Adaptive robust control for active suspension systems: targeting nonholonomic reference trajectory and large mismatched uncertainty

Abstract

We formulate control goals of nonlinear underactuated active suspension systems (ASSs) with uncertainties as a class of constraints, which may be holonomic or nonholonomic. The uncertainties in the sprung mass, the external disturbance, the control input, the spring force and dam** force are (possibly fast) time-varying, unknown but bounded. However, the bounds are unknown. The task is to design control which drives systems to approximately obey the constraints (hence constraint-following control) under the uncertainties effect. Since these uncertainties do not all fall within the range space of the input matrix, we creatively decompose uncertainties into matched portion and mismatched portion that may be large. Consequently, this decomposition is able to ensure the mismatched portions of uncertainties to “disappear,” which indicates that the control design can be based on only matched portions. To emulate a constant design parameter vector that may be relevant to bounds of the uncertainties, we construct a continuous adaptive law, which call it smooth-zone adaptive law. This sets a base for an adaptive robust control design. We prove the uniform boundedness and the uniform ultimate boundedness of ASSs under the proposed control by the Lyapunov approach. Furthermore, experimental and numerical simulation results on a 2-DOF nonlinear ASS with uncertainties are presented for demonstrations of the proposed control.

Appendix

To render the sprung mass displacement and velocity in an ASS to follow the sprung mass displacement and the velocity of the skyhook model, the sliding mode surface is adopted

$$\begin{aligned} s_c= & {} d_{c1}e_{c1}+d_{c2}e_{c2}, \end{aligned}$$

(89)

$$\begin{aligned} e_{c1}= & {} z_{s}-z_{sr},\quad e_{c2}=\dot{z}_s-\dot{z}_{sr}, \end{aligned}$$

(90)

where $d_{c1}$ and $d_{c2}$ are control parameters; $e_{c1}$ is the difference between the sprung mass displacement of the ASS and the sprung mass displacement of the skyhook model; and $e_{c2}$ is the difference between the sprung mass velocity of the ASS and the sprung mass velocity of the skyhook model.

By (89), the derivative of the sliding mode surface is obtained as

$$\begin{aligned} \dot{s}_c=d_{c1}e_{c2}+d_{c2}\dot{e}_{c2}. \end{aligned}$$

(91)

Substituting equations (66) and (90) into (91), the derivative of the sliding mode surface is rewritten as

$$\begin{aligned} \dot{s}_c= & {} d_{c1}\dot{z}_s+d_{c2}f_c/m_s+D_c-(d_{c1}\dot{z}_{sr}+d_{c2}\ddot{z}_{sr}),\nonumber \\ \end{aligned}$$

(92)

$$\begin{aligned} D_c= & {} d_{c2}(-f_s-f_d-\eta _{sd}-f_e-m_1\ddot{z}_s+\hat{d}f_c), \end{aligned}$$

(93)

where $D_c$ is defined as real disturbances in the ASS.

By using Lyapunov stability theory, the control input should satisfy the following condition

$$\begin{aligned} s_c\dot{s}_c<0. \end{aligned}$$

(94)

With (92) and (94), the control is constructed as

$$\begin{aligned} f_c= & {} -m_s(\mu _c+\mu _s+\mu _k-d_{c1}\dot{z}_{sr}-d_{c2}\ddot{z}_{sr}), \end{aligned}$$

(95)

$$\begin{aligned} \mu _c= & {} d_{c1}\dot{z}_s+k_cs_c, \end{aligned}$$

(96)

$$\begin{aligned} \mu _s= & {} l_csat(s_c), \end{aligned}$$

(97)

$$\begin{aligned} \mu _k= & {} \hat{D}_c, \end{aligned}$$

(98)

where “sat($\cdot$)” in (97) is the saturation function and described as

$$\begin{aligned} \begin{aligned} sat(s_c)=\left\{ \begin{array}{rcl} -1\quad &{}s_c\le -\epsilon _c,\\ s_c\quad &{}|s_c|\le \epsilon _c,\\ 1\quad &{}s_c\ge \epsilon _c. \end{array} \right. \end{aligned} \end{aligned}$$

(99)

In (96), (97) and (99), the parameters $k_c$, $l_c$ and $\epsilon _c$ are positive constants. The control parameter $k_c$ regulates the rate of arriving at the sliding mode surface. The control parameter $l_c$ is the switch coefficient of the sliding mode surface. $\hat{D}_c$ is the estimated disturbances in the ASS and obtained by using the disturbance observer.

To design the disturbance $\hat{D}_c$, the derivative of the real disturbances is assumed to be bounded and described as

$$\begin{aligned} |\dot{D}_c|<\theta _c, \end{aligned}$$

(100)

where $\theta _c$ is a positive constant.

By [12], the estimated disturbances obtained from the disturbance observer are written as

$$\begin{aligned} \hat{D}_c=\int _{0}^{t}-q_c(-k_cs_c-l_csat(s_c))\mathrm{d}t+q_cs_c, \end{aligned}$$

(101)

where $q_c$ is the control parameter of the disturbance observer.

In this paper, the control parameters of the SMC-DO are selected as: $d_{c1}=4, d_{c2}=1, k_c=5, l_c=0.05, q_c=10 and \epsilon _c=0.02$.