Strojniški vestnik - Journal of Mechanical Engineering 64(2018)10, 579-589 © 2018 Journal of Mechanical Engineering. All rights reserved. D0l:10.5545/sv-jme.2018.5339 Original Scientific Paper Received for review: 2018-03-07 Received revised form: 2018-05-29 Accepted for publication: 2018-07-02 Force Control of Hydraulic Actuators using Additional Hydraulic Compliance Job Angel Ledezma Perez - Edson Roberto De Pieri - Victor Juliano De Negri* Federal University of Santa Catarina, Brazil This paper presents an approach for improving the performance of hydraulic force control systems by adding hydraulic compliance. Typically, force control systems require a flexible coupling system, such as a spring between the actuator and the load, to achieve a non-oscillatory response. To avoid the use of springs, this study proposes the use of special hoses to add compliance to the system hydraulically. With this approach, a direct connection between the actuator and the load is feasible, simplifying the mechanical assembly and saving physical space in the axial direction. An analytical model is proposed to estimate the appropriate hydraulic stiffness value and to select a commercial hose based on catalogue data. Moreover, a robust controller based on quantitative feedback theory is designed to improve the stability, performance and disturbance rejection of the force control system. The system performance is demonstrated by dynamic simulation and experimental results. Keywords: hydraulic force control, hydraulic compliance, high expansion volumetric hose, QFT-based control Highlights • A hydraulic force control system with added hydraulic compliance is presented. • Passive compliance is provided with the use of hydraulic hoses instead of mechanical springs. • An analytical procedure for selecting and sizing hydraulic hoses is proposed. • The presented achievements are validated by dynamic simulation and experimental results. 0 INTRODUCTION Several industrial applications require controlled force for handling materials or machining tasks in which the end effector is in contact with the piece or equipment. In most cases, position/trajectory control is required at the same time as contact force control. Assembly of mechanical components, polishing and grinding of complex pieces, milling, and endurance tests are some examples for which there are demands for force-controlled actuators in substitution to manual work [1] and [2]. As discussed in [3], the problem of force control is more complex than the problem of position control, which often makes its practical implementation more difficult. In contrast, to force control systems, the dynamics of the load is not included in the closed-loop in the case of positioning systems, and thus good performance and disturbance rejection are more easily achieved. One way to overcome the inherent problems associated with force control systems is by increasing the compliance. System compliance can be controlled or modified in active or passive ways [4] and [5]. Active compliance is obtained with force-based feedback control (software), allowing more flexibility to perform tasks for which compliance is required. Typically, this approach requires additional measuring for control and collision detection [6]. In contrast, passive compliance is typically obtained by introducing mechanical elastic components (hardware) between the actuator and the environment. This approach enables energy storage, and it is appropriate for safety applications [7]. The series elastic actuator (SEA) represents one well-known example of a passive compliance actuation system [8]. This kind of actuator has been applied to robotics and biomechanics due to its simplicity and good performance in force control [9] and [10]. In an SEA, a spring is placed between the actuator and the load, providing good force fidelity, low output impedance, and shock tolerance capability. However, the system bandwidth is reduced due to the use of this compliant element [8]. In addition to the use of a spring to increase the system compliance and improve the system performance, researchers have employed a spring as a simplified representation of the load. Good force responses have been obtained using control techniques such as nonlinear PD (proportional-derivative), fuzzy PID (proportional-integral-derivative), QFT (quantitative feedback theory), and