Design of Controllers using Fast Output Sampling Feedback – FOS for 1 & 2-link FMs

Vol. 28, No. 7, (2019), pp. 124-135

124

Design of Controllers using Fast Output Sampling Feedback – FOS for 1 & 2-link FMs

Joshi Shubangi Milind

USN: 7VC17PED01, Ph.D. Research Scholar – Part Time, Dept. of ECE, VTU RRC-Belgaum, Karnataka

Dr. Arunkumar G.

Associate Professor & Head, ECE Dept.,

JSS Academy of Tech. Education (JSSATE), Noida, Uttar Pradesh

Dr. T.C.Manjunath

Professor & Head, ECE Dept., Dayananda Sagar College of Engg. (DSCE), Bangalore, Karnataka Email : [email protected]

Abstract—The highlight of this research article is the design & development of Controllers using FOS for 1 &

2-link FMs is being presented for the control of a 2-link flexible robotic manipulator in the 3 Dimensional Euclidean space along with the simulation results. A 2- link FM with 2-DOF is being considered in this research article and the base motor is attached to the single flexible link & the other link being attached @ the far end of the 1st link to which the shoulder motor is connected and the entire set-up is a single structure one and 2 motors are being used for actuation purposes & is being controlled / regulated using the developed sliding mode controller, i.e., the dual links are revolving around the base, further the link-2 is rotating w.r.t. the shoulder axes & the joint axes of the two are parallel and perpendicular w.r.t. the robot work surface. Hence, the entire system can be called as a planar mechanism, i.e., the 2 links are moving parallel to the plane of the work surface (x – y plane). There are 2 joints, viz., base joint & the shoulder joint along with the end-effector being the tip of the 2-link manipulator. The small errors that occurs in the control signal u is due to the changes during the set-point. Simulink model is being developed in the Simulink-MATLAB environment & the results of the MATLAB simulation are observed after the model is being run, from the obtained results thus showing the efficacy of the methodology developed by us.

Keywords—Robot/s, Flexible, Manipulator, F O S, Sampling, Mode Control, Simulink, MATLAB, Simulink, Simulation, Run time, Result.

1. Introduction

A brief introduction about the controller design using the theory of the fast output sampling feedback is dealt with aftermath being used to control the various parameters of the 1-link & 2-link flexible manipulator.

Once, the controller is designed using the FOS concepts (which is a particular type of multi-rate output feedback controller/s), the developed controller when it is connected in feedback loop with the model of the plant (1- link or 2-link flexible manipulator) and the control strategy developed is tested for its effectiveness.

Matlab tool is being used to develop the control algorithm and observe the results. The research work done

is compared with some of the works of the yesteryears.

Simulink model for the FOS designed controller is also developed & the results are observed. The chapter concludes with the results of the simulation being discussed in brief & followed by the final conclusions of this proposed research presented in this research article.

The paper is organized in the following manner. A brief introduction to the paper is presented in the section I followed by a brief insight into the development of the FOS f/b Control Design for FMs in the section II. Next, the mathematical modelling development is depicted in the section III. The entire control design for 1-link FRM& a dual link FM is presented in subsequent sections IV & V respectively one after the other. The development of the FOS f/b simulink model is presented in section VI. The 7th section finally is going to conclude this article followed by an exhaustive list of references.

2. A brief insight into the development FOSController Design for FMs

Control of adaptable controllers finds a ton of uses in the cutting edge world, especially in the field of avionics, robotics & the smart intelligent flexible systems. The need for a sophisticated control arises from the output feedback concepts, which has gained a lot of advantages over the traditional control schemes for controlling the various parameters of the plants. One such output feedback algorithm is the fast output sampling feedback abbreviated as FOS. The output of the plant, i.e., the ‗displacement‘, velocity, acceleration (each one obtained by the differentiation of the earlier ones) is considered as the parameter for control purposes, in our case, the displacement is being considered as the control variable to be tracked upon with. A brief review of which follows in the following paragraphs.

This special multiratecontrol algorithmis developed to cater for the flexibility in the robotic arms, and these algorithms are not that complex, considering that the system is non-linear as generally all flexible systems even though they are non-linear will be linearized about an operating point.

Vol. 28, No. 7, (2019), pp. 124-135 In the work considered, POF is used for controlling

the dynamic motion of the FRMs & take to a present location and also to track the set-points, curb down the vibrations with less errors. In order to develop the FOS controller, first, a SS model (mathematical-A B C D) is to be developed first from using the first principles starting from the M, B, K parameters.

2 separate cases of the flexible robotic manipulator are presented here with, viz., the SISO (1-link) control, second-the MIMO (2-link) control, in the sense the former has one actuator – so, 1 input & 1 output, whereas the latter has two actuators – so, 2 inputs & 2 outputs. Open loop (OL) & closed loop (CL) responses are also observed with SFB gain F& with the FOS gain L. Simulations are carried out with & without the FOS controller in order to depict the profoundness of the control techniques which are proposed in this research article thus showing the effectivity of our work in comparision with the work done by other researchers.

