Discrete approximate optimal vibration control for nonlinear vehicle active suspension

2.1. Discrete nonlinear vehicle active suspension model

Consider a simplified nonlinear vehicle suspension shown in Fig. 1 [3, 4, 8, 9]. The dynamic equation can be expressed as:

where $m_s$ and $m_u$ denote the sprung mass and unsprung mass, receptively; $b_s$ and $k_s$ are the stiffness and damping of vehicle passive suspension; $k_t$denotes the compressibility of the pneumatic tire; ${\epsilon }_1$ and ${\epsilon }_2$ are the nonlinear parameters. $z_s\left(t\right)$ and $z_u\left(t\right)$ stand for the displacements of the sprung mass and unsprung mass, respectively. $z_r\left(t\right)$ denotes the external road displacement acting on the active suspension, and $u\left(t\right)$ represents the active control force generated from the hydraulic pressure or electromagnetic equipment.

As it is well known, the performance criteria of ride comfort, road holding ability, and suspension deflection must be taken into consideration while designing the vibration control strategy for nonlinear vehicle active suspension. The sprung mass acceleration ${\ddot{z}}_s\left(t\right)$, suspension deflection $z_s\left(t\right)-z_u\left(t\right)$, and tire deflection $z_u\left(t\right)-z_r\left(t\right)$ must be small enough to improve the ride comfort, prevent from the excessive suspension bottoming, and obtain the good road holding, respectively. Therefore, the controlled output $y_c\left(t\right)$ can be defined as:

Introducing the following state variables as:

and defining:

the space state expression of nonlinear active vehicle suspension can be constructed as:

where:

Fig. 1Simplified vehicle active suspension

Road disturbance $v\left(t\right)$ is considered as vibration caused by road roughness [3-5, 8]. As stated in [3, 8], the road disturbances can be derived from a random process with ground displacement power spectral density (PSD):

where $\mathrm{\Omega }$ denotes a spatial frequency, ${\mathrm{\Omega }}_0=1/2\pi$ is a reference frequency. The value of $S_g\left({\mathrm{\Omega }}_0\right)$ denotes the measurement for the road roughness, where $n_1$ and $n_2$ are the roughness constants.

The road displacement input $z_r\left(t\right)$ from road irregularities in Eq. (1) can be written as:

where ${\varphi }_n=\sqrt{2S_g\left(2n\pi \mathrm{\Omega }/l\right)\times \left(n\pi \mathrm{\Omega }/l\right)}$, $l$ is the length of a road segment, ${\theta }_n\in \left(0\right.,\left.2\pi \right]$ is a random variable, ${\mathrm{\Omega }}_0=2\pi /\left(l{v}_0\right)$, and $p$can restrict the frequency range of a vehicle body. Then, the road disturbance $\stackrel{-}{v}\left(t\right)$ can be described as:

The following state space vectors are defined as:

Then the road disturbance input $v\left(t\right)$ can be formulated as the output of a random linear exogenous system with an unknown initial value:

where $w\left(t\right)\in {\mathbb{R}}^{2p}$ is the state vector with unknown initial value $w_0\text{,}$ the expressions of $\stackrel{-}{G}\in {\mathbb{R}}^{2p\times 2p}$ and $F\in {\mathbb{R}}^{1\times 2p}$ are represented as:

where ${\omega }_0=2\pi v_{0}l\text{,}$ and $v_0$ is the constant horizontal velocity. Positive integer $p$ limits the considered frequency band lower than 20 Hz in road disturbances Eq. (8).

Defining $T$ as the sampling period, the discrete expression of Eq. (5) can be given by:

where $A=e^{\stackrel{-}{A}T}$, $B={\int }_0^T{e}^{\stackrel{-}{A}T}\stackrel{-}{B}dt$, $D={\int }_0^T{e}^{\stackrel{-}{A}T}\stackrel{-}{D}dt$, and $f\left(x\left(k\right)\right)={\int }_0^T{e}^{\stackrel{-}{A}T}dt\stackrel{-}{f}\left(x\left(t\right)\right)$.

Meanwhile, the discrete form of exogenous system Eq. (11) can be described as:

where $G=e^{\stackrel{-}{G}T}$.

