关于$k$ -正线性系的对角稳定性
On the Diagonal Stability of $k$-Positive Linear Systems
摘要：
我们考虑$k$ -正线性系，也就是说，系统映射的一组向量与$k-1$的符号变化本身。对于$k=1$，这简化为正线性系。众所周知，稳定的正线性时不变（LTI）系统允许一个对角李雅普诺夫函数。这一性质具有许多重要的意义。一个自然的问题是稳定的$k$ -正系统是否也允许对角李雅普诺夫函数。本文表明，在一般情况下，答案是否定的。然而，对于连续时间和离散时间$n$ -维系统是$(n-1)$ -积极的，我们提供了一个充分条件对角稳定。
英文摘要：
We consider $k$-positive linear systems, that is, systems that map the set of vectors with up to $k-1$ sign variations to itself. For $k=1$, this reduces to positive linear systems. It is well-known that stable positive linear time invariant (LTI) systems admit a diagonal Lyapunov function. This property has many important implications. A natural question is whether stable $k$-positive systemsalso admit a diagonal Lyapunov function. This paper shows that, in general, the answer is no. However, for both continuous-time and discrete-time $n$-dimensional systems that are $(n-1)$-positive we provide a sufficient condition for diagonal stability.
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