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Mistuning-based Control Design to Improve Closed-Loop Stability of Vehicular Platoons

Prabir Barooah, Prashant G. Mehta, Joao P Hespanha

arXiv:0812.1030v1math.OC

TL;DR

Large decentralized vehicle platoons lose closed-loop stability as their size increases, creating a need for effective local control design. The paper uses a PDE approximation of the discrete platoon dynamics to analyze this loss and design mistuned gains, showing improved eigenvalue scaling with numerical corroboration.

  • Problem

    Symmetric decentralized bidirectional control makes the least stable closed-loop eigenvalue decay as platoon size increases, while decentralized control lacks sufficient design methods.

  • Method

    The paper derives and analyzes a PDE model of the decentralized bidirectional platoon, then uses the analysis to design small forward-backward asymmetries in controller gains.

  • Results

    O(1/N^2) becomes O(1/N) for the least stable eigenvalue with arbitrarily small mistuning, with PDE predictions corroborated by state-space calculations.

  • Takeaways & Limitations

    Small mistuning improves the asymptotic closed-loop stability behavior and also significantly reduces sensitivity to external disturbances.

Abstract

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We consider a decentralized bidirectional control of a platoon of N identical vehicles moving in a straight line. The control objective is for each vehicle to maintain a constant velocity and inter-vehicular separation using only the local information from itself and its two nearest neighbors. Each vehicle is modeled as a double integrator. To aid the analysis, we use continuous approximation to derive a partial differential equation (PDE) approximation of the discrete platoon dynamics. The PDE model is used to explain the progressive loss of closed-loop stability with increasing number of vehicles, and to devise ways to combat this loss of stability. If every vehicle uses the same controller, we show that the least stable closed-loop eigenvalue approaches zero as O(1/N^2) in the limit of a large number (N) of vehicles. We then show how to ameliorate this loss of stability by small amounts of "mistuning", i.e., changing the controller gains from their nominal values. We prove that with arbitrary small amounts of mistuning, the asymptotic behavior of the least stable closed loop eigenvalue can be improved to O(1/N) All the conclusions drawn from analysis of the PDE model are corroborated via numerical calculations of the state-space platoon model.

I. INTRODUCTION

The paper addresses stability degradation in large decentralized bidirectional vehicle platoons using PDE-based analysis and mistuned controller gains. It shows that small forward-backward gain asymmetries improve the scaling of the least stable eigenvalue and reduce disturbance sensitivity.

  • Control architecture: Decentralized bidirectional control uses local measurements from each vehicle and its nearest neighbors, avoiding the continual communication overhead of centralized architectures.Each vehicle is modeled as a double integrator with local state feedback.
  • Stability challenge: O(1/N^2) is the scaling of the least stable closed-loop eigenvalue under symmetric bidirectional control, independent of controller gains.The real part of the least stable eigenvalue measures the stability margin.
  • PDE analysis: PDE analysis provides a controller-independent stability conclusion and explains the mechanism of progressive stability loss as platoon size increases.The PDE approximation is used to analyze eigenvalues and recover the discrete dynamics upon discretization.
  • Mistuned design: Arbitrarily small mistuning changes the symmetric controller gains to introduce forward-backward asymmetry in the information weights.The asymmetry assigns different gains to information from preceding and following vehicles.
  • Disturbance sensitivity: Mistuning also significantly reduces sensitivity to external disturbances, measured by the H∞ norm from disturbances to spacing errors.The benefit is reported for bidirectional architectures and is supported by numerical computation.

III. PDE MODEL OF PLATOON CLOSED LOOP DYNAMICS

The paper develops a continuous PDE approximation of the spatially discrete platoon dynamics using a scaled coordinate x ∈[0, 2π].

  • The PDE model is formulated with respect to the scaled spatial coordinate x ∈[0, 2π].

A. PDE derivation

The PDE is constructed by replacing spatial differences in the discrete platoon model with derivatives, while continuous fields and gains approximate their vehicle-level counterparts. Boundary conditions encode the endpoint vehicle scenarios, and the hyperbolic PDE describes finite-speed disturbance propagation with damping and possible directional asymmetry.

