Electrohydraulic force control design of a hardware-in-the-loop load emulator using a nonlinear QFT technique

Control Engineering Practice 20 (2012) 598–609

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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac

Electrohydraulic force control design of a hardware-in-the-loop load emulator using a nonlinear QFT technique Mark Karpenko 1, Nariman Sepehri n Fluid Power Research Laboratory, Department of Mechanical Engineering, The University of Manitoba, Winnipeg, Manitoba, Canada R3T 5V6

a r t i c l e i n f o

a b s t r a c t

Article history: Received 21 October 2010 Accepted 6 February 2012 Available online 13 March 2012

This paper presents the design of a robust force control system for an electrohydraulic load emulator utilized as part of a hardware-in-the-loop flight simulation experiment. In this application, the force controlled hydraulic actuator is used to artificially recreate in-service loads upon a second hydraulic flight actuator operated in closed-loop position control. Electrohydraulic force control is more difficult than electrohydraulic position tracking because the load dynamics influence the force transfer function in a way that makes it challenging to develop an accurate force tracking system using simple feedback control. Nonlinear quantitative feedback theory (QFT) is applied in this paper to address this issue. First, an effective and robust feedback controller is designed by nonlinear QFT to desensitize the force control loop to nonlinear servovalve flow/pressure effects and typical system uncertainties. A secondary compensator is also designed within the QFT framework to extend the force tracking bandwidth with respect to the load motion. Experiments demonstrate acceptable force tracking performance within the scope of a representative flight-simulation experiment. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Hydraulic actuators Force control Quantitative feedback theory Hardware-in-the-loop simulator Equivalent time-varying frequency response functions

1. Introduction Force tracking hydraulic actuators are important components in applications such as robotics, vibration isolation, and active suspensions. Force controlled hydraulic actuators are also often employed as loading devices in mechanical and structural testing installations, when it is necessary to artificially recreate complex dynamic loading conditions (Plummer, 2010). Other examples where hydraulic force control is important include testing of fluid power systems such as hydraulic mill stands (Kim & Lee, 2006), or aerospace flight actuators (Karpenko & Sepehri, 2009). In such applications, the secondary force controlled actuators are used to generate the expected in-service loads. In many cases, the electrohydraulic loading actuator is coupled to a non-stationary environment whose motion creates disturbances that can impact the ability of the force control system to properly recreate the desired load profiles. This paper presents a practical and easy-toimplement fixed-gain force control scheme for such an electrohydraulic loading actuator that can maintain small force tracking errors, despite large movements of the environment. As far as previous research on hydraulic actuator force control is concerned, feedback linearization has been employed together n

Corresponding author. E-mail address: [email protected] (N. Sepehri). 1 Presently at: Jack Baskin School of Engineering, University of California, Santa Cruz, CA 95064, USA 0967-0661/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2012.02.004

with a Lyapunov-based adaptation scheme for force tracking control against a stationary environment (Alleyne & Liu, 2000; Liu & Alleyne, 2000). Alleyne and Liu (2000) showed that their control algorithm could be reduced, in its simplest form, to a force error feedback control system with an adaptive velocity feedback term. Niksefat, Wu, and Sepehri (2001) used Lyapunov control concepts to design a switching controller for contact task control. Lyapunov-based force control was investigated in (Sekhavat, Wu & Sepehri, 2005) for hydraulic actuator impact stabilization. A backstepping scheme based on Lyapunov theory was also used by Nakkarat and Kuntanapreeda (2009) to design a full-state nonlinear feedback controller for force control of a single-rod electrohydraulic actuator. Velocity feedback, with adaptive valve flow/pressure compensation, was employed in (Jacazio & Balossini, 2007) for the development of an electrohydraulic loading system for an advanced aerospace test rig. An approach for dynamic force control of electrohydraulic actuators using added compliance and a Smith predictor with displacement compensation feedback was studied in (Sivaselvan, Reinhorn, Shao, & Weinreber, 2008). Other researchers have developed variable structure and sliding mode force controllers for hydraulic actuators (Jerouane, Sepehri, & Lamnabhi-Lagarrigue, 2004; Boumhidi & Mrabit, 2006). Model predictive control has also been successfully applied for electrohydraulic force control (Marusak & Kuntanapreeda, 2011). Electrohydraulic force control has also been studied from a robust control perspective. Robust control techniques, such as HN and quantitative feedback theory (QFT), compensate plant nonidealities by selecting the gains of a fixed-gain controller

