Embed Size (px)
Citation preview
Advanced Mechatronics Engineering
Islam S. M. Khalil
German University in Cairo
September 3, 2016
Islam S. M. Khalil Linear Systems
Outline
Motivation
Agenda
Linear systems
State transition matrix
Islam S. M. Khalil Linear Systems
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
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
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
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
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
Motivation
Pantograph mechanism
High precision motion control.
Figure: Pantograph mechanism for micromachining and microassembly.
Islam S. M. Khalil Linear Systems
Motivation
Linear Motion Stage
High precision motion control.
Figure: Linear motion stage for micromachining and microassembly.
Islam S. M. Khalil Linear Systems
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
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
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
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
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
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
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
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
Thank You
Thank You!Questions please
Islam S. M. Khalil Linear Systems
