Optimal design for robust control parameter for active roll control system: a fuzzy approach

Abstract

In this paper, we investigate the dynamical model of an active roll control system (ARCS) which can impose an anti-roll moment quickly by active actuators to prevent a vehicle rolling when the vehicle generates the roll tendency and effectively enhances the vehicle dynamic performance without sacrificing ride comfort. In the dynamic model of the ARCS, we consider the sprung mass of the vehicle which is (possibly) time-varying and the initial conditions are the uncertain parameters which are described by fuzzy set theory. A new optimal robust control which is deterministic and is not the usual if–then rules-based control is proposed. The desired controlled system performance is twofold: one deterministic, which includes uniform boundedness and uniform ultimate boundedness, and one fuzzy, which enhances the cost consideration. We then formulate an optimal design problem associated with the control as a constrained optimization problem. The resulting control design is systematic and is able to guarantee the deterministic performance and minimize the average fuzzy performance. Numerical simulations show that the control design renders the ARCS practically stable and achieves constraints following maneuvering.

Keywords

Active roll control system system uncertainty fuzzy set theory robust control optimal design

1. Introduction

For customer perception of satisfaction, ride and handling are two of the key attributes in a vehicle. When a vehicle is cornering, it is obvious to find that the vehicle body leans towards outside due to the centrifugal force and too much of this roll motion will make passengers feel uncomfortable. Basically, most passenger vehicles today are equipped with the passive anti-roll bars which are attached at the front and rear axles so as to reduce the body roll acceleration and roll angle during single wheel lifting and cornering maneuver. But these conventional passive anti-roll bars are difficult to satisfy the requirements of ride comfort and handling stability at the same time, and can’t adjust the roll stiffness of the suspension in real-time.

The active roll control system (ARCS) has become increasingly popular for researchers and engineers to deal with the issues of tradeoffs between ride and handling. The ARCS can impose anti roll moment quickly to prevent the vehicle rolling when the vehicle generates the roll tendency. It can also reduce the roll angle and roll angle velocity of the vehicle body greatly, increase the tire normal force of the independent suspension and improve the adhesion conditions of the wheels and the road. Hence the ARCS can effectively enhance the vehicle dynamic performance without sacrificing the ride comfort (Gosselin-Brisson et al., 2009; Cronje and Els, 2010).

According to the actuator types and the actuator locations, there are two different types of ARCS in passenger vehicles (Zulkarnain et al., 2012). The first type, which is the most popular actuator in ARCS today, is the hydro-pneumatic and hydraulic system (Konik, 2002; Kim et al., 2012). BMW company developed an active anti-roll bar system called Dynamic Drive with hydraulic system in 2001 (Strassberger and Guldner, 2004). However, the obvious drawbacks of these hydraulic actuators are manufacturing cost, power consumption, and their slow response on various roads and steering inputs. The second type, which today is attracting more and more attention, is the electric ARCS (Kim and Lee, 2002; Ohta et al., 2006; Buma et al., 2010; Jeon et al., 2012), which is more fuel efficient, energy saving, and environmentally friendly than conventional hydraulic ARCS. Also, this electric ARCS is more suitable for today’s new energy vehicles.

In this paper, we aim to deal with the active roll control problem considering system uncertainties which include uncertain parameter that is (possibly) time-varying but bounded, and uncertain initial conditions. Applying fuzzy set theory (Zadeh, 1965) to handle the uncertainties is a salient feature of this paper. We stress that these are different from the very popular Takagi–Sugeno model or other fuzzy if–then rules-based models (Takagi and Sugeno, 1985; Huang and Lin, 2003; Chen et al., 2007; Soltanpour et al., 2016). In this study, we shall attempt to pursue optimal robust control design which is deterministic and is not if–then rules-based for fuzzy dynamical systems.

The main contributions of this paper are threefold. First, we take the lead in applying fuzzy set theory to the dynamic model of the electric ARCS. We consider the sprung mass which is (possibly) time-varying and the initial conditions are uncertainties that are described by fuzzy set theory (see Appendix). Second, a robust control which is deterministic and not if–then rules-based is proposed to render the ARCS to achieve the deterministic system performance: uniform boundedness and uniform ultimate boundedness. Third, a performance index including average fuzzy system performance and control effort is proposed based on the fuzzy information. The optimal design problem associated with the control can then be solved by minimizing the performance index. Hence, the resulting control is able to guarantee the deterministic performance as well as minimizing the cost.

2. Fuzzy dynamical system

Consider the following uncertain system:

\begin{matrix} \overset{\cdot}{x} (t) = Ax (t) + (B + Δ B (x (t), σ (t), t) u (t) \\ + B (h (t) + v (x (t), σ (t), t)) \\ x (t_{0}) = x_{0} \end{matrix}

(1)

where

t \in R

is the time,

x (t) \in R^{n}

is the state, x₀ is the uncertain initial state,

u (t) \in R^{m}

is the control,

σ (t) \in R^{p}

is an unknown time-varying parameter (which may include system parameter and input disturbance),

A \in R^{n \times n}

and

B \in R^{n \times m}

are known constant matrices,

h (t) \in R^{m}

is a known vector which depends on t, and

Δ B (x (t), σ (t), t) \in R^{n \times m}

and

v (x (t), σ (t), t) \in R^{m}

are matrix, respectively, vector, which depend on x, t, and the uncertain parameter σ. The functions

Δ B (\cdot, t)

and

v (\cdot, t)

are continuous. The functions

h (\cdot)

Δ B (x, σ, \cdot)

and

v (x, σ, \cdot)

are Lebesgue measurable.

Assumption 1

The pair $(A, B)$ is stabilizable.

Assumption 2

There exists a scalar constant $\bar{h}$ such that

{max}_{t} ‖ h (t) ‖ \leq \bar{h}

(2)

Assumption 3

(i) For each entry of x₀, namely $x_{0 i}, i = 1, 2, \dots, n$ , there exists a fuzzy set $X_{0 i}$ in a universe of discourse $Ξ_{i} \subset R$ characterized by a membership function $μ_{Ξ_{i}} : Ξ_{i} \to [0, 1]$ . That is,

X_{0 i} = {(x_{0 i}, μ_{Ξ_{i}} (x_{0 i})) | x_{0 i} \in Ξ_{i}}

(3)

Here, $Ξ_{i}$ is known and compact. (ii) For each entry of σ, namely $σ_{i}, i = 1, 2, \dots, p$ , the function $σ_{i} (\cdot)$ is Lebesgue measurable. (iii) For each $σ_{i}$ , there exists a fuzzy set S_i in a universe of discourse $Σ_{i} \subset R$ characterized by a membership function $μ_{i} : Σ_{i} \to [0, 1]$ . That is,

S_{i} = {(σ_{i}, μ_{Σ_{i}} (σ_{i})) | σ_{i} \in Σ_{i}}

(4)

Remark

Assumption 3 imposes fuzzy description on the uncertainty x₀ and $σ (t)$ . Equation (1) under this fuzzy description of uncertainty is called the fuzzy dynamical system.

Consider the following Riccati equation:

A^{T} P + PA - 2 {PBR}^{- 1} B^{T} P + Q = 0

(5)

where

R > 0

and

Q > 0

. The solution

P > 0

exists and is unique if

(A, B)

is stabilizable.

Assumption 4

(i) There are fuzzy numbers a and b such that for all $(x, t) \in R^{n} \times R$

| | v (x, σ, t) | | \leq a | | x | | + b

(6)

(ii) There exists a matrix $E (x, σ, t)$ such that $Δ B (x, σ, t) = BE (x, σ, t)$ . In addition, there are a scalar constant $ρ_{E 1}$ and a fuzzy number $ρ_{E 2}$ such that for all $(x, t) \in R^{n} \times R$

\begin{matrix} \frac{1}{2} λ_{min} (E (x, σ, t) R^{- 1} + R^{- 1} E^{T} (x, σ, t)) \\ \geq ρ_{E 1} > - λ_{min} (R^{- 1}) \end{matrix}

(7)

| | E (x, σ, t) | | \leq ρ_{E 2}

(8)

Throughout the paper, unless otherwise stated, vector norms are Euclidean and matrix norms are the corresponding induced ones.

