Robust fixed-order dynamic output feedback controller design for nonlinear uncertain suspension system

Mechanical Systems and Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Robust fixed-order dynamic output feedback controller design for nonlinear uncertain suspension system Pouya Badri, Amir Amini, Mahdi Sojoodi n Advanced Control Laboratory, School of Electrical and Computer Engineering, Tarbiat Modares University, Tehran, Iran

a r t i c l e i n f o

abstract

Article history: Received 31 December 2015 Received in revised form 4 March 2016 Accepted 15 April 2016

This paper deals with designing a robust fixed-order non-fragile dynamic output feedback controller for active suspension system of a quarter-car, by means of convex optimization and linear matrix inequalities (LMIs). Our purpose is to design a low-order controller that keeps the desired design specifications besides the simplicity of the implementation. The proposed controller is capable of asymptotically stabilizing the closed-loop system and developing H∞ control, despite model uncertainties and nonlinear dynamics of the quarter-car as well as the norm bounded perturbations of controller parameters. Furthermore, controller parameters are prevented from taking very large and undesirable amounts through appropriate LMI constraints. Finally, a numerical example is presented to show the effectiveness of the proposed method by comparing it with similar works. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Convex optimization Robust stability Uncertainty Linear matrix inequality (LMI) Fixed order controller Non-fragile controller Active suspension system

1. Introduction A vehicle's suspension system guarantees a smooth ride over rugged roads damping disturbances caused by the road surface, while ensuring the tyres remain in contact with the road surface and that body roll is minimized. Also, a suspension system allows the car to turn without rolling by changing the car's center of gravity to maintain balance. Therefore a comfortable and safe ride could be achieved [1]. Passive suspension systems which include conventional springs and dampers with fixed stiffness and damping coefficients are proved incapable of damping the road disturbances in all frequency ranges. Recently, controlled suspension systems, comprise of semi-active and active suspensions have been emerged due to unavoidable limitations of conventional suspensions. Semi-active suspension uses a controlled dissipative component rather than the damper and no energy input into the system is needed. While an active suspension needs to use an active actuator, and an adequate energy input. The advantages of active suspension systems compared with passive ones are presented in [2]. Design of automobile suspension system from the ride comfort point of view is reviewed in [3]. Subsequently a large number of studies have been performed to use the most appropriate control strategies acquiring different performance objectives. In [4] two approaches are introduced in order to design centralized H∞ controllers for an active magnetic suspension system. Semi-active H∞ control of vehicle suspension is studied in [5]. In [6] the problem of robust sampled-data control for n

Corresponding author. E-mail address: [email protected] (M. Sojoodi). URL: http://www.modares.ac.ir/
sojoodi (M. Sojoodi).

http://dx.doi.org/10.1016/j.ymssp.2016.04.020 0888-3270/& 2016 Elsevier Ltd. All rights reserved.

Please cite this article as: P. Badri, et al., Robust fixed-order dynamic output feedback controller design for nonlinear uncertain suspension system, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.04.020i

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P. Badri et al. / Mechanical Systems and Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

active suspensions considering input delay was investigated. Models utilized in [5,6] are considered to be linear and because of the non-linear nature of the springs used in the suspension system, this approximation can lead to inaccuracy. In [7] robust controller design with actuator saturation for a comprehensive nonlinear system model with parameter uncertainty in the presence of external disturbance was investigated. The proposed method therein, uses a nonlinear sign function for limiting the nonlinear active force. In [8] the concentration is also on sliding mode algorithms within the Lyapunov stability theorem, in which different practical control objectives are also considered. A robust fault-tolerant H∞ controller with multiple control objectives for an active suspension system of a full-car model is designed in [9]. A strategy utilizing the proportional-integral sliding mode control scheme in controlling the active suspension system is presented in [10]. In [11] a control strategy based on model predictive control (MPC) methods for semi-active suspensions in road vehicles is designed. A fuzzy logic technique was used to control an active suspension system in [12], the ride comfort is also guaranteed by reducing the body acceleration. Practical feasibility of a new hybrid control method applied to an active suspension system of a quarter car model by skyhook and adaptive neuro active force control was investigated in [13]. An estimation of control voltage input to the magnetorheological damper is made in [14] by using neural network in order to reduce the vibrations. A non-fragile static output feedback controller for active suspension systems, considering input delay, is investigated in [15], where the resulted constraints are in the form of bilinear matrix inequalities (BMIs), requiring iterative methods to be solved. In [16] and [17] state feedback is used to design constrained H∞ control for quarter-car active suspension system. However, using state feedback entails having all states which may sound difficult and costly in some cases. Static output feedback controllers for a quarter-car active suspension model are designed in [18]. Also in [19] a robust non-fragile static output feedback controller is designed for a quarter-car with active suspension. However dynamic feedback is always preferable due to its more effective control performances. A great deal of the controller design methods lead to high order controllers. Such systems have high implementation cost, poor reliability, high fragility, numerical error, and potential problems in maintenance. Plant or controller reduction techniques do not always guarantee that the closed-loop performance is preserved. Therefore, a challenging problem is to design directly a low-, fixed-order controller for a system [20]. Thus, by using a fixed order controller design method one can obtain simplicity and desired performance together. There is no result on the LMI-based designing of a linear robust fixed-order dynamic output feedback controller with model uncertainty and nonlinear dynamics, for suspension systems of quarter-car models. In our proposed method we proved and showed that for the special case of active suspension system, with a full linear controller, desired specifications are obtainable through LMI optimization even with considering a time-varying delay in the control input. Besides, the approach for saturating the active force is also within an LMI viewpoint and thus remains linear. Furthermore, to the best of our knowledge considering the most complete model of linear controller with direct feedthrough parameter, is not available on earlier similar researches. It is worth noting that norm-bounded perturbations of controller parameters can also be tolerated using the proposed method. The organization of the remainder of this paper is as follows. Problem statement as well as primary backgrounds such as introducing the model and performance objectives is provided in Section 2. Section 3 is devoted to the main results of the paper including problem formulation and controller design in the form of a theorem and a corollary. Some numerical examples are provided in Section 4 to demonstrate the effectiveness of the proposed method. Eventually, conclusions are discussed in Section 5.

