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Order-Adaptive Distributed Integral Control
Fei Chen
TL;DR
Unknown target complexity forces a choice between low-order controllers that may leave tracking error and high-order controllers that burden every agent with unnecessary dynamics. The paper proposes OADIC, which adds integral states only when distributed error monitoring finds the current order inadequate, using nested linear controllers. It proves stability and finite-time rejection of insufficient orders, and reports reduced integral-state time or tail RMS error in numerical comparisons.
Problem
Unknown target complexity leaves fixed-order distributed controllers choosing between persistent tracking error at low order and unnecessary integral states at high order.
Method
OADIC begins with proportional feedback, monitors local relative errors, and activates additional integral states through nested linear controllers while preserving existing states and gains.
Results
OADIC rejects every insufficient nonsaturated order after finitely many decision intervals, while numerical comparisons report 23.3% lower integral-state time than fixed order three and approximately 95.8% lower tail RMS error than fixed order two.
Takeaways & Limitations
OADIC obtains necessary tracking capability with purely linear active controllers without carrying the full controller order throughout operation.
Abstract
from arXiv · showhide
We address a structural tradeoff in distributed dynamic coordination: when the target complexity is unknown, a low controller order saves states but may leave a persistent tracking error, whereas a high order improves tracking but may burden every agent with unnecessary dynamics. To remove this choice without resorting to computationally more involved nonlinear feedback or chattering-prone nonsmooth feedback, we develop an order-adaptive distributed integral controller (OADIC). Specifically, we start with proportional feedback and add integral states only when locally measurable relative errors show that the current order is inadequate. Meanwhile, we organize the candidate controllers in a nested form, thereby preserving the existing states and gains and maintaining continuous control inputs during order transitions. To provide a theoretical basis for this design, we first characterize the consistency of prescribed relative displacements on the augmented agent--target graph. Next, we construct gains that stabilize all admissible fixed-order subsystems and establish a uniform input-to-state stability bound for the variable-dimension closed loop. Furthermore, we prove that OADIC rejects every insufficient order after finitely many decision intervals and explicitly bound the rejection time of the critical order.
I. INTRODUCTION
The paper develops OADIC to resolve the tradeoff between low-order controllers that may leave tracking error and high-order controllers that impose unnecessary dynamics on every agent. It adapts nested linear integral controllers using distributed error monitoring while establishing stability and finite-time rejection guarantees.
- Motivation: Unknown target complexity creates a tradeoff between low controller order with unmodeled dynamics and high order with unnecessary integral states at every agent.The replicated controller dynamics make this cost especially restrictive in large-scale networks.
- Controller design: OADIC starts with proportional feedback and activates one additional integral state when a distributed monitor indicates inadequate tracking progress.The monitor uses locally measurable relative errors.
- Controller design: Nested controllers preserve existing states and gains while maintaining continuous control inputs during order transitions.This avoids rebuilding the controller when its order increases.
- Theoretical guarantees: A constructive nested gain condition stabilizes every admissible fixed-order subsystem and supports a uniform input-to-state stability bound for the variable-dimension closed loop.The stability analysis accounts for continuous target forcing and reset maps.
- Paper scope: The paper characterizes consistency of prescribed relative displacements and analyzes finite-time detection of insufficient controller orders.The paper also includes numerical comparisons with fixed-order and nonlinear controllers.
A. Notation
This section establishes notation for scalar, vector, and matrix spaces, norms, asymptotic orders, switching limits, and eigenvalue conventions.
- Basic notation: R and Z+ denote the real numbers and positive integers, while R^n and R^(n×m) denote vector and matrix spaces.The notation applies throughout the paper.
- Basic notation: The Euclidean norm is used for vectors and the induced Euclidean norm for matrices.The same norm symbol is used when the dimension is clear.
- Switching notation: At a switching instant, ξ(t_s^−) denotes the left-hand limit of signal ξ(t).This convention describes signal values immediately before switching.
- Asymptotic notation: The notation f(t)=O(g(t)) bounds the norm of f by a constant multiple of |g|, whereas f(t)=o(g(t)) requires their ratio to vanish.The relevant limit is stated explicitly unless clear from context.
B. Auxiliary lemmas
The auxiliary results provide a polynomial-root condition used in the paper’s subsequent analysis.
- Auxiliary lemma: For a symmetric matrix M, Lemma 1 supplies an auxiliary spectral result for the analysis.The supplied passage introduces the lemma but does not include its displayed inequality.
- Auxiliary lemma: A polynomial P_p with p≥2 and positive coefficients a_l is considered under a stated condition.The condition itself is not fully present in the supplied passage.
- Auxiliary lemma: When the stated condition holds, all roots of P_p are real and distinct.This is the lemma’s explicit conclusion.
