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Control of Homodirectional and General Heterodirectional Linear Coupled Hyperbolic PDEs

Long Hu, Florent Di Meglio, Rafael Vazquez, Miroslav Krstic

arXiv:1504.07491v1math.OC

TL;DR

The paper tackles control of coupled hyperbolic PDEs beyond restricted heterodirectional configurations, including arbitrary numbers of equations in either direction and homodirectional systems. It uses PDE backstepping to design boundary controllers and obtains finite-time stabilization, output-feedback control, trajectory planning, and tracking results.

  • Problem

    Existing stabilization research primarily treated restricted heterodirectional systems, while the fully general case and coupled homodirectional control remained unresolved.

  • Method

    The paper uses PDE backstepping with a cascade target system and Volterra integral transformation to construct boundary-control designs.

  • Results

    The designs provide finite-time convergence to zero for arbitrary initial conditions and address full-state, observer-based output feedback, trajectory planning, and trajectory tracking.

  • Takeaways & Limitations

    A single-boundary actuation framework covers arbitrary heterodirectional dimensions and extends control design to homodirectional systems with arbitrary local coupling.

Abstract

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Research on stabilization of coupled hyperbolic PDEs has been dominated by the focus on pairs of counter-convecting ("heterodirectional") transport PDEs with distributed local coupling and with controls at one or both boundaries. A recent extension allows stabilization using only one control for a system containing an arbitrary number of coupled transport PDEs that convect at different speeds against the direction of the PDE whose boundary is actuated. In this paper we present a solution to the fully general case, in which the number of PDEs in either direction is arbitrary, and where actuation is applied on only one boundary (to all the PDEs that convect downstream from that boundary). To solve this general problem, we solve, as a special case, the problem of control of coupled "homodirectional" hyperbolic linear PDEs, where multiple transport PDEs convect in the same direction with arbitrary local coupling. Our approach is based on PDE backstepping and yields solutions to stabilization, by both full-state and observer-based output feedback, trajectory planning, and trajectory tracking problems.

I. Introduction

The paper addresses general coupled hyperbolic PDE control, extending beyond restricted heterodirectional systems to arbitrary homodirectional and heterodirectional configurations. PDE backstepping provides boundary-control designs for stabilization, trajectory planning, and tracking.

  • Literature: Prior stabilization results focused mainly on heterodirectional systems with restricted numbers of PDEs and boundary actuation configurations.Earlier methods could not be extended to the case m > 1.
  • Problem setting: Homodirectional systems convect in the same direction and are inherently stable, but coupling can cause undesirable transients and make trajectory planning non-trivial.The transport velocities may have distinct speeds and arbitrary local coupling.
  • Problem setting: Heterodirectional systems contain PDEs traveling in opposite directions, whose coupling may cause instability.The paper considers arbitrary numbers m and n of PDEs in the two directions.
  • Contribution: The paper derives control designs for arbitrary heterodirectional dimensions using actuation at only one boundary on all downstream-convecting PDEs.The general problem is approached through the previously unsolved homodirectional control problem.
  • Contribution: The designs use PDE backstepping and address stabilization, full-state and observer-based output feedback, trajectory planning, and trajectory tracking.A Volterra integral transformation and a cascade target-system structure support the designs.
  • Contribution: For heterodirectional systems, the paper combines full-state feedback with an observer using measurements at a single boundary to obtain implementable output feedback.The full-state law would require full distributed measurements, which is not realistic in practice.

II. System description

The paper formulates a general linear hyperbolic system with arbitrary downstream and upstream transport states, boundary coupling, and local inter-state coupling. It then seeks an invertible backstepping map to a target system with desirable stability properties.