The FOS controller algo actually requires the plant system to be fully observable&controllable, thus giving the actual performances even in the amidst of the un- measured noises or disturbances. Next, the performance appreciation of the fast o/psampling f/b controller/s will be implemented on the 1 & 2 link FRMs& could be generalized for a n-link flexible manipulator case also, which could be treated as future works.

The FOS control design proposes a methodical rule w.r.t. the robust controller, wherein the controllingeffortu is going to be constrained to be a linear f/n of most of the recent o/p measurements of the plant (plant output), but the output is generally permitted to be varying at discrete moments of time between the estimations.In this FOS controller, the controlling input, i.e.,u at a certain moment of the time t will depend on the plant model‘so/py at a time prior to that juncture, actually at the start of sampling time . In the case of a SISO system, one control variable u₁ is sufficient, where as in a MIMO case, 2 control variables u₁ &u₂ are required.

FOS concepts can be connected to structure the controller design in which the i/p info u is changed a few times in a single o/p interval of sampling time. The problem of control design using piecewise constant fast output sampling feedback is considered here, since complete assigning of the poles in the s-plane is absurd (ruled) oututilizing SOFB. Such a control calculation can balance out an a lot bigger class of dynamical systems than static o/p f/b. Since the criticism gains are piecewise steady, this technique can be effectively actualized to get wanted outcomes. The strategy won‘t require any type of assuming any data except for a general supposition that the dynamical system ought to be totally observable &

controllable condition. (all states should be in a position to be controlled & all the states should be seen, i.e., output measurements can be made).

A standard outcome in controller design theory says that assigning of the system poles (OL or CL) of a LTI controllable dynamical system can be discretionarily doled out by state f/b i/p. By and large, the whole state vector isn‘t straightforwardly accessible for the purpose of f/b.

Subsequently, it is attractive to go for a o/p f/b based design for the controlling of a particular parameter in a dynamical system. The SOP issue is a standout amongst the most examined issues in the control hypothesis and its applications.

One motivation behind why the static o/p f/b has gotten so much consideration in the control literature is that it speaks to the very fundamental of controlling any system (simple control scheme) control that can be acknowledged in useful circumstances especially practically. In any case, no outcomes are accessible till today which demonstrate that total shaft task is conceivable utilizing static o/p f/b. Rehearsing state i/p and ideal f/b controllers needs cautious thought in keen structure application zone like the aerospace structures &

smart structure designs, in light of the fact that the state f/b controller will need the accessibility of the whole state vector or the help of a estimating block in the f/b loop.

when the SFB case of a dynamical system is considered, the ideal controlling law requires the design of a state onlooker/observer. This builds the execution &

building cost and decreases the unwavering quality of the controlling & the controlled system. Another hindrance of the eyewitness based control system is that even slight varieties of the model parameters from their ostensible qualities may result into critical degeneration of the CL execution/performance.

Proposed flow-chart for designing of the controller using the FOS methodology is depicted in Fig. 1.

3. Mathematical modelling development

The SOF requires just the estimation of the o/p of the system, however there is no assurance w.r.t. the CLCS's stability point of view. In spite of the fact that the CLCS‘s stability can be ensured utilizing the SFB concepts, the equivalent isn't correct utilizing SOF. Thus, if a dynamical system, for instance, intelligent cantilever shaft, for this situation, must be balanced out utilizing just the o/p f/b (states may not be accessible for estimation purposes), one can depend on FOS f/b concepts, which is static in nature also, ensures the CLCS stability. Here, the estimation of the contribution at a specific minute relies upon the yield an incentive at once preceding this minute (specifically toward the start of the period).

The issue of F O S f/b concepts was examined by Werner and Furuuta for LTI dynamical frameworks with inconsistent perceptions & looks. They have demonstrated that the poles (roots of denominator polynomial of the TF) of the DTCS could be alloted discretionarily (inside the normal limitation that they ought to be found symmetrically as for the real axis of the s-plane) utilizing

Vol. 28, No. 7, (2019), pp. 124-135

126 the concepts of FOS method. Since the FOS f/b gains are

piecewise constants, their technique could undoubtedly be executed and showed another plausibility. Such a control law can balance out an a lot bigger class of frameworks.

