An Optimization Framework for the Design of Noise Shaping Loop Filters with Improved Stability Properties

Abstract

A framework using semidefinite programming is proposed to enable the design of sigma delta modulator loop filters at the transfer function level. Both continuous-time and discrete-time, low-pass and band-pass designs are supported. For performance, we use the recently popularized Generalized Kalman–Yakubovič–Popov (GKYP) lemma to place constraints on the $\mathcal {H}_\infty$ norm of the noise transfer function (NTF) in the frequency band of interest. We expand the approach to incorporate common stability criteria in the form of $\mathcal {H}_2$ and $\ell _1$ norm NTF constraints. Furthering the discussion of stability, we introduce techniques from control systems to improve the robustness of the feedback system over a range of quantizer gains. The performance-stability trade-off is examined using this framework and motivated by simulation results.

Derivation of Matrix Inequalities with One Non-Convex Term

1.1 Derivation of GKYP Inequality with Arbitrary $\mathcal {D}$

Theorem A.1

Equation (6) from Sect. 3.1 is equivalent to the following:

$$\begin{aligned} \begin{bmatrix} -\varXi _{11} + aa^{\mathrm{T}} &{}\quad -\varXi _{12} + a &{}\quad -\mathcal {C}_q^{\mathrm{T}} - a\mathcal {D}_{qp}^{\mathrm{T}} \\ -\varXi _{12}^{\mathrm{T}} + a^{\mathrm{T}} &{}\quad -\varXi _{22} + 1 &{}\quad -\mathcal {D}_{qp}^{\mathrm{T}} \\ -\mathcal {C}_q - a^{\mathrm{T}}\mathcal {D}_{qp} &{}\quad -\mathcal {D}_{qp} &{}\quad \gamma _\infty \end{bmatrix} \ge 0 \end{aligned}$$

(28)

where (28) contains just one nonlinear term in variable a, and:

$$\begin{aligned} \begin{bmatrix} \varXi _{11} &{}\quad \varXi _{12} \\ \varXi _{12}^{\mathrm{T}} &{}\quad \varXi _{22} \end{bmatrix}= & {} \begin{bmatrix} I &{}\quad a \\ 0 &{}\quad 1 \end{bmatrix} \begin{bmatrix} \mathcal {A} &{}\quad \mathcal {B}_p \\ I &{}\quad 0 \end{bmatrix}^{\mathrm{T}} \left( \varPhi \oplus P_\gamma + \varPsi \oplus Q_\gamma \right) \begin{bmatrix} \mathcal {A} &{}\quad \mathcal {B}_p \\ I &{}\quad 0 \end{bmatrix} \begin{bmatrix} I &{}\quad a \\ 0 &{}\quad 1 \end{bmatrix}^{\mathrm{T}} \nonumber \\ P_\gamma= & {} \gamma _\infty ^{-1}P \qquad \qquad Q_\gamma = \gamma _\infty ^{-1}Q. \end{aligned}$$

(29)

Proof

Starting from (6), we follow the procedure mentioned in Sect. 3.4.2 to eliminate non-convex products in the first term of the QMI [16, Th. 1]:

$$\begin{aligned} -\begin{bmatrix} I &{}\quad a \\ 0 &{}\quad 1 \end{bmatrix} \begin{bmatrix} \mathcal {A} &{}\quad \mathcal {B}_p \\ I &{}\quad 0 \end{bmatrix}^{\mathrm{T}} f\left( \varPhi , \varPsi , P, Q\right) \begin{bmatrix} \mathcal {A} &{}\quad \mathcal {B}_p \\ I &{}\quad 0 \end{bmatrix} \begin{bmatrix} I &{}\quad a \\ 0 &{}\quad 1 \end{bmatrix}^{\mathrm{T}} + \ldots \ge 0 \end{aligned}$$

(30)

and introduce the notation $\varXi _{ij}$ for this linear part:

$$\begin{aligned} -\begin{bmatrix} \varXi _{11} &{}\quad \varXi _{12} \\ \varXi _{12}^{\mathrm{T}} &{}\quad \varXi _{22} \end{bmatrix} + \ldots \ge 0. \end{aligned}$$

