Robust Control design of a Hydraulic Actuator using the QFT method

R. Nandakumar, G.D. Halikias and A. Zolotas

Abstract— In this paper a control design study of a non-linear hydraulic actuator is presented based on the QuantitativeFeedback Theory (QFT). The model is linearized around anoperating point and the uncertainty in the nominal plant isquantified. Using the robust design specifications, a robust QFTcontroller is designed using a novel fixed structure optimizationalgorithm. The resulting feedback controller has low complexityand, as shown via numerous simulations, is successful inmeeting the robust stability and performance specificationsof the design. The design illustrates the effectiveness of theproposed algorithm in achieving tight loop-shaping for a realsystem of medium complexity.

I. I NTRODUCTION

The paper presents the results of a case study involvingthe robust control design of a non-linear hydraulic actuator.The design method which is followed is based on Quanti-tative Feedback Theory (QFT) [1],[7],[8]. The model of theactuator represents a real system which is fully described in[10], [11]. The model is linearized around an operating pointand the uncertainty in the nominal plant is quantified. Thisinvolves ten uncertain parameters which are assumed to varyindependently over their corresponding ranges. Using therobust design specifications of [10], a robust QFT controlleris designed using a novel fixed-structure optimization method[6],[9]. This is followed by the design of a pre-filter usinga systematic procedure based on linear programming. Theresulting feedback controller has low complexity and, asshown via numerous simulations, is successful in meetingthe design objectives.

The paper is organized as follows: Section2 providesa brief background of recent research on force control ofhydraulic actuators, describes the main issues involved intheir design (e.g. non-linearities and model uncertainty) andvarious control methods that have been proposed in theliterature. Section3 outlines the modelling procedure of [10]for a hydraulic actuator interacting with an uncertain envi-ronment. Here emphasis is placed on highlighting the mainassumptions and for developing an appropriate parametricLTI model, together with a quantitative uncertainty modelwhich is used extensively in the sequel. Section4 definesthe design objectives and outlines the steps of the QFT-basedprocedure for designing a feedback controller and a pre-filterof low complexity. The design of the feedback controller is

R. Nandakumar, G.D. Halikias are with the Control EngineeringResearch Centre, School of Engineering and Mathematical Sciences,City University, Northampton Square, London EC1V 0HB, U.K.; A.Zolotas is with the Department of Electronic and Electrical Engi-neering, Loughborough University, Loughborough, Leicestershire, LE113TU, UK ramnath.nandakumar@teercoatings.co.uk ,g.halikias@city.ac.uk , A.C.Zolotas@lboro.ac.uk

based on a fixed-structure optimization method developed inprevious work [6], while the design of the pre-filter is carriedout via linear programming. Both steps are successful indesigning the overall compensation scheme, as is illustratedvia numerous simulations and analysis of the results. Thefeedback controller is compared with that of [10] whichrelied on a manual QFT loop-shaping procedure and has beenvalidated experimentally. The two controllers exhibit similarresponses, which is not surprising as the design specificationsare tight. It is thus possible to conclude that the proposedfixed-structure optimization algorithms can help to automatethe loop-shaping step of the QFT design for realistic designproblems. The overall conclusions of the paper appear inSection5.

II. BACKGROUND AND GENERAL DESIGN OBJECTIVES

A wide range of engineering problems involve the controlof hydraulic actuators interacting with uncertain environ-ments. These include flight control, robot position con-trol, manufacturing systems, etc. Numerous control strate-gies have been proposed in the literature for the designof such systems, including combinations classical velocityfeed-forward/output feedback control, observer-based con-trol, sliding mode control, adaptive and switched controlschemes, generalized predictive control, robust control meth-ods based onH∞ optimization, etc. The main challengeof the design problem arises from model non-linearities,uncertainty of the environment and actuator parameters,and poor industrial control architectures, which limit theimplementation of complex algorithms [10]. Thus the designof simple controllers, ideally of a fixed structure, becomesan important design issue.

In this paper QFT techniques are employed to designan fixed-structure force control scheme for an industrialhydraulic actuator model. It is intended that the fixed gaincontroller:

1) Is low order and easy to implement.2) Is robust against uncertainties in both environmental

stiffness and actuator functions, and3) Does not require precise knowledge of the systems

parameters.