Now, to implement this proposed flow-chart shown in the Fig. 1 for the controller design using the FOS concepts on the 1-link & 2-link flexible robotic manipulator, a dynamical plant which is best modelled as a LTI SS model

&given by the Eq. (1) as

START



LTI state space model

     

     

x t Ax t Bu t y t Cx t Du t

 

 





Sample the CT system with a sampling interval of  = 0.004 secs



Divide sampling interval into 10 sub-intervals



^N10





Obtain the



  ^,  ^,C



, DT system (tau system) of CT system sampled at a rate of 1

/ 



tau systems to be controllable & observable



Obtain the stabilizing state feedback gains for the tau system&Eigenvalues of



_i _iFi



to lie inside the unit

circle



Obtain the closed loop impulse response with the state feedback gain F



Sample CT system with a sampling interval of  = 0.004 secs



Divide sampling interval into 10 subintervals



N10





Obtain the



^,,C



, DT system (delta system) of CT system sampled at a rate of 1

/ 

, i.e., 

 /

N



delta system to be controllable & observable



obtain the FOS gain matrix

L

with the help of solving the linear equation

LC  F



L M I Optimisation method to be used



obtain the optimized FOS gain matrix

L



Obtain the CLIR with the FOS f/b gain L

 END

Fig. 1 : Proposed flow-chart for the controller design using the FOS methodology

     

     

x t Ax t Bu t

y t Cx t Du t

 

 

 (1)

where the different parameters are belonging to the different spaces such as ….

x  

ⁿ_,

x  

^m_,

x  

^p_,

n

A  

n^ _,

B  

^nm_,

C  

^pn, the state space matrices

A

,

B

&C are constant matrices and it can be assumed that

( A , B )

is controllable and

( C , A )

is observable. Assume that output measurements are available at time instants,

t k  

, where k = 0, 1, 2, 3,

…..It is to be noted that A, B, C, D,x, u,y are the plant system matrix, i/p & o/p matrix, transmission matrix, state

& i/p variable & the o/p variable of the plant (1-link & 2- link flexible system) & are constant matrices having the correct dimensions. The next step is to construct a DT LTI digital system & from these o/p measurements/readings, which is obtained by sampling at a rate of



1 (sampling time interval given by



secs). The discrete system so obtained could be denoted as the tau



system & is given by Eqn. (2) as

), ( )

(

), ( )

( )

) 1 ((









 _{ }

k x C k

y

k u k

x k

x













(2) Here, the matrices _,_,C are matrices having proper dimensions. It is well assumed that the dynamical plant which is to be controlled using a PC or a laptop with a sampling period of



secs & using a ZOH circuit. A sampled data SFB designis to be done in order to find SFBgain

F

so that the CLCS

 k    

_{ }

F  x ⁽ k  ⁾

x     

(3)

will be having desirable properties. Here,



 e

^A



(4)

&







_ ^

0

B ds e

^As

. (5)

On the other way, if a state observer is not being used to design the controller, then a sampled data control system can be designed &used to realize the effect of the SFB gain

F

by using o/p f/b.

Vol. 28, No. 7, (2019), pp. 124-135

Fig. 2 : Graphical illustration of FOS feedback method

Let us consider delta, i.e.,

N

 



and then finally consider the control effort u(t) as

 

,

) (

: :

) (

) (

...

)

( _{0 1 1}

































 _











k y

k y

k y

L L

L t

u _N (6)

i.e.,

yk

L t

u( ) (7)

for kt(k1) _.

Here, the matrix blocks

L

_jdenotes the o/p f/b gains

& the notation

L, y

_khas been incorporated in this equation for the sake of convenience. It has to be noted that



1

is the rate at which the f/b loop is closed in the

designed controller, but the o/p samples will be taken as

N

- times faster rate of the parameter

 1.

Fig. 3 : Block-diagrammatic representation of FOS feedback method of control strategy

The FOS f/b control algo is illustrated as shown in Fig. 2 along with the proposed block diagrammatic representation of the FOS control strategy in the Fig. 3. To demonstrate how a FOS f/b controller in Eq. (7) can be

intended to understand the given inspected SD SFB gain for a observable & controllable framework, we develop an invented, lifted framework/system for which the Eq. (6) can be deciphered as a SOFB case. Consider the



,,C



DT system as given in Eq. (1) and which is sampled @ the rate of

1 be called the deltasystem. Let us now consider the DT system @the time interval given by

t  k 

, the control input beu_k u(k



)& the state of the DT system be

x

_k

 x ( k  )

& the o/p

y

_kbe represented as

, ,

0 0

1 1

k k

k

k k

k

u x

y

u x

x

D

C 















 

(8) Here,

 

 





 

 







 

1 0

C

N

C C

C 

,

 

 







 

 

















^

 2

0 0

0

N

j

C

j

C

D 

. (9)

Assume that the state feedback gain

F

^{has been}

designed such that

 

_

 

_

F 

has no Eigen values at the origin. Then, assuming that in the interval

) (  

  t  k 

k

_,

) ( )

( t F x k 

u 

, (10)

one can define the fictitious measurement matrix,



_{0 0}

 

¹

) ,

( F N  C  D F 

_

 

_

F

^

C

, (11)

which satisfies the fictitious measurement equation

k

k

x

y  C

. (12)

For

L

to realize the effect of

F

, it must satisfy the equation

 F

LC

. (13)

In the next step of designing the FOS f/b controller we express a variable called as



which will give the observability index of the DT system

  ,  , C 

. Next, it can be proved that - for

N  

, the matrix

C

has got a full column rank, so that any SFB gain could be realized using fa FOS f/b gain represented as

L

. Suppose for ex., the starting state (initial) of the system is unknown, then, an error in the input

 u

_k

 u

_k

 F x

_k exists, which could be used to construct the control effort signal using the concepts of SFB. Then, finally, the research can verify that the dyanmics of the CLCS would be governed by

 

 





 



 















 

 

 







_



k k k

k

u x F

u x









0 1

1

0 LD

^{. (14)}

The following below mentioned coordinate transformation could be used in equation no. (14) as

Vol. 28, No. 7, (2019), pp. 124-135

128

 

 



 

I F

T I 0

(15) to the Eq. (16)



 





 



 



 





 









k k k

k

u x u

x

0 0 1

1

LD LC



 (16)

& incorporatethe Eq. (11).

