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International Conference on Systemics, Cybernetics and Informatics
Stable Design Of A Bionic Eye Control System Achintya Das
Rimi Ghosh
Rupa Das
Professor and Head Electronics and Communication Engineering Kalyani Goverment Engineering College P.O: Kalyani Dist:Nadia WestBengal. [email protected]
Faculty member Electronics and Instrumentation Engineering J.I.S College of .Engineering, P.O:Kalyani, Dist:Nadia West Bengal [email protected]
M.Tech Project Scholar Electronics and Communication Engineering Kalyani Goverment Engineering College P.O: Kalyani, Dist:Nadia WestBengal. [email protected]
brain, closing the loop.. Figure.3(a), is a block diagram of the control loop of the artificial eye. The compensator of Figure.3(b). simulate the lens/detector portion of Figure.3(a).
Abstract The present work describes the design and analysis of a control system, which is a simulation of an artificial eye or bionic eye in an efficient way. In accordance with the principle of artificial vision introduced, a gimbaled mirror rotates to follow the motion of an otherwise blind eye. The mirror, in turn, causes a camera to detect light from the direction toward which the eye is pointed. That light proceeds to an artificial retina, to simulate brain activity. The total control system in this work is considered to be composed of Gimballed mirror and compensator system as operated in a closed loop manner with unity negative gain in the closed loop feedback path. Hence, the overall open loop transfer function in the forward path is simply the product of the transfer functions of the Gimballed mirror, and the compensator system. The controllability and the observability of the system are tested, and the system is found to be controllable & observable. MATLAB 7.1 software is suitably used in the concerned analysis and design.
Figure 1. Normal Vision
Figure.2. Bionic Vision(with an artificial eye).
Index Terms— Bionic eye, Controllability, observability. 1. Introduction Control system science and biomedical instrumentation form a partnership that has the promise of restoring sight to those without central vision.[1]. Bio-Electronic eye-An electronic device which replaces functionality of a part or whole of eye. It is used for replacing functionality or adding functionality to the eye. Figure.1. shows a block diagram of normal human vision. When a person wants to view an object, the brain causes the muscle to rotate the eye’s central line to rotate the object. The image from that object passes through the eye’s lens to the retina, sending impulses back to the brain, closing the loop. In Figure.2, the bionic artificial eye system, the brain attempts to view the object by pointing the poorly functioning eye along some angle. An artificial eye orients a camera along the same angle as the eye. The image from the object enters the camera and is then routed to some components that elicit response in the
Figure.3(a). Artificial eye control loop dynamics. The control loop.
Figure.3(b). Simulation of Figure.3(a).
2. Conversion From S-Domain To Z-Domain [5] The transformation of z transform helps in the analysis & design of sample data control system, as Laplace transform does in the analysis and design of continuous data control system. The z-transform F(z) of a sample data control signal f(KT) is defined by the relation:
Copyright © 2010 Paper Identification Number: CC-1.1 This peer-reviewed paper has been published by the Pentagram Research Centre (P) Limited. Responsibility of contents of this paper rests upon the authors and not upon Pentagram Research Centre (P) Limited. Copies can be obtained from the company for a cost.
∞
F(z)=
∑ f ( KT ) z K =0
97
−K
…….. (1)
Stable Design Of A Bionic Eye Control System
The above relation is derived from the Laplace transformation as applied to sample data control signal, assuming esT= z as the concerned transformation variable in Laplace transformation. We have, sT=lnz i.e, s-1=
T ln z
For example, the eye should be able to change focus from one object to another object without any steady state angular positioning error. Thus, there should be zero steady state step error. Similarly, the eye should be able to scan a scene or track a moving object with smooth response. That property will follow if the artificial eye provides zero steady state ramp error. Therefore the control loop should be type 2. and Gp(s) each have a pole at s=0 as Figure.4(c), then the steady state properties are assured in that the control loop is clearly type 2.[1]. The close loop Transfer function is
……………………(2).