Remark

The fuzzy numbers a and b depend on σ. Their associated membership functions can be determined via $μ_{i} (\cdot)$ , the fuzzy arithmetic, and the decomposition theorem. Both $ρ_{E 1}$ and $ρ_{E 2}$ can be evaluated since the universes of discourse, $Σ_{i} (\cdot)$ , are known.

We introduce the following desirable deterministic dynamical system performance.

Definition 1

Consider a dynamical system

\overset{\cdot}{ζ} (t) = f (ζ (t), t), ζ (t_{0}) = ζ_{0}

(9)

The solution of the system (suppose it exists and can be continued over

[t_{0}, \infty)

) is uniformly bounded if for any

r > 0

with

| | ζ_{0} | | \leq r

, there is

d (r) > 0

with

| | ζ (t) | | \leq d (r)

for all

t \geq t_{0}

. It is uniformly ultimately bounded if for

| | ζ_{0} | | \leq r

, there are

\bar{d} (r) > 0

and

T (\bar{d} (r), r) \geq 0

such that

| | ζ (t) | | \leq \bar{d} (r)

for all

t \geq t_{0} + T (\bar{d} (r), r)

3. Deterministic robust control design

We now propose the control u as follows:

\begin{matrix} u (t) = - R^{- 1} B^{T} Px (t) - h (t) \\ - γ R^{- 1} B^{T} Px (t) (δ_{1} | | x (t) | | + δ_{2})^{2} \end{matrix}

(10)

where the design parameters

δ_{1, 2}

and γ are strictly positive scalar constants.

Theorem 1

Subject to Assumptions 1–4, suppose that the control of equation (10) is applied to the system represented by equation (1). Then, the solution of the controlled system is uniformly bounded and uniformly ultimately bounded. In addition, the size of the ultimate boundedness region can be made arbitrarily small by suitable choices of the design parameters.

Proof

We prove this via the Lyapunov minimax approach which is a general approach to the robust control problem (Corless, 1993; Leitmann, 1993). Consider the Lyapunov function candidate

V (x) = x^{T} Px

(11)

where P is the solution of the Riccati equation (equation (5)). For any admissible

σ (\cdot)

, the time derivative of V along the trajectory of the controlled system (equation (1)) is given by

\begin{matrix} \overset{\cdot}{V} = 2 x^{T} P [Ax + B (I + E) \\ \times (- R^{- 1} B^{T} Px - h - γ R^{- 1} B^{T} Px (δ_{1} | | x | | + δ_{2})^{2}) \\ + B (h + v)] \\ = x^{T} (A^{T} P + PA - 2 {PBR}^{- 1} B^{T} P) x \\ + 2 x^{T} PBE (- R^{- 1} B^{T} Px) + 2 x^{T} PBE (- h) \\ + 2 x^{T} PB (I + E) (- γ R^{- 1} B^{T} Px (δ_{1} | | x | | + δ_{2})^{2}) \\ + 2 x^{T} PBv \end{matrix}

(12)

Regarding the first term on the right-hand side (RHS) of equation (12), by using the Riccati equation (equation (5)) and the Rayleigh’s principle, we have

\begin{matrix} x^{T} (A^{T} P + PA - 2 {PBR}^{- 1} B^{T} P) x \\ = - x^{T} Qx \leq - λ_{min} (Q) | | x | |^{2} \end{matrix}

(13)

Let $\hat{α} : = B^{T} Px$ . Regarding the second term on the RHS of equation (12), by using equation (8) we have

2 x^{T} PBE (- R^{- 1} B^{T} Px) \leq 2 | | \hat{α} | | ρ_{E 2} | | R^{- 1} B^{T} P | | | | x | |

(14)

Regarding the third term on the RHS of equation (12), by using equations (2) and (8), we have

2 x^{T} PBE (- h) \leq 2 | | \hat{α} | | ρ_{E 2} | | h | | \leq 2 | | \hat{α} | | ρ_{E 2} \bar{h}

(15)

Regarding the fourth term on the RHS of equation (12), by using the Rayleigh’s principle and equation (7), respectively, we have

\begin{matrix} 2 x^{T} PB (- γ R^{- 1} B^{T} Px (δ_{1} | | x | | + δ_{2})^{2}) \\ = - 2 γ ({\hat{α}}^{T} R^{- 1} \hat{α}) (δ_{1} | | x | | + δ_{2})^{2} \\ \leq - 2 γ λ_{min} (R^{- 1}) | | \hat{α} | |^{2} (δ_{1} | | x | | + δ_{2})^{2} \end{matrix}

(16)

and

\begin{matrix} 2 x^{T} PBE (- γ R^{- 1} B^{T} Px (δ_{1} | | x | | + δ_{2})^{2}) \\ = 2 γ ({\hat{α}}^{T} {ER}^{- 1} \hat{α}) (δ_{1} | | x | | + δ_{2})^{2} \\ = - γ {\hat{α}}^{T} ({ER}^{- 1} + R^{- 1} E^{T}) \hat{α} (δ_{1} | | x | | + δ_{2})^{2} \\ \leq - γ λ_{min} ({ER}^{- 1} + R^{- 1} E^{T}) | | \hat{α} | |^{2} (δ_{1} | | x | | + δ_{2})^{2} \\ \leq - 2 γ ρ_{E 1} | | \hat{α} | |^{2} (δ_{1} | | x | | + δ_{2})^{2} \end{matrix}

(17)

Regarding the fifth term on the RHS of equations (12), by using equation (6) we have

2 x^{T} PBv \leq 2 | | \hat{α} | | (a | | x | | + b)

(18)

Combining equations (13)–(18), we have

\begin{matrix} \overset{\cdot}{V} \leq - λ_{min} (Q) | | x | |^{2} + 2 | | \hat{α} | | ρ_{E 2} | | R^{- 1} B^{T} P | | | | x | | \\ + 2 | | \hat{α} | | ρ_{E 2} \bar{h} - 2 γ λ_{min} (R^{- 1}) | | \hat{α} | |^{2} (δ_{1} | | x | | + δ_{2})^{2} \\ - 2 γ ρ_{E 1} | | \hat{α} | |^{2} (δ_{1} | | x | | + δ_{2})^{2} + 2 | | \hat{α} | | (a | | x | | + b) \\ = - λ_{min} (Q) | | x | |^{2} + 2 | | \hat{α} | | \\ \times (ρ_{E 2} | | R^{- 1} B^{T} P | | | | x | | + ρ_{E 2} \bar{h} + a | | x | | + b) \\ - 2 | | \hat{α} | |^{2} (ρ_{E 1} + λ_{min} (R^{- 1})) γ (δ_{1} | | x | | + δ_{2})^{2} \end{matrix}

(19)

Since the second and third terms on the RHS of equation (19) are linear and quadratic, respectively, in $| | \hat{α} | |$ , a lengthy but straightforward algebra shows that

\begin{matrix} 2 | | \hat{α} | | (ρ_{E 2} | | R^{- 1} B^{T} P | | | | x | | + ρ_{E 2} \bar{h} + a | | x | | + b) \\ - 2 | | \hat{α} | |^{2} (ρ_{E 1} + λ_{min} (R^{- 1})) γ (δ_{1} | | x | | + δ_{2})^{2} \\ \leq \frac{(ρ_{E 2} | | R^{- 1} B^{T} P | | | | x | | + a | | x | | + ρ_{E 2} \bar{h} + b)^{2}}{2 (ρ_{E 1} + λ_{min} (R^{- 1})) γ (δ_{1} | | x | | + δ_{2})^{2}} \\ = \frac{1}{2 (ρ_{E 1} + λ_{min} (R^{- 1})) γ} \frac{(\bar{a} | | x | | + \bar{b})^{2}}{(δ_{1} | | x | | + δ_{2})^{2}} \\ \leq \frac{1}{2 (ρ_{E 1} + λ_{min} (R^{- 1})) γ} [max (\frac{\bar{a}}{δ_{1}}, \frac{\bar{b}}{δ_{2}})]^{2} \end{matrix}

