Design and experimental evaluation of a robust force controller for an electro-hydraulic actuator via quantitative feedback theory

Transcription

1 Control Engineering Practice 8 (2000) 1335}1345 Design and experimental evaluation of a robust force controller for an electro-hydraulic actuator via quantitative feedback theory N. Niksefat, N. Sepehri* Experimental Robotics and Teleoperation Laboratory, Department of Mechanical and Industrial Engineering, The University of Manitoba, Winnipeg, Manitoba, Canada R3T 5V6 Received 17 September 1999; accepted 4 May 2000 Abstract This paper presents the design and experimental evaluation of an explicit force controller for a hydraulic actuator in the presence of signi"cant system uncertainties and nonlinearities. The nonlinear version of quantitative feedback theory (QFT) is employed to design a robust time-invariant controller. Two approaches are developed to identify linear time-invariant equivalent model that can precisely represent the nonlinear plant, operating over a wide range. The "rst approach is based on experimental input}output measurements, obtained directly from the actual system. The second approach is model-based, and utilizes the general nonlinear mathematical model of a hydraulic actuator interacting with an uncertain environment. Given the equivalent models, a controller is then designed to satisfy a priori speci"ed tracking and stability speci"cations. The controller enjoys the simplicity of "xed-gain controllers while exhibiting robustness. Experimental tests are performed on a hydraulic actuator equipped with a low-cost proportional valve. The results show that the compensated system is not sensitive to the variation of parameters such as environmental sti!ness or supply pressure and can equally work well for various set-point forces Elsevier Science Ltd. All rights reserved. Keywords: Hydraulic actuators; Force control; Uncertain dynamic systems; Nonlinear systems; Robust control 1. Introduction Hydraulic systems are potential choices for modern industries due to their sti!ness, compactness and high payload capability. Their application scope ranges from heavy-duty manipulators to precision machine tools. Hydraulic actuators are able to maintain their loading capacity inde"nitely, something that would usually cause excessive heat generation in electrical components (Alleyne, 1996). They are also advantageous for applications requiring environmental interactions because of their high force-to-weight ratio and fast response time. The utilization of hydraulic systems, however, is not without a cost. Hydraulic systems are complex and pose nonlinearities, which make the modelling and design of feedback controllers challenging. The nonlinearities are mainly due to servovalve #ow-pressure characteristics, ori"ce area openings, variations of #uid volume under compression and in part, to cavitation and seal friction. * Corresponding author. Tel.: # ; fax: # address: nariman@cc.umanitaba.ca (N. Sepehri). Aside from the nonlinearities, hydraulic systems contain large extent of model uncertainties (Yao, Bu, Reedy & Chiu, 1999). The uncertainties can originate from #uctuation in supply pump pressure, variation of some hydraulic parameters such as bulk modulus and for force control tasks, changes in the environmental sti!ness. To overcome these di$culties much research has been undertaken. This paper addresses the problem of robust force control for hydraulic actuators that experience uncertainties in both the environment and the hydraulic functions. Force control in hydraulic actuators is a di$cult problem. Conventional PID controllers do not yield reasonable performance over a wide range of operating conditions (Alleyne, Liu & Wright, 1998; Niksefat and Sepehri, 1999). Several researchers have, therefore, considered the use of adaptive and sliding mode control techniques. To name a few, Chen, Lee and Tseng (1990) designed a variable structure force controller for a single-rod hydraulic cylinder. Using position, velocity, acceleration, force and pressure feedback signals, the variable structure controller proved to be capable in both static and dynamic force control tasks. The controller, however, exhibited steady-state errors for step-inputs /00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S ( 0 0 )

