Control & dynamics
29 Aug 2023

Linear System Theory （Part II, Stability)

INTRODUCTION TO LINEAR SYSTEM THEORY

Keypoints:

This file is based on ME 530.616 Introduction to Linear Systems Theory by Prof. Louis Whitcomb

Stability: Basic

  • Convergence(Limit) of a sequence: Given an infinite sequence ${x_k}$, the limit of the sequence is $x^{\ast}$ if and only if $\forall \epsilon>0$, $\exists N\in \mathbb{N}$ such that $\forall k>N$, $|x_k-x^*|<\epsilon$.

  • Stability of linearized system: the real parts $\lambda_i$ of the eigenvalues of $A$ determine the stability of the linearized system.
    • All negative: asymptotically stable
    • Some zero: indeterminate
      • $Re(\lambda_i)=0$, and algebraic multiplicity=geometric multiplicity: marginally stable
    • Any positive: unstable

  • Lyapunov stability: Given a system $\dot{x}=f(x(t)), x(t_0)=x_0$, the origin is Lyapunov stable if and only if $\forall \epsilon>0$, $\exists \delta>0$ such that $\forall x_0\in \mathbb{C}^n$, $|x_0|<\delta$ implies $|x(t)|<\epsilon$ for all $t\geq 0$.

  • Asymptotic stability for LTI systems:
    • Definition: Given an LTI system with zero-input ($u(t)=0$), $\dot{x}=Ax$, the origin is asymptotically stable if and only if $\forall x_0\in \mathbb{C}^n$, the response $x(t)$ satisfies $\lim_{t\rightarrow \infty}x(t)=0$.
    • Theorem: The origin is asymptotically stable if and only if all eigenvalues of $A$ have negative real parts, i.e. $\Re(\sigma(A))<0$. iamge


  • Exponentially stable: The system $\dot{x}=f(x(t)), x(t_0)=x_0$ is exponentially stable if and only if $\exists \alpha>1$ and $\beta>0$ such that $|x(t)|\leq \alpha |x_0|e^{-\beta t}$, $\forall x_0\in \mathbb{C}^n$ and $\forall t\geq 0$. iamge

Stability: Linear Lyapunov Theorem

  • Linear Lyapunov Stability Theorem:
    • (a) Given $A\in \mathbb{R}^{n\times n}$, and two positive definite symmetric matrices $P\in \mathbb{R}^{n\times n}$ and $Q\in \mathbb{R}^{n\times n}$, such that $A^TP+PA=-Q$, then $Re(\sigma(A))<0$.
    • (b) If $Re(\sigma(A))<0$, then there exists a positive definite symmetric matrix $P\in \mathbb{R}^{n\times n}$ such that $A^TP+PA=-Q$, given by $Q=\int_0^\infty e^{A^T\tau}Me^{A\tau}d\tau$.

Stability: Uniform Bounded Input-Bounded Output Stability (UBIBOS)

  • The supremum/least upper bound of $u(t): \mathbb{R}\rightarrow \mathbb{R}$, written $v=\sup u(t)$, is the smallest finite number $v$ such that $u(t)\leq v$ for all $t\in \mathbb{R}$. If no such $v$ exists, then $v=\infty$.

  • The state equation with input $u(t)$ and output $y(t)$ is UNIFORMLY BOUNDED INPUT-BOUNDED OUTPUT STABLE (UBIBOS) if $\exists \eta>0$ such that for any input $u(t)$, the zero-initial condition $x(0)=0$ output satisfies $\sup \vert y(t) \vert \leq \eta \sup \vert u(t)\vert$.

    Note: Bound is independent of the shape of the input $u(t)$, depend only on $\sup \vert u(t)\vert$.

  • UBIBOS absolute integrablity theorem: Given a system $\dot{x}=Ax+Bu$, $y=Cx+Du$, the system is UBIBOS if and only if $Ce^{At}B$ is absolutely integrable, i.e. $\int_0^\infty \vert Ce^{At}B\vert dt<\infty$.

  • UBIBOS Transfer function: Given a system $\dot{x}=Ax+Bu$, $y=Cx+Du$, the system is UBIBOS if and only if all poles of the transfer function $G(s)=C(sI-A)^{-1}B+D$ have negative real parts.
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