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in a way that desensitizes the control loop to a priori defined variations and uncertainties in the plant dynamics. This is in contrast to the other control approaches cited above, which deal with system nonlinearities and parametric uncertainties in an online fashion using controller adaptation. Laval, M’Sirdi and Cadiou (1996) designed an HN force controller for a double rod actuator by linearizing the actuator dynamics around an operating point. Stability and performance tradeoffs were discussed in their paper. HN together with linearized plant models was also used to design a force controller for a hydraulic load simulator for ground testing of hydraulic flight actuators (Wang, Liang, Zhang, Ye, & Li, 2007). Although flight actuator motion was not explicitly considered in the controller design, experiments showed that the resulting control system was reasonably effective at accommodating slow movements of the environment. Niksefat and Sepehri (2001) used QFT to design a force control for a servovalve driven hydraulic actuator based on a linearized model of the actuator operating against a variable stiffness load. Experimental results verified the robustness of the force controller operating against a spring-type load, despite a wide range of system uncertainties. Nam and Hong (2002) applied QFT for electrohydraulic force control in an aerodynamic load simulator. The efficacy of the QFT controller, which was designed based on a simplified model of the hydraulic actuator, was verified by nonlinear simulations that included aircraft dynamics. More recently, Ahn and Chau (2007) designed a QFT force controller for an electro-hydrostatic actuator comprised of a pump-controlled actuator manipulating a spring-type load. QFT force control of electrohydraulic actuators has also been employed for the design of single and multi-axis dynamic road simulators (Kim, Xuan, & Kim, 2008; Kim, Xuan, & Kim, 2011). In all of the robust force control designs cited above, the nonlinear hydraulic actuator dynamics are modeled by first developing a linearized model of the hydraulic system. The linearization approach approximates the nonlinear system only over a limited region of operation. Thus, it is necessary to consider many time-invariant linearized operating points in order to properly model the dynamics of the electrohydraulic force control system. A significant shortcoming of this approach is the fact that the parametric uncertainty associated with the linearized operating points is observed over the entire control bandwidth when, in fact, the uncertainty is time-varying in nature. Although it is possible, in some instances, for the controller to overcome this additional time-invariant uncertainty the price that must be paid is that of unnecessary conservatism and controller over-design. The resulting high-gain controllers may perform well in practice, due to their inherent stiffness, but they are in contrast to the fundamental doctrine of QFT: controller gain should be minimized throughout the entire control bandwidth. These points have been illustrated in the context of electrohydraulic and pneumatic position control as part of the authors’ previous fluid power research (Karpenko & Sepehri, 2006; Karpenko & Sepehri, 2008; Karpenko & Sepehri, 2010). In particular, it has been shown (Karpenko & Sepehri, 2008) that by identifying a family of equivalent, time-varying, frequency response functions by Fourier transformation of representative plant input–output data, controller design conservatism can be minimized for electrohydraulic position control while retaining the desired robustness to system nonidealities. These identified time-varying models form the basis for control system design using a nonlinear QFT design approach (Horowitz, 1993). This particular approach has not yet been applied for the challenging problem of electrohydraulic force control design. The objective of this paper is to utilize equivalent time-varying frequency response modeling and the nonlinear QFT design framework to develop a practical electrohydraulic force control

599

system. In the application under consideration, the electrohydraulic loading actuator is used to recreate in-service loads as part of a hardware-in-the-loop flight actuator test environment. Control system development for this type of electrohydraulic force control problem is fundamentally different than electrohydraulic position tracking because the poles of the load are manifest as zeros in the open-loop force transfer function (Alleyne & Liu, 1999). Due to this structure, a bandwidth limitation results that makes it challenging to develop an accurate force tracking system using simple, single-degree-of-freedom, feedback control strategies such as proportional or proportional–integral control. The nonlinear QFT approach taken in this paper addresses the inherent difficulties of the electrohydraulic force tracking problem in a novel fashion. First, a robust feedback controller is designed by nonlinear QFT to desensitize the force control loop to nonlinear servovalve flow/pressure effects as well as typical uncertainties in the system parameters. The use of time-varying frequency response functions to model the nonlinear electrohydraulic loading system, as opposed to conventional small-signal linearization, enables the controller gain to be minimized over a large region of operation of the actuator. This enables the force control bandwidth to be maximized while avoiding saturation of the servovalve input. A displacement compensator that monitors the movement of the hydraulic actuator under test is then designed, within the framework of QFT, to further reduce the closed-loop sensitivity of the electrohydraulic force control system and extend the useful force tracking bandwidth. The results of several experimental tests clearly illustrate the performance of the designed electrohydraulic force control system. In particular, the accuracy of the loading system in reproducing the in-service load profile is demonstrated when the loading system is utilized as part of a hardware-in-the-loop flight actuator testbed.

2. Hydraulic test bench 2.1. Experimental setup A photograph and schematic of the hardware-in-the-loop load emulator is shown in Fig. 1. The test bench is comprised of two independent hydraulic actuator circuits that are supplied from a common hydraulic power unit. The hydraulic circuit on the left represents the closed-loop position controlled hydraulic actuator. The second hydraulic system, installed on the right hand side of the test bench, is the loading actuator: its purpose is to recreate the desired force profile upon the first position controlled hydraulic actuator. Thus, from the perspective of the loading actuator, the position controlled actuator is viewed as an environment. In this paper, a robust force controller is designed for closing the loop around the loading actuator that can recreate the desired force profile despite large movements of the position controlled actuator. The two hydraulic actuators are connected using a flexible coupling (see Fig. 1b), which acts as nearly a pure stiffness. The loading actuator is a servovalve driven double-rod type having a 38.1-mm bore, 25.4-mm rods and a 203-mm stroke. The servovalve is a Moog 31 closed-center nozzle-flapper valve that has a 26 L/min (6.8 GPM) flow capacity at 21 MPa (3000 psi) supply pressure. The actuator is interfaced to a desktop computer workstation running the Windows XP operating system via a DAS16F input–output board and a 2-axis M5312 quadrature encoder card. The nominal sampling rate of the interface and control software is approximately 1 kHz. The DAS16F is used to digitize all analog instrumentation, including a 22 kN (5000 lb) load cell that is used to measure the load force applied on the position controlled actuator. A cable driven optical rotary

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Fig. 1. Experimental setup: (a) photograph of experimental test bench and (b) schematic of the test bench architecture.

encoder, interfaced to the M5312 card, is used to measure the location of the position controlled actuator. The servovalve coils are driven by a Moog N121-132 servo-controller operating in open-loop current control mode. The servovalve saturates when the demanded control valve current exceeds 10 mA. 2.2. Mathematical modeling A schematic of the electrohydraulic loading actuator for mathematical modeling is shown in Fig. 2. The nonlinear state equations that describe the motion of the piston, xp, in response to changes in the command signal, u, can be written as (Merritt, 1967) x_ p ¼ vp  1 AP 1 AP2 bvp þ F a v_ p ¼ m  bh  P_ 1 ¼ Q 1 Avp Axp þ V 1   bh Q 2 þ Avp P_ 2 ¼ AðLxp Þ þ V 2 x_ v ¼ vv v_ v ¼ o2v xv þ 2zv ov vv þ ksp o2v u