2 1336 N. Niksefat, N. Sepehri / Control Engineering Practice 8 (2000) 1335}1345 Nomenclature A, A piston areas β e!ective bulk modulus c ori"ce discharge coe$cient d actuator' viscous damping coe$cient valve ori"ce area f measured force f actuator force k environmental sti!ness k valve gain (s) loop transmission function m mass of moving part p return pressure p, p line pressures p pump pressure P set of plants q, q #uid #ow rates S(s) sensitivity function ¹(s) closed-loop system transfer function ¹, ¹ lower and upper tracking bounds, respectively u input signal <, < #uid volumes x actuator displacement x spool displacement W nonlinear plant set Y acceptable plant output set ρ hydraulic #uid density τ valve time constant and the control signal was discontinuous. Alleyne and Hedrick (1995) applied an adaptive force control for an active suspension system driven by a double-rod hydraulic cylinder. The nonlinear dynamics of the electrohydraulic actuator was considered and was used to formulate a nonlinear control law. Vossoughi and Donath (1995) formulated a feedback linearization method for the control of an electro-hydraulic servo system. Their approach took into account nonlinearities associated with asymmetric actuation, variations in the trapped #uid volume, pressure-#ow characteristics and valve underlapping. Since, feedback linearization incorporates auxiliary measurements of some states, the method relies on measurements of at least the load's position, velocity and the hydraulic line pressures. This makes their method less appealing for industrial implementation. Wu, Sepehri and Ziaei (1998) implemented a generalized predictive force control algorithm to a hydraulic actuator. The controller was experimentally evaluated for various environment sti!nesses. The method, however, relies heavily on on-line parameter estimation and consequently demands large computational time. Conrad and Jensen (1987) also addressed the problem of force regulation in a single hydraulic piston. Combination of velocity feedforward and a Luenberger observer were implemented in simulation and experiment. However, the variations of load and supply pressure were not considered in their study. Robust control algorithms have also been studied to design hydraulic force controllers. Laval, M'Sirdi and Cadiou (1996) used an H linear design approach to control the force exerted by a double-acting symmetric hydraulic cylinder. The importance of environmental uncertainties and hydraulic function nonlinearities, on the control system's performance, were highlighted and limited test results, demonstrating the achievement of a stability/performance trade-o!, were presented. H -based controllers, however, have been reported to be normally of high-orders which make them di$cult for practical implementation (Landau, Rey, Karimi, Voda & Franco 1995). This paper utilizes the nonlinear version of quantitative feedback theory (QFT) to design a time-invariant force controller for a hydraulic actuator. Only a brief discussion of QFT is given here, since a detailed discussion can be found in books by Horowitz (1992) or Yaniv (1999). QFT o!ers a direct frequency-domain design methodology for satisfying robust performance objectives in uncertain plants. QFT can be distinguished from other frequency-domain methods, such as H optimal control or LQG/LTR, at least in its ability to deal simultaneously with di!erent types of uncertainty models and speci"cations (Chait, 1997). The method has been successfully applied to many applications ranging from a #ight control problem (Pachter, Houpis & Kang, 1997; Thompson, Pruyn & Shukla, 1999) to robot positioning (Bossert, Lamont, Leahy & Horowitz, 1990). However, for a long time and even today, its e!ectiveness is not widely recognized (Jayasuriya, 1993). In particular, the nonlinear QFT control approach has received little attention in the literature. Thus, there is still very little experience in its use and there is very little known about its utility and range of applicability (Banos & Bailey, 1996). Part of the objective of this paper is to evaluate, for the "rst time, the application of the nonlinear QFT approach to the design of hydraulic force control systems. The approach taken in this study is to "rst identify linear time-invariant equivalent models that can precisely represent the nonlinear plant under a wide range of operation. Thus, the nonlinear QFT methodology is not simply a linearization of the nonlinear plant about an equilibrium point. Two methods are employed to "nd the equivalent models. The "rst approach utilizes experimental input}output data obtained directly from the acceptable performance of the actual system. Generally, this approach is applicable to linear or nonlinear plants for which no analytical model is available (Horowitz, 1992). For example, this approach has been applied to

