGO into the suspension system, the proposed controller is designed to saturate properly without affecting system performance attributes. The stability of the suspension system and the performance requirements of the system output are strictly proven. Finally, the comparative simulations are conducted to validate the effectiveness of the proposed method for improving suspension performance.

Appendices

Appendix I

Proof of Lemma 4

Proof

The zero-dynamic system consists of unsprung subsystem (9c). In order to conduct zero-dynamic system, let $s_1(t)\equiv s_2(t)\equiv 0$, which implies $\vartheta _{ji}=0,j=1,2, i=1,2,3.$ Considering further (13), then $x_{13}$ and $x_{23}$ can be rewritten as follows:

$$\begin{aligned} x_{13}= & {} M\ddot{y}_{1r}+F_{d1}+F_{d2}+F_{s1}+F_{s2}-\varDelta F_{1}\nonumber \\ x_{23}= & {} I\ddot{y}_{2r}+a(F_{d1}+F_{s1})-b(F_{d2}+F_{s2})-\varDelta F_{2}\nonumber \\ \end{aligned}$$

(42)

By substituting (2) and (42) into unsprung subsystem (9c), the zero dynamics system can be obtained in compact form:

$$\begin{aligned} \dot{\bar{x}}^T=A\bar{x}^T+Bz^T+\omega ^T \end{aligned}$$

(43)

where $\bar{x}=[x_4, x_5, x_6, x_7], z=[z_{r1}, \dot{z}_{r1}, z_{r2}, \dot{z}_{r2}]$

$$\begin{aligned} \begin{aligned} A&=\left[ \begin{array}{cccc} 0 &{} 1 &{} 0 &{} 0 \\ -\frac{k_{t1}}{m_1} &{} -\frac{k_{b1}}{m_1} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} -\frac{k_{t2}}{m_2} &{} -\frac{k_{b2}}{m_2} \\ \end{array} \right] \\ B&=\left[ \begin{array}{cccc} 0 &{} 0 &{} 0 &{} 0 \\ \frac{k_{t1}}{m_1} &{} \frac{k_{b1}}{m_1} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} \frac{k_{t2}}{m_2} &{} \frac{k_{b2}}{m_2} \\ \end{array} \right] , \omega =[0, \frac{\omega _1}{m_1},0, \frac{\omega _2}{m_2}]\\ \end{aligned} \end{aligned}$$

(44)

where

$$\begin{aligned} \begin{aligned} \omega _1&=-\frac{bM\ddot{y}_{1r}+I\ddot{y}_{2r}-b\varDelta F_{1}-\varDelta F_{2}}{a+b}\\ \omega _2&=-\frac{aM\ddot{y}_{1r}-I\ddot{y}_{2r}-a\varDelta F_{1}+\varDelta F_{2}}{a+b} \end{aligned} \end{aligned}$$

It is verified that the matrix A is Hurwitz, which implies there exist positive matrices P, Q such that $A^TP+PA=-Q$. Moreover, z and $\omega$ are bounded because of the boundedness of $M, I, \ddot{y}_{jr}, \varDelta F_{j}, z_{rj}, \dot{z}_{rj}, j=1,2$, i.e. $\Vert z\Vert<\bar{z}, \Vert \omega \Vert <\bar{\omega }$. Define a Lyapunov function as $V=\bar{x}P\bar{x}^T$, whose derivative is calculated as following:

$$\begin{aligned} \begin{aligned} \dot{V}&=\bar{x}(A^TP+PA)\bar{x}^T+2zB^TP\bar{x}^T+2\bar{x}P\omega ^T\\&\le -\lambda V+\eta \end{aligned} \end{aligned}$$

(45)

where $\eta =\mu (\bar{z}^2+\bar{\omega }^2);~\lambda =\lambda _{\textrm{min}}(P^{-\frac{1}{2}}QP^{-\frac{1}{2}})-\frac{1}{\mu }(\lambda _{\textrm{max}} (P)+\lambda _{\textrm{max}}(P^{\frac{1}{2}}BB^TP^{\frac{1}{2}}));$ a appropriately designed parameter $\mu >(\lambda _{\textrm{max}}(P)+\lambda _{\textrm{max}}(P^{\frac{1}{2}}BB^TP^{\frac{1}{2}}))/P^{-\frac{1}{2}}QP^{-\frac{1}{2}}$ comes from the Young’s inequality $ab\le \frac{a^2}{2\mu }+\frac{\mu b^2}{2}$, used to the terms $2zB^TP\bar{x}^T$ and $2\bar{x}P\omega ^T$. By integrating both sides of Eq. (45), it yields $V(t)\le e^{-\lambda t}V(0)+\eta /\lambda$, which means that $|x_i|\le \sqrt{(V(0)+\eta /\lambda )/\lambda _{\textrm{min}}(P)}, i=4 \ldots 7$. This completes the proof of lemma 4. $\square$