Using power series expansion of lnz, the above equation becomes:
1
1
4
44
3 5 s-1= T [ − u − u u ]…….….(3), 45 945 2 u 3
T ( s) =
where u=
1 − z −1 1 + z −1
k +k Gc ( s) = 1 + 1 2 s 60 G p ( s) = s( s + 60)
In general, for any positive integral value n, we have, n
s- n = T 1 − 1 u − 4 u 3 − 44 u 5 45 945 2 u 3
Gc ( s)G p (s ) 1 + Gc (s )G p ( s )
n
….(4)
By using binomial expansion in the above equation for various values of n, we may have the transformation from s to z domain.
……………(5)
………………(6) ………….……(7)
Under present situation, overall transfer function of the system is given by the relation:
Y ( s) = T (s ) R( s ) 60 s + 60( k1 + k 2 ) T ( s) = 3 s + 60 s 2 + 60 s + 60( k1 + k 2 )
3. Controllability & Observability [2] 3.1 Controllability A system described by the matrices (A,B) can be said to be controllable, if there exists an unconstrained control u that can transfer any initial state x(0) to any other described location x(t). For the system, the state equation is represented as: x’=Ax + Bu. We can determine whether the system is controllable by examining the algebraic condition: Rank[B AB A2B……An-1B]=n. For a single input, single output system, the controllability matrix Pc is described in terms of A &B. Pc= [B AB A2B……An-1B], which is an nxn matrix. Therefore if the determinant of Pc is nonzero, the system is controllable[2]. 3.2 Observability Observability refers to the ability to estimate a state variable. .A system is observable, if and only if, there exists a finite time t such that the initial state x(0) can be determined from the observation history y(t) given the control u(t)[12]. Consider the single input single output system represented by the state space equation: x’=Ax + Bu, and y=Cx, C is a row vector & x is a column vector. This system is observable, when the determinant of Q is nonzero where Q= C CA ……CAn-1 which is a nxn matrix.
…..(8)
Thus for the entire system the characterstic equation is …… (9). s 3+60s2+60s+60(k1+k2) =0 For this characteristics equation to make the design problem with stability, the Routh array is constructed as below: S3 1 60 S2 60 60(k1+k2) S1 60 -( k1+k2) S0 60(k1+k2) For stability (k1+k2) >0 (k1+k2) <60 Considering k1 = 1 and k2 = 20, overall transfer function of the system becomes: 1260 + 60 s ………..(10) T (s ) = 3 s + 60s 2 + 60 s + 1260
5. Methods And Material Normally control system is considered in continuous data control system (continuous time domain ↔ Laplace domain), the system analysis and study get restricted for any change in the system parameter, or input variation for easy and ready study. To circumvent this problem sample data(s.d.) control system makes study and analysis easy and ready available with variation in the system parameter and also the input. For this reason the system is also studied in sample data control model. The stability of the present system is tested by Jury’s stability test which guarantees the stability of the overall system. Needless to mention, any stable system when operated in s.d. mode, the system is not necessarily to be guaranteed to remain stable in the s.d. mode also, there being
4. System Design The overall control system under study consists of one appropriate compensator in cascade with the gimbaled mirror system in closed loop manner with unity negative feedback path. To design the compensator Gc(s) and gimbaled mirror Gp(s), we need to think about the properties of normal vision this electronic eye should reproduce.
98
International Conference on Systemics, Cybernetics and Informatics
the enhancement of the order of the system as introduced by the sampler. As any control system deserves to reach its steady state by which the system finally runs, and follows the input at that state, the designed parameter k1 and k2 is accordingly decided. The other desirable characteristic performances being also available in the system.
4.
Jury stability criteria are satisfied.
