Robust Fault Detection of Uncertain Lipschitz Nonlinear Systems with Simultaneous Disturbance Attenuation Level and Enhanced Fault Sensitivity and Lipschitz Constant

Abstract

In this paper, the problem of optimal robust fault detection (FD) for uncertain Lipschitz nonlinear systems is considered. A robust active fault detection approach for a class of the Lipschitz nonlinear systems in the presence of disturbances and parametric uncertainties is proposed, wherein the Lipschitz constant is assumed as one of the optimization parameters in the observer design. In addition to disturbance attenuation level, the fault sensitivity criterion based on $H_-$ index is also defined in the FD system design. Different criteria are defined as a weighted multi-objective linear matrix inequality optimization problem, and the optimal variables of the FD system are derived based on a newly defined cost function. A numerical example is provided to demonstrate the effectiveness of the proposed FD system. The results show the robustness of the proposed method against parametric uncertainty and nonlinear uncertainty as well.

Appendix

In this section, the proposed method is applied to an active suspension system as a real-world system. The configuration of the system is depicted in Fig. 9.

Fig. 9

A quarter car model [15]

State space representation of the system is given by

$$\begin{aligned} \dot{x}(t)=\left( \begin{array}{cccc} 0 &{}\quad 0 &{}\quad 1 &{}\quad -\,1 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ -\frac{k_s}{m_s} &{}\quad 0 &{}\quad -\frac{C_s}{m_s} &{}\quad \frac{C_s}{m_s} \\ \frac{k_s}{m_u} &{}\quad -\frac{k_t}{m_u} &{}\quad \frac{C_s}{m_u} &{}\quad -\frac{C_s+C_t}{m_u} \\ \end{array} \right) x(t)+ \left( \begin{array}{c} 0\\ -\,1\\ 0\\ \frac{C_t}{m_u}\\ \end{array} \right) w(t)+\left( \begin{array}{c} \psi _1 \\ \psi _2 \\ \frac{1}{m_s}u+\psi _3 \\ -\frac{1}{m_u}+\psi _4\\ \end{array} \right) \end{aligned}$$

(61)

$$\begin{aligned} y(t)=\left( \begin{array}{cccc} 1 &{}\quad 0 &{}\quad 0 &{} \quad 0 \\ 0 &{}\quad \frac{k_t}{(m_s+m_u)g} &{}\quad 0 &{}\quad 0\\ \end{array} \right) x(t). \end{aligned}$$

The states of the system are defined as suspension deflection, tire deflection, sprung mass speed and unsprung mass speed, respectively, which are represented as the following equations.

$$\begin{aligned} x_1(t)= & {} z_x(t)-z_u(t), \end{aligned}$$

(62)

$$\begin{aligned} x_2(t)= & {} z_u(t)-z_r(t), \end{aligned}$$

(63)

$$\begin{aligned} x_3(t)= & {} \dot{z}_s(t), \end{aligned}$$

(64)

$$\begin{aligned} x_4(t)= & {} \dot{z}_u(t). \end{aligned}$$

(65)

Disturbance is considered as

$$\begin{aligned} w(t)=\dot{z}_r(t), \end{aligned}$$

(66)

which is simulated as follows.

$$\begin{aligned} w(t)=0.05 \sin t. \end{aligned}$$

(67)

The nonlinear term of the state space equation is defined as the subsequent equation.

$$\begin{aligned} \psi (t)=\left( \begin{array}{c} 0.09 \sin t \\ 0.04 \cos t \\ 0.07 \sin t \\ 0.01 \sin t \\ \end{array} \right) . \end{aligned}$$

(68)

According to [15], the parameters of the system, their definition and considered values of the active suspension system are tabulated in Table 3.