In order to trade off among the various performance requirements, the following quadratic average performance index is chosen as:

where $Q\in {\mathbb{R}}^{3\times 3}$ is a positive semi-definite matrix, $\stackrel{-}{R}>0$ is a constant.

In what follows, we will discuss how to design the DAOVC for system Eq. (5) subject to road disturbances (14) such that the quadratic average performance index Eq. (15) reaches its minimum value. The following assumptions and lemma are given first.

Assumption 1: The triple $\left(\begin{array}{ccc}A& B& C\end{array}\right)$ is completely controllable-observable.

Assumption 2: The nonlinear vector function$f\left(\bullet \right):C^1\left({\mathbb{R}}^n\right)\to {\mathbb{R}}^n$ in Eq. (5) is piecewise continuous in $t$ with $f\left(0\right)\equiv 0$ satisfying Lipschitz conditions on ${\mathbb{R}}^n$ and $‖f\left(x\right(\infty \left)\right)‖<\gamma$, where $\gamma$ is a positive integer related to $v\left(\infty \right)$.

Assumption 3: The exogenous system Eq. (14) is stable, i.e., ${\lambda }_i\le 1$ for any ${\lambda }_i\in \sigma \left(G\right)$, where $\sigma (\bullet )$ denotes the eigenvalues of matrix $G$.

Lemma 1: [16] Let $A_1\in {\mathbb{R}}^{n\times n}\text{,}$$B_1\in {\mathbb{R}}^{m\times m}\text{,}$$C_1\in {\mathbb{R}}^{n\times m}\text{,}$ and $X\in {\mathbb{R}}^{n\times m}\text{.}$ The matrix equation:

has a unique solution $X$ if and only if:

where ${\lambda }_i\left(\bullet \right)$ are eigenvalues of matrix.

3.1. Original nonlinear TPBV problem

Applying the maximum principle to the vibration control problem for Eqs. (13) and (14) with respect to the performance index Eq. (15), a nonlinear TPBV problem can be formulated as:

where $f_x^T=\partial f\left(x\right)/\partial x^T$ denotes a Jacobian matrix of $f_x$ with respect to $x$, $R=E^{T}QE+\stackrel{-}{R}$, $A_1=A-B{R}^{-1}\left(E^{T}QC\right)$, and $Q_1=C^{T}QC-\left(C^{T}QE\right)R^{-1}\left(E^{T}QC\right).$

Then, the optimal vibration controller can be expressed as:

However, it is difficult to obtain the analytical solution of TPBV problem Eq. (18) caused by nonlinear items $f\left(x\left(k\right)\right)$ and $f_x^T\left(x\left(k\right)\right)$. Therefore, we will design a procedure for solving the nonlinear TPBV problem Eq. (18) to obtain DAOVC.

3.2. Reformation of original nonlinear TPBV problem

By introducing a sensitive parameter $\epsilon$, the nonlinear TPBV problem Eq. (18) can be formulated as:

Meanwhile, the optimal vibration control law $u^{*}\left(k\right)$ Eq. (19) can be reconstructed as:

It is assumed that $u\left(k,\epsilon \right)$, $x\left(k,\epsilon \right)$, $\lambda \left(k+1,\epsilon \right)$, $f\left(x\left(k,\epsilon \right)\right)$, and $f_x^T\left(x\left(k,\epsilon \right)\right)$ are differentiable at $\epsilon =0$, and their Maclaurin series are described as:

where ${(\bullet )}^{\left(i\right)}={\partial }^i(\bullet )/\partial {\epsilon }^i{|}_{\epsilon =0}$. While $\epsilon =$ 1, the formulated TPBV problem Eq. (20) is equivalent to nonlinear TPBV problem Eq. (18).

Then, the formulated TPBV problem Eq. (20) can be rewritten as:

The following sequence of TPBV problems can be obtained:

From Eq. (20) and Eq. (21), one yields:

It should be pointed that DAOVC $u\left(k\right)$ can be obtained by integrating $u^{\left(0\right)}\left(k\right)$ with Eq. (24), $u^{\left(i\right)}\left(k\right)$ with Eqs. (25), and (26).

