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Optimal Output Consensus of High-Order Multi-Agent Systems with Embedded Technique
Yutao Tang, Zhenhua Deng, Yiguang Hong
TL;DR
The paper studies how high-order MIMO minimum-phase agents can achieve consensus on the optimizer of a global cost function. It constructs an optimal signal generator and embeds it in feedback, yielding state- and output-feedback algorithms whose optimal output consensus converges asymptotically or exponentially under standard assumptions.
Problem
The paper addresses optimal output consensus for high-order MIMO minimum-phase agents, extending output consensus with an optimization requirement beyond settings that mainly consider single-integrator agents.
Method
The approach constructs an optimal signal generator and embeds it in a feedback loop to separate virtual single-integrator optimization from high-order reference tracking.
Results
The proposed algorithms achieve optimal output consensus asymptotically or exponentially under standard assumptions.
Takeaways & Limitations
The embedded technique provides a constructive framework for solving optimal output consensus in high-order multi-agent systems with state or output feedback.
Takeaways & Limitations
Open cases include practical and nonlinear agents, uncertainties from communication or the environment, and different optimization constraints.
Abstract
from arXiv · showhide
In this paper, we study an optimal output consensus problem for a multi-agent network with agents in the form of multi-input multi-output minimum-phase dynamics. Optimal output consensus can be taken as an extended version of the existing output consensus problem for higher-order agents with an optimization requirement, where the output variables of agents are driven to achieve a consensus on the optimal solution of a global cost function. To solve this problem, we first construct an optimal signal generator, and then propose an embedded control scheme by embedding the generator in the feedback loop. We give two kinds of algorithms based on different available information along with both state feedback and output feedback, and prove that these algorithms with the embedded technique can guarantee the solvability of the problem for high-order multi-agent systems under standard assumptions.
1 Introduction
The paper addresses optimal output consensus for high-order MIMO minimum-phase agents, extending distributed optimization beyond the predominantly single-integrator setting. It proposes an embedded design that separates optimal consensus generation from high-order reference tracking and proves asymptotic or exponential solvability under standard assumptions.
- Motivation: Most existing distributed optimization results consider single-integrator agents, while practical coordination systems require high-order dynamics.This gap motivates treating physical agent dynamics directly rather than assuming idealized integrators.
- Problem and contribution: The paper formulates optimal output consensus for groups of MIMO minimum-phase agents with vector relative degrees.The problem combines high-order output consensus with distributed optimization over convex objective functions.
- Method: An embedded controller combines an optimal signal generator with a reference-tracking controller, separating optimization for virtual single integrators from high-order tracking.This separation is intended to simplify the design despite nonlinearities introduced by general convex gradients.
- Results: Under standard assumptions, the proposed algorithms achieve optimal output consensus asymptotically and even exponentially.The paper reports consistency with existing high-order output-consensus results and gives examples illustrating the algorithms.
- Algorithms: The framework provides two distributed gradient-based algorithm classes for different gradient-information settings, with both state-feedback and output-feedback versions.Local observers are included because high-order state variables may not be directly measurable.
2 Preliminaries
The paper formulates optimal output consensus for high-order minimum-phase multi-agent systems, requiring bounded trajectories and outputs to converge to the optimizer of a global convex cost function under distributed information constraints.
- Agent model: The agents are continuous-time linear MIMO systems assumed to be minimum-phase with a vector relative degree.Vector relative degree describes the input-output differentiation structure used in the formulation.
- Problem formulation: Each agent knows only its differentiable local cost function, which cannot be shared globally, and exchanges information with neighboring agents.The distributed controller must use local data and exchanged neighbor information.
- Problem formulation: The problem requires bounded agent trajectories and output convergence to the optimizer of the global cost function.The control objective is posed for high-order agent dynamics rather than only integrator models.
- Problem formulation: Optimal output consensus extends output consensus by requiring the common output to equal the optimal solution of a convex cost function.For single-integrator agents, the requirement coincides with distributed optimization.
- Assumptions: The communication graph is undirected and connected, while local cost functions are strictly convex; stronger assumptions impose strong convexity and Lipschitz gradients.Connectivity supports information propagation, strict convexity implies uniqueness, and the stronger conditions facilitate exponential convergence.
- Motivation: High-order dynamics and the nonlinear structure of gradient-based optimization make the problem more challenging than optimal consensus for single- and double-integrator agents.The paper motivates an embedded control framework to address this design difficulty.
3 Embedded Control Scheme
The embedded scheme separates optimization from high-order plant control by generating an optimal reference for virtual integrators and tracking it through transformed agent dynamics. Under stated stability and convexity conditions, the generator achieves optimal consensus, including exponential convergence under stronger assumptions.
