Black-Box Control in Theory and Applications RMIT University, 19.08.2016 Arie Levant School of Mathematical Sciences, Tel-Aviv University, Israel Homepage: http://www.tau.ac.il/~levant/
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Control problems The task is to make a process behave as we want. Mathematical control appears only when we succeed to quantify the problem. Mathematical control theory usually requires a mathematical model of the process.
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Contr. problems which cannot be addressed here Control of war and peace, geopolitics, Long-term climate control, Public opinion control Contr. problems which maybe can be addressed Finances: Macro-economic control by state bank, Taxes control, etc Short-term climate control (?) Contr. problems which are addressed Air condition, auto-pilots, keeping bicycle balance, targeting, tracking, orientation, hormonal levels in blood, etc. 3
General Control Problem as Black-Box control
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Tracking deviation: s =
y - yc ( t ) The goal: s = 0
We need some PSEUDO-MODEL 5
Main "principles" System model is a mathematical model which adequately describes the input-output relations. Whatever it means … No model is exact. The control goal is to make the output s satisfy some requirements by a proper choice of the control u in real time. Any solution of the problem should be feasible and robust. 6
Models & approaches to "Black Box" 1. Sliding-Mode Control (here): r d s = h( t ) + g ( t ) u , r dt r Î ¥, h Î [-C, C ], g Î [ K m , K M
]
2. Model-free control (Fliess, Join, Lafont, et al) "Ultra-local model" r d s = F + Ku , r = 1, 2 , F , K = const r dt PID (proportional, integral, derivative) control
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Some names and notation r
d s = h( t ) + g ( t ) u r dt r is called the relative degree r d s = s( r ) , d s = s & r dt dt
x Î [ a, b] Û a £ x £ b
s = O(e) of the order of e, i.e. roughly proportional 8
In order to control a Black Box (r ) s Î [-C, C ] + [ K m , K M ]u one should at least identify r. r is called the Practical Relative Degree (PRD) In the framework by Fliess r
= 1, 2
We also want some nice features: smooth / Lipschitzian bounded control 9
Start with control of a smooth system x& = f (t , x, u), s = s(t , x) n
x Î R , u, s Î R
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Systems non-affine in control x& = f(t,x,u), xÎR , n
output s(t,x) (tracking error), input u ÎR
l
The goal: s º 0
Nonlinearity in control and its discontinuity Þ v = u& is taken as a new control, æ x& ö æ f (t , x, u ) ö æ 0 ö + ç ÷v ç u& ÷ = ç ÷ 0 è ø è ø èIø
The new system is affine in control, u(t) is differentiable.
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From that moment the system is x& = a(t , x ) + b(t , x )u, s = s(t , x) ¥
n
a, b, s Î C , x Î R , u, s(t , x ) Î R
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Relative Degree (RD) x& = a(t , x ) + b(t , x )u, xÎR , s, u ÎR n
Informally: RD is the number r of the first total derivative of s where the control explicitly appears with a not-vanishing coefficient. s
(r )
= h( t , x ) + g ( t , x ) u , g ¹ 0
1 && Newton law: x = m F , RD=2 13
In my practice the relative degrees r = 2, 3, 4, 5 mechanical systems, Newton law, integrators
But the solution will be valid for any r.
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Increasing the relative degree Black-Box Control problem: s ® 0
s s
( r +1)
(r )
= h( t , x ) + g ( t , x ) u
= h% (t , x ) + g% (t , x )u + g (t , x )u&
v = u&, s
( r +1)
Î [-C1 , C1 ] + [ K m , K M ]v
Remark: u is to be kept bounded … 15
Any relative degree is possible (example by Isidori)
q1 K J1 q&&1+F1 q&1- (q2- ) = u, N N q1 J2 q&&2 +F2 q&2 +K(q2- )+mgd cos q2 = 0 N The output : q2, The input: u. The relative degree r = 4
u& = v
The output : q2, The input: v. The relative degree r = 4+1=5
Any relative degree can be got in such a way. 16
Inevitable BAD subproblem z&0 = z1 , z&1 = z2 , ..., z&r - 2 = zr -1 , z&r -1 = u , output: y = z0
The goal: s = y (t ) - f (t ) = 0 s
(r )
= f
(r )
(t ) + u compare s Let | f
(r )
Î [-C , C ] + [ K m , K M ]u
(r )
( t ) |£ C (i )
If s º 0 then zi = f (t ) , i = 0,1,..., r - 1 Exact differentiation is included! 17
Main idea Black-Box Control problem: s ® 0
s
(r )
= h(t , x (t )) + g (t , x (t ))u is replaced with
s
(r )
Î [-C , C ] + [ K m , K M ]u Assumptions h Î [ -C , C ], g Î [ K m , K M ] 18
Solution method s
(r )
Î [-C , C ] + [ K m , K M ]u
u = aU r (s, s& ,..., s or
s
( r +1)
( r -1)
)
Î [-C1 , C1 ] + [ K m , K M ]u&
u& = a1U r +1 (s, s& ,..., s
(r )
)
Continuous control cannot solve the problem
U r , U r +1 are discontinuous, but bounded 19
Sliding mode (SM) (not a math. definition) Any system motion mode existing due to highfrequency (theoretically infinite-frequency) control switching is called SM.
