Singular Perturbation Theory for a Finite-Dimensional, Discrete-Time Chi Nonlinear System

TL;DR

This paper develops a singular perturbation theory tailored for a specific discrete-time nonlinear system modeling a digitally controlled buck converter with saturation effects.

Contribution

It introduces a novel singular perturbation framework for analyzing a finite-dimensional nonlinear system in digital power electronics.

Findings

Provides a new theoretical approach for system analysis.

Enhances understanding of digital control in power converters.

Lays groundwork for improved control strategies.

Abstract

This paper develops the singular perturbation theory for a particular discrete-time nonlinear system which models the saturating inductor buck converter using cycle-by-cycle digital control.

Equations

$$x[n+1]$$

$$z[n+1]$$

$$x_s[n+1]$$

$$z_s[n]$$

$$y[n+1]=$$

$$+h(x_s[n]),0)-h(x_s[n]).$$

$$\Vert x[n]-x_s[n]\Vert \le \lambda \mu .$$

$$\Vert x[1]-x_s[1]\Vert =\Vert f(\mu z_0)\Vert \le L\Vert z_0\Vert \mu .$$

$$\Vert x[k+1]-x_s[k+1]\Vert$$

$$=$$

$$\le$$

$$=$$

$$\Vert x[n]-x_s[n]\Vert =O(\mu ).$$

$$\Vert z[n]-h(x_s[n])-y[n]\Vert$$

$$=$$

$$-g(x_s[n-1],y[n-1]$$

$$+h(x_s[n-1]),0)+h(x_s[n-1])\Vert$$

$$\le$$

$$-g(x[n-1],z[n-1],0)\Vert$$

$$+\Vert g(x[n-1],z[n-1],0)-g(x_s[n-1],z[n-1],0)\Vert$$

$$+\Vert h(x_s[n-1])-h(x_s[n])\Vert$$

$$\le$$

$$+L_h{L}_f\mu \Vert x_s[n-1]\Vert$$

$$\le$$

$$+L_g\Vert x[n-1]-x_s[n-1]\Vert +L_h{L}_f\mu \Vert x_s[n-1]\Vert$$

$$\le$$

$$z[n,\mu ]-h(x_s[n])-y[n]=O(\mu ).$$

$$z[n]-h(x[n])=O(\mu )$$

$$\Vert y[n]\Vert \le \Vert y[n_1]\Vert =ϵ\Vert y[0]\Vert e^{\theta n_1}=ϵ\Vert y[0]\Vert \mu .$$

$${\gamma }_1{\Vert x\Vert }^2\le V(x)\le {\gamma }_2{\Vert x\Vert }^2,$$

$$V(x[n+1])\le {\sigma }_{1}V(x[n]),$$

$${\gamma }_3{\Vert y\Vert }^2\le W(y)\le {\gamma }_4{\Vert y\Vert }^2,$$

$$W(y[n+1])\le {\sigma }_{2}W(y[n]),$$

$$\text{min}\{{\gamma }_1,{\gamma }_3\}{\Vert [x,y]\Vert }^2\le \nu (x,y)\le \text{max}\{{\gamma }_2,{\gamma }_4\}{\Vert [x,y]\Vert }^2,$$

$$\nu (x[n+1],y[n+1])\le \text{max}\{{\sigma }_1,{\sigma }_2\}\nu (x[n],y[n]).$$

Full text

††publicationid: pubid:

I Introduction

Singular perturbation theory [1] is a well-known method for studying nonlinear systems with two well-separated time scales. The $\chi$ nonlinear system is a finite-dimensional, discrete-time nonlinear system which can be used to model a broad class of power electronics systems.

Although there exist several discrete versions of singular perturbation theory in the literature [2, 3, 4, 5, 6], to the best of authors’ knowledge, there is no such theorem which can be directly applied to the $\chi$ nonlinear system. Therefore, in this paper, we establish singular perturbation theory for the $\chi$ nonlinear system.

II System description

The $\chi$ nonlinear system is a finite-dimensional, discrete-time nonlinear system whose state-space representation can be expressed as

[TABLE]

where $\mu$ is called perturbation parameter.

Direct quote from [7]:

We assume the following assumptions hold for all $(n,x,z)\in[0,\infty)\times D_{x}\times D_{z}$ containing the origin for some domain $D_{x}\subset\mathbb{R}^{n}$ and $D_{z}\subset\mathbb{R}^{m}$:

(a) the matrix $\partial g/\partial z-I_{m}$ is invertible;

(b) the function $f$ and $g$ are Lipschitz in $(x,z)$ with Lipschitz constant $L_{f}$ and $L_{g}$;

(c) $f(0)=0$, $g(0,0,0)=0$.

