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100: ð31ÞEnergy stored in the hydraulic accumulator was estimated usingEq. (32):Ea ¼ ∫Qapdt: ð32Þ3. Design of the control system3.1. Design of a hierarchal control systemThe overall control structure was designed as shown in Fig. 5.Asmentioned above, the control scheme of the supervisory controlwas simple. The supervisory controller generated the upper andlower limit pressures (pup,plow) for the primary controller, the digitalsignals (V11,V12) for the valve controller, and the speed reference forthe secondary controller. The digital signal V11 was set to one duringacceleration or driving, while V12 was one during deceleration orbraking; however, the valves were never simultaneously set to one.In practical applications, such as vehicle driving and braking, signalsare generated by different pedals, controlled by the driver, so the above strategy was acceptable. In addition, the supervisory controllergenerated the activated signal and speed reference for the brake con-troller without using regenerative braking. Adjusting the hydraulicpressure via a proportional reducing valve controlled this frictionbrake. A traditional PID controller was employed for the brake con-troller, and the valve controller only amplified the signals (V11,V12)to engage or disengage the valve solenoids. The primary controllerwas designed using the law in Fig. 6, in which u1 was the displace-ment control signal of P1. To achieve high efficiency, pump P1 only op-erated at the maximum or zero displacement.3.2. Adaptive fuzzy sliding mode control (AFSMC) for the secondarycontrollerIn the proposed control scheme, the controlled plant included theflywheel, the secondary unit PM2, and the displacement controlmechanism. Model uncertainties included the inaccuracy of modelingbased on the mathematical model and the variation in the operatingpressure in the supervisory control scheme. The external disturbancewas generated by the friction brake system and the mechanical lossesin the secondary unit. To apply AFSMC, we do not need to know all ofthe parameters, but we should know the general form and order ofthe system. When the speed of the secondary unit and the voltageof the displacement control mechanism were considered as the out-put and input, respectively, the controlled system was a second-order system. From Eqs. (17), (29), and (30), where x1=ω and _ x1 ¼x2 with some manipulations, the system was re-expressed usingEq. (33):( _ x 1 ¼ x2_ x2 ¼ −Cτx1− τJþ Jτ x2 þ ΔpKsvτηTu þ Tex:ð33Þ For simplicity, Eq. (33) could be expressed in general form as:( _ x 1 ¼ x2_ x2 ¼ fx1;x2 þ gu þ d;ð34Þwhere g ¼ ΔpKsvτ ηT,fx1; x2 ðÞ¼ − Cτ x1 þ τJþ Jτ x2 ,d=Tex, and ηT isthe mechanical efficiency of the secondary unit. In the system,Δp>0 while driving, but Δpb0 while braking. Thus, g≠0 but wasnot only positive while working, and |f|≤L(x1,x2). The aim of the sys-tem control was to determine a proper control law so that the fly-wheel could track the reference. The tracking error is defined asfollows:e ¼ e1;e2hiT¼ x1−xd;1; x2−xd;2hiT¼ ω−ωr; _ ω− _ ωr ½ T¼ e; _ e ½ T: ð35ÞA block diagram of the controller is shown in Fig. 7. In this controlscheme, the fuzzy controller had only one input, the sliding surface.The process of the controller design was as follows: first, a sliding sur-face was selected asst ðÞ¼ k1e þ _ e; ð36Þwhere k1 and ωr were the positive gain and the reference speed, re-spectively. For a traditional sliding mode control SMC,Eqs. (37)–(40) expressed the control law [27], where ueq was thefeedback linearization controller and ur was the reaching controller.u ¼ ueq þ ur; ð37Þueq ¼ 1g−f−d−k1 _ e þ _ x2r ðÞ; ð38Þur ¼ −η1gsgn s ðÞ; ð39Þsgn s ðÞ¼1ifs > 00ifs ¼ 0−1ifs b 0:8<: ð40ÞHowever, the feedback linearization ueq was difficult to imple-ment in the practical application. Thus we proposed a direct adaptivefuzzy sliding mode with bound estimation on scale with the uncer-tainties and the chattering control effort. The design procedure ofthe proposed controller was as follows. We assumed that all parameters of the system were well known. Then, an ideal controllercould be obtained using Eqs. (41)–(43).u ¼ u eq þ u r; ð41Þu eq¼ 1g−f−d−k1 _ e þ € ωr ðÞ; ð42Þu r¼ −η1gsgn s ðÞ: ð43ÞSince the system was not exactly known, the ideal controller wasapproximated as a fuzzy system as follows. The fuzzy rules weregiven in this study in the following form [23]:ithrule : ifswasLi; thenuwasqi;where Li is the linguistic variable of the sliding surface and θi is thesingleton value of the output fuzzy membership function. The mem-bership function of the sliding mode surface input was in the follow-ing form:μLis ðÞ¼ exp − s−δiσi 2 ; ð44Þwhere δi and σi are the center and width of the membership function,respectively. The fuzzy controller was expressed in the form^ ufz s; θ ðÞ ¼ θTξ; ð45Þwhere θ=[θ1,θ2,θ3,…,θm]Tis a parameter vector and ξ=[ξ1,ξ2,ξ3,…,ξm]Tis a regressive vector with ξi defined as:ξi¼ wiP mi¼1wi; ð46Þwherewi is the firing weight of the ith rule. According to the universalapproximation theorem [28], there existed an optimal fuzzy control-ler u*fz(s,θ*)=θ*Tξ such thatu eq ¼ u fz þ ε; ð47Þwhere ε is the estimated error and was assumed to be bounded by|ε|≤E. Employing a fuzzy ^ ufzcontroller in Eq. (45) approximatedufz*, where θ is the estimated vector of θ⁎.