Consequently, one can say that the Eigen-estimations of the CLCS under a FOS f/b controlled algorithm as given in Eqn. (11) are those of

 

_

 

_

F 

together with the eigen estimations of

 LD

₀

 

_



. From this previous estimations, it could be observed the SFB F has to be designed in such a way that the 2 discrete systems given by

 ^

_

^{ }

_

^F 

^&

 LD

0

^{ F }

_



, their stabilities will be ensured. It can also be observed that the dynamical system given in the Eq. (16) will be stable if &

only if

F

stabilizes

 

_

, 

_



& the matrix

 LD

₀

 F 

_



will be having all the Eigenestimated values inside the circle of unit radius.

The issue with the digital controllers got along these lines is that, despite the fact that they are settling and accomplish the ideal CL behaviour in the o/p sampling times, they may cause an over the huge oscillations which exists b/w different sampling times. The FOS f/b gains got might be of very high value. In order to reduce this impact, we loosen up the condition that precisely fulfill the straight condition, which is a linear equation given in (13) & then incorporate a limitation on the gain (set up a constraint or a limit). Along these lines, we touch the base at the accompanying in Eqs. (17)- ( 20).



1

 L

,

2 0

 F 

_

 

LD

,

3



F

LC . (17)

The above 3 equations could be designed in LMI form as given by

0

2

1 



 









I I LT



L _{, (18)}



0



⁰

0 2

2 

























I F

F I

 T





LD

LD , (19)

 

⁰

2

3 



 













I F

F I

LC T



LC

. (20)

Considering the previous equations, by incorporating all of them, the LMI optimization toolbox could be used for the design of the FOS f/b gain L. Thus, finally by using the LMI toolbox of Matlab, the optimized value of the gain L can be found out. It should be noted that in this context, the optimized value of this FOS gain L requires only piecewise constants and will definitely be easier to implement in the RT applications.

4 Control design for Single link flexible manipulator

It should be noted that in this case, a single flexible link is connected to the hub where a single actuator is used for actuation purpose. Refer Figs. 14-16(appendix) for the diagrammatic representation of the flexible system with its specifications in Table 1 (appendix). The actuator is actuated upon by a control effort u such that the desired set-point is reached. In the sense, when the single link gets actuated, the motor as well as the flexible link starts vibrating. At the tip of the flexible link, one displacement sensor is placed, which is used to give the feedback to the system and could be used for tracking purpose. This displacement x (), which is nothing but the output of the system ycould be considered as one of the state variable, which is used for tracking and bringing back to the desired position in no time. In other words, there is one actuator (motor)-input & one sensor (used to obtain the displacement of the end-effector-output as such the 1-link flexible system can be considered as a SISO system.

The sampling time given by () could be first issue that could be taken up in the designing of the FOS f/b controller. An external force v (voltage is applied to the actuator) for a particular duration of t sec at the hub of the flexible link & once the motor is actuated, the system starts to move & hence is subject to mechanical vibrations

& the OLIP (i.e., the plot of tip o/p‘ss y as a f/n of timet) of the plant is observed. The max^m BW for all the tip displacement sensor &motor-actuator locations (at the base) on the flexible dynamical system are found out (here, the 2nd mode of the overall flexible dynamical plant) &at that point by utilizing the current exact principles for choosing the examining interims dependent on the BW of the system, roughly ten times of the max^m second vibrational mode of the adaptable plant is chosen,i.e.,  is selected as  = 0.004 secs& the no. of sub-intervals Nwas chosen as equal to 10.

The FOS control law is graphically displayed in the Fig. 2 with its block diagrammatic counterpart in the Fig.

3. Next, the given CT system is discretized & the

&systems are obtained. Finally, it can be observed that both the &discrete systems are observable&controllable. In our case, the controllability index is given by 4 & the observability index is also obtained as 4, in the sense that all the states are controllable & observable. Next, the stabilizing FOS feedback gains Lare obtained (after obtaining the SFB gainF) such that the Eigen estimated values of the  - DT system is placed within circle of unit radius and the dynamical system is stable.

Next the  - DT dynamical system is considered and the FOS gain L is found out be solving the control law equation after taking the displacement output from the sensor place at the end of the end-effector of the FM, followed by the LMI optimization concepts such that we

Vol. 28, No. 7, (2019), pp. 124-135 get an optimal value of the FOS gain so that the amplitude

of the control signaluis reduced before taking the end of the flexible link to the desired position. The closed loop response, i.e., the displacement y () is observed. It could be seen that the CLIR settles at a quicker rate contrasted with the OLIR.