  • PDE derivation: The continuous model maps vehicle velocities and controller gains to spatial fields evaluated at each vehicle's scaled position.The mappings include v(x, t) to vehicle velocities and continuous approximations of b_i, k(f)_i, and k(b)_i.
  • PDE derivation: The resulting PDE models the discrete platoon dynamics, assuming a positive continuous front gain k(+)(x).
  • PDE derivation: Finite-difference discretization of the PDE recovers the original ordinary differential equations for the discrete vehicles.
  • PDE derivation: Dirichlet boundary conditions represent fictitious lead and follow vehicles, whereas Neumann-Dirichlet conditions represent only a fictitious lead vehicle.
  • PDE derivation: For N = 25 vehicles, PDE and discrete-platoon eigenvalues show an excellent match, particularly for the least stable eigenvalues.The comparison uses numerically computed state-space and Galerkin/Fourier PDE eigenvalues.
  • PDE derivation: The hyperbolic PDE predicts finite-speed upstream and downstream disturbance propagation, with damping and advection modifying its temporal decay and directional behavior.

B. Eigenvalue comparison

The PDE approximation reproduces the closed-loop platoon eigenvalues and explains how the least stable eigenvalue deteriorates as the platoon grows. With symmetric bidirectional control, this eigenvalue approaches zero as O(1/N^2), consistent with numerical state-space calculations.

  • PDE approximation: Constant symmetric gains reduce the platoon PDE to a damped wave equation whose disturbances propagate equally in both directions.The constant-gain case uses k(f)(x) = k(b)(x) = k0 and b(x) = b0.
  • PDE approximation: The PDE eigenvalues closely match those of the discrete platoon model, including the least stable eigenvalues.For N = 25, the PDE eigenvalues were computed with a Galerkin method and showed an excellent match with the state-space model.
  • Asymptotic stability: O(1/N^2) is the asymptotic decay rate of the lth less-stable PDE eigenvalue as N tends to infinity.The result applies to the Dirichlet-Dirichlet and Neumann-Dirichlet boundary-condition scenarios.
  • Asymptotic stability: O(1/N^2) is also the decay rate of the least stable eigenvalue under symmetric bidirectional control.The stability margin is measured by the real part of the least stable closed-loop eigenvalue.
  • Numerical comparison: Numerical evaluation confirms that the PDE predictions track the least stable eigenvalues of the state-space platoon model across increasing N.Figure 3 compares the PDE, discrete-platoon, and asymptotic predictions for the two boundary-condition scenarios.
  • Eigenvalue behavior: As N increases, eigenvalues move toward zero, with a critical collision marking the transition from complex to real eigenvalues.For N > Nc, the eigenvalue s+_1 approaches zero-related asymptotic behavior on the real axis.

V. REDUCING LOSS OF STABILITY BY MISTUNING

The paper uses small, spatially varying changes in forward and backward position-feedback gains to counteract the stability loss caused by symmetric bidirectional control. The design exploits forward-backward asymmetry, which introduces a first-order transport contribution capable of improving the asymptotic stability scaling.

  • Design objective: Mistuning changes the controller gains slightly from the symmetric design to minimize the least-stable eigenvalue.The design problem is formulated using the mistuned PDE and gain functions k(f)(x) and k(b)(x).
  • Mechanism: O(1/N^2) arises in the symmetric case because the coefficient of the first spatial-derivative term is zero.Forward-backward asymmetry creates nonzero k(−)(x), adding an O(1/N) contribution.
  • Mechanism: O(1/N) is the potential stability-margin scaling enabled by judicious forward-backward asymmetry.The mistuning amplitude is represented by a small parameter ǫ, while k_m(x) and k_s(x) capture perturbations from the nominal gain.
  • Design framework: The resulting framework systematically designs control gains by introducing small changes to the symmetric architecture.The objective is to choose k_m(x) and k_s(x) to improve the stability margin as N increases.