(20)

where

\bar{a} : = ρ_{E 2} | | R^{- 1} B^{T} P | | + a \bar{b} = ρ_{E 2} \bar{h} + b

, Let

δ^{\frac{1}{2}} = \frac{\bar{a}}{δ_{1}} + \frac{\bar{b}}{δ_{2}}

(21)

\bar{δ} = \frac{δ}{2 (ρ_{E 1} + λ_{min} (R^{- 1}))}

(22)

Then, we have

\overset{\cdot}{V} \leq - λ_{min} (Q) | | x | |^{2} + \frac{\bar{δ}}{γ}

(23)

This in turn means that $\overset{\cdot}{V}$ is negative definite for all $| | x | |$ such that

| | x | | > \sqrt{\frac{\bar{δ}}{γ λ_{min} (Q)}}

(24)

Since all universes of discourses, $Σ_{i} (\cdot)$ , are compact (hence closed and bounded), $\bar{δ}$ is bounded. In addition, both γ and $λ_{min} (Q)$ are crisp. Thus, $\overset{\cdot}{V}$ is negative definite for sufficiently large $| | x | |$ and, therefore, the solution of the controlled system is uniformly bounded and uniformly ultimately bounded (Chen and Leitmann, 1987). Q.E.D.

4. Optimal gain design

Section 3 presents a system performance which can be guaranteed by a deterministic control scheme. From equation (24), we know that the size of the uniform ultimate boundedness region decreases as γ increases. As γ approaches to infinity, the size approaches to zero. This rather strong performance is accompanied by a (possibly) large control effort, which is reflected by γ. From the practical design point of view, it is interesting and necessary for the designer to seek an optimal choice of γ for a compromise among various conflicting criteria. This is associated with the minimization of a performance index.

We first explore more on the deterministic performance of the uncertain system. By the Rayleigh’s principle

λ_{min} (P) | | x | |^{2} \leq x^{T} Px \leq λ_{max} (P) | | x | |^{2}

(25)

and hence

- | | x | |^{2} \leq - \frac{1}{λ_{max} (P)} V

(26)

Inserting this into equation (23), we have

\overset{\cdot}{V} (t) \leq - \frac{λ_{min} (Q)}{λ_{max} (P)} V (t) + \frac{\bar{δ}}{γ}

(27)

where

V_{0} = V (t_{0}) = x_{0}^{T} {Px}_{0}

. This is a differential inequality, analysis of which can be made according to Chen (1996) as follows. According to Chen (2011), the solution of this differential inequality can be given as:

\begin{matrix} V (t) \leq (V_{0} - \frac{λ_{max} (P)}{λ_{min} (Q)} \frac{\bar{δ}}{γ}) exp [- \frac{λ_{min} (Q)}{λ_{max} (P)} (t - t_{0})] \\ + \frac{λ_{max} (P)}{λ_{min} (Q)} \frac{\bar{δ}}{γ} \end{matrix}

(28)

for all $t \geq t_{0}$ . By the same argument, we also have, for any t_s and $τ \geq t_{s}$

\begin{matrix} V (τ) \leq (V_{s} - \frac{λ_{max} (P)}{λ_{min} (Q)} \frac{\bar{δ}}{γ}) exp [- \frac{λ_{min} (Q)}{λ_{max} (P)} (τ - t_{s})] \\ + \frac{λ_{max} (P)}{λ_{min} (Q)} \frac{\bar{δ}}{γ} \end{matrix}

(29)

where

V_{s} = V (t_{s}) = x^{T} (t_{s}) Px (t_{s})

. The time t_s is when the control scheme of equation (10) starts to be executed. It does not have to be t₀.

For each $τ \geq t_{s}$ , let

η (\bar{δ}, γ, τ, t_{s}) : = (V_{s} - \frac{λ_{max} (P)}{λ_{min} (Q)} \frac{\bar{δ}}{γ}) exp [- \frac{λ_{min} (Q)}{λ_{max} (P)} (τ - t_{s})]

(30)

η_{\infty} (\bar{δ}, γ) : = \frac{λ_{max} (P)}{λ_{min} (Q)} \frac{\bar{δ}}{γ}

(31)

Notice that for each $\bar{δ}, γ, t_{s}$ , $η (\bar{δ}, γ, τ, t_{s}) \to 0$ as $τ \to \infty$ .

Remark

$η (\bar{δ}, γ, τ, t_{s})$ and $η_{\infty} (\bar{δ}, γ)$ can be regarded as the transient portion and the steady state portion of the system performance respectively. Since there is no knowledge of the exact value of the uncertainty, it is only realistic to refer to $η (\bar{δ}, γ, τ, t_{s})$ and $η_{\infty} (\bar{δ}, γ)$ , while analyzing the system performance. We also notice that both $η (\bar{δ}, γ, τ, t_{s})$ and $η_{\infty} (\bar{δ}, γ)$ are dependent on $\bar{δ}$ . The value of $\bar{δ}$ is not known except that it is characterized by a membership function.

Definition 2

Consider a fuzzy set

N = {(ν, μ_{N} (ν)) | ν \in N}

(32)

For any function

f : N \to R

, the D-operation

D [f (ν)]

is given by

D [f (ν)] = \frac{\int_{N}^{x} f (ν) μ_{N} (ν) d ν}{\int_{N}^{x} μ_{N} (ν) d ν}

(33)

Remark

In a sense, the D-operation $D [f (ν)]$ takes an average value of $f (ν)$ over $μ_{N} (ν)$ . In the special case that $f (ν) = ν$ , this is reduced to the well-known center-of-gravity defuzzification method (Klir and Yuan, 1995). If N is crisp (i.e., $μ_{N} (ν) = 1$ ) for all $ν \in N$ , then $D [f (ν)] = f (ν)$ .

Lemma 1

For any crisp constant $a \in R$

\begin{matrix} D [af (ν)] = \frac{\int_{N}^{x} af (ν) μ_{N} (ν) d ν}{\int_{N}^{x} μ_{N} (ν) d ν} \\ = a \frac{\int_{N}^{x} f (ν) μ_{N} (ν) d ν}{\int_{N}^{x} μ_{N} (ν) d ν} \\ = aD [f (ν)] \end{matrix}

(34)

For any t_s, let

J (γ, t_{s}) : = \int {no}_{t_{s}}^{\infty} η^{2} (\bar{δ}, γ, τ, t_{s}) d τ + α η_{\infty}^{2} (\bar{δ}, γ) + β γ^{2}

(35)

where α and

β > 0

are scalars.