(31)

Equation (31) may undergo a congruent transformation by $\gamma _\infty ^{-\frac{1}{2}}I$ introducing a commutable factor of $\gamma _\infty ^{-1}$ to every element. For the first summation term, we absorb the factor with redefinition (36) yielding:

$$\begin{aligned} -\begin{bmatrix} \varXi _{11} &{}\quad \varXi _{12} \\ \varXi _{12}^{\mathrm{T}} &{}\quad \varXi _{22} \end{bmatrix} - \begin{bmatrix} I &{}\quad a \\ 0 &{}\quad 1 \end{bmatrix} \begin{bmatrix} \mathcal {C}_q &{}\quad \mathcal {D}_{qp} \\ 0 &{}\quad I \end{bmatrix}^{\mathrm{T}} \begin{bmatrix} \gamma _\infty ^{-1} &{}\quad 0 \\ 0 &{}\quad -1 \end{bmatrix} \begin{bmatrix} \mathcal {C}_q &{}\quad \mathcal {D}_{qp} \\ 0 &{}\quad I \end{bmatrix} \begin{bmatrix} I &{}\quad a \\ 0 &{}\quad 1 \end{bmatrix}^{\mathrm{T}} \ge 0. \end{aligned}$$

(32)

Multiplying the inner factors in the second term of (32) leads to:

$$\begin{aligned} -\begin{bmatrix} \varXi _{11} &{}\quad \varXi _{12} \\ \varXi _{12}^{\mathrm{T}} &{}\quad \varXi _{22} \end{bmatrix}^{\mathrm{T}} - \begin{bmatrix} I &{}\quad a \\ 0 &{}\quad 1 \end{bmatrix} \begin{bmatrix} \gamma _\infty ^{-1}\mathcal {C}_q^{\mathrm{T}}\mathcal {C}_q &{}\quad \gamma _\infty ^{-1}\mathcal {C}_q^{\mathrm{T}}\mathcal {D}_{qp} \\ \gamma _\infty ^{-1}\mathcal {D}_{qp}^{\mathrm{T}}\mathcal {C}_q &{}\quad \gamma _\infty ^{-1}\mathcal {D}_{qp}^{\mathrm{T}}\mathcal {D}_{qp} - 1 \end{bmatrix} \begin{bmatrix} I &{}\quad a \\ 0 &{}\quad 1 \end{bmatrix}^{\mathrm{T}} \ge 0 \end{aligned}$$

which can be expanded into:

$$\begin{aligned}&-\begin{bmatrix} \varXi _{11} &{}\quad \varXi _{12} \\ \varXi _{12}^{\mathrm{T}} &{}\quad \varXi _{22} \end{bmatrix} - \begin{bmatrix} I &{}\quad a \\ 0 &{}\quad 1 \end{bmatrix} \begin{bmatrix} I &{}\quad a\mathcal {D}_{qp}^{\mathrm{T}} \\ 0 &{}\quad \mathcal {D}_{qp}^{\mathrm{T}} \end{bmatrix} \begin{bmatrix} \mathcal {C}_q^{\mathrm{T}} \\ 1 \end{bmatrix} \gamma _\infty ^{-1} \begin{bmatrix} \mathcal {C}_q^{\mathrm{T}} \\ 1 \end{bmatrix}^{\mathrm{T}} \begin{bmatrix} I &{}\quad a\mathcal {D}_{qp}^{\mathrm{T}} \\ 0 &{}\quad \mathcal {D}_{qp}^{\mathrm{T}} \end{bmatrix}^{\mathrm{T}} \begin{bmatrix} I &{}\quad a \\ 0 &{}\quad 1 \end{bmatrix}^{\mathrm{T}} \nonumber \\&\quad +\begin{bmatrix} I &{}\quad a \\ 0 &{}\quad 1 \end{bmatrix} \begin{bmatrix} 0 &{}\quad 0 \\ 0 &{}\quad 1 \end{bmatrix} \begin{bmatrix} I &{}\quad a \\ 0 &{}\quad 1 \end{bmatrix}^{\mathrm{T}} \ge 0. \end{aligned}$$

