energies Article
Robust Longitudinal Speed Control of Hybrid Electric Vehicles with a Two-Degree-of-Freedom Fuzzy Logic Controller Jau-Woei Perng and Yi-Horng Lai * Department of Mechanical and Electro-Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan;
[email protected] * Correspondence:
[email protected]; Tel./Fax: +886-7-525-3021 Academic Editor: Dirk Söffker Received: 6 January 2016; Accepted: 7 April 2016; Published: 16 April 2016
Abstract: This paper proposes a new robust two-degree-of-freedom (DoF) design method for controlling the nonlinear longitudinal speed problem of hybrid electric vehicles (HEVs). First, the uncertain parameters of the HEV model are described by fuzzy α-cut representation, in which the interval uncertainty and the possibility can be simultaneously indicated by the fuzzy membership function. For the fuzzy parametric uncertain system, the maximum uncertainty interval can be translated into the weighting matrix Q of the linear quadratic tracking problem to guarantee that the designed feedback controller is robust. Second, the fuzzy forward compensator is incorporated with a robust feedback controller to enhance the system tracking response. The simulation results demonstrate that the proposed controller has higher tracking performance compared to the single-DoF self-tuning fuzzy logic controller or conventional optimal H8 controller. Keywords: two-degree-of-freedom (DoF) design; fuzzy parametric uncertain system; fuzzy α-cut representation
1. Introduction Recently, increasing concern about a cleaner environment and fuel conservation has made hybrid electric vehicles (HEVs) an indispensable next-generation technology. HEVs coordinate both electric machines and internal combustion engines (ICEs) to deliver propulsion power. Many previous studies have focused on the energy management and optimal power flow of HEV dynamics [1–4]. However, HEV speed control techniques are drastically different than those of a conventional vehicle, because HEVs typically move in electric mode, and ICEs can be operated at higher speeds. Therefore, the drive performance and wide-range speed control algorithm of HEVs are also key concerns [5]. The practical application of speed control includes adaptive cruise control, intelligent collision avoidance and car following [6–8]. In the traditional ICE propulsion system, the vehicle speed and engine power are directly controlled by the mechanical throttle control system (MTCS). However, the hybrid powertrain request does not go directly to the engine in HEVs; instead, the MTCS is replaced by an electronic throttle control system (ETCS) [9–11]. The ETCS system is a complex engine mechanism that utilizes a DC servo motor to regulate the throttle position. With ETCS, the desired torque and a wide range of vehicle speeds can be achieved in HEVs [12]. In addition, HEV speed control must combine ETCS with nonlinear vehicle dynamics. Because of factors such as the uncertainty parameters of nonlinear elements and instability from environmental disturbances, designing an algorithm for HEV speed control is challenging [13]. Longitudinal speed control belongs to a group of wide-range and cyclic operations. The goal of HEV speed response is to track a desired speed under any operating condition. A robust and Energies 2016, 9, 290; doi:10.3390/en9040290
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adaptive internal model control algorithm was used to track speed and reject road grade disturbance in [14]. The sliding mode control, incorporated with an adaptive proportional-integral-derivative (PID) controller, was proposed to solve the uncertain speed servo problem [15]. The speed control performance among the state feedback controller, intelligent control techniques and adaptive controller was compared in [16]. However, these previous studies have two limitations. First, the uncertainty of the dynamic system model can generally be described as a bounded interval model that considers each operational point with equal probability. Compared to the interval approach, the fuzzy α-cut representation of uncertainties can use the fuzzy membership function to indicate the possibilities and intervals of variations. Such a fuzzy parametric uncertain system (FPUS) can be viewed as an extension of interval systems and has attracted considerable attention from researchers [17–19]. In [20], the problem of designing a robust controller for FPUS was converted into an optimal linear quadratic regulator (LQR) control approach. The optimal LQR controller, which was designed for the worst-case condition (α “ 0q, can stabilize all systems for various values of α P r0, 1s. Second, previous studies have addressed control schemes in one-degree-of-freedom (1-DoF) controllers, which may address them for specific types of performance, but be compromised in others. Two-degree-of-freedom (2-DoF) controllers can fulfill another performance requirement by adding a feed-forward controller or prefilter. For example, combining a forward fuzzy prefilter and a feedback controller for hydraulically-actuated robotic mechanisms was studied in [21]. Because the prefilter can compensate the effects of the dead-zone of the electromagnetic proportional control valve, the 2-DoF controller has quite good tracking trajectories compared to the conventional 1-DoF controller. In [22], the 2-DoF integral-P (IP) controller for electrical drives was shown to have good reference tracking and load-torque rejection performance. Coordinating the inner loop observer to reject disturbances and an outer loop tracking controller to achieve control performances was successful implemented in robust yaw stability control of electric vehicles [23]. Therefore, 2-DoF control systems can be used to enhance wide-range operations. This paper investigates the nonlinear longitudinal speed control model of HEVs with fuzzy parametric uncertain systems and proposes a new robust 2-DoF design method of speed control systems for HEVs. The design procedure consists of two steps. In the first step, the different loads of HEV components are described according to a fuzzy α-cut number, and the maximum uncertainty interval of the system is translated into the weighting matrix Q of the linear quadratic tracking (LQT) servo problem, to guarantee that the designed feedback optimal controller is robust under the worst-case condition. In the second step, a fuzzy forward compensator is incorporated with a robust feedback controller to enhance the system response. The robust property of the proposed controller can track a desired speed at a wide range of vehicle speeds with varying road grades. In addition, the fast dynamic response has significant effects on engine performance, fuel consumption and pollution emission, especially in the transition mode of the hybrid operating system of HEVs. The effectiveness of this longitudinal speed controller has been demonstrated in simulation studies. The remainder of this paper is organized as follows. Section 2 details the fuzzy parametric uncertain system for HEV speed control. In Section 3, the methodology for the synthesis of the 2-DoF fuzzy controller and the stability analysis of the proposed controller are presented. The simulation implementation of the proposed controller, with other controllers, such as the H8 and 1-DoF self-tuning fuzzy PID controllers, are described in Section 4. Finally, Section 5 presents the conclusions of this study. 2. Problem Formulation and Longitudinal Speed Control Modeling This paper focuses on the speed control of a small HEV with an uncertain parameter of ETCS and a nonlinear vehicle dynamic model, the control scheme for which is shown in Figure 1. The uncertainty component parameters are listed in Table 1 [12].
