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Arab J Sci Eng (2011) 36:311327DOI 10.1007/s13369-010-0022-8

R E S EA R C H A R T I CL E - M E C H AN I C AL E N G I NE E R IN G

Sohail Iqbal Aamer Iqbal Bhatti

Load Varying Polytopic Based Robust Controller Designin LMI Framework for a 2DOF Stabilized Platform

Received: 23 April 2009 / Accepted: 21 December 2009 / Published online: 27 January 2011 King Fahd University of Petroleum and Minerals 2011

Abstract This paper tackles the problem of payload uncertainties through polytopic system formulation and

robust controllerdesign fora 2DOF parallel manipulator. Typically, such platforms areused as a base fordiffer-ent payloads, e.g., satellite antenna and camera in oceangoing crafts. Traditionally, these kinds of manipulatorsare modeled through a time varying nonlinear model, thus providing the rationale for a nonlinear or adaptivecontroller. Uncertainties due to load variations present a significant challenge for robust control design. In thispaper, the authors have proposed a novel and practical approach to solve the variant payload dilemma for thestabilized platform. The novelty lies in extracting different linear models with distinct load conditions usingthe system identification method and quantifying them into a convex hull to formulate a polytopic system.A regulator is then designed by mixed H2/H synthesis with pole-placement constraints in a linear matrixinequality (LMI) frameworkto compensate outputdisturbances.Theresults are compared witha Riccati-basedH loop shaping controller. It is shown through simulations and experiments that an LMI design is a betterchoice for achieving robustness as well as performance. The hallmark of this work is the successful testing ofthe control strategy on a stabilized platform with heavy asymmetric satellite antenna to reject the tides effectin a deep, turbulent sea.

Keywords Parallel robot Stabilized platforms System identification Robust control application

S. Iqbal (B) A. I. BhattiControl and Signal Processing Research Group, Department of Electronic Engineering,Mohammad Ali Jinnah University, 74-E Jinnah Avenue, Blue Area, Islamabad, PakistanE-mail: siayubi@gmail.com

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1 Introduction

The management companies of oceangoing luxury cruises go to great lengths to isolate critical systems fromthe slow persistent vibrations caused by sea waves. Systems like satellite tracking antennas or critical itemssuch as surgery tables or even billiard tables need to be isolated from the sea wave-related pitching and rollingof the ocean liners. Stabilized platforms are used as mounts for systems which are meant to be decoupled from

sea waves.A two-degree-of-freedom (2DOF) parallel manipulator stabilized platform system is constructed to reject

such torque disturbances and keep its top plate level with respect to the horizontal axis. The schematic diagramof the stabilized platform is shown in Fig. 1.

The stabilized platform has a top-plate and a base-plate linked by two variable-length electro-mechanicalactuators with the help of spherical joints. The angular motion of the top plate with respect to the base plate isproduced by reducing or extending the actuators length to reject disturbances at the base. The outputs of thestabilized platform are top-plate angular position monitored by highly precise sensors. These outputs are againfed back to a controller for error adjustment. The controller produced two separate voltage commands for eachactuator. The input ability of the power amplifier through the data acquisition card (DAC) is 10 V, whichcorresponds with the actuator position. The maximum voltage commands produced by the controller should liewithin this limit. This sensor-controller-actuator package allows the stabilized platform to be self-correcting.The detailed block diagram of the plant is given in Fig. 2.

The mechanical limits of the stabilized platform, both in roll and pitch, are 10

. The stabilized platformcannot compensate for disturbances beyond these limits and become saturated. The corresponding sensors are

Fig. 1 The schematic diagram of the stabilized platform

TorquePWM

Errors

RPM

Roll&

Pitch

Control

Action in

Volts

Roll & Pitch

References- Controller

Power

Amplifier # 1Motor # 1 Actuator # 1 Payload

Power

Amplifier # 2Motor # 2 Actuator # 2 Axis # 2

Axis # 1

Fig. 2 Block diagram of the stabilized platform

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Table 1 Stabilized platform and payload characteristics

Platform weight 550kgPlatform max roll range 10

Platform max pitch range 10

Platform dimensions (length width height) 1.7 1.4 1 mTop plate radius 0.7 mPayload weight capacity 650kg

not limited by these mechanical constraints. The key characteristics of the stabilized platform are given inTable 1.