3 N. Niksefat, N. Sepehri / Control Engineering Practice 8 (2000) 1335} design a controller to automate a welding process and the input}output data set was generated by experienced welders (Bentley, 1990, 1992). This approach is suitable for implementation on complex hydraulic systems, such as excavator machines, for which the derivation of mathematical models and the determination of many parameters are di$cult and time-consuming. The second approach is model-based and employs the general nonlinear mathematical model of the system. The acceptable input}output set is then derived analytically or through numerical simulations. This approach has been successfully applied to design robust #ight controllers based on nonlinear model of the aircraft (Horowitz, Golubev & Kopelman, 1980; Miller, Horowitz, Houpis & Bar"eld, 1992). In the present work, nonlinear mathematical model of a hydraulic actuator interacting with an environment is used. Uncertainties are included in the model by allowing large variations of some hydraulic parameters. The input}output data set, obtained from either the experimental or the model-based approach, is then used within an algorithm proposed by Golubev and Horowitz (1982), to derive the equivalent time-invariant plant models. Once the equivalent time-invariant plant models are derived, a robust controller is designed to provide desirable performance for the entire set members. With the proposed design method, many nonlinearities such as nonlinear valve ori"ce area openings or variations in the trapped #uid volume, are incorporated into the design procedure. These nonlinearities are normally overlooked in conventional linearization methods. Moreover the presented method does not need precise knowledge of the plant parameters. For example, in this study the environmental sti!ness as well as the pump pressure are considered explicitly unknown but bounded, both in the modelling and in the design procedure. Overall, the present work makes the following contributions: (i) a robust force controller is designed based on nonlinear QFT methodology that overcomes many of the nonlinearities and uncertainties present in an experimental industrial hydraulic actuator; (ii) a numerical algorithm is developed for derivation of the input control signals given plant outputs in a hydraulic actuator interacting with the environment; and (iii) the force controller is implemented on a hydraulic actuator to assert the validity of nonlinear QFT methodology. 2. Experimental setup With reference to Fig. 1, the experimental test station consists of a hydraulic single-rod cylinder, a 486/66- based PC, equipped with a Metrabyte M5312 quadrature incremental encoder card and a DAS-16 analog/digital (A/D) conversion card and, a hydraulic power supply that provides constant operational supply pressure. The Fig. 1. Schematic diagram of hydraulic actuator test stand. hydraulic valve is a low-cost, closed-centre four-way proportional valve. The positioning of the valve spool (x )is based on the pulse width modulation principle. The valve operates within the range $1.8 V input signal and has a dead-band of +$0.15 V within which the actuator does not move. The reaction time of the valve, from neutral position to the maximum spool travel, is rated at 120 ms. A spring is used to represent the environment. Replacing the spring can change the sti!ness (k ) of the environment. The force transducer, with the capacity of 1000 lb (4.5 kn), is inserted between the environment and the piston rod. The 12-bit analog-to-digital conversion allows a force resolution of 10 N when the full range is used. The control signal is converted to an analog signal by the A/D card and is transmitted to the hydraulic valve ampli"er. 3. Outline of robust controller design The structure of a two-degree-of-freedom QFT control system is shown in Fig. 2. The controller, G(s), and the pre"lter, F(s), are to be designed to meet frequency bound speci"cations. w3w describes any member of the nonlinear and/or time-varying plants. Let set Y contain all acceptable plant outputs, y. The linear time-invariant equivalent of W is de"ned as P" p p " [y ] [x ], y 3Y, x "w(y ), w3w, (1) where [..] denotes the Laplace transform. It is essential that Y consists of all the desired outputs the system is to deliver, and W contains all system uncertainties and