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Figure 1. 1. The The speed speedcontrol controlscheme scheme HEV with an electronic throttle control system Figure of of thethe HEV with an electronic throttle control system (ETCS)(ETCS) and a and a nonlinear vehicle dynamic system. nonlinear vehicle dynamic system. Table 1. The parameters of of the the HEV. HEV. Table 1. The nominal nominal values values and and uncertainty uncertainty parameters
ETCS ETCS Descriptions Symbol Nominal Value (SI unit) Uncertainty Bounds Descriptions Symbol Nominal Value (SI unit) Uncertainty 𝑅𝑎 Armature resistance 2 Ω [1.5,2.5]Bounds Armature resistance 2Ω [1.5,2.5] 𝐿𝑎 R a Armature inductance 0.003 H [0.002,0.004] Armature inductance L 0.003 H [0.002,0.004] a Back electromotive force (EMF) 0.110.11 Vs/rad [0.07,0.15] Back electromotive force (EMF) constant 𝐾𝑏 Kb Vs/rad [0.07,0.15] constant Gear ratio N 4 [2,6] 𝑁 Kt Gear ratioconstant 4 N m/A [2,6] Motor torque 0.1 [0.08,0.12] 𝐾𝑡 Ksp Throttle spring constant ms/rad [0.3,0.5] Motor torque constant 0.10.4 NN m/A [0.08,0.12] Equivalent inertia [0.009,0.0502] kgm2 𝐾𝑠𝑝 J Throttle spring constant 0.4 N0.021 ms/rad [0.3,0.5] Damping constant B 0.482 N ms/rad [0.082,1.443] 𝐽 Equivalent inertia 0.021 kgm2 [0.009,0.0502] Vehicle Dynamic Model 𝐵 Damping constant 0.482 N ms/rad [0.082,1.443] Descriptions Symbol Nominal Vehicle Dynamic ModelValue (SI unit) Uncertainty Bounds Descriptions Symbol m Nominal Value Uncertainty Bounds Vehicle mass 1000(SI kg unit) [750,1250] 2 [0.4,0.56] Drag coefficient α 0.48 N/pm{sq 𝑚 Vehicle mass 1000 kg [750,1250] 2 Engine force coefficient γ 12500 N [10000,15000] 𝛼 Drag coefficient 0.48 N/(m/s) [0.4,0.56] Engine idle force Fi 6400 N [5500,7300] γ Engine force coefficient 12500 N [10000,15000] Engine time constancy τe 0.5 s [0.2,0.8] 𝐹𝑖 bw Engine idle force 6400 [5500,7300] Bearing damping coefficient 0.035 N N ms/rad [0.03,0.04] 𝜏𝑒 rtire Engine time constancy 0.570 s mm [0.2,0.8] Radius of tire [50,90] coefficient 0.011 [0.01,0.012] 𝑏𝑤 µ Bearing Friction damping coefficient 0.035 N ms/rad [0.03,0.04] ˝ ˝ Road slope/grade Variable [˘2 , ˘30 ] 𝑟𝑡𝑖𝑟𝑒 β Radius of tire 70 mm [50,90] 𝜇 Friction coefficient 0.011 [0.01,0.012] 2.1. The Architecture of the HEV Model 𝛽 Road slope/grade Variable [±2° , ±30° ] The speed control architecture of the HEV includes an engine with ETCS and a nonlinear vehicle 2.1. The Architecture the HEV longitudinal motion of model. TheModel ETCS uses a DC servo motor to adjust the throttle, as expressed in the governing differential Equations (1)–(3) The speed control architecture of [9]. the HEV includes an engine with ETCS and a nonlinear ˆ ˙ motor to adjust the throttle, as vehicle longitudinal motion model. The DC servo di a 1 ETCS uses a dθ m “ ´R a i a(1)–(3) ´ Kb [9]. ` Ea (1) expressed in the governing differential dt Equations La dt 𝑑𝑖𝑎 1 𝑑𝜃𝑚 = (−𝑅𝑎 𝑖𝑎 − 𝐾𝑏 + 𝐸𝑎 ) 𝑑𝑡 𝐿𝑎 𝑑𝑡
(1)
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˙ ˆ d2 θ m 1 dθm “ ´ T ` T ´B m m L Jm dt dt2 ˙ ˆ 1 d2 θ dθ “ ´ Ta ` Tg ´Tsp ´ Bt Jg dt dt2
(2) (3)
where i a is armature current (A), Tsp is the spring torque and θm and θ are the angular position (rad) of the armature and throttle plate, respectively. R a and L a represent the armature resistance and inductance, respectively. The back electromotive force constant is Kb . The parameters Bm and Bt are defined as the motor shaft and throttle viscous damping coefficients, respectively. Ta is the torque due to airflow. Motor inertia and throttle inertia are defined as Jm and Jg , respectively. Assume the gear ratio N and the motor torque Tm can be expanded as: N“
Tg θm “ θ TL
(4)
Tm “ Kt i a
(5)
where Tg is the torque transmitted from gears, TL is the load torque and Kt is motor torque constant. Equations (1) and (2) can be expressed in terms of throttle plate angular rotation as: di a 1 “ dt La
ˆ
dθ ` Ea ´R a i a ´ Kb N dt
˙
´ ¯ dθ d2 θ 1 2 “ ´T ´ pN B ` B ´ Ta ` NTm q sp m t dt dt2 N 2 Jm ` Jg
(6)
(7)
For simplicity, let spring torque Tsp “ 2Ksp ˆ θ, equivalent inertia J “ N 2 Jm ` Jg and damping constant B “ N 2 Bm ` Bt . Taking the Laplace transform of Equations (6) and (7) into the s-domain, we obtain: θ psq “ Ea psq s3 `
R a J ` BL a 2 s La J
`
NKt {L a J R a B` N 2 Kt Kb `2Ksp L a 2K R s ` Lspa J a La J
(8)
From Table 1, the nominal transfer function of ETCS is given as: θ psq 6349 “ 3 2 Ea pSq s ` 689.7s ` 1.82 ˆ 104 s ` 2.54 ˆ 104
(9)
The dynamics of the nonlinear HEVs is given as [12,16]: m
? dV bω V dFe pθq “ Fi ` γ θ ´ τe ´ µmg pcosβq ´ αV 2 ´ ´ mg psinβq dt dt rtire
(10)
