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

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)

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

Bernoulli beam element

Strain Energy Strain energy: Then,

Kinetic Energy Kinetic Energy: Then,

Assembly and modes extraction Characteristic Equation:

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

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

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

Mathematical solution With boundary condition,

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 ]

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)

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

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

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

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

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

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

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

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

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

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

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

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

Filter design

Performance: validation of the 15th-order controller

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

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

Simulink model

Simulation results: (average) consensus

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

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

G. Other literature study Bassam Bamieh's framework: Distributed Control of Spatially Invariant Systems. IEEE Transactions on Automatic Control, Vol. 47, pp. 1091-1107, 2002. Raffaello D'Andrea's framework: Distributed Control Design for Spatially Interconnected Systems. IEEE Transactions on Automatic Control, Vol. 48, pp. 1478-1495, 2003. Simple control law A. Positive position feedback (PPF) B. Direct velocity feedback C. Acceleration feedback D. Integral force feedback E. Piezoelectric Shunt damping

H. Directions?