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Distributed Control of Positive Systems
Anders Rantzer
TL;DR
Large-scale control methods often lack scalability, while positive systems offer simpler stability and performance certificates. The paper develops distributed verification and synthesis methods based on positive and positively dominated models, obtaining scalable global-performance analysis and stronger positive-system KYP results.
Problem
Large-scale multivariable control suffers from computational complexity and controllers lacking distributed structure, motivating scalable methods for positive systems.
Method
The paper uses linear Lyapunov and storage-function conditions, distributed linear programming, positively dominated transfer functions, and positive-system KYP generalizations.
Results
Stability and performance verification and controller synthesis can scale linearly with the number of interconnections, with distributed certificates for global optimality.
Takeaways & Limitations
Positive and positively dominated system structure enables scalable distributed control analysis and synthesis with optimal input-output performance.
Abstract
from arXiv · showhide
A system is called positive if the set of non-negative states is left invariant by the dynamics. Stability analysis and controller optimization are greatly simplified for such systems. For example, linear Lyapunov functions and storage functions can be used instead of quadratic ones. This paper shows how such methods can be used for synthesis of distributed controllers. It also shows that stability and performance of such control systems can be verified with a complexity that scales linearly with the number of interconnections. Several results regarding scalable synthesis and verfication are derived, including a new stronger version of the Kalman-Yakubovich-Popov lemma for positive systems. Some main results are stated for frequency domain models using the notion of positively dominated system. The analysis is illustrated with applications to transportation networks, vehicle formations and power systems.
I. INTRODUCTION
Positive-system structure addresses scalability problems in large-scale control by enabling simpler Lyapunov-based verification and distributed controller synthesis. The paper extends these ideas to frequency-domain models and local control-loop structures.
- Motivation: Positive-system dynamics permit stability and performance verification with Lyapunov functions containing only n linear terms instead of n^2 quadratic terms.This reduction applies when closed-loop matrices have nonnegative off-diagonal entries.
- Scope: Scalable conditions arise naturally in stochastic systems, economics, transportation networks, chemical reactions, power systems, and ecology.
- Extensions: The positive-system property extends to frequency-domain models through positively dominated transfer functions.
- Background: Positive-system theory builds on nonnegative-matrix and monotone-dynamics results used in economics, Markov chains, compartment models, and population dynamics.
- Paper roadmap: The paper develops stability criteria, input-output analysis, distributed linear-programming synthesis, positively dominated transfer-function methods, and scalable Lyapunov-inequality verification.
II. NOTATION
The notation defines nonnegative vectors and matrices, matrix positivity properties, stability classes, and rational stable transfer-function spaces used throughout the paper.
- Basic notation: R+ denotes the set of nonnegative real numbers, while |x| denotes the element-wise absolute value.
- Basic notation: The vector 1 has all entries equal to one; X > 0 and X ≥ 0 mean element-wise positive and nonnegative, respectively.
- Matrix properties: For symmetric X, X ≻ 0 means positive definite; a matrix is Metzler when all off-diagonal entries are nonnegative.
- Stability classes: Hurwitz matrices have eigenvalues with negative real parts, while Schur matrices have eigenvalues strictly inside the unit circle.
- Transfer functions: RH∞ denotes rational real-coefficient functions without poles in the closed right half plane, and RH^{n×m} contains matrices of such functions.
III. DISTRIBUTED STABILITY VERIFICATION
For Metzler systems, stability has equivalent vector, matrix, and inverse-characterizations that support distributed verification and scalable feasible-vector search. Examples show each node or vehicle can check inequalities using local and neighboring information.
- Stability criteria: For a Metzler matrix A, Hurwitz stability is equivalent to the existence of ξ > 0 with Aξ < 0, z > 0 with z^T A < 0, a diagonal P ≻ 0 satisfying A^T P + PA ≺ 0, or a nonnegative -A^-1.
- Stability criteria: These conditions correspond respectively to rectangular, linear, and quadratic Lyapunov functions.
- Scalability: Verification of Aξ < 0 requires scalar operations proportional to the number of nonzero entries in A, and feasible ξ can be generated by integrating ξ̇ = Aξ.
- Transportation network: In the transportation example, each buffer verifies one inequality using only parameters at its node and neighboring nodes, without a global model.
- Vehicle formation: In the vehicle-formation example, the first and fourth vehicles can stabilize independently, while the second and third rely on distance measurements; verification remains distributed.
- Discrete time: The discrete-time counterpart characterizes Schur stability through positive vector inequalities, diagonal quadratic inequalities, and a nonnegative resolvent.
IV. INPUT-OUTPUT PERFORMANCE OF POSITIVE SYSTEMS
Positive-system input-output performance can be characterized through induced norms and state-space inequalities. Under nonnegativity, induced norms are determined by static gain, enabling tractable distributed performance analysis.
- Static-gain characterization: For positive impulse responses with Hurwitz A, induced p-norms equal the static-gain norm for p = 1, 2, and ∞; scalar systems satisfy this equality for all p ∈ [1, ∞].
- Static-gain characterization: The equality follows because nonnegative impulse responses make the frequency-domain maximum occur at zero frequency, with endpoint cases extended by convexity.
- ∞-norm performance: Theorem 4 gives equivalent state-space conditions for Hurwitz stability and induced ∞-norm gain below γ, together with bounds on all states under bounded disturbances.
- Discrete time: The continuous-time results have discrete-time counterparts that preserve state bounds and induced-performance characterizations.
V. DISTRIBUTED CONTROL SYNTHESIS BY LINEAR PROGRAMMING
The paper formulates distributed controller synthesis for positive systems as linear programs, enabling stability and performance design through local inequalities and distributed computation.