ð1Þ

In (1), m, b, A, and bh are the mass of the piston, the effective viscous damping of the actuator, the piston annulus area, and the bulk modulus of the hydraulic fluid, respectively. Pressures P1 and P2 denote the hydraulic pressures in each of the two actuator chambers, and parameter L denotes the length of the actuator stroke. V 1 and V 2 are the volumes of the lines connecting the servovalve to the actuator. The relationship between the control

Fig. 2. Schematic of hydraulic actuator for mathematical modeling.

signal, u, and the position of the servovalve spool, xv, is modeled as a second-order lag (Thayer, 1965) having undamped natural frequency ov, damping ratio zv, and valve spool position gain, ksp. The servovalve control flows are denoted by variables Q1 and Q2. The magnitudes of the control flows are dependent on the square root of the pressure differential across each valve port and are computed from the following equations, which are valid for both extending and retracting strokes Q 1 ¼ K v wxv

Q 2 ¼ K v wxv

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   P s P r P s þP r þ sgnðxv Þ P1 2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   P s P r Ps þ Pr þ sgnðxv Þ P 2  2 2

ð2Þ

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electrohydraulic force control can be derived:

Table 1 Parameter values of electrohydraulic loading actuator. Parameter

F a ðjoÞ ¼ ½WðjoÞUðjoÞT X ðsÞX c ðjoÞ

Value

Supply pressure, Ps (MPa) Mass of piston and rods, m (kg) Viscous damping coefficient, b (N s/m) Coupling stiffness, kc (kN/m) Piston annulus area, A (mm2) Actuator stroke, L (mm) Line volumes, V 1 ,V 2 (cm3) Fluid bulk modulus, bh (MPa) Valve flow coefficient, Kv (m3/kg)1/2  102 Valve orifice area gradient, w (mm2/mm) Valve spool position gain, ksp (m/mA)  105 Valve natural frequency, ov (Hz) Valve damping ratio, zv

Min

Nominal

Max

13.8 3.4 200 1000 – – –

17.2 3.8 250 1200 633 203 14.7

18.6 4.2 300 1400 – – –

345 – – 2.84 – 0.5

689 2.92 13.21 3.15 150 0.9

1030 – – 3.47 – 1.0

T X ðjoÞ ¼

T F ðjoÞ ¼ ð3Þ

v

Pressures Ps and Pr in (2) refer to the hydraulic supply and return pressures. Constants Kv and w are the flow coefficient and orifice area gradient of the servovalve. The approximate values of the loading actuator parameters are listed in Table 1 along with the various parametric uncertainties against which the force controller is designed to be robust.

kc DT F ðjoÞ DT F ðjoÞ þkc NT F ðjoÞ

ð4Þ

In (4), functions N T F ðjoÞ and DT F ðjoÞ represent the numerator and denominator polynomials of transfer function T F ðjoÞ ¼ NT F ðjoÞ=DT F ðjoÞ. Transfer function (4) emphasizes one of the key challenges in the design of electrohydraulic force control systems. In particular, the poles of the position controlled actuator response (seen by the loading actuator as a flexible environment) are manifest as zeros in the open-loop force transmission. This issue has been observed to place a fundamental limitation on achievable performance of force tracking control systems when simple compensators, such as a PID controller, are used to close the loop (Alleyne and Liu, 1999). The extent to which the presence of these undesirable zeros complicates the electrohydraulic force control problem, however, must be studied on a case-by-case basis. In order to investigate this aspect of the electrohydraulic force control problem, the nominal frequency responses, TX(jo) and TF(jo), of the nonlinear position controlled actuator were identified by numerical transformation of actuator input–output time histories to the frequency domain. Details on the design of the position control system and the frequency response identification technique are given in (Karpenko & Sepehri, 2010; Karpenko & Sepehri, 2008), respectively. The Bode magnitudes of identified transfer functions TX(jo) and TF(jo) are given in Fig. 4. These frequency responses can be modeled by the following equivalent linear transfer functions

Fig. 3. Open-loop block diagram of electrohydraulic force control system.

where sgnðÞ denotes the sign function 8 if xv 4 0 > <1 if xv ¼ 0 sgnðxv Þ ¼ 0 > : 1 if x o 0

601

m ðjo þ20Þðjo þ 100ÞððjoÞ þ125jo þ 1:56  10 Þ m 3:12  107

4

2

m

0:088jo 2

5

ðjo þ 26ÞððjoÞ þ 98jo þ 2:4  10 Þ N

ð6Þ

Since the complex conjugate poles of (6) are manifest as complex conjugate zeros of the open-loop force transfer function, the ability of the electrohydraulic loading actuator to reproduce the desired load force around the anti-resonance frequency will be poor (Sivaselvan et al., 2008). This fact is demonstrated in Fig. 5, which shows the effective stiffness of the load as seen by the electrohydraulic loading actuator, i.e. 1/kcUkc/(1þkcTF(s)). Referring to Fig. 5, it is observed that the effective stiffness of the load is

3. Electrohydraulic force control Fig. 3 shows a block diagram of the open-loop electrohydraulic loading actuator that considers the effects of motion and flexibility of the position controlled actuator. In Fig. 3, WðjoÞ denotes a family of time-varying magnitude and phase responses that represent the nonlinear dynamic behavior of the loading actuator between the control current, u, and the loading actuator position, xp. The set of nonlinear frequency response functions, WðjoÞ, accounts for the uncertainty in the parameters of the loading actuator and will be determined later as part of the controller design process. The motion, xe, of the position controlled actuator perturbs the loading actuator. The influence of the dynamics of the position controlled actuator can be modeled by the superposition of two transfer functions, TX(jo) and TF(jo). The first transfer function, TX(jo), describes the displacement that results from the application of command xc, assuming that Fa ¼const. Transfer function TF(jo), on the other hand, defines the response of the position controlled actuator with constant input xc to the applied load force, Fa, generated by the electrohydraulic loading actuator. By manipulating the blocks of Fig. 3, the relevant transfer function for

ð5Þ

Fig. 4. Nominal frequency responses of position controlled actuator.