In practical situations, the stabilized platform may be used with different payloads at distinct times. Thestructure of these payloads can either be symmetric or asymmetric. The mass and moment of inertia (MOIs)of these payloads will always be indeterminate. The controller design for these platforms is really a greatchallenge. One may model the platform using nonlinear differential equations and considering variation inmass and MOIs of payloads as uncertainties. However, as the authors have shown in [1], these uncertaintiesare adequately large and can degrade the system performance. In the current paper, the authors have provideda novel and easy approach to handle these types of problems. The various linear models of stabilized platformare obtained by system identification with distinct symmetric and asymmetric payloads. Subsequently, theseidentified models are used to formulate a polytopic model. After that, the polytope is employed to design robustcontroller using Mixed H2/H synthesis with pole-placement constraints [2,3]. At the end of this paper, acomparison of proposed control techniques is made with a Riccati-based H loop-shaping compensator [4]on the stabilized platform. The simulation results and experimental field trials demonstrated that the controllerdesign through LMI optimization results in a better performance.

System and parameter identification for a parallel manipulator is a relatively new trend; this topic wasalmost nonexistent a decade ago. A significant work on parameter identification for a fully parallel manipula-tor can be found in [5]. The weighted least square method was used in that paper for a rigid multi-body modelalong a closed loop trajectory. In [6], a systematic load identification procedure was used with recursive leastsquare method for parallel robot machine (PRM) through periodic excitation signals. In 2005, Abdellatif andGrotjahn [7] presented an identification strategy for uncertain parameters of 6DOF parallel mechanism. Theappropriate excitation signals were used in that research for identification. In [8], the authors employed systemidentification for extracting a linear model of a 3DOF parallel manipulator without any payload. In 2006,

Wiens and Hardage [9] identified structural dynamic parameters for parallel kinematic machines (PKM). Therigid-body and flexible-body dynamics were combined in that study by using a component-based approach.In this way, a set of linear ordinary differential equation (ODE) was yielded to solve the problem. So far,the system identification method has not been used for the construction of a load varying polytopic model.Moreover, the above-mentioned references only concentrate on the modeling aspect without proceeding to thesubsequent controller design.

H loop-shaping combines classical loop shaping and notion of bandwidth with H robustness. Thistechnique extends the traditional controller parameter-space and maps its frequency response onto controllerparameter-space. It also fastens the controller to accommodate H robustness and performance simulta-neously [10,11]. This method was first proposed by McFarlane and Glover [12] in 1992. It is now widely usedin the industry. Sugie and Shibukawa [13] proposed a robust coordinative H loop-shaping control design formultiple actuators. In 2004, Fite et al. [14] experimentally demonstrated the frequency-domain loop-shapingcontrol methodology to provide transparency, stability, and robustness for a single degree of the freedom

master-slave telemanipulator system. In [4], the authors identified a linear system model for a ship-mounted2DOF parallel manipulator using the auto-regressive exogenous (ARX) technique. This model was then usedto design an H loop-shaping regulator based on Riccati equations to reject disturbance caused by sea waves.

Mixed H2/H synthesis with pole-placement constraints and LMI optimization provides the advantagesof classical, optimal, and robust control for noise isolation and disturbance rejection. The H2-based control-ler can achieve greater noise attenuation against random disturbances. The H controller can attain bettertracking and robust stabilization. The pole-placement technique is suitable to accomplish desired response andexcellent closed loop damping. Pole placement is also helpful to prevent the fast controller dynamics that cancause system instability. In [15], Amato et al. proposed a tracking controller for a flexible manipulator thatcombined sliding mode control (SMC) and LMI-based optimization. The control law recommended in thatresearch was successfully applied on a two link robotic manipulator, but exponential stability at the origin

