Advanced Mechatronics Engineering - MNRLab€¦ · Pantograph mechanism High precision motion...
Transcript of Advanced Mechatronics Engineering - MNRLab€¦ · Pantograph mechanism High precision motion...
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Advanced Mechatronics Engineering
Islam S. M. Khalil
German University in Cairo
September 3, 2016
Islam S. M. Khalil Linear Systems
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Outline
Motivation
Agenda
Linear systems
State transition matrix
Islam S. M. Khalil Linear Systems
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Motivation
Targeted Drug Delivery
Wireless motion control of microrobots under the influence ofcontrolled magnetic fields (delicate retinal surgeries).
Figure: Electromagnetic system for the wireless control of drug carriers(Khalil et al., Applied Physics Letters, 2013).
Islam S. M. Khalil Linear Systems
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Motivation
Targeted Drug Delivery
Motion control of drug carriers through the spinal cord.
Figure: Electromagnetic system for the wireless control of drug carriers.
Islam S. M. Khalil Linear Systems
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Motivation
Targeted Drug Delivery
Wireless motion control of self-propelled microjets.
Figure: Self-propelled microjets (Image courtesy of Oliver G. Schmidt).
Islam S. M. Khalil Linear Systems
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Motivation
Biological Cells Characterization and Manipulation
Transparent bilateral control systems are used to characterizebiological cell and perform surgeries with minimal interventions.
Figure: Drug injection in a cell using a bilateral control system.
Islam S. M. Khalil Linear Systems
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Motivation
Delta Robot
Relatively high speeds and reasonable rigidity are combined.
Figure: Delta robot with three active and three passive joints.
Islam S. M. Khalil Linear Systems
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Motivation
Pantograph mechanism
High precision motion control.
Figure: Pantograph mechanism for micromachining and microassembly.
Islam S. M. Khalil Linear Systems
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Motivation
Linear Motion Stage
High precision motion control.
Figure: Linear motion stage for micromachining and microassembly.
Islam S. M. Khalil Linear Systems
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Agenda
Week Topics
1 Similarity transformations, diagonal and Jordan forms, ...2 Lyapunov equation, quadratic form and +/- definiteness, ...3 Singular value decomposition, norms of matrices, ...4 Controllability, observability, canonical decomposition, ...5 Teleoperation using 2-channel control architectures, ...6 Qualitative behavior near equilibrium points, limit cycles, ...7 Lyapunov stability, ...8 Input output stability, ...9 Feedback system: The small gain theorem, ...
10 Passivity, memoryless functions, state models, ...11 Passivity theorem, absolute stability, circle criterion, ...12 Bilateral control of nonlinear teleoperation, ...13 Real-time operating systems, deadlock, ...14 Schedulability tests, hard and soft real-time, ...
Islam S. M. Khalil Linear Systems
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Linear System
Consider the scalar case
x(t) = ax(t). (1)
Taking the Laplace transform of (1), we obtain
sX (s)− x(0) = aX (s), (2)
X (s) =x(0)
s − a= (s − a)−1x(0). (3)
Finally, inverse Laplace transform of (3) yields
x(t) = eatx(0). (4)
Islam S. M. Khalil Linear Systems
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State Transition Matrix
Now consider the following homogenous state equation
x(t) = Ax(t). (5)
sX(s)− x(0) = AX(s), (6)
X(s) = (sI− A)−1x(0). (7)
The inverse Laplace transform yields
x(t) = L−1[(sI− A)−1
]x(0) = eAtx(0). (8)
Therefore, the state transition matrix (eAt) is given by
eAt = L−1[(sI− A)−1
]. (9)
Islam S. M. Khalil Linear Systems
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State Transition Matrix
Calculate the state transitionmatrix of the following system
[x1x2
]=
[−1 02 −3
] [x1x2
](10)
[sI− A] =
[(s + 1) 0−2 (s + 3)
](11)
[sI− A]−1 =
[(s+3)
(s+1)(s+3) 02
(s+1)(s+3)(s+1)
(s+1)(s+3)
]
=
[ 1(s+1) 0(
1(s+1) −
1(s+1)
)1
(s+3)
]
eAt = L−1[(sI− A)−1
], (12)
eAt =
[e−t 0
(e−t − e−3t) e−3t
].
Islam S. M. Khalil Linear Systems
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State Transition Matrix
Calculate the state transitionmatrix of the following system
[x1x2
]=
[0 1−2 −3
] [x1x2
](13)
[sI− A] =
[s −12 (s + 3)
](14)
[sI− A]−1 =
[(s+3)
(s+1)(s+2)1
(s+1)(s+2)−2
(s+1)(s+2)s
(s+1)(s+2)
]
eAt = L−1[(sI− A)−1
], (15)
=
[2et − e−2t e−t − e−2t
−2e−t + 2e−2t −e−t + 2e−2t
].
Islam S. M. Khalil Linear Systems
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State Transition Matrix
If the matrix A can be transformed into a diagonal form, then thestate transition matrix eAt is given by
eAt = PeDtP−1 = P
eλ1t 0 . . . 0
0 eλ2t . . . 0... . . .
. . . 00 . . . 0 eλnt
P−1, (16)
where P is a digonalizing matrix for A. Further, λi is the itheigenvalue of the matrix A, for i = 1, . . . , n.
Islam S. M. Khalil Linear Systems
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State Transition Matrix
Derivation: Consider the following homogenous state equation
x = Ax, (17)
and the following similarity transformation:
x = Pξ , x = Pξ. (18)
Substituting (18) in (17) yields
ξ = P−1APξ = Dξ. (19)
Solution of (19) isξ(t) = eDtξ(0), (20)
using (18)
x(t) = Pξ(t) = PeDtξ(0) , x(0) = Pξ(0). (21)
Thereforex(t) = PeDtP−1x(0) = eAtx(0). (22)
Islam S. M. Khalil Linear Systems
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State Transition Matrix
Calculate the state transitionmatrix of the following system
[x1x2
]=
[0 10 −2
] [x1x2
](23)
The eigenvalues of A are λ1 = 0 andλ2 = −2. A similarity transformationmatrix P is
P =
[1 10 −2
]. (24)
Using (16) to calculate the statetransition matrix
eAt = PeDtP−1 (25)
=
[1 10 −2
] [e0 00 e−2t
] [1 1
20 −1
2
]eAt =
[1 1
2(1− e−2t)0 e−2t
]. (26)
Islam S. M. Khalil Linear Systems
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Thank You
Thank You!Questions please
Islam S. M. Khalil Linear Systems