The designed FOS f/b controller is put in the f/b loop with the simulated dynamical plant and the CL step response, i.e. response and change of the control i/p signal

‗u‘ w.r.t ‗t‘ is observed.Pole placement technique is used to obtain the value of the SFB gain F (by using the place command in Matlab). Fis got so that the eigen estimated values of the CLCS of the dynamical plant given by

 



 



^F 

is placed within the circle of unit radius &

the system will be having a very good settling time w.r.t.

the o/p response, i.e., stabilizing SFB gains are got for the tau systems so that the eigen-values of the discrete system



_ _F



will be placed inside the circle of unit radius.

It should be noted that the effect of the state feedback is realized by a MROF f/b gain L. The SFG, F is obtained as

 

=  0.2134 0.3354  0.4659 0.5758

F

(21)

The CLIR of the 1-link flexible manipulator system with the SFB gain Fcan also be observed. The obtained SFB gain F is further used to develop the FOS f/b controller for the original CT system. Then, the FOS gain L can be determined by solving the linear gain equation

 F

LC

. But, the magnitude of this FOS gain obtained L is very large and hence Actually, the gains obtained by the exact solutions will be more & they may cause huge variation of the magnitudes of the i/p signals during output sampling times, which may require more control effort u exceeding the hardware limitations and cause more SNR to occur.

 

=  0.7134 9.3358 10.4699 10.4699

L 1 ⁽²²⁾

Moreover with large feedback gains, the flexible system becomes more sensitive to noise. So, an LMI optimization procedure is used to get the optimized value of the FOS gain L. The value of the performance matrices in the LMI concepts, i.e., ₁, ₂&₃are tuned properly to get the optimized FOS gain vector L, where the gains are having less values so that input saturation will not occur.

This gain vector L is having lesser magnitude & further it is found that the CLIR of the original dynamical system with FOSf/b gain controller gain Lgives a stable and desired optimal behavior.

Fig. 4 : Open loop & closed loop response of a 1-link flexible manipulator with SFB gain F& FOS gain L with

control input u

These FOS gains L are having very small magnitude compared to the FOS f/b gain obtained before the optimization procedure. Obviously the control effort u is very much reduced and the change of the i/p during an o/p sampling times is also less, thus not taxing the system, further involving less computations. The FOS f/b controller proposed by the methodology will require only piecewise constant gains & its RT implementation will be very much easy.

 ^{0.0352 0.1346 0.1346 0.6357} 

 

L

(23)

The value of 1, 2&3 chosen (tuned) to obtain the optimal response in the performance index calculation of norms of  was 1 = 0.01, 2 = 0.002 &3 = 0.0003 respectively. Codes are developed in the Matlab environment as .m files. The developed .m files are run, the test input is given as the input to the developed Matlab code & after running the simulation, the simulation results are observed for the proposed methodology and are presented as shown in the Figs. 4 & 5 respectively for the 1-link FRM case. Also, the bode plot in the frequency domain with & without the control effort is also shown here in this context for the sake of authentication to show the control effect. It has to be noted in this case that the gains are of dimensions (1 × 4) as it is a SISO case (single input single output as there is only one motor & one output displacement) & the state space model (A) is of (4 × 4).

From the frequency plot, it can be observed that the dB value is reduced with control.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-50 0 50

o/p disp tip-1 (deg)

OL resp.

CL resp. with F

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-5 0 5

Control Signal [V]

Control i/p

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-50 0 50

Time [secs]

o/p disp tip-1 (deg)

Open Loop, Closed Loop Responses with F, L & Control Input....FOS Simulation Results (1-link FM)

OL resp CL resp with L

Vol. 28, No. 7, (2019), pp. 124-135

130 Fig. 5 : Bode diagram of a 1-link flexible manipulator to

show effectiveness before & after the FOS control

5Control design for dual link flexible manipulator

It should be noted that in this case, a dual / double / 2- flexible links are connected to obtain the overall plant structure so that the 1^st motor or the 1^st actuator is connected to the hub, which is termed as the base motor or base actuator to which the link 1 is connected and w.r.t. to rear end of the 1^st link of the FRM, another 2^nd actuator or the 2^nd motor is connected, which is termed as the link- 2.Refer Fig. 14-16(appendix)for the diagrammatic representation of the flexible system with its specifications in Table 1 (appendix). Our ROI (region of interest) is the tip of the link 2, which is to be controlled. The actuator/s are actuated upon by a control effort u₁&u₂ such that the desired set-points are reached.

