Source-linked AI summary
The Cooperative Output Regulation Problem of Discrete-Time Linear Multi-Agent Systems by the Adaptive Distributed Observer
Jie Huang
TL;DR
The paper addresses cooperative output regulation for discrete-time linear multi-agent systems when followers should not need prior knowledge of the leader matrix S. It develops an adaptive distributed observer and a discrete adaptive regulator-equation algorithm, then establishes solvability using both state-feedback and output-feedback adaptive distributed control laws.
Problem
The problem is to achieve cooperative output regulation in discrete-time linear multi-agent systems without requiring every follower to know the leader matrix S.
Method
The paper combines a discrete adaptive distributed observer with an adaptive difference-equation algorithm that estimates S and computes regulator-equation solutions and feedforward gains.
Results
Under the stated assumptions, the cooperative output regulation problem is solvable using adaptive distributed control laws with state feedback and with measurement output feedback.
Takeaways & Limitations
The resulting approach removes the requirement that each follower know the leader system matrix while retaining distributed cooperative output regulation.
Abstract
from arXiv · showhide
In this paper, we first present an adaptive distributed observer for a discrete-time leader system. This adaptive distributed observer will provide, to each follower, not only the estimation of the leader's signal, but also the estimation of the leader's system matrix. Then, based on the estimation of the matrix S, we devise a discrete adaptive algorithm to calculate the solution to the regulator equations associated with each follower, and obtain an estimated feedforward control gain. Finally, we solve the cooperative output regulation problem for discrete-time linear multi-agent systems by both state feedback and output feedback adaptive distributed control laws utilizing the adaptive distributed observer.
I. INTRODUCTION
The paper develops a discrete-time adaptive distributed observer for cooperative output regulation, addressing the need for followers to estimate both the leader’s signal and system dynamics without knowing the leader matrix S. It then combines this observer with an adaptive regulator-equation algorithm and state- or output-feedback control laws.
- Motivation: Cooperative output regulation requires distributed control that tracks leader-generated references and rejects leader-generated disturbances.The problem extends classical output regulation to multi-agent systems and generalizes leader-following consensus by including disturbance rejection.
- Motivation: Distributed observers estimate the leader’s signal, but conventional designs require every follower to know the leader matrix S.The adaptive observer removes this information requirement by estimating the leader’s dynamics as well.
- Approach: The paper proposes a discrete counterpart of an adaptive distributed observer for a discrete-time leader system.The observer is designed to provide followers with estimates of both the leader signal and its dynamics.
- Approach: The proposed adaptive scheme estimates S, computes solutions to each follower’s regulator equations, and obtains an estimated feedforward control gain.These steps form the basis for the subsequent adaptive cooperative regulation design.
- Contributions: The paper designs both state-feedback and output-feedback adaptive distributed control laws for discrete-time linear multi-agent systems.The contributions also include a stability result supporting existence conditions for the adaptive observer.
II. PROBLEM FORMULATION
The problem formulation models a discrete-time leader with N follower subsystems connected through a directed graph and seeks a distributed controller achieving bounded trajectories and asymptotic regulation. Solvability is stated under spectral, stabilizability, detectability, regulator-equation, and graph-connectivity assumptions.
- System model: The leader generates reference and disturbance signals, while each follower has state, input, regulated-output, and measurement-output dynamics.The multi-agent system consists of one leader and N follower subsystems.
- System model: The communication graph permits each agent to use the state or output of neighboring agents for control.The leader is node 0, followers are nodes 1 through N, and an edge indicates available information flow.
- Control objective: The distributed control law depends only on neighbor measurements and includes state feedback as a special case of output feedback.State feedback is recovered when each follower’s measurement output equals its state.
- Control objective: The regulation objective requires closed-loop trajectories to exist for all t ≥ 0, remain bounded for bounded v, and satisfy lim t→∞ e_i(t) = 0.The last condition applies to every follower i = 1, . . . , N.
- Assumptions: The formulation assumes leader eigenvalues have modulus at most 1, follower pairs are stabilizable and detectable, regulator equations have unique solutions, and the graph has a leader-rooted spanning tree.These assumptions provide the stated solvability conditions for the cooperative output regulation problem.