III. M ODELLING OF HYDRAULIC ACTUATOR

INTERACTING WITH ENVIRONMENT

The model of the linearised actuator and the accompanyinguncertainty model is based on references [10], [11]. Here, abrief outline is given with particular emphasis on various

modelling assumptions and simplifications and the origin ofmodel uncertainty.

A schematic of the hydraulic actuator is shown in Figure 1.Uncertainty in the model arises from variation in operating-point dependent parameters, changes in the environment andchanges in the hydraulic actuator’s functions.

ds

ksma me

de

ke

xex

d

xsp

AiAo

pipo

qi qo

Pe PePs

Fig. 1. Diagram of the hydraulic actuator (based on [10])

The model of the overall system can be decomposed intothree parts. These are: (i) Electric relay/valve, (ii) Actuator,and (iii) Manipulator-sensor-environment.

(i) Electric relay/Valve

The model for the electric relay/valve relates the appliedcontrol voltageV (t) to spool displacementxsp(t). By sup-plying power the lever inside the choke moves controllingthe valve’s openings. This part of the system is modelled asa first-order lag:

Xsp(s)V (s)

=ksp

1 + sτ

where ksp is a gain parameter andτ is the effective timeconstant.

(ii) ActuatorBy spool displacement, the effective openings of the valve

change. We thus have different pressuresPi, Po on the twosides of the piston and so a differential forceFa is developedwhich moves the piston. The model for this part of the systemis derived by linearising the non-linear fluid-flow equationsaround the nominal operating point, while neglecting leakageflow across the piston. The variation of the operating-pointdependent parameters is included in the model as uncertainty.Assuming that the system has the same response in extensionand retraction, the incremental flow variables can be writtenseparately in these two cases as:

Case (i), xsp ≥ 0 (extension):

qi = Kisxsp −Ki

ppi (1)

qo = Kosxsp + Ko

ppo (2)

Case (ii), xsp < 0 (retraction):

qi = Kisxsp + Ki

ppi (3)

qo = Kosxsp −Ko

ppo (4)

wherepi and po denote the increments in input and outputpressure, respectively and where the sensitivity parametersKi

s, Kip, Ko

s andKop takepositivevalues.

Next, consider the force developed on the piston due tothe presence of different pressures on its two sides. This isgiven as:

Fa = AiPi −AoPo

whereAi and Ao are the effective areas (inner and outer),and Pi, Po represent the inner and outer line pressures,respectively. The inward/outward flows may be expressedas:

qi = Aidx

dt+

Vi

β

dPi

dt

andqo = Ao

dx

dt− Vo

β

dPo

dt

where x denotes the displacement of the piston,β is theeffective bulk modulus of the hydraulic fluid andVi, Vo

represent the volumes of the fluid at the two sides of thepiston, which are functions of the piston displacement, i.e.Vi = Vi(x) and Vo = Vo(x). Now, for small displacementsx,

Vi = V i + xAi, Vo = V o + xAo

whereV i and V o represent the initial volumes of the fluidtrapped on the two sides of the piston. Also, assuming smallpiston displacements in the vicinity of the mid-stroke, thefollowing approximations can be made:

Vi(x)β

≈ Vo(x)β

≈ V i + V o

2β:= C

and thus,

qi = Aidx

dt+ C

dPi

dt, qo = Ao

dx

dt− C

dPo

dt

Using equations (1)-(4) we can write:

Pi(s) =Ais

Cs + Kip

X(s) +Ki

s

Cs + Kip

Xsp(s)

Po(s) =Aos

Cs + Kop

X(s) +Ko

s

Cs + Kop

Xsp(s)

Thus, the force acting on the piston can be written as:

Fa(s) =(

KsAo

Cs + Kp+

KsAi

Cs + Kp

)Xsp(s)

−(

A2i s

Cs + Kp+

A2os

Cs + Kp

)X(s)

Any variation in parametersKis (Ko

s ) andKip (Ko

p ) can beincluded as uncertainty in the new-defined parametersKs

andKp, respectively.

(iii) Manipulator-sensor-environment

This part of the system involves the model of themanipulator-sensor-environment which is coupled to the hy-draulic actuator dynamics. A schematic diagram of this partof the system is shown in Figure 2 below:

Fama me

x xe

Ks Ke

dds de

Fig. 2. Manipulator-sensor-environment

Here Ks and Ke represent sensor and environment stiff-ness,ds and de represent sensor and environment dampingand d models relative damping betweenms (sensor mass)and me (environment mass). The sensed force is measuredvia the elongation (contraction) of springKs,i.e.

