Stability Properties of the Impulsive Goodwin's Oscillator in 1-cycle

TL;DR

The paper tackles stability analysis for the simplest periodic solution (the 1-cycle) of the Impulsive Goodwin's Oscillator (IGO), a hybrid system with pulse-modulated, amplitude- and frequency-dependent feedback. It reduces the hybrid dynamics to a discrete-time map $Q(X)=\mathrm{e}^{A\,\Phi(CX)}(X+F(CX)B)$ and shows that a 1-cycle corresponds to a fixed point $X=Q(X)$ with period $T=\Phi(y_0)$ and impulse weight $\lambda=F(y_0)$. The main contribution is a necessary-and-sufficient linear stability criterion: for $0<a_1<a_2<a_3$, if $F'(y_0)\le 0$ and $\Phi'(y_0)\ge 0$, then $Q'(X)$ is Schur stable iff $\det(-I - Q'(X))<0$, equivalently $ $C(I+e^{\Phi(y_0)A})^{-1}(F'(y_0)J+\Phi'(y_0)D)>-1$, where $J=e^{AT}B$ and $D=AX$; also, a positive real eigenvalue lies in $[e^{-a_3T}, e^{-a_1T}]$. Numerical examples with drug-dosing-inspired parameters illustrate how the stability boundary governs convergence to the 1-cycle and how unstable 1-cycles can lead to higher-period attractors (e.g., 2-cycles) while remaining near the output corridor. Overall, the work provides a practical, linear-in-slopes stability test to guide impulsive controller design in IGO-like systems and highlights the role of negative feedback encoded via impulsive modulation.

Abstract

The Impulsive Goodwin's Oscillator (IGO) is a mathematical model of a hybrid closed-loop system. It arises by closing a special kind of continuous linear positive time-invariant system with impulsive feedback, which employs both amplitude and frequency pulse modulation. The structure of IGO precludes the existence of equilibria, and all its solutions are oscillatory. With its origin in mathematical biology, the IGO also presents a control paradigm useful in a wide range of applications, in particular dosing of chemicals and medicines. Since the pulse modulation feedback mechanism introduces significant nonlinearity and non-smoothness in the closedloop dynamics, conventional controller design methods fail to apply. However, the hybrid dynamics of IGO reduce to a nonlinear, time-invariant discrete-time system, exhibiting a one-to-one correspondence between periodic solutions of the original IGO and those of the discrete-time system. The paper proposes a design approach that leverages the linearization of the equivalent discrete-time dynamics in the vicinity of a fixed point. A simple and efficient local stability condition of the 1-cycle in terms of the characteristics of the amplitude and frequency modulation functions is obtained.

Figures (5)

• Figure 1: Spectral radius of $Q^\prime(X)$ and condition \ref{['eq.stability-slopes']} as function of $\Phi^\prime(y_0)$ and $F^\prime(y_0)$. Red line -- stability border \ref{['eq.stability-slopes']}. Blue dot -- stable 1-cycle with $F^\prime(y_0)=-1$, $\Phi^\prime(y_0)=4$. Red dot -- unstable 1-cycle with $F^\prime(y_0)=-1$, $\Phi^\prime(y_0)=5.5$.
• Figure 2: The 1-cycle with $\lambda=300, T=20$ stabilized by the modulation function slopes $F^\prime(y_0)=-1$, $\Phi^\prime(y_0)=4$. The initial condition on the continuous block is $x(0)=X$. Output corridor values for the linear and nonlinear output are marked.
• Figure 3: The 2-cycle with $\lambda=300, T=20$ stabilized by the modulation function slopes $F^\prime(y_0)=-1$, $\Phi^\prime(y_0)=5.5$. The initial condition on the continuous block is $x(0)\ne X$. Output corridor values for the linear and nonlinear output in the designed 1-cycle are marked.
• Figure 4: The 1-cycle with $\lambda=300, T=20$ stabilized by the modulation function slopes $F^\prime(y_0)=-0.1$, $\Phi^\prime(y_0)=0.29$. The initial condition on the continuous block is far from the fixed point $X$. The Jacobian eigenvalues are $\sigma(Q^\prime(X))=\{0.2348, 0.1814, 0.0003\}$. Output corridor values for the linear and nonlinear output are marked.
• Figure 5: The 1-cycle with $\lambda=300, T=20$ stabilized by the modulation function slopes $F^\prime(y_0)=-1$, $\Phi^\prime(y_0)=4$. The initial condition on the continuous block is far from the fixed point $X$. The Jacobian eigenvalues are $\sigma(Q^\prime(X))=\{ -0.4757 \pm 0.2343i, -0.0000\}$. The spectral radius is $\rho(Q^\prime(X))=0.5302$. Output corridor values for the linear and nonlinear output are marked.

Theorems & Definitions (5)

• Definition 1
• Proposition 1
• Theorem 1
• Lemma 1
• Corollary 1