Let $κ : = λ_{max} (P) / λ_{min} (Q),$ $ɛ : = ρ_{E 1} + λ_{min} (R^{- 1})$ . Regarding the first term on the RHS of equation (35), by using equations (21) and (22), we have

\begin{matrix} \int {no}_{t_{s}}^{\infty} η^{2} (\bar{δ}, γ, τ, t_{s}) d τ = (V_{s} - \frac{λ_{max} (P)}{λ_{min} (Q)} \frac{\bar{δ}}{γ})^{2} \\ \times \int {no}_{t_{s}}^{\infty} exp [- 2 \frac{λ_{min} (Q)}{λ_{max} (P)} (τ - t_{s})] d τ \\ = (V_{s} - \frac{λ_{max} (P)}{λ_{min} (Q)} \frac{\bar{δ}}{γ})^{2} (\frac{1}{- 2 \frac{λ_{min} (Q)}{λ_{max} (P)}}) \\ \times exp [- 2 \frac{λ_{min} (Q)}{λ_{max} (P)} (τ - t_{s})] | \begin{matrix} \infty \\ t_{s} \end{matrix} = (V_{s} - κ \frac{\bar{δ}}{γ})^{2} \frac{κ}{2} \\ = \frac{κ}{2} V_{s}^{2} - \frac{κ^{2}}{γ} V_{s} \bar{δ} + \frac{κ^{3}}{2 γ^{2}} {\bar{δ}}^{2} \\ = \frac{κ}{2} V_{s}^{2} - \frac{κ^{2}}{γ} V_{s} \frac{(\frac{\bar{a}}{δ_{1}} + \frac{\bar{b}}{δ_{2}})^{2}}{2 (ρ_{E 1} + λ_{min} (R^{- 1}))} \\ + \frac{κ^{3}}{2 γ^{2}} \frac{(\frac{\bar{a}}{δ_{1}} + \frac{\bar{b}}{δ_{2}})^{4}}{4 (ρ_{E 1} + λ_{min} (R^{- 1}))^{2}} \\ = \frac{κ}{2} V_{s}^{2} - \frac{κ^{2}}{γ} (\frac{V_{s} {\bar{a}}^{2}}{2 δ_{1}^{2} ɛ} + \frac{2 V_{s} \bar{a} \bar{b}}{2 δ_{1} δ_{2} ɛ} + \frac{V_{s} {\bar{b}}^{2}}{2 δ_{2}^{2} ɛ}) \\ + \frac{κ^{3}}{2 γ^{2}} (\frac{{\bar{a}}^{4}}{4 δ_{1}^{4} ɛ^{2}} + \frac{4 {\bar{a}}^{3} \bar{b}}{4 δ_{1}^{3} δ_{2} ɛ^{2}} + \frac{6 {\bar{a}}^{2} {\bar{b}}^{2}}{4 δ_{1}^{2} δ_{2}^{2} ɛ^{2}} \\ + \frac{4 \bar{a} {\bar{b}}^{3}}{4 δ_{1} δ_{2}^{3} ɛ^{2}} + \frac{{\bar{b}}^{4}}{4 δ_{2}^{4} ɛ^{2}}) \end{matrix}

(36)

Similarly, regarding the second term on the RHS of equation (35), we have

\begin{matrix} η_{\infty}^{2} (\bar{δ}, γ) = (\frac{λ_{max} (P)}{λ_{min} (Q)})^{2} (\frac{\bar{δ}}{γ})^{2} \\ = \frac{κ^{2}}{γ^{2}} \frac{(\frac{\bar{a}}{δ_{1}} + \frac{\bar{b}}{δ_{2}})^{4}}{4 (ρ_{E 1} + λ_{min} (R^{- 1}))^{2}} \\ = \frac{κ^{2}}{γ^{2}} (\frac{{\bar{a}}^{4}}{4 δ_{1}^{4} ɛ^{2}} + \frac{4 {\bar{a}}^{3} \bar{b}}{4 δ_{1}^{3} δ_{2} ɛ^{2}} \\ + \frac{6 {\bar{a}}^{2} {\bar{b}}^{2}}{4 δ_{1}^{2} δ_{2}^{2} ɛ^{2}} + \frac{4 \bar{a} {\bar{b}}^{3}}{4 δ_{1} δ_{2}^{3} ɛ^{2}} + \frac{{\bar{b}}^{4}}{4 δ_{2}^{4} ɛ^{2}}) \end{matrix}

(37)

Based on equations (35)–(37), we now propose the following performance index $J_{D} (γ, t_{s})$ : For any t_s, let

\begin{matrix} J_{D} (γ, t_{s}) = \frac{κ}{2} D [V_{s}^{2}] - \frac{κ^{2}}{γ} \\ \times (\frac{D [V_{s}] D [{\bar{a}}^{2}]}{2 δ_{1}^{2} ɛ} + \frac{2 D [V_{s}] D [\bar{a}] D [\bar{b}]}{2 δ_{1} δ_{2} ɛ} + \frac{D [V_{s}] D [{\bar{b}}^{2}]}{2 δ_{2}^{2} ɛ}) \\ + \frac{κ^{3}}{2 γ^{2}} (\frac{D [{\bar{a}}^{4}]}{4 δ_{1}^{4} ɛ^{2}} + \frac{4 D [{\bar{a}}^{3}] D [\bar{b}]}{4 δ_{1}^{3} δ_{2} ɛ^{2}} + \frac{6 D [{\bar{a}}^{2}] D [{\bar{b}}^{2}]}{4 δ_{1}^{2} δ_{2}^{2} ɛ^{2}} \\ + \frac{4 D [\bar{a}] D [{\bar{b}}^{3}]}{4 δ_{1} δ_{2}^{3} ɛ^{2}} + \frac{D [{\bar{b}}^{4}]}{4 δ_{2}^{4} ɛ^{2}}) \\ + α \frac{κ^{2}}{γ^{2}} (\frac{D [{\bar{a}}^{4}]}{4 δ_{1}^{4} ɛ^{2}} + \frac{4 D [{\bar{a}}^{3}] D [\bar{b}]}{4 δ_{1}^{3} δ_{2} ɛ^{2}} \\ + \frac{6 D [{\bar{a}}^{2}] D [{\bar{b}}^{2}]}{4 δ_{1}^{2} δ_{2}^{2} ɛ^{2}} + \frac{4 D [\bar{a}] D [{\bar{b}}^{3}]}{4 δ_{1} δ_{2}^{3} ɛ^{2}} + \frac{D [{\bar{b}}^{4}]}{4 δ_{2}^{4} ɛ^{2}}) + β γ^{2} \\ = : κ_{1} - \frac{κ_{2}}{γ} + \frac{κ_{3}}{γ^{2}} + α \frac{κ_{4}}{γ^{2}} + β γ^{2} \\ = : J_{1 D} (γ, t_{s}) + α J_{2 D} (γ) + β J_{3 D} (γ) \end{matrix}

(38)

where

κ_{1} = \frac{κ}{2} D [V_{s}^{2}]

\begin{matrix} κ_{2} = κ^{2} (\begin{matrix} \frac{D [V_{s}] D [{\bar{a}}^{2}]}{2 δ_{1}^{2} ɛ} + \frac{2 D [V_{s}] D [\bar{a}] D [\bar{b}]}{2 δ_{1} δ_{2} ɛ} \\ + \frac{D [V_{s}] D [{\bar{b}}^{2}]}{2 δ_{2}^{2} ɛ} \end{matrix}), \\ κ_{3} = \frac{κ^{3}}{2} (\begin{matrix} \frac{D [{\bar{a}}^{4}]}{4 δ_{1}^{4} ɛ^{2}} + \frac{4 D [{\bar{a}}^{3}] D [\bar{b}]}{4 δ_{1}^{3} δ_{2} ɛ^{2}} + \frac{6 D [{\bar{a}}^{2}] D [{\bar{b}}^{2}]}{4 δ_{1}^{2} δ_{2}^{2} ɛ^{2}} \\ + \frac{4 D [\bar{a}] D [{\bar{b}}^{3}]}{4 δ_{1} δ_{2}^{3} ɛ^{2}} + \frac{D [{\bar{b}}^{4}]}{4 δ_{2}^{4} ɛ^{2}} \end{matrix}) \\ κ_{4} = κ^{2} (\begin{matrix} \frac{D [{\bar{a}}^{4}]}{4 δ_{1}^{4} ɛ^{2}} + \frac{4 D [{\bar{a}}^{3}] D [\bar{b}]}{4 δ_{1}^{3} δ_{2} ɛ^{2}} + \frac{6 D [{\bar{a}}^{2}] D [{\bar{b}}^{2}]}{4 δ_{1}^{2} δ_{2}^{2} ɛ^{2}} \\ + \frac{4 D [\bar{a}] D [{\bar{b}}^{3}]}{4 δ_{1} δ_{2}^{3} ɛ^{2}} + \frac{D [{\bar{b}}^{4}]}{4 δ_{2}^{4} ɛ^{2}} \end{matrix}) \\ J_{1 D} (γ, t_{s}) = κ_{1} - \frac{κ_{2}}{γ} + \frac{κ_{3}}{γ^{2}} and J_{2 D} (γ) = \frac{κ_{4}}{γ^{2}}, \\ J_{3 D} (γ) = γ^{2} \end{matrix}

Remark

The performance index $J_{D} (γ, t_{s})$ consists of three parts. The first parts, $J_{1 D} (γ, t_{s})$ may be interpreted as the average (via the D-operation) of the overall transient performance (via the integration) from time t_s. The second part, $J_{2 D} (γ)$ may be interpreted as the average (via the D-operation) of the steady state performance. The third part, $J_{3 D} (γ)$ is due to the control cost. Both α and β are weighting factors. The weighting of $J_{1 D} (γ, t_{s})$ is normalized to be unity.