where Fi and Fe are the engine idle force and engine force. γ, µ, α and bω are the coefficients of engine force, friction, drag and bearing damping, respectively. τe is the engine time constant. β is the road slope. The Simulink model of the nonlinear vehicle dynamic Equation (10) is depicted in Figure 1. ˝ The procedure of linearization of the nominal model [12] is as follows: First, β = 10 is considered. Then, by using the MATLAB linearization command “linmod”, the numerical transfer function of the vehicle which is linearized around the nominal value of Table 1 is given as: V psq 7906 “ 2 θ pSq s ` 2s ` 0.001
(11)
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Combining Equations (9) and (11) and referring to Table 1, the nominal transfer function and the lowest and top-most bounds of the overall speed control system are given as: V psq Ea p S q
“ GN psq “
5ˆ107 s5 `691.7s4 `1.95ˆ104 s3 `6.17ˆ104 s2 `5.08ˆ104 s`25.4
GL psq “
1.87ˆ108 s5 `764s4 `1.19ˆ104 s3 `9.1ˆ104 s2 `2.5ˆ105 s`200
GU psq “
1.7ˆ107 s5 `655s4 `2.2ˆ104 s3 `3.9ˆ104 s2 `1.56ˆ104 s`5.5
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(12)
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The transfer function of the system under parametric uncertainty can be described as the plant with six interval parameters, , qr0 , qr1parametric , qr2 , qr3 and uncertainty qr4 . Theuncertain transfer function of the systempr0under can be described as the plant with six uncertain interval parameters, 𝑝̃0 , 𝑞̃0 , 𝑞̃1 , 𝑞̃2 , 𝑞̃3 and 𝑞̃4 . pr0 G ps, pr, qrq “ 5 (13) 𝑝̃0 4 3 r r r 2 r r 𝐺(𝑠, 𝑝̃, 𝑞̃) = s ` (13) 5 q4 s 4` q3 s3 ` q22s ` q1 s ` q0 𝑠 + 𝑞̃4 𝑠 + 𝑞̃3 𝑠 + 𝑞̃2 𝑠 + 𝑞̃1 𝑠 + 𝑞̃0
2.2. Fuzzy Parametric α-Cut Representation of the Uncertain HEV Model 2.2. Fuzzy Parametric α-Cut Representation of the Uncertain HEV Model The interval uncertainty representation assumes all of the parameters have the same probability. The interval uncertainty representation assumes all study, of the the parameters the same However, this is not true in practical applications. In this uncertain have parameters are probability. by However, this is not truemembership in practicalfunction applications. uncertain represented a fuzzy number qri with α “ µ pIn qri q this P r0,study, 1s. Thethe membership ) ∈ [0,1]. parameters by a fuzzy membership number 𝑞̃𝑖 function, with membership function 𝛼 = 𝜇(𝑞̃𝑖endpoint. q canrepresented function µ pqriare be any nonsymmetrical but decreases to the interval ) The fuzzy membership function 𝜇(𝑞̃𝑖 can beisany nonsymmetrical membership function, but decreases to The parametric uncertainty α-cut defined as: the interval endpoint. The fuzzy parametric uncertainty 𝛼-cut “ ‰ is defined as: ` qi pαi q “ q´ pα q , q pα q (14) i i − i +i 𝑞𝑖 (𝛼𝑖 ) = [𝑞𝑖 (𝛼𝑖 ), 𝑞𝑖 (𝛼𝑖 )]
(14)
where q´ p.q is an increasing function and q` p.q is a decreasing function. Let αi be the membership where i𝑞𝑖− (. ) is an increasing function iand 𝑞𝑖+ (. ) is a decreasing function. Let 𝛼𝑖 be the level of qi , as shown in Figure 2; we obtain: membership level of 𝑞𝑖 , as shown in Figure 2; we obtain: ´ ` + `+ −´ + q` p1q 0 “ q0 − q , q − p0q p1q “ q´ i 𝑖 (0) 𝑖 i(1) = 𝑞𝑖 (1)i = 𝑞𝑖 i p0q 𝑞“ i = 𝑞𝑖i , 𝑞𝑖 (0)“=q𝑞i 𝑖 , 𝑞q
(15) (15)
Figure 𝛼-cut representation representationof ofthe theuncertain uncertainparameter. parameter. Figure 2. Fuzzy Fuzzy α-cut
Consider the model Equation (13). (13). Assume that thethat onlythe information available Consider theuncertain uncertaininterval interval model Equation Assume only information for the values of the uncertain parameters p ̃ , q ̃ , q ̃ , q ̃ , q ̃ and q ̃ , is the linguistic information r r r r r r available for the values of the uncertain0 parameters p30 , q0 , 4q1 , q2 , q3 and q4 , is the 0 1 2 “around the nominal value of Table 1”,nominal by using value intervalofarithmetic (affine linearization) [17,19,20]; the linguistic information “around the Table 1”, by using interval arithmetic linguistic information can be represented as a fuzzy set with triangular membership functions, where (affine linearization) [17,19,20]; the linguistic information can be represented as a fuzzy set ` ˘ 4 4 7 , 5 ˆ 10 p̃0 = tri(1.7 × 107membership , 5 × 107 , 1.87 × 108 ) , q̃0 where = tri(5.5,25.4,200) q̃1 ˆ=10 tri(1.56 ×710 , 5.08 × r0 “ tri , 1.7 r, 02.5 “ with triangular functions, p , 1.87 ˆ 10×8 10 , q ` ˘ ` ˘ 5 4 44 4 4 4 44 44 5 4 10p5.5, ) , q̃25.4, tri(3.9q 10 , 6.17 × 10 , 9.1 × 10 ) , ,q̃2.5 10tri , 1.95 andˆ 10 q̃4 =, r× r2× “ tri ˆ 10 10 , q 3.9× ˆ10 10 ,, 2.2 6.17׈10 10), 9.1 2 = 200q, 3 =ˆtri(1.19 1 “ tri 1.56 ˆ 10 , 5.08 ` ˘ 4 (Figure 3).4 For α-cut 4 =and 1, we a nominal condition; 0, we= obtain rtri(655,691.7,764) r4 obtain q q “ tri p655, 691.7, 764q (Figurefor3).α-cut For =α-cut 1, we 3 “ tri 1.19 ˆ 10 , 1.95 ˆ 10 , 2.2 ˆ 10 maximum uncertainty. The fuzzy numbers their ownuncertainty. confidence level can be obtain a nominal condition; for α-cut = 0,correspond we obtain to maximum The α-cut fuzzyand numbers interpreted as distributions. Finally, nonlinear HEV system translatedistributions. into a fuzzy correspond to possibility their own confidence level α-cutthe and can be interpreted ascan possibility parametric uncertain system with a degree of confidence of α ∈ [0,1]. 𝐺𝑁 (𝑠, 𝑞𝑖 (𝛼 = 1)) =
