A logarithmic sliding mode controller for stochastic active suspension systems

Abstract

In this paper, a new approach for stochastic logarithmic variable structure control has been proposed to create a compromise between reaching phase duration and response rate of the stochastic suspension system. The asymptotic stability with probability one for the closed-loop suspension system has been analyzed by Lyapunov method. Some practical considerations in proving theorems are uncertain parameter variations, environmental matched disturbances, actuator degradation, stochastic environmental mismatched disturbances, which is a representative of the change in road profile with time, and unmodeled dynamics. Finally, by regulating a specific parameter, the simulation results corroborate the advantage of the proposed controller in managing and reducing the sensitivity of the closed-loop system. Also, compared with other sliding surfaces of past researches, the designed sliding variable minimizes a certain performance function and therefore provides better performance for the closed-loop system, which is also verified in the simulation results.

Keywords

Sliding mode stochastic system reaching phase sensitivity

Introduction

Suspension systems provide road holding and isolation of vehicle bodies from the irregularities of road surfaces (Yagiz et al., 2008). Unlike passive suspension systems, semiactive and active ones can continuously change the vibration energy of the vehicle body induced by road excitation; therefore, they have a potential to improve the ride comfort and vehicle maneuverability. In the field of mechatronics, theoretical and experimental investigations of several types of suspension systems, including passive (Williams, 1997), semiactive (Ahmadian and Pare, 2000; Verros et al., 2005; van der Sande et al., 2016; Yin et al., 2016), and active suspensions (Dong et al., 2016; Göhrle et al., 2015; Huisman et al., 1993; Rath et al., 2017; Sun et al., 2013) have been performed. The dynamics of the active suspensions possess inherent complexity due to some kind of uncertainties and stochastic surface roughness. Hence, over the past decades, robust control strategies such as control (Rizvi et al., 2018; Wang et al., 2015) and particularly sliding mode control (SMC) (Al-Holou et al., 2002; Chen et al., 2021; Chen and Huang, 2005, 2006; Ho et al., 2021; Le et al., 2016; Lin et al., 2009; Wijaya et al., 2010) have been extensively studied in different types of active suspensions. Most important features of SMC include fast response and good transient performance, insensitivity to parameter variation, and external disturbance in sliding mode, and robustness against a large class of perturbations or model uncertainties. Considering the sliding surface selection points, past research could be divided into three categories.

In the first category, the structure and all parameters of the sliding variable are specified from the beginning, and the stability of the sliding mode can be deduced without any complex analysis (Al-Holou et al., 2002; Chen and Huang, 2005, 2006; Le et al., 2016). In Al-Holou et al.’s study (2002), in order to enhance ride and comfort, authors have designed a sliding mode neural network interference fuzzy logic controller with a simple PD sliding variable. In Chen and Huang’s study (2005), an adaptive sliding controller with a function estimation technique has been used via an ordinary PD sliding function. Using negative stiffness structure, Le et al. (2016) have proposed an adaptive SMC algorithm to improve isolation effectiveness. In another study, an SMC controller was designed while hydraulic actuator dynamics were uncertain (Chen and Huang, 2006). Sliding variable in their paper is the difference between the actuator output and the actuator-desired output. This sliding surface is a simple linear one, too. In Lin et al. (2009) and Wijaya et al. (2010), the authors have proposed a PD and a logarithmic sliding manifold, respectively, both of which are considered in the first category of choosing the sliding surface, too.

In the second class of sliding manifold selection, even though the initial formation of the sliding function is indicated, some of its parameters are not determined from the beginning. This category itself could be divided into two groups. In the first one, the unknown parameters are designed only for substantiation of a stable dynamics on the sliding surface (Chen and Chang, 2000), and in the second one, they are designed at a higher level to achieve both sliding mode stability and some good operational features (Acosta, 2014; Choi and Park 1992; Deepak et al., 2012; Saeedi and Beheshti 2009; Veselić et al., 2014; Yagiz et al., 2008). For example, Yagiz et al. (2008) used a fuzzy logic to modify the slope of sliding surface, and Saeedi and Beheshti (2009) proposed a time-varying sliding variable. These methods were presented for reducing the reaching phase duration and, consequently, increasing robustness.

In the third category, at a higher level than those of the two former ones, there is no restriction on the structure of the sliding manifold from the beginning; therefore, the sliding variable could be searched in a wider space, resulting in a controller with a better performance. In other words, in this type, a purposeful approach to obtain the sliding manifold is suggested to meet certain operational purposes. Integral sliding surface is a rare well-developed example of this category (Anbalagan and Joo, 2022; Cao et al., 2021; Chen et al., 2013, 2022; Li et al., 2022a, 2022b; Xu, 2016; Zhang et al., 2016). This manifold is the outcome of an optimization procedure in the sliding mode. Except for this surface, there is scant research on such a systematic approach.

Although a car suspension system has a stochastic nature due to road bumps, except in some cases like Moghadam and Kebriaei (2019), Azizi and Mobki (2021), and Wang (2021), our survey interestingly showed that robust controller, rather than a robust stochastic controller, has been designed to activate a car suspension system. In other words, in the process of controller design, the disturbance due to surface roughness has been considered as an uncertainty, rather than a stochastic process, with known or unknown bound which might cause a conservative controller.

Motivated by the two aforementioned points, this study aims to propose a sliding function obtained from a purposeful procedure for stochastic variable structure control of a quarter car active suspension system with practical considerations, including parameter variations, environmental disturbances, actuator degradation, and unmodeled dynamics in the stochastic part of the system. The proposed control law has a main advantage in that both the duration of reaching phase (related to robustness) and the response rate, as two highly important factors in control of active suspensions, could be well managed.

