advanced mechatronics engineering - mnrlab€¦ · pantograph mechanism high precision motion...

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