5 × 107 𝑠 5 + 691.7𝑠 4 + 1.95 × 104 𝑠 3 + 6.17 × 104 𝑠 2 + 5.08 × 104 𝑠 + 25.4
𝐺𝐿 (𝑠, 𝑞𝑖 (𝛼 = 0)) = 𝐺𝑈 (𝑠, 𝑞𝑖 (𝛼 = 0)) =
𝑠5
+
764𝑠 4
+ 1.19 ×
1.87 × 108 + 9.1 × 104 𝑠 2 + 2.5 × 105 𝑠 + 200
104 𝑠 3
1.7 × 107 𝑠 5 + 655𝑠 4 + 2.2 × 104 𝑠 3 + 3.9 × 104 𝑠 2 + 1.56 × 104 𝑠 + 5.5
(16)
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Finally, the nonlinear HEV system can translate into a fuzzy parametric uncertain system with a degree of confidence of α P r0, 1s. GN ps, qi pα “ 1qq “
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5ˆ107 s5 `691.7s4 `1.95ˆ104 s3 `6.17ˆ104 s2 `5.08ˆ104 s`25.4
GL ps, qi pα “ 0qq “
1.87ˆ108 s5 `764s4 `1.19ˆ104 s3 `9.1ˆ104 s2 `2.5ˆ105 s`200
GU ps, qi pα “ 0qq “
1.7ˆ107 s5 `655s4 `2.2ˆ104 s3 `3.9ˆ104 s2 `1.56ˆ104 s`5.5
(16) 6 of 14
r00, , q̃q r00, q rq̃11,, q rq̃2 ,2 ,q r3q̃and r4 .q̃4 . Figure3.3.Membership Membershipfunction functionfor forp̃p q Figure 3 and
Thispaper paperfocuses focuses problem of longitudinal speed control HEV The systems. The This on on the the problem of longitudinal speed control for HEVfor systems. designed designed controller only stabilize theparametric fuzzy parametric uncertain system, but satisfy also satisfy controller must not must only not stabilize the fuzzy uncertain system, but also the the performance requirements. performance requirements. 3. Controller Design Design 3. Controller 3.1. 3.1. Optimal-Based Optimal-Based Robust Robust Feedback Feedback Controller ControllerDesign Design Nonlinear Nonlinear dynamic dynamic equations equations can can be be represented represented as as linear linear models models at at specific specific operating operating points. points. When a nonlinear system can be stabilized at different operating points, it is equivalent to stabilizing When a nonlinear system can be stabilized at different operating points, it is equivalent to the parametric uncertain linear model. Consider an uncertain system represented as a system with stabilizing the parametric uncertain linear model. Consider an uncertain system represented as a fuzzy parametric uncertainty, described by following transfer function: system with fuzzy parametricas uncertainty, as the described by the following transfer function: 𝑛−1 (𝛼)𝑠 ̃1 (𝛼)𝑠 + 𝑝̃0s(𝛼) pαq sn´1 +`⋯¨ ¨+¨ 𝑝` ` pr0 pαq prn´𝑝̃1𝑛−1 pr1 pαq 𝐺(𝑠, 𝑝 ̃(𝛼), 𝑞 ̃(𝛼)) = 𝑛 G ps, pr pαq , qr pαqq “ n 𝑠 + 𝑞̃𝑛−1 (𝛼)𝑠n𝑛−1 (𝛼)𝑠 + ⋯ + 𝑞 ̃ + 𝑞̃ (𝛼) 1 s ` qrn´1 pαq s ´1 ` ¨ ¨ ¨ ` qr1 pαq0s ` qr0 pαq
(17) (17)
where 𝑝̃𝑖 (𝛼), 𝑞̃𝑖 (𝛼) represents the fuzzy interval number. The 𝛼-cut confidence is given as 𝛼 ∈ [0,1]. whereFurthermore, pri pαq, qri pαq represents fuzzy interval number. Theisα-cut confidence is given representation as α P r0, 1s . the fuzzy the parametric uncertain system realized in state-space Furthermore, the fuzzy parametric uncertain system is realized in state-space representation by a by a controllable canonical form: controllable canonical form: 0 1 ⋯ 0 0 » fi ⋮ » fi ⋮ ⋮ ⋱ ⋮ 𝑥̇0= [ 01 ] 𝑥 + [0] 𝑢 0 0 1 0 ¨ ¨ ¨⋯ — — ffi (18) .. −𝑞̃0 (𝛼) .. −𝑞̃1 (𝛼). . ⋯ 𝑞̃𝑛−1.. (𝛼) ffi ffi 1 — ... ffi — . . . . . ffi — ffi u — x ` x“— ffi — ffi fl – – 0 fl 0 𝑦 = [𝑝̃00(𝛼) 𝑝̃1¨(𝛼) ¨ ¨ ⋯ 𝑝1̃𝑛−1 (𝛼)]𝑥 (18) r r r pαq ´ pαq ¨ ¨ ¨ pαq q q 1 ´ q 0 1 n ´ 1 The compact representation of Equation (18) is: ı ” y “ pr0 pαq 𝑥̇ = pαq ¨ ¨ +¨ 𝐵𝑢prn´1 pαq x pr1𝐴(𝑞 ̃(𝛼))𝑥 y=C (𝑝̃(𝛼))𝑥
(19)
The compact representation of Equation (18) is: Assume that there exists a nominal value 𝑞𝑛𝑜𝑚 ∈ 𝑞̃(𝛼), such that (𝐴(𝑞𝑛𝑜𝑚 ), 𝐵) is stable; there exists a matrix 𝜙(𝑞̃(𝛼)). The uncertainty in A is represented as: 𝐴(𝑞̃(𝛼))𝑥 − 𝐴(𝑞𝑛𝑜𝑚 )𝑥 = 𝐵 𝜙(𝑞̃(𝛼))
(20)
The fuzzy parametric uncertain system can then be rewritten as: 𝑥̇ = 𝐴(𝑞
)𝑥 + 𝐵𝜙(𝑞̃(𝛼)) + 𝐵𝑢
(21)
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.
x “ A pqr pαqq x ` Bu
(19)
y “ C p pr pαqq x
Assume that there exists a nominal value qnom P qr pαq, such that pApqnom q, Bq is stable; there exists a matrix φpqr pαq). The uncertainty in A is represented as: A pqr pαqq x ´ Apqnom qx “ Bφ pqr pαqq
(20)
The fuzzy parametric uncertain system can then be rewritten as: .
x “ Apqnom qx ` Bφ pqr pαqq ` Bu
(21)
The problem of designing a robust controller for a system with fuzzy parametric uncertainty lies in finding a feedback control law u “ ´ kx such that the closed loop system: .
