1. Progress Report during secondment in Prague
Xueji Zhang / March 11, 2015
Progress Report during secondment in Prague

2. Outline Research Modeling of a clamped-clamped beam
Finite Element Method
Analytical solution: standing wave equation
LQR control of FE model of the beam
Collocated control with root-locus method
Robust control of FE model of the beam (FSC course)
𝐾𝑖𝑟𝑐ℎℎ𝑜𝑓𝑓−𝐿𝑜𝑣𝑒 plate equation derivation with Hamilton’s principle
Ph.D. Training
Cooperative Control of multi-agent systems
(Flexible Structure Control: included in the research section)
(Linear Matrix Inequality in Control: to share after finalizing)

3. A.1. Euler-Bernoulli beam: Finite Element Method

4. Bernoulli beam element

5. Strain Energy
Strain energy: Then,

6. Kinetic Energy
Kinetic Energy: Then,

7. Assembly and modes extraction
Characteristic Equation:

8. C-C beam mode shapes (Solved with MATLAB)
# of elements = 40; First 4 modes: Natural frequency: Mode Shapes:

9. A.2. Euler-Bernoulli beam: Partial Differential Equation (PDE) model with standing wave solution

10. Analytical solution
Governing equation of motion for Bernoulli beam: Standing wave equation: Clamped-Clamped Boundary Condition:

11. Mathematical solution
With boundary condition,

12. Shape of 𝝓 𝑛
𝜙 𝑛 𝑥 = cos 𝑘 𝑛 𝑥 − cosh 𝑘 𝑛 𝑥 − cos 𝑘 𝑛 − cosh 𝑘 𝑛 sin 𝑘 𝑛 − sinh 𝑘 𝑛 [ sin 𝑘 𝑛 𝑥 − sinh 𝑘 𝑛 𝑥 ]

13. B. LQR control of the beam(SPIE2015 paper)
𝜕 2 𝑤 𝜕 𝑡 2 𝑥,𝑡 =− 𝜕 4 𝑤 𝜕 𝑡 4 𝑥,𝑡 +𝑢 𝑥,𝑡 , 𝑤 𝑥,0 = 𝑤 0 𝑥 , 𝜕𝑤 𝜕𝑡 𝑥,0 = 𝑤 0 (𝑥), 0<𝑥<1, 𝑡≥0. B.C. : 𝜕𝑤 𝜕𝑥 0,𝑡 = 𝜕𝑤 𝜕𝑥 1,𝑡 =0=𝑤 0,𝑡 =𝑤(1,𝑡)

14. LQR formulation
min 𝑢(∙,𝑡) 𝐽 𝑧 0 ;𝑢(∙,𝑡 )= 0 ∞ (<𝑦,𝑦>+ <𝑢,ℛ𝑢>)𝑑𝑡 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑧 ∙,𝑡 =𝒜𝑧 ∙,𝑡 +ℬ𝑢(∙,𝑡) 𝑦 ∙,𝑡 =𝒞𝑧 ∙,𝑡 𝑧 ∙,0 = 𝑧 0 𝑧 ∙,𝑡 = 𝑤 ∙,𝑡 𝑤 ∙,𝑡 , 𝒜= 𝑂 𝐼 − 𝑑 4 𝑑 𝑥 4 0 , ℬ= 𝑂 𝐼 , 𝒞=I

15. Results:
Convolution Kernel 𝓚(𝒙,𝝃) : 𝑢 𝑥,𝑡 =− 0 1 𝓚(𝒙,𝝃) 𝑤 𝜉,𝑡 𝑑𝜉 Decentralization properties are shown in next slides

16. Numerical evaluation: 𝓚 𝒙,𝝃 , 𝒙= 𝟏 𝟑 , 𝑵=𝟓, 𝟏𝟎, 𝟏𝟓, 𝟐𝟎 𝓚 𝒙,𝝃 ≈ 𝒏=𝟏 𝑵 𝜶 𝒏 𝝓 𝒏 (𝒙) 𝝓 𝒏 (𝝃)
𝑢 𝑥,𝑡 =− 0 1 𝓚(𝒙,𝝃) 𝑤 𝜉,𝑡 𝑑𝜉

17. Numerical evaluation: 𝓚 𝒙,𝝃 , 𝒙= 𝟏 𝟐 , 𝑵=𝟓, 𝟏𝟎, 𝟏𝟓, 𝟐𝟎 𝓚 𝒙,𝝃 ≈ 𝒏=𝟏 𝑵 𝜶 𝒏 𝝓 𝒏 (𝒙) 𝝓 𝒏 (𝝃)
𝑢 𝑥,𝑡 =− 0 1 𝓚(𝒙,𝝃) 𝑤 𝜉,𝑡 𝑑𝜉