4 1338 N. Niksefat, N. Sepehri / Control Engineering Practice 8 (2000) 1335}1345 Fig. 2. Two-degree-of-freedom QFT control system. plant variations. The "rst step in designing a robust QFT controller is to generate sets of acceptable input}output time histories over a wide range of operation. Two approaches are used in this paper for generating such a set. The "rst approach uses only experimental input}output data obtained directly from the experimental test stand. In the second approach, a complete nonlinear mathematical model of the hydraulic system is used to derive input}output time histories. For the hydraulic system under investigation, the coupled and complex nonlinear equations hamper any utilization of an analytical method. Therefore, a novel inverse model is developed, which is solved for the required input signal, given any acceptable plant output trajectory. Once the acceptable plant input}output set is determined, experimentally or numerically, Golubev's method, an algorithm developed by Golubev and Horowitz (1982), is applied. Golubev's method takes the time histories of input}output signal pairs and numerically determines a linear transfer function relating the two. The method is powerful since (i) it could work when plant input}output data is only available over "nite time intervals, i.e. [0, ¹]; (ii) it determines the transfer function directly, without calculating the Laplace transforms of the input and output signals separately, and (iii) it involves only integration of the data and is therefore highly immune to noise. Once the equivalent models are derived, QFT method is employed to develop a single linear robust controller. If a controller provides acceptable responses for the linear time-invariant equivalent models, then Schauder's "x-point theorem guarantees the same controller provides acceptable responses for the original nonlinear plant (for proof see Horowitz, 1976; Banos & Bailey, 1996). 4. Derivation of linear time-invariant equivalent models 4.1. Experimental approach In this section the equivalent linear time-invariant models are derived based on the experimental data obtained from the test stand shown in Fig. 1. The input is the voltage applied to the proportional valve and the output is the reading of the force sensor. To extract the acceptable plant outputs, a proportional controller and a second-order pre"lter are applied, similar to the con"guration shown in Fig. 2. The pre"lter is inserted to shape the reference input and to prevent the control signal from saturation. For each operating condition, the proportional controller gain is adjusted by trial-and-error to produce responses with acceptable design speci"cations. The design speci"cations are to have the settling time of +1 s and maximum percent overshoot less than 5% for the compensated system. The acceptable input}output signals are recorded over the following range of parameter changes: Environmental sti!ness 25}100 kn/m, pump supply pressure 250}400 psi, desired contact force 500}1000 N For practical cases, a limited number of combinations is su$ce. However, these combinations should include the extreme working conditions. Typical experimental responses are shown in Fig. 3. Due to the existence of valve dead-band, the results show steady-state errors. Hence, only the portion of input}output set beyond the dead-band is used for derivation of equivalent models. For each input}output pair, Golubev's method is applied to derive directly, the plant rational transfer function. Using this method, the hydraulic actuator in contact with the environment, can be represented by a family of second-order transfer functions in the following form: K P (s)" (1#s/α )(1#s/β ), (2) where K 310[1.5, 10], α 3[0.04, 0.2] and β 3[15, 28]. Bode plots of the above transfer functions are shown in Fig Model-based approach This approach, unlike the experimental one, is based on the mathematical model of the system. First the general nonlinear equations describing the hydraulic actuator interacting with the environment, are determined. Then, a numerical algorithm is presented, which given the desired output, calculates the required input Mathematical model The schematic diagram of the system under consideration is shown in Fig. 1. The environment, is represented by a pure sti!ness, k, which is consistent with the experimental set-up. This type of environment has enjoyed popularity among many researchers (Seraji & Colbaugh, 1997). Certainly, this is the simplest model that represents the environment. A more complete model may include the e!ect of environmental damping and/or inertia. The methodology presented here for designing the controller,