Notations: $M_{ij}$ , $M_{i}$ , $M^{j}$ , and $m_{i}$ indicate the ith row and jth column element of $M$ , the ith row of $M$ , jth column of $M$ , and ith element of $m$ , respectively. $mi n_{i} (M_{ii})$ , $m_{\min (\max)}$ , and $trace [M]$ show the minimum diagonal element of $M$ , minimum (maximum) element of $m$ , and the sum of diagonal elements of $M$ , respectively. $| . |$ , $| | . | |$ , and $| | . | |_{1}$ denote absolute value, the Euclidean norm of a vector or its induced matrix norm and 1th norm of a vector, respectively. For a real matrix, $M > 0$ means that $M$ is symmetric and positive definite, and $I$ is used to represent an identity matrix of appropriate dimensions. $mi n_{i} (| λ_{i} (.) |)$ shows the minimum Eigenvalue of a matrix, regardless of its sign. (Ω, F, P) is a probability space, with Ω the sample space, F the σ-algebra of subsets of the sample space, and P the probability measure. Matrices, if not explicitly stated, are assumed to have compatible dimensions.

The structure of the article is as follows: in the second part, the uncertain stochastic faulty active suspension dynamics is extracted. Then, in the third section, the sliding surface is designed based on an optimization process, and in the next part, the design of the control system is done based on the Lyapunov method. Finally, the simulation results are extracted and analyzed.

Uncertain stochastic faulty active suspension dynamics

In this part of the article, after six steps, the uncertain stochastic faulty model for a quarter car active suspension system has been extracted.

Nominal active suspension dynamics

Generally speaking, quarter car active suspension is modeled by movable sprung and unsprung masses, a damper modeling shock absorber, a spring modeling rigidity coefficient of the tire, another spring for absorbing energy, and a force production source whose output is the control signal. Figure 1 shows a schematic view of this system.

The nominal quarter car active suspension (without considering bump effects) can be formulated from Newton’s Law as follow

m_{s} {\overset{\cdot\cdot}{x}}_{s} = - k_{s} (x_{s} - x_{u}) - c_{s} ({\overset{\cdot}{x}}_{s} - {\overset{\cdot}{x}}_{u}) - F

(1)

m_{u} {\overset{\cdot\cdot}{x}}_{u} = k_{s} (x_{s} - x_{u}) + c_{s} ({\overset{\cdot}{x}}_{s} - {\overset{\cdot}{x}}_{u}) - k_{t} x_{u} + F

(2)

where $m_{s}$ and $m_{u}$ are sprung and unsprung masses, $k_{s}$ is the rigidity coefficient of the spring, $c_{s}$ is the damping coefficient of shock absorber, and $k_{t}$ is the rigidity coefficient of the tire.

Moreover, $x_{s}$ and $x_{u}$ are the displacement of the sprung and unsprung masses, respectively, and $F$ as the input control is the output of the actuator (for example, a hydraulic or pneumatic type).

Let $x_{1} = x_{s}$ , $x_{2} = {\overset{\cdot}{x}}_{s}$ , $x_{3} = x_{u}$ , $x_{4} = {\overset{\cdot}{x}}_{u}$ , and $u = F$ . Then, the state equations for nominal quarter car active suspension system in the matrix form can be represented as follows

dx = (Ax + Bu) dt

(3)

where

\begin{matrix} x = {[x_{1} x_{2} x_{3} x_{4}]}^{T} \\ A = [\begin{matrix} 0 & 1 & 0 & 0 \\ - \frac{k_{s}}{m_{s}} & - \frac{c_{s}}{m_{s}} & \frac{k_{s}}{m_{s}} & \frac{c_{s}}{m_{s}} \\ 0 & 0 & 0 & 1 \\ \frac{k_{s}}{m_{u}} & \frac{c_{s}}{m_{u}} & - \frac{k_{s} + k_{t}}{m_{s}} & - \frac{c_{s}}{m_{u}} \end{matrix}], B = [\begin{matrix} 0 \\ \frac{1}{m_{s}} \\ 0 \\ - \frac{1}{m_{u}} \end{matrix}] \end{matrix}

(4)

Uncertain parameter variations

In practice, suspension system parameters may change over the time as a result of various factors such as depreciation of springs and shock absorbers, the number of passengers and the mass of luggage. In such conditions, the nominal model can be modified as an uncertain model as follows

dx = [(A + Δ A) x + Bu] dt

(5)

where $Δ A$ is the uncertainty matrix which can describe the unknown parameter variations.

Assumption 1. It is assumed that $Δ A$ satisfies

Δ A = EF (t) H

(6)

where E and H are known real matrices, and $F (t)$ is an unknown function satisfying $F^{T} (t) F (t) \leq I$ .

Uncertain environmental matched disturbances

In practice, environmental matched disturbances such as the vertical component of wind power may degrade the performance of the suspension system. Therefore, their influence can be described as follows

dx = [(A + Δ A) x + B (u + f)] dt

(7)

where $f$ represents the matched uncertain disturbance.

Assumption 2. $f$ is an unknown time-varying function and $| | f (t) | | \leq r$ where $r \geq 0$ is a known scalar.

Uncertain actuator degradation

In the practical process, various malfunctions or defective behaviors always occur in the normal operations resulting from the unexpected variations in external surroundings and abrupt changes in signals, and so on. This kind of phenomenon is considered as a sensor (actuator) fault. In the active suspension system, it is assumed that partial actuator degradation may occur, which can be modeled as follows:

u_{F} = (I - Λ) u (t)

(8)

where $Λ = diag {μ_{1}, μ_{2}, \dots, μ_{m}}$ with $μ_{i}$ satisfying $0 \leq {\underline{μ}}_{i} \leq μ_{i} \leq {\bar{μ}}_{i} < 1$ . $μ$ and $\bar{μ}$ are the vectors including under and upper bound of each $μ_{i}$ , respectively.