rth-order sliding mode (r-SM) (not a math. definition) r-SM is a SM keeping s º 0 for RD = r by means of high(infinite)-frequency switching of u. 20
Some abbreviations till now SM - sliding mode, r-SM – rth order SM SMC – sliding mode control RD – relative degree PRD – practical relative degree
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Preliminary conclusions SMC theoretically "almost" solves the classical Black-Box control problem.
It includes exact robust differentiation of any order and robustness to small sampling/model noises, delays and disturbances (also singular). 22
Special power functions (standard notation) g g g
ë sù = s
@ s sign s
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The following controllers exactly robustly and in finite time provide for sº0 for the simplest model s
(r )
Î [-C , C ] + [ K m , K M ]u 24
Simplest r-SM controllers §s¨
g
(Ding, Levant, Li, Automatica 2016) g " d > 0 , $b0 ,..., bn- 2 > 0 @ s sign s ,
Relay-polynomial homogeneous r-SMC d d dù é ( r -1) ¬ 1 ( r - 2) ¬ 2 © © ê u = -a sign ª« s -® + bn - 2 ª« s -® + L + b0 §s¨ r ú êë úû
Quasi-continuous polynomial homogeneous r-SMC u = -a
d d d © ( r -1) ¬ 1 © ( r - 2) ¬ 2 ª« s -® +bn - 2 ª« s -® +L +b0 s r
§ ¨
d d d s( r -1) 1 + bn - 2 s( r - 2) 2 +L +b0 s r
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Quasi-continuous control u = U (s, s& ,..., s
( r -1)
)
is called quasi-continuous (quasi-smooth), provided it remains a continuous (smooth) function whenever
(s, s& ,..., s
Example: u = -a
( r -1)
) ¹ (0,0,...,0)
d d d © ( r -1) ¬ 1 © ( r - 2) ¬ 2 s +b s +L +b0 s r ª« -® n-2 ª « ®
§ ¨
d d d s( r -1) 1 + bn - 2 s( r - 2) 2 +L +b0 s r
d > kr Þ quasi k-smooth
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List of controllers, d = r r = 1,2,3,4,5
1. u = -a sign s, 2 2. u = -a sign(§s& ¨ + s),
&& + §s& ¨ + s) , 3. u = -a sign(s 3 2
3
&& ¨ + 2§s& ¨ + s) , 4. u = -a sign(§&&& s ¨ + 2§s 5 5 5 (4) ¬5 © && ¨3 + 3§s& ¨4 + s). 5. u = -a sign(ª« s -® + 6§&&& s ¨ 2 + 5§ s 4
2
4 3
a is to be taken sufficiently large. 27
Quasi-continuous controllers, d = r 1. u = -a sign s, 2 §s& ¨ +s 2. u = -a 2 , s& +|s|
3. u = -a 4. u =
3 s + s& 2 +s , 3 &&|3 +|s& | 2 +|s| |s
&&3
§ ¨
4 &&& && + 2 s& 3 +s s +2 s -a , 4 &&& && 2 + 2s& 3 +|s| s4 + 2s 5 5 5 © (4) ¬5 && 3 + 3 s& 4 +s s 2 +5 s ª s - + 6 &&&
5. u = -a
§ ¨
§ ¨
4
«
2
®
§ ¨
§ ¨
§ ¨
§ ¨
5 5 5 &&| 3 +3|s& | 4 +|s| s(4) |5 +6|&&& s| 2 +5|s
. 28
Another family (Levant 2005) quasi-continuous controller r = 2 s& + | s | sign s u=-a 1/ 2 | s& | + | s | 1/ 2
quasi-continuous controller r = 3 u=-a
s && + 2
( s& + |s|2 / 3 sign s )