F = ks(x− xe)

The force Fa is applied to the massma. The slidingfriction between the massesma andme surface is assumednegligible. The system equation can be written in terms ofthe two displacementsx andxe. The mechanical network isdrawn by connecting the terminals of the elements that havethe same displacement. So, we will have two sub-networks,and the equations of motion can be written for each one ofthem.

Force balancing for the first sub-system gives:

Fa = max + ds(x− xe) + dx + ks(x− xe)

Taking Laplace transforms gives:

Fa(s) = [mas2 + (d + ds)s + ks]X(s)− (dss + ks)Xe(s)

Force balancing for the second subsystem gives:

mexe − ds(x− xe) + dexe − ks(x− xe) + kexe = 0

and hence,

X(s) =mes

2 + (ds + de)s + ks + ke

dss + ksXe(s)

(iv) Overall model

Combining the various equations obtained so far allows usto determine the overall model of the system. After a numberof simplifying assumptions the transfer function betweenapplied voltageV (s) and measured contact forceF (s) isobtained in the form:

G(s) =ksp

τs + 1

[Kske(Ai + Ao)

(Kp + Cs)(mas2 + ds + ke) + (A2i + A2

o)s

]

It has been assumed that the stiffness of the force sensorand the piston rod are high compared with the environmentalstiffness and the hydraulic compliance and have setds =de = 0 andme = 0.

IV. U NCERTAINTY MODELLING AND QFT CONTROL

DESIGN

The transfer function of the hydraulic actuator obtained atthe end of the last section will be used for control design. Thevariables involved in the transfer function are summarizednext for convenience: Variablesks and ds represent sensorstiffness and damping, respectively. The sensor connects theactuator’s piston of massma to the environment, representedby a massme, stiffnesske and dampingde. Further,Ai andAo represent the effective inner and outer areas of the piston,τ and ksp are parameters describing the valve dynamics,while Ks andKp are load and pressure-dependent variables,respectively. Finally, parameterC is a constant arising fromthe linearisation of the fluid flow equations, is defined as:

C =1β

(V i + V o

2

)

whereV i andV o represent the initial volumes of hydraulicfluid trapped in the blind and the rod sides of the piston andβ is the effective bulk modulus of the fluid.

A list of all parameters defining the linearised transferfunction is given in Table 1. For each parameter a minimum,maximum and nominal value is given. The parameters areassumed to vary independently between their correspondingextreme values.

Parameter Nominal value Rangeke 75 (KN/m) 50− 100Ks 0.375 (m3/pa.s) 0.25− 0.5Kp 2.5× 10−12 (m2/s) 0− 5× 10−12

C 1.5× 10−11 (m3/pa) 1× 10−11 − 3× 10−11

d 700 (N/m/s) 600− 800ma 20 (Kg) 19.9− 20.1Ai 0.00203 (m2) 0.00193− 0.00213Ao 0.00152 (m2) 0.00144− 0.00160ksp 0.0012 (m/V ) 0.0011− 0.0013τ 35 (ms) 30− 40

Table 1: Operating values and parameter ranges

The uncertainty inKs and Kp reflects variations in theoperating point (especially the non-linearity arising at theinterface between positive and negative spool displacements),supply pressure and orifice area gradient. Uncertainty in pa-rameterske andd model variations in environmental stiffnessand damping, while uncertainty in valve characteristics ismodelled by variation is parametersτ andksp [10]. Variationin parameterC reflects changes in the fluid bulk modulusand the volumes of the fluid trapped at the two sides of thepiston. All these parameters are known to affect the dynamicstability of the system.

Following [10], the design frequencies were chosen asΩ = 0.01, 0.05, 0.1, 0.5, 15, 10, 50, 70, 100 rads/s. The ro-bust tracking bounds are defined by the magnitude frequencyresponse of the two systems:

Bu(s) =s

2.8 + 1(s4 + 1

) (s7 + 1

) (s8 + 1

)

and

Bl(s) =1(

s4.8 + 1

) (s80 + 1

) (s2

50 + 9.6s50 + 1

)

These were obtained by step-response figures of merit relatedto rise-time, percent overshoot and settling time [10]. Thusthe design specifications for the feedback controllerK(s)are:

maxp∈P

∣∣∣∣G(p, jωi)K(jωi)

1 + G(p, jωi)K(jωi)

∣∣∣∣dB

≤ |Bu(jωi)|dB−|Bl(jωi)|dB

which should hold for all10 design frequencies inΩ and forevery possible combination of the ten uncertain parametersvarying over their respective ranges specified in Table 1.The frequency response of the tracking specification boundsBu(jω) andBl(jω) are displayed in Figure 3 below (designfrequencies are indicated by circles).