- Distributed synthesis: Linear programming replaces bilinear controller–certificate products by auxiliary variables such as µ12 := ℓ12ξ2, µ32 := ℓ32ξ2, and µ23 := ℓ23ξ3.The resulting program searches jointly over positive ξ and nonnegative µ variables.
- L∞ synthesis: Theorem 7 characterizes existence of a controller minimizing the L∞-induced gain through equivalent linear inequalities when F ≥ 0.The controller is recovered from a feasible ξ and µ satisfying µ = LFξ + LH1.
- Distributed verification: Each row of the synthesis inequalities can be verified separately, producing a distributed performance test for graph-structured matrices.The paper also describes distributed algorithms in which nodes use local variables and exchange information only with neighbors.
- L1 synthesis: Theorem 8 provides the dual formulation for minimizing the L1-induced gain under nonnegative closed-loop matrix conditions.Feasible p and q yield controllers satisfying q = LETp + LGT1.
- Vehicle formation: The vehicle-formation example uses Theorem 8 to stabilize the formation and minimize disturbance-to-state gain by selecting feedback gains ℓij ∈ [0, 1].Three disturbance scenarios are illustrated; one reported solution has (ξ1, ξ2, ξ3, ξ4) = (0.5, 0.5, 1.69, 0.87) and gains ℓ12 = 1, ℓ32 = 1, ℓ23 = 0.
VI. POSITIVELY DOMINATED SYSTEMS
The paper extends positive-system analysis from state-space models to positively dominated transfer matrices, where frequency-domain conditions support scalable performance verification and synthesis.
- Motivation: Verifying nonnegative impulse responses directly is NP-hard, motivating positively dominated systems as a tractable alternative definition.The paper cites hardness results for both discrete-time and continuous-time problems.
- Positive domination: A positively dominated transfer matrix satisfies |Gjk(iω)| ≤ Gjk(0) for every entry and all real ω.The defining frequency inequality can be tested by semidefinite programming through a sum-of-squares condition.
- Distributed synthesis: Theorem 11 gives equivalent conditions for selecting diagonal L so a positively dominated closed loop is stable and has L1-induced gain below γ.The controller can be recovered from feasible p and q satisfying q = LE(0)Tp.
- Vehicle formation with inertia: In the inertial vehicle-formation example, Theorem 11 finds optimal spring constants under conditions ensuring the relevant transfer matrices are positively dominated.The constants ℓij and ℓji are optimized separately even though symmetry requires equality at optimum.
VII. SCALABLE VERIFICATION OF THE LYAPUNOV INEQUALITY
The section shows that Lyapunov and related semidefinite conditions for positive systems can be decomposed into local tests, making verification scale with nonzero interconnections.
- Motivation: Input-output gains are extended beyond L1- and L∞-based measures to scalable verification of semidefinite inequalities such as AT P + PA ≺ 0.The goal is to decompose these tests linearly with the number of nonzero matrix entries.
- Decomposition: A symmetric Metzler matrix with m nonzero entries above the diagonal is negative semidefinite exactly when it is a sum of m negative semidefinite matrices with four nonzero entries each.This converts a global matrix property into interconnection-sized components.
- Positive quadratic programming: Positive quadratic programming replaces the relevant quadratic constraints with trace constraints over a structured matrix set without changing the optimal value.Under an interior-feasibility condition, strong duality yields a finite dual characterization using nonnegative multipliers.
- Scalability: The structured relaxation remains scalable because only entries corresponding to nonzero entries of the Metzler matrices need to be considered.The condition X ∈ X has a natural decomposition aligned with the sparsity pattern.
VIII. THE KYP LEMMA FOR POSITIVE SYSTEMS
The section develops a stronger KYP lemma for positive systems, relating frequency-domain, time-domain, diagonal-storage, and linear-programming conditions. It also applies the result to optimal power flow, where minimum resistive losses can be attained with constant voltages and currents.
- Assumptions: Strict inequalities remove the need for the stabilizability or anti-stabilizability assumption in the corresponding equivalences.A counterexample shows that stabilizability of (−A, B) is essential without this strictness condition.
- KYP lemma: The positive-system KYP result generalizes earlier versions through non-strict inequalities, more general Q, and a fourth equivalent linear-programming condition.The theorem is stated for continuous and discrete time.
- KYP lemma: The theorem establishes equivalence among frequency-domain, time-domain, and diagonal-matrix conditions under Metzler, Hurwitz, and sign-structure assumptions.The proof proceeds through the implications (14.1) ⇒ (14.2) ⇒ (14.3) ⇒ (14.1) and the equivalence (14.2) ⇔ (14.4).
- Optimal power flow: Minimal losses can be attained with constant voltages and currents after relaxing the original optimization problem and applying the positive-system theory.The relaxed problem includes the constant-variable case, so its optimum establishes the stated attainment result.
- Optimal power flow: For the power-flow application, the system models line currents as states and voltage differences as inputs while enforcing generation, load, transmission, and voltage constraints.The objective is to minimize resistive power losses in the network.
IX. CONCLUSIONS
The paper identifies scalable verification and synthesis of distributed control systems as a key benefit of positive and positively dominated systems. It also highlights open directions involving monotone nonlinear systems and dynamic-controller optimization.
- Scalable verification and synthesis provide optimal input-output performance for distributed control systems.The paper attributes these benefits to the monotonicity properties of positive and positively dominated systems.
- Linear programming solutions provide certificates for distributed verification of global optimality without requiring access to a global model.
- Extending scalable verification and synthesis methods to monotone nonlinear systems remains an open research problem.
- Extending controller optimization to scalable optimization of dynamic controllers remains an open research problem.