backstepping [11] to [14]. References [15] and [16] included additional load motion measurements in order to cancel the disturbance characteristic and obtain better force tracking. In contrast, the fluid compressibility in hydraulic actuators introduces a spring effect that is *Corr. Author's Address: Federal University of Santa Catarina, Mechanical Engeeniring Department, LASHIP, Campus Universitario, Trindade, 88040-900, Florianopolis, SC, Brazil, victor.de.negri@ufsc.br 579 Strojniski vestnik - Journal of Mechanical Engineering 64(2018)10, 579-589 characterized by the hydraulic stiffness. This can be shaped by using hydraulic capacitive components such as flexible hoses or accumulators. The use of accumulators connected to the cylinder chambers has been described in conference papers and patents [17] to [19]. However, these publications do not take into account the mathematical modelling to determine the hydraulic compliance needed to achieve the force control requirements and select the hydraulic capacitive component. In this context, the present paper introduces a pure hydro-elastic actuator (PHEA) that uses only hydraulic components to reduce the transmission stiffness. Mathematical modelling and analysis are carried out, and a procedure for sizing the required hydraulic components in order to increase the compliance is determined. Instead of accumulators, the use of hoses with high volumetric expansion (HVE) is investigated. Considering the effects of compliance addition and the possible parametric variations of the system, this paper considers the use of a robust controller based on the quantitative feedback theory (QFT) technique [20]. Furthermore, the load is assumed to be continuously in contact with the actuator, and the load-displacement acts as a disturbance over the force control system. The following section provides a brief description of the hydraulic force control system. The nonlinear and linear modelling is presented in Section 2 followed by the proposed method for determining the hydraulic stiffness. The force controller based on the QFT technique is then designed, and simulation and experimental results are discussed. The last section presents the conclusions. 1 THE FORCE CONTROL SYSTEM A hydraulic actuation system is very stiff due to the low fluid compressibility. Direct contact between the cylinder and the load would cause the entire system to be stiffer, and even a small control signal sent to a servovalve could generate large force variations due to the high open-loop gain. Reference [21] emphasizes the advantages of including a spring in a force control system. In [8], the procedure of the spring selection for the SEA was improved by using two types of linear actuators: an electromechanical and a hydraulic actuator. In both cases, the introduction of a compliant element between the actuator and the load decreases the output impedance. Additionally, the measurement of the spring compression is used to calculate indirectly the force applied over the load (Fig. 1a). a) b) Fig. 1. Compliance addition in a hydraulic force system using a) spring and b) HVE hoses The use of the spring limits the rate of the force applied by the hydraulic actuator, ensuring some degree of isolation between the hydraulic system and the movement of the load (environment) and maintaining a stable and robust applied force. The use of a hydraulic compliant component instead of a spring is proposed. Fig. 1b shows an option using high volumetric expansion (HVE) hoses between the servovalve and cylinder. The use of accumulators instead of hoses could also be possible for similar purposes. 2 MATHEMATICAL MODELING Two mathematical models are explained in this section: a nonlinear model, which is used for simulation in order to obtain force responses; and a linear model, used for the controller design and hose sizing. Both models are based on Fig. 1b. 2.1 Nonlinear Modelling of the Hydraulic System 2.1.1 Servovalve Modelling In the design of the hydraulic circuit, a symmetrical servovalve is used (Fig. 2) [22]. The dynamic relationship between the input control signal (Uc) and the spool displacement, represented by an equivalent voltage (^Csp), can be approximated by a second-order function: UC = 1 d2Ucsp , l^v dU, Csp dr dr U Csp: (1) where ^v is the natural frequency of the valve and represents the damping ratio of the