(|s& |+ |s|2 / 3 )1 / 2 2 / 3 1/ 2
|s && | +2(| s& | + | s |
)
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Discontinuous Differential Equations Filippov Definition
x& = f(x) Û x& Î F(x)
x(t) is an absolutely continuous function
F ( x) =
II
e> 0mN = 0
convex_closure f (Oe ( x) \ N )
Filippov DI:
F(x) is non-empty, convex, compact, upper-semicontinuous. Theorem (Filippov 1960-1970): Þ Solutions exist for Filippov DIs, and for any locally bounded Lebesgue-measurable f(x). Non-autonomous case: t& = 1 is added. 30
Discontinuous Differential Equations Filippov Definition
When switching imperfections (delays, sampling errors, etc) tend to zero usual solutions uniformly converge to Filippov solutions 31
2-sliding mode
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Robust differentiation problem Unbounded derivatives st
Bounded 1 derivatives &ˆ | f |£ L nd
Bounded 2 derivatives &&ˆ | f |£ L
Arzela Theorem: Bounded functions with bounded derivative of the order k constitute a compact set in C. "Solution": Take the closest function fˆ (t ) ! 33
Landau-Kolmogorov inequalities
Landau: k = 1, ¡ + , 1912; Kolmogorov: k > 1, 1935 There exist such constants b jk ³ 1 , k = 1,2,…, j = 0,1,…,k+1, b0 k = bk +1,k = 1 that for any function j j : ¡ ® ¡ (or j : ¡ + ® ¡), "t : | j(t ) |£ e; (k ) ( k +1) j (t ) is a Lipschitz function, i.e. a.e. | j (t ) |£ L implies ( j)
"t : | j
(t ) |£ b jk
j k +1- j Lk +1 e k +1
b jk cannot be decreased and are realizable. 34
Kolmogorov constants j : ¡ ® ¡ | j(t ) |£ e; | j "t : | j
( j)
(t ) |£ b jk L
( k +1)
(t ) |£ L
j /( k +1) ( k +1- j )/( k +1)
e
1 £ b jk < p / 2
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kth-Order Differentiation Problem Parameters of the problem: k Î ¥, L > 0 Measured input: f(t) = f0(t) + h(t), | h | < e f0 ,h, e are unknown, h(t) - Lebesgue-measurable function, (k+1) known: |f0 (t)| £ L (k) (or |Lipschitz constant of f0 | £ L ) The goal: (k) & & & real-time estimation of f 0 (t), f 0 (t), ..., f0 (t) 36
Best worst differentiation error (k+1)
Suppose both f(t) , f0(t) satisfy: |f0 ( k +1)
Then | h
(t)|, |f0
(k+1)
(t)| £ L
(t ) |£ 2 L , | h(t ) |£ e , "h is possible
The worst possible error in the jth derivative is not ( j)
less than sup | h
(t ) |£ b jk
j j k +1- j 2 k +1 Lk +1 e k +1
In particular for n = j = 1 get b11 = 2 sup | f& (t ) - f&0 (t ) |£
1 1 2 L2 e 2 37
Differentiator (Levant 1998, 2003)
z& = Dk ( z , f (t ), L), | f z&0 = -l k
1 Lk +1
z&1 = -l k -1
1 Lk
... z&k -1 = -l1
1 L2
( k +1)
|£ L
§ z0 - f (t )¨ + z1, k -1 § z1 - z&0 ¨ k + z2 , k k +1
§ zk -1 - z&k -2 ¨
z&k = -l 0 L sign ( zk - z&k -1 ),
1 2
+ zk , zi - f
(i )
® 0.