  • System formulation: Boundary conditions couple the upstream boundary values to v(t, 0) and the downstream boundary values to u(t, 1) plus the control input.The boundary relations are u(t, 0) = Q0v(t, 0) and v(t, 1) = R1u(t, 1) + U(t).
  • System formulation: The system contains m upstream states with negative speeds and n downstream states with positive speeds, ordered by transport velocity.The speeds satisfy −µ1 < · · · < −µm < 0 < λ1 ≤ · · · ≤ λn.
  • Assumptions: The analysis assumes constant coupling coefficients and transport velocities for readability, while allowing extension to spatially varying coefficients.The extension is described as straightforward but technically more involved.
  • Isotachic states: States sharing a transport speed are called isotachic and are handled through an invertible block-diagonal change of coordinates.Each isotachic group receives an independently computed block transformation that removes its internal coupling while introducing spatially varying coupling terms.
  • Backstepping design: The general stabilization design maps the original system to a target system using an invertible backstepping transformation.The target-system matrices include triangular coupling structures whose coefficients are determined by the backstepping design.

2) Stability of the target system:

The target-system stability analysis exploits cascade structures and characteristic solutions. The backstepping kernel equations are posed on a triangular domain, with well-posedness established through a unique bounded solution and a novel boundary-design freedom.

  • 2) Stability of the target system:: The α-system reaches zero after a finite time because its ordered transport structure permits recursive characteristic-based analysis.The proof uses the absence of zero transport velocities and a change in the roles of time and space.
  • 2) Stability of the target system:: The β-system is a cascade that can be solved recursively along characteristic lines, leading each β_j to vanish after a finite time.The recursive construction proceeds from β1 through βm by mathematical induction.
  • Kernel equations: The backstepping transformation and its kernel equations are defined on a triangular domain and are derived by differentiating the transformation and matching the target dynamics.The design includes kernel PDEs, boundary conditions, and additional artificial boundary conditions to ensure well-posedness.
  • Kernel equations: The equations for C− form Volterra equations of the second kind, while C+ is explicitly determined from C− and K when the kernels are bounded.This reduces the design of C− and C+ to the kernel construction.
  • Well-posedness: A boundary condition for selected Lij kernels introduces a control-design degree of freedom not present in previous backstepping designs.Its effect on closed-loop transient behavior remains an open question outside the paper’s scope.
  • Well-posedness: Theorem 3.2 guarantees a unique K- and L-kernel solution in L∞(T), with all kernel boundary traces in L∞(0, 1).This well-posedness proof is identified as the paper’s main technical difficulty.

C. Control law and main stabilization result

The control law is obtained by evaluating the inverse backstepping transformation at the actuated boundary, yielding finite-time stabilization for arbitrary initial conditions. An observer-based output-feedback design uses measurements of the v states at the left boundary.

  • C. Control law and main stabilization result: The main stabilization result reaches the zero equilibrium at finite time t = tF for every initial condition in (L∞(0, 1))(n+m)×(n+m).The time tF is defined by equation (17).
  • C. Control law and main stabilization result: Finite-time convergence of the transformed variables implies finite-time convergence of the original states through the inverse Volterra transformation.The proof invokes the finite-time stability lemma for (α, β) and then applies the inverse transformation.
  • Observer design: The observer uses measurements of the v states at the left boundary and feeds its estimates into the control law.Observer gains P+(·) and P−(·) are designed so the estimation-error system is finite-time stable.
  • Observer design: The observer equations inject the output discrepancy v̂(t, 0) − v(t, 0) through spatially distributed gains in both transport subsystems.The error equations contain the corresponding P+(x) and P−(x) injection terms.
  • Observer design: Because the output is injected directly at the left boundary, potential sensor noise is filtered only throughout the spatial domain.The design therefore places the measurement injection at the boundary rather than distributing the measurement itself.

B. Target system and backstepping tranformation

The target-system and controller constructions rely on Volterra backstepping transformations whose kernels satisfy coupled equations on a triangular domain. Their well-posedness supports finite-time convergence and an observer construction with equivalent kernel-system well-posedness.