Remark 1. In this paper, it can be seen that if $μ_{i} = 1$ for some i, then the proposed control signal might not satisfy the nonsingularity condition, that is, a general assumption in the design of a controller. Hence, it is totally reasonable to consider only the partial actuator degradation in this design.

Active suspension system, under actuator degradation is expressed as

\begin{matrix} dx = [(A + Δ A) x + B (u_{F} + f)] \\ dt = [(A + Δ A) x + B ((I - Λ) u (t) + f)] dt . \end{matrix}

(9)

Stochastic environmental mismatched disturbances

In the car suspension system, road bumps are stochastic environmental disturbances whose effect on the dynamic is appeared as a mismatched part. Considering bump effects, Equation (2) can be modified as

m_{u} {\overset{\cdot\cdot}{x}}_{u} = k_{s} (x_{s} - x_{u}) + c_{s} ({\overset{\cdot}{x}}_{s} - {\overset{\cdot}{x}}_{u}) - k_{t} x_{u} + k_{t} x_{r} + F

(10)

where $x_{r}$ is the change in road profile with time. $x_{r}$ has a stochastic nature and is considered as a white noise. As we know, in limiting conditions, Brownian process is equivalent to the white noise. In other words, when using Ito’s formula, in notation, the derivative of Brownian process can be used instead of a white noise or $x_{r}$ can be replaced by $de / dt$ where $e (t)$ is a one-dimensional Brownian motion.

Therefore, by Ito’s formula, the state equations in matrix form can be rewritten as follows

dx = [(A + Δ A) x + B ((I - Λ) u (t) + f)] dt + Dde (t)

(11)

where

D = [\begin{matrix} 0 \\ 0 \\ 0 \\ \frac{k_{t}}{m_{u}} \end{matrix}]

(12)

Unmodeled dynamics in stochastic part

Considering the unmodeled multiplicative dynamics in stochastic part of the system, the uncertain car active suspension system which will be considered in this paper can be Modeled by the It $\hat{o}$ stochastic differential equation as follows

dx = [(A + Δ A) x + B ((I - Λ) u (t) + f)] dt + DG (x, t) de (t)

(13)

where $G (x, t)$ denotes a nonlinear uncertainty in stochastic part of the model.

Assumption 3. The following matrix $G (x, t)$ , satisfies

t r a c e [G {(x, t)}^{T} G (x, t)] \leq {‖ N x (t) ‖}^{2}

(14)

where $N$ is a real constant matrix.

Without loss of generality, it is assumed that the pair of $(A, B)$ is controllable. It should be noted that, although the quarter car active suspension is a single-input system, for the possibility of extension to multi-input systems, controller design in this article has been performed based on multi-input systems which are modeled by Equation (13).

Logarithmic sliding surface design

Consider the following SMC law (with the continuous switching function $\tanh (s)$ instead of sign(s))

u (t) = Zx (t) + ρ (t) \tanh (s)

(15)

where $u (t) \in R^{m}$ is the control input vector, $Z$ is the state feedback gain matrix, $ρ (t)$ is the scalar switching gain which is designed later, and $\tanh (s)$ is the vector of continuous switching function which includes $\tanh (s_{i})$ for $i = 1, \dots, m$ , where m is the dimension of the control input. In other words, we will have $\tanh (s) = [\begin{matrix} \tanh (s_{1}) \\ \tanh (s_{2}) \\ \dots \\ \tanh (s_{m}) \end{matrix}]$ .

Suppose the control signal in Equation (15) is applied to the following nominal system

dx (t) = (Ax (t) + Bu (t)) dt

(16)

where $x (t)$ is the state vector, $A$ is the state matrix, and $B$ is the input matrix with appropriate dimensions.

Then the result is as follows

\begin{matrix} dx (t) = [Ax + B (Zx + ρ (t) \tanh (s))] \\ dt = [(A + BZ) x + B ρ (t) \tanh (s)] dt \end{matrix}

(17)

With considering the following definitions

\begin{matrix} \tilde{A} = A + BZ \\ v (t) = ρ (t) \tanh (s) \end{matrix}

(18)

Equation (17) can be rewritten as

dx (t) = (\tilde{A} x (t) + Bv (t)) dt

(19)

In the above system, for compromising between the rate of disturbance rejection and reaching phase duration, we choose the following performance criterion.

J = \int_{0}^{\infty} {x^{T} (t) Q x (t) + v {(t)}^{T} R v (t)} d t

(20)

where $Q \in R^{n * n}$ is a positive-definite symmetric matrix, and $R = diag {r_{1}, r_{2}, \dots, r_{m}}$ is a positive-definite diagonal matrix.

As we know from the Linear Quadratic Regulator (LQR) theorem, the answer to this optimization problem is as follows

v (t) = - R^{- 1} B^{T} Px (t)

(21)

where $P$ satisfies: ${\tilde{A}}^{T} P + P \tilde{A} + Q - PB R^{- 1} B^{T} P = 0$

Comparing Equation (18), Equation (21) gives

ρ (t) \tanh (s (t)) = - R^{- 1} B^{T} Px (t)

(22)

Therefore, one can obtain

s_{i} (t) = \frac{1}{2} Ln \frac{ρ (t) - {(R^{- 1} B^{T} Px (t))}_{i}}{ρ (t) + {(R^{- 1} B^{T} Px (t))}_{i}}

(23)

The abovementioned method is only for understanding how we obtain this new sliding surface, and the notations such as $u (t)$ should not be confused with the controller applied to the system in the next section.