18. Numerical evaluation: 𝓚 𝒙,𝝃 , 𝒙= 𝟐 𝟑 , 𝑵=𝟓, 𝟏𝟎, 𝟏𝟓, 𝟐𝟎 𝓚 𝒙,𝝃 ≈ 𝒏=𝟏 𝑵 𝜶 𝒏 𝝓 𝒏 (𝒙) 𝝓 𝒏 (𝝃)
𝑢 𝑥,𝑡 =− 0 1 𝓚(𝒙,𝝃) 𝑤 𝜉,𝑡 𝑑𝜉

19. LQR formulation for FE model
min 𝑢(𝑡) 0 ∞ [ 𝑦 𝑡 𝑇 𝑦 𝑡 + 𝑢 𝑡 𝑇 𝑅𝑢(𝑡)]𝑑𝑡 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑥 𝑡 =𝐴𝑥 𝑡 +𝐵𝑢(𝑡) 𝑦 𝑡 =𝐶𝑥 𝑡 𝑥 0 = 𝑥 0 . Herein 𝑥(𝑡)= 𝒒 𝒎 (𝑡) 𝒒 𝒎 (𝑡) ⟹ Static feedback: 𝑢=− 𝑅 −1 𝐵 𝑇 𝑃𝑥=−𝐹𝑥

20. Extracted from 𝐹 (𝑢=− 𝑅 −1 𝐵 𝑇 𝑃𝑥=−𝐹𝑥)
𝑓 1𝑧 : 𝑤⟼𝑓𝑜𝑟𝑐𝑒

21. Extracted from 𝐹 (𝑢=− 𝑅 −1 𝐵 𝑇 𝑃𝑥=−𝐹𝑥)
𝜏 2𝜃 : 𝜃 ⟼𝑡𝑜𝑟𝑞𝑢𝑒

22. Simulations (Simulation time = 1 s)
Distributed Sensors Decentralized control

23. C. MIMO control of beam with root-locus (IEEE CDC2015 in processing)
Improve FE model (compared with model in SPIE paper)
Element Nr=1000;
Model Order Reduction: modal truncation up to first 20 modes
Open-loop video: Open-loop.avi
Vel Feedback in the middle position: OneVelFb.avi
Angular Vel in the middle position: OneAngularVelFb.avi

24. C. MIMO control of beam with root-locus (IEEE CDC2015 in processing)
Improve FE model (compared with model in SPIE paper)
Element Nr=1000;
Model Order Reduction: modal truncation up to first 20 modes
Open-loop video: OpenLoop.avi
Vel Feedback in the middle position: OneVelFb.avi
Angular Vel in the middle position: OneAngularVelFb.avi
Technical findings:
One sensor can ‘eliminate’ at most 1 vibration mode;
Placement of one single sensor depends on which modes need to damp
Distributed Vel FB damp lower modes first: VelFb_every_node_rootlocus.avi
Distributed Angular Vel damp higher modes first: AngularVelFb_every_node_rootlocus.avi
Research ongoing: density and scalability

25. D. Robust Control of a clamped-clamped beam (FSC)
Parameter uncertainty in FE model
Additive uncertainty model: 𝐺= 𝐺 0 +∆ 𝑊 𝑎 , 𝐺 0 nominal plant

26. Filter design

27. Performance: validation of the 15th-order controller

28. E. 𝐾𝑖𝑟𝑐ℎℎ𝑜𝑓𝑓−𝐿𝑜𝑣𝑒 plate governing equation derivation: Hamilton’s principle
Dynamics_Hamiton_Splitted.pdf

29. F. Ph.D. course: Cooperative control of multi-agent systems
5-agent integrator model: 𝑥 𝑖 = 𝑢 𝑖
Graph topology (info flow): 𝐸=
Consensus protocol: 𝑢 𝑖 = 𝑗 𝑒 𝑖𝑗 ( 𝑥 𝑗 − 𝑥 𝑖 )
Laplacian matrix:
𝐿= −1 −1 0 − − −1 −
Global dynamics: 𝒙 =−𝐿𝒙, 𝒙=[ 𝑥 1 𝑥 2 𝑥 3 𝑥 4 𝑥 5 ] 𝑇

30. Simulink model

31. Simulation results: (average) consensus

32. Small toy-project: mass-spring-damper system synchronization

33. More advanced project (only if time permitting)
Parallel parking for 5 mobile cars: nonlinear dynamics involved
I/O feedback linearization
Leader-following stabilization problem

34. G. Other literature study
Bassam Bamieh's framework: Distributed Control of Spatially Invariant Systems. IEEE Transactions on Automatic Control, Vol. 47, pp , 2002.
Raffaello D'Andrea's framework: Distributed Control Design for Spatially Interconnected Systems. IEEE Transactions on Automatic Control, Vol. 48, pp , 2003.
Simple control law
A. Positive position feedback (PPF) B. Direct velocity feedback C. Acceleration feedback D. Integral force feedback E. Piezoelectric Shunt damping

35. H. Directions?