III. PROBLEM FORMULATION
The paper formulates dynamic coordination for single-integrator agents communicating over an undirected graph with partial target access and prescribed relative displacements. The desired constraints are represented on an augmented agent–target graph and solved using linear distributed feedback based on locally available relative information.
- Agent model: Each agent follows single-integrator dynamics, isolating the role of controller order from additional agent dynamics.The scalar formulation extends to higher-dimensional states through Kronecker products.
- Communication graph: Agents communicate over an undirected graph whose adjacency, neighbor sets, degree matrix, and Laplacian encode network connectivity.Undirectedness gives symmetric adjacency entries.
- Target access: Target-access indicators identify which agents can measure the dynamic target state.The indicators are collected in a diagonal matrix.
- Assumptions: The standing assumption requires a connected communication graph and access to the target by at least one agent.Under this assumption, L+A_0 is symmetric and positive definite.
- Coordination constraints: Prescribed agent–agent and agent–target displacements are unified by treating the target as node 0 in an augmented graph.Each augmented edge specifies a desired relative displacement.
- Coordination objective: The problem asks for linear distributed controls using only local relative errors, prescribed displacements, and target measurements available to informed agents.The formulation includes leader-following consensus, formation control, and formation tracking as special cases.
IV. CONSISTENCY
The section identifies consistency as the solvability condition for prescribed relative displacements and characterizes it algebraically on the augmented agent–target graph. When consistent, the desired offsets are unique and coordination is equivalent to tracking the resulting desired trajectories.
- Motivation: Arbitrary desired relative displacements can be inconsistent, making the coordination problem infeasible.In the example, ζ̄12 = ζ̄32 = 1 implies x3 − x1 = 0, contradicting ζ̄31 = 2.
- Tracking equivalence: When consistent offsets exist, coordination is solved if and only if each agent trajectory converges to x̄i(t) = x0(t) + d̄i.
- Definition: Consistency means that constants d̄0,...,d̄n with d̄0 = 0 reproduce every prescribed relative displacement.
- Algebraic characterization: The desired relative displacements are consistent if and only if the stacked edge-displacement vector lies in the range of the reduced augmented-graph incidence matrix.
- Graph interpretation: Equivalently, the signed sum of desired displacements along every augmented-graph cycle must equal zero.For the illustrated cycle, ζ̄12 − ζ̄32 + ζ̄31 = 0 is required.
- Uniqueness: Under the connectivity assumption, consistent displacements determine a unique offset vector d̄.
V. ALGORITHM
OADIC begins with proportional feedback and increases controller order when relative-error improvement is insufficient. Nested states and gains preserve prior controller components, while distributed monitoring enables synchronized order decisions and bumpless transitions.
- Adaptive order rule: OADIC starts at the lowest order and activates an additional integral state when relative error fails to decrease sufficiently over a decision interval.The monitor evaluates relative improvement rather than absolute error and stops increasing at qmax.
- Controller structure: At order q, each controller contains q − 1 integral states, with q = 1 representing proportional feedback and q = 2 activating the first integral state.
- Performance measurement: The reduction percentage is evaluated between consecutive decision instants, with rk < 0 indicating error growth and rk = 1 assigned at zero error.
- Distributed monitor: The network-wide indicator J(t) is equivalent to the global tracking error and can be computed from local regulation quantities and distributed maximum consensus.J(t) = 0 if and only if the global error is zero.
- State transfer: An order increase retains existing integral states and initializes only the newly activated state, producing a bumpless change in the control input.Continuity of the tracking error supports continuity across the switch.
- Nested gains: A single nested gain sequence is reused across orders, so switching activates the next gain without replacing previously used gains.The sequence is constructed to provide a common sufficient stability condition for every fixed-order subsystem.
VI. STABILITY
The stability analysis treats each active order as a stable continuous subsystem connected by reset maps. Under the stated assumptions and gain conditions, the switched system has a uniform input-to-state stability bound, and the adaptation eventually settles at a finite order.
- Fixed-order stability: Under consistent displacements, sufficient regularity, connectivity, and gains satisfying (24), every fixed-order state matrix Aq is Hurwitz.
- Switched-system bound: The complete variable-dimension dynamics, including continuous evolution and order-increase resets, are uniformly input-to-state stable with state η(t) and input w(t).The resulting constants account for every admissible order and reset.
- Spectral analysis: The transformed error coordinates decouple the dynamics, reducing stability verification to characteristic polynomials whose roots are strictly negative.
- Order convergence: Because the integer-valued order is nondecreasing and bounded by qmax, it converges to a limiting order q∞ after finitely many order changes.
