Harmonic Modeling and Control

Abstract

Harmonic modeling involves transforming a periodic system into an equivalent time-invariant model of infinite dimension. The states of this model, also referred to as phasors, are represented by coefficients obtained through a sliding Fourier decomposition process. This chapter aims to present a unified and coherent mathematical framework for harmonic modeling and control. By adopting this framework, the analysis and design processes become significantly simplified, as all the methods established for time-invariant systems can be directly applied. Within this framework, we explore the application of these methods to tackle the task of designing control laws based on harmonic pole placement for Linear Time-Periodic (LTP) systems. Additionally, we delve into the computational aspects associated with these control designs.

Appendix

We provide here some results concerning operator norms used in this chapter. The missing proofs can be found in [10] (Part V, pp. 562–574).

The norm of an operator M from $\ell ^p$ to $\ell ^q$ is given by

$$\begin{aligned}\Vert M\Vert _{\ell ^p,\ell ^q}:=\sup _{\Vert X\Vert _{\ell ^p}=1}\Vert MX \Vert _{\ell ^q}.\end{aligned}$$

This operator norm is sub-multiplicative, i.e., if $M: \ell ^p \rightarrow \ell ^q$ and $N: \ell ^q \rightarrow \ell ^r$ then $\Vert NM\Vert _{\ell ^p,\ell ^r} \le \Vert M\Vert _{\ell ^p,\ell ^q} \Vert N\Vert _{\ell ^q,\ell ^r}$. If $p=q$, we use the notation: $\Vert M\Vert _{\ell ^p}:=\Vert M\Vert _{\ell ^p,\ell ^p}$.

Definition 13.6.1

Consider a vector $x(t)\in L^2([0 \ T],\mathbb {C}^n)$ and define $X:=\mathcal {F}(x)$ with its symbol X(z). The $\ell ^2 -$norm of X(z) is given by

$$\begin{aligned}\Vert X(z)\Vert _{\ell ^2}:=\Vert X\Vert _{\ell ^2},\end{aligned}$$

where $\Vert X\Vert _{\ell ^2}:=\left( \sum _{k\in \mathbb {Z}}|X_k|^2\right) ^{\frac{1}{2}}$.

Theorem 13.6.2

Let $A\in L^2([0 \ T],\mathbb {C}^{n\times m})$. Then, $\mathcal {A}:=\mathcal {T}(A)$ is a bounded operator on $\ell ^2$ if and only if $A\in L^{\infty }([0\ T],\mathbb {C}^{n\times m} )$. Moreover, we have

• 1. the operator norm induced by the $\ell ^2$-norm satisfies

  $$\begin{aligned}\Vert A(z)\Vert _{\ell ^2}=\Vert \mathcal {A}\Vert _{\ell ^2}=\Vert A\Vert _{L^{\infty }}\end{aligned}$$

• 2. the operator norm of the semi-infinite Toeplitz matrix satisfies: $\Vert \mathcal {T}_s(A)\Vert _{\ell ^2}=\Vert \mathcal {A}\Vert _{\ell ^2}$

• 3. the operator norm of the Hankel operators $\mathcal {H}(A^+)$, $\mathcal {H}(A^-)$ satisfies $\Vert \mathcal {H}(A^-)\Vert _{\ell ^2}\le \Vert A\Vert _{L^{\infty }}$ and $\Vert \mathcal {H}(A^+)\Vert _{\ell ^2}\le \Vert A\Vert _{L^{\infty }}$

• 4. the operator norm related to the left and right $m-$truncations satisfies: $\Vert \mathcal {A}_{m^+}\Vert _{\ell ^2}=\Vert \mathcal {A}_{m^-}\Vert _{\ell ^2}=\Vert \mathcal {A}\Vert _{\ell ^2}=\Vert A\Vert _{L^\infty }$.

Proposition 13.6.3

Let $P(\cdot )$ be a matrix function in $L^\infty ([0 \ T],\mathbb {C}^{n\times n})$. Define $\textbf{P}:=\mathcal {F}(P)$ and $\mathcal {P}:=\mathcal {T}(P)$. If $\Vert \textrm{col}(\textbf{P})\Vert _{\ell ^2}\le \epsilon$ then $\Vert \mathcal {P}\Vert _{\ell ^2}\le \epsilon$.

Proof

Using Riesz–Fischer theorem, we have

$$\begin{aligned} \Vert \textrm{col}(\textbf{P})\Vert _{\ell ^2}&=\Vert \textrm{col}( P)\Vert _{L^2}=(\sum _{i,j=1}^n\Vert P_{ij}\Vert ^2_{L^2})^{1/2}=\Vert P\Vert _F, \end{aligned}$$

where $\Vert P(t)\Vert _F$ stands for the Frobenius norm. As $P \in L^\infty ([0 \ T],\mathbb {C}^{n\times n})$, H$\ddot{\text{ o }}$lder’s inequality implies $Px\in L^2([0 \ T],\mathbb {C}^{n})$ for any $x\in L^2([0 \ T],\mathbb {C}^{n})$. Thus, the result follows from the following relations between operator norms:

$$\begin{aligned} \Vert \mathcal {P}\Vert _{\ell ^2}&=\sup _{\Vert X\Vert _{\ell ^2}=1}(<\mathcal {P}X,\mathcal {P}X>_{\ell ^2})^{1/2}\\ &=\sup _{\Vert x\Vert _{L^2}=1}(<Px,Px)>_{L^2})^{1/2}\\ &\le (\text {trace}( P^*P))^{1/2}= \Vert P\Vert _F, \end{aligned}$$

where $<\cdot ,\cdot >$ stands for the scalar product.    $\square$

Theorem 13.6.4

Let $A(t)\in L^\infty ([0 \ T],\mathbb {C}^{n\times n})$. $\mathcal {A}$ is invertible if and only if there exists $\gamma >0$ such that the set $\{t: |\det (A(t))|<\gamma \}$ has measure zero. The inverse $\mathcal {A}^{-1}$ is determined by $\mathcal {T}(A^{-1})$. In addition, $\mathcal {A}$ is invertible if and only if $\mathcal {A}$ is a Fredholm operator [10], or equivalently in this setting if and only if there exists $c > 0$ such that

$$\begin{aligned}\Vert \mathcal {A}x\Vert _{\ell ^2} > c \Vert x\Vert _{\ell ^2},\text { for any } x \in \ell ^2.\end{aligned}$$