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cannot be achieved. In [16], nonlinear matrix inequality (NLMI) control design based on Hamilton Jacobean(HJ) was employed on a 2DOF planner manipulator. In that study, first the robot dynamics were transformedinto a nonlinear affine system, and then HJ inequality was used to solve NLMI optimization. In [17], Yu et al.combined gain scheduling with H theory and used linear parameter variation (LPV) synthesis on an n-jointrigid robots. In that paper, the nonlinear time-varying robotic manipulator was modeled into the LPV convexpolytopic system and then applied to the LPV convex decomposition filter. In this way, the subsequent LMIs

were solved to achieve the desired H performance and pole-placement constraints. In [18], a polytopic gainscheduling H controller with pole placement was proposed for a two-link planner robot. In that investiga-tion, the robot model was transformed into the LPV system, with respect to the equilibrium manifold to applyLMI optimization. So far, according to the authors best knowledge, no paper has presented the load varyingpolytopic controller synthesis with LMI constraints for parallel manipulators.

In this paper, Sect. 2 formulates the polytopic model for the stabilized platform with distinct payloads withthe help of system identification and LMI optimization. Section 3 deals with mixed H2/H controller synthe-sis with pole placement constraints in the LMI framework. Simulation studies of Riccati- and LMI-based Hcontroller designs are discussed in Sect. 4. The experimental results of both techniques on the actual platformin laboratory and under deep turbulent sea are duly reported in Sect. 5. Conclusions are drawn in Sect. 6.

2 Problem Formulation

The polytopic systems are formulated by different uncertain LTI systems as its vertices. Such systems in theLMI framework cover a long range of uncertainties via convex hull formulation. These convex hulls containLTI models which are developed traditionally by two methods. Firstly, linearization of a nonlinear modelabout different operating points gives distinct LTI models for polytope construction. Secondly, a nonlinearmodel gives LTI systems by considering a range value of uncertain affine parameters. The twomethods dependevidently upon static operating points and numeric values of uncertain parameters, respectively. These twotechniques also do not cover dynamic uncertain scenarios, such as moving payload on a platform. In this work,to account for such a situation, a novel approach is proposed to formulate a polytopic system, i.e., systemidentification is employed to develop LTI models under various payload operating conditions.

The stabilized platform is used as a case study to validate the proposed technique. Mainly, these types ofplatforms are served as a base for distinct payloads at different times. These payloads could be satellite anten-nas, tracking cameras, billiard tables, etc. with symmetric or asymmetric structures in nature, so the mass and

MOIs of payloads would always be uncertain. The uncertainties in payload affect the dynamical behavior ofthe overall system. To muddle through these situations, a controller should be robust enough to deal with thesetypes of uncertainties. For this purpose, distinct models for a polytopic system are obtained through systemidentification for the stabilized platform with different load conditions. Subsequently, this polytopic system isused to design controller with mixed H2/H synthesis and pole placement constraints in the LMI framework.In this section, system identification is first presented for linear models extraction of stabilized platform withvarious payloads and then provided the mathematical formulation of LMI-based polytopic system.

2.1 System Identification

The system identification is used to acquire the state space models for the stabilized platform under threedifferent load conditions. The first model is extracted without any payload. The second model is identified

with a symmetric payload of volume 1.7 1.4 1 m3 and 550kg weight. The third model is obtained with500kg asymmetric satellite antenna. In this section, the system identification is discussed only for one payloadcondition for brevity. The other two models are identified in the same way.

The stabilizedplatform has two input and two output axes, i.e., roll and pitch. Both are orthogonal and haveno effect on each other. The single input single output (SISO) system identification can easily be applicableon both axes separately. Only the pitch axis identification process is explained here in detail.

System identification includes data acquisition, model or structure determination, parameter estimation,and model verification [19,20]. Common choices for input signal for the identification are Gaussian whitenoise signal, Chirp signal, or a pseudo random binary signal (PRBS) [21,22]. Due to discontinuities in nature,it is not recommended to apply white noise and PRBS to the stabilized platform because of its sensitive andsluggish mechanical structure. Chirp signal is, therefore, selected as an input; it is a sinusoid with continuously

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Fig. 3 Input output signals of the stabilized platform

varying frequency over a definite band : 1 2 for a certain time period 0 t M [22], i.e.,

u(t) = A cos

1t +

(2 1) t2

2M

(2.1)