- Framework: The framework combines a precompensator, an optimal signal generator, and a reference-tracking controller.The precompensator handles vector relative degrees, the generator solves the optimization problem, and the tracker forces the agent output to follow the generated reference.
- Optimal signal generator: The optimal signal generator applies the same optimization problem to virtual single integrators and produces signals that asymptotically reproduce the optimal solution.These signals become output references for the high-order agents through an embedded feedback loop.
- Precompensator: The precompensator transforms minimum-phase MIMO agents into a normal form with decoupled inputs and outputs and homogeneous relative degrees.This reduces the plant-design problem to a high-order integrator form.
- Design considerations: The choice of signal generator must account for robustness and compatibility with the high-order plant because not every candidate yields a solvable augmented tracking system.The generator design is therefore constrained by system composition and embedded implementation.
- Stability analysis: The stability lemma establishes asymptotic stability generally and global exponential stability when the nonlinear term satisfies the stronger Lipschitz and positivity conditions.The proof uses a Lyapunov function and LaSalle’s invariance principle.
- Optimal signal generator: The generator is based on distributed primal-dual dynamics and solves the optimal consensus problem under undirected connected graphs and strictly convex local costs.Its equilibrium enforces consensus and the first-order optimality condition for the sum of local costs.
- Stability and convergence: Under strong convexity and Lipschitz-gradient conditions, the generated outputs approach the optimal solution exponentially.The result is stated for the virtual-agent generator and is inherited by the embedded output-consensus design under the corresponding tracking construction.
4 Optimal Output Consensus of High-order Agents
The section develops embedded controllers for optimal output consensus in high-order agents, separating optimization from dynamics and covering two gradient-information cases with state and output feedback. Under the stated assumptions, the resulting schemes achieve asymptotic, and in specified cases exponential, convergence to the global optimum.
- The design treats known gradients and real-time gradients as two cases, with state-feedback and output-feedback controllers proposed for both.The section introduces a convergence lemma for cascaded systems before presenting the two cases.
- The optimal signal generator can be implemented independently, placing optimization and high-order tracking in a cascade.Strict convexity suffices for solvability, while Assumption 4 gives exponentially fast convergence.
- Case I: Under Assumptions 1–3, the state-feedback designs solve optimal output consensus for any ε > 0.For the corresponding theorem, Assumption 4 additionally yields exponential convergence of each output to y∗.
- Case II: The output-feedback designs use observer-based high-gain embedding and likewise achieve optimal consensus, with exponential convergence under Assumption 4.The analysis establishes asymptotic stability for strictly convex local costs and global exponential stability under the stronger assumption.
- The framework decouples optimization complexity from high-order tracking, reducing controller design to parameter tuning and allowing flexible signal-generator choices.The authors state that this may be favorable for large-scale networks and more complex agent dynamics.
- For general linear minimum-phase agents, solvability is guaranteed for ε ∈ (0, ε∗), with exponential convergence under the stated assumptions.Theorems 5 and 6 provide the corresponding state-feedback and output-feedback results.
5 Simulations
Two simulations illustrate the embedded designs on robot rendezvous and a higher-order multi-agent system with more complex objective functions. In both examples, the outputs converge to the global optimal point.
- Example 1: With c0 = 4, c1 = 8, ε = 1, and random initial conditions in [−10, 10]8, all robots achieve the optimal rendezvous at y∗.The simulation marks local minimizers by diamonds and the global optimum by a circle.
- Example 2: The second example tests the embedded design on a high-order system modified from prior work with more complex objective functions.The system is transformed into the paper’s required form and satisfies the stated assumptions, including relative degree (2, 1).
- Example 2: Using only real-time gradients, the output-feedback controller solves the second optimal consensus problem.The simulation uses c0 = 4, c1 = 8, l1 = 4, l2 = 8, ε = 1, and random initial conditions in [−10, 10]14.
- Example 2: The second simulation shows all outputs converging to the global optimal point.
6 Conclusions
The paper addresses optimal output consensus for high-order minimum-phase multi-agent systems using an embedded control scheme based on an optimal signal generator. The proposed algorithms are proved to converge to the optimal solution under different conditions, while practical, nonlinear, uncertain, and differently constrained cases remain open.
- The study targets optimal output consensus in high-order minimum-phase multi-agent systems.
- The proposed embedded control scheme introduces an optimal signal generator to solve the problem.
- The proposed algorithms converge to the optimal solution asymptotically or exponentially under different conditions.
- Open problems include practical and nonlinear agents, uncertainties from communication or the environment, and different optimization constraints.
- The paper illustrates the approach with state-feedback robot rendezvous and output-feedback phase-portrait and component-profile simulations.