layout: true class: typo, typo-selection --- class: nord-dark, middle, center Lecture 04b - 🛡️ Robust Geometric Programming 📐 ============================ @luk036 👨‍💻 · 2026 📅 --- Outline 📝 ------- .pull-left[ - Problem Definition for Robust Analog Circuit Sizing ⚙️ - Robust Geometric Programming 📈 - Ellipsoid Method - Affine Arithmetic ➕➖ - Example: CMOS Two-stage Op-Amp 💻 - Numerical Result - Conclusions 🏁 ] .pull-right[  ] --- Robust Circuit Sizing Problem ⚖️ ------------------------------------ - Given a circuit topology and a set of specification requirements: .font-sm.mb-xs[ | Constraint | Spec. | Units | | -------------- | ---------- | ---------- | | Device Width | $\geq 2.0$ | $\mu$m | | Device Length | $\geq 1.0$ | $\mu$m | | Estimated Area | minimize | $\mu$m$^2$ | | $\vdots$ | $\vdots$ | $\vdots$ | | CMRR | $\geq 75$ | dB | | Neg. PSRR | $\geq 80$ | dB | | Power | $\leq 3$ | mW | ] - Find the worst-case design variable values that meet the specification requirements and optimize circuit performance 🎯. --- Robust Optimization Formulation 🧮 ------------------------------- - Consider $$\begin{array}{ll} \text{minimize} & \sup_{q \in \mathbb{Q} } f_0({\color{darkmagenta}x},q), \\ \text{subject to} & f_j({\color{darkmagenta}x},q) \leq 0 \qquad \\ & \forall q \in \mathbb{Q} \; \text{and} \; j = 1,2,\cdots,m, \\ \end{array}$$ where - ${\color{darkmagenta}x} \in \mathbb{R}^n$ represents a set of design variables (such as $L$, $W$), - $q$ represents a set of varying parameters (such as $T_{OX}$) - $f_j \leq 0$ represents the $j$th specification requirement (such as phase margin $\geq 60^\circ$). --- Geometric Programming in Standard Form 📜 -------------------------------------- - We further assume that $f_i({\color{darkmagenta}x},q)$'s are convex for all $q \in \mathbb{Q}$. - Geometric programming is an optimization problem that takes the following standard form: $$\begin{array}{lll} \text{minimize} & p_0(y) & \\ \text{subject to} & p_i(y) \leq 1, & i=1,\ldots,l \\ & g_j(y) = 1, & j=1,\ldots,m \\ & y_k > 0,& k=1,\ldots,n , \end{array}$$ where - $p_i$'s are posynomial functions and $g_j$'s are monomial functions. --- Posynomial and Monomial Functions 🔢 --------------------------------- - A monomial function is simply: $$g(y_1,\ldots,y_n) = c y_1^{ {\alpha}_{1} } y_2^{ {\alpha}_{2} } \cdots y_n^{ {\alpha}_{n} }, \quad y_k > 0.$$ where - $c$ is non-negative and ${\alpha}_{k}\in {\mathbb{R} }$. - A posynomial function is a sum of monomial functions: $$p(y_1,\ldots,y_n) = \sum_{s=1}^{T}{c_s y_1^{ {\alpha}_{1,s} } y_2^{ {\alpha}_{2,s} } \cdots y_n^{ {\alpha}_{n,s} } }, \quad y_k > 0 ,$$ - A monomial can also be viewed as a special case of posynomial where there is only one term of the sum. --- Geometric Programming in Convex Form 🔄 ------------------------------------ - Many engineering problems can be formulated as a GP 🤔. - On Boyd's website there is a Matlab package "GGPLAB" and an excellent tutorial material 📚. - GP can be converted into a convex form by changing the variables ${\color{darkmagenta}x}_k = \log(y_k)$ and replacing $p_i$ with $\log p_i$: $$\begin{array}{lll} \text{minimize} & \log p_0(\exp({\color{darkmagenta}x})) & \\ \text{subject to} & \log p_i(\exp({\color{darkmagenta}x})) \leq 0, & i=1,\ldots,l \\ & a_j^\mathsf{T} {\color{darkmagenta}x} + b_j = 0, & j=1,\ldots,m \end{array}$$ where - $\exp({\color{darkmagenta}x}) = (e^{ {\color{darkmagenta}x}_1}, e^{ {\color{darkmagenta}x}_2}, \cdots, e^{ {\color{darkmagenta}x}_n})$ - $a_j = (\alpha_{1,j}, \cdots, \alpha_{n,j})$ - $b_j = \log(c_j)$ --- ### 🏗️ Geometric Programming Workflow .mermaid[ <pre> graph LR A[Design Variables] --> B[Posynomial Constraints] B --> C[GP Solver] C --> D[Optimal Solution] style A fill:#4caf50 style B fill:#ff9800 style C fill:#2196f3 style D fill:#9c27b0 </pre> ] --- Robust GP 🛠️ --------- .pull-left[ - GP in the convex form can be solved efficiently by interior-point methods 📈. - In robust version, coefficients $c_s$ are functions of $q$ 🔄. - The robust problem is still convex. Moreover, there is an infinite number of constraints 🔁. - Alternative approach: Ellipsoid Method 🔳. ] .pull-right[  The streetlights are bright, but are you sure this is where you dropped your key? 🔦🧐 ] --- class: nord-dark, middle, center .pull-left[ # Q&A 🎤 ] .pull-right[  ]