To obtain the linear model of the stabilized platform, the chirp signal should contain enough frequencies, so itcan cover the whole bandwidth of the system to exalt overall dynamics. As mentioned earlier, voltages throughDAC are inputs to the stabilized platform and top plate angular positions, measured in degrees, are the desiredoutputs. The input limit is 10V and output tends to saturate beyond 10 both in roll and pitch axes. Thefrequency range for the chirp signal is swept between 0.04 and 2Hz. This range is selected after studying theresponse of the system experimentally and a rough cut-off frequency is founded. Therefore, the optimum value

of input voltage for identification is set as 3 V to keep the system in operating range for this experiment. Ifthese limits are crossed then the actuators of the system become saturated and it may lead to loss of importantdynamics of the plant. The selected input is given to the platform and the corresponding output is measured.Figure 3 shows the input/output data graphs.

The data collected from this experiment is used for model estimation. To choose a best model structurewhich is suitable for identification is perhaps the most difficult decision that has to be made [22]. This choicemust be based both on an understanding of the identification procedure and insights of the system. An underparameterized model may be very inaccurate and an over parameterized model can lead to unnecessary com-putations [20]. Before model estimation, it is also necessary to treat the raw data so that it becomes appropriatefor the identification process. In the first step, a treatment is done to remove means and trends from the rawdata. In the next step, the processed data are fitted into the ARX model. The ARX model relates the currentoutput to a finite number of past outputs and inputs for model estimation.

The estimated model has three poles and two zeros. The delay observed is too small so it is neglected to

avoid complication. This model structure is chosen on the basis of best fits between the identified models andthe actual systems outputs. If the order of the system is increased, then poles are canceled with zeros, whichcould lose controllability and observability of the model. Hence, the model has three states, but these statesdont have any physical meaning. The equation of the identified model can be written as follows:

y(k) = a1y(k 1) a2y(k 2) a3y(k 3) + b1u(k) + b2u(k 1) (2.2)

The parameters of three different load conditions for the above equation are given in Table 2.Forvalidation purposes a signaldifferent from the signal used in estimation is recommended [22]. A cosine

signal with 0.04 Hz frequency is chosen for the validation process. The validation frequency is set to the worstcase scenario for sea dynamics, as the observed sea waves frequency was found to be less than the specifiedfrequency during experimentation. The responses of the sine signal were obtained from the actual platform as

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Table 2 Parameters of the three identified models

Model a1 a2 a3 b1 b2

Without payload 1.6432 0.7629 0.1197 0.0152 0Symmetric payload 1.4203 0.6370 0.2166 0.0246 0Asymmetric payload 1.6364 1.5107 0.8741 0 0.0176

Fig. 4 Model validation on test data

Fig. 5 Prediction error over test data

well as from the identified model. Figure 4 shows the comparison results and Fig. 5 shows prediction errors.Theerrorsare almostnegligibleand the identified model outputalmost overlaps the actualoutputof thesystem.

The three models of the stabilized platform, one without payload, another with symmetric payload, andthe third one with asymmetric payload, were extracted in the same manner. The continuous-time counterparts

for these models are obtained by TUSTIN approximation. Pole-zero plots for these systems are shown inFig. 6. These plots illustrate that the identified models have very slow dynamics and also have non-minimumphase behavior. The damping ratios of these systems are 0.961, 0.686, and 0.0714, respectively. Moreover, forasymmetric payload, poles are shifted towards the origin. This shows a slow response and more oscillatorybehavior is expected with these types of payloads.

It is obvious from the identification process that the system model is sensitive to size and shape of thepayload being used. These models, i.e., without payload, symmetric payload, and asymmetric payload, havesignification parameter variations due to different load conditions. The main characteristics of three identifiedmodels of stabilized platform are given in Table 3.