2. Problem statement and preliminaries A vehicle can be modeled as two sub-systems including sprung mass and unsprung mass joined together with some elastic and dissipative elements such as suspensions and tyres. Sprung mass refers to chassis mass imposed to springs and dampers, while unsprung mass is the mass of wheels, axles, and connections placed between springs and road surface. Aforesaid two degree-of-freedom model is influenced by disturbances and inputs caused by road surface. Suspension dynamics of an active quarter-vehicle can be modeled with the relationship of mass spring damper components as depicted in Fig. 1. Moreover, required parameters of a quarter car suspension system are provided in Table 1. In this model the wheel is represented by the tyre, with spring and damping characteristics. The tyre also plays a low-pass filter role in the suspension system reducing the high-frequency shakes caused by road surface [21]. Assuming no wheel lift-off and no slippage, one can obtain the dynamic equations of vehicle suspension system [22].

ms x¨ s (t ) + cs [ẋs (t ) − xu̇ (t )] + ks [xs (t ) − xu (t )] + k ns [xs (t ) − xu (t )]3 = u (t − τ (t )) mu x¨ u (t ) + cs [xu̇ (t ) − ẋs (t )] + ks [xu (t ) − xs (t )] + k ns [xu (t ) − xs (t )]3 + kt [xu (t ) − xr (t )] + ct [xu̇ (t ) − xṙ (t )] = − u (t − τ (t ))

(1)

In which τ (t ) is a time-varying delay by the condition of τ ̇ (t ) < τ¯d < 1. Undoubtedly the principal task of the suspension system is to minimize the vertical acceleration sensed by the rider which brings about ride comfort and less depreciation. Therefore the body acceleration x¨S (t ) is taken as the first control Please cite this article as: P. Badri, et al., Robust fixed-order dynamic output feedback controller design for nonlinear uncertain suspension system, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.04.020i

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Fig. 1. Quarter-vehicle model [17].

Table 1 Quarter car suspension system terminology. Parameter

Description

xS xu xr u ms mu cs ks kns ct kt

Movement of sprung mass Movement of unsprung mass Road displacement input Active input Sprung mass Unsprung mass Damping coefficient of the suspension system Linear stiffness coefficient of the suspension system Nonlinear stiffness coefficient of the suspension system Damping coefficient of the suspension tyre Stiffness coefficient of the suspension tyre

output and one of the design objectives is to minimize it. Besides comfort, security is one of the most important objectives of the suspension system. The dynamic tyre load should not exceed the static tyre load in the way that the contact between the wheels and the road is preserved, that is kt (xu (t ) − xr (t )) ≤ (mS + mu ) g . In addition to the structural constraint on the maximum deflection, a predetermined limitation on the power of the actuator is also considered u (t ) ≤ umax , in which umax is the maximum possible value of the actuator control force. The body acceleration x¨S (t ) is assumed to be performance control output, while constrained control output x2 (t ) is supposed to be the suspension stroke xS (t ) − xu (t ) and relative dynamic tyre load

kt (xS (t ) −xr (t ) ) . (mS +mu ) g

Besides, y (t ) is taken to

designate the measured output vector. State variables and disturbance input values and descriptions are tabulated in Table 2. Defining x (t ) = [ x1 (t ) x2 (t ) x3 (t ) x 4 (t )]T as the state vector and using Table 2 definitions one can write state space equations of suspension system as follows [23]:

Table 2 State space variables terminology. Parameter

Value

Description

x1 x2 x3 x4 w

xS (t ) − xu (t ) xu (t ) − xr (t ) xṠ (t ) xu̇ (t ) xṙ (t )

Suspension deflection Tyre deflection Sprung mass speed Unsprung mass speed Disturbance input

Please cite this article as: P. Badri, et al., Robust fixed-order dynamic output feedback controller design for nonlinear uncertain suspension system, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.04.020i

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ẋ (t ) = Ax (t ) + Bu u (t − τ (t )) + Bω ω (t ) + h (t , x) y (t ) = Cx (t )

(2)

in which:

⎡ 0 0 1 ⎢ 0 0 ⎢ 0 ⎢ ks c 0 − s A = ⎢− ms ms ⎢ ⎢ ks k cs − t − ⎢⎣ m mu mu u

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ cs + ct ⎥ mu ⎥⎦ −1 1 cs ms

T ⎡ 1 1 ⎤ − Bu = ⎢ 0 0 ⎥ , ms mu ⎦ ⎣

⎡ ct ⎤T Bω = ⎢ 0 − 1 0 mu ⎥⎦ ⎣

C = ⎡⎣ 1 1 1 0⎤⎦ ,

(3)

Considering that the proposed suspension system model is not linear and also contains high order terms, it is necessary to restrain and control the influences of these terms in a systematic context. Furthermore any proposed method should make it possible to design a robust controller in the presence of uncertainties in suspension systems parameters. Therefore, h (t , x ) is a piecewise continuous vector function including all possible nonlinearities and probable uncertainties of the system which satisfies the following quadratic constraint in its domain of continuity [24]:

¯ h (t , x)T h (t , x) ≤ α¯ 2xT H¯ T Hx

(4)

Where α¯ > 0 is a bounding parameter which should be maximized and H¯ is a constant v × n matrix. Let Γ¯ = γ¯Iv with γ¯ = α¯ −2. Then, it is always possible to find matrices H and Γ such that [24]:

¯ ≤ xT H T Γ −1Hx h (t , x)T h (t , x) ≤ xT H¯ T Γ¯ −1Hx

(5)

where H is a v × n matrix and:

Γ = γIv

,

γ>0

(6)

The sufficient condition for satisfying (5) is [6]:

λmax (H¯ T H¯ ) γ¯ ≤ γ λmin (H T H )

(7)

3. Main results The main objective of this paper is to design a non-fragile robust controller that asymptotically stabilizes the suspension system. Therefore a dynamic output feedback controller is proposed to meet the aforesaid objectives. It is likely that the controller applied to the system is sensitive to uncertainties in controller parameters due to implementation inaccuracies [25]. Therefore the following non-fragile dynamic output feedback controller is presented to satisfy abovementioned objective besides preventing unfavorable closed loop performance caused by accomplishment inaccuracies.