valve. The flow rate through the valve, including the effects of internal leakage can be described by the 580 Ledezma, J.A. - De Pieri, E.R. - De Negri, V.J. Strojniski vestnik - Journal of Mechanical Engineering 64(2018)10, 579-589 following equations [23], where a control signal range of -10 V to +10 V is assumed: for Ucsp > 0: IvA = 4vB = U, Csp KVP U V UCn - K„; f UC K +K (3) for Ucsp < 0: ( IvA =■ K„ U Csp UC ■ Kv„: 4pä~ Pt +kvinpvPs - Pa > ( (4) qvB = K„ U Csp I U - K, Pb Pt: (5) where qvA and qvB are the flow rates at ports A and B, respectively; pA andpB are the pressures in lines A and B, respectively; pS and pT represent the supply and the reservoir pressures, respectively; UCn is the nominal control signal. Kvp is the partial flow coefficient of the valve; and Kvinp is the internal partial leakage coefficient. «»Al IflvB Pa T t pB èé .....ta.......ffls *-> ,i rü iw P s1 'PI Fig. 2. F/ow rates and pressures at the servovalve 2.1.2 Cylinder Modelling Considering Fig. 3 and applying the continuity equation for each cylinder chamber yields: dxp dt and q = A ^ + Ya^a. qvA = Aa - + ße dt ' (6) q _ A dip -ViêPL qvB _ Ab dt - (7) Pe dt where Aa and AB are the piston areas in chambers A and B, respectively; xp represents the piston displacement, and fie is the effective bulk modulus. Fig. 3. Variables and parameters of the cylinder The displacement of the rod can be modelled using Newton's second law: F - Ffr - F = M d2 A dt2 (8) where FH = pAAA - p^4B is the hydraulic force, Ffr represents the friction force, Fe is the applied force, and M is the piston mass. In turn, the force applied over the load can be represented as: Fe = KS (p - ) (9) such that KS is the load cell stiffness and xL represents the load displacement. 2.2 Linear Modelling of the Hydraulic System 2.2.1 Servovalve Modelling The flow rate through the valve (Eqs. (2) to (5)), linearized for any UCspi andpLi, can be represented as: where: KqUi ~ qvC = KqUiUCsp - KciPL f K.„ Ï (10) V UCn y ^{ps -Pt -sgn(UCsp)Pu) 12> (11) and Km U vp I Cspi I V8 UCn \ Ps - Pt - sgn(UCspi )Pl, Kv V8 Ps - Pt - sgn(UCsp,)Pl, Ps - pt+sgn(ucsp,)pl, (12) such that KqUi is the flow-voltage gain at operating point i; Kci the flow-pressure coefficient at operating point i; pL = pA - pB the load pressure; and qvC the control flow rate considered as the average of the qvA and qvB. Force Control of Hydraulic Actuators using Additional Hydraulic Compliance 581 p 1 + 1 + 1 Strojniski vestnik - Journal of Mechanical Engineering 64(2018)10, 579-589 2.1.2 Cylinder Modelling Since the total variation of volume is the same for the two chamber volumes, it is possible to substitute VA and VB in Eqs. (6) and (7) with the parametric uncertainty V. This approach is suitable for the use of the quantitative feedback theory (QFT). Therefore, the hydraulic stiffness (KH) can be generalized as: Kh = M.+M=2M Va Vb V (13) and the linear model of the symmetrical cylinder results: . dxP A dpL qvC = 4, + , v ^ dt kh dt (14) where Au = Aa = AB is the useful area of the piston. The friction force is a nonlinear function of velocity and can be described by a variable viscous friction coefficient (f) [24]. This coefficient can be considered as an uncertain parameter, and this assumption allows the motion equation to become linear, yielding: dx d2 xp APl - fv - F = M —f. (15) dt dt 2.3 Open Loop Transfer Function For the open-loop (OL) transfer function, the values of KqUi and Kci are considered to be evaluated at null operating point (i = 0), where UCsp0 = 0, qvC0 = 0, and Plo = Combining Eqs. (1), (9), (10) and (14) with (15), the OL transfer function that relates the output force (Fe) to the input control signal (UC) and the load displacement (xL) is: Fe( S) =- C0Uc( s) KS (b4s4 + b3s3 + b2s2 + b1s + b0 )sXL (s) as5 + as4 + a3s3 + a2 s2 + + a0 (16) where: a0 =®nv2Kc0KHKS> ai =®nv2Au2(KH + Ks) + KcoKH(2^v®nvKs + fflnv2fv), a2 =2iv®nvAu2(KH + Ks) + ®nv2Au2fv + KcoKh(2^v®dv/v + ®nv2M + Ks), a3 =2ivfflnvAu2fv + ®nv2Au2M + K^h^^M + fv) + Au2(Kh + Ks), a4 =2^fflnvAu2M + Au2fv+ KcoKhM, a5 =Au2M, bo =®nv2Au2KH + ®nv2fvKcoKH, b =2^v®nvAu2KH + fflnv2Au2fv + K^K^^n/v + ®nv2M), b2 =2ivfflnvAu2fv + ®nv2Au2M + Au2KH + KC0KH(2&fflnvM+f), b3 = 2^vmavAu2M + Au2fv + KC0KhM, b4 =Au2M, c0 =AuKHKqU0KS®nv2. 