A. Mistuning-based design for scenario I

For scenario I, the paper derives an asymptotically optimal antisymmetric mistuning profile under Dirichlet boundary conditions. Even arbitrarily small mistuning improves damping from the symmetric O(1/N^2) behavior to O(1/N), with numerical confirmation using ±10% gain changes.

  • Asymptotic analysis: Theorem 1 gives an asymptotic formula for each mistuned eigenvalue pair as ǫ approaches zero and N becomes large.The result applies to the mistuned PDE with Dirichlet boundary conditions for scenario I.
  • Gain design: Only k_m(x) affects the leading stability improvement, so the design sets k_s(x) ≡ 0 and uses opposite forward and backward perturbations.This choice gives k(f,purt)(x) = −k(b,purt)(x) and k_m(x) = 2k(f,purt)(x).
  • Gain design: Corollary 2 identifies the optimal scenario-I mistuning profile under an L∞ norm constraint.The profile is defined using the Heaviside function and antisymmetric forward and backward perturbations.
  • Stability improvement: O(1/N) is the mistuned asymptotic scaling, compared with O(1/N^2) for the symmetric case.The improvement holds with an arbitrarily small mistuning amplitude ǫ.
  • Numerical confirmation: ±10% gain variation produces a large eigenvalue improvement over the nominal case, especially for large N.The PDE and platoon state-space eigenvalues also match accurately over the tested range of N.
  • Mechanism: Mistuning exchanges damping from the more stable eigenvalue to the less stable one while preserving their total damping.The sum satisfies s+_1 + s−_1 = −b0.

B. Mistuning-based design for scenario II

For scenario II, the paper derives a corresponding mistuning design under Neumann-Dirichlet boundary conditions. The design improves the least-stable eigenvalue from O(1/N^2) to O(1/N), and numerical results show an order-of-magnitude gain with only ±10% variation.

  • Asymptotic analysis: Theorem 2 provides the asymptotic formula for mistuned eigenvalue pairs under Neumann-Dirichlet boundary conditions.The formula is valid for each mode as ǫ approaches zero and N becomes large.
  • Gain design: Corollary 3 gives the optimal scenario-II profile under the stated L∞ constraint with opposite forward and backward perturbations.The specified profile is k(f,purt)(x) = 1 and k(b,purt)(x) = −k(f,purt)(x).
  • Stability improvement: O(1/N) is the mistuned scaling instead of O(1/N^2) in the symmetric case.The result establishes improved closed-loop stability with arbitrarily small mistuning ǫ.
  • Numerical confirmation: An order-of-magnitude improvement occurs with only ±10% gain variation in scenario II.The numerical comparison uses least stable eigenvalues from both the PDE and the platoon state-space model.
  • Robustness: The mistuning design remains effective under small local discrepancies from the optimal gains when the relevant first-harmonic integral is nonzero.The robustness argument applies similarly to scenario II.

C. Simulations

Simulations compare nominal symmetric and mistuned bidirectional control for a 20-vehicle platoon. Mistuning reduces initial position errors faster and agrees with the predicted stability-margin improvement.

  • Simulation setup: N = 20 vehicles are simulated in scenario I with desired gap ∆ = 1 and desired velocity Vd = 5.All vehicles initially have the desired velocity; only the first vehicle has a nonzero initial relative position error of 0.5.
  • Nominal control: Symmetric control uses identical gains k(f)i = 1 and bi = 0.5 for all 20 vehicles.
  • Mistuned control: Mistuned control uses the gains from Figure 5(a), with maximum and minimum gains within ±10% of nominal values.
  • Results: The mistuned design reduces initial-condition errors faster than the nominal symmetric design, consistent with improved closed-loop stability margin.

VI. DISCUSSION ON MISTUNING DESIGN

The discussion explains how mistuning is implemented, validated for small platoons, and connected to stability and disturbance sensitivity. It also identifies robustness to gain errors and unresolved analysis of disturbance reduction.