In the sense, when both the link gets actuated, the motor/s as well as the flexible links starts vibrating. At the tip of the flexible link-2, one displacement sensor is placed, which is used to give the feedback to the system and could be used for tracking purpose. Also, at the tip of the link-1 another displacement sensor-1 is attached for feedback purposes. This displacement x₁() of the link 1

&x₂() of the link 2 could be considered as the state variables, which is used for tracking and bringing back to the desired position in no time as a result of which the entire system becomes a 2 input, 2 output system or a MIMO system. In other words, there are two actuators, viz., (motor 1)-base motor & (motor 2)-shoulder motor.

These displacement x₁ () = y₁&x₂ () = y₂ are nothing but the outputs of the 2-link flexible manipulator system.

The sampling time () is the preliminary step taken during the FOS f/b controller designing. An external force v (voltage is applied to the actuator) for a particular duration of t sec at the hub of the flexible link & once the motor is actuated, the system starts to move & hence will be subject to mechanical vibrations & the OLIR (graph of tip outputy₁ as a f/n of t – end of link 1 & plot of tip output y₂ as a function of t – end of link 2) of the plant is observed. The max^mBW for all of the tip displacement sensor & motor-actuator locations (at the base) on the flexible system are obtained (in this context, the 2^nd vibrational mode of the overall flexible plant) and then by using the existing empirical rules for selecting the sampling interval based on bandwidth, approximately 10

times of the maximum second mode of the flexible plant is chosen, i.e.,  is selected as  = 0.004 s & the no. of sub- intervals Nwastaken to be equal to 10.

The FOS control law is graphically displayed in the Fig. 2 with its block diagrammatic approach in the Fig. 3.

Next, the given CT system is discretized and the &

systems are obtained. It is observed that both the

&discrete systems are observable&are controllable. In our case, the controllability index is given by 4 & the observability index is also obtained as 4, in the sense that all the states are controllable & observable. Next, the stabilizing o/p injection gain L is obtained so that the Eigen values of the  - DT system are placed within the circle of unit radius & will definetly be in stable condition.

Next the  - DT system is considered and the FOS gain L is found out be solving the control law equation after taking the sensor output, followed by the LMI optimization concepts such that we get an optimal value of the FOS gain so that the magnitude of the control effort uwill be drastically reduced before taking the end of the flexible link to the desired position. The closed loop response, i.e., the displacement y1 () … end of link 1 &

the displacement y₂ () … end of link 2 is observed. It could be seen from the graphical simulations that the CLIR settles at a much faster rate compared to the OLIR.

The designed FOS f/b controller is connected in the f/b loop with the simulated dynamical plant and the closed loop step response, i.e. response and change of the control signal or the effort ‗u1‘ & ‗u2‘ w.r.t. ‗t‘ is observed. We need 2 control efforts as there are 2 actuators or 2 motors to control the 2 links & take it to the desired position. Pole placement technique is used to obtain the SFB gainF (use the place command in Matlab). F = [F1 F2]^Tis obtained so that the Eigen values of the CLCS‘s



_ _F



lies will be placed within the circle of unit radius & the system will be having a very good settling time, i.e., stabilizing SFB gains are obtained for the tau system so that the eigen- values of the discrete system



_ _F



are placed within the circle of unit radius. It should be noted that the effect of the state feedback is realized by a MROP f/b gain L. The SFG, F is obtained as

0.2134 0.3354 0.4659 0.5758

= 0.3534 0.2314 0.5578 0.4569

 

 

   

 

F ⁽²⁴⁾

The closed loop impulse response of the 2-link flexible manipulator system with SFB gainFcan also be observed. The obtained SFB gain F is further used to design the FOS controller for the original CT system.

Then, the FOS gain L can be determined by solving the gain equation

LC  F

. But, the magnitude of this FOS gain obtained L is very large and hence Actually, the gains obtained by the exact solutions are of huge magnitude and they may cause large changes of inputs during an o/p sampling times, which may also require more control

-150 -100 -50 0

Magnitude (dB)

10⁰ 10¹ 10² 10³

0 45 90 135 180 225

Phase (deg)

Bode Diagram for 1-link FM with & w/o control

Frequency (rad/s)

with control w/o control

Vol. 28, No. 7, (2019), pp. 124-135 effort u exceeding the hardware limitations and cause

more SNR to occur.

 

0.7134 9.3358 10.4699 8.5658

= 9.9534 0.2319 8.5568 0.4669

 

 

   

 

L 1 L₁ L₂

1 1

(25)

Fig. 6 : OL & CL response w.r.t. a 2-link flexible robotic manipulator with SFB gain F₁& FOS gain L₁ with control

input u₁, output displacement of tip-1, end of link-1 (shoulder joint), y₁& the zoomed version of the control i/p

u.

Moreover with large feedback gains, the flexible system becomes more sensitive to noise and saturation may occur. So, an L M I optimization procedure is used to get the optimized value of the FOS gain L. The value of the performance matrices in the L M I concepts, i.e., ₁,

2&3 are tuned properly to get the optimized FOS gain vector L, where the gains are having less values so that input saturation will not occur. This gain vector L is having lesser magnitude & further it is found that the CL responses of the original CT system with FOS controller gain L gives a stable and desired optimal behavior.