3 SELECTION OF THE HYDRAULIC HOSE In this section, a procedure for sizing the hydraulic hose required to add hydraulic compliance to the system is proposed. It consists of a few analytical expressions that allow the selection of commercial HVE hoses in a simple way. To demonstrate its effectiveness, dynamic responses using the nonlinear model as well as experimental results are reported in Section 5. 3.1 Selecting the Desired Hydraulic Stiffness Based on Eq. (16), and expressing the input control signal UC(s) as G(s)(Fref (s) - Fe(s)), where G(s) and Fref(s) represent the controller transfer function and the reference force, respectively, the closed-loop (CL) transfer function results in: G( s) N(s) Fc(s) = D(s) + G( s) N,(s ) N2( s) D( s) + G (s) N1( s) F-ef (s) ^l(S), (17) where: N(s) = C0, N2(s) = KS (b4s4 + b3s3 + b2s2 + bjs + b0) s, D(s) = a5s5 + a4s4 + a3s3 + a2s2 + axs + a0. The first part of Eq. (17) determines the system performance in response to a reference force. The second part of Eq. (17) describes how the system behaves when a disturbance input XL(s) occurs. Therefore, it is related to the output impedance and can be considered as a measure of the system's capability against external disturbances. The hydraulic stiffness KH affects both parts of Eq. (17) as reported below. Therefore, the hose selection will be based on a trade-off between performance and disturbance rejection. 3.1.1 Output Impedance The output impedance can be analysed considering the transfer function Fe(s)/XL(s) presented in the second part of Eq. (17). The presence of a zero at the origin introduces an implicit disturbance rejection. However, 582 Ledezma, J.A. - De Pieri, E.R. - De Negri, V.J. Strojniski vestnik - Journal of Mechanical Engineering 64(2018)10, 579-589 based on the initial value theorem, an initial overshoot equal to the equivalent stiffness (KHKS/(KH+KS)) times a disturbance amplitude is expected. Therefore, increasing the rejection capability against load movement requires that KH is as low as possible. Nevertheless, this value must be greater than or equal to the hydraulic stiffness required to achieve the required performance, which is discussed next. 3.1.2 Desired Tracking Control Ratio An approximation of the first part of Eq. (16) expressed by first and second order terms can be written as: ( V V / A V J A FM. Uc(s) KqU0 Keq / Au s + K co Keq/ Au2 s + dls + d0 j (18) e0 ei where: d0 = (Kh + Ks)/M, di =f/M, = O 2 Keq = KsKH/(Ks+KH), representing the equivalent stiffness. The first term is equivalent to a linear modelling of the open-loop system neglecting mass, friction and valve dynamics. The middle term is related to Newton's second law including the effect of the hydraulic stiffness. Finally, the last term represents the servovalve dynamics. It is noteworthy that, using the parameters presented in the Appendix, Eq. (18) is an almost exact approximation of Eq. (16), with a maximum difference between curves of 0.2 dB in magnitude and 0.003° in phase. The open loop poles of Eq. (18) are shown in Table 1. for different hydraulic stiffness values for which KHn corresponds to the nominal stiffness when using rigid pipes (instead of hoses). The real pole related to the hydraulic subsystem is -Kc0Keq/Au2, and it is dominant over the other two complex poles. This fact becomes more noticeable as the hydraulic stiffness is reduced. The damped natural frequency of the mechanical subsystem is loosely influenced by the hydraulic stiffness variation. Due to the dominance of the real pole, the first term in Eq. (16) can be substituted by the first order term in Eq. (18) and the corresponding closed-loop transfer function is: Fe(ss) = G(s)(KquoKeq / Au) Fref (s) S + (Kc0Keq / Au2) + G(sXKqU0Keq / AJ Table 1. Pole locations for different values of KH .(19) Hydraulic Mechanical Servovalve dynamics dynamics dynamics Kh [N/m] KqV0Keq/ Au d0 e0 (s + Kc0KCq/ Au' ) (s2 + djS + d0 ) (i2 + e1s + e0 ) Kh = 1.5X107 -2.963 -11.6 ± 2.93 x103i -989 ± 479i 0.1 Kh„ -0.336 -11.4 ± 2.75 x103i -989 ± 479i 0.01 Kh„ -0.034 -11.4 ± 2.73x103i -989 ± 479i A desired tracking control ratio for the system can be specified by: Fe( s)/ Ffs) = Kss/(rds +1), (20) where KSS is the steady-state gain, and Td is the desired time constant. Comparing Eqs. (19) and (20) and assuming a proportional controller, a correlation between the required hydraulic stiffness and the proportional gain (Kp) can be obtained: Kh = Ks Au2 KsT(Kc0 + AuKpK:qm) - Au 2 (21) 3.2 Selecting a Commercial