  • PDE validation: The PDE is intended for large N, but numerical comparisons show quantitatively correct predictions even for relatively small platoons.PDE eigenvalues match the least stable and other dominant discrete-platoon eigenvalues, although the PDE has infinitely many eigenvalues.
  • Stability improvement: ±10% mistuning improves the stability margin for N = 20 by 150% in scenario I and 400% in scenario II.The real part of the least stable eigenvalue changes from −0.0491 to −0.1281 in scenario I and from −0.012 to −0.05 in scenario II.
  • Information requirements: Scenario I implementation requires the mistuning amplitude and front-half membership information, whereas scenario II requires only the mistuning amplitude.
  • Robustness: Errors in identifying front-half membership can produce non-optimal gains, but small gain deviations do not greatly affect the stability improvement.This membership issue does not arise in scenario II.
  • Eigenvalue behavior: Sturm-Liouville theory rules out eigenvalue cross-over, supporting the continued least-stable status of the relevant eigenvalue under mistuning.Figure 10 numerically depicts the six eigenvalues closest to zero for both PDE and discrete models.
  • Disturbance sensitivity: For N = 20, 10% mistuning reduces the H∞ norm from 6.69 to 3.38, approximately a 50% reduction in disturbance sensitivity.The H∞ norm measures sensitivity from external disturbances to inter-vehicle spacing errors.
  • Conclusions: The PDE analysis predicts O(1/N^2) decay for symmetric control and O(1/N) decay with arbitrarily small mistuning, with state-space calculations confirming the predictions.
  • Open questions: Detailed analysis of mistuning’s disturbance-sensitivity benefits remains future work.The paper also identifies PDE models for two- or three-dimensional vehicle fleets as future research.

A. Solution properties of PDE (15).

This section establishes well-posedness of the PDE through semigroup theory. The proof reformulates the PDE as an evolution equation, characterizes its spectrum through an elliptic operator, and treats mistuning as a bounded perturbation.

  • Evolution formulation: The PDE is rewritten as a first-order evolution equation generated by a linear operator A on Z = L2 × L2.The initial/boundary-value problem is represented as ẋ = Az with an initial state z0.
  • Proof strategy: The analysis uses the Hille-Yosida theorem to establish solution properties through operator closure, resolvent characterization, and a resolvent bound.
  • Spectral characterization: The spectrum of A is characterized through an associated elliptic operator whose coefficients satisfy k0(x) > 0.Spectral bounds are used to establish Real[s] < α for some α < 0 and thus a right-half-line subset of the resolvent set.
  • Resolvent properties: For k1(x) = 0, the resolvent analysis gives [0, ∞) ⊂ ρ(A) for any positive k0(x).
  • Mistuned operator: For general k1(x), the operator is decomposed into A0 plus a bounded perturbation, and semigroup existence follows from a perturbation theorem.

B. Proof of Theorem 1

The proof analyzes mistuned PDE eigenvalues by Laplace transforming the boundary-value problem and applying perturbation expansions. A resonance condition determines the first-order eigenvalue shifts.

  • Perturbed PDE: The spatially varying mistuning coefficients destroy Fourier diagonalization, so the nominal eigenvalue methods require modification.
  • Eigenvalue problem: Dirichlet eigenvalues are obtained by finding s values for which the homogeneous Laplace-transformed PDE has a nontrivial solution.
  • Perturbation expansion: The eigenfunction and eigenvalue are expanded as η(x) = η0(x) + ǫη1(x) + O(ǫ^2) and s = r0 + ǫr1 + O(ǫ^2).The term ǫr1 represents the mistuning-induced perturbation of the nominal eigenvalue r0.
  • Solvability condition: A resonance condition is required for η1 to exist, placing the forcing term in the range space of the linear operator.For the self-adjoint operator, this range space is the complement of its null space.
  • Eigenvalue shifts: The resulting perturbation formulas determine the shifts of the eigenvalues s_l^± from their nominal values s_l^±(0).
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