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relatively flat up to o E100 rad/s. In the low frequency range (o o100 rad/s), the stiffness, kc, of the mechanical coupling between the two hydraulic actuators is dominant. Therefore, the disturbance force generated by movement of the secondary actuator will be proportional to the relative displacement between the position controlled actuator and the loading actuator. At frequencies above 100 rad/s, on the other hand, the characteristics of the complex conjugate pole/zero pair are observed to significantly alter the effective stiffness of the load. The influence of the complex conjugate pole/zero pair on the force response can, however, be neglected if these dynamics are purposefully not excited by the loading system or by the motion of the position controlled actuator. This assumption is examined below. The largest force that can be produced by the actuator is governed by the nominal pressure of the hydraulic supply. Considering the fact that maximum power transfer to the load occurs when the pressure drop across the servovalve is 2/3Ps (Merritt, 1967), the maximum force output for the hydraulic actuator configuration under investigation is approximately Fmax ¼7.3 kN. Generating forces of this amplitude, however, can only be realized over a finite range of frequencies, referred to as the large force bandwidth of the actuator (Robinson and Pratt, 2000). The frequency, oo, above which the maximum force amplitude can no longer be achieved may be estimated by assuming a sinusoidal load profile, F a ðtÞ ¼ F max sinðoo tÞ, and

evaluating the actuator slew rate   dF a ðtÞ ¼ F max oo dt max

ð7Þ

where, due to the presence of the coupling spring, ðdF a ðtÞ= dtÞmax ¼ kc ðx_ p Þmax . The maximum velocity of the actuator, ðx_ p Þmax , occurs with the control valve saturated, so (7) can be rewritten as

oo ¼

1 3Q max kc ðx_ p Þmax ¼ kc F max 2P s A2

ð8Þ

In (8), Q max is the rated flow of the servovalve at supply pressure, Ps, and A is the annulus area of the actuator piston. Substituting kc ¼1200 kN/m, Q max ¼ 3:9  104 m2 =s and the other relevant nominal system parameters (see Table 1) into (8) gives oo E100 rad/s. Since it is desirable to design the force control system to produce the maximum force amplitude without saturating the control valve input, the nominal closed-loop force control bandwidth should be set to a value that is less than the value given by (8). The closed-loop force control bandwidth is also approximately one order of magnitude lower than the resonant frequency of the position controlled actuator. Therefore, it is reasonable to conclude that the dynamics of transfer function TF(s) will have very little impact on the overall closed-loop force tracking response. Thus, for the particular actuator configuration considered here, TF(s)E0 can be assumed in order to alleviate the need to consider the influence of the flexibility of the position controlled actuator on the response of the loading actuator. The effective load stiffness is therefore essentially equal to the value of kc over the entire control bandwidth.

4. Controller design 4.1. Overview of nonlinear QFT design technique A schematic of the selected control structure is shown in Fig. 6. The proposed control system has three degrees of freedom, namely controller G(s), prefilter F(s), and displacement compensator FX(s). By reducing the blocks, the applied force, Fa, can be written in terms of the commanded force, Fc, and the position of the actuator under test, xe, as follows

Gkc W kc W kc Fc þ FX  Xe Fa ¼ F ð9Þ 1 þGkc W 1 þ Gkc W 1 þ Gkc W

Fig. 5. Effective load stiffness.

By inspection of (9), it is evident that each degree of freedom has a specific role to play in tuning the performance of the electrohydraulic force control system. The objective of controller, G(s), is to manage the complementary sensitivity of the closed-loop output,

Fig. 6. Block diagram of closed-loop electrohydraulic force control system.

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1. Identifying the equivalent linear plant set, P, and constructing the associated QFT uncertainty templates. 2. Nominal loop shaping and prefilter design to obtain compensators, G(s) and F(s). 3. Designing a displacement compensation filter, FX(s), by an extension of the original QFT design procedure. 4. Validating the controller design and satisfaction of the robust performance specifications through nonlinear simulations.

4.2.1. Identification of equivalent linear plant set and construction of plant templates The family of acceptable closed-loop responses was selected as (Horowitz, 1993):

  ao2n F a ðsÞ ¼ F c ðsÞTðsÞ ¼ F c ðsÞ ð10Þ ðs þ aÞðs2 þ 2zon s þ o2n Þ with on ¼125, z ¼ 0.68 and aA[45,95]. Using (10), the acceptable closed-loop bandwidth for step changes in the commanded force varies between 45 and 88 rad/s, which is compatible with the expected value of the large force bandwidth, oo E100 rad/s, given by (8). Although it is possible to set the parameters of (10) to achieve a larger closed-loop bandwidth, the servovalve would be forced to operate in the input saturation region for large changes in the force setpoint. This mode of operation is undesirable and the control system will be designed specifically to avoid this. Typical acceptable closed-loop responses are shown in Fig. 7a for the set of command inputs, F c , consisting of steps from 1 to 7.3 kN in magnitude. The plant drive signals required to generate the acceptable force outputs are shown in Fig. 7b over the extreme ranges of the parameter uncertainties listed in Table 1. The time histories of the control valve spool displacements were computed by simulating an inverse model of the nonlinear hydraulic actuator dynamics. An approach for computing the inverse hydraulic actuator model can be found in (Karpenko and Sepehri, 2008). The time histories of the hydraulic actuator input–output signal pairs were then used to identify a frequency response function, P(jo), for each equivalent linear plant in set P. Each P(jo) frequency response was computed as the product, PðjoÞ ¼ PV ðjoÞ

F a ðjoÞ X v ðjoÞ

ð11Þ

where PV ðjoÞ ¼ ksp o2v =ððjoÞ2 þ 2zv ov ðjoÞ þ o2v Þ refers to the second-order linear transfer function between the control valve input signal, u, and the valve spool displacement, xv. Ratio Fa(jo)/Xv(jo) is the ratio of the Laplace transform of the actuator output force, Fa, to the Laplace transform of the servovalve spool displacement. The frequency response of each xv(t)-Fa(t) pair was computed by numerically evaluating the truncated, continuous-time, Fourier integral. The frequency response of Fa(t), for 8 force (kN)

