[Paper Review] Block Backstepping for Isotachic Hyperbolic PDEs and Multilayer Timoshenko Beams

Guangwei Chen, Rafael Vázquez+2 more|arXiv (Cornell University)|Oct 17, 2023

Stability and Controllability of Differential Equations4 citations

TL;DR

This paper introduces a block backstepping design for isotachic hyperbolic PDEs—where multiple states share the same transport speed—to enable rapid stabilization of N-layer Timoshenko beams with anti-damping and anti-stiffness at uncontrolled boundaries. By transforming the beam model into a 1-D hyperbolic PIDE-ODE system via Riemann transformation and applying block backstepping, the method achieves arbitrarily fast $L^2$-stability with exponential decay rates adjustable via control parameters.

ABSTRACT

In this paper, we investigate the rapid stabilization of N-layer Timoshenko composite beams with anti-damping and anti-stiffness at the uncontrolled boundaries. The problem of stabilization for a two-layer composite beam has been previously studied by transforming the model into a 1-D hyperbolic PIDE-ODE form and then applying backstepping to this new system. In principle this approach is generalizable to any number of layers. However, when some of the layers have the same physical properties (as e.g. in lamination of repeated layers), the approach leads to isotachic hyperbolic PDEs (i.e. where some states have the same transport speed). This particular yet physical and interesting case has not received much attention beyond a few remarks in the early hyperbolic design. Thus, this work starts by extending the theory of backstepping control of (m + n) hyperbolic PIDEs and m ODEs to blocks of isotachic states, leading to a block backstepping design. Then, returning to multilayer Timoshenko beams, the Riemann transformation is used to transform the states of N-layer Timoshenko beams into a 1-D hyperbolic PIDE-ODE system. The block backstepping method is then applied to this model, obtaining closed-loop stability of the origin in the L2 sense. An arbitrarily rapid convergence rate can be obtained by adjusting control parameters. Finally, numerical simulations are presented corroborating the theoretical developments.

Motivation & Objective

• Address the lack of systematic control design for isotachic hyperbolic PDEs, where multiple states share identical transport speeds, a common case in laminated structures with repeated layers.
• Extend the backstepping methodology to handle blocks of isotachic states, overcoming limitations of prior general backstepping designs that assume distinct transport speeds.
• Stabilize N-layer Timoshenko beams with destabilizing boundary conditions (anti-damping and anti-stiffness) using boundary feedback control.
• Achieve arbitrarily rapid stabilization (exponential decay with user-defined rate) for multilayer beams via a novel block backstepping framework.
• Demonstrate the effectiveness of the controller through numerical simulations on a two-layer beam with specified decay rate and kernel computations.

Proposed method

• Apply Riemann transformation to convert the N-layer Timoshenko beam model into a 1-D hyperbolic PIDE-ODE system with $ (m+n) $ hyperbolic PIDEs and $ m $ ODEs.
• Develop a block backstepping design by diagonalizing the system into isotachic blocks, where states within each block share the same transport speed.
• Construct a Volterra-type transformation with block-structured kernel functions to map the original system into a target system with desired stability properties.
• Derive and solve a system of 48 coupled kernel equations with triangular domains, including discontinuities in kernels due to block structure, using a power series method.
• Design a boundary feedback controller using the transformation kernel values at the boundary, with independent actuation for forces and torques at each layer’s end.
• Ensure closed-loop stability by verifying that the transformation satisfies the target system’s stability condition with a decay rate $ C_2 = 6 $, set via initial gain matrix $ oldsymbol{ ilde{ heta}}(0) $.

Experimental results

Research questions

• RQ1 How can backstepping control be generalized to systems with isotachic hyperbolic PDEs, where multiple states share the same transport speed?
• RQ2 Can the block backstepping framework stabilize N-layer Timoshenko beams with anti-damping and anti-stiffness at uncontrolled boundaries?
• RQ3 What is the role of the Riemann transformation in enabling the application of backstepping to multilayer Timoshenko beams?
• RQ4 How can arbitrarily fast stabilization be achieved in such systems through controller parameter tuning?
• RQ5 What numerical challenges arise in solving the kernel equations for isotachic systems, and how can they be addressed?

Key findings

• The block backstepping method successfully stabilizes N-layer Timoshenko beams with anti-damping and anti-stiffness at uncontrolled boundaries, achieving $ L^2 $-stability of the origin.
• The closed-loop system exhibits exponential decay with a user-defined rate; a decay rate of $ C_2 = 6 $ was achieved by setting $ oldsymbol{ ilde{ heta}}(0) $ with specific values.
• Numerical simulations confirm the theoretical results: open-loop states diverge, while closed-loop states converge to zero over time.
• The gain kernels $ K_{ij}(1,y) $ and $ L_{ij}(1,y) $ for $ i,j = 1,2,3,4 $ are computed and shown to have discontinuities along lines due to block structure, requiring domain splitting.
• The controller requires independent actuation of forces and torques at each layer’s boundary, which is essential for rapid stabilization and not achievable with a single common actuator.
• The transformation kernel matrix $ oldsymbol{ ilde{ heta}}(1) $ is computed as a 4×4 matrix with specific entries, confirming the feasibility of the controller design.