[Paper Review] Diagonal Stability of Discrete-time $k$-Positive linear Systems with Applications to Nonlinear Systems

Chengshuai Wu, Michael Margaliot|arXiv (Cornell University)|Feb 3, 2021

Control and Stability of Dynamical Systems24 references4 citations

TL;DR

This paper introduces discrete-time $k$-diagonal stability as a necessary condition for diagonal stability in $k$-positive linear systems, generalizing the well-known result that stable positive systems (k=1) are diagonally stable. It demonstrates that this new notion enables stability analysis of a class of discrete-time nonlinear systems via wedge products and compound matrices, extending prior results beyond the $k=1$ case.

ABSTRACT

A linear dynamical system is called $k$-positive if its dynamics maps the set of vectors with up to $k-1$ sign variations to itself. For $k=1$, this reduces to the important class of positive linear systems. Since stable positive linear time-invariant (LTI) systems always admit a diagonal quadratic Lyapunov function, i.e. they are diagonally stable, we may expect that this holds also for stable $k$-positive systems. We show that, in general, this is not the case both in the continuous-time (CT) and discrete-time (DT) case. We then focus on DT $k$-positive linear systems and introduce the new notion of DT $k$-diagonal stability. It is shown that this is a necessary condition for standard DT diagonal stability. We demonstrate an application of this new notion to the analysis of a class of DT nonlinear systems.

Motivation & Objective

• To investigate whether stable $k$-positive linear time-invariant (LTI) systems are diagonally stable, generalizing the known result for $k=1$ (positive systems).
• To define and analyze a new stability concept—discrete-time $k$-diagonal stability—for $k$-positive systems.
• To establish a geometric framework using wedge products and compound matrices to analyze the asymptotic behavior of nonlinear systems with cyclic dynamics.
• To demonstrate that $k$-diagonal stability ensures global asymptotic stability of a class of discrete-time nonlinear systems, extending the classical diagonal stability approach.

Proposed method

• Introduce the notion of $k$-positive systems as those mapping vectors with at most $k-1$ sign variations into the same set.
• Define discrete-time $k$-diagonal stability via the existence of a diagonal matrix $D \succ 0$ such that $(A^{(k)})^T D A^{(k)} \prec D$, where $A^{(k)}$ is the $k$th compound matrix of $A$.
• Use wedge products to represent the $k$-dimensional volume (content) of parallelepipeds spanned by $k$ state vectors.
• Construct a Lyapunov function $V(y(j)) = y(j)^T D y(j)$, where $y(j) = \wedge_{i=1}^k x(j, a^i)$, to analyze the evolution of $k$-volumes.
• Prove that if $A$ is $k$-diagonally stable, then $V(y(j))$ decreases monotonically, implying convergence of $k$-volumes to zero.
• Apply the framework to discrete-time nonlinear systems of the form $x(j+1) = A\phi(x(j))$, showing global asymptotic stability under $k$-diagonal stability and $\ell$-content preserving nonlinearity.

Experimental results

Research questions

• RQ1 Is every stable $k$-positive discrete-time LTI system diagonally stable, generalizing the $k=1$ case where stable positive systems are always diagonally stable?
• RQ2 What is the appropriate generalization of diagonal stability for $k$-positive systems, and how does it relate to standard diagonal stability?
• RQ3 Can the geometric structure of $k$-volumes (via wedge products) be used to analyze the stability of nonlinear systems with cyclic dynamics?
• RQ4 Under what conditions does $k$-diagonal stability of a matrix $A$ imply global asymptotic stability of the associated nonlinear system $x(j+1) = A\phi(x(j))$?
• RQ5 How does the compound matrix $A^{(k)}$ and its spectral properties relate to the stability of the $k$-volume dynamics?

Key findings

• Stable $k$-positive systems for $k>1$ are not necessarily diagonally stable in discrete time, showing that the $k=1$ result does not generalize directly.
• The paper introduces discrete-time $k$-diagonal stability as a necessary condition for standard diagonal stability in $k$-positive systems.
• For a class of discrete-time nonlinear systems with $\ell$-content preserving nonlinearities, $k$-diagonal stability of $A$ ensures global asymptotic stability of the system.
• The $k$-volume of any $k$-tuple of trajectories converges to zero under $k$-diagonal stability, implying no nontrivial limit cycles and convergence to a line in the state space.
• The method is demonstrated numerically in Example 4, where $V(y(j)) = y(j)^T D y(j)$ decreases over time despite $A$ not being Schur, due to $A^{(2)}$ being Schur and $A$ being $SR_2$ with $\epsilon_2 = 1$.
• The framework provides a geometric interpretation using wedge products, showing that the dynamics of $k$-volumes evolves according to the compound matrix $A^{(k)}$, enabling stability analysis through $A^{(k)}$'s spectral properties.