5 N. Niksefat, N. Sepehri / Control Engineering Practice 8 (2000) 1335} Fig. 3. Typical experimental input}output responses. and the hydraulic compliance, in this work. Hence, their dynamics are not excited during the contact, and they are considered lumped together as a rigid body. The governing equations that describe the nonlinear valve #ow characteristics can be written, as (Merritt, 1967). for extension (x *0): q "c (x ) 2 ρ (p!p ), q "c (x ) 2 ρ (p!p ), (6a) Fig. 4. Plant frequency domain responses obtained from experimental approach. for retraction (x (0): q "c (x ) 2 ρ (p!p ), however, is independent of the environmental model used. The dynamic equations of the actuator combined with the environment, modelled as a pure sti!ness, are described by mxk #dx #k x"f, (3) f"k x, (4) where f is the sensed force. x, m and d are the actuator's displacement, inertia of the moving part and the equivalent viscous damping coe$cient, respectively. The force originating from the hydraulic actuator, f,is f "p A!p A, (5) where A and A are the e!ective piston areas. p and p are the input and output line pressures, respectively. Note, the sti!ness of the force sensor and piston rod are too high, in comparison with the environmental sti!ness q "c (x ) 2 ρ (p!p ), (6b) where q and q represent #uid #ows into and out of the valve, respectively. c is the ori"ce coe$cient of the discharge, ρ is the mass density of the #uid, p is the pump pressure and p is the return (exit) pressure. (x ) represents a nonlinear function that relates the spool displacement, x, to the valve ori"ce area. For the round port valve in our system, this relationship is shown in Fig. 5, which matched well with experimental observations (Ziaei, 1998). Continuity equations for the oil #ow through the cylinder are dx q "A dt #< (x) dp β dt, dx q "A dt!< (x) dp β dt, (7a) (7b)

6 1340 N. Niksefat, N. Sepehri / Control Engineering Practice 8 (2000) 1335} Derivation of input}output set Towards derivation of equivalent plant set, "rst a set of acceptable output responses is generated. Then a numerical algorithm is developed, based on the mathematical nonlinear model, to determine the corresponding input set. A simple means for generating acceptable plant responses, y (t)"f (t), is to set > (s)" [y (t)]"¹(s)r(s), (10) where R(s)" k s k3[500, 1000] (11) Fig. 5. Valve ori"ce area versus spool displacement. β is the e!ective bulk modulus of the hydraulic #uid and, < (x) and < (x) are the volumes of the #uid trapped at the sides of the actuator. They are expressed as < (x)"<m #xa, (8a) < (x)"<m!xa, (8b) where <M and <M are the initial #uid volumes trapped in the blind and the rod side of the actuator. Finally, the relationship between the spool displacement, x, and the input voltage, u, to the proportional valve can be expressed by the following "rst-order di!erential equation (Ziaei, 1998): u" τ dx k dt # 1 x, (9) k where τ and k are gains describing the valve dynamics. Eqs. (3)}(5), (6a), (6b), (7a), (7b), (8a), (8b) and (9) express the relationship between the contact force, f, and the input control voltage, u. and 1 ¹(s)", a3[5, 25]. (12) (1#s/a)(1#9.6s/50#s/50) Note that the upper and lower bounds of the output responses, generated by (10), satisfy the speci"cations previously set (i.e., 1 s settling time and 5% maximum overshoot). Fig. 6a shows the acceptable generated output derived from (10). Assuming, a desired contact force trajectory, f (t), de"ned by (10) on a "nite interval [0, ¹], Eqs. (3)}(5) are applied to produce the corresponding actuator displacement, x (t), and the force required by the actuator, f (t), x (t)" f (t) "φ(t), (13) k f (t)"p (t)a!p (t)a "mφk (t)#dφ (t)#k φ(t)"ψ(t). (14) From (14) one can obtain p (dropping t, that denotes the time dependency, for simplicity), p " p A!ψ. (15) A Fig. 6. Acceptable plant input}output histories.