Remark

The steady state performance reflects the size of the uniform ultimate boundedness region which decreases as γ increases. As γ approaches to infinity, the size approaches to zero. This is rather strong performance. However, with γ increasing, the control cost $J_{3 D}$ increases. In practice, we want to obtain a small size of the uniform ultimate boundedness region without too much cost. Thus, there is a trade-off between the performance and the control cost.

The optimal design problem is then equivalent to the following constrained optimization problem: For any t_s

{min}_{γ} J_{D} (γ, t_{s}) subjectto γ > 0

(39)

For any $t_{s}$ , taking the first order derivative of J_D with respect to γ

\begin{matrix} \frac{\partial J_{D}}{\partial γ} = \frac{κ_{2}}{γ^{2}} - 2 \frac{κ_{3}}{γ^{3}} - 2 α \frac{κ_{4}}{γ^{3}} + 2 β γ \\ = \frac{1}{γ^{3}} (κ_{2} γ - 2 κ_{3} - 2 α κ_{4} + 2 β γ^{4}) \end{matrix}

(40)

That

\frac{\partial J_{D}}{\partial γ} = 0

(41)

leads to

κ_{2} γ - 2 κ_{3} - 2 α κ_{4} + 2 β γ^{4} = 0

(42)

κ_{2} γ + 2 β γ^{4} = 2 (κ_{3} + α κ_{4})

(43)

Equation (43) is a quartic equation.

Theorem 2

Suppose $D [\bar{a}], D [\bar{b}] \neq 0$ . For given $κ_{1}, κ_{2}, κ_{3}, κ_{4}$ , the solution $γ > 0$ to equation (43) always exists and is unique, which globally minimizes the performance index of equation (38).

Proof

Let $θ (γ) : = κ_{2} γ + 2 β γ^{4}$ . Then $θ (0) = 0$ and $θ (\cdot)$ is continuous in γ. In addition since $κ_{2} \geq 0$ and $β > 0$ , $θ (\cdot)$ is strictly increasing in γ. Since $D [\bar{a}], D [\bar{b}] \neq 0$ , we have $D [\bar{a}] > 0, D [\bar{b}] > 0, κ_{3}, κ_{4} > 0$ , and, therefore, $2 (κ_{3} + α κ_{4}) > 0$ (notice that $α, κ > 0$ ). As a result, the solution $γ > 0$ to equation (43) always exists and is unique. For the unique solution $γ > 0$ that solves equation (43)

\begin{matrix} \frac{\partial^{2} J_{D}}{\partial γ^{2}} = - \frac{3}{γ^{4}} (κ_{2} γ - 2 κ_{3} - 2 α κ_{4} + 2 β γ^{4}) + \frac{1}{γ^{3}} (κ_{2} + 8 β γ^{3}) \\ = \frac{1}{γ^{3}} (κ_{2} + 8 β γ^{3}) > 0 \end{matrix}

(44)

Therefore, the positive solution

γ > 0

of the quartic equation (43) solves the constrained minimization problem of equation (39). Q.E.D.

Remark

Combining the results of Sections 3 and 4, the robust control scheme of equation (10) using the optimal design of $γ > 0$ renders the solution of the closed-loop system uniformly ultimately bounded (with the initial state $x (t_{s})$ ). In addition, the performance index J_D in equation (38) is globally minimized.

5. Dynamical model of the ARCS

Figure 1 shows the behavior of the vehicle suspension and ARCS. When the vehicle enters a corner, a rolling force is generated by the centrifugal force. Consequently, the lower arm on one side of the suspension is pulled up, and the other is pulled down. The ARCS aims to control the torsion angle of the stabilizer bar by utilizing the roll reaction force generated by the actuators and the roll reaction force obtained from the roll stiffness of the suspension to offset the roll moment caused by lateral acceleration. Figure 2 illustrates the dynamic model of the ARCS. Key variables and parameters of the ARCS are listed in Table 1.

Figure 1.

Behavior of the active roll control system.

Figure 2.

Dynamic model of the active roll control system.

Table 1.

Nomenclature of the active roll control system.

m_s	sprung mass	∅	Roll angle
I_s	moment of inertia of sprung mass	$\overset{\cdot}{Ø}$	roll rate
$C_{Ø}$	total roll damping	$Ø^{\overset{··}{}}$	roll acceleration
$C_{Ø f}$	front roll damping	h_s	distance from roll axis to CG of sprung mass
$C_{Ø r}$	rear roll damping	g	gravity constant
$C_{c Ø f}$	front coil spring damping	a_y	lateral acceleration
$C_{c Ø r}$	rear coil spring damping	$ξ$	front roll stiffness ratio
$K_{Ø}$	total roll stiffness	$M_{af}$	front active roll moment
$K_{Ø f}$	front roll stiffness	$M_{ar}$	rear active roll moment
$K_{Ø r}$	rear roll stiffness	$γ_{f}$	front transmission ratio
$K_{c Ø f}$	front coil spring rate	$γ_{r}$	rear transmission ratio
$K_{c Ø r}$	rear coil spring rate	$m_{af}$	front actuator active torque
$K_{t Ø f}$	front torsion spring rate	$m_{ar}$	rear actuator active torque
$K_{t Ø r}$	rear torsion spring rate

The following equation expresses the moment equilibrium around the sprung roll center:

I_{s} \overset{··}{Ø} + C_{Ø} \overset{\cdot}{Ø} + K_{Ø} Ø + M_{af} + M_{ar} = m_{s} h_{s} a_{y} + m_{s} {gh}_{s} Ø

(45)

where

I_{s} = m_{s} h_{s}^{2}

(46)

C_{Ø} = C_{Ø f} + C_{Ø r} = 2 C_{c Ø f} + 2 C_{c Ø r}

(47)

K_{Ø} = K_{Ø f} + K_{Ø r} = 2 K_{c Ø f} + 2 K_{t Ø f} + 2 K_{c Ø r} + 2 K_{t Ø r}

(48)

We wish the roll angle ∅ of the vehicle to follow a target roll angle $Ø_{d}$ which is dependent on the lateral acceleration a_y. Let

e (t) : = Ø (t) - Ø_{d} (t)

(49)

and hence

\overset{\cdot}{e} (t) : = \overset{\cdot}{Ø} (t) - {\overset{\cdot}{Ø}}_{d} (t)

\overset{··}{e} (t) : = \overset{··}{Ø} (t) - {\overset{··}{Ø}}_{d} (t)

. Therefore, the system of equation (45) can be written as

\begin{matrix} I_{s} (\overset{··}{e} + {\overset{··}{Ø}}_{d}) + C_{Ø} (\overset{\cdot}{e} + {\overset{\cdot}{Ø}}_{d}) + K_{Ø} (e + Ø_{d}) + M_{af} + M_{ar} \\ = m_{s} h_{s} a_{y} + m_{s} {gh}_{s} (e + Ø_{d}) \end{matrix}

(50)

Remark

In this paper, we consider the sprung mass of the vehicle is uncertain parameter: $m_{s} = {\bar{m}}_{s} + Δ m_{s} (t)$ , due to the variation in the vehicle load. Here ${\bar{m}}_{s} > 0$ is the constant nominal value and $Δ m_{s} (t)$ is the time-varying uncertain portion.