Hydraulic Hose The hose diameter can be calculated based on the maximum flow rate (qvmax) and the recommended fluid velocity in the hose (voil) according to: Dho =V 4 (23) (24) A, L + 2Vho where Vho represents the volume of oil trapped in the hose coupled to each cylinder chamber, and L is the 0 Force Control of Hydraulic Actuators using Additional Hydraulic Compliance 583 Strojniski vestnik - Journal of Mechanical Engineering 64(2018)10, 579-589 piston stroke. For simplification, henceforth Eq. (24) will be used for the calculation of KH. The effective bulk modulus can be represented as: r PoPhoSS ße = ßoßho ßo +ßho ßo + r ßh, (25) where ßo is the fluid bulk modulus, ßho represents the dynamic hose bulk modulus, ßhoSS is the static hose bulk modulus, and rß is the ratio ßho/ßhoss- According to [27], the hose bulk modulus changes during the system operation, i.e., dynamic bulk modulus (ßho) is higher than static bulk modulus (ßhoSS) due to a phenomena called dynamic hardening. Those authors state that the rß is about 4 to 5 for nylon braid hoses, which is the case of the HVE hoses used in this paper. According to the experimental results presented in [27], the static hose bulk modulus can be expressed by: V dp oho y AVo (26) where AVho represents the change in the hose volume, V0ho is the initial volume of the hose, and p is the working pressure. This equation can be rewritten in order to calculate fihoSS as a function of hose catalogue data, yielding: (Ao + E) Ko2 + 4E) ßhoSS _ " 4s (27) where E represents the volumetric expansion of the hose, defined as AVho/Lho, being Lho the hose length; Dho is the hose diameter, and s corresponds to dE/dp. E is a parameter determined at the working pressure (p), and s can be assumed constant for a specific hose, corresponding to the slope of straight lines in graphs of E versus p provided by manufacturers [25] and [26]. Combining Eqs. (24), (25) and (27) yields: 4 = 8 P0 (npj + 4 *) _ (28) 0 nDjKh (( + r (nDj + 4*)) nDho2' Eq. (28) shows that to obtain smaller hydraulic stiffness Kh, longer hose Lho is required. Furthermore, the greater the hose's volumetric expansion, the lower the hose length to achieve the same hydraulic stiffness. If necessary, the process of hose selection using Eqs. (21) and (28) can be iterative to avoid obtaining long hoses, which may not be suitable for implementation. For example, if a calculated hydraulic stiffness results in a long hose, the proportional gain can be decreased in Eq. (21), resulting in a higher KH and smaller Lho. Therefore, the final value of hydraulic stiffness is obtained from a trade-off between KH, Kp, and Lho. In contrast, Eq. (28) can also be used to calculate Kh based on a pre-selected hose with the required Dho (from Eq. (22)) and the desired Lho. After that, the proportional gain to achieve the required closed loop response can be calculated by isolating it in Eq. (21). The use of a proportional controller in this design stage allows obtaining a very good approximation for the required hydraulic stiffness. A specific controller design and the system dynamic analysis using a nonlinear model can then be carried out as shown in Sections 5 and 6. 3.3 Example of Hose Selection Consider a Pure Hydro-Elastic Actuator (PHEA), as shown in Fig. 1b, able to apply forces up to 9000 N. The desired tracking control ratio is specified according to Eq. (20) with a time constant rd equal to 50 ms. The system parameters are according to the Appendix, with KqU0 and Kc0, calculated at null operating based on Eqs. (11) and (12). Considering the maximum flow rate equal to the test rig pump supply (1.6*10-4 m3/s) and fluid velocity in the line equal to 2 m/s, a commercial hose diameter of 12.7 mm (1/2 in) was selected using Eq. (22). Analysing HVE hose catalogues, the EATON Synflex ® 3130-08 was selected [25]. The hose volumetric expansion is 1.56*10-5 m3/m @ 7*106 Pa (4.7 cc/ft @ 1000 psi), resulting in a static bulk modulus (Aoss) of 6.36*107 Pa. Based on Eq. (26), an rp value equal to 5 was considered. As discussed before, the system performance must be determined by a trade-off between KH, Kp, and Lho. In this study, a Lho equal to 1.5 m was specified and using Eq. (28) the resulting KH is 2.24x106 N/m and, based on Eq. (21), Kp is 2.7 X10-4. The hydraulic force control system according to the configuration shown in Fig. 1b was assembled