7 N. Niksefat, N. Sepehri / Control Engineering Practice 8 (2000) 1335} The relations between q and q from Eqs. (6a) and (6b) are q "q (p!p ) (p!p ) q "q (p!p ) (p!p ) (x *0), (16a) (x (0). (16b) Inserting q and q from (7) into (16), substituting p with (15) and rearranging (16) to derive p, yields, (φ)ψq #βa p "< A φ p!p!βa φ p A!ψ!p A < (φ) p A!ψ!p # < (φ)a p!p A A "Ω(p, φ, φ, ψ, ψq ), (x *0). (17a) p "< (φ)ψq A #βa φ p!p!βa φ p!p A!ψ < (φ) p!p A!ψ A # < (φ)a p!p A "Ω(p, φ, φ, ψ, ψq ), (x (0). (17b) Eqs. (17a) and (17b) can be solved numerically given any pre-speci"ed contact force trajectory, f (t) shown in (13). Consequently, p, q and q are obtained from Eqs. (15), (7a) and (7b), recursively. The spool displacement is then derived from Eqs. (6a) and (6b) as follows: x " x " q c ) 2 ρ (p!p q c ) 2 ρ (p!p A (x *0), (18a) (x (0). (18b) Finally the required input signal, to achieve the desired contact force, is determined from (9): Environmental sti!ness 25}100 kn/m, pump supply pressure 250}400 psi, desired contact force 500}1000 N, bulk modulus (β) 0.5(10)}1.5(10) Pa. Note that the above variations are similar to the experimental approach (see Section 4.1), except for the variation of the #uid bulk modulus is also included. The e!ective bulk modulus of a hydraulic system could significantly change under various load conditions, oil temperature and air contents in the oil (Yu, Chen & Lu, 1994). Acceptable plant inputs, corresponding to acceptable plant outputs generated by Eq. (10), are shown in Fig. 6b. For each input}output pair, Golubev's method is applied to directly derive a rational transfer function. It was found that the hydraulic actuator in contact with the environment, can be represented by a family of secondorder transfer functions having the following form: K P (s)" (1#s/α )(1#s/β ), (20) where K 310[1.5,9.5], α 3[!0.05,#0.03] and β 3[16, 25] The Bode plots of the family of transfer functions are shown in Fig. 7. This section concludes with some observations about the results obtained for the equivalent plant sets, based on the two methods. With reference to Eqs. (2) and (20), both methods have produced similar transfer functions. The range of the open-loop pole, α, found by the modelbased approach, however, includes small positive values. These unstable poles were found, when Golubev's method was applied to high velocity responses with overshoots (see Fig. 6a). It is mainly caused by the nonlin- u" τ k dx dt # 1 k x. (19) The above algorithm based on Eqs. (17a), (17b), (18a), (18b) and (19), was examined for di!erent desired output forces, and the uniqueness of the solutions was checked by solving Eqs. (3)}(5), (6a), (6b), (7a), (7b), (8a), (8b) and (9) straightforwardly. There was an excellent agreement between the results of the simulation program and the acceptable plant outputs. Various combinations of the system parameter ranges as shown below, were used: Fig. 7. Plant frequency domain responses obtained from model-based approach.

8 1342 N. Niksefat, N. Sepehri / Control Engineering Practice 8 (2000) 1335}1345 earity in the valve opening areas (see Fig. 5). At output velocities above certain values, hydraulic systems with overlapped or round ports valves, give an increasing #ow gain characteristic that can cause instablity (Merritt, 1967). Note that, high-velocity responses did not appear in the experiments described here due to the utilization of a proportional controller and the limitation for its adjustment. A QFT controller is now designed. The family of transfer functions found by the model-based approach is used for the controller synthesis. 5. QFT controller synthesis The family of transfer functions, P (s), presented by Eq. (20), is now used to design a robust force controller for the hydraulic actuator. A strictly proper controller, G(s), and a strictly proper pre"lter, F(s), are to be designed having the following properties: (i) Closed-loop robust stability: Associated QFT robust constraint is given by )M"1.4 ω3[0,r), (21) 1# (iω) where (s), the open-loop transfer function, is de"ned as (s)"p (s)g(s). (22) (ii) Robust tracking of reference input: For tracking performance requirement, the controller should satisfy the following inequality: ¹ (iω))¹(iω))¹ (iω) ω3[0,r), (23) where ¹(s)" F(s)G(s)P (s) 1#G(s)P (s), (24) and 1 ¹ (s)" (25a) (1#s/5) (1#(9.6/50)s#s/50) 1 ¹ (s)" (25b) (1#s/25) (1#s/80) (1#(9.6/50)s#s/50) ¹ (s) and ¹ (s) are the upper and lower tracking bounds, de"ned based on (12). The frequency domain plots of these bounds are given in Fig. 9. (iii) Closed-loop disturbance attenuation: For disturbance rejection at plant output, an upper tolerance is imposed on the sensitivity function. Here, only a constant upper bound is considered to limit the peak value of the disturbance ampli"cation as follows: S(iω) " 1 )M (ω)"1.2 ω3[0,r). 