Gkc W=ð1 þ Gkc WÞ, to nonidealities in the hydraulic functions. After the sensitivity has been reduced to an acceptable level using the controller, the prefilter, F(s), is designed to shape the overall reference tracking performance. Since the bandwidth of the position controlled actuator is similar to that of the large force bandwidth (see the dominant poles in Eq. (6)), the force tracking performance will be poor when displacement commands are applied to the position controlled actuator. The role of the displacement compensation filter, FX(s), is to alleviate this issue by further reducing the closed-loop sensitivity of the electrohydraulic force control system. This will extend the useful force tracking bandwidth. The design of each of the three compensators is accomplished in a sequential fashion by the nonlinear QFT synthesis technique. Using the nonlinear QFT design methodology, the control system is designed in a way that attempts to force the closed feedback loop around the original nonlinear plant set, kc WðjoÞ, to behave similarly to a linear feedback system designed around an equivalent linear plant set PðjoÞ. Thus, in order to design the force control system, it is necessary to first identify a set of transfer functions, PðjoÞ, between servovalve input, u, and the actuator force, Fa, that accurately captures the uncertain nonlinear plant dynamics in terms of an equivalent linear form. The original nonlinear design problem is recast as an equivalent linear design problem in the following way (Horowitz, 1993): (i) A family of outputs, F a ðtÞ A F a , is defined to represent acceptable responses of the nonlinear closed-loop system for different command inputs, F c ðtÞ A F c . A set of acceptable closed-loop responses must be specified since given fixed compensators, FX(s), F(s) and G(s), it is impossible to realize a single system transfer function owing to plant nonlinearities and parameter variations. (ii) A set of unique plant drive signals, uðtÞ A U, that generate each F a ðtÞ A F a , is solved for every member of set W. The drive signals are computed as u(t)¼W  1Fa(t), where W  1 denotes the inverted dynamics of the nonlinear plant. Finally the Laplace transform Lfg, is evaluated for each input–output signal pair to obtain the equivalent linear, timevarying, frequency response function, PðjoÞ ¼ LfF a ðtÞg=LfuðtÞg. Since the robust performance of the original nonlinear control system is determined via control system design based on a set of equivalent linear system models, it is imperative that the family of linear frequency response functions captures the nonlinear characteristics of the hydraulic system as accurately as possible. Otherwise, the control system may not perform as expected when implemented around the original nonlinear system. Having replaced the original set of uncertain nonlinear plants, W, by equivalent linear set, P, the QFT design technique documented in detail elsewhere (e.g. Horowitz, 1993) can be used to complete the control system synthesis. The remaining steps include:

603

6 4 2 0 0

0.05

0.1

0.15

0.2

time (sec)

4.2. Controller synthesis Details on the synthesis of the force control system are now presented in a step-by-step fashion. Emphasis has been placed on explaining the procedure used for identifying the hydraulic actuator equivalent plant set P. The design of the displacement compensation filter is also discussed. Aspects of the QFT design implementation that have been reported before are covered only briefly.

Fig. 7. Time histories of acceptable plant input-output responses for uncertainties in Table 1 (a) acceptable force responses (output) and (b) corresponding valve spool displacements (input).

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Fig. 9. Templates of the identified equivalent linear plant set, PðjoÞ, at selected frequencies, o rad/s.

Fig. 8. Computed frequency response ratios, Fa(jo)/Xv(jo).

example, was obtained as follows: F a ðjoÞ ¼

Z

T

F a ðtÞejot dt

ð12Þ

0

A simple numerical algorithm for calculating (12) was derived by first subdividing the integration interval [0,T] into N panels of length h. Then, by using a Filon-like approach (Filon, 1929) together with the assumption that term Fa(t) is constant over each panel, (12) was approximated as F a ðjoÞ ¼

1   1cosðohÞ þjsinðohÞ NX F a ðkhÞ cosðokhÞjsinðokhÞ jo k¼0

ð13Þ When computing the frequency response of Fa(t), the Fourier integral was truncated at T¼0.3 s, which is after the system reached the steady-state (see Fig. 7). However, the fact that Fa(T) a0 introduces an error in the computed value of Fa(jo). Therefore, (13) was applied instead to the time derivative F_ a ðtÞ ¼ d=dtðF a ðtÞÞ, which is zero for t ZT. Frequency response Fa(jo) was then recovered as F a ðjoÞ ¼ F_ a ðjoÞ=jo. A similar approach was taken to evaluate the frequency responses of xv(t). A more thorough description of the procedure for evaluating the nonlinear frequency response of hydraulic actuators has been given elsewhere (Karpenko & Sepehri, 2008). The computed frequency response ratios are shown in Fig. 8. With reference to Fig. 8, it is observed that the identified frequency responses properly capture two important features of the hydraulic actuator behavior. The first is the inherent integration characteristic of the actuator in the low-frequency range, o o10 rad/s. The second feature is the resonant mode in the high-frequency range, o E1500 rad/s, which results from the interaction between the piston/rod mass and the compressibility of the hydraulic oil in the loading actuator. A decrease in phase lag is also observed for some frequency responses in the midband, 5o o o500 rad/s, due to the time-varying nature of the equivalent linear transfer function models. The variation in actuator phase lag over this range of frequencies is due to the changes in the servovalve pressure gain, which can vary widely during the actuator step response. This effect is captured by the equivalent transfer function modeling technique, but is not seen in conventional linearization because in the latter approach, the