Fig. 7 : Open loop & closed loop response of a 2-link flexible robotic manipulator with SFB gain F₂& FOS gain L₂ with control input u₂, output displacement of tip-2, end

of link-2 y₂

These FOS gains L are having very small magnitude in comparision with the FOS gains which are obtained before the optimization procedure. Obviously the control effort u is very much reduced and the variation of the control signal u during an o/p sampling time is also very less, thus not taxing the system, further involving less computations. RT implementation of the proposed FOS f/b controller is very easier as the process requires only constant gains. It should be noted that in the 2-link case, the L is a matrix of (2 × 4) size, C is a matrix of size (4 × 2), hence, F would be a matrix of (2 × 4) size. The frequency response plot of the 2-link FM system is shown in the Fig. 8.

 

^{0.0352 0.1346 0.3756 0.6357}

= 0.2436 0.0536 0.5456 0.7357

 

 

    

L L₁ L₂ (26)

The value of ₁, ₂&₃ chosen (tuned) to obtain the optimal response in the performance index calculation of norms of  was 1 = 0.02, 2 = 0.004 &3 = 0.0006 respectively. Codes are developed in the Matlab environment as .m files. The developed .m files are run, the test input is given as the input to the developed Matlab code & after running the simulation, the simulation results are observed for the considered research work &are presented as shown in the Figs. 6 &7 for the 1 & 2-link FRM system case. Also, the bode plot in the frequency domain with & without the control effort is also shown here in this context in Fig. 8 for the sake of authentication to show the control effect. It has to be noted in this case that the gains are of dimensions (2 × 4) as it is a MIMO case (2 input 2 output as there are 2 motors & 2 output displacements), but the state space model (A) is of (4 × 4)

& only i/p & the o/p vectors get changed in the SS model of the system.

0 1 2 3 4 5 6 7 8 9 10

-200 0 200

OL, CL y1 responses with F1 and L1 w ith control input u1 for joint-1 of 2 link FM using FOS

o/p disp tip-1 (deg)

OL resp.

CL resp. with F

0 1 2 3 4 5 6 7 8 9 10

-20 0 20

Control Signal [V]

Control i/p

0 1 2 3 4 5 6 7 8 9 10

-200 0 200

Time [secs]

o/p disp tip-1 (deg)

OL resp CL resp with L

0.1 0.2 0.3 0.4 0.5 0.6 0.7

-6 -4 -2 0 2 4 6 8

Control Signal [V]

0 1 2 3 4 5 6 7 8 9 10

-100 0 100

OL, CL responses y 2 with F

2 and L

2 with control input u

2 for joint-1 of 2 link FM using FOS

o/p disp tip-2 (deg)

OL resp.

CL resp. with F 2

0 1 2 3 4 5 6 7 8 9 10

-10 0 10

Contr Sig u2 [V]

Contr i/p u 2

0 1 2 3 4 5 6 7 8 9 10

-100 0 100

Time [secs]

o/p disp tip-2 (deg)

OL resp CL resp with L

2

Vol. 28, No. 7, (2019), pp. 124-135

132 Fig. 8 : Bode diagram of a 2-link flexible manipulator

system to show effectiveness before & after the FOS control

Fig. 9 : OL & CL response of the link-1 of a 2-link FRM with FOS gain L₁& control u₁

Till the previous cases, the simulations were carried out for a stepped excitation, in the next case, simulation is carried out for a sinusoidal excitation & the same designed FOS controller is used to get the output responses of the link-1 & link-2 (rear end displacements) of a 2-link FRM system. The sim results are shown in the Figs. 9 &10 respectively for the 2-link FM case. From these sim results, it could be inferred that the magnitude of the control signal which is required for controlling the dynamics of the link-2 (outer 10 V) is lesser compared to that of the link-1 (inner 20 V) as the base has to bear the entire weight of the flexible system.

Fig. 10 : OL & CL response of the link-2 of a 2-link FRM with FOS gain L₂& control u₂

6 Development of the FOS simulink model

Fig. 11 : Simulink model for design of a FOS controller for a 1-link & 2-link flexible robotic manipulator system

Fig. 12 : Scope results of OL & CL for the base motor (link-1) … amplitude more

The FOS controller is also developed in the Matlab- Simulink scene as displayed in the Fig. 11 for a 2-link case in which there are 2 sub-systems. It has to be noted that the model is the same for the 1-link case in which there will be one sub-system. The simulink model is constructed using sub-systems, sources, scopes, sinks, comparators, gain blocks, sample and hold circuits, multiplier blocks & the connectors. All these mentioned blocks are available in the simulink modelling library.