1# (iω) (26) Fig. 8. QFT bounds on Nichols chart and nominal loop. Fig. 9. Closed-loop frequency responses with QFT controller over range of parameter uncertainties. Inequalities (21), (23) and (26) impose constraints on the allowable nominal loop gain, ( (s)"p (s)g(s) and, P (s) is the nominal plant transfer function), for every phase (!2π))0) at each frequency. The above constraints can be displayed as a lower bound or a forbidden region, on the Nichols chart at each frequency. The synthesized (s) should meet these bounds as close as possible. The generated bounds by constraints (21), (23) and (26) are shown in Fig. 8, for selected frequencies. The nominal open-loop transfer function, (s), must satisfy all these bounds at each frequency. Furthermore, for the industrial hydraulic actuator under investigation, valve dead-band produces steady-state errors in the system responses in the absence of an integrator factor. To remove the steady-state errors, should contain an integrator. Hence, a possible loop can be obtained by cascading an integrator. The "nal loop shaping of the system without violating the bounds, is shown in Fig. 8.

9 The following controller is proposed: 0.08(1#s/4.5) (1#s/35) G(s)" s(1#(320/77 500)s#s/77 500). (27) The pre"lter is designed such that the closed-loop frequency responses lie between the tracking boundaries, ¹ (s) and ¹ (s). The suitable pre"lter is determined as (see D'Azzo & Houpis, 1988 for details), F(s)" (1#s/25) (1#s/14) (1#s/7). (28) N. Niksefat, N. Sepehri / Control Engineering Practice 8 (2000) 1335} Fig. 9 shows the closed-loop frequency responses for the equivalent transfer function family. Note that, the closed-loop frequency responses all satisfy the design speci"cation for the equivalent transfer function family. To evaluate the controller, it should be implemented on the original nonlinear system. The following experimental tests are then performed for this propose. Fig. 11. Step force responses for di!erent reference forces (experiment). 6. Experimental results The goal of the experiments presented here was to demonstrate the feasibility of the QFT force controller and to illustrate the capability of the proposed approach for designing controllers for hydraulic systems with signi"cant uncertainties. Therefore, the controller shown in (27) was implemented on the experimental test rig shown in Fig. 1. The e!ects of variations in environmental sti!- ness, pump pressure and loading were particularly evaluated. First, the variations in environmental sti!ness was tested. Three di!erent environmental sti!nesses of 25, 75 and 100 kn/m were used. The test results are shown in Fig. 10. In spite of an environmental sti!ness variation of about 300% and, large valve dead-band (+10%), the system responses remained insensitive and the steadystate errors are small. With reference to Fig. 10, experimental results exhibit initial delays mainly due to the Fig. 10. Step force responses with various environmental sti!nesses (experiment). Fig. 12. Step force responses with di!erent supply pressures (experiment). valve dead-band. Consequently the system responses do not precisely "t the design speci"cation bounds shown in dotted lines. Indeed, the e!ect of valve dead-band was not considered in the QFT design procedure and was only handled by cascading an integrator in the controller. Valve dead-band also caused the responses to have overshoots that did not dampen quickly, since the control signal had to sweep the entire dead-band range to change the sign. Fig. 11 compares the test results given three di!erent set-point forces (500, 800 and 1000 N). In spite of changing the loading condition by 100%, the system's performances, i.e. rise time and overshoot, did not change considerably as depicted in Fig. 11. The ability of the controller to cope with pump pressure variations was also tested. Typical results are shown in Fig. 12, where the pump pressure was varied by 60%. Finally, the ability of the control system to follow a step-input square-wave function was investigated. With reference to Fig. 13, the system displays good tracking performance. It is, however, seen that the response contains steady-state errors in retraction strokes. This phenomenon is mainly due to