- Endpoint convergence: Decision-interval contraction yields endpoint convergence when the required reductions satisfy ∑k εk = ∞.For the prescribed schedule, this holds if ε∞ > 0, or if ε∞ = 0 and 0 < α ≤ 1.
- Scope of guarantees: If q∞ < qmax, repeated monitor success does not by itself establish an exact internal model, and endpoint contraction does not exclude larger inter-interval errors.
- Final-order behavior: Reaching q∞ = qmax does not imply instability; the remaining tracking error depends on the unmodeled derivative x0^(qmax) and the final stable subsystem’s induced gain.Polynomial targets of degree d are tracked asymptotically whenever q∞ ≥ d + 1.
VII. DETECTION OF INSUFFICIENT ORDERS
OADIC rejects every insufficient order in finite time: orders below the target degree produce growing errors, while the critical order leaves a bounded nonzero residual. The critical-order rejection time is explicitly bounded.
- Orders below the target degree: For a polynomial target of degree d, every active order q<d eventually yields negative relative improvement and is rejected when q<qmax.The unmatched polynomial forcing makes the error grow polynomially, so the monitored reduction eventually falls below the positive threshold.
- Critical order: At the critical order q=d, the error converges to a nonzero steady residual, causing the monitored relative improvement to converge to zero.Because εk≥ε∞>0, the switching condition is eventually met whenever d<qmax.
- Finite-time rejection: Theorem 3 proves that OADIC successively rejects every active order q≤d after finitely many decision intervals, provided the maximum order has not been reached.This combines the finite-time result for q<d with the critical-order argument for q=d.
- Rejection-time bound: The critical-order rejection time is no later than one decision interval after the first decision instant at or beyond tkd+τd.Equivalently, the bound uses tkd+⌈τd/Td⌉Td.
- Rejection-time bound: Weak connectivity, slow modal decay, a small threshold, or a large reset transient can delay detection of the critical order.These factors appear in the explicit rejection-time bound.
VIII. SIMULATION
The simulation compares OADIC with fixed-order and nonlinear controllers on a ten-agent quadratic-target coordination problem, showing adaptive state allocation alongside tracking behavior.
- Setup: The ten-agent simulation uses a quadratic target and compares global tracking errors, OADIC's selected order, and cumulative integral-state time per agent.The network is an undirected ring augmented with links {1,6} and {3,8}.
- Tracking accuracy: The proportional controller cannot follow the accelerating target, while order two removes the velocity-dependent component but retains an acceleration-induced residual.Both order-three controllers contain the required internal model and drive the error toward zero.
- Tracking accuracy: NFOIC achieves the smallest tail error through signed-power feedback, but retains the full order-three internal model and all associated dynamic states.Its nonlinear enhancement does not reduce the prescribed order or number of dynamic states.
- Adaptive order selection: OADIC activates its first integral state at t = 8 and second at t = 20, then reaches the same order-three internal model as the full-order controllers.Because activation is delayed, its finite-horizon tail error is larger, trading transient speed for online complexity allocation.
- Accuracy–complexity tradeoff: 23.3% reduction in integral-state time is achieved relative to either full-order benchmark, while tail RMS error falls 95.8% relative to fixed order two.OADIC has average active integral-state count 1.5333 over the 60-unit horizon, compared with 2 for full-order controllers.
IX. CONCLUSION
The conclusion presents OADIC as a nested linear controller that adapts internal-model order to observed tracking need while preserving state and gain continuity. The paper establishes feasibility, stability, finite rejection, and numerical accuracy–complexity guarantees, while identifying robustness to noise and topology changes as future work.
- Controller design: OADIC adds integral states when observed relative-error progress is insufficient, moving among nested linear controllers without fixing a conservative high order.Existing states and gains are preserved during transitions, and new states are initialized to maintain continuous control inputs.
- Theoretical guarantees: The theory derives a cycle-consistency condition for feasible relative displacements, nested gains stabilizing every admissible fixed-order subsystem, and a uniform input-to-state stability bound.The bound accounts for both target motion and order transitions.
- Theoretical guarantees: OADIC rejects every insufficient nonsaturated order after finitely many decision intervals, including stable fixed-order subsystems that retain nonzero tracking error.The conclusion also reports an explicit rejection-time bound for the critical order in the paper's theoretical development.
- Numerical conclusions: 23.3% lower integral-state time than fixed order three and approximately 95.8% lower tail RMS error than fixed order two illustrate the accuracy–complexity tradeoff.NFOIC attains a smaller finite-horizon tail error but retains both integral states throughout and uses signed-power feedback.
- Scope and future work: The numerical results do not establish uniform superiority in transient speed; future work will study noisy measurements and changing network topologies.The supported conclusion is that necessary tracking capability can be obtained with purely linear active controllers without carrying full order throughout operation.