ẋC (t ) = (AC + ΔAC ) xC (t ) + (BC + ΔBC ) y (t ) u (t ) = (CC + ΔCC ) xC (t ) + (DC + ΔDC ) y (t )

(8)

with xC (t ) ∈ R nC where nc is the arbitrary order of the controller and AC , BC , CC , and DC are the controller parameters. Controller parameters are supposed to be perturbed by uncertainties in the form below:

ΔAC ≤ δ AC ,

ΔBC ≤ δ BC ,

ΔCC ≤ δCC ,

ΔDC ≤ δ DC

(9)

The closed loop system including (2) and (8) is as follows:

^ ^ ẋCl (t ) = (E + ΔE ) xCl (t ) + (F + ΔF ) xCl (t − τ (t )) + H (t , x) + Bω ω (t )

(10)

where: ⎡ x (t ) ⎤ x Cl (t ) = ⎢ ⎥, ⎣ x C (t )⎦

⎡ A 0⎤ E=⎢ ⎥, ⎣ BC C A C ⎦

⎡ B ΔD C B u ΔC C ⎤ ΔF = ⎢ u C ⎥, ⎣ 0 0 ⎦

⎡ 0 0 ⎤ ΔE = ⎢ ⎥, ⎣ ΔB C C ΔA C ⎦

⎡ h (t , x )⎤ ^ H (t , x ) = ⎢ ⎥, ⎣ 0 ⎦

⎡B ⎤ ^ Bω = ⎢ ω ⎥ ⎣ 0⎦

⎡ B D C Bu C C ⎤ F=⎢ u C ⎥ ⎣ 0 0 ⎦

(11)

Please cite this article as: P. Badri, et al., Robust fixed-order dynamic output feedback controller design for nonlinear uncertain suspension system, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.04.020i

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Definition 1. [26] Hα class: For any prescribed α and matrix H with suitable dimension, Hα is a class of piecewise-continues functions described by:

Hα = {h (t , x) h ∈ R n, hT h ≤ xT H T αT αHx

in the domain of continuity}

(12)

Therefore h ∈ Hα indicates that h (t, 0) = 0, which means x = 0 is the equilibrium point of system (10). Definition 2. [26] Robust stability with scalar degree α : system (10) with bounded nonlinear uncertainty provided in (5) is robustly stable with degree α if the equilibrium x = 0 is globally asymptotically stable for all h (t , x ) ∈ Hα . Theorem 1. Considering closed-loop system in (10) and given positive matrices r and q as prescribed symmetric positive definite weighting matrices and predetermined scalars pmc , pms , kT , kL , kN , and kG , if positive definite matrices P = PT in the form of:

P = diag (PS , PC )

(13)

and Q = QT alongside with matrices T , L, N , G , and positive scalars τi for i = 1 , … , 5 exist such that the following minimization problem becomes feasible:

minimize γω + γ ⎡ Π11 Π12 ⎤ subject to ⎢ ⎥<0 ⎣* Π22 ⎦ ⎡ PC I ⎤ ⎢ ⎥ > 0, ⎣ * pmc I ⎦

⎡ PS I ⎤ ⎢ ⎥>0 ⎣ * pms I ⎦

⎡ −kT I T ⎤ ⎢ ⎥ < 0, ⎣* −I ⎦

⎡ −kL I L ⎤ ⎢ ⎥ < 0, ⎣* −I ⎦

⎡ −kN I N ⎤ ⎢ ⎥ <0, ⎣* −I ⎦

⎡ −kG I G ⎤ ⎢ ⎥<0 ⎣* −I ⎦

(14)

in which:

Π11 = Φ11 + Q + Q˜ + Δ11, ⎡ ⎡ B ⎤ ⎡ B ⎤ ⎡ I ⎤ ⎡ 0⎤ ⎡ 0⎤ ⎡ B ⎤ ⎡ H ⎤⎤ Π12 = ⎢ Φ12 P ⎢ u ⎥ P ⎢ u ⎥ P ⎢ ⎥ P ⎢ ⎥ P ⎢ ⎥ P ⎢ ω ⎥ τ 3 ⎢ ⎥⎥, ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ 0 ⎦⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ 0 I I 0 0 0 ⎣ Π22 = diag ( −(1 − τ¯d ) Q + Δ22 , − τ1I , − τ 2 I , − τ 3 I , − τ4 I , − τ 5 I , − γω , − Γ )

(15)

with:

⎡ P A + AT P C T LT ⎤ S ⎥, Φ11 = ⎢ S ⎣ T + TT⎦ *

( Δ22 = diag ( τ2

Q˜ = diag (r , q),

), τ1 δC2 I ) ,

Δ11 = diag τ 5 C δ B2C I , τ4 δ A2C I C δ D2C I ,

C

⎡ GC N ⎤ Φ12 = ⎢ , ⎣ 0 0 ⎥⎦ Γ = γ Iv

(16)

then, the norm bounded non-fragile dynamic output feedback controller parameters of

AC = PC−1T , BC = PC−1L, CC = Bu↑ PS−1N , DC = Bu↑ PS−1G

(17)

make the closed loop system in (10) robustly stable and guarantee external disturbance attenuation with H∞ performance, where Bu↑ is the pseudo inverse of Bu . Proof The main idea of proving Theorem 1 is to formulate the robust stabilization problem of the system (10) in an LMIbased optimization structure. The following inequality must be satisfied according to the H∞ disturbance attenuation per^: formance index proposed in [27], with respect to ω

˜ Cl (t ) − ωT (t ) γ ω (t ) ≤ 0 V̇ (xCl (t ), t ) + xClT (t ) Qx ω

(18)

In which V (xCl (t ) , t ) is a positive quadratic Lyapunov–Krasovskii candidate function, and Q˜ as defined in (16) is a weighting matrix containing predefined matrices r and q for the states of system and those of the controller, respectively. The Lyapunov quadratic candidate function is considered as follows:

V (xCl (t ), t ) = xClT (t ) PxCl (t ) +

t

∫t − τ (t)

xClT (s ) QxCl (s ) ds ,

(19)

Please cite this article as: P. Badri, et al., Robust fixed-order dynamic output feedback controller design for nonlinear uncertain suspension system, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.04.020i

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Then, the derivative of V̇ (xCl (t ) , t ) can be expressed by:

̇ T (t ) PxCl (t ) + xClT (t ) PẋCl (t ) V̇ (xCl (t ), t ) = xCl + xClT (t ) QxCl (t ) − (1 − τ ̇ (t )) xClT (t − τ (t )) QxCl (t − τ (t ))

(20)

According to (2), V̇ (xCl (t ) , t ) can be expanded as:

V (xCl (t ), t ) ≤

T

( (E + ΔE)x (t) + (F + ΔF )x (t − τ (t)) + H^ (t, x) + B^ ω (t)) Px (t) ^ ^ + x (t ) P ( (E + ΔE ) x (t ) + (F + ΔF ) x (t − τ (t )) + H (t , x) + B ω (t ) ) Cl

Cl

T Cl

+

Cl

xClT (t ) QxCl (t )

− (1 −

Cl

ω

Cl

τ¯d xClT (t

)

ω

− τ (t )) QxCl (t − τ (t ))

(21)

then:

⎛ ⎡ B z + Bu z1⎤ ⎞T ⎡ z3 ⎤ + FxCl (t − τ (t )) + ⎢ u 2 V̇ (xCl (t ), t ) ≤ ⎜ ExCl (t ) + ⎢ ⎥ ⎟ PxCl (t ) ⎥ ⎣ z5 + z 4 ⎦ ⎣ ⎦⎠ 0 ⎝ ⎛ ⎡ B z + Bu z1⎤ ⎞ ⎡ z3 ⎤ + xClT (t ) P ⎜ ExCl (t ) + ⎢ + FxCl (t − τ (t )) + ⎢ u 2 ⎥⎟ ⎣ z5 + z 4 ⎥⎦ ⎣ ⎦⎠ ⎝ 0 ⎡ B ⎤T ⎡B ⎤ + ωT (t ) ⎢ ω ⎥ PxCl (t ) + xClT (t ) P ⎢ ω ⎥ ω (t ) + xClT (t ) QxCl (t ) ⎣ 0⎦ ⎣ 0⎦ − ( 1 − τ¯d ) xClT (t − τ (t )) QxCl (t − τ (t ))

(22)

where:

z1 = ΔCC xC (t − τ (t )), Introducing Ψ =

⎡x T ( t ) ⎣ Cl

z2 = ΔDC Cx (t − τ (t )), xClT

(t − τ (t ))

z1T

z2T

z3 = h (t , x) z3T

z4T

z5T

z 4 = ΔAC xC , T ωClT (t ) ⎤⎦ ,

z5 = ΔBC x

(23)

the inequality (21) is rearranged.

V (xCl (t ), t ) ≤ ⎡ ⎡ B ⎤ ⎡ B ⎤ ⎡ I ⎤ ⎡ 0 ⎤ ⎡ 0 ⎤ ⎡ B ⎤⎤ ⎢ E T P + PE + Q PF P ⎢ u ⎥ P ⎢ u ⎥ P ⎢ ⎥ P ⎢ ⎥ P ⎢ ⎥ P ⎢ ω ⎥⎥ ⎣ 0 ⎦ ⎣ 0 ⎦ ⎣ 0 ⎦ ⎣ I ⎦ ⎣ I ⎦ ⎣ 0 ⎦⎥ ⎢ ⎢ ⎥ 1 Q 0 0 0 0 0 0 ⎥ * − ( − τ ) ¯ d ⎢ ⎢ 0 0 0 0 0 0 ⎥ * * ΨT ⎢ 0 0 0 0 0 ⎥Ψ * * * ⎢ ⎥ 0 0 0 0 ⎥ * * * * ⎢ ⎢ 0 0 0 ⎥ * * * * * ⎢ ⎥ 0 0 * * * * * * ⎢ ⎥ ⎣ 0 ⎦ * * * * * * *

(24)

Now substituting (24) into (18), the subsequent matrix inequality is obtained:

⎡ ⎡ B ⎤ ⎡ B ⎤ ⎡ I ⎤ ⎡ 0 ⎤ ⎡ 0 ⎤ ⎡ B ⎤⎤ ⎢ E T P + PE + Q˜ + Q PF P ⎢ u ⎥ P ⎢ u ⎥ P ⎢ ⎥ P ⎢ ⎥ P ⎢ ⎥ P ⎢ ω ⎥⎥ ⎣ 0 ⎦ ⎣ 0 ⎦ ⎣ 0 ⎦ ⎣ I ⎦ ⎣ I ⎦ ⎣ 0 ⎦⎥ ⎢ ⎢ 0 0 0 0 0 0 ⎥⎥ * − (1 − τ¯d ) Q ⎢ ⎢ 0 0 0 0 0 0 ⎥ * * ⎢ 0 0 0 0 0 ⎥<0 * * * ⎢ ⎥ 0 0 0 0 ⎥ * * * * ⎢ ⎢ 0 0 0 ⎥ * * * * * ⎢ ⎥ 0 0 * * * * * * ⎢ ⎥ * * * * * * * − γω ⎦ ⎣

(25)

According to (5) it is easy to conclude:

z3T z3 ≤ xT H T Γ −1Hx

(26)

Moreover, parameters z1, z2, z3, and z 4 are considered to meet the following norm bounds:

z1T z1 ≤ δC2C xCT (t − τ (t )) xC (t − τ (t )),

z2T z2 ≤ ‖C‖δ D2C xT (t − τ (t )) x (t − τ (t ))

z4T z 4 ≤ δ A2C xCT xC ,

z5T z5 ≤ ‖C‖δ B2C xT x

(27)

where ‖C‖ denotes the Euclidean norm of C . Consequently the combination of Eqs. (26) and (27) with respect to Ψ will result Please cite this article as: P. Badri, et al., Robust fixed-order dynamic output feedback controller design for nonlinear uncertain suspension system, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.04.020i

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in:

⎡⎡ 2 ⎢ ⎢ C δ BC I ⎢⎢ ⎢ ⎣0 ⎢ ⎡ −1 ⎢ + ⎢ HΓ H ⎢ ⎣0 ⎢ ⎢ * ΨT ⎢ ⎢ ⎢ * ⎢ ⎢ * ⎢ * ⎢ * ⎢ ⎢ * ⎢ ⎣ *

⎤ ⎥ 2 ⎥ δ AC I ⎦

0

0

0⎤ ⎥ 0⎦

⎡ C δ2 I 0 ⎤ DC ⎢ ⎥ ⎢⎣ 0 δ C2C I ⎥⎦ * * * * * *

⎤ ⎥ ⎥ 0 0 0 0 0 0⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 0 0 0 0 0 0⎥ Ψ ≥ 0 ⎥ ⎥ − I 0 0 0 0 0⎥ * − I 0 0 0 0⎥ ⎥ * * − I 0 0 0⎥ * * * − I 0 0⎥ * * * * − I 0⎥ ⎥ * * * * * 0⎦