l0 = 1.1, l1 = 1.5, l2 = 2, l3 = 3, l4 = 5, l5 = 8, … 38
The differentiation accuracy e = 0 (no noise) Þ in a finite time zi º f
(i )
, i = 0,...,k
In the presence of the noise with the magnitude e, and sampling with the step t: $m j ³ 1 | zj -
( j) f0
|£ m j Lr
k +1- j
, r = max( t
k +1- j
,
k + 1- j e k +1
(L)
),
The asymptotics with respect to noise cannot be improved! (Kolmogorov, » 1935)
t = 0 Þ | zj - f
( j)
j k +1
|£ m kj L e
k +1- j k +1
, m kj
³
j 2 k +1 39
In particular the kth derivative has the worst-case accuracy k 1 (k ) k +1 k +1 | zk - f |£ m kk L e For k = 5,6,...: m kk ³ 3 -6 (5) k = 5, L = 1, e = 10 , error of f > 0.3 -16 Digital round up: e = 5 ×10 k = 5 : error : 0.01; k = 6 : error : 0.02 It is bad, but it cannot be improved! 40
Universal controller for any RD r s
(r )
Î [-C, C ] + [ K m , K M ]u u = -aY r ( z ), z = Dr -1 ( z, s, L )
L ³ C + aK M , a is sufficiently large Accuracy: |noise| ≤ e, sampling step ≤ t |s
( j)
|£ n j r
n +1- j
, r=
t=e=0Þsº0
1 r max( t ,| e | n +1 ),
in finite time
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EXAMPLES
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5th-order differentiator, | f é ê ê ê ê ê 5 ê4 ê ê ê ê ê ë
é ê ê é ê ê ê3 ê é ê ê ê 2 ê ê ê ê ê ê1 é ê ê ê ê êë êë ë êë
z&0 =
1 -12 L6
z&1 =
1 -8 L5
z&2 =
1 -5L4
z&3 =
1 -3L3
(6)
§ z0 - f (t )¨
§ z1 - z&0 ¨
4 5
§ z2 - z&1 ¨
3 4
§ z3 - z&2 ¨
2 3
|£ L.
5 6
+ z1 ,
+ z2 , + z3 , + z4 ,
z&4 = § z4 - z&3 ¨ + z5 , z&5 = -1.1L sign( z5 - z&4 ) 1 -1.5L2
1 2
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5th-order differentiation f(t) = sin 0.5t + cos 0.5t, L =1
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Example: car control
x& = V cos j, y& = V sin j, j& = (V/l)tan q, q& = u
RD = 3
x, y are measured. The task: real-time tracking y = g(x) V = const = 10 m/s = 36 km/h, l = 5 m, x = y = j = q = 0 at t = 0 Solution: s = y - g(x), r = 3
3-sliding controller (N°3), a = 2, L = 100 45
3-sliding car control s = y - g(x). Simulation: g(x) = 10 sin(0.05x) + 5, x = y = j = q = 0 at t = 0.
u = 0, 0 £ t<1, The controller:
u=
s2 + 2
( s1 +|s|2/3 sign s )
(|s1|+|s|2/3 )1/2 -2 | s2 | +2(| s1 | + | s |2/3 )1/2
, t ³1
Differentiator:
s&0 = -9.28§ s0 - s¨ + s1, 2 3
s& = D2 ( s, s,100) , L = 100: s& = -15 s - s& 12 + s , § 1 0¨ 2 1 s&2 = -110sign( s2 - s&1 )
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3-sliding car control
-4
t = 10 Þ -5 t = 10 Þ
|s| £ 5.4×10 , | s & | £ 2.5×10 , | s && | £ 0.04 -10 |s| £ 5.6×10 , | s & | £ 1.4×10-5, | s && | £ 0.004 -7
-4
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Practical Relative Degree PRD NO MODEL AT ALL 48
Practical Relative Degree Definition Nothing is known on the system. r Î ¥ is called the PRD, if $ls = 1 or -1: e, dt, aM, am, L, Lm > 0, am £ aM, Lm £ L,: 1. For any (measurable) u(t), |u-u0|£ UM: Output: s% = s + h, |h| £ e, (r-1) s ÎLip(L) 2. w ls s: If t ³ t0 aM ³ u(t) - u0 ³ am (-aM £ u(t) - u0 £ -am), then t ³ t0+ dt: (r) (r) w ³ Lm (w £ -Lm) 49
Naming
u0 is the reference input, in the following u0 = 0 ls is the influence direction parameter, in the following ls = 1 dt is the delay parameter e is the approximation parameter. Local Practical Relative Degree Definition $ t1, t2, T, t1 < t2, dt < T, such that requirement 1 is true over the time interval [t1, t2 + T]; requirement 2 is true for each t0 Î [t1, t2] over [t0, t0 + T]. 50
Graphical interpretation - 1
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Graphical interpretation – 2
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Remarks The function s does not necessarily need to have any real meaning. It can be just an output of some smoothing filter, in particular, of a tracking differentiator. Local practical relative degree is used for temporary output regulation. Keeping s º 0 is not possible under these conditions. 53
Control u = -aY r ( z ), z& = Dr -1 ( z , s, L) , am £ a £ aM Differentiator parameters li are properly chosen Theorem. b1, …, br-1 (coefficients of the r-SM homogeneous controller): Accuracy:
s = O(max[e, dt ]) r
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Important remark The differentiator is the just a dynamic part of the controller. Its outputs do not have any physical meaning, because PRD ≠ RD. Without the differentiator the controller loses its robustness. An attempt to replace the differentiator with any sensors or other observers is very dangerous!