The system of equation (50) can be put into the state space form (with $x_{1} = e, x_{2} = \overset{\cdot}{e}$ )

\begin{matrix} [\begin{matrix} {\overset{\cdot}{x}}_{1} \\ {\overset{\cdot}{x}}_{2} \end{matrix}] = [\begin{matrix} 0 & 1 \\ - \frac{K_{Ø} - m_{s} {gh}_{s}}{I_{s}} & - \frac{C_{Ø}}{I_{s}} \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] + [\begin{matrix} 0 \\ \frac{1}{I_{s}} \end{matrix}] (u + \tilde{ϑ}) \\ = : \tilde{A} x + \tilde{B} (u + \tilde{ϑ}) \end{matrix}

(51)

where

u = - (M_{af} + M_{ar})

\tilde{ϑ} = m_{s} h_{s} a_{y} - I_{s} {\overset{··}{Ø}}_{d} - C_{Ø} {\overset{\cdot}{Ø}}_{d} - (K_{Ø} - m_{s} {gh}_{s}) Ø_{d}

Decompose $m_{s}^{- 1} = ({\bar{m}}_{s} + Δ m_{s})^{- 1}$ . Let $\bar{z} = {\bar{m}}_{s}^{- 1}$ , $Δ z = ({\bar{m}}_{s} + Δ m_{s})^{- 1} - {\bar{m}}_{s}^{- 1}$ . It can be checked that $m_{s}^{- 1} = \bar{z} + Δ z$ . Hence, we can obtain

\begin{matrix} \tilde{A} = [\begin{matrix} 0 & 1 \\ - \frac{K_{Ø} - m_{s} {gh}_{s}}{I_{s}} & - \frac{C_{Ø}}{I_{s}} \end{matrix}] \end{matrix}

(52)

\begin{matrix} = [\begin{matrix} 0 & 1 \\ - m_{s}^{- 1} \frac{K_{Ø}}{h_{s}^{2}} + \frac{g}{h_{s}} & - m_{s}^{- 1} \frac{C_{Ø}}{h_{s}^{2}} \end{matrix}] \\ = [\begin{matrix} 0 & 1 \\ - \bar{z} \frac{K_{Ø}}{h_{s}^{2}} + \frac{g}{h_{s}} & - \bar{z} \frac{C_{Ø}}{h_{s}^{2}} \end{matrix}] + [\begin{matrix} 0 & 1 \\ - Δ z \frac{K_{Ø}}{h_{s}^{2}} & - Δ z \frac{C_{Ø}}{h_{s}^{2}} \end{matrix}] \end{matrix}

\tilde{B} = [\begin{matrix} 0 \\ \frac{1}{I_{s}} \end{matrix}] = [\begin{matrix} 0 \\ m_{s}^{- 1} \frac{1}{h_{s}^{2}} \end{matrix}] = [\begin{matrix} 0 \\ \bar{z} \frac{1}{h_{s}^{2}} \end{matrix}] + [\begin{matrix} 0 \\ Δ z \frac{1}{h_{s}^{2}} \end{matrix}]

(53)

Let $\tilde{A} = A + Δ A$ , $\tilde{B} = B + Δ B$ , where

\begin{matrix} A = [\begin{matrix} 0 & 1 \\ - \bar{z} \frac{K_{Ø}}{h_{s}^{2}} + \frac{g}{h_{s}} & - \bar{z} \frac{C_{Ø}}{h_{s}^{2}} \end{matrix}], Δ A = [\begin{matrix} 0 & 1 \\ - Δ z \frac{K_{Ø}}{h_{s}^{2}} & - Δ z \frac{C_{Ø}}{h_{s}^{2}} \end{matrix}] \end{matrix}

(54)

\begin{matrix} B = [\begin{matrix} 0 \\ \bar{z} \frac{1}{h_{s}^{2}} \end{matrix}], Δ B = [\begin{matrix} 0 \\ Δ z \frac{1}{h_{s}^{2}} \end{matrix}] \end{matrix}

(55)

The following matching condition is met with $Δ A$ and $Δ B$

Δ A = [\begin{matrix} 0 \\ \bar{z} \frac{1}{h_{s}^{2}} \end{matrix}] [\begin{matrix} - {\bar{z}}^{- 1} Δ {zK}_{Ø} & - {\bar{z}}^{- 1} Δ {zC}_{Ø} \end{matrix}] = : B \tilde{D}

(56)

Δ B = [\begin{matrix} 0 \\ \bar{z} \frac{1}{h_{s}^{2}} \end{matrix}] [{\bar{z}}^{- 1} Δ z] = : BE

(57)

Similarly, $\tilde{ϑ}$ can be decomposed as

\begin{matrix} \tilde{ϑ} = m_{s} h_{s} a_{y} - I_{s} {\overset{··}{Ø}}_{d} - C_{Ø} {\overset{\cdot}{Ø}}_{d} - (K_{Ø} - m_{s} {gh}_{s}) Ø_{d} \\ = ({\bar{m}}_{s} + Δ m_{s}) h_{s} a_{y} - ({\bar{m}}_{s} + Δ m_{s}) h_{s}^{2} {\overset{··}{Ø}}_{d} \\ - C_{Ø} {\overset{\cdot}{Ø}}_{d} - (K_{Ø} - ({\bar{m}}_{s} + Δ m_{s}) {gh}_{s}) Ø_{d} \\ = {\bar{m}}_{s} h_{s} a_{y} - {\bar{m}}_{s} h_{s}^{2} {\overset{··}{Ø}}_{d} - C_{Ø} {\overset{\cdot}{Ø}}_{d} - (K_{Ø} - {\bar{m}}_{s} {gh}_{s}) Ø_{d} \\ + Δ m_{s} h_{s} a_{y} - Δ m_{s} h_{s}^{2} {\overset{··}{Ø}}_{d} + Δ m_{s} {gh}_{s} Ø_{d} \\ = : h + ϑ \end{matrix}

(58)

where

h = {\bar{m}}_{s} h_{s} a_{y} - {\bar{m}}_{s} h_{s}^{2} {\overset{··}{Ø}}_{d} - C_{Ø} {\overset{\cdot}{Ø}}_{d} - (K_{Ø} - {\bar{m}}_{s} {gh}_{s}) Ø_{d}

and

ϑ = Δ m_{s} h_{s} a_{y} - Δ m_{s} h_{s}^{2} {\overset{··}{Ø}}_{d} + Δ m_{s} {gh}_{s} Ø_{d}

Thus, the system of equation (51) can be expressed in the form of equation (1) with $A, B, Δ B, h, \tilde{D}, E$ given above:

\begin{matrix} \overset{\cdot}{x} = \tilde{A} x + \tilde{B} (u + \tilde{ϑ}) \\ = (A + Δ A) x + (B + Δ B) (u + h + ϑ) \\ = Ax + B \tilde{D} x + (B + Δ B) u + B (I + E) h + B (I + E) ϑ \\ = Ax + (B + Δ B) u + B [h + (\tilde{D} x + Eh + (I + E) ϑ)] \end{matrix}

(59)

where

v = \tilde{D} x + Eh + (I + E) ϑ

The required compensating roll moment can be distributed to the front and rear components by using the roll stiffness distribution to the front axle ration:

ξ = \frac{M_{af}}{M_{af} + M_{ar}}

(60)

The active torque $m_{af}$ and $m_{ar}$ for the actuator positions in the active suspension system can be obtained by

m_{af} = γ_{f} M_{af}

(61)

m_{ar} = γ_{r} M_{ar}

(62)

Remark

$γ_{f}$ and $γ_{r}$ are constants related to the geometry of the front and rear stabilizer bars respectively. Due to the limit of the space layout of the vehicle, the structure of the front and rear stabilizer bars are usually different. Hence, $γ_{f}$ and $γ_{r}$ are correspondingly different.