8. Output Plots
6. Matlab Output Files [6] Transfer function: G= Transfer function: 60 s + 1260 -------------------------s^3 + 60 s^2 + 60 s + 1260 Gain Margin and Phase Margin G(m)=infinite P(m)= 22.2 deg (at 6.55 rad/sec) A= -60.0000 -3.7500 -9.8438 16.0000 0 0 0 8.0000 0 B=4 0 0 C = 0 0.9375 2.4609 D=0 >> rank(M) ans = 3 >> length(A) ans = 3 >> rank(N) ans = 3 Sampling Time T=0.02 G’= 0.0096115 z^2 + 0.0011566 z - 0.0049097 -------------------------------------------------z^3 - 2.2837 z^2 + 1.5908 z - 0.30119 for jury test a0= 1 a1= -2.2837 a2= 1.5908 a3= -0.30119 b0= -0.902972 b1= 1.80456 b2= - 0.9092845 c0= -0.011368 c1= -0.01141623 G’(1)>0 G’(-1) is negative │ a3│< │ a0│ satisfied │ b2│> │b0│ satisfied │ c1│> │c0│ satisfied
Plot 1. Bode plot
Plot 2. Root Locus plot Impulse Response 5 4 3
Amplitude
2
0 -1 -2 -3
7. Results 1. 2. 3.
1
GAIN MARGIN= infinite PHASE MARGIN = 22.2 deg (at 6.55 rad/sec) Rank of the controllability & observability matrix=3, as same as the order of the system & hence controllable & observable.
-4
0
2
4
6
8
10
12
14
Time (sec)
Plot 3. Impulse Response plot
99
16
18
Stable Design Of A Bionic Eye Control System
margin of value 22.2 degree. Since the system is stable in both continuous and sample data control system, controllable, observable, having appropriate gain margin and phase margin, the design of an artificial eye (Bionic eye) system becomes feasible as done in this present paper
Step Response 2 1.8 1.6 1.4
Amplitude
1.2
10. References
1
1.
Stefani Shahian Savant Hostetter “Design of feedback control system”, 4rth ed, Oxford University press, 2002, pp.237-240. 2. R.C.Dorf, Bishop,”Modern Control system”8th ed, Addison Wesley, 1999, pp 286- 286, P5.16. 3. S.C Biswas,A.Das,P.Guha pp 15- 17”Mathematical Model of Cardiovascular system by transfer function Method”, Calcutta Medical Journal. Vol-103,No.4 July-Aug2006. 4. William F Ganong,pp63-64”Review of Medical edition, Prentice –Hall Physiology”,14th International. 5. Achintya Das, pp 96-98”Advance Control system”,3rd ed,Matrix Educare,Feb2009. 6. MATLAB 7.0. 7. R.N Clark ,Introduction to Automatic control system, John wiley& sons,New York,1962,pp 115124. 8. R.C.Dorf, Electric circuits, 3rd ed. John Wiley&sons,New York,1996. 9. Athans ,M;”The status of optimal control theory & Applications for Deterministic systems” IEEE Trans. Autom. Control (July 1996). 10. Sage .A.P,and White,C.C.3;Optimum system control, Englewood cliffs, NJ :Prentice Hall,1977. 11. W.J .Rough, Linear system Theory ,2nd ed, Prentice Hall, Englewood Cliffs,N.J.,1996. 12. C.M close & D.K Fredrick, Modeling & Analysis of Dynamic systems,2nd ed, Houghton Mifflin, Boston, 1993.
0.8 0.6 0.4 0.2 0 0
2
4
6
8
10
12
14
16
18
Time (sec)
Plot 4. Step Response plot Step Response 2 1.8 1.6 1.4
Amplitude
1.2 1 0.8 0.6 0.4 0.2 0
0
50
100
150
200
250
300
350
Time (sec)
Plot 5. Step Response plot (sample data)
9. Conclusion The system offers an in depth sight for the biomedical engineering application of an artificial eye (Bionic eye) system. As it is analyzed using MATLAB software, the overall system is found to have infinite gain margin and phase
100
Stable Design Of A Bionic Eye Control System Achintya Das
Rimi Ghosh
Rupa Das
Professor and Head Electronics and Communication Engineering Kalyani Goverment Engineering College P.O: Kalyani Dist:Nadia WestBengal. [email protected]
Faculty member Electronics and Instrumentation Engineering J.I.S College of .Engineering, P.O:Kalyani, Dist:Nadia West Bengal [email protected]
M.Tech Project Scholar Electronics and Communication Engineering Kalyani Goverment Engineering College P.O: Kalyani, Dist:Nadia WestBengal. [email protected]
brain, closing the loop.. Figure.3(a), is a block diagram of the control loop of the artificial eye. The compensator of Figure.3(b). simulate the lens/detector portion of Figure.3(a).
Abstract The present work describes the design and analysis of a control system, which is a simulation of an artificial eye or bionic eye in an efficient way. In accordance with the principle of artificial vision introduced, a gimbaled mirror rotates to follow the motion of an otherwise blind eye. The mirror, in turn, causes a camera to detect light from the direction toward which the eye is pointed. That light proceeds to an artificial retina, to simulate brain activity. The total control system in this work is considered to be composed of Gimballed mirror and compensator system as operated in a closed loop manner with unity negative gain in the closed loop feedback path. Hence, the overall open loop transfer function in the forward path is simply the product of the transfer functions of the Gimballed mirror, and the compensator system. The controllability and the observability of the system are tested, and the system is found to be controllable & observable. MATLAB 7.1 software is suitably used in the concerned analysis and design.
Figure 1. Normal Vision
Figure.2. Bionic Vision(with an artificial eye).
Index Terms— Bionic eye, Controllability, observability. 1. Introduction Control system science and biomedical instrumentation form a partnership that has the promise of restoring sight to those without central vision.[1]. Bio-Electronic eye-An electronic device which replaces functionality of a part or whole of eye. It is used for replacing functionality or adding functionality to the eye. Figure.1. shows a block diagram of normal human vision. When a person wants to view an object, the brain causes the muscle to rotate the eye’s central line to rotate the object. The image from that object passes through the eye’s lens to the retina, sending impulses back to the brain, closing the loop. In Figure.2, the bionic artificial eye system, the brain attempts to view the object by pointing the poorly functioning eye along some angle. An artificial eye orients a camera along the same angle as the eye. The image from the object enters the camera and is then routed to some components that elicit response in the
Figure.3(a). Artificial eye control loop dynamics. The control loop.
Figure.3(b). Simulation of Figure.3(a).
2. Conversion From S-Domain To Z-Domain [5] The transformation of z transform helps in the analysis & design of sample data control system, as Laplace transform does in the analysis and design of continuous data control system. The z-transform F(z) of a sample data control signal f(KT) is defined by the relation:
Copyright © 2010 Paper Identification Number: CC-1.1 This peer-reviewed paper has been published by the Pentagram Research Centre (P) Limited. Responsibility of contents of this paper rests upon the authors and not upon Pentagram Research Centre (P) Limited. Copies can be obtained from the company for a cost.
∞
F(z)=
∑ f ( KT ) z K =0
97
−K
…….. (1)
Stable Design Of A Bionic Eye Control System
The above relation is derived from the Laplace transformation as applied to sample data control signal, assuming esT= z as the concerned transformation variable in Laplace transformation. We have, sT=lnz i.e, s-1=
T ln z
For example, the eye should be able to change focus from one object to another object without any steady state angular positioning error. Thus, there should be zero steady state step error. Similarly, the eye should be able to scan a scene or track a moving object with smooth response. That property will follow if the artificial eye provides zero steady state ramp error. Therefore the control loop should be type 2. and Gp(s) each have a pole at s=0 as Figure.4(c), then the steady state properties are assured in that the control loop is clearly type 2.[1]. The close loop Transfer function is
……………………(2).
Using power series expansion of lnz, the above equation becomes:
1
1
4
44
3 5 s-1= T [ − u − u u ]…….….(3), 45 945 2 u 3
T ( s) =
where u=
1 − z −1 1 + z −1
k +k Gc ( s) = 1 + 1 2 s 60 G p ( s) = s( s + 60)
In general, for any positive integral value n, we have, n
s- n = T 1 − 1 u − 4 u 3 − 44 u 5 45 945 2 u 3
Gc ( s)G p (s ) 1 + Gc (s )G p ( s )
n
….(4)
By using binomial expansion in the above equation for various values of n, we may have the transformation from s to z domain.
……………(5)
………………(6) ………….……(7)
Under present situation, overall transfer function of the system is given by the relation:
Y ( s) = T (s ) R( s ) 60 s + 60( k1 + k 2 ) T ( s) = 3 s + 60 s 2 + 60 s + 60( k1 + k 2 )
3. Controllability & Observability [2] 3.1 Controllability A system described by the matrices (A,B) can be said to be controllable, if there exists an unconstrained control u that can transfer any initial state x(0) to any other described location x(t). For the system, the state equation is represented as: x’=Ax + Bu. We can determine whether the system is controllable by examining the algebraic condition: Rank[B AB A2B……An-1B]=n. For a single input, single output system, the controllability matrix Pc is described in terms of A &B. Pc= [B AB A2B……An-1B], which is an nxn matrix. Therefore if the determinant of Pc is nonzero, the system is controllable[2]. 3.2 Observability Observability refers to the ability to estimate a state variable. .A system is observable, if and only if, there exists a finite time t such that the initial state x(0) can be determined from the observation history y(t) given the control u(t)[12]. Consider the single input single output system represented by the state space equation: x’=Ax + Bu, and y=Cx, C is a row vector & x is a column vector. This system is observable, when the determinant of Q is nonzero where Q= C CA ……CAn-1 which is a nxn matrix.
…..(8)
Thus for the entire system the characterstic equation is …… (9). s 3+60s2+60s+60(k1+k2) =0 For this characteristics equation to make the design problem with stability, the Routh array is constructed as below: S3 1 60 S2 60 60(k1+k2) S1 60 -( k1+k2) S0 60(k1+k2) For stability (k1+k2) >0 (k1+k2) <60 Considering k1 = 1 and k2 = 20, overall transfer function of the system becomes: 1260 + 60 s ………..(10) T (s ) = 3 s + 60s 2 + 60 s + 1260
5. Methods And Material Normally control system is considered in continuous data control system (continuous time domain ↔ Laplace domain), the system analysis and study get restricted for any change in the system parameter, or input variation for easy and ready study. To circumvent this problem sample data(s.d.) control system makes study and analysis easy and ready available with variation in the system parameter and also the input. For this reason the system is also studied in sample data control model. The stability of the present system is tested by Jury’s stability test which guarantees the stability of the overall system. Needless to mention, any stable system when operated in s.d. mode, the system is not necessarily to be guaranteed to remain stable in the s.d. mode also, there being
4. System Design The overall control system under study consists of one appropriate compensator in cascade with the gimbaled mirror system in closed loop manner with unity negative feedback path. To design the compensator Gc(s) and gimbaled mirror Gp(s), we need to think about the properties of normal vision this electronic eye should reproduce.
98
International Conference on Systemics, Cybernetics and Informatics
the enhancement of the order of the system as introduced by the sampler. As any control system deserves to reach its steady state by which the system finally runs, and follows the input at that state, the designed parameter k1 and k2 is accordingly decided. The other desirable characteristic performances being also available in the system.
4.
Jury stability criteria are satisfied.