Remark 1: For the $i\text{th}$ step, the nonlinear item $f^{\left(i-1\right)}\left(x\left(k\right)\right)$ and $i\sum _{j=0}^{i-1}{C}_{i-1}^j{f_x^T}^{(i-1-j)}\left(x\left(k\right)\right){\lambda }^{\left(j\right)}\left(k+1\right)$ can be calculated from the $(i-1)$th step. Therefore, the series $u^{\left(i\right)}\left(k\right)$ Eq. (26) can be calculated from the sequence of Eqs. (24) and (25).

3.3. Design of DAOVC

In this subsection, we will propose the designed DAOVC and prove its existence and uniqueness. We give out the following theorem.

Theorem 1: Consider the vibration control problem for discrete nonlinear vehicle active suspension Eq. (13) subject to road disturbance Eq. (14) with respect to the quadratic average performance index Eq. (15), the DAOVC is existent and unique, which can be expressed as:

where $S={(R+B^T{P}_{1}B)}^{-1}$; $P_1$ is the unique positive-definite solution of the Riccati equation:

$P_2$ is the unique solution of the following Stein equation:

and the nonlinear compensation $g_i\left(k\right)$ can be obtained from:

where the sequence of $\left\{x^{\left(i\right)}\left(k\right)\right\}$ can be calculated from:

Proof: First, we will discuss the solution of $u^{\left(0\right)}\left(k\right)$. Defining ${\lambda }^{\left(0\right)}\left(k\right)$ as:

and substituting Eqs. (32) into (26), one gets:

Then the $u^{\left(0\right)}\left(k\right)$ in the 0th sequence can be described as:

From Eqs. (24), (32), and (34), we have:

Substituting Eq. (35) into the second formula in Eq. (24), one gets:

Comparing the coefficients between Eq. (32) and Eq. (36), Riccati Eq. (28) and Stein Eq. (29) can be obtained. Then, the $u^{\left(0\right)}\left(k\right)$ can be computed.

Next, we will discuss the solution of$u^{\left(i\right)}\left(k\right)$. Introducing the following ${\lambda }^{\left(i\right)}\left(k\right)$ as:

and substituting the first formula of Eq. (21) into ${\lambda }^{\left(i\right)}(k+1)=P_1{x}^{\left(i\right)}\left(k+1\right)+g_i\left(k+1\right)$, one gets:

From Eqs. (26) and (37), the $u^{\left(i\right)}\left(k\right)$in the $i$th sequence can be described as:

Then one yields:

Based on Eq. (40) and the second formula of Eq. (25), we have:

Comparing the coefficients between Eqs. (37) and (41), the nonlinear compensation $g_i\left(k\right)$ Eq. (30) can be obtained. From $u^{\left(0\right)}\left(k\right)$ of Eq. (34), $u^{\left(i\right)}\left(k\right)$ of (39), and $u^{*}\left(k\right)={\sum }_{i=0}^{\infty }\left(1/i!\right)u^{\left(i\right)}\left(k\right)$, the DAOVC can be obtained as Eq. (27).

Finally, the existence and uniqueness of DAOVC Eq. (27) will be proved. Actually, it is equivalent to prove the existence and uniqueness of the matrices $P_1$ and$P_2$, respectively.

Based on the Assumption 1 that the triple $\left(\begin{array}{ccc}A& B& C\end{array}\right)$ is completely controllable-observable and the optimal control theory, the positive-definite matrix $P_1$ is the existence and uniqueness solution of Eq. (29). The solution of nonlinear compensation Eq. (30) can be described as:

Based on Assumption 1, one has:

From Assumption 2 and Eq. (43), we have:

Then$P_2$ is the existent and unique solution of Stein Eq. (30) based on Lemma 1. Therefore, the proposed DAOVC Eq. (27) is the existence and uniqueness. The proof is completed.

3.4. Computational realizability of DAOVC

We can easily discover that it is impossible to obtain the vibration control law Eq. (27) caused by the infinite item $\sum _{i=1}^{\infty }\frac{1}{i!}{g}_i(k+1)$. In order to make the result more practical, a positive integer $M$ can be decided to replace the $\infty$ in Eq. (27). Then DAOVC Eq. (27) can be reformed as:

An iteration formula for DAOVC can be obtained: