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Design of the Impulsive Goodwin's Oscillator: A Case Study

Alexander Medvedev, Anton V. Proskurnikov, Zhanybai T. Zhusubaliyev

TL;DR

The paper addresses how to design an Impulsive Goodwin's Oscillator (IGO) to admit a predefined $1$-cycle by treating the continuous plant as known and the impulsive feedback parameters as design degrees of freedom, with the impulse-to-impulse map $Q$ yielding a unique positive fixed point that defines the cycle. It develops an analytical framework using the Opitz formula and divided differences to compute matrix functions, and shows that the stability of the cycle can be achieved through a static-output-feedback–like design by tuning the slopes $F'(z_0)$ and $\Phi'(z_0)$, ensuring a Schur-stable Jacobian $Q'(X)$. A concrete design algorithm is proposed to specify $(\lambda,T)$, compute the fixed point $X$ and $z_0$, shape the modulation functions, and verify stability while enforcing $F(z_0)=\lambda$ and $\Phi(z_0)=T$. A numerical example demonstrates the procedure, while bifurcation analysis reveals how the designed $1$-cycle evolves under parameter variations, including period-doubling; the work lays the groundwork for dosing applications in chemical and biomedical contexts by providing a principled method to realize robust, predefined periodic behavior.

Abstract

The impulsive Goodwin's oscillator (IGO) is a hybrid model composed of a third-order continuous linear part and a pulse-modulated feedback. This paper introduces a design problem of the IGO to admit a desired periodic solution. The dynamics of the continuous states represent the plant to be controlled, whereas the parameters of the impulsive feedback constitute design degrees of freedom. The design objective is to select the free parameters so that the IGO exhibits a stable 1-cycle with desired characteristics. The impulse-to-impulse map of the oscillator is demonstrated to always possess a positive fixed point that corresponds to the desired periodic solution; the closed-form expressions to evaluate this fixed point are provided. Necessary and sufficient conditions for orbital stability of the 1-cycle are presented in terms of the oscillator parameters and exhibit similarity to the problem of static output control. An IGO design procedure is proposed and validated by simulation. The nonlinear dynamics of the designed IGO are reviewed by means of bifurcation analysis. Applications of the design procedure to dosing problems in chemical industry and biomedicine are envisioned.

Design of the Impulsive Goodwin's Oscillator: A Case Study

TL;DR

The paper addresses how to design an Impulsive Goodwin's Oscillator (IGO) to admit a predefined -cycle by treating the continuous plant as known and the impulsive feedback parameters as design degrees of freedom, with the impulse-to-impulse map yielding a unique positive fixed point that defines the cycle. It develops an analytical framework using the Opitz formula and divided differences to compute matrix functions, and shows that the stability of the cycle can be achieved through a static-output-feedback–like design by tuning the slopes and , ensuring a Schur-stable Jacobian . A concrete design algorithm is proposed to specify , compute the fixed point and , shape the modulation functions, and verify stability while enforcing and . A numerical example demonstrates the procedure, while bifurcation analysis reveals how the designed -cycle evolves under parameter variations, including period-doubling; the work lays the groundwork for dosing applications in chemical and biomedical contexts by providing a principled method to realize robust, predefined periodic behavior.

Abstract

The impulsive Goodwin's oscillator (IGO) is a hybrid model composed of a third-order continuous linear part and a pulse-modulated feedback. This paper introduces a design problem of the IGO to admit a desired periodic solution. The dynamics of the continuous states represent the plant to be controlled, whereas the parameters of the impulsive feedback constitute design degrees of freedom. The design objective is to select the free parameters so that the IGO exhibits a stable 1-cycle with desired characteristics. The impulse-to-impulse map of the oscillator is demonstrated to always possess a positive fixed point that corresponds to the desired periodic solution; the closed-form expressions to evaluate this fixed point are provided. Necessary and sufficient conditions for orbital stability of the 1-cycle are presented in terms of the oscillator parameters and exhibit similarity to the problem of static output control. An IGO design procedure is proposed and validated by simulation. The nonlinear dynamics of the designed IGO are reviewed by means of bifurcation analysis. Applications of the design procedure to dosing problems in chemical industry and biomedicine are envisioned.
Paper Structure (12 sections, 5 theorems, 49 equations, 5 figures)

This paper contains 12 sections, 5 theorems, 49 equations, 5 figures.

Table of Contents

  1. INTRODUCTION
  2. BACKGROUND
  3. The Impulsive Goodwin's Oscillator
  4. Solution Properties
  5. DESIGN
  6. Divided differences and the Opitz formula
  7. Fixed point
  8. Stability of $1$-cycle
  9. Design algorithm
  10. DESIGN EXAMPLE
  11. BIFURCATION ANALYSIS
  12. CONCLUSIONS

Key Result

Proposition 1

System eq:1, eq:2 has one and only one (positive) $1$-cycle, that is, eq:1-cycle has a unique solution $X>0$. The cycle parameters $\lambda$, $T$, and $z_0$ can be evaluated by solving the following system of algebraic equations The key idea of proving Theorem th:1-cycle in Aut09 is to rewrite eq:1-cycle in terms of the output variable $z=CX=x_3$ as which equation is subsequently reduced to the

Figures (5)

  • Figure 1: The plot of function $\nu(x)$ for $x<0$.
  • Figure 2: The designed 1-cycle ($\Gamma$, in red) corresponding to the fixed point ($\mathcal{O}=X$). Trajectories converging to $\Gamma$ are in blue.
  • Figure 3: The convergence of the sequence $F(z_k)$ to the desired $\lambda$. Since all the multipliers are negative $-1<\rho_i<0$, $1\leqslant i\leqslant 3$, the convergence is non-monotonous. To highlight the evolution, the point $F(z_{k-1})$ is connected to the next one $F(z_{k})$ (blue lines).
  • Figure 4: Bifurcation analysis: (a),(b) - for $T=66.7502$, $\lambda = 4.66$, (c),(d) - for $T=65.4542$, $\lambda=4.7273$.
  • Figure 5: (a) Variation of $\tau$ and $\rho_2$ on $-0.6<F'<0.0$ and $\Phi'=-\dfrac{k_2}{k_4}F'$ for $a_3=0.3005$0. (b) Variation of $\tau$ and $\rho_2$ on $-0.6<F'<0.0$, $\Phi'=-\dfrac{k_2}{k_4}F'$ for $a_3=0.2505$0. Here 1 denotes the convergence time $\tau$ and 2 marks $\rho_2$. $T=66.7502$, $\lambda=4.6625$.

Theorems & Definitions (8)

  • Proposition 1: Aut09
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Remark 1
  • Lemma 1: Theorem 3.1, FGL98
  • Proposition 4