10−3

10−2

10−1

100

101

102

103

−160

−140

−120

−100

−80

−60

−40

−20

0

20Closed loop tracking specifications

Angular frequency rads/s

dB

Fig. 3. Closed-loop design tracking specifications, upper and lower bounds

It is further required that the open-loop frequency response(for any permissible combination of parameters) should notenter the M = 1.4 circle, which gives the design anapproximate gain margin of3 dB. Finally, since for hydraulicactuators of this type the valve dead-band typically producesa steady-state-error in the system response [10], [11], integralaction is required from the feedback controller to eliminatethe steady-state error.

The uncertainty templates of the model were first obtainedby using a three-point grid for each uncertain parameter(nominal, minimum and maximum value). This resulted in310 = 59049 uncertain points in the Nichols chart for eachtemplate. To reduce the number of subsequent calculations,the convex hull of each uncertainty template was alsoobtained and used to derive the Horowitz bounds at eachdesign frequency; this process introduces some measure ofconservativeness to the design, as the uncertainty templatesneed not be convex, which in this case, however, is minimal.The uncertainty template of the plant at the sixth design

frequency (along with its convex hull) is shown in Figure4 below.

−102.5 −102 −101.5 −101 −100.5 −100 −99.5 −99 −98.5 −98 −97.560

62

64

66

68

70

72

74

76

78

Phase DEG

Gai

n D

B

Uncertainty template at design frequnecy 6

Fig. 4. Uncertainty template, sixth design frequency

The Horowitz bounds (robust performance trackingbounds) were next calculated numerically at the ten designfrequencies, with a gain tolerance of0.1 dB and a phase stepof 1. This was followed by the construction of theU-contour(universal high-frequency contour), corresponding to anM -circle withM = 1.4 and a high-frequency uncertainty spreadof V∞ = 11.03 dB, calculated analytically from the model.The corresponding contours are shown in Nichol’s chart(Figure 5), together with the nominal frequency responseof the plant (the ten design frequencies being marked witha circle). Note that only the first seven design frequenciescorrespond to open Horowitz templates.

−400 −350 −300 −250 −200 −150 −100 −50 0−40

−20

0

20

40

60

80

100

120

Phase DEG

Gai

n D

B

ω=0.01

ω=0.05

ω=0.1

ω=0.5

ω=1

ω=5

ω=10 U−contour

Nominal−plant FR

Fig. 5. Nominal-plant frequency response, Horowitz templates andU -contour

The most challenging step of the QFT design procedure is“loop shaping” [5], [2], [3], which involves the design of afeedback controller so that the nominal open-loop response:(i) does not encircle the critical (−1) point [4] (for nominalopen-loop stable systems), and (ii) lies in the high-frequencyregion of the Nichols chart (as defined by the open Horowitztemplates) at each design frequency and avoids enteringthe universalU -contour at every frequency. To avoid over-designing the system, it is desirable to meet the above twoobjectives with the minimum amount of gain.

In this case study the loop-shaping design procedure wasbased on an optimization algorithm reported in earlier work[9], [6], which is based on the following result involving PIDcontrollers:

Theorem 1: [6] Let K(s) = kp + kds + ki

s with kp,kd, ki real parameters. Suppose thatarg K(jωi) = ψi andarg K(jωj) = ψj whereωi 6= ωj . Then the matrix:

Aij =

1 − 1

ω2i

− tan(ψi)ωi

1 − 1ω2

j− tan(ψj)

ωj

has full (row) rank. Let(V ij) = [V ij1 V ij

2 V ij3 ]′ ∈ R3 be

a (real) non-zero vector in the (one-dimensional) kernel ofAij . Then,

kd

ki

kp

= λ

V ij1

V ij2

V ij3

:= λ

ωi tan ψi−ωj tan ψj

ω2i−ω2

jωiωj(ωj tan ψi−ωi tan ψj)

ω2i−ω2

j

1

whereλ is an arbitrary real constant. Moreover, the gain andphase of the controller at any frequencyω is given by:

|K(jω)|2 = |λ|21 + (ω(ωi tan ψi − ωj tan ψj)