Bode plot of three models are shown in Fig. 7. As revealed from Fig. 3, the output signal is at 90 leadwith respect to input signal. Thus, the actual phase difference is 270. The Bode diagram shows that the gain

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Pole-Zero Map

Real Axis

Ima

ginaryAxis

-50 -40 -30 -20 -10 0 10 20 30 40 50-25

-20

-15

-10

-5

0

5

10

15

20

25

-50 -40 -30 -20 -10 0 10 20 30 40 50-25

-20

-15

-10

-5

0

5

10

15

20

25Pole-Zero Map

Real Axis

Ima

ginaryAxis

(a) (b)

Pole-Zero Map

Real Axis

ImaginaryAxis

-50 -40 -30 -20 -10 0 10 20 30 40 50-30

-20

-10

0

10

20

30

(c)

Fig. 6 Pole-zero plots of three identified models. a Without payload. b Symmetric payload. c Asymmetric payload

Table 3 Characteristics of the system with different payloads

Stabilized platform models Bandwidth (Hz) Damping ratio

Without payload 0.105 0.961Symmetric payload 0.09 0.686Asymmetric payload 0.0445 0.0714

decreases and the phase lag increases with payloads. Moreover, it is interesting to note that the phase margindrastically increased when the asymmetric payload was used. The Bode plots reveal that tracking turns outto be poorer for the loaded system while noise sensitivity is decreased, whereas the bandwidth of the systemwith symmetric and asymmetric loads also decreases.

The root locus plots of three identified models are shown in Fig. 8. It can easily be seen from these plotsthat the asymmetric load model has the lowest natural frequency, and it is closer to instability.

The rank of the controllability and observability matrices are full, so the systems are fully controllable andobservable. The state-space matrices of three identified models are given in the Appendix. These models areused as vertices of the convex hull to formulate a polytopic system. The results are to be used for the robust

controller design. The next section will discuss mixed H2/H synthesis with pole-placement constraints onthe basis of LMI optimization.

2.2 Polytopic Formulation

Theresulting identifiedstate-space models of the stabilizedplatform, i.e., without payload, symmetric payload,and asymmetric payload can be written as

x = Aix + Bi u

y = Cix + Di u

i = 1, 2, 3 (2.3)

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Fig. 7 Bode plot of three identified models

Fig. 8 Root locus of three identified models

The system matrices of the above dynamical system can be written as follows:

i (t) =

Ai (t) Bi (t)

Ci (t) Di (t)

i = 1, 2, 3 (2.4)

The combination of these system matrices can be constructed into a polytopic system as follows:

(t) Co(1, 2, 3) =

k

i =1

i i : i 0,

ki =1

i = 1

(2.5)

where Co() is the convex hull and nonnegative numbers are i called the polytopic system coordinate of.

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As stated above, the stabilized platform is a regulation problem with uncertainties in mass and MOIs ofpayloads. The friction of joints and sensors noise can be incorporated by disturbance variable d and whitenoise variable n. These disturbance and noise elements are members of uncertainty vector (t). Let z be theregulation error of the system and z2 is the combination of state vector and control law. Both z2 and z shouldbe minimized in order to achieve LMI optimization. Therefore, the LMI constraints can be written as

= [d n]T

z = e

z2 = [x u]T

(2.6)

The new A, B, C and D of the augmented system can be composed as follows:

A = A

B =

B1 B2

11 12

0 0 0 0

0 0

0 1

D D

D =

where D22 =

0 0 0 1T

Here the polytopic formulation for the stabilized platform is completed.

3 The Controller Synthesis

This section deals with controller synthesis. A brief overview about the pole-placement region is given. In thesecond part, the LMI formulation of the proposed pole placement region is illustrated.

3.1 Pole-Placement Region

The required pole-placement region can be designed using half-plane, disk, and conic sector regions charac-terized by S = (, r, ) . A minimum delay rate , a minimum damping ratio = cos , and a maximumundamped natural frequency = rsin is ensured when such a region is proposed. The values used for theregion are as follows: the left haft-plane = 10, the disk center is at zero and the radius r = 200, and theconic sector origin is at zero, and the half inner angle = 3/2, as shown in Fig. 9.

The poles are chosen such that the settling time is less than one second. The rationale for this settlingtime specification comes from the observed quasi-frequency of ocean waves (taken as disturbance) as it canbe seen in the observed experimental data shown in Fig. 17. A wide variation in this disturbance is expectedwith seasonal and geographic changes and is also due to the tracking orientation of the asymmetric payload.A safety factor of ten is assumed for this purpose. For the mentioned rise time, the desired natural frequencycomes out to be 2 rad/s. If the lower limit of the desired Eigenvalue region is kept at 2 rad/s, then the closedloop dynamics at the rig are observed to be too slow, so this is set at 10rad/s. Similarly, the angle theta forthe chosen conic sector is kept at 3/4, which corresponds to a damping ratio of 0.707, which is realisticfor the mechanical structure being considered. The upper bound on the Eigenvalues comes from the need forkeeping the actuators in the linear region, away from saturation. Confining the closed-loop poles between theseconstraints ensures the desired performance. This pole placement minimizes the overshoot and oscillation inthe system response and decreases its rise time and settling time.