(33)

The 3 outer factors multiplied with $\gamma _\infty ^{-1}$ in the middle term of (33) are then combined together and the last summation term is also multiplied through, resulting in the following:

$$\begin{aligned} -\begin{bmatrix} \varXi _{11} &{}\quad \varXi _{12} \\ \varXi _{12}^{\mathrm{T}} &{}\quad \varXi _{22} \end{bmatrix} -\begin{bmatrix} \mathcal {C}_q^{\mathrm{T}} + a\mathcal {D}_{qp}^{\mathrm{T}} \\ \mathcal {D}_{qp}^{\mathrm{T}} \end{bmatrix} \gamma _\infty ^{-1} \begin{bmatrix} \mathcal {C}_q^{\mathrm{T}} + a\mathcal {D}_{qp}^{\mathrm{T}} \\ \mathcal {D}_{qp}^{\mathrm{T}} \end{bmatrix}^{\mathrm{T}} + \begin{bmatrix} aa^{\mathrm{T}} &{}\quad a \\ a^{\mathrm{T}} &{}\quad 1 \end{bmatrix} \ge 0. \end{aligned}$$

(34)

The last summation term of (34) is then added with the linear part $\varXi$. Because $\gamma _\infty> 0 \leftrightarrow \gamma _\infty ^{-1} > 0$, a Schur complement taken around $\gamma _\infty$ allows (34) to be written as the single matrix inequality (28). $\square$

1.2 Derivation of $\mathcal {H}_2$ and $\ell _1$ Inequalities

Theorem A.2

Equations (10) from Sect. 3.2 and (17) from Sect. 3.3 are equivalent to the following:

$$\begin{aligned} \begin{bmatrix} -\varXi _{11} + aa^{\mathrm{T}} &{}\quad -\varXi _{12} + a \\ -\varXi _{12}^{\mathrm{T}} + a^{\mathrm{T}} &{}\quad -\varXi _{22} + 1 \end{bmatrix} > 0 \end{aligned}$$

(35)

where (35) contains just one nonlinear term in variable a, and:

$$\begin{aligned} \begin{bmatrix} \varXi _{11} &{}\quad \varXi _{12} \\ \varXi _{12}^{\mathrm{T}} &{}\quad \varXi _{22} \end{bmatrix} = \begin{bmatrix} I &{}\quad a \\ 0 &{}\quad 1 \end{bmatrix} \begin{bmatrix} \mathcal {A} &{}\quad \mathcal {B}_p \\ I &{}\quad 0 \end{bmatrix}^{\mathrm{T}} f\left( \varPhi , P_\gamma , \alpha \right) \begin{bmatrix} \mathcal {A} &{}\quad \mathcal {B}_p \\ I &{}\quad 0 \end{bmatrix} \begin{bmatrix} I &{}\quad a \\ 0 &{}\quad 1 \end{bmatrix}^{\mathrm{T}} \end{aligned}$$

$$\begin{aligned} f\left( \varPhi , P_\gamma , \alpha \right)= & {} {\left\{ \begin{array}{ll} \varPhi \oplus P_\gamma &{}\quad \text {for the }\mathcal {H}_2\text { case} \\ \left( \varPhi + \begin{bmatrix} 0 &{}\quad 0 \\ 0 &{}\quad \alpha \end{bmatrix}\right) \oplus P_\gamma&\quad \text {for the }\ell _1\text { case} \end{array}\right. } \end{aligned}$$

(36)

$$\begin{aligned} P_\gamma= & {} \gamma _\infty ^{-1}P. \end{aligned}$$

(37)

Proof

Starting from either (10) or (17), we follow the procedure mentioned in Sect. 3.4.3 to eliminate non-convex products in the first term of the QMI independent of $f\left( \varPhi , P_\gamma , \alpha \right)$. The second summation term is the same in both QMIs and simplifies as shown in (24). Combining these results in the matrix inequality (35). $\square$