servovalve pressure gain is assumed to be constant over the entire step response. Some templates of the identified equivalent plant set PðjoÞ ¼ fP V ðjoÞðF a ðjoÞ=X v ðjoÞÞg are shown in Fig. 9 at selected frequencies, o. The plant templates describe, in terms of the open-loop gain and phase variations on the Nichols chart, the changes in the dynamics of the hydraulic actuator that are attributable to the considered system uncertainties. The templates also inherently account for the effect of the nonlinear terms in the hydraulic functions in the form of an equivalent amount of linear gain-phase variation that is lumped together with the variations due to the plant parametric uncertainty at each frequency. Referring to Fig. 9, it is observed that at small frequencies, o o 10, the templates are close to vertical lines. This indicates that only the gain of the equivalent plants is impacted by the hydraulic nonlinearity and parameter uncertainties. The lowfrequency magnitude variation results primarily from changes in the servovalve flow gain, uncertainty in the stiffness of the flexible actuator coupling, and the estimated range of variation in the valve spool position gain, ksp . As o increases from 10 to 500 rad/s, the templates widen due to the effects of the servovalve spool excursions and the uncertainty in the servovalve spool dynamics. The plant templates are the largest around the actuator resonant frequency, o Z 800 rad/s, because of the relationship between the characteristics of the actuator resonant mode and the uncertainty in the actuator mass, viscous damping coefficient, hydraulic fluid bulk modulus, and the changes in the actuator cylinder volumes that occur as the actuator is stroked. Beyond the resonant frequency, o 4 2000 (not shown in Fig. 9), the template widths decrease in size and the template points merge again into vertical lines.

4.2.2. Nominal loop shaping and prefilter design Given the templates of uncertain plant set P, the next step in the QFT design process is to synthesize functions F(s) and G(s) in Fig. 6 to realize the desired force tracking function. In other words, it is required to solve F(s) and G(s) such that function T¼Fa/Fc ¼F(GP/1þGP) lies within set T ¼ F a =F c for all P A P. QFT provides guidelines for accomplishing this task. In particular, the controller and prefilter should be designed to satisfy the following frequency domain tolerance on reference tracking performance

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(Houpis, Rasmussen, and Garcia-Sanz, 2006):    GP  9T L 9 r F r 9T U 9 1 þ GP 

605

ð14Þ

where T L ðsÞ ¼

T U ðsÞ ¼

1:56  1010 ðs þ42:5Þðsþ 2:34  104 Þðs2 þ 170s þ1:56  104 Þ 1:86  104 ðs þ 35Þ ðs þ 40Þðs þ 125Þðs þ 130Þ

ð15Þ

ð16Þ

are prescribed lower and upper bounding functions whose step responses form a rough envelope encompassing the acceptable set F a (Horowitz, 1993). Robust stability properties of the closed-loop force control system are shaped by constraining the peak magnitude of the closed-loop frequency responses as follows:    GP    ð17Þ 1 þ GP  r 1:9 dB In (17), the 1.9 dB specification is chosen to be roughly compatible with the tracking envelope and is equivalent to prescribing 5 dB and 481 gain/phase margins for the closed-loop system. Note that satisfaction of (17) will only be rigorous for the members of the family of the equivalent linear frequency response functions that have been used to model the nonlinear hydraulic system. For this reason, it is generally very difficult to prove the stability of the original nonlinear system. As such, no rigorous claim regarding the closed-loop stability of the original nonlinear system can be made here. However, a good correlation between the stability margins implied by (17) and the actual closed-loop performance of the nonlinear hydraulic system can be obtained if the equivalent frequency response functions very closely match the actual nonlinear frequency response of the hydraulic actuator. An appropriate way to obtain a good equivalent representation of the nonlinear dynamics is to follow the approach for identifying the hydraulic actuator frequency response that was presented earlier in this section. To avoid saturating the servovalve input for large changes in the force setpoint, a constraint on the control effort can also be specified. The control effort constraint is given in the frequency domain as      G  umax 1   r U ð18Þ 1 þ GP  F  max T U ðsÞ The constraint given by the right hand side of (18) recognizes that the upper tracking bound given in (16) is the quickest possible response that is to be produced by the closed-loop force control system. Consequently, any response closely following this bound will also demand the largest amount of effort from the servovalve. Thus, the control effort constraint provides a means of shaping the controller gains across the bandwidth in order to maintain the control output less than maximum value of umax ¼10 mA for force setpoints up to F max ¼ 7:3 kN. To design the controller, Eqs. (14), (17), and (18) are used to formulate point-wise design bounds on the nominal system loop transmission, Lo(jo) ¼G(jo)Po(jo), as well as design bounds on F(jo). The bounds on Lo(jo) are computed by first decoupling the roles of F(s) and G(s) to form the following inequality, which implies the existence of filter F(s):      GP   GP  9T U 9    log ð19Þ log rlog 1þ GP  1 þ GP max 9T L 9 min The QFT bounds, BðoÞ, that delineate the regions of the Nichols chart that simultaneously satisfy (17), (18), and (19) are then solved either graphically or numerically at each design frequency

Fig. 10. QFT bounds, BðoÞ and the designed nominal loop transmission, Lo(jo).

using the plant templates. In this work, the QFT bounds were solved using the numerical procedure documented in (Chait & Yaniv, 1993). To economize on computation time, only plant magnitude-phase points lying on the perimeter of each template were used, as (17), (18), and (19) will be satisfied automatically for any magnitude-phase points lying inside the template boundary. The QFT bounds are shown in Fig. 10 along with the designed nominal loop transmission. Lo(jo) was shaped so that it lies above all of the open tracking bounds, which are single valued functions of the phase angle and outside the closed stability bounds that encircle the (  1801, 0 dB) critical point. To satisfy the constraint on the control effort, it was also necessary to ensure that Lo(jo) lies below the control effort bounds. Otherwise, the servovalve input may become saturated during the operation of the force control system. The dominant control effort bounds, which are shown for o ¼50 and o ¼100 rad/s in Fig. 10, form a practical limit on the maximum force control bandwidth. This limitation exists because increasing the gain crossover frequency can only be accomplished by an accompanying increase in controller gain which, in turn, would cause the control effort bounds to become violated. Thus, the nominal loop shape given in Fig. 10 strikes a reasonable balance between achieving the closed-loop sensitivity reduction that is necessary for robust tracking and the desire to avoid saturating the servovalve input. After manipulating the nominal loop transmission to satisfy the bounds, compensator G(s) was solved as the ratio Lo(jo)/Po(jo) and has the following transfer function. GðsÞ ¼