$R$ in this sliding variable plays an important role in that, by increasing $R$ , the reaching phase duration and response rate might decrease.

It should be emphasized that the ambiguity about occurring any singularity in Equation (23) will resolve in future remarks.

It should be noted that from Equations (13) and (23), under the condition $B^{T} PD = 0$ , $s (x, t)$ is not explicitly involved in Brownian motion $e (t)$ , which simplifies the reachability proving of sliding surfaces. This condition will be ensured in Theorem 1.

Control scheme

The variable structure controller is synthesized as follows

u = (I - \hat{Λ})^{- 1} Zx + ρ (0)^{2} ρ (t) sign (s)

(24)

where $\hat{Λ} = diag {{\hat{μ}}_{1}, {\hat{μ}}_{2}, \dots, {\hat{μ}}_{m}}$ and ${\hat{μ}}_{i}$ represent the estimation of effectiveness loss value of the ith actuator (Liu et al., 2015), and $sign (s) = [\begin{matrix} sign (s_{1}) \\ sign (s_{2}) \\ \dots \\ sign (s_{m}) \end{matrix}]$ is the vector of switching function.

The adaptation laws for $ρ$ and ${\hat{μ}}_{i}$ are given as

\overset{\cdot}{ρ} (t) = ξ^{- 1} (κ (t) - ρ (t))

(25)

where

ξ = \frac{1}{1 - {\bar{μ}}_{\max}} ‖ R {(B^{T} PB)}^{- 1} R ‖

(26)

κ (t) = \frac{ϖ + r ‖ R ‖}{1 - {\bar{μ}}_{\max}} + ‖ Kx (t) ‖

(27)

with $ϖ = \max (r, {mi n_{i} (R_{ii})}^{- \frac{1}{2}})$ , $K = R^{- 1} B^{T} P$ .

There is the following restriction on the initial condition

ρ (0) > κ (0)

(28)

For regulating ${\hat{μ}}_{i}$ , the following adaptation law is suggested

\begin{matrix} \dot{{\hat{μ}}_{i}} = 0 If {\hat{μ}}_{i} = {\underline{μ}}_{\min} & θ_{i} < 0 or {\hat{μ}}_{i} = {\bar{μ}}_{\max} & θ_{i} \geq 0 \\ \dot{{\hat{μ}}_{i}} = θ_{i} elsewhere \end{matrix}

(29)

where

θ_{i} = - γ_{i} x^{T} (PB)^{i} {(1 - {\hat{μ}}_{i})}^{- 1} Z_{i} x

(30)

and $γ_{i} > 0$ is the arbitrary updating gain, and $P > 0$ will be designed in Theorem 1.

Substituting Equation (24) into Equation (13) one obtains

\begin{matrix} dx = [(A + Δ A) x (t) + B {(I - Λ) [(I - \hat{Λ})^{- 1} Zx (t) \\ + ρ (t) sign (s (t))] + f (x, t)}] dt \\ + DG (x, t) de (t) \end{matrix}

(31)

Definition 1 ( Khasminskii, 1980): The equilibrium solution, $x_{t} = 0$ of the stochastic differential Equation (13) with $u (t) = 0$ is said to be globally asymptotically stable with probability one if for any $τ$ ≥ 0 and ε > 0

lim_{‖ x ‖ \to 0} P {sup_{τ < t} ‖ {x_{t}}^{τ, x} ‖ > ε} = 0, P {lim_{t \to \infty} ‖ {x_{t}}^{τ, x} ‖ = 0} = 1

(32)

where ${x_{t}}^{τ, x}$ denotes the solution at time $t$ of a stochastic differential equation starting from the state $x$ at time $τ$ for $τ < t$ .

Lemma 1 (Lin and Cai 1995): The trivial solution of the stochastic differential equation

dx (t) = a (t, x) dt + b (t, x) de (t)

(33)

with $a (t, x)$ and $b (t, x)$ sufficiently differentiable maps, is globally asymptotically stable with probability one, if there exists a positive-definite radially unbounded function $V (t, x)$ and satisfies

\begin{matrix} L V = \frac{\partial V}{\partial t} + \frac{\partial V}{\partial x} a (t, x) + \frac{1}{2} trace {b {(t, x)}^{T} \frac{\partial^{2} V}{\partial x^{2}} b (t, x)} < 0 \\ for x \neq 0 \end{matrix}

(34)

Remark 2. The adaptation law Equation (25) and $ρ (0) > κ (0)$ gives

ρ (t) = \frac{ϖ + r ‖ R ‖}{1 - {\bar{μ}}_{\max}} + ‖ Kx (t) ‖ - ξ \overset{\cdot}{ρ} (t)

(35)

Equation (35) and the condition $mi n_{i} (| λ_{i} (A + BZ) |) >> ξ^{- 1}$ makes $ρ (t)$ to reduce more slowly than $x (t)$ , which means $ρ (t) \geq κ (t)$ for all $t \geq 0$ , yielding $\overset{\cdot}{ρ} (t) \leq 0$ and $ρ (t) > | | Kx (t) | |$ . This leads to avoidance of any singularity in Equation (23). Considering the above description, we can conclude that $ρ_{ss} = \frac{(ϖ + r | | R | |)}{(1 - {\bar{μ}}_{\max})}$ where $ρ_{ss}$ is the steady-state value of $ρ (t)$ .