x “ Apqnom qx ` Bφ pqr pαqq ´ Bkx
(22)
is stable for all α P r0, 1s. For the system with fuzzy parametric uncertainty in Equation (19), let the cost function be designed as: ż8´ ¯ J“ x T Fx ` x T x ` u T Ru dt (23) 0
where F is an upper bound on the uncertainty. Now, the aforementioned robust control problem can be translated into an optimal control problem by using an LQR approach. Generally, the weighting matrices Q and R are often determined arbitrarily or based on trial and error. In this study, we assume the uncertain system φpqr pαq) is bounded; the upper bound on F can be written as: φ pqr pαqqT φ pqr pαqq ď F
(24)
When Q “ rF ` Is, the cost function is rewritten as: ż8´ ¯ J“ x T Qx ` u T Ru dt
(25)
0
The LQR optimal control problem involves finding the optimal feedback gain u “ ´ kx that minimizes the cost function. When there exists a feedback control law u “ ´kx, such that Equation (22) is stable for all qr pαq , α P r0, 1s, the design of a robust controller is completed. For a system with fuzzy parametric uncertainty, the solution to the LQR problem is the solution to the robust control problem. The following proposition demonstrates how to determine the weighting matrix Q in the LQR problem. “ `‰ For α “ 0, consider the maximum uncertainty described by qi P q´ i , qi . For α P r0, 1s, the “ ´ ‰ “ ‰ ´ ` uncertainty qi pαi q , q` i pαi q can be written ” as any value in qi , qıi . For the sake of demonstration, assume that the nominal value is qnom “ system Bφ pqrq can be written as: » — — — — –
0 .. . 0 ´qr0
1 .. . 0 ´qr1
¨¨¨ 0 .. .. . . ¨¨¨ 1 ¨ ¨ ¨ ´qrn´1
fi
»
ffi — ffi — ffi ´ — ffi — fl –
0 .. . 0 ´q´ 0
q´ 0
1 .. . 0 ´q´ 1
q´ 1
¨¨¨
q´ n ´1
¨¨¨ 0 .. .. . . ¨¨¨ 1 ¨ ¨ ¨ ´q´ n ´1
The maximum uncertainty φ pqrq is bounded by:
fi
. In Equation (21), the uncertain
»
ffi — ffi — ffi “ — ffi — fl –
0 .. . 0 1
fi » ffi — ffi — ffi — ffi — fl –
r q´ 0 ´ q0 r1 q´ ´ q 1 .. . q´ ´ qrn´1 n ´1
fiT ffi ffi ffi ffi fl
(26)
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» — — F “ φT φ “ — –
“ ` ‰“ ` ‰ q0 ´ q´ q0 ´ q´ ¨¨¨ 0 0 .. .. . . ” ı“ ` ´ ` ´‰ q n ´1 ´ q n ´1 q 0 ´ q 0 ¨¨¨
ı “ ` ‰” ` ´ q0 ´ q´ q ´ q 0 n ´1 n ´1 .. ” ı.” ı ` ´ ´ q n ´1 ´ q n ´1 q ` n ´1 ´ q n ´1
fi ffi ffi ffi fl
(27)
Let Q “ rF ` Is, and designate Q as the cost function for the LQR optimal control problem. With the feedback control law u “ ´ kx, the characteristic equation of the closed loop system in Equation (22) can be written as: sn ` rk n ` qrn´1 s sn´1 ` ¨ ¨ ¨ ` rk2 ` qr1 s s ` rk1 ` qr0 s “ 0
(28)
Kharitonov’s theorem can be used to determine whether the interval polynomial is stable. If and only if all four Kharitonov extreme characteristic polynomials have roots in the left-half plane (LHP), the optimal feedback controller design is a solution to stabilize all of the systems for various values of α P r0, 1s. For the linear quadratic tracking (LQT) problem, we cannot use the aforementioned algorithm . directly. We must augment another system state. Define the augment system state x I ptq “ e ptq “ r ptq ´ y ptq, and augment Equation (18) as: «
.
x . xI
ff
« “
A ´C
0 0
ff «
x ptq x I ptq « X“
ff
« `
.
x . xI
Bu 0
ff
« u ptq `
0 I
ff r ptq
ff
Then, the cost function in Equation (25) will be rewritten as: ż8´ ¯ T J“ X QX ` u T Ru dt
(29)
(30)
0
The tracking problem can be transformed into a stabilization problem. The block diagram for the 8 of 14 HEV longitudinal speed control with a robust feedback controller is shown in Figure 4.
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Figure4.4.Block Blockdiagram diagramfor forthe theHEV HEVlongitudinal longitudinalspeed speedcontrol control with with robust robust feedback feedback controller. controller. Figure
3.2. Fuzzy Fuzzy Logic Logic Forward Forward Compensator Compensator Design Design 3.2. For accurate accurate speed that cancan exhibit robustness in stability, as well For speed tracking, tracking,we weneed needa acontroller controller that exhibit robustness in stability, as as in tracking performance. In Section 3.1, we complete the design of the optimal linear quadratic (LQ) well as in tracking performance. In Section 3.1, we complete the design of the optimal linear feedback controller with robustness. However, neglecting the nonlinearities of the system mayof result quadratic (LQ) feedback controller with robustness. However, neglecting the nonlinearities the in an LQ controller with poor tracking performance. Combining the forward and feedback controllers system may result in an LQ controller with poor tracking performance. Combining the forward and can therefore satisfy can the performance requirements. feedback controllers therefore satisfy the performance requirements. 3.2.1. Forward Forward Compensator Compensator Fc Fc Design Design 3.2.1. As shown about thethe architecture is that the response to thetoreference input As shown in in Figure Figure4,4,a aconcern concern about architecture is that the response the reference is driven only by the integrated error. There is no forward path from the reference input to the system, input is driven only by the integrated error. There is no forward path from the reference input to
the system, and the transient response may be slow. This drawback can be mitigated by adding the forward compensator 𝐹𝐶 shown in Figure 5. The control law for the revised implementation can be written as: 𝑢 = −𝐾(𝑥̌ + 𝐹𝑐 × 𝑒) − 𝐾𝐼 𝑋𝐼
(31)
feedback controllers can therefore satisfy the performance requirements. 3.2.1. Forward Compensator Fc Design As shown in Figure 4, a concern about the architecture is that the response to the reference Energiesis 2016, 9, 290only by the integrated error. There is no forward path from the reference input 9 ofto 15 input driven the system, and the transient response may be slow. This drawback can be mitigated by adding the forward compensator 𝐹𝐶 shown Thedrawback control law revised by implementation can be and the transient response may in beFigure slow. 5. This canfor bethe mitigated adding the forward written as: compensator F shown in Figure 5. The control law for the revised implementation can be written as: C
𝑢 = −𝐾(𝑥̌ + 𝐹𝑐 × 𝑒) − 𝐾𝐼 𝑋𝐼
(31)
u “ ´K pxˇ ` Fc ˆ eq ´ K I X I
(31)
Figure 5. 5. Block Blockdiagram diagramfor forthe theHEV HEVlongitudinal longitudinal speed speed control control with with forward forward compensator. compensator. Figure
Note Note that that the the tracking tracking error error includes includes the the output output of of the the integrator integrator and and the the state state feedback feedback components. Thus, this thistype type of 2-DoF approach haspotential the potential to enhance the performance components. Thus, of 2-DoF approach has the to enhance the performance problems problems in the original implementation. The process entire design process controller of the forward identified identified in the original implementation. The entire design of the forward (FC ) is controller ) is described as follows. described (𝐹 as𝐶follows. Assume state XX of ofthe theHEV HEVsystem systemininFigure Figure 4 can partitioned into 𝑥 and 𝑥̌. 𝑥the areparts the ˇ x are Assume the state 4 can bebe partitioned into x and x. parts weabout care about for tracking (𝑇𝑋)we that we assume are directly available = 𝑥 and = 𝐶𝑋, andthe x̌ we care for tracking (TX) that assume are directly available from y from “ x “𝑦 CX, xˇ are are not for care about (xfor tracking (𝑥̌ = 𝑇̅𝑋).ofThe matrices 𝑇 and 𝑇̅ selectors can be ˇ “ TX). partsthe weparts do notwe caredoabout tracking The matrices T and T can beofconsidered considered selectors diagonals of one andalways zero, but not Assume always take thiscomplementary form. Assume with diagonals of onewith and zero, but they do not takethey thisdo form. T is the ` ˘ ̅ is theofcomplementary 𝑇matrix matrix 𝑇 (𝑇 +speed) 𝑇̅ = 𝐼)x and state (vehicle speed) .sT q; of T T ` T “ I and stateof(vehicle is part of state vector Xprx,xv,is. .part letstate e “ rvector ´ CX, 𝑋([𝑥, 𝑣, … ]𝑇 ); let 𝑒u = − 𝐶𝑋, the(31) control input 𝑢as:in Equation (31) is rewritten as: the control input in𝑟Equation is rewritten = −𝐾(𝑥 𝐶𝑋 + 𝐹𝑐 × 𝑟) − 𝐾𝐼 𝑋𝐼 u “ 𝑢´K pxˇ ´̌ −Fc𝐹𝑐ˆ×CX ` Fc ˆ rq ´ K I X I To proceed, define 𝐹𝐶 × 𝐶 = −𝑇; then: To proceed, define FC ˆ C “ ´ T; then:
(32)
(32)
𝑥̌ − 𝐹𝑐 × 𝐶𝑋 = 𝑇̅𝑋 − (−𝑇)𝑋 = 𝑋
(33)
xˇ ´ Fc ˆ CX “ TX ´ p´Tq X “ X Finally, the control input becomes:
(33)
Finally, the control input becomes: 𝑢 = −𝐾(𝑋 + 𝐹𝑐 × 𝑟) − 𝐾𝐼 𝑥𝐼
(34)
u “ ´K pX ` Fc ˆ rq ´ K I x I
(34)
Using T “ ´FC ˆ C ensures avoiding double counting in the feedback. Without loss of generality, we can use: Fc “ ´γC T , γ ă 1 (35) The entire closed-loop dynamics of the 2-DoF controller system is expressed as: «
.