1:22ðs þ48Þðs þ 500Þ ðs þ35Þðs2 þ 700s þ 2:5  105 Þ

ð20Þ

To design (20), the proportional loop gain was first adjusted so that the low-frequency bound, Bð1Þ, was satisfied. Then, a lag filter formed by the pole-zero pair, (sþ48)/(sþ 35), was implemented to bring the Lo locus closer to the mid-frequency bounds Bð50Þ and Bð100Þ and reduce design conservatism. Finally, a real zero at s¼  500 and an underdamped complex pole-pair were added to roll off the high-frequency controller gain and low-pass filter the noise originating from the load cell. As is evident from (19), controller G can only reduce the variation in the magnitudes of GP=ð1 þ GPÞ to be less than the amount dictated by the lower and upper tracking bounds. Therefore, the prefilter must be designed to ensure that the closed-loop step responses all lie within the acceptable frequency response envelope. The transfer function of the prefilter was designed using a straight forward process involving Bode approximations

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25

magnitude (dB)

0

-25 Fig. 12. Magnitude plots of set, kc =PðjoÞ and designed filter, FX(jo).

-50

-75 101

102 103 frequency (rad/sec)

104

Fig. 11. Magnitude plot of the nominal closed-loop force transfer function.

(Houpis et al., 2006) and was found to be FðsÞ ¼

1:75  104 ðs þ100Þðs þ 175Þ

ð21Þ

The magnitude of the nominal closed-loop system function, 9F(GPo/1þ GPo)9, is shown in Fig. 11 along with the design bounds used in the development of the QFT control law. The nominal bandwidth of the force control system is observed to extend to approximately 80 rad/s, which is slightly smaller than the large force bandwidth o E100 rad/s, predicted by (8). Also since the controller was designed to satisfy a constraint on the control effort, it is expected that the maximum load force can be generated throughout the control bandwidth without saturating the control valve input. The QFT controller also ensures that the closed-loop frequency response remains within the design bounds over the entire range of actuator parametric uncertainty. Since the frequency response of the position controlled actuator has significant frequency components up to 100 rad/s, however, force tracking performance will be poor when the position controlled actuator moves against the load. The inherent bandwidth limitation of electrohydraulic force control, therefore, motivates the design of a displacement compensation filter to extend the force tracking bandwidth.

4.2.3. Design of displacement compensator, FX(s) A displacement compensation filter is now designed to extend the tracking bandwidth of the force control system. This will improve the ability of the force control system to maintain the desired load profile despite the motion of the secondary position controlled actuator. The displacement compensator (transfer function FX(s) in Fig. 6) accomplishes this task by generating a secondary command input to drive the servovalve that is derived by measuring the motion of the position controlled actuator. The filter is designed to approximately zero the second term on the right hand side of (9), which is herewith rearranged slightly and rewritten in terms of equivalent plant set P as F X ðjoÞ

PðjoÞ kc F a ðjoÞ  ¼ 0 1 þ GðjoÞPðjoÞ 1 þGðjoÞPðjoÞ X e ðjoÞ

ð22Þ

Considering the effects of uncertainty in the hydraulic actuator dynamics, (22) can be manipulated to give the following inequality that is used to guide the design of the displacement

Fig. 13. Normalized closed-loop sensitivity with (solid line) and without (dashed line) displacement compensation filter.

compensation filter        kc   kc     r F X ðjoÞ r  PðjoÞ PðjoÞmax min

ð23Þ

A plot of the set of kc =PðjoÞ magnitudes is shown in Fig. 11 along with the magnitude plot of the designed displacement compensation filter, which has transfer function F X ðsÞ ¼

1:6  104 s s þ 1000

ð24Þ

Referring to Fig. 12, filter FX(s) was designed to closely follow the locus of 9kc =PðjoÞ9min up to 1000 rad/s. Selecting 9F X ðjoÞ9  9kc =PðjoÞ9min gives a well-damped regulating response. Transfer function (24) shows that the designed displacement compensation filter is essentially a differentiator on the motion of the position controlled actuator, up to a certain frequency. This type of filter structure improves the performance of the force control system by approximately canceling the inherent cylinder velocity feedback (Kosuge et al., 1996; Alleyne & Liu, 1999; Sivaselvan et al., 2008). Fig. 13 shows the normalized nominal closed-loop sensitivities of the hydraulic force control system with and without the displacement compensation filter. Referring to Fig. 13, the addition of the displacement compensation filter extends the range of frequencies over which the closed-loop sensitivity is less than 0 dB from approximately 80 rad/s without the displacement compensation filter (solid line) to 400 rad/s with the displacement compensation filter (dashed line). Thus, the design of the displacement compensation filter has an important role to play in alleviating disturbances caused by movement of the position controlled actuator.

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normalized force

4.2.4. Controller verification The performance of the designed force control system in the time-domain should be evaluated prior to implementation on the experimental hydraulic test bench because the closed-loop QFT design inequalities are only satisfied in the frequency domain. To this end, the closed-loop control system was simulated using the original nonlinear state Eq. (1). Several hundred simulations were performed using the extreme combinations of the uncertain system parameter values given in Table 1 and for different step command inputs ranging from 1 to 7.3 kN. Normalized step responses with xc(t) ¼0, i.e. the position controlled actuator held in a fixed position, are shown in Fig. 14a. The acceptable step response envelope defined by (15) and (16) is included for reference. The corresponding control signals are shown in Fig. 14b. Referring to Fig. 14a, the step responses all fall well within the specified acceptable response envelope, which confirms the validity of the control system design in the time domain.