3.2 LMI Formulation

This section gives the review of LMI-based robust controller design methods found in the existing literaturewith due references so that the current treatise could be self-explanatory. The contents presented here can befound in [2,3,23]

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Fig. 9 The pole placement region

The LMI theory offers multi-objective mixed H2/H synthesis with pole-placement constraints to solvepractical problems. The H2 achieves greater noise rejection against random disturbance. H mostly dealswith robust stability and does not allow placing the poles in desired regions. In contrast, pole placement helpsto achieve satisfactory time response and closed-loop damping. In this section, a regulator based on LMI isdesigned for the stabilized platform.

Here the output z is associated with H performance. The output z2 is related with H2 performance.The Tz and Tz2 relates the closed loop transfer function from to z and from to z2, respectively. Theuncertainties due to load variation, friction, and sensor noise can be incorporated into the disturbance vector(t). Equation (2.3) can be extended to polytopic formulation with state-feedback control as

x = Ax + B1w + B2uz = C1x + D11w + D12uz2 = C2x + D22u

(2.7)

The state-feedback control law u = K x guarantees

to place the closed-loop pole in some prescribed LMI stability regions. H performance such that Tz < where > 0. H2 performance such that Tz2 < v where v > 0.

Equation (2.7) can be further simplified as

x = (A + B2 K)x + B1wz = (C1 + D12 K)x + D11wz2 = (C2 + D22 K)x

(2.8)

The closed loop root means square (RMS) satisfying gain from to z cannot exceed if and only if thereexists a matrix Y := K X and a symmetric matrix X := X such that

A X + X AT + B2Y + YTBT2 B1 XCT1 + YTDT12B T1 1 DT11C1X + D12Y D11 I

< 0 (2.9)

Moreover, the closed loop H2 norm ofTwz2 cannot exceed v if there exist two symmetric matrices X and Qsuch that

Q C2X + D22Y

X CT2 + YTDT22 X

> 0, (2.10)

Trace(Q) < v20 , (2.11)

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w

u3

z1

x2

KFig. 10 The polytopic state feedback control

where

2 < 20 . (2.12)

The closed-loop poles that lie in the LMI region are given by the equation

D = {z C : L + M z + MT Z < 0} (2.13)

where L = L T =

i j

1i,j mand M = MT =

i j

1i,j m

and the M and L are fixed real matrices.

The closed-loop poles can lie in the region if and only if there exists a symmetric matrix Xpol satisfying thefollowing condition:

i j + i j (A X + B2Y) Xpol + j i

X AT + YTBT2

1i,j m

< 0 (2.14)

The final closed-loop polytopic design principle for stabilized platform can be represented as shown in Fig. 10.

4 Simulation Studies

This section pertains to simulation testing of the design controller. For benchmark purposes, the performance

of the designed controller is compared with a robust H loop-shaping controller [4]. Since the stabilizedplatform is a regulation problem, an impulse-like disturbance is first given to test the performance of bothcontrollers. The impulse responses are shown in Fig. 11.

The test shows that LMI-based controller performance is better than loop-shaping in terms of settling timeand overshoot. The settling time of the LMI controller is 0.5s, while the loop shaping controller takes 2s tosettle down about its steady state. The results are summarized in Table 4.

A second simulation set is performed to judge the system behavior against continuous disturbance. Forthis purpose, sine wave of 0.1Hz is given to the system. Figures 12 and 13 describe the simulation results ofboth controllers.

The LMI base regulator attenuates the disturbance 92%, while the loop-shaping regulator suppresses itonly 80%.