In the following theorem, we have presented the conditions sufficient for global asymptotic stability of the stochastic active suspension system Equation (31).

Theorem 1. Consider the uncertain stochastic active suspension system Equation (13). The SMC law is synthesized as Equation (24) by updating laws (25), (29), and (30). If there is matrix $P > 0$ and scalars $ε > 0$ , $δ > 0$ satisfying the following linear matrix inequalities (LMIs) with equality constraint

Ω = [\begin{matrix} T & PE \\ E^{T} P & - δ I \end{matrix}] < 0

(36)

D^{T} PD - ε I < 0

(37)

B^{T} PD = 0

(38)

where

T = P (A + BZ) + {(A + BZ)}^{T} P + δ H^{T} H + ε N^{T} N

(39)

then closed-loop stochastic active suspension system is globally asymptotically stable with probability one.

Proof: Let

V_{1} = x^{T} Px + \sum_{k = 1}^{m} {γ_{k}}^{- 1} {\tilde{μ}}_{k}^{2} + {\tilde{ρ}}^{2} (t)

(40)

where ${\tilde{μ}}_{k} = {\hat{μ}}_{k} - μ_{k}$ is the estimate error of $μ_{k}$ and $\tilde{ρ} (t) = ρ_{ss} - ρ (t)$ .

By Itô’s formula, the infinitesimal generator $L V_{1}$ associated with the system (31) is obtained as

\begin{matrix} L V_{1} (t) = 2 x^{T} PAx + 2 x^{T} P Δ Ax + 2 x^{T} PB (I - Λ) (I - \hat{Λ})^{- 1} Zx \\ + 2 x^{T} PB (I - Λ) ρ (t) sign (s) + 2 x^{T} PBf (x, t) \\ + trace (G^{T} D^{T} PDG) \\ + 2 \sum_{k = 1}^{m} {γ_{k}}^{- 1} {\tilde{μ}}_{k} {\overset{\cdot}{\tilde{μ}}}_{k} + - 2 \tilde{ρ} (t) \overset{\cdot}{ρ} (t) \end{matrix}

(41)

If the expression $(I - Λ) (I - \hat{Λ})^{- 1} = I + \tilde{Λ} (I - \hat{Λ})^{- 1}$ , where $\tilde{Λ} = \hat{Λ} - Λ$ , is utilized, one may have

\begin{array}{l} ℒ V_{1} (t) = 2 x^{T} P A x + 2 x^{T} P Δ A x + 2 x {(t)}^{T} P B Z x (t) \\ + 2 x {(t)}^{T} P B (I - Λ) ρ (t) s i g n (s (t)) \\ + 2 x^{T} P B f (x, t) + t r a c e (G^{T} D^{T} P D G) + 2 x {(t)}^{T} P B \tilde{Λ} \\ \times {(I - \hat{Λ})}^{- 1} Z x (t) \\ + 2 \sum_{k = 1}^{m} {γ_{k}}^{- 1} {\tilde{μ}}_{k} {\dot{\tilde{μ}}}_{k} - 2 \tilde{ρ} (t) \dot{ρ} (t) \end{array}

(42)

First, using Remark 2, we will have: $- 2 \tilde{ρ} (t) \overset{\cdot}{ρ} (t) \leq 0$

Then, taking into account the definition of the sliding function in section “Logarithmic sliding surface design,” and utilizing $| X^{T} Y | \leq | | X | |_{2} | | Y | |_{2}$ and $| | X | |_{2} \leq | | X | |_{1}$ , we will have

\begin{array}{l} 2 x {(t)}^{T} P B (I - Λ) ρ (t) s i g n (s (t)) + 2 x^{T} P B f (x, t) \\ = - 2 ρ^{2} t a n h {(s)}^{T} R (I - Λ) s i g n (s) - 2 ρ t a n h {(s)}^{T} R f (x, t) \\ = - 2 ρ^{2} \sum_{i = 1}^{m} r_{i} (1 - μ_{i}) t a n h (s_{i}) s i g n (s_{i}) - 2 ρ t a n h {(s)}^{T} R f (x, t) \\ \leq - 2 ρ^{2} (1 - {\bar{μ}}_{\max}) \sum_{i = 1}^{m} r_{i} | t a n h (s_{i}) | + 2 ρ | t a n h {(s)}^{T} R f (x, t) | \\ \leq 2 ρ (t) {- ρ (t) (1 - {\bar{μ}}_{\max}) {‖ R t a n h (s (t)) ‖}_{1} + {‖ R t a n h (s (t)) ‖}_{2} {‖ f (x, t) ‖}_{2}} \\ \leq 2 ρ {- ρ (1 - {\bar{μ}}_{\max}) {‖ R t a n h (s) ‖}_{2} + {‖ R t a n h (s) ‖}_{2} r} \end{array}

(43)

Using Remark 2, one may have

ρ (t) > \frac{r}{1 - {\bar{μ}}_{\max}}

(44)

Therefore, one has

- 2 ρ^{2} (1 - {\bar{μ}}_{\max}) ‖ R \tanh (s) ‖_{2} + 2 ρ ‖ R \tanh (s) ‖_{2} r \leq 0

(45)

Moreover, if ${\hat{μ}}_{i} = {\underline{μ}}_{\min} and θ_{i} < 0 or {\hat{μ}}_{i} = {\bar{μ}}_{\max} and θ_{i} \geq 0$ we obtain from Equation (29), ${\hat{μ}}_{i}^{.} = 0$ and ${\tilde{μ}}_{i} θ_{i} \geq 0$ which implies that the following expression (46) is satisfied. On the contrary, if ${\hat{μ}}_{i}^{.} = θ_{i}$ , the expression (46) is also satisfied. Thus, it can be concluded that the following expression is ensured by Equations (29) and (30)