-200 -150 -100 -50 0

Magnitude (dB)

10⁰ 10¹ 10² 10³

-90 -45 0 45 90

Phase (deg)

Bode Diagram for a 2-link FM using FOS method w ith & w/o control effect

Frequency (rad/s)

w/o control with control

0 2 4 6 8 10 12 14 16 18 20

-20 -10 0 10 20

Control Signal [V]

FOS simulations for a sinusoidal excitation for link-1 base motor of a 2-link FM Control i/p u₁

0 2 4 6 8 10 12 14 16 18 20

-200 -100 0 100 200

Time [secs]

o/p disp link-1

OL resp CL resp with L₁

0 2 4 6 8 10 12 14 16 18 20

-10 -5 0 5 10

Control Signal [V]

FOS simulations for a sinusoidal excitation for link-2 shoulder motor of a 2-link FM Control i/p u₂

0 2 4 6 8 10 12 14 16 18 20

-100 -50 0 50 100

Time [secs]

o/p disp link-2 (deg)

OL resp CL resp with L₂

Vol. 28, No. 7, (2019), pp. 124-135

Fig. 13 : Scope results of OL & CL for the shoulder motor (link-2) … amplitude less

Apart from these, various toolboxes such as control system tool box, optimization tool box, signal processing tool boxes available in the simulink library is being used.

Various parameters are to be set in the different blocks that are used in the development of the simulink model.

Simulation time taken is 0 to 10 seconds (varying-any time can be taken). For the discretization purposes, the time, i.e., sample time used in the simulation is 1 ms. The developed simulink model is the same for both the types of flexible systems & only the number of sub-systems will be different. The model is run for the given requisite simulation time & the same results obtained as shown in the Matlab Simulation outputs is observed for a 2-link case

& the results are not shown here for a 1-link case here for the sake of convenience.

7 Conclusions

Research was conducted on the set-point control, end- point displacements, control of the joints of the tip-1 &

tip-2 of the single link and double link flexible robotic manipulator. CT SS dynamical model of the 1 & 2 link FRM system was developed. The CTSS model was discretized @ tau & delta sampling intervals and the discretized systems was obtained for developing the controller. FOS f/b controllers were developed for the flexible manipulators to control the joint 1 of



link 1 of 1-link manipulator & the joints 1 & 2 of – the 2 link flexible manipulator. The different results / graphs of the plant parameters aregot for each of the SS models of the 2 individual FRM systems.

The developed .m file was run & the simulation results were observed. Through the results of the Matlab simulations, it is construed that when the FRM plant system is set with this planned/proposed FOS f/b controller, the plant performs extremely nice,the output tip displacements reaches the set-point quickly in the form of the closed loop response performing much better than the OLIR. From the responses of the double link flexible manipulator, it is observed that the output displacement of the link-1 is more compared to that of the 2-link case as it has to drive the link-1 & 2 plus the motor-2 weight also along with the end-effector payload, which is housed @ the shoulder level point. It has to be noted that the flexible

manipulator taken into consideration is a planar 2-link one which moves in the x – y plane.

In the 1-link case, the control effort is 5 units, whereas in the 2-link case, the effort is 20 units for the same base motor. It could be also noted that in the 2-link case, the control effort required for the base motor is 20 units as it has to drive a less load (only the link-2) plus the end-effort mass of the payload, whereas for the link-1 it is

10 units. It can also be noted that control exertion u required from the FOS controller gets decreased if the actuator is moved far from the fixed end to the end- effector's ROI. A little extent of the control effort u is adequate to control the joint-2of the 2-link case (10 units).

ASFB gain F for each discrete system models (tau &

delta systems) of the 2 flexible systems is got so that the poles (roots of the denominator of the TF) are placed within the circle of unit radius at requisite positions & the CLflexible system will be having a very good time of settling w.r.t. the o/p response curves. A very very small amplitude of the of control effortuwill is chosen to control the actuator as it is moved away from the base in a 2-link case as a result of which less effort has to be put by the controller (since the link-2 motor has to take care of only the link-2, whereas the link-1 motor has to bear both the link-1 & link-2 motor). The amplitude of the CLIR of both the CT & the lifted system, i.e., the intermediate system, i.e., with SFB gain Fwill be very less in comparision to their open loop counterparts.

The close loop impulse response performances with the gains w.r.t.F&L are found to bethe best & settle out quickly. Thus, the observations are made with and without the controller to show the control effect.

Considering the simulations, it was seen that w/o control the transient response characteristics was not satisfactory

& was taking much time to settle & with the control effort, the closed loop response is less satisfactory. Unlike the SFB case, the FOS f/bcontrol schemewill definitely guarantee the stability of the CLCS, which could be seen from the results of the simulation.

The proposed FOS f/b controller requires steady gains and thus might be simpler to actualize progressively in the RT applications. The controllability & observability index obtained is 4 so that all the states are controllable &

observable. One excellent advantage of the proposed methodology developed is that the computation time required for processing & getting the output is just within 3-4 seconds, which shows the advantage of our proposed method over the others [21] - [25]. One advantage of the 2-link mechanisms is that it can cover a greater area of the work-space & the tip or the end-effector can be tracked in a bigger dimension, i.e., R² area as the flexible manipulator works in the planar environment (x – y place).

The work is carried out for both the step & sinusoidal excitations & here only the Matlab & Simulink‘s