X . XI
ff
« “
A ´ BK ´C
ff « ´BK I 0
X XI
ff
« `
ff ´BKFc I
r
(36)
Because of the nonlinear properties of HEV speed control, through Equations (16) and (35), the matrix C is not constant, but the fuzzy parametric uncertainty. To obtain optimal system performance, the weighting (BKFC ) of forward compensator FC should not be constant, either. The next section discusses how to use fuzzy logic controllers (FLCs) to tune the BKFC of forward compensator FC . 3.2.2. The Weighting (BKFc) of Forward Compensator Tuning by FLC An FLC is used to design the tuning of forward compensator FC . The inputs to the FLC are the error (e “ r ´ y) and change in error (ce), and the output variable is the BFC . The input triangular
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membership functions are designed according to five linguistic terms: positive big (PB), positive (P), zero (Z), negative (N) and negative big (NB). The output linguistic levels are assigned as small (S), medium small (MS), medium (M), medium big (MB) and big (B). Twenty-five rules for using the trial and error method are shown in Table 2. The entire system diagram is depicted in Figure 6. Table 2. The rule base of the FLC: positive big (PB), positive (P), zero (Z), negative (N), negative big (NB), small (S), medium small (MS), medium (M), medium big (MB) and big (B). Error
NB
N
Z
P
PB
MS M MB MB B
M MB MB B B
Change in Error NB N Z P PB
S S MS MS M
S MS MS M MB
MS MS M MB MB
Figure 6. Block diagram for the HEV longitudinal speed control with the 2-DoF fuzzy controller. FLC, fuzzy logic controller.
3.3. Design Procedure We summarize the design procedure for the proposed robust 2-DoF controller as follows. 1. 2.
3. 4. 5. 6.
Step 1: Linearize the nonlinear HEV model at specific operating points and represent as an uncertain interval model. Step 2: The uncertain interval parameters are represented by a fuzzy number qr with membership function α “ µ pqrq P r0, 1s. Translate the uncertain interval system into the fuzzy parametric uncertain system. Step 3: For α “ 0, the maximum uncertain interval of the system is translated into the weighting matrix Q of the linear quadratic tracking (LQT) servo problem. Step 4: Design an optimal controller for α “ 0, which can be considered as the worst case condition. Step 5: Use Kharitonov’s theorem to test whether the optimal feedback controller is a solution to stabilize all of the systems for various values of α P r0, 1s. Step 6: Design the FLC-based forward compensator to satisfy the performance requirements.
4. Simulation Results This section illustrates how to design robust longitudinal speed control systems for HEVs based on a 2-DoF design method. Simulation of Optimal Based Robust Feedback Controller In Equation (16), the transfer function of the HEV system with parametric variation is expressed as:
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r psq “ G
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r1.87 17s ˆ 108 s5 ` r655 764s s4 ` r1.19 2.2s ˆ 104 s3 ` r3.9 9.1s ˆ 104 s2 ` r1.56 25s ˆ 104 s ` r5.5 200s
(37)
For convenience, we translate this equation into its controllable canonical form, and the transfer function of the parametric uncertain system in Equation (21) is expressed as: fi » fi 0 0 1 0 0 0 ffi — ffi 0 0 1 0 0 — 0 ffi ffi ffi — ffi 0 0 0 1 0 ffix ` — 0 ffiu ffi — ffi – 0 fl fl 0 0 0 0 1 4 4 4 1 ´ r5.5 200s ´ r1.56 25s ˆ 10 ´ r3.9 9.1s ˆ 10 ´ r1.19 2.2s ˆ 10 ´ r655 764s loomoon loooooooooooooooooooooooooooooooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooooooooooooooooooooooooooooooon
» — — .— x— — –
Bu
A
” C“
r1.87
(38)
17s ˆ 108
ı 0 0
0 0
In order to satisfy the requirement of LQT, we augment the system Equations (38) to (29). Using Equation (20), the maximum uncertainty φ pqrq is bounded by: ” φ pqrq “
1.3 ˆ 108
ı 194.5
234, 400
52, 000
10, 100
109
(39)
The upper bound of the uncertainty in Equation (24) can be expressed as F “ φ T φ. Then, the LQT weighting matrix Q can be written as rF ` Is. Considering R “ 1000 and solving the feedback control gain by using the LQT approach, we obtain K LQT as follows: ” ı K LQT “ 1.38 ˆ 108 1.45 ˆ 107 8.99 ˆ 105 29096.8 98.5 ´3.45 (40) With the feedback control law, all four of Kharitonov’s extreme characteristic polynomials over all of the operating conditions can be expressed as: k1 psq “ s6 ` 753.5s5 ` 40, 997s4 ` 9.9 ˆ 105 s3 ` 1.47 ˆ 107 s2 ` 1.38 ˆ 108 s ` 5.85 ˆ 107 k2 psq “ s6 ` 862.5s5 ` 51, 097s4 ` 9.3 ˆ 105 s3 ` 1.45 ˆ 107 s2 ` 1.38 ˆ 108 s ` 6.45 ˆ 108 k3 psq “ s6 ` 862.5s5 ` 40, 997s4 ` 9.3 ˆ 105 s3 ` 1.47 ˆ 107 s2 ` 1.38 ˆ 108 s ` 5.85 ˆ 107
(41)
k4 psq “ s6 ` 753.5s5 ` 51, 097s4 ` 9.9 ˆ 105 s3 ` 1.45 ˆ 107 s2 ` 1.38 ˆ 108 s ` 6.45 ˆ 108 We verify the conditions for robust stability in Equation (41) and find the entire fuzzy uncertain system to be stable. Furthermore, to demonstrate that the proposed feedback approach can consider robustness, we evaluate the proposed approach by comparing H8 methods [12] with different cruise-tracking regimes. The step responses of the system with two kinds of feedback controller are shown in Figure 7. The comparison of the time domain performance indices in terms of the overshoot (OS), rise time (RT), delay time (DT) and settling time (ST) that correspond to transient state characteristics is shown in Table 3. Both control schemes stabilize the entire uncertain system. The simulation results show higher overshoot and longer settling times in the H8 controller. However, the proposed robust feedback controller yields longer response times.