10 1344 N. Niksefat, N. Sepehri / Control Engineering Practice 8 (2000) 1335}1345 Fig. 13. System response to square wave force set-point (experiment). di!erent piston cross-sectional areas, which results in di!erent plant gains for extraction and retraction strokes. Indeed, such a large retraction stroke in the square-wave step-input was not included when arriving at the acceptable plant input}output sets in the design procedure. 7. Conclusions This paper documented the design and experimental evaluation of a force controller for an industrial hydraulic actuator with several uncertainties, within the framework of the nonlinear quantitative feedback theory (QFT). The design methodology proceeded with the generation of linear time-invariant equivalent models using two methods. The "rst method was based on experimental input}output measurement of acceptable system responses. In the second method, the general nonlinear mathematical model was used for the derivation of input}output histories. Many plant nonlinearities including servovalve #ow-pressure characteristics, valve opening areas and variations in the trapped #uid volume as well as plant uncertainties such as pump pressure and environmental sti!ness, were included in the model. The extension method by Golubev and Horowitz (1982) was then employed to determine the family of linear transfer functions, which equivalently represents the original nonlinear system. The results indicated that a second-order transfer function family could represent the hydraulic actuator under investigation. Furthermore, there was a strong correlation between the results of the two approaches, indicating that the mathematical model developed here could accurately represent the system under investigation. The QFT design technique was then employed to design a robust controller which satis"es a priori de"ned tracking and stability performance. The designed linear controller is of low-order and incorporates only measured contact force as feedback, which makes it attractive for industrial implementation. The designed controller was implemented on an industrial hydraulic actuator equipped with a low-cost proportional valve. Several tests were performed under di!erent conditions including: variations up to 300% in environmental sti!ness, 60% in supply pressure and 100% in the reference force. The experimental results demonstrated the robustness of the QFT controller to real parameter variations and good performance in spite of signi"cant actuator dynamics. The results of this paper clearly showed that the nonlinear QFT approach could provide an e!ective tool for the control design of hydraulic actuators. Moreover, in this paper, nonlinear QFT approach as well as the Golubev's method was experimentally evaluated on a real industrial problem, which has been rarely reported in the literature. The contributions of this study have impact on the control of outdoor hydraulic manipulators such as those used in mining or underwater explorations/inspections. These manipulators must constantly interact with uncertain environments and have considerable non-ideal dynamics. The application of the methodology presented here is currently being investigated in the automation of complex hydraulic excavator machines that are used in primary industries. Due to the need for improved technology, these industries have started to employ advanced technologies on their existing machines. In this case, an expert operator can generate acceptable responses for a wide range of machine operations that will then be utilized to design a QFT controller. Acknowledgements The authors wish to acknowledge the support of the Natural Science and Engineering Research Council (NSERC) of Canada. The authors also wish to thank Mr. Al Lohse, a support sta! of the Mechanical and Industrial Engineering Department, in implementing the hardware and sensor packages. References Alleyne, A. (1996). Nonlinear force control of an electro-hydraulic actuator. Proceedings of the Japan/USA symposium on yexible automation, Boston, MA. Vol. 1 (pp. 193}200). Alleyne, A., & Hedrick, J. K. (1995). Nonlinear adaptive control of active suspensions. IEEE Transaction on Control Systems Technology, 3, 94}102. Alleyne, A., Liu, R., & Wright, H. (1998). On the limitation of force tracking control for hydraulic active suspensions. Proceedings of the American control conference, Philadelphia, PA (pp. 43}47). Banos, A., & Bailey, F.N. (1996). Linear control of uncertain nonlinear plants. Proceedings of the Third IFAC nonlinear control system design symposium, Tahoe City, CA (pp. 813}818). Bentley, A. E. (1990). Feedback control of pinch welding using QFT. Report No. Sand , Sandia National Labs., Livermore, CA.

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