  • B. Target system and backstepping tranformation: The target-system construction maps the original variables to a system whose solutions converge to zero in finite time.Proposition 4.1 states finite-time convergence, with the final time tF defined by equation (17).
  • B. Target system and backstepping tranformation: The homodirectional target dynamics form a cascade from slow states into fast states, enabling the stability analysis.The rigorous proof follows the same steps as the proof of Lemma 3.1.
  • B. Target system and backstepping tranformation: The transformation uses Volterra kernels M and N defined on a triangular domain, with kernel equations and boundary conditions obtained by differentiating it.Artificial boundary conditions are added for selected Nij entries to ensure well-posedness.
  • B. Target system and backstepping tranformation: The kernel construction includes artificial boundary conditions for Nij(i < j), followed by formulas for the remaining coefficients.These conditions supplement the natural boundary conditions of the kernel equations.
  • B. Target system and backstepping tranformation: The observer kernel system has the same well-posedness structure as the controller kernel system after an alternate-variable transformation.The controller system’s well-posedness is assessed in Theorem 3.2.
  • B. Target system and backstepping tranformation: The observer kernel system inherits the controller kernel system’s well-posedness result because the transformed equations have the exact same structure.The correspondence is established through the alternate variables and their boundary conditions.

C. Output feedback controller

The observer-controller design combines backstepping transformations with an output-feedback law whose state and observer estimates converge to zero in finite time. The homodirectional case also supports finite-time motion planning and tracking through an additional boundary degree of freedom.

  • Output feedback controller: Under the stated control law, the plant states and observer estimates converge in finite time to zero.The control law uses kernels K and L defined by equations (36)–(41).
  • Output feedback controller: The observer estimates converge to zero after a finite time, enabling the observer-controller scheme to apply the finite-time stabilization result.The observer error states converge for t ≥ tF1, after which the observer system follows the controller result.
  • Output feedback controller: For t ≥ 2tF, the observer and plant state variables are identically zero.This follows after the observer errors vanish and the observer system reaches zero.
  • Homodirectional systems: Homodirectional systems are inherently finite-time stable because transport information flows in only one direction.With U(t)=0, the method of characteristics gives u(t,x) ≡ 0 after the relevant transport time.
  • Motion planning: Motion planning seeks a control U(t) that makes v(t,0) equal a prescribed signal Φ(t) after a finite planning time.The construction introduces B(t) as an extra degree of freedom in the target-system boundary conditions without changing the backstepping transformation.
  • Motion planning: The motion-planning construction yields finite-time tracking, which is stronger than pure motion planning.The paper notes that pure motion planning would require eliminating the state dependence through explicit target and inverse transformations; that result is omitted for lack of space.

C. An explicit motion planning example

The explicit m=2 example illustrates how coupled motion-planning kernels are constructed and why direct characteristic-based input design is not implementable. The backstepping-based construction nevertheless provides inputs achieving the prescribed boundary outputs.

  • Problem setup: For m=2, the objective is to design U1(t) and U2(t) so that v1(t,0)=Φ1(t) and v2(t,0)=Φ2(t).The target behavior is required after a finite planning time tM.
  • Difficulty of direct design: Directly solving the coupled equations produces a feedback law requiring future values of v1 and v2, making it non-causal and non-implementable.Thus, the direct approach fails even in the m=2 case.
  • Backstepping construction: The example solves the motion-planning problem using inputs obtained from Theorem 5.1.The kernels L11, L12, L21, and L22 satisfy the associated kernel equations.
  • Explicit kernels: The explicit kernel solutions involve modified and regular Bessel functions of orders 0 and 1.The resulting kernels are given by equations (106)–(109).
  • Kernel behavior: For μ1=1, μ2=0.2, σ12=2, and σ21=5, L11(1,ξ) and L12(1,ξ) are monotone while L21(1,ξ), L21(ξ,0), and L22(1,ξ) are oscillatory.The whole-domain plot also shows L12 discontinuous along a characteristic line, whereas L11 is continuous there.
  • Kernel behavior: The artificial boundary function l21 is non-trivial and is determined by extending the kernel domain rather than prescribing it initially.This construction supplies the boundary condition needed to find l21.