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Figure 7. Step response for the HEV speed control (a) with the 𝐻∞ controller and (b) with the Figure 7. for HEV speed control (a) with the H and (b) with robust 8 controller Figure 7. Step Step response response forthe the HEV speed control (a) with the 𝐻∞ controller and (b)the with the robust feedback controller. feedback controller. robust feedback controller. Table 3. The comparison of the time domain performance index: overshoot (OS), rise time (RT), Table 3. comparison ofofthe time domain performance index: overshoot (OS),(OS), rise time Table 3. The The comparison the time domain performance index: overshoot rise (RT), time delay (RT), delay time (DT) and settling time (ST). time (DT) and settling time (ST). delay time (DT) and settling time (ST).
Controller Controller Controller
Condition Condition Condition Nominal Nominal Nominal 𝐻∞ controller Lower Bound 𝐻∞ Hcontroller Lower LowerBound Bound 8 controller Upper Bound UpperBound Bound Upper Nominal Nominal Nominal Proposed feedback controller Lower Bound Proposed feedback controller Lower LowerBound Bound Proposed feedback controller Upper Bound Upper Bound Upper Bound
OS (%) OS (%) OS (%) 1.369 1.369 1.369 0 0 25.13 25.13 25.13 0 0 13.37 13.37 13.37 0 00
RT (s)
DT (s)
ST (s)
1.59 1.59 2.86 2.86 2.86 1.92 1.92 1.92
0.63 0.63 0.2 0.2 0.2 1.13 1.13 1.13
1.7 1.7 0.31 0.31 0.31 5.27 5.27
0.61
2.172.17 5.27 5.275.27 8.99 8.998.99
0.61 0.61 0.24 0.24 0.24 1.67 1.67 1.67
2.8 2.80.72 0.720.72 8.77 8.77 8.77
RT (s) DT DT (s) RT (s) (s)(s) ST ST (s) 1.59 0.63 2.17
1.7
5.27
2.8
Because the speed does not generally change stepwise in practical applications, trapezoidal speed not generally change stepwise in tracking practical applications, trapezoidal speed Because the speed does not generally change stepwise in practical trapezoidal speed profilesthe are useddoes instead. In this case, the accurate of theapplications, reference input during profilesprofiles are used instead. In this theAgain, accurate tracking the reference during acceleration speed are used instead. In this case, the accurate tracking ofand theinput reference input during acceleration and deceleration is case, crucial. the robust of properties tracking performance of and deceleration is crucial. Again, the robust properties and tracking performance of the two control acceleration and deceleration is crucial. Again, the robust properties and tracking performance of the two control schemes are also assessed. The scenario of wide-range cruise-tracking performance schemes are alsoschemes assessed. scenario of wide-range performance is set as follows: the two areThe also assessed. The of cruise-tracking performance is set as control follows: the vehicle speed is 20 m/s forscenario the firstcruise-tracking 6 s;wide-range there is then an acceleration at the rate 2 the vehicle speed is 20 m/s for the first 6 s; there is then an acceleration at the rate of 6 m/s ; thence, is set as follows: the vehicle speed is 20 m/s for the first 6 s; there is then an acceleration at the rate 2 of 6 m/s ; thence, the vehicle runs at a constant speed of 32 m/s for 8 to 12 s. For a time interval of 12 2; thence, the614 vehicle runs atthe aruns constant speed m/s for 8 to4 12 a for time interval of 12 tofrom 14interval s, the vehicle of m/s vehicle atofa 32 constant speed of s. 32 m/s 8 24 to 12 s. For a time of 2,For to s, the vehicle in a runs decelerating mode at m/s reaching m/s; finally, 14 to 2012 s, 2 , reaching 24 m/s; finally, 2 runs in a decelerating mode at 4 m/s from 14 to 20 s, the vehicle runs at to 14 s, the vehicle runs in a decelerating mode at 4 m/s , reaching 24 m/s; finally, from 14 to 20 s, the vehicle runs at a constant speed of 24 m/s. The wide-range cruise-tracking responses of thea constant speed 24am/s. The wide-range of the system with two types of the vehicle runs at constant speed of 24cruise-tracking m/s. wide-range cruise-tracking responses of the system with twooftypes of feedback controllers are The shown inresponses Figure 8. feedback controllers areofshown in Figure 8. system with two types feedback controllers are shown in Figure 8.
Figure 8. Wide range cruise tracking response for the HEV speed control (a) with the 𝐻∞ controller Figure Wide cruise tracking response for the HEV speed control (a) with the 𝐻∞ controller and (b)8. with therange robust feedback controller Figure 8. Wide range cruise tracking response for the HEV speed control (a) with the H8 controller and (b) with the robust feedback controller and (b) with the robust feedback controller.