2 x^{T} PB \tilde{Λ} (I - \hat{Λ})^{- 1} Zx + 2 \sum_{k = 1}^{m} {γ_{k}}^{- 1} \tilde{μ} \dot{\tilde{μ}} \leq 0

(46)

Note that from Equation (6), for $δ > 0$

2 x^{T} P Δ Ax \leq δ^{- 1} x^{T} PE E^{T} Px + δ x^{T} H^{T} Hx

(47)

Thus, the following inequality can be written by using Equations (14), (42), (45), (46), and (47).

\begin{matrix} L V_{1} (t) \leq x^{T} P (A + BZ) x + x^{T} {(A + BZ)}^{T} Px + δ^{- 1} x^{T} PE E^{T} \\ Px + δ x^{T} H^{T} Hx + x^{T} ε N^{T} Nx \\ = x^{T} Σ x \end{matrix}

(48)

where

Σ = P (A + BZ) + {(A + BZ)}^{T} P + δ^{- 1} PE E^{T} P + δ H^{T} H + ε N^{T} N

(49)

Hence if $Σ < 0$ , we have

L V_{1} (t) < 0 x (t) \neq 0

(50)

By Lemma 1, it can be concluded that the closed-loop active suspension system (31) is globally asymptotically stable with probability one. On the contrary, $Σ < 0$ is equivalent to the condition (36). This concludes the proof.

In the sequel, an algorithm is given to solve LMIs with equality constraint as Equations (36)–(38).

Consider the following matrix inequality for $β > 0$

(B^{T} PD)^{T} (B^{T} PD) \leq β I

(51)

By Schur’s complement, Equation (38) is equivalent to

[\begin{matrix} - β I & D^{T} PB \\ B^{T} PD & - I \end{matrix}] \leq 0

(52)

Now, we define the following minimization problem

Min β subject to Equations (36) - (37) and Equation (52)

(53)

This is a minimization problem involving linear objective and LMI constraints, which can be solved by using LMI toolbox in MATLAB. It can be seen that if the global infimum of problem (52) equals zero, the corresponding solutions $P$ , $ε$ and $δ$ will satisfy Equations (36)–(38).

Remark 3. Suppose that we define the following diagonal matrix $M$ .

M = diag {\frac{1}{ρ^{2} - {(Kx)}_{1}^{2}}, \frac{1}{ρ^{2} - {(Kx)}_{2}^{2}}, \dots, \frac{1}{ρ^{2} - {(Kx)}_{m}^{2}}}

(54)

By remark 2, we will have $M > 0$ and $| | M^{- 1} | | \leq ρ (0)^{2}$

Also, adaptation law (25), gives

ρ (t) = κ (t) - ξ \overset{\cdot}{ρ} (t)

(55)

Then with the Remark 2, we have

ρ (t) > ‖ Kx (t) ‖ + {{min}_{i} (R_{ii})}^{- \frac{1}{2}}

(56)

which yields $(M^{- 1} R)_{ii} \geq 1$

Remark 4. Remark 2, gives $ρ (t) > ϖ$ and Remark 3, gives

\begin{matrix} \frac{d}{dt} {M^{- 1}} = diag {2 ρ \overset{\cdot}{ρ} - 2 (Kx)_{1} (K \overset{\cdot}{x})_{1}, 2 ρ \overset{\cdot}{ρ} - 2 (Kx)_{2} \\ (K \overset{\cdot}{x})_{2}, \dots, 2 ρ \overset{\cdot}{ρ} - 2 (Kx)_{m} (K \overset{\cdot}{x})_{m}} \end{matrix}

Let,

‖ Kx ‖ ‖ K \overset{\cdot}{x} ‖ \leq - ϖ \overset{\cdot}{ρ}

(57)

Therefore, one can obtain

- ρ (t) \overset{\cdot}{ρ} (t) \geq - ϖ \overset{\cdot}{ρ} (t) \geq ‖ Kx ‖ ‖ K \overset{\cdot}{x} ‖ \geq - {(Kx)}_{i} {(K \overset{\cdot}{x})}_{i}

(58)

that yields

{(\frac{d}{dt} {M^{- 1}})}_{ii} = 2 ρ (t) \overset{\cdot}{ρ} (t) - 2 {(Kx (t))}_{i} {(K \overset{\cdot}{x})}_{i} \leq 0

(59)

Due to the fact that one of the key features of SMC is the absence of sensitivity to external disturbances in the sliding phase, it is necessary to prove that the sliding phase will arrive eventually. The sliding phase is a mode in which the state trajectories are placed on the sliding surface or $s (t) = 0$ . Theorem 2 proves that the proposed controller (24) can convey state trajectories to the sliding manifold.

Theorem 2. Consider the active suspension system (13). The sliding variable is designed as $s_{i} = \frac{1}{2} Ln \frac{(ρ (t) - R^{- 1} B^{T} Px (t))}{(ρ (t) + R^{- 1} B^{T} Px (t))}$ where matrix $P > 0$ is the solution of matrix inequalities and constrain equality (36)–(38). The reachability of the sliding surfaces $s_{i}$ for all i is guaranteed under the sliding-mode controller Equation (24).

Proof. Choose the Lyapunov function as

V = \frac{1}{2} s^{T} ρ (0)^{- 2} (JP J^{T})^{- 1} s

(60)

where $J = M R^{- 1} B^{T}$ .