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Continuous controller based on quasi-continuous controller
u = - aF(||z||h)Yr(z) (SM regularization)
ì 1 with || z ||h > g max[e, d ], ï F (|| z ||h ) = í 1 2 r || || with || || £ g max[ e , d ], z z h h t r ï max[e, d ] î t 2 2/ r 2/ ( r -1) 2 || z ||h = z0 + z1 + ... + zr -1 r t
The accuracy is the same. 56
Simulation
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Perturbed car model
x& = Vcos f, y& = Vsin f, &f& = -4sign(f-j)-6f& , Þ Rel. degree does not exist! V j& = D tan q, q& = z1, Actuator: input u, output z1 3 && & z1= -100(2 (z1- u) +0.01 z1) - 100( z1- u)- 2 z& 1, Sensor: s % = z2+0.01 z& 2 - g(x) + h(t), h is a noise, |h| £ 0.01. &&& z2 = - 100(z2 - y) - 2 z& 2 -0.02&& z 2, z2= -10, z& 2 = 2000, &&z 2 = -80000, z1= z& 1= f = f& = 0 at t = 0, If the system were smooth the new RD were 10 58
Practical rel. degree = 3
Differentiator of the order 3 is used with L = 100. 59
System performance
|s| £ 0.16 60
APPLICATION Blood Glucose Control
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Body reaction to glucose concentration increase
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Different models
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The simplest model
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Sorensen model
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3-sliding QC control (BeM)
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3-sliding QC control (SoM)
The same parameters
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PID control (SoM)
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Experiments on rats
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Rat 1
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Rat 2
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Rat 3
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Conclusions In practice the system relative degree is a design parameter. Systems of uncertain nature can be effectively controlled, provided their practical relative degree is identified. A system can have a few generalized PRDs! That is why the considered control is universal. 74
Hypothesis Humans (and animals) have universal controllers embodied for PRD ≤ 2 (3?).
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Thank you very much!
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Applications
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Pitch Control Problem statement. A non-linear process is given by a set of 42 linear approximations d dt
& q = q, 3 xÎR , q, q, uÎR, t
t
(x,q,q) = G(x,q,q) + Hu,
x1, x2 -velocities, x3 - altitude The Task: q ® qc(t), qc(t) is given in real time. G and H are not known properly Sampling Frequency: 64 Hz, Measurement noises Actuator: delay and discretization. dq/dt does not depend explicitly on u (relative degree 2) Primary Statement: Available: q, qc, Dynamic Pressure and Mach. & q& are measured Main Statement: also q, c
The idea: keeping 5(q
- qc) + ( q& - q& c ) = 0 in 2-sliding mode
(asymptotic 3-sliding) 79
Flight Experiments
qc(t), q(t)
q& c = qc(t), q& = q(t)
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Example: practical pitch control Actuator (server-stepper) output, Levant et al., 2000
Switch from Linear ( H ¥ ) control to 3-SM control 81
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On-line calculation of the angular motor velocity and acceleration (data from Volvo Ltd)
Experimental data, t = 0.004
1st derivative. 2nd order differentiation, L = 625 83
On-line 2nd order differentiation Volvo: comparison with optimal spline approximation
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Image Processing: Crack Elimination
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Edge Detection
3 successive lines of a grey image
zoom 86
Edge Detection
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