in a test rig as shown in Fig. 4. The system comprises two hoses interconnecting each cylinder chamber with the servovalve. A load cell is fixed at the cylinder rod, and it will be in contact with a metal block attached to the test rig frame. The parameter values of this experimental setup are presented in the Appendix. For the QFT controller design and dynamic simulation, KS equal to 2*107 N/m was used, corresponding to the d s 584 Ledezma, J.A. - De Pieri, E.R. - De Negri, V.J. Strojniski vestnik - Journal of Mechanical Engineering 64(2018)10, 579-589 equivalent stiffness resulting from the load cell and environment compliances. Fig. 4. Hydraulic test bench used for experiments 4 CONTROLLER DESIGN USING LINEAR QFT The QFT allows designing a linear controller for a nonlinear system, assuming it as linear under the effect of disturbances and parametric variations [20]. In this study, the controller design was carried out using the QFT frequency domain control design toolbox for use with Matlab [28]. The objective of the QFT-based control design is to synthesize a prefilter (F(s)) and a controller (G(s)) such that the force responses of the system always fall within a predefined time-domain tolerance described by upper and lower limits. Typically, these limits are based on second-order specifications, such that the time constant value (rd = 50 ms) used at the hose selection procedure (Section 3.1.2) is converted on a settling time (?si% = 200 ms). In this study, the upper and lower limits were defined as: • Upper Limit (Bu(t)): settling time (tsi%) of 0.1 s and maximum overshoot of 1 %. • Lower limit (BlW): settling time (tsi%) of 0.3 s without overshoot. Translating the specifications from the time domain to the frequency domain yields: 2116 s" + 73.6s + 2116 A 75 s + 75 and TlS = 400 s2 + 40s + 400 A s + 50 50 (29) (30) The addition of a zero and a pole in Eqs. (29) and (30), respectively, relax the requirements for the controller design in the frequency domain without affecting the responses in the time domain. Table 2 shows the parametric uncertainties assumed in the plant model described by Eq. (16). Table 2. Parameter uncertainties at Eq. (16) Description Nominal value Range fv [Ns/m] 100 100 to 3x104 Kqu [m3/(s-V)] 5.4x10-5 3.23x10-5 to 5.61x10-5 Kci [m3/(s-Pa)] 6.42x10-13 6.42x10-13 to 7x10-11 Kh [N/m] 2.24x106 2.24x106 to 4.11x106 œnv [rad/s] 1099 879 to 1319 & 0.9 0.72 to 1.08 To define the uncertainties, the following considerations were assumed: • fv varies from a minimum value related to the Coulomb friction to a maximum value representing the stiction, which was obtained experimentally, • the range of parametric uncertainty for Kqu was calculated using Eq. (11), assuming no load (PLi=0) and the operating point at the maximum power (PLi=2ps/3). The nominal value was obtained through experiments according to ISO 10770-1 [29], • the range for Kci was calculated using Eq. (12). The nominal value was obtained based on the results of the internal leakage measurement test according to ISO 10770-1 [29], • Kh varies from the minimum to maximum hydraulic stiffness calculated by Eqs. (23) and (24), • Finally, a ±20% variation around the nominal values extracted from the catalogue is assumed for ^nv and tv. The other parameter values used for the controller design are shown in the Appendix, with the exception of KS that was changed in order to represent the equivalent stiffness instead of just the load cell characteristic. Based on the measurement of the environment deflection, the resulting Ks was 2*107 N/m. For the QFT boundary generation, robust stability, disturbance rejection and reference tracking criteria are considered in this study. The robust stability bounds are defined through [20]: TiM P(jö)G (jö) 1 + P(jö)G(jö) 1 + L(jö) < Ôi(w) = 1.3, ©gQ15 (31) Force Control of Hydraulic Actuators using Additional Hydraulic Compliance 585 Strojniski vestnik - Journal of Mechanical Engineering 64(2018)10, 579-589 where P(]oS) represents the plant, G(j^) represents the controller, L(]oS) is known as the loop transmission function, and d\(a>) is a constant constraint calculated assuming a gain margin of 5 dB. The subset of analysis frequencies Q is {0.01, 0.1, 1, 5, 10, 50, 100, 150} rad/s. For the disturbance rejection bounds, the constraint function represents the disturbance control ratio: TJ = Y (J®) 1 D(jw) 1 + Lj)