(28)

Repeatedly using S-procedure ([28]) for (25) and (28) gives the following inequality:

(29)

ΨT ΩΨ < 0 where:

⎡ Ω11 Ω12 ⎤ Ω=⎢ ⎥ ⎣ * Ω22 ⎦

(30)

in which:

⎡ HΓ −1H 0⎤ Ω11 = E T P + PE + Q + Q˜ + Δ11 + τ 3 ⎢ ⎥, ⎣ 0 0⎦ ⎡ ⎡ B ⎤ ⎡ B ⎤ ⎡ I ⎤ ⎡ 0 ⎤ ⎡ 0 ⎤ ⎡ B ⎤⎤ Ω12 = ⎢ PF P ⎢ u ⎥ P ⎢ u ⎥ P ⎢ ⎥ P ⎢ ⎥ P ⎢ ⎥ P ⎢ ω ⎥⎥ ⎣ 0 ⎦ ⎣ 0 ⎦ ⎣ 0 ⎦ ⎣ I ⎦ ⎣ I ⎦ ⎣ 0 ⎦⎦ ⎣ Ω22 = diag ( −(1 − τ¯d ) Q + Δ22 , − τ1I , − τ 2 I , − τ 3 I , − τ4 I , − τ 5 I , − γω )

(31)

with:

⎡ P A + AT P C T PC BC ⎤ S S ⎥, E T P + PE = ⎢ ⎢⎣ * PC AC + ACT PC ⎥⎦ ⎡ P B D C PS Bu CC ⎤ PF = ⎢ S u C ⎥. ⎣ 0 0 ⎦

(32)

However the matrix Eq. (32) are not linear due to several multiplications of variables. Hence, by changing variables as follows:

T = PC AC ,

L = PC BC ,

N = PS Bu CC ,

G = PS Bu DC

(33)

and finally using Schur complement [29], Π in (14) is obtained. Thus controller parameters can be achieved using (17). It should be noted that there is no bounding constraint on the obtained control parameters in this stage. Therefore, it is necessary to apply appropriate constraints to controller parameters in order to prevent them from taking very large and undesirable amounts. For instance, applying bounding constraint on the ‖AC ‖ in order to prevent AC from becoming very large requires ‖T‖ and ‖PC−1‖ to be bounded as its components. For this purpose ‖T‖ can be bounded as follows:

⎡− k I TT ⎤ T ⎢ ⎥<0 ⎣ T − I⎦ in which kT is a scalar LMI variable. Applying Schur-complement to (34) leads to the obvious constraint ‖T‖ <

(34) kT . Similarly,

bounding ‖PC−1‖ is also possible using the following inequality:

⎡ PC I ⎤ ⎢ ⎥>0 ⎣ I k PC I ⎦

(35)

where kT is a scalar parameter. Consequently the latter constraint is equivalent to ‖PC −1‖ < k PC . Continuing in a similar procedure, we can bound ‖BC ‖, ‖CC ‖, and ‖DC ‖ as well according to their components [30]. This completes the proof. □ Please cite this article as: P. Badri, et al., Robust fixed-order dynamic output feedback controller design for nonlinear uncertain suspension system, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.04.020i

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Corollary 1. Consider open-loop system (2) and the controller (8) without uncertainties, i.e. ΔAC = 0 , ΔBC = 0, ΔCC = 0, and ΔDC = 0. Given matrices r and q described in Theorem 1 and predetermined scalars pmc , pms , kT , kL , kN , and kG , if positive definite matrix P = PT of the form (13) and Q = QT alongside with matrices T , L, N , G , and positive scalar τ1, exist such that the following minimization problem becomes feasible: 4

minmize

γω +

∑ γ^i i=1

⎡ ⎢ Φ11 + Q + Q˜ Φ12 ⎢ ⎢* −(1 − τ¯d ) Q subject to ⎢ ⎢* * ⎢ * ⎢* ⎢⎣ * * ⎡ PC I ⎤ ⎢ ⎥ >0, ⎣ * pmc I ⎦

⎡I⎤ ⎡B ⎤ ⎡ H ⎤⎤ P ⎢ ⎥ P ⎢ ω ⎥ τ1 ⎢ ⎥ ⎥ ⎣ 0⎦ ⎣ 0 ⎦ ⎣ 0 ⎦⎥ ⎥ 0 0 0 ⎥<0 ⎥ − τ1 I 0 0 ⎥ * −γω 0 ⎥ * * −Γh ⎥⎦

⎡ PS I ⎤ ⎢ ⎥>0 ⎣ * pms I ⎦

⎡ −kT I T ⎤ ⎡ −kL I L ⎤ ⎡ −kN I N ⎤ ⎡ −kG I G ⎤ ⎢ ⎥ <0, ⎢ ⎥<0 , ⎢ ⎥ <0, ⎢ ⎥<0 ⎣* ⎣* ⎣* ⎣* −I ⎦ −I ⎦ −I ⎦ −I ⎦

(36)

where necessary parameters are all defined in Theorem 1, then the output feedback controller parameters given in (17) make the overall closed-loop system robustly stable and simultaneously guarantee H∞ performance. Proof The proof procedure is similar to that of Theorem 1, assumingΨ = [xClT xClT (t − τ (t )) z1T ωCl ]T .