Following the dynamics of the ARCS with the parameter uncertainty, the controller design will be presented in the next section to render the vehicle equipped with the ARCS to achieve the target roll angle in a given driving condition.

6. Controller design

6.1. Target roll angle of the ARCS

The roll angle is used as an evaluation index in the development of a vehicle equipped with the ARCS. Based on the results of roll attitude angle and a subjective evaluation, a target roll angle of 1° for a lateral acceleration of $5 m / s^{2}$ is set, which is half that of a vehicle equipped with conventional passive stabilizer bar system for the same lateral acceleration. Figure 3 shows the relationship between the target roll angle according to the lateral acceleration, i.e.,

Ø_{d} = 0.2 a_{y} (deg) = \frac{0.2 π}{180} a_{y} (rad)

(63)

Figure 3.

Target roll angle of the ARCS.

It should be noted that taking the vehicle’s driving safety into account, the lateral acceleration is usually bounded. Here we assume that the upper bound of the lateral acceleration is ${\bar{a}}_{y}$ . Hence, it is reasonable to assume that the upper bounds of $Ø_{d}, {\overset{\cdot}{Ø}}_{d}, {\overset{··}{Ø}}_{d}$ are ${\bar{Ø}}_{d}, {\bar{\overset{\cdot}{Ø}}}_{d}, {\bar{\overset{··}{Ø}}}_{d}$ .

6.2. Controller design

From the system of equation (59), we have

\begin{matrix} | | v | | \leq | | \tilde{D} | | | | x | | + | | E | | | | h | | + | | (I + E) \\ | | | | ϑ | | \leq | | \tilde{D} | | | | x | | + ρ_{E 2} \bar{h} + (1 + ρ_{E 2}) \bar{ϑ} \end{matrix}

(64)

where

\bar{h} = {\bar{m}}_{s} h_{s} {\bar{a}}_{y} + {\bar{m}}_{s} h_{s}^{2} {\bar{\overset{··}{Ø}}}_{d} + C_{Ø} {\bar{\overset{\cdot}{Ø}}}_{d} + (K_{Ø} + {\bar{m}}_{s} {gh}_{s}) {\bar{Ø}}_{d}

\bar{ϑ} = \bar{Δ m_{s}} h_{s} {\bar{a}}_{y} + \bar{Δ m_{s}} h_{s}^{2} {\bar{\overset{··}{Ø}}}_{d} + \bar{Δ m_{s}} {gh}_{s} {\bar{Ø}}_{d}

\bar{Δ m_{s}}

is the upper bound of the uncertain parameter

Δ m_{s}

To evaluate $| | \tilde{D} | |$ , we invoke the norm inequality that for a matrix $Ψ \in R^{p \times q}$ , $\frac{1}{\sqrt{p}} | | Ψ | |_{1} \leq | | Ψ | |_{2} \leq \sqrt{q} | | Ψ | |_{1}$ . This means $| | \tilde{D} | |_{2} \leq \sqrt{2} | | \tilde{D} | |_{1}$ . Here, to avoid confusions we use subscript 1 and 2 to denote two classes of induced matrix norms. Since

\begin{matrix} | | \tilde{D} | |_{1} = {max}_{j} \sum_{i} | d_{ij} | \\ = max {{\bar{z}}^{- 1} | Δ z | K_{Ø}, {\bar{z}}^{- 1} | Δ z | C_{Ø}} \end{matrix}

(65)

where

d_{ij}

denotes the

ij

entry of

\tilde{D}

, we choose the sum of the two entries to the upper bound

| | \tilde{D} | |_{1}

, and therefore

| | \tilde{D} | |_{2} \leq \sqrt{2} ({\bar{z}}^{- 1} | Δ z | K_{Ø} + {\bar{z}}^{- 1} | Δ z | C_{Ø})

(66)

Thus, fuzzy numbers a and b in equation (6) can be chosen as $a = \sqrt{2} ({\bar{z}}^{- 1} | Δ z | K_{Ø} + {\bar{z}}^{- 1} | Δ z | C_{Ø})$ , $b = ρ_{E 2} \bar{h} + (1 + ρ_{E 2}) \bar{ϑ}$ .

In the Riccati equation (5), we choose $Q = I$ and $R = I$ . To verify Assumption 3(ii), we note that

\begin{matrix} \frac{1}{2} λ_{min} (E (x, σ, t) R^{- 1} + R^{- 1} E^{T} (x, σ, t)) \\ = \frac{1}{2} λ_{min} (E + E^{T}) = {\bar{z}}^{- 1} Δ z \end{matrix}

(67)

Since $m_{s} > 0, {\bar{m}}_{s} > 0$ , we have $m_{s} / {\bar{m}}_{s} > 0$ and $(m_{s} / {\bar{m}}_{s})^{- 1} > 0$ . As a result

\begin{matrix} {\bar{z}}^{- 1} Δ z = {\bar{m}}_{s} (m_{s}^{- 1} - {\bar{m}}_{s}^{- 1}) = (\frac{m_{s}}{{\bar{m}}_{s}})^{- 1} - 1 \\ > - 1 = - λ_{min} (R^{- 1}) \end{matrix}

(68)

Thus, equation (7) is met. In addition, we have

| | E | | = {\bar{z}}^{- 1} | Δ z |

(69)

Hence, we choose $ρ_{E 2} = {\bar{z}}^{- 1} | Δ z |$ to guarantee equation (8).

The nominal system parameters of the vehicle are chosen as follows:

\begin{matrix} {\bar{m}}_{s} = 1762 kg, h_{s} = 0.452 m, \\ C_{c Ø f} = C_{c Ø r} = 3000 Nm / (rad / s), \\ K_{c Ø f} = K_{c Ø r} = 10, 440 Nm / rad, \\ K_{t Ø f} = K_{t Ø r} = 14, 760 Nm / rad, \\ g = 9.8 m / s^{2}, ξ = 0.52, γ_{f} = 0.301, γ_{r} = 0.195 \end{matrix}

(70)

By solving the Riccati equation (5), we can obtain $λ_{max} (P) = 3.9543$ , $κ = λ_{max} (P) / λ_{min} (Q) = 3.9543$ , and $| | R^{- 1} B^{T} P | | = 4.2172 \times 10^{- 5}$ . For the uncertainty, there is a direct correspondence between $Δ z$ and $Δ m_{s}$ . Thus, the following choices are directly made for simulations: $Δ m_{s}$ is varying from $0$ to $100 kg$ , and $| Δ z |$ is “close to $1.524 \times 10^{- 5}$ ”. The corresponding membership function is (triangular)

\begin{matrix} μ_{| Δ z |} = {\begin{matrix} \frac{1}{1.524 \times 10^{- 5}} ν & 0 \leq ν \leq 1.524 \times 10^{- 5} \\ - \frac{1}{1.524 \times 10^{- 5}} ν + 2 & {\begin{matrix} 1.524 \times 10^{- 5} \leq ν \\ \leq 3.048 \times 10^{- 5} \end{matrix}} \end{matrix} \end{matrix}

(71)

For the uncertain initial conditions, we assume that $x_{1} (0)$ and $x_{2} (0)$ are “close to 0”. They are governed by the membership functions (all triangular)

\begin{matrix} μ_{x_{1} (0)} = {\begin{matrix} \frac{1}{0.00872} ν + 1 & - 0.00872 \leq ν \leq 0 \\ - \frac{1}{0.00872} ν + 1 & 0 \leq ν \leq 0.00872 \end{matrix} \end{matrix}

(72)

\begin{matrix} μ_{x_{2} (0)} = {\begin{matrix} \frac{1}{0.01047} ν + 1 & - 0.01047 \leq ν \leq 0 \\ - \frac{1}{0.01047} ν + 1 & 0 \leq ν \leq 0.01047 \end{matrix} \end{matrix}

(73)

In this paper, we choose $ρ_{E 1} = - \frac{100}{1862}$ , ${\bar{a}}_{y} = 1.5 g m / s^{2}$ , ${\bar{Ø}}_{d} = 0.2 * \frac{π}{180} * 1.5 g rad$ , ${\bar{\overset{\cdot}{Ø}}}_{d} = 0.2 * \frac{π}{180} * 1.5 g * 0.5 rad / s$ , ${\bar{\overset{··}{Ø}}}_{d} = 0.2 * \frac{π}{180} * 1.5 g * 0.25 rad / s^{2}$ .