607

Moreover, since the servovalve control signal always remains below the 10 mA saturation level, the simulation results have verified that large changes in the actuator force output can indeed be achieved without saturating the control valve input. The performance of the force control system with the position controlled actuator in motion was examined next. In the

1 0.8 0.6 acceptable response envelope

0.4 0.2 0 0

0.05

0.1

0.15

0.2

time (sec)

Fig. 16. Representative experimental step responses of closed-loop force control system: (a) normalized force and (b) control signal.

control signal (mA)

10 8 6 4 2 0 0

0.05

0.1

0.15

0.2

time (sec) Fig. 14. Representative normalized step responses of closed-loop force control system (nonlinear simulations): (a) normalized force and (b) control signal.

force error (kN)

0.5 0.25 0 −0.25 −0.5 0

0.1

0.2

0.3

0.2

0.3

time (sec)

force error (kN)

0.5 0.25 0 −0.25 −0.5 0

0.1 time (sec)

Fig. 15. Closed-loop force regulating errors arising from movement of the position controlled actuator: (a) with filter FX(s) and (b) without filter FX(s).

Fig. 17. Typical experimental force regulating errors arising from motion of position controlled actuator: (a) displacement response of position controlled actuator; (b) force control system response with filter FX(s) and (c) force control system response without filter FX(s).

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Fig. 16 shows a set of representative experimental closed-loop step responses, and the corresponding control signals, covering a range of force setpoints from 1 to 7.3 kN. In the experiments, the commanded displacement of the position controlled actuator was zero, i.e. xc(t)¼0. The experimental results are normalized to facilitate comparison with the specified acceptable response envelope. As expected, the experimental step responses fall within the design bounds. Comparing the experimental responses of Fig. 16 with the nonlinear simulation results shown in Fig. 14, it is observed that there is a good degree of consistency between the two sets of data. This confirms the validity of the proposed modelbased control system design approach and verifies the efficacy of the numerical procedure used to identify the equivalent linear plant set, P, for the controller design.

Typical experimental responses under excitation of the position controlled actuator are shown in Fig. 17. In the experiments commanded force was held at different constant values, in the range 1–7.3 kN, and a 5-mm step command was applied to the position controlled actuator. Fig. 17a shows the resulting displacement responses of the position controlled actuator under the various applied loads. The force errors, Fe ¼Fa  Fc, arising due to the motion of the position controlled actuator are shown in Fig. 17b with the displacement compensation filter active in the control loop. Fig. 17c shows similar closed-loop regulating responses, without using FX(s), for comparison. Referring to Fig. 17b, it is seen that the force control system can maintain the force error within 7 400 N of the force setpoint and zero the force error in less than 0.1 s. This result is consistent with the trends observed in the nonlinear simulations (see Fig. 15) and, when further compared with the regulating responses of Fig. 17c, confirms the efficacy of the designed displacement compensation filter. The ability of the electrohydraulic loading actuator to track a dynamic force profile is demonstrated in Fig. 18. The commanded force profile employed in the test (see Fig. 18a) represents a realworld aerodynamic load disturbance that acts on an electrohydraulic flight actuator during an in-flight manoeuvre (Karpenko and Sepehri, 2009). The corresponding motion of the flight actuator during the manoeuvre is shown in Fig. 18b. Referring to Fig. 18a, it is observed that the experimental electrohydraulic loading actuator can track the commanded force profile with an excellent degree of accuracy despite the over 70-mm of variation in the position of the flight actuator. As shown in Fig. 18c, the peak value of the force error never exceeded 0.3 kN.

Fig. 18. Results of force tracking experiment with filter Fx(s) active: (a) measured and commanded force; (b) displacement of the position controlled actuator and (c) force error.

Fig. 19. Results of force tracking experiment with filter Fx(s) disabled: (a) measured and commanded force; (b) displacement of the position controlled actuator and (c) force error.

simulations, the displacement of the position controlled actuator was varied by applying a 5-mm step command to the system. Simulated force errors, Fe ¼Fa Fc, are shown in Fig. 15, for various combinations of the uncertain model parameters and for different force set points. With reference to Fig. 15, it is observed that the use of the displacement compensation filter enables the peak values of the force error to be reduced by approximately 50%. Moreover, the 2% settling time of the regulating response is improved by more than 30% through the use of the displacement compensation filter. Thus, the implementation of filter FX(s) greatly improves the ability of the force control system to accommodate the motion of the position controlled actuator.

5. Experimental results

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Fig. 19 confirms that without the displacement compensation, the force tracking performance of the loading actuator is significantly degraded. This is especially true for quick changes in the position of the flight actuator, which are observed between 5 and 10 s (see Fig. 19b). With reference to Fig. 19c, the peak value of the force error is about 0.85 kN, which is nearly 3 times larger than the peak force error recorded when FX(s) was active in the force control loop.

6. Conclusions This paper presented the design of a robust force control scheme for an electrohydraulic loading actuator utilized as part of a hardware-in-the-loop flight simulator. The loading actuator is used to recreate the aerodynamic load disturbance acting on a second, position controlled, hydraulic flight actuator during simulated flight manoeuvres. This electrohydraulic force control problem is challenging because the hydraulic actuator hardware and the dynamics of the flight actuator influence the force transfer function in a way that makes it difficult to develop an accurate force tracking system using simple feedback control. These issues were addressed in a step-by-step fashion by using nonlinear quantitative feedback theory (QFT) to design the electrohydraulic force control system. The QFT-based feedback controller was designed to desensitize the control loop to uncertainty in the actuator dynamics caused by system nonlinearities and typical variations in the actuator parameters. A prefilter and a displacement compensation filter were also designed to further shape the reference tracking and regulating responses. Experimental test results demonstrated the robustness of the force control system against the servovalve flow nonlinearity as well as the ability of the loading system to maintain small force tracking errors despite large motion of the position controlled flight actuator.

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