VI. Proof of Theorem 3.2: well-posedness of the kernel equations

The well-posedness proof transforms the kernel PDEs into integral equations and establishes their solvability through successive approximations.

  • Well-posedness proof: The kernel equations are classically transformed into integral equations and solved using the method of successive approximations.The proof is more involved because controlled homodirectional states generate additional homodirectional kernel PDEs.
  • Well-posedness proof: The proof strategy follows similar arguments developed for less general systems.The paper specifically points to earlier proofs for related systems.

A. Method of characteristics

The method-of-characteristics analysis constructs characteristic curves for each kernel equation, integrates along them, and converts the resulting relations into a successive-approximation scheme.

  • K-kernel characteristics: For K kernels, characteristic lines originate at (x,ξ), terminate on the hypotenuse, and yield integral relations after imposing boundary conditions.These characteristics correspond to equations (36) and are depicted in Figure 3.
  • L-kernel characteristics: For L kernels, characteristic lines are constructed separately for i>j, i=j, and i<j.The three cases correspond to Figures 4–6 and have characteristics terminating on different parts of the triangular domain boundary.
  • L-kernel characteristics: The L-kernel characteristics may terminate on ξ=0, the hypotenuse, or x=1, determining which boundary conditions enter the integral equations.The boundary conditions used are (39), (40), and (41).
  • Successive approximations: The kernel vector H is reordered and stacked, then used to define linear operators and an iterative sequence.If the sequence limit exists, it solves the original hyperbolic system.
  • Successive approximations: The iteration tracks increments ΔHq=Hq−Hq−1 and proves convergence of the resulting series in L∞.The increment equation follows from linearity of the functional Φ.

C. Convergence of the successive approximation series

The convergence proof establishes recursive bounds for the successive approximation terms and uses them to show convergence of the kernel series and associated estimates.

  • Lemma and proof strategy: The proof introduces a crucial lemma to establish inequalities for the kernel-related functions used in the successive approximation analysis.The argument begins with a change of variables and applies the definitions of the auxiliary functions.
  • Proposition 6.1: The proof repeatedly transforms the left-hand sides of the governing inequalities and verifies the required conditions using the ordering of the µ_i and the definition of ϵ.Several intermediate inequalities are established before the proof is concluded.
  • Proposition 6.1: The successive approximation terms satisfy uniform bounds over T, including |∆H_i(x, ξ)| ≤ ¯φ_M q(x − (1 − ξ))^q for the indicated indices.This proposition provides the estimate used to control the approximation terms.
  • Theorem 3.2: Proposition 6.1 directly leads to Theorem 3.2 because the successive approximation series converges and the resulting terms obey the stated bounds.The theorem follows by procedures corresponding to those used in the cited prior analyses.

VII. Concluding remarks

The paper presents boundary control designs for a general class of linear first-order hyperbolic systems, including stabilization and tracking designs for heterodirectional and homodirectional systems. The concluding remarks identify transient-performance dependence on kernel boundary values and several directions for further work.

  • Contributions: The paper presents boundary control designs for a general class of linear first-order hyperbolic systems.The designs form the main contribution summarized in the concluding remarks.
  • Contributions: The results include an output-feedback stabilization law for heterodirectional systems and a tracking controller for motion planning in homodirectional systems.These designs cover both stabilization and trajectory-tracking objectives.
  • Relation to prior work: The explicit designs bridge a gap with results proving null or weak controllability for heterodirectional states without providing a corresponding design.The comparison concerns systems with (n + m) states.
  • Open problems: The control-design degree of freedom in Equation (41) raises another important question for future investigation.The remarks also point toward observer, disturbance-rejection, parameter-identification, adaptive-control, and quasilinear-system problems.
  • Practical considerations: The boundary values of the kernels have a non-trivial effect on closed-loop transient performance, which is important for applications.The concluding remarks also identify this dependence as a practical concern.
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