ω2i − ω2

j

− ωiωj(ωj tan ψi − ωi tan ψj)

ω(ω2i − ω2

j ))2

and

arg K(jω) = arctan(ω(ωi tan ψi − ωj tan ψj)

ω2i − ω2

j

− ωiωj(ωj tan ψi − ωi tan ψj)

ω(ω2i − ω2

j ))

respectively. ¤The Theorem shows that fixing the phase of the PID

controller at two frequencies, will fix the phase of the con-troller at every frequency. Thus the controller is determinedup to scaling and it is straightforward to determine theminimum gain required to meet the QFT robust stability andperformance specifications. The procedure can be repeatedby considering phase pairs over a discretized grid to result inthe optimal (minimum gain) design consistent with the QFTconstraints. See [6] for full details. Similar techniques canbe applied to other controller structures (e.g. phase lead/lag,PDD2, etc).

The specifications of the present design indicate thatintegral action must be introduced via the feedback con-troller. First, it was attempted to design an optimal PID

controller using the method of Theorem 1, which provedto be infeasible. The reason in clear from Figure 5 whichindicates that a large amount of phase advance (exceeding90) should be introduced in the mid-high frequency range.Thus the controller structure was modified as:

K1(s) =k1 + k2s + k3s

2

s(

s130 + 1

)

The s-term in the denominator provides the required inte-gral action, while the numerator is aPDD2 (proportional-derivative-double-derivative) term providing sufficient phase-advance (up to180 at high frequencies). The additional poleat s = −130 was introduced to ensure that the controller isproper. Next, the denominator term ofK1(s) was absorbedto the nominal plant, and the three-parametersk1, k2 andk3

were optimized using the algorithm based on the variationof Theorem1 appropriate toPDD2 control structures [6].The cost-function chosen for optimization was the open-loopasymptotic gain (controlled byk3). The new optimisationproblem proved feasible and resulted in an optimalPDD2-controller with

k?1 = 0.004, k?

2 = 0.002 k?3 = 4.9778× 10−5

The resulting open-loop system is shown in Figure 6.

−240 −220 −200 −180 −160 −140 −120 −100 −80 −60 −40−40

−20

0

20

40

60

80

100

120

Phase DEG

Gai

n D

B

ω=0.01

ω=0.05

ω=0.1

ω=0.5

ω=1

ω=5

ω=10

ω=50

ω=70 ω=100

OL frequency response

U−contour

Fig. 6. Nominal open-loop response withPDD2-optimal controller

It can be seen that the design specifications at all tendesign frequencies are satisfied, the first seven points lyingon or above the corresponding templates, the last three (highfrequencies corresponding to the closed Horowitz contours)lying outside or on theU-template. It may be seen, howeverthat the nominal frequency response penetrates theU-contourbetween two consecutive design frequencies (ω = 10 andω = 50 rads/s). This is a common problem with QFT designwhich is based on a discrete set of design frequencies. Atypical remedy is to define a more dense set of designfrequencies or tighten the specifications. Here a simplertechnique was followed by adding an additional first-order

lag term to the controller to modify the open loop responsein the offending frequency range10 ≤ ω ≤ 50 rads/s. Theoverall controller is:

K(s) =(0.004 + 0.002s + 4.9778× 10−5s2)(0.0623s + 1)

s(s/130 + 1)(0.1295s + 1)

The corresponding open-loop response is shown in Figure 7.The nominal open-loop system has a cross-over frequencyof 16.91 rads/s and3-dB closed-loop bandwidth equal to28.21 rads/s. The shaped open-loop response is also shownin 8, along with the uncertainty templates at the ten designfrequencies. It may be seen that all design specifications aremet at the first seven design frequencies, with the nominalopen loop system lying above the corresponding performancebounds (Horowitz templates). The fact that five of theseseven points lie (almost) on the bounds is a consequence ofthe optimisation method that was used to design the feedbackcontroller and indicates that the design is in agreement withthe general spirit of the QFT philosophy, in the sense thatthe minimum possible gain is used, sufficient to meet therobust performance specifications. In can also be seen thatthe uncertainty templates for the last three design frequencieslie outside theM = 1.4 circle (equivalently the nominalfrequency response at these three frequencies lies outsidethe U-contour), so that the robust stability objectives ofthe design have also been met. Again, the three templatesare close to theM -circle boundary which indicates thatthe enforced stability margins are tight, for at least certaincombinations of the uncertain parameters. Robust stabilityis an important consideration in this case, as the systemmanifests a high-frequency resonance peak. This can beclearly seen from the Bode plots, or indirectly inferred fromthe shape of the uncertainty templates, whose phase spreadincreases at high frequencies, as the resonance frequencyshifts with parameter uncertainty.