III. ADAPTIVE DISTRIBUTED OBSERVER
The adaptive distributed observer estimates both the leader’s system matrix and signal, removing the need for every follower to know S. Under stated stability and graph conditions, these estimation errors converge exponentially to zero.
- Motivation: The discrete distributed observer estimates the leader signal, but requires every follower to know the matrix S.The observer is called a distributed observer when its associated estimation-error system is asymptotically stable.
- Adaptive observer: The adaptive observer updates each follower’s matrix estimate and signal estimate using information exchanged over the network.Only followers that are children of the leader know S directly.
- Stability analysis: A stability lemma shows that a Schur nominal matrix with exponentially vanishing perturbations yields exponentially vanishing state trajectories.The proof uses a quadratic Lyapunov function and input-to-state stability.
- Convergence result: Under the stated assumptions, suitable μ1 makes the matrix-estimation error converge exponentially, while a Schur condition on the signal-error dynamics yields exponential signal convergence.The argument treats the decaying matrix-estimation error as a vanishing input to the signal-estimation dynamics.
- Convergence result: The combined result establishes exponential convergence of the adaptive observer’s estimation errors for arbitrary initial estimates under the required assumptions.The proof combines exponential matrix convergence with the stability lemma for the signal-estimation error.
IV. MAIN RESULT
The paper develops a discrete adaptive algorithm that estimates regulator-equation solutions from each follower’s estimate of S, then uses these estimates in state-feedback and output-feedback controllers to solve the regulation problem.
- Adaptive regulator-equation calculation: Because some followers do not know S, the paper adaptively calculates regulator-equation solutions using each follower’s estimate S_i.This removes the need to directly use the regulator-equation solution when S is unavailable to non-child followers of the leader.
- Supporting stability result: A supporting lemma extends difference-equation methods for solving linear algebraic equations to the case where the matrix is unknown but estimated exponentially.The result provides the convergence mechanism used by the adaptive regulator-equation calculation.
- Adaptive regulator-equation calculation: The difference-equation algorithm has bounded solutions, and its estimated matrices converge exponentially to the corresponding regulator-equation matrices.The convergence follows from the nonsingularity condition on Q_i and exponential convergence of G_i(t) to Q_i.
- State feedback: The adaptive observer and regulator-equation algorithm are combined with stabilizing state feedback to yield a controller solving the cooperative output regulation problem.The controller uses K_xi to make A_i+B_iK_xi Schur and an adaptive feedforward gain K_ηi(t).
- Output feedback: The same adaptive observer and regulator-equation algorithm also solve the problem under measurement output feedback when each follower’s relevant pair is detectable.An observer gain L_i makes A_i+L_iC_mi Schur, while the state-estimation error decays exponentially.
V. AN EXAMPLE
A four-follower example applies the proposed observer and controllers to a discrete-time leader system. The reported simulations show estimation of the leader signal and satisfactory tracking under state feedback.
- Setup: The example considers a cooperative output regulation problem with four followers and a leader system specified by the paper’s discrete-time model.The communication graph and follower dynamics are selected so the stated assumptions can be verified.
- Setup: The chosen feedback gain K_xi=[0.2, 0] places the follower closed-loop eigenvalues at {0.447, −0.447} for all four followers.This verifies the desired Schur stability condition for the state-feedback design.
- Simulation results: Figures 2 and 3 display estimation errors for the first and second components of the leader’s signal produced by the adaptive distributed observer.The example uses the leader trajectory generated from the stated initial condition.
- Controller parameters: The parameter choices µ1=0.3, µ2=0.4, and µ3i=0.1 satisfy the stated bounds, enabling the state-feedback and output-feedback controller designs.The bounds reported are 0<µ1<0.778, 0<µ2<1.414, and 0<µ3i<0.615.
- Simulation results: Figure 4 shows satisfactory tracking performance for the four followers under the state-feedback control law.The paper reports this outcome after presenting the observer estimation-error plots.
VI. CONCLUSION
The paper studies adaptive cooperative output regulation for discrete-time linear multi-agent systems without requiring each follower to know the leader’s system matrix. It identifies communication delays as a natural extension for future work.
- The approach solves adaptive cooperative output regulation for discrete-time linear multi-agent systems using an adaptive distributed observer.
- Unlike existing distributed-observer approaches, each follower need not know the leader system’s matrix.
- Communication delays remain a stated direction for extending the problem.