From assumption (a) above and the implicit function theorem [8], the equation $z=g(x,z,0)$ has explicit solution $z=h(x)$. We assume the function $h$ is Lipschitz in $x$ with Lipschitz constant $L_{h}$. We define the reduced model by

[TABLE]

The reduced model describes trajectories of $x$ and $z$ which an observer sees in the slow time frame when $\mu$ approaches 0. We define the boundary-layer model by

[TABLE]

where $y[n]=z_{f}[n]-z_{s}[n]$. The $z_{f}[n]$ is the trajectory of $z$ which an observer sees in the fast time frame when $\mu$ approaches 0. The boundary-layer model describes the difference between the trajectory of $z$ which an observer sees in the fast time frame and that in the slow time frame. We note that $y(n)=0$ is a solution for the boundary-layer model.

III Theory

The following Theorem 1 shows the relationship between the trajectory of the original system and that of the reduced model as well as the boundary-layer model.

Direct quote from [7]:

Theorem 1**.**

If $x_{s}=0$ is an exponentially stable equilibrium of system (3) and $y=0$ is an exponentially stable equilibrium of system (II), uniform in $x_{s}$, then there exists a positive constant $\mu^{*}$ such that for all $n\geq 0$ and $0<\mu\leq\mu^{*}$, the singular perturbation problem (1) and (2) has a unique solution $x[n,\mu]$, $z[n,\mu]$ on $[0,\infty)$, and $x[n,\mu]-x_{s}[n,\mu]=O(\mu),z[n,\mu]-h(x_{s}[n,\mu])-y[n]=O(\mu)$ hold uniformly for $n\in[0,\infty)$, where $x_{s}[n,\mu]$ and $y[n]$ are the solutions of the system (3) and system (II). Furthermore, there exists $n_{1}>0$, such that $z[n,\mu]-h(x_{s}[n])=O(\mu)$ holds uniformly for $n\in[n_{1},\infty)$.

Proof.

We use mathematic induction to prove 111In this paper, we assume the 2-norm $\|\cdot\|=\|\cdot\|_{2}$.

[TABLE]

Equation (6) holds when $n=1$ because

[TABLE]

We suppose that there exists $\lambda$ satisfying $\mathinner{\!\left\lVert x[n]-x_{s}[n]\right\rVert}\leq\lambda\mu$ when $n=k$. The following derivations prove (6) when $n=k+1$:

[TABLE]

By defining $\lambda^{*}=(\lambda+L_{f}\beta)$, we show $\mathinner{\!\left\lVert x[k+1]-x_{s}[k+1]\right\rVert}=O(\mu)$.

By mathematic induction, we conclude that

[TABLE]

Then we prove $z[n,\mu]-h(x_{s}[n])-y[n]=O(\mu)$.

[TABLE]

We conclude that

[TABLE]

Finally, we prove that there exists $n_{1}>0$, such that

[TABLE]

holds uniformly for $n\in[n_{1},\infty)$.

Let $n_{1}=-\text{ln}(\mu)/\theta$, for all $n>n_{1}$,

[TABLE]

We proved $\mathinner{\!\left\lVert y[n]\right\rVert}=O(\mu)$ for all $n>n_{1}$. Therefore $z[n]-h(x[n])=O(\mu)$ holds uniformly for $n\in[n_{1},\infty)$. ∎

The following Theorem 2 shows the relationship between the stability of the original system and that of the reduced model as well as the boundary-layer model.

Theorem 2**.**

There exists $\mu^{*}>0$ such that for all $\mu\leq\mu^{*}$, then $x=0$, $z=0$ is an exponentially stable equilibrium of the singular perturbation problem (1) and (2).

Proof.

The exponential stability of the $x=0$ of system (3) is equivalent to

[TABLE]

where $\gamma_{1}>0$, $\gamma_{2}>0$, $0<\sigma_{1}<1$. The exponential stability of the equilibrium $y=0$ of system (II) requires the following equations to hold uniformly in $x$

[TABLE]

where $\gamma_{3}>0$, $\gamma_{4}>0$, $0<\sigma_{2}<1$.

We use $\nu=V+W$ as a Lyapunov function candidate for the system (3) and (II). From (14), (15), (16) and (17)

[TABLE]

From (18) and (19), the equilibrium $[x,y]=0$ of the systems (3) and (II) is exponentially stable. Considering Theorem 1, we can conclude that the equilibrium $[x,y]=0$ of the systems (1) and (2) is exponentially stable. A more rigorous proof can be performed by following the same methods as the proofs of Proposition 8.1 and Proposition 8.2 in [9]. ∎

IV Conclusion

The theoretical contribution of this paper is developing singular perturbation theory for the $\chi$ nonlinear system.