8. Output Plots
6. Matlab Output Files [6] Transfer function: G= Transfer function: 60 s + 1260 -------------------------s^3 + 60 s^2 + 60 s + 1260 Gain Margin and Phase Margin G(m)=infinite P(m)= 22.2 deg (at 6.55 rad/sec) A= -60.0000 -3.7500 -9.8438 16.0000 0 0 0 8.0000 0 B=4 0 0 C = 0 0.9375 2.4609 D=0 >> rank(M) ans = 3 >> length(A) ans = 3 >> rank(N) ans = 3 Sampling Time T=0.02 G’= 0.0096115 z^2 + 0.0011566 z - 0.0049097 -------------------------------------------------z^3 - 2.2837 z^2 + 1.5908 z - 0.30119 for jury test a0= 1 a1= -2.2837 a2= 1.5908 a3= -0.30119 b0= -0.902972 b1= 1.80456 b2= - 0.9092845 c0= -0.011368 c1= -0.01141623 G’(1)>0 G’(-1) is negative │ a3│< │ a0│ satisfied │ b2│> │b0│ satisfied │ c1│> │c0│ satisfied
Plot 1. Bode plot
Plot 2. Root Locus plot Impulse Response 5 4 3
Amplitude
2
0 -1 -2 -3
7. Results 1. 2. 3.
1
GAIN MARGIN= infinite PHASE MARGIN = 22.2 deg (at 6.55 rad/sec) Rank of the controllability & observability matrix=3, as same as the order of the system & hence controllable & observable.
-4
0
2
4
6
8
10
12
14
Time (sec)
Plot 3. Impulse Response plot
99
16
18
Stable Design Of A Bionic Eye Control System
margin of value 22.2 degree. Since the system is stable in both continuous and sample data control system, controllable, observable, having appropriate gain margin and phase margin, the design of an artificial eye (Bionic eye) system becomes feasible as done in this present paper
Step Response 2 1.8 1.6 1.4
Amplitude
1.2
10. References
1
1.
Stefani Shahian Savant Hostetter “Design of feedback control system”, 4rth ed, Oxford University press, 2002, pp.237-240. 2. R.C.Dorf, Bishop,”Modern Control system”8th ed, Addison Wesley, 1999, pp 286- 286, P5.16. 3. S.C Biswas,A.Das,P.Guha pp 15- 17”Mathematical Model of Cardiovascular system by transfer function Method”, Calcutta Medical Journal. Vol-103,No.4 July-Aug2006. 4. William F Ganong,pp63-64”Review of Medical edition, Prentice –Hall Physiology”,14th International. 5. Achintya Das, pp 96-98”Advance Control system”,3rd ed,Matrix Educare,Feb2009. 6. MATLAB 7.0. 7. R.N Clark ,Introduction to Automatic control system, John wiley& sons,New York,1962,pp 115124. 8. R.C.Dorf, Electric circuits, 3rd ed. John Wiley&sons,New York,1996. 9. Athans ,M;”The status of optimal control theory & Applications for Deterministic systems” IEEE Trans. Autom. Control (July 1996). 10. Sage .A.P,and White,C.C.3;Optimum system control, Englewood cliffs, NJ :Prentice Hall,1977. 11. W.J .Rough, Linear system Theory ,2nd ed, Prentice Hall, Englewood Cliffs,N.J.,1996. 12. C.M close & D.K Fredrick, Modeling & Analysis of Dynamic systems,2nd ed, Houghton Mifflin, Boston, 1993.
0.8 0.6 0.4 0.2 0 0
2
4
6
8
10
12
14
16
18
Time (sec)
Plot 4. Step Response plot Step Response 2 1.8 1.6 1.4
Amplitude
1.2 1 0.8 0.6 0.4 0.2 0
0
50
100
150
200
250
300
350
Time (sec)
Plot 5. Step Response plot (sample data)
9. Conclusion The system offers an in depth sight for the biomedical engineering application of an artificial eye (Bionic eye) system. As it is analyzed using MATLAB software, the overall system is found to have infinite gain margin and phase
100