\overset{\cdot}{V} = s^{T} ρ (0)^{- 2} (JP J^{T})^{- 1} \overset{\cdot}{s} + s^{T} ρ (0)^{- 2} (\overset{\cdot}{J} P J^{T})^{- 1} s

(61)

By Equation (23), we will have

{\overset{\cdot}{s}}_{i} = \frac{\overset{\cdot}{ρ} (t) {(Kx (t))}_{i} - ρ (t) {(K \overset{\cdot}{x} (t))}_{i}}{ρ {(t)}^{2} - {(Kx (t))}_{i}^{2}}

(62)

which can be rewritten without index i as follows:

\overset{\cdot}{s} = M {\overset{\cdot}{ρ} (t) Kx (t) - ρ (t) K \overset{\cdot}{x} (t)}

(63)

By substituting Equation (63) into Equation (61), one can obtain

\begin{matrix} \overset{\cdot}{V} = s^{T} ρ {(0)}^{- 2} {(JP J^{T})}^{- 1} M {\overset{\cdot}{ρ} (t) Kx (t) - ρ (t) K \overset{\cdot}{x} (t)} + s^{T} ρ (0)^{- 2} \dot{(M^{- 1})} R {(B^{T} PB)}^{- 1} R M^{- 1} s \\ = s^{T} ρ {(0)}^{- 2} {(JP J^{T})}^{- 1} MK {\overset{\cdot}{ρ} (t) x (t) - ρ (t) [(A + Δ A) x (t) + B {(I - Λ) [{(I - \hat{Λ})}^{- 1} Zx (t) \\ + ρ {(0)}^{2} ρ (t) sign (s (t))] + f (x, t)}]} + s^{T} ρ (0)^{- 2} \dot{(M^{- 1})} R {(B^{T} PB)}^{- 1} R M^{- 1} s \\ = s^{T} ρ {(0)}^{- 2} {M^{- 1} R {(B^{T} PB)}^{- 1} [RKx \overset{\cdot}{ρ} - ρ B^{T} PAx - ρ B^{T} P Δ Ax] \\ + ρ M^{- 1} R [- (I - Λ) {(I - \hat{Λ})}^{- 1} Zx - ρ (I - Λ) sign (s) - f] + \dot{(M^{- 1})} R {(B^{T} PB)}^{- 1} R M^{- 1} s} \end{matrix}

(64)

By Remarks 2 and 3, one may have

\begin{matrix} \overset{\cdot}{V} \leq ρ {- ‖ R {(B^{T} PB)}^{- 1} R ‖ ‖ s ‖ \overset{\cdot}{ρ} + ‖ R {(B^{T} PB)}^{- 1} B^{T} PEH ‖ ‖ x ‖ ‖ s ‖ \\ + (1 - {\underline{μ}}_{\min}) {(1 - {\bar{μ}}_{\max})}^{- 1} ‖ RZ ‖ ‖ x ‖ ‖ s ‖ - ρ (1 - {\bar{μ}}_{\max}) ‖ s ‖_{1} \\ + r ‖ R ‖ ‖ s ‖ \\ + s^{T} ρ {(0)}^{- 2} \dot{(M^{- 1})} R {(B^{T} PB)}^{- 1} R M^{- 1} s \end{matrix}

(65)

Now, we define the following domain

Φ = {x (t) | η ‖ x (t) ‖ < ϖ and ‖ Kx ‖ ‖ K \overset{\cdot}{x} ‖ \leq - ϖ \overset{\cdot}{ρ}}

(66)

where

\begin{matrix} η = ‖ R {(B^{T} PB)}^{- 1} B^{T} PA ‖ + ‖ R {(B^{T} PB)}^{- 1} B^{T} PEH ‖ \\ + (1 - {\underline{μ}}_{\min}) {(1 - {\bar{μ}}_{\max})}^{- 1} ‖ RZ ‖ \end{matrix}

(67)

By the result in Theorem 1, we can obtain that the state trajectories will enter the domain Φ in finite time.

That is, the relation $η | | x | | < ϖ$ and $| | Kx | | | | K \overset{\cdot}{x} | | \leq - ϖ \overset{\cdot}{ρ}$ can be ensured in finite time.

In this domain, by Remark 4, one has

s^{T} ρ {(0)}^{- 2} \dot{(M^{- 1})} R {(B^{T} PB)}^{- 1} R M^{- 1} s \leq 0

(68)

Therefore, in this region, the following inequality can be written

\dot{V} \leq ρ ‖ s ‖ {- ‖ R {(B^{T} P B)}^{- 1} R ‖ \dot{ρ} + ϖ - ρ (1 - {\bar{μ}}_{\max}) + r ‖ R ‖}

(69)

Finally, the updating law (25) gives

\overset{\cdot}{V} \leq - ρ (1 - {\bar{μ}}_{\max}) ‖ s ‖ ‖ Kx ‖ \leq 0

(70)

Hence, it follows from Equation (70) that the reachability of all the sliding surfaces can be ensured in finite time. This concludes the proof.▪

Simulation results

In this section, two following cases are considered to verify the advantage of the proposed purposeful sliding manifold extraction in compromising between reaching phase duration (robustness) and response rate, both of which have key roles in stabilizing active suspension systems.