−250 −200 −150 −100 −50

−40

−20

0

20

40

60

80

100

120

Phase DEG

Gai

n D

B

ω=0.01

ω=0.05

ω=0.1

ω=0.5

ω=1

ω=5

ω=10

ω=50

ω=70 ω=100

U−contour

Shaped OL response

Fig. 7. Nominal open-loop response with modifiedPDD2-optimalcontroller

−350 −300 −250 −200 −150 −100 −50 0

−20

0

20

40

60

80

100

120

Phase DEG

Mag

DB

Shaped Open−loop response with templates

Fig. 8. Shaped open-loop response and uncertainty templates

The controller is quite similar to the one designed in [10]which has been validated experimentally. This suggests thatthe design specifications in this case are tight.

The last step of the QFT design is to design a pre-filter. Here the following procedure was used: First, themagnitude frequency responses of35 = 243 open-loopuncertain systems were plotted (see 9). These correspond tothe five more important parameters (in terms of uncertaintytemplate spread), the remaining five parameters being fixedto their nominal value. Next, the maximum and minimumgains were recorded at the ten design frequencies. The valuesobtained are summarised in the table below.

10−1

100

101

102

103

−160

−140

−120

−100

−80

−60

−40

−20

0

20

Angular Frequency rad/s

Mag

nitu

de D

B

Closed loop responses and specifications (No prefilter)

Fig. 9. Closed-loop frequency responses - No pre-filter

Frequency (rad/s) Upper bound (dB) Lower bound (dB) )0.01 0.0000 0.00000.05 0.0003 −0.00040.10 0.0013 −0.00180.50 0.0300 −0.04371.00 0.1030 −0.17335.00 −1.0882 −3.8959

10.00 −6.1353 −14.051450.00 −30.1052 −55.794070.00 −35.8489 −65.5845

100.00 −41.9907 −76.4891

Table 2: Closed-loop specifications

Frequency (rad/s) Max gain (dB) Min gain (dB)0.01 0.0000 0.00000.05 0.0008 0.00020.10 0.0032 0.00070.50 0.0777 0.01771.00 0.2796 0.06315.00 2.2479 0.5656

10.00 2.7065 1.386450.00 −4.7396 −17.777870.00 −8.3369 −21.1456

100.00 −7.1198 −23.9876

Table 3: Closed-loop minimum and maximum gain

As expected, the spread in closed-loop gain is withinthe required tolerances; thus all responses can be broughtbetween the specified lower and upped bounds by designinga pre-filter which essentially provides frequency-dependentscaling. In this case, the pre-filter must provide adequateattenuation at high frequencies.

The filter was designed via an optimization procedurebased on linear programming. First, the difference of themaximum gain from the upper bound was recorded atthe design frequencies, together with the difference of theminimum gain from the lower frequency bound. The twodifferences were then averaged and this defined the requiredattenuation of the pre-filter (at the ten design frequencies).This procedure was followed so as to bring the responsesin the middle of the specified region in the Bode diagram.With the magnitude frequency-response of the filter specifiedat the ten design frequencies, the next task was to fit astable rational function to approximate the response. Varioustechniques (e.g. least-squares) can be used for this purpose,but the one that was followed was based on Matlab’s routinefitmaglp.m(from theµ-control toolbox) in which fitting canbe performed interactively over various filter orders, whilethe results are graphically displayed (target and achievedfrequency response). The function formulates the problemas a linear programme using weights to emphasize the fit atthe required frequency-ranges. In this example equal weightswere used for all (ten) frequencies. A filter order equalto three was found to give a good compromise betweenaccuracy and complexity. The transfer function of the filterwas obtained as:

F (s) =(1 + s/16.25)(1 + s/(20.08± j145.74))(1 + s/104.36)(1 + s/(4.52± j2.09))

so thatF (s) is both stable and minimum-phase (as guaran-teed by this method). The specified and achieved responsesof the filter at the design frequencies are summarised in theTable below:

Frequency (rad/s) Target gain (dB) Achieved gain (dB)0.01 0.0000 0.00000.05 −0.0005 −0.00060.10 −0.0021 −0.00220.50 −0.0528 −0.05461.00 −0.2120 −0.20655.00 −4.8403 −3.8989

10.00 −12.2030 −12.139750.00 −31.8440 −31.690970.00 −36.7536 −35.9755

100.00 −43.9554 −43.6862

Table 4: Target and achieved pre-filter gains

The closed-loop responses of the system (with pre-filter)are shown in Figure 10. As expected, these are all containedwithin the specified upper and lower bounds. The35 (unit)step responses of the system are finally shown in Figure 11.

10−1

100

101

102

103

−160

−140

−120

−100

−80

−60

−40

−20

0

20

Angular Frequency rad/s

Mag

nitu

de D

BClosed loop responses and specifications (with prefilter)

Fig. 10. Closed-loop frequency responses - With pre-filter

V. CONCLUSIONS

In this paper the robust control design of a hydraulicactuator has been undertaken based on the QFT designmethodology. The nominal model of the actuator based on[10] has been obtained via a linearisation procedure and anumber of simplifying assumptions. This procedure has alsoquantified the uncertainty of the model’s parameters, andthis information has been used to define the robust stabilityand performance objectives of the design. The optimization-based loop shaping techniques developed in [6] has provedsuccessful in designing a low-degree robust QFT controller,despite the fact that the chosen specifications are tight. Thedesign procedure is systematic, fast and almost completelyautomated, although a slight modification of the feedback

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

Time sec

Mag

nitu

deClosed loop step responses

Fig. 11. Closed-loop step responses - With pre-filter

controller was needed at the last step of the design to accountfor the fact that a small number of design frequencies hadbeen chosen. A pre-filter has been designed using a sys-tematic optimization-based procedure (linear programming).The effectiveness of the design was illustrated via numeroussimulations.

REFERENCES

[1] D’Azzo J. and C. Houpis,Feedback control systems analysis andsynthesis”, Prentice-Hall, 1998.

[2] G.F. Bryant and G.D. Halikias,“Optimum loop-shaping for systemswith large parameter uncertainty via linear programming”, Int. Jour-nal of Control, Vol. 62(3):557-568, 1995.

[3] Y. Chait, Chen, Q. and C. V. Hollot,“Automatic Loop- Shaping ofQFT Controllers Via Linear Programing”, ASME Journal of DynamicSystems, Measurement and Contro1, 121:351-357, 1999.

[4] W. Chen and D.J. Ballance,“Stability analysis on the Nichols Chartand its application in QFT”, preprint, Centre for Systems and Control,Dept. of Mechanical Engineering, University of Glasgow, August1997.

[5] A. Gera and I. Horowitz,“Optimization of the Loop Transfer Func-tion” , Int. J. of Control, 31:389-398, 1980.

[6] G.D. Halikias, A. Zolotas and R. Nandakumar,“An optimisationalgorithm for designing fixed-structure controllers using the QFTmethod”, submitted, 2006.

[7] I.M. Horowitz, “Synthesis of Linear Systems”, Academic Press, 1973.[8] I.M. Horowitz and M. Sidi, “Synthesis of linear systems with large

plant ignorance for prescribed time-domain tolerances”, Int. J. Con-trol, Vol. 16, pp. 287-309, 1972.

[9] R. Nandakumar, G.D. Halikias and A. Zolotas,“An optimisationalgorithm for designing fixed-structure controllers using the QFTmethod”, IEEE CCA02-CACSD, Glasgow, UK, September 2002.

[10] N. Niksefat and N. Sepehri,“Designing robust force control ofhydraulic actuators despite system and environmental uncertainties”,IEEE Control Systems Magazine, April 2001.

[11] N. Niksefat and N. Sepehri,“Robust force controller design for anelectro-hydraulic actuator based on a nonlinear model”, Proc. IEEEInt. Conf. on Robotics and Automation, Detroit, Michigan, USA, 200-206, 1999.

[12] O. Yaniv and M. Nagurka,“Robust PI Controller Design SatisfyingSensitivity and Uncertainty Specifications”, IEEE Aut. Cont., Vol. 48,No. 11, 2069-2072, 2003.

[13] A.C. Zolotas and G.D. Halikias,Optimal design of PID controllersusing the QFT method, IEE Proc.-Control Theory Appl., Vol 146, No6, November 1999.