4. Simulation example In this section, to validate the effectiveness of the proposed theorem and its corresponding corollary, some design examples are presented. Several effective convex optimization Parsers and Solvers can be used to assess the feasibility of the optimization problem and obtain design parameters. In this paper, we employ YALMIP parser [31], which comprises many solvers and can be implemented as a toolbox in Matlab [32], including SDPT3 [33], which we used. Model parameters of suspension system presented in [16] are provided in Table 3. Bump response and random response of the suspension system in (2) is studied in order to evaluate the aforementioned control objectives. Also h (t , x ) containing model nonlinearities and uncertainties can be considered as follows:

⎡ ⎤T k Δks Δkt Δcs Δct k Δks Δcs x 4 + ns x13⎥ h (t , x) = ⎢ 0 0 − x1 + (x 4 − x3 ) − ns x13 x1 − x2 + (x3 − x 4 ) − ms Δmu mu ⎦ ⎣ Δms Δms Δmu Δmu Δmu

(37)

which satisfies the bounding inequality (4). Unavoidable bumps and potholes on the road surface can always cause undesirable shakes and movements, assumed as external disturbances introduced into suspension system. The transient response of suspension system is studied in the presence of the following road disturbance proposed in [23]:

⎧ AπV 2π V L ⎪ ⎪ L sin L t 0 ≤ t ≤ V W (t ) = xṙ (t ) = ⎨ L ⎪ 0 t> ⎪ ⎩ V

(38)

in which A and L are the height and the length of the bump and the vehicle forward velocity is represented by V which assumed to have the amounts of A = 60 mm , L = 5 m , and v = 25 (km/h). In this section, bump responses of the active suspensions system (2) in the presence of Gaussian disturbance with zero mean and unit variance ([34]) through the proposed method in Theorem 1 and Corollary 1 are obtained. The nonlinear model of suspension system is considered to be certain, however the second and the third order controllers with uncertain and certain parameters, as two possible scenarios are assumed. Let H = I , r = 0.5I , and q = 2I for both Table 3 Numerical values of suspension system. ms

mu

ks

kns

kt

cs

ct

973 kg

114 kg

42720 N/m

8220 N/m3

101115 N/m

1095 Ns/m

14.6 Ns/m

Please cite this article as: P. Badri, et al., Robust fixed-order dynamic output feedback controller design for nonlinear uncertain suspension system, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.04.020i

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Fig. 2. Bump responses of the suspension system for third order controllers obtained via Theorem 1 and Corollary 1.

investigated methods. As indicated in [23], controller delay is also supposed to be 20 ms for all subsequent scenarios. Predetermined scalars pmc = 4 × 10−3, pms = 3.2 × 10−1, kT = 2.5 × 1015, kL = 8.7 × 1016 , kN = 2.4 × 105, and kG = 9 × 102 for Corollary 1 are also considered. Thanks to these bounding constraints, the obtained controllers are applicable to the system, just as the bump response of the suspension system depicted in Fig. 2 confirms this claim. Nevertheless, inappropriate selection of aforementioned bounds may result in an infeasible problem. Furthermore, the following parameter uncertainties for perturbed dynamic output feedback controllers, with uncertainty bounds δ AC = 10, δ BC = 1000, δCC = 10, and δ DC = 0.5 in Theorem 1, are assumed:

⎡ sin (5t ) cos (3t )⎤ ⎡ sin (2t ) ⎤ ΔA C = δ A C ⎢ ⎥, ΔBC = δ BC ⎢ ⎥ ⎣ sin (t ) cos (2t )⎦ ⎣ cos (3t )⎦ ΔCC = δCC ⎡⎣ cos (t ) sin (4t )⎤⎦ ,

ΔDC = δ DC cos (3t ).

(39)

and for the third order controller:

ΔAC = δ AC

⎡ sin (2t ) cos (4t) cos (4t )⎤ ⎡ cos (4t )⎤ ⎢ ⎥ ⎢ ⎥ ⎢ cos (t ) sin (t ) sin (2t ) ⎥, ΔBC = δ BC ⎢ sin (3t ) ⎥ ⎢⎣ sin (4t ) sin (5t ) cos (t ) ⎥⎦ ⎢⎣ cos (2t )⎥⎦

ΔCC = δCC ⎡⎣ cos (2t ) sin (2t ) sin (3t )⎤⎦ ,

ΔDC = δ DC sin (t ).

(40)

The obtained controller parameters using Theorem 1 and Corollary 1, are listed in Table 4 for nC = 2, 3. Please cite this article as: P. Badri, et al., Robust fixed-order dynamic output feedback controller design for nonlinear uncertain suspension system, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.04.020i

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Table 4 Controller parameters using Theorem 1 and Corollary 1 for nC = 2, 3. Order of controllers (nC )

Ac

Bc

Cc

Dc

2

Theorem 1

⎡− 3.6383 − 2.0037 ⎤ × 104 ⎣⎢− 1.9543 − 2.8724 ⎦⎥

⎡− 2.634 ⎤ × 105 ⎣⎢ − 2.587 ⎦⎥

⎡1.3265⎤T ⎥ × 103 ⎢⎣ 1.0451⎦

−323.43

Corollary 1

⎡ − 3.5377 − 1.8868⎤ ⎥ × 104 ⎢⎣ − 1.9004 − 3.0069 ⎦

⎡− 2.737⎤ ⎥ × 105 ⎢⎣ − 2.431⎦

⎡1.0056 ⎤T ⎥ × 103 ⎢⎣ 0.9310 ⎦

−172.24

Theorem 1

⎡− 3.2942 − 1.3926 − 1.3023⎤ ⎢ ⎥ 4 ⎢− 1.3572 − 2.4744 − 1.4187⎥ × 10 ⎣− 1.3032 − 1.0348 − 2.1875⎦

⎡− 3.059 ⎤ ⎢ ⎥ 5 ⎢ − 2.307 ⎥ × 10 ⎣− 1.194 ⎦

⎡1.2494 ⎤T ⎢ ⎥ 3 ⎢1.5538 ⎥ × 10 ⎣1.3984 ⎦

−58.76

3

Table 5 Body acceleration and active force performances for second and third order controllers obtained via Theorem 1 and Corollary 1 for nC = 2, 3. Order of controllers (nC )

2

3

Different cases of control perturbations

Controllers' perturbation

Body acceleration

Active force

δ AC

δ BC

δCC

δ DC

Settling time 5% (s)

Maximum amplitude

Settling time 5% (s)