Table 3 Parameters of active suspension system

By using the values in the table, the matrices of the system can be obtained as:

$$\begin{aligned} A=\left( \begin{array}{cccc} 0 &{}\quad 0 &{}\quad 1&{}\quad -\,1 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ -\,43.905 &{}\quad 0 &{} \quad -\,1.125 &{} \quad 1.125 \\ 374.736 &{} \quad -\,886.973 &{} \quad 9.605 &{} \quad -\,9.733 \\ \end{array} \right) . \end{aligned}$$

The disturbance and fault distribution matrices are defined as:

$$\begin{aligned} D_1= & {} \left( \begin{array}{c} 0.3 \\ 0.6 \\ 0.1 \\ 0.4 \\ \end{array} \right) , Q_1=\left( \begin{array}{cc} 0 &{} \quad 0.4 \\ 0.8 &{} \quad 0.8 \\ 0.7 &{} \quad 0 \\ 0.3 &{} \quad 0.6 \\ \end{array} \right) \\ D_2= & {} \left( \begin{array}{c} 0.4 \\ 0.5 \\ \end{array} \right) , Q_2=\left( \begin{array}{cc} 0.7 &{} 0.5 \\ -\,0.6 &{} 0.5 \\ \end{array} \right) . \end{aligned}$$

Two actuator and sensor faults are considered for the system. The disturbance distribution matrices are assumed as $D_1$ and $D_2$ that is different from (61), which is considered for more strict conditions wherein disturbance can influence all states of the system instead of $x_2$ and $x_4$. It is also worth noting that the selection of disturbance and fault distribution matrices has a direct effect on the disturbance attenuation level, fault sensitivity and achievable Lipschitz constant as well.

The parametric uncertainties in the matrices of the system in (61) are assumed as follows.

$$\begin{aligned} M_1= & {} \left( \begin{array}{cccc} 0.11 &{}\quad -\,0.04 &{} \quad 0.02 &{} \quad 0.11 \\ 0.15 &{} \quad 0.13 &{} \quad 0.12 &{} \quad 0.02\\ 0.0.1 &{} \quad -\,0.11 &{} 0.03 &{} \quad 0.04 \\ 0.03 &{} \quad -\,0.08 &{} \quad 0.11 &{} \quad -\,0.06 \\ \end{array} \right) \\ N_1= & {} \left( \begin{array}{cccc} 0.063 &{} \quad -\,0.055 &{} \quad 0.074 &{} \quad 0.094 \\ 0.015 &{} \quad 0.012 &{} \quad 0.011 &{} \quad 0.064\\ 0.073 &{} \quad -\,0.101 &{} \quad 0.043 &{} \quad 0.045 \\ 0.034 &{} \quad -\,0.028 &{} \quad 0.083 &{} \quad -\,0.061 \\ \end{array} \right) \\ M_2= & {} \left( \begin{array}{c} 0.38 \\ 0.1\\ \end{array} \right) , N_2=\left( \begin{array}{cccc} 0.15 &{} \quad 0.1 &{} \quad 0.13 &{} \quad 0.1 \\ \end{array} \right) . \end{aligned}$$

The parameters for the observer design are considered as:

$$\begin{aligned} \beta = 0.29, H=\left( \begin{array}{cc} -\,0.18 &{}\quad 0.35 \\ -\,0.35 &{} \quad 0.28 \\ -\,0.35 &{} \quad 0.70 \\ 0.35 &{} \quad 0.11 \\ \end{array} \right) . \end{aligned}$$

The obtained results of the theorem for different weighting factors of the optimization problem are given in Figs. 10 and 11.

Fig. 10

Different criteria of the optimization problem versus weighting factor

Fig. 11

Optimal trade-off curves

The optimal value for the weighing factor is obtained as $W=0.9$. One can obtain the following optimal values by solving the LMI problem.

$$\begin{aligned} \mu =0.926 , \gamma =0.091 , \beta =0.237. \end{aligned}$$

The observer gain is achieved as:

$$\begin{aligned} L=\left( \begin{array}{cc} -\,0.193 &{} \quad 0.750 \\ 0.243 &{} \quad 1.411 \\ 1.669 &{} \quad 0.119 \\ 2.365 &{} \quad -\,0.582 \\ \end{array} \right) . \end{aligned}$$

The given values are obtained for $k_1=k_2=k_3=1$. The optimal values of these coefficients and validation of the proposed method for an experimental setup are considered as future works of this study.