Although both the 𝐻∞ feedback Although the 𝐻∞ feedback because of highboth nonlinearities in the because of high nonlinearities in the
controller and proposed LQ feedback controller are robust controller andwhen proposed LQ feedback controller are robust HEV system, the range of operation is complex, the HEV system, when the range of operation is complex, the
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Although both the H8 feedback controller and proposed LQ feedback are robust single-type feedback controller deteriorates in tracking performance. Figure 8controller shows that the 1-DoF Energies 2016, 9, 290 nonlinearities in the HEV system, when the range of operation is complex, 12 of 14the because of high fixed-value controller cannot yield adequate tracking performance. To improve the slow transient single-type feedback controller deteriorates in tracking performance. Figure 8 shows that theit1-DoF response and enhance the tracking performance of the 1-DoF robust feedback can be single-type feedback controller deteriorates in tracking performance. Figure 8 showscontroller, that the 1-DoF fixed-value controller cannot yield adequate tracking performance. To improve thecontroller slow transient paired with the forward compensator 𝐹 in Figure 6. The proposed 2-DoF fuzzy has 𝐶 fixed-value controller cannot yield adequate tracking performance. To improve the slow transient an response and enhance the tracking performance of the 1-DoF robust feedback controller, it can be additional component for the effects of 1-DoF the response. response and enhance to thecompensate tracking performance of the robust feedback controller, it can be pairedBased with the forward 6. Theand proposed 2-DoF fuzzy controller has an C in Figure compensator 6, the FLC F𝐹is a Figure two input one output system. For successful paired with on the forward compensator 𝐶 in Figure 6. The proposed 2-DoF fuzzy controller has an additional component to compensate for the effects of the response. implementation of thetofuzzy forward weresponse. must estimate the maximum excursion of additional component compensate forcompensator, the effects of the Based on Figure 6, the FLC is a two input and one output system. For successful implementation the input the fuzzy error, the universe of discourse Basedand onoutput Figuresignals 6, theof FLC is a controller. two inputUsing and trial one and output system. For successful ofin the fuzzy forward compensator, we must estimate the maximum excursion of the input and covers output implementation of the function fuzzy forward compensator, we must estimate maximum of fuzzy membership designed for the error, the change of the error and the excursion output a signals of the fuzzy controller. Using trial and error, the universe of discourse in fuzzy membership the input and output signals of[0.25,7.5], the fuzzy respectively. controller. Using trial and error, the universe of discourse range of [–10,20], [–25,25] and function designed for the error, the of the error andthe the output of covers aand range [–10,20], [–25,25] in fuzzy membership function designed for error, change the output covers The tracking performance of change the designed 2-DoF controller is error analyzed by of comparing it toa that and [0.25,7.5], respectively. range [–10,20], [–25,25] [0.25,7.5], respectively. of the ofself-tuning fuzzy and logic PID (STF-PID) controller in [12]. The integral error performance The tracking performance of the designed 2-DoF is comparing to that that The of The tracking performance of the designed 2-DoF controller is analyzed analyzed by comparing itit to indices are used to obtain greater insight into HEV controller tracking performance with two controllers. the fuzzyfuzzy logic PID (STF-PID) controller in [12]. The integral error performance indices arein ofself-tuning the self-tuning logic PID (STF-PID) in [12]. The error performance wide-range cruise-tracking responses of the controller system with two typesintegral of controllers are shown used to obtain greater insight into HEV tracking performance with two controllers. The wide-range indices are used to obtain greater insight into HEV tracking performance with two controllers. The Figure 9. The comparison of input energy for the period from t =5 s to t = 16 s in the nominal wide-range cruise-tracking ofwith the system withoftwo types ofare controllers areFigure inThe cruise-tracking of responses the system two types controllers shown in condition are responses shown in Figure 10. Table 4 shows the performance index analysis ofshown the 9. integral Figure 9. The comparison of energyfrom for the t =5 to t = 16condition s in the nominal comparison of input forinput the period =5period s to t =from 16 s in thes nominal are shown squared error (ISE) energy and integral absolute error t(IAE). shown in Figure 10. Table 4 shows performance index analysis oferror the integral incondition Figure 10.are Table 4 shows the performance indexthe analysis of the integral squared (ISE) and squared error (ISE) and integral absolute error (IAE). integral absolute error (IAE).
Figure 9. Wide range operation response for the HEV speed control (a) with STF-PID and (b) with Figure Wide range operationresponse responsefor forthe theHEV HEVspeed speedcontrol control(a) (a)with withSTF-PID STF-PIDand and(b) (b)with withthe Figure 9.9.Wide range operation the 2-DoF robust controller. the 2-DoF robust controller. 2-DoF robust controller. Corresponding change in input energy (Nominal Condition) 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1
Corresponding change in input energy (Nominal Condition)
STF-PID STF-PID 2 DoF Controller 2 DoF Controller
Ea Ea(V) (V)
0.05
0.05
0
0
-0.05 -0.05 -0.1 -0.1 -0.15 -0.15 -0.2 -0.2 -0.25 -0.25 66
8 8
10 10 12 12 Time (s) (s) Time
14 14
16
16
Figure Corresponding change in input input energy energyfor forthe theperiod periodfrom fromt =t = s to t = 16 s. Figure10. 10. 5t s5 Figure 10. Corresponding Correspondingchange changeinin input energy for the period from =to 5 st =to16t =s.16 s.
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Table 4. The comparison of the integral error performance index. IAE, integral absolute error; ISE, integral squared error; STF, self-tuning fuzzy logic. IAE
ISE
Controller
Nominal
Lower
Upper
Nominal
Lower
Upper
STF-PID 2-DoF controller
13.58 7.35
12.94 6.59
13.27 8.15
86.80 44.59
68.77 49.07
88.18 45.84
The proposed 2-DoF controller uses more input energy to adjust the throttle position quickly to satisfy the performance requirement. Although the input energy of the proposed controller (0.0343) is bigger than that of STF-PID controller (0.0078), the performance of the proposed controller is higher than that of the STF-PID controller, as shown in Figure 9 and Table 4. Compared to the STF-PID controller, the maximum tracking errors (IAE, ISE) of the 2-DoF controller are smaller and reach 8.15 and 49.7, respectively. From the practical point of view, the accuracy of speed control is the key technology to improve many important applications of HEVs (E.g. adaptive cruise control, intelligent collision avoidance or car following). Besides, the accuracy of throttle position control can upgrade the fuel economy and reduce the pollutant emission. Furthermore, if the performance requirements concern the factor of input energy, based on the LQT character, the proposed methodology still has the flexibility to adjust the LQT weighting matrix R to reduce the magnitude of input energy. 5. Conclusions In this paper, a 2-DoF robust fuzzy controller is successfully applied to the nonlinear uncertain HEV longitudinal speed control model. First, using the proposed algorithm, the uncertainty intervals of HEV dynamic systems are approximated by fuzzy α-cut coefficients. Subsequently, the maximum uncertainty interval is then translated into the weighting matrix Q of the LQT problem to guarantee that the designed optimal feedback controller is robust under various values of α P[0,1]. The robust stability of the longitudinal speed control is analyzed using Kharitonov’s theorem. Finally, to compensate for the longer response time of the single-type feedback controller, the forward compensator is connected to enhance tracking performance. In contrast to many previously-proposed nonlinear controllers, our controller is easy to understand and implement. The proposed 2-DoF method was successfully applied in speed tracking control of HEVs and can also be extended to general servo control design. Acknowledgments: This paper was supported by the Ministry of Science and Technology, Taiwan (No. MOST 104-3011-E-110-001). Author Contributions: Yi-Horng Lai wrote this manuscript and contacted the editor. Jau-Woei Perng was in charge of the research architecture and revising the English editing in this manuscript. Conflicts of Interest: The authors declare no conflict of interest.
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6. 7. 8. 9. 10.
11. 12. 13. 14. 15.
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