Total ITSE

Case I

20

3 × 103

40

1

1.355

1.449

1.623

Case II

20

4 × 103

40

1.5

1.341

1.688

1.635

857605.8

Case III

15

50

2

1.752

1.233

1.662

647575.8

Corollary 1 Case I

0 20

6 × 103 0 1 × 103

0 10

0 1

1.306 1.363

1.754 0.9698

1.205 1.353

867432.7 1019306.5

845409.4

Case II

20

2 × 103

20

1

1.345

0.9478

1.282

981546.4

Case III

10

10

1.5

1.345

0.9478

1.493

1019306.5

Corollary 1

0

3 × 103 0

0

0

1.316

1.291

1.408

980739.6

Fig. 2 illustrate bump responses of the suspension system (2), including body acceleration, suspension deflection, relation dynamic tyre load, and active force, for the third order controllers obtained via Theorem 1, with three different cases of controller parameter perturbations, and also via Corollary 1. Besides, in order to have a numerical and analytical comparison between the different cases of proposed methods in Theorem 1 and Corollary 1 settling time and maximum amplitude of body acceleration along with settling time and total ITSE [35] of active force for the obtained second and third order controllers are provided in Table 5. It can be seen from Fig. 2 and Table 5 that the bump responses of suspension system (2) via proposed controllers through Theorem 1, affected by Gaussian disturbances with amplitudes even up to 50 percent of nominal control parameters, have acceptable differences with those of controllers obtained from Corollary 1. In this scenario bump responses of active suspension system (2), in the presence of model uncertainties and nonlinearities through the proposed method in Corollary 1 are obtained. Fig. 3 demonstrates the bump responses of the suspension system for different cases of uncertainties listed in Table 6. In order to have an explicit comparison, Case I is considered to be certain. It is obvious from Fig. 3 that considerable amounts of uncertainties can be tolerated using the proposed method in Corollary 1. Furthermore, simulation results obtained by MATLAB are compared with those of obtained by CarSim 8.1 mechanical simulator as a more real-like car model, and also with results of the method presented in [23] to evaluate the effectiveness of proposed method. Since the controller design method presented in [23] does not consider the model uncertainties and perturbations of the controller parameters, its performances are compared with those of the proposed controllers in Corollary 1 without model uncertainties. The obtained controller parameters using Corollary 1 are listed in Table 7 for nC = 1, 2, 3, 4. Bump responses of suspension system (2) via proposed method in Corollary 1 and the presented method in [23] are depicted in Fig. 4. Comparing these responses indicates that with adequate actuator force and suspension deflection, one can obtain smaller body acceleration as the first and of course the most important control objective of the suspension system, even with second order controller. In addition, smaller relation dynamic tyre load is obtained using proposed controllers with nC = 2, 3, 4. As demonstrated in Fig. 4, the performances of the proposed 4th order controller by MATLAB are better than that of obtained by CarSim 8.1 due to more realistic nature of the CarSim simulator, but still the performances of 4th order controller obtained by Corollary 1 and via CarSim 8.1 is acceptable. To have a more precise comparison, Integral Square Error (ISE), Integral Absolute Error (IAE), Integral Time Square Error (ITSE), and also Integral Time Absolute Error (ITAE) methods in [35] are provided in Table 8 as active force indices for proposed method in Corollary 1 with two models by MATLAB and CarSim 8.1 and [23]. Settling time and maximum absolute value of body acceleration, suspension deflection, relation dynamic tyre load, and active force of the bump responses of suspension system (2) via proposed method in Corollary 1 and the presented method in [23] is provided in Table 9. Where settling time refers to the time in which the amplitude of each control performance Please cite this article as: P. Badri, et al., Robust fixed-order dynamic output feedback controller design for nonlinear uncertain suspension system, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.04.020i

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Fig. 3. Bump responses of suspension system via proposed method in Corollary 1 with system uncertainties and nonlinearities.

Table 6 Different cases of system perturbations for third order controller via Corollary 1. Different cases of system perturbations

Case Case Case Case

I II III IV

System perturbation

Δms (%)

Δmu (%)

Δks (%)

Δkt (%)

Δcs (%)

Δct (%)

0 5 3 10

0 5 4 7

0 5 7 2

0 5 3 3

0 5 6 8

0 5 6 5

reaches to the five percent of total range of each group. Comparing settling time of each performance, one can conclude that in the most cases bump responses of proposed controllers with nC = 2, 3, 4, show better attenuation behavior than the controller proposed in [23]. We know that the modeling of CarSim is done more precisely, with that in mind the results of the CarSim modeling is still acceptable compared with MATLAB modeling results. Please cite this article as: P. Badri, et al., Robust fixed-order dynamic output feedback controller design for nonlinear uncertain suspension system, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.04.020i

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Table 7 Controller parameters using and Corollary 1 for nC = 1, 2, 3, 4. Controller order

Ac

Bc

Cc

1

−5.2571 × 104 ⎡ − 3.5377 − 1.8868⎤ ⎥ × 104 ⎢⎣ − 1.9004 − 3.0069 ⎦

−1.418 × 105 ⎡− 2.737⎤ ⎥ × 105 ⎢⎣ − 2.431⎦

2.5297 × 103

408.02

⎡1.0056 ⎤T × 103 ⎣⎢ 0.9310 ⎦⎥

−172.24

3

⎡− 3.0946 − 1.3980 − 1.3096 ⎤ ⎢ ⎥ 4 ⎢− 1.3694 − 2.4058 − 1.0224 ⎥ × 10 ⎣− 1.2990 − 1.0152 − 2.2588 ⎦

⎡− 2.489 ⎤ ⎢ ⎥ 5 ⎢− 1.959 ⎥ × 10 ⎣− 1.814 ⎦

⎡1.6040 ⎤T ⎢ ⎥ 3 ⎢ 1.3121⎥ × 10 ⎣1.2074 ⎦

−33.48

4

⎡− 2.8228 ⎢ ⎢− 1.5937 ⎢− 1.2742 ⎢⎣− 1.2857

⎡ 4.492 ⎤ ⎢ ⎥ ⎢ 2.902 ⎥ × 105 ⎢− 1.826 ⎥ ⎢⎣ − 5.033⎥⎦

⎡− 6.2240 ⎤T ⎢ ⎥ ⎢− 3.6492⎥ × 103 ⎢ 3.4816 ⎥ ⎢⎣ 8.6441 ⎥⎦

365.54

2

− 1.5940 − 2.6274 − 1.1821 − 1.2017

− 1.2980 − 1.2046 − 2.0862 − 9.6128

− 1.3062⎤ ⎥ − 1.2211⎥ × 104 − 9.5855⎥ ⎥ ⎦ − 2.0859

Dc

Fig. 4. Bump responses of suspension system presented in [17] and proposed method in Corollary 1.

Please cite this article as: P. Badri, et al., Robust fixed-order dynamic output feedback controller design for nonlinear uncertain suspension system, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.04.020i

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Table 8 Active force indices for proposed method in Corollary 1 and [17]. Control method

Controller order

ISE

IAE

ITSE

ITAE

Corollary 1 by MATLAB

1 2 3 4 4

1166891.6 1439946.2 1839694.5 2037257.7 1784367.3

1107.6 1151.5 1293.9 1430.7 1169.5

768911.9 867432.7 980739.6 1068528.0 841891.1

836.8 767.2 812.9 957.2 793.3

4

1043627.2

991.3

697257.8

725.34

Model by CarSim 8.1 proposed method in [23]

Table 9 Settling time (5%)(I) and absolute maximum value (II) of bump response performances for proposed method in Corollary 1 and [17]. Control method

Corollary 1 by MATLAB

Model by CarSim 8.1 Proposed method in [23]

Controller order

1 2 3 4 4 4

Body acceleration

Suspension deflection

Relation dynamic tyre load

Active force

I

II

I

II

I

II

I

II

1.512 1.250 1.112 0.943 1.634 1.397

2.117 1.754 1.304 1.058 1.561 1.921

1.447 1.040 1.198 1.350 1.216 0.948

0.0524 0.0490 0.0545 0.0579 0.0623 0.0328

1.473 1.241 1.127 0.944 1.138 1.4042

0.2034 0.1728 0.1344 0.1139 0.1549 0.1823

1.571 1.162 1.283 1.427 1.227 1.305

1471 1841 2263 2472 2123 1592

Fig. 5. Power spectral density (PSD) of body acceleration.

Table 10 RMS body acceleration for 4th order controller via Corollary 1 and [23] with different road roughness coefficients. Road roughness coefficient

Corollary 1

Proposed method in [23]

16 × 10−6 m3

0.0029

0.0041

64 × 10−6 m3

0.0059

0.0081

256 × 10−6 m3

0.0120

0.0164

1024 × 10−6 m3

0.0261

0.0331

Please cite this article as: P. Badri, et al., Robust fixed-order dynamic output feedback controller design for nonlinear uncertain suspension system, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.04.020i

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Table 11 RMS suspension deflection for 4th order controller via Corollary 1 and [23] with different road roughness coefficients. Road roughness coefficient

Corollary 1

Proposed method in [23]

16 × 10−6 m3

8.3246 × 10−5

6.9215 × 10−5

64 × 10−6 m3

1.6729 × 10−4

1.3132 × 10−4

256 × 10−6 m3

3.3781 × 10−4

2.6750 × 10−4

1024 × 10−6 m3

8.003 × 10−4

5.4571 × 10−4

Table 12 RMS relative dynamic tyre load for 4th order controller via Corollary 1 and [23] with different road roughness coefficients. Road roughness coefficient

Corollary 1

Proposed method in [23]

16 × 10−6 m3

2.6765 × 10−4

3.8624 × 10−4

64 × 10−6 m3

256 × 10−6 m3

5.3873 × 10−4 0.0011

7.6119 × 10−4 0.0015

1024 × 10−6 m3

0.0023

0.0031

In addition to bumps and potholes, suspension systems are affected by random vibrations specified as a random process in which ground displacement power spectral density (PSD) is defined as follows:

⎛ n ⎞−c Gq (n) = Gq (n0 ) ⎜ ⎟ ⎝ n0 ⎠

(41)

where n stands for spatial frequency and n0 = 0.1 (1/m) is the reference spatial frequency. Gq (n0 ) and c = 2 denote the road roughness coefficient and road roughness constant respectively. By f = nV time frequency is determined in which V is the car forward velocity. PSD ground displacement can be obtained by:

V . f2

Gq (n) = Gq (n0 ) n02

(42)

Therefore, PSD ground velocity is achieved as follows:

G q̇ (n) = (2πf )2Gq (f ) = 4πGq (n0 ) n02 V

(43)

With fixed forward velocity, the ground velocity is a white-noise signal. Also, PSD body acceleration can be calculated for linearized system by:

G x1 (f ) = G (jω) G q̇ (f ) = G (jω) 4πGq (n0 ) n02 V

(44)

in which G x1 (f ) is the transfer function of body acceleration to disturbance which is calculated by the third row of the closedloop matrix (10). Vehicle forward velocity is supposed to be V = 25 (km/h). Furthermore, four different types of road disturbances by road roughness coefficients Gq (n0 ) = 16 × 10−6 m3, Gq (n0 ) = 64 × 10−6 m3, Gq (n0 ) = 256 × 10−6 m3, and Gq (n0 ) = 1024 × 10−6 m3 which are respectively related to B Grade (Good), C Grade (Average), D Grade (Poor), and E Grade (Very Poor), are assumed to evaluate the random response of the suspension system. Fig. 5 demonstrates PSD body acceleration for these four different types of roads disturbances. In addition, the RMS values of the body acceleration, suspension deflection, and relative dynamic tyre load achieved by the 4th order controller proposed in Corollary 1 and by the method presented in [23], for the abovementioned different road roughness coefficients are tabulated in Tables 10–12. In which the RMS value of the signal

u (t ) , is RMSu =

(1/T ) ∫

T

0

uT (t ) u (t ) dt with T = 100 s [23].Comparing the RMS values of the control performances of 4th order

controller proposed in Corollary 1 with that of the controller obtained by the method in [23], validates the performances of the proposed controller under the random road disturbances.

5. Conclusions In this paper a fixed order non-fragile dynamic output feedback controller design method based on LMI approach is proposed to robustly control bump response and random response of a quarter car active suspension system. Obviously, a low-order controller that keeps the desired specifications and simultaneously can be implemented simply is desirable. Please cite this article as: P. Badri, et al., Robust fixed-order dynamic output feedback controller design for nonlinear uncertain suspension system, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.04.020i

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Model uncertainties and nonlinear dynamics of the quarter-car are considered as well. Despite assuming a direct feedthrough parameter for the proposed controller, LMI approach of developing H∞ control is preserved. Moreover, appropriate LMI constraints are considered in order to prevent controller parameters from taking very large and undesirable amounts. The proposed procedure of controller design is organized in such a way that it can tolerate norm bounded perturbations of controller parameters. Since the considered model is influenced by road surface disturbance, an H∞ performance index is used to attenuate external disturbance. Eventually, a numerical example is presented to show the effectiveness of the proposed controller design method.

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Please cite this article as: P. Badri, et al., Robust fixed-order dynamic output feedback controller design for nonlinear uncertain suspension system, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.04.020i