5 Experimental Results

For controller validation, a state-of-the-art test platform is set up with a variety of actuators and sensors. In thestabilized platform, two ball-screw actuators (Duff-Norton ball skew) are deputed for roll and pitch movementof payload. The actuators used AC servo motors (Mitsubishi HC-SFS152B) driven by two 50 Hz PWM gen-erator amplifiers (Mitsubishi MR-J2S-200A). The optical and magnetic sensors (motion sensor OCTANS andAHRS 400CA, respectively) are utilized to sense outputs of the system. The closed-loop control algorithmis implemented in DSP board (Trio motion MC206x). The actual routines have been written in Trio Basiclanguage. The stabilized platform of 550kg weight is used to stabilize a 500kg payload in deep turbulent sea.The details of the hardware configuration are outlined in Table 5.

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Fig. 11 Impulse response of both controllers

Table 4 Summary of controllers characteristics by simulations

Parameters H controller LMI controller

Disturbance rejection time (s) 2

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Fig. 13 Simulation LMI-based H controller

Table 5 Hardware configurations of stabilized platform

Actuator (Duff-Norton Ball Skew)Force 4595NCapacity 5 tonsMax speed 27.2 inch/min

Servo Motor (Mitsubishi HC-SFS152B)Max PRM 3000RPMMax torque 21.6Nm

Servo Amplifier (Mitsubishi MR-J2S-200A)Voltage 3 Phase 200230V ACFrequency 50/60 Hz

Controller (Trio motion MC206x)

Technology 32-bit DSPSoftware language Motion perfect

Sensor (AHRS400 CA)Range roll, pitch and heading 180, 90 and 180

Update rate 100 HzResolution

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Fig. 14 Disturbance rejection with loop-shaping controller and LMI based controller on actual platform

Fig. 15 Disturbance rejection on rig with LMI and loop-shaping controllers

5.2 Sea Trials

The stabilized platform is also tested with a 500 kgsatellite antenna in deep turbulent sea to carry out the perfor-mance analysis of the actual tidal effect. The satellite tracking dish was supposed to be rotated in elevation andazimuth to follow the tracking commands to track the satellite trajectory. Placing the antenna directly on theship deck is satisfactory if the ship deck is considered still. However, the ship deck is continuously perturbedby the pitching and rolling motion of the ship due to tidal effects. This pitching and rolling motion is causedby the tidal waves bouncing the ship around both axes. During low sea states, the problem may be mitigated.However, during rough seas or stormy conditions, the ship deck undergoes pitching and rolling due to tidalwaves, causing the subsequent attitude changes in the desired precise maneuvering of the satellite antenna.This makes optimum satellite tracking impossible. In order to make the tracking performance independent of

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Fig. 18 Stabilization after zooming

6 Conclusion

Uncertainties in mass and MOIs of stabilized platform due to payload variations are catered to with the helpof the polytopic model formulation using the system identification method. An H robust controller based onLMI optimization is designed to reject the disturbance in roll and pitch caused by water waves in oceangoingcrafts. Results based on simulations and in the real environment show that controller-based LMI optimizationperforms better than Riccati-based optimization in terms of performance as well as robustness.

Acknowledgments The authors would like to thank the Higher Education Commission (HEC), Pakistan for their financial sup-port of this work. The authors are also thankful to the research fellows at the Control and Signal Processing Research (CASPR)group, Department of Electronic Engineering, Mohammad Ali Jinnah University, Islamabad, Pakistan for their useful help andsuggestions.

Appendix

The state space models Ai , Bi , Ci , Di i = 1, 2, 3 of identified systems are given as

A1 =

17.3100 22.6900 22.6900

20.0261 19.9739 59.9739

2.7156 2.7156 37.2844

A2 =

15.5646 24.4354 24.4354

20.8593 19.1407 59.1407

5.2933 5.2933 34.7067

A3 =

24.0672 15.9328 15.9328

37.9951 2.0049 42.0049

13.9261 13.9261 26.0739

B1 =

0.0086

0.0076

0.0010

B2 =

0.0151

0.0129

0.0033

B3 =

0.0070

0.0184

0.0290

C1 =

11.3450 11.3450 11.3450

123

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Arab J Sci Eng (2011) 36:311327 327

C2 =

12.2177 12.2177 12.2177

C3 =

7.9664 7.9664 7.9664

D1 = [0.0043] D2 = [0.0075] D3 = [0.0035]

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