Case 1. Suspension system with $R = 10^{- 4}$

Case 2. Suspension system with $R = 10^{- 2}$

Computer simulations have been performed with the following plant parameters.

\begin{matrix} m_{s} = 400 (kg), m_{u} = 40 (kg), k_{s} = 18, 000 (N / m), \\ c_{s} = 1000 (Ns / m), and k_{t} = 200, 000 (N / m) \end{matrix}

and therefore, one can obtain

\begin{matrix} A = [\begin{matrix} 0 & 1 & 0 & 0 \\ - 45 & - 2.5 & 45 & 2.5 \\ 0 & 0 & 0 & 1 \\ 450 & 25 & - 5450 & 25 \end{matrix}], B = [\begin{matrix} 0 \\ 0.0025 \\ 0 \\ - 0.0025 \end{matrix}], \\ D = [\begin{matrix} 0 \\ 0 \\ 0 \\ 5000 \end{matrix}] \end{matrix}

and other parameters have been selected as follows

\begin{matrix} E = [\begin{matrix} 0.2 \\ 0.2 \\ 0.3 \\ 0.2 \end{matrix}], F = \sin (t), \\ H = [0.1 0.2 0.2 0.2], f = \sin (x_{1} (t)), G = 0.1 x_{1} (t), \end{matrix}

N = [0.30.30.3]

Λ = 0.3 and {\underline{μ}}_{\min} = 0.2, {\underline{μ}}_{\max} = 0.5

In the controller design process, we have selected the following parameters

r = 1 \geq ‖ f (t) ‖

Z = [\begin{matrix} 13, 790 & - 110 & 163, 340 & 1310 \end{matrix}]

where

A + BZ

is Hurwitz.

After solving Equation (53), we will have

P = [\begin{matrix} 0.1294 & 0.0078 & - 0.1383 & - 0.0016 \\ 0.0078 & 0.001 & - 0.157 & - 0.0002 \\ - 0.1383 & - 0.0157 & 0.3232 & 0.006 \\ - 0.0016 & - 0.0002 & 0.006 & 0 \end{matrix}]

Suppose that one realization of $x_{r} (t)$ is as Figure 2 which is a concave-convex as a typical type of road profile. The simulation results in two cases have been displayed in Figures 3 and 4. They show the displacement of sprung mass and sliding variable, in two cases, respectively. As expected, the comparison between two cases indicates that as $R$ increases, the rate of disturbance effect rejection decreases while the reaching phase duration becomes shorter. As previously explained, the reaching phase is a mode in which there is sensitivity to parameter variations and disturbances. This means that the positive effect of increasing $R$ leads to a reduction in the sensitivity of the closed-loop system to parameter variations and disturbances, which is an important issue in control systems research.

More specifically, in Figures 5 and 6, the change of the responses in cases 1 and 2 are compared when there is a pure delay of 30 ms in all system output measurements. In Case 1, after applying delay, the closed-loop system has been unstable which indicates that the delay margin is less than 30 ms, while in Case 2, the response has changed a little only which shows that the delay margin is more than 30 ms. This comparison shows that the parameter $R$ could be increased to reduce the sensitivity. However, increasing $R$ has a negative effect: the settling time in two cases is approximately 1.2 and 3 s, respectively, which shows a decrease in the rate of disturbance effect rejection with an increase in $R$ .

Figure 7 shows control signals in two cases. Figure 8 shows how the switching gain is updated.

In the sequel, in Figures 9 and 10, we have extracted the curves showing the effect of changing R which can help a designer to adjust the parameter $R$ according to design goals for the active suspension system.

Also, in order to show the improvement of system performance in comparison with other sliding surfaces, two examples of important and widely used control surfaces, namely linear sliding surface and proportional–integral one, have been selected. The performance criterion is defined as

J = 100 \int_{0}^{t} {x^{T} (t^{'}) Qx (t^{'}) + v {(t^{'})}^{T} Rv (t^{'})} {dt}^{'}

Two selected linear sliding variables are $s = Cx$ with $C = (B^{T} B)^{- 1} B^{T}$ (Veselić et al., 2014)

and $s = Cx$ with $C = B^{T}$ (Acosta, 2014)

Two selected integral sliding variables are $s = Cx - \int_{0}^{\infty} C (A + BZ) x$ with $C = B^{T}$ (Lian et al., 2012)

and $s = Cx - \int_{0}^{\infty} C (A + BZ) x$ with $C = B^{T} P$ (Niu et al., 2007)

Changes of performance function over time are shown in Figures 11 and 12. Table 1 shows the steady-state values of the cost function and the improvement percentages of the proposed method.

Table 1.

Comparison of different sliding surfaces with the proposed sliding surface.

Sliding variable	Parameter	The final value of the index	The improvement percentage of the newmethod compared to this method
$s = Cx$	$C = (B^{T} B)^{- 1} B^{T}$	$263$	96.6
$s = Cx$	$C = B^{T}$	$52$	82.7
$s = Cx - \int_{0}^{\infty} C (A + BZ) x$	$C = B^{T}$	$28$	67.9
$s = Cx - \int_{0}^{\infty} C (A + BZ) x$	$C = B^{T} P$	$13$	30.8
$s = \frac{1}{2} Ln \frac{ρ (t) - R^{- 1} B^{T} Px (t)}{ρ (t) + R^{- 1} B^{T} Px (t)}$	$R = 10^{- 4}$	9	–

As the results show, the final value of the performance index in the proposed method is 9. Compared to the most common existing methods, the improvement percentage of the system has reached about 30% in the worst case and even 96% in the best case.

Conclusion

In this paper, we have presented a new SMCler for active suspension, which can manage the robustness and sensitivity of the close-loop system. Lyapunov theorem has been used to design such a controller. Furthermore, a robust stochastic controller has been designed due to uncertainties in the model and stochastic nature of road bumps. The simulation results confirmed the expected performance of the control system. Considering the novel perspective presented in this paper to design sliding the manifold, further research could address the development of this method for other dynamic systems.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Seyed Ali Zahiripour

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