Based on the parameters design above, we can obtain

\bar{a} = {\bar{z}}^{- 1} | Δ z | | | R^{- 1} B^{T} P | | + \sqrt{2} ({\bar{z}}^{- 1} | Δ z | K_{Ø} + {\bar{z}}^{- 1} | Δ z | C_{Ø})

(74)

\bar{b} = 2 {\bar{z}}^{- 1} | Δ z | \bar{h} + (1 + {\bar{z}}^{- 1} | Δ z |) \bar{ϑ}

(75)

For the control design, we choose $δ_{1} = 1762,$ $δ_{2} = 18277$ . The controller is designed as

\begin{matrix} u (t) = - R^{- 1} B^{T} Px (t) - h (t) \\ - γ R^{- 1} B^{T} Px (t) (δ_{1} | | x (t) | | + δ_{2})^{2} \end{matrix}

(76)

The active torques $m_{af}$ and $m_{ar}$ for the front and rear actuators are given as

m_{af} = - γ_{f} ξ u

(77)

m_{ar} = - γ_{r} (1 - ξ) u

(78)

By using the fuzzy arithmetic, decomposition theorem, and the D-operation, we can obtain $κ_{1} = 1.5562 \times 10^{- 8}$ , $κ_{2} = 3 \times 10^{- 3}$ , $κ_{3} = 672.6396$ , and $κ_{4} = 340.1988$ .

To solve the quartic equation (43), we choose seven sets of weighting parameters α and β. Their values and the corresponding

γ_{opt}

and

J_{Dmin}

are summarized in Table 2.

Table 2.

Weighting, optimal gain, and minimum cost.

$(α, β)$	$γ_{opt}$	$J_{Dmin}$
(1,0.1)	10.0319	20.1277
(1,1)	5.6414	63.6497
(1,10)	3.1724	201.2788
(1,100)	1.7840	636.5008
(0.1,1)	5.1559	53.1656
(10,1)	7.9895	127.6652
(100,1)	13.6477	372.5183

7. Numerical simulations

The vehicle equipped with the ARCS is assumed to run under the given case with $a_{y} = 6 sin 0.5 t m / s^{2} .$ Hence, the target roll angle according to this lateral acceleration is calculated as

Ø_{d} = \frac{π}{150} sin 0.5 t rad

(79)

Differentiating equation (79) with respect to time once and twice respectively yields

{\overset{\cdot}{Ø}}_{d} = \frac{0.5 π}{150} cos 0.5 t rad / s

(80)

and

{\overset{··}{Ø}}_{d} = - \frac{0.25 π}{150} sin 0.5 t rad / s^{2}

(81)

The initial conditions of the constraints are given as follows:

\begin{matrix} Ø (0) = 0.287 deg = 0.005 rad, \\ \overset{\cdot}{Ø} (0) = 0.573 \frac{deg}{s} = 0.01 rad / s \end{matrix}

(82)

For numerical simulation, we choose $t_{s} = 0$ , $Δ m_{s} = 40 | cos (0.1 t) |$ . For comparison, we select the standard LQG (linear-quadratic-Gaussian) control. Figure 4 shows the tracking error of the roll angle e (i.e., $φ - φ_{d}$ ) for five $γ_{opt}$ values (by using different ( $α, β$ ) combinations) and LQG control. The e-trajectory with the proposed control enters a small region around 0 after some time (hence uniformly bounded). While, with the LQG control, the final error fluctuation is much heavier.

Figure 4.

Comparison of system performances e under different γ_opt and LQG control.

Figure 5 shows the tracking error norm $| | x | |$ (i.e., $x = [e \overset{\cdot}{e}]^{T}$ ) for five $γ_{opt}$ values and LQG control. The $| | x | |$ -trajectory with the proposed control also enters a small region around 0 after some time (hence uniformly bounded). While, with the LQG control, the final $| | x | |$ fluctuation is heavier.

Figure 5.

Comparison of system performances ||x|| under different γ_opt and LQG control.

Figures 6 and 7 depict the accumulative tracking error of e-trajectory and $| | x | |$ -trajectory in Figures 4 and 5 for five $γ_{opt}$ values and LQG control. As $γ_{opt}$ increases, the accumulative tracking error $Area (e)$ and $Area (| | x | |)$ both decrease. While, with the LQG control, the accumulative tracking error $Area (e)$ and $Area (| | x | |)$ are much more than those of the proposed robust control.

Figure 6.

Comparison of the accumulative tracking error Area(e) under different γ_opt and LQG control.

Figure 7.

Comparison of the accumulative tracking error Area(||x||) under different γ_opt and LQG control.

Figure 8 presents the control effort u for five $γ_{opt}$ values. Under these five different $γ_{opt}$ values, the control efforts u are almost the same.

Figure 8.

Comparison of the control efforts u under different γ_opt.

Figures 9 and 10 display the control torques of the front and rear actuators (i.e., $m_{af}$ and $m_{ar}$ ) under $γ_{opt} = 13.6477$ .

Figure 9.

The control torque of the front actuator m_af under γ_opt = 13.6477.

Figure 10.

The control torque of the rear actuator m_ar under γ_opt = 13.6477.

Figure 11 shows the influence of the weighting parameters α and β on $γ_{opt}$ . $γ_{opt}$ gradually increases, subject to the increasing of α and decreasing of β. Figure 12 shows the influence of the weighting parameters α and β on $J_{Dmin}$ . $J_{Dmin}$ gradually increases, subject to the increasing of α and increasing of β.

Figure 11.

γ_opt vs. log α and log β.

Figure 12.

J_Dmin vs. log α and log β.

In comparison with the results of Han et al. (2017), the main feature of this paper is that we take the lead in applying fuzzy set theory to the dynamic model of the electric ARCS and propose a robust control scheme with fuzzy optimal design for the ARCS.

8. Conclusion

In this paper, we successfully deal with the active roll control problem with system uncertainties which are (possibly) time-varying but bounded. Fuzzy set theory is employed to describe the system uncertainties in ARCS and we incorporate fuzzy uncertainty and fuzzy performance into the control design. A new robust control scheme which is deterministic and is not the usual if–then rules-based control is proposed to guarantee the deterministic performance (including uniform boundedness and uniform ultimately boundedness). The resulting controlled system is stable, proved via the Lyapunov minimax approach. A performance index including average fuzzy system performance and control effort is proposed based on the fuzzy information. The optimal design problem associated with the control can then be solved by minimizing the performance index. The solution of the optimal gain is unique. Hence, the resulting control is able to guarantee the deterministic performance as well as minimizing the cost.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant number 51505116), the Natural Science Foundation of Anhui Province (grant number 1508085SME221), the Fundamental Research Funds for the Central Universities (grant number JZ2016HGTB0716), and the China Postdoctoral Science Foundation (grant number 2016M590563).

Appendix: Fuzzy mathematics

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