layout: true class: typo, typo-selection --- count: false 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 $$\begin{array}{ll} \text{minimize} & \sup_{{\color{coral}q} \in \mathbb{Q} } {\color{purple}f_0}({\color{darkblack}x},{\color{coral}q}), \\ \text{subject to} & {\color{purple}f_j}({\color{darkblack}x},{\color{coral}q}) \leq 0 \qquad \\ & \forall {\color{coral}q} \in \mathbb{Q} \; \text{and} \; j = 1,2,\cdots,m, \\ \end{array}$$ where - ${\color{darkblack}x} \in \mathbb{R}^n$ represents a set of design variables (such as ${\color{blue}L}$, ${\color{blue}W}$), - ${\color{coral}q}$ represents a set of varying parameters (such as ${\color{green}T_{OX}}$) - ${\color{purple}f_j} \leq 0$ represents the $j$th specification requirement (such as phase margin $\geq 60^\circ$). --- ### 📜 Geometric Programming in Standard Form - We further assume that ${\color{purple}f_i}({\color{darkblack}x},{\color{coral}q})$'s are convex for all ${\color{coral}q} \in \mathbb{Q}$. - Geometric programming is an optimization problem that takes the following standard form: $$\begin{array}{lll} \text{minimize} & {\color{purple}p_0}({\color{darkblack}y}) & \\ \text{subject to} & {\color{purple}p_i}({\color{darkblack}y}) \leq 1, & i=1,\ldots,l \\ & {\color{purple}g_j}({\color{darkblack}y}) = 1, & j=1,\ldots,m \\ & {\color{green}y}_k > 0,& k=1,\ldots,n , \end{array}$$ where - ${\color{purple}p_i}$'s are posynomial functions and ${\color{purple}g_j}$'s are monomial functions. --- ### 🔢 Posynomial and Monomial Functions - A monomial function is simply: $${\color{purple}g}({\color{darkblack}y}_1,\ldots,{\color{darkblack}y}_n) = {\color{orange}c} {\color{darkblack}y}_1^{ {\color{blue}\alpha}_{1} } {\color{darkblack}y}_2^{ {\color{blue}\alpha}_{2} } \cdots {\color{darkblack}y}_n^{ {\color{blue}\alpha}_{n} }, \quad {\color{darkblack}y}_k > 0.$$ where - ${\color{orange}c}$ is non-negative and ${\color{blue}\alpha}_{k}\in {\mathbb{R} }$. - A posynomial function is a sum of monomial functions: $${\color{purple}p}({\color{darkblack}y}_1,\ldots,{\color{darkblack}y}_n) = \sum_{s=1}^{T}{{\color{orange}c_s} {\color{darkblack}y}_1^{ {\color{blue}\alpha}_{1,s} } {\color{darkblack}y}_2^{ {\color{blue}\alpha}_{2,s} } \cdots {\color{darkblack}y}_n^{ {\color{blue}\alpha}_{n,s} } }, \quad {\color{darkblack}y}_k > 0 ,$$ - 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 ${\color{darkblack}x}_k = \log({\color{darkblack}y}_k)$ and replacing ${\color{purple}p_i}$ with $\log {\color{purple}p_i}$: $$\begin{array}{lll} \text{minimize} & \log {\color{purple}p_0}(\exp({\color{darkblack}x})) & \\ \text{subject to} & \log {\color{purple}p_i}(\exp({\color{darkblack}x})) \leq 0, & i=1,\ldots,l \\ & {\color{blue}a_j}^\mathsf{T} {\color{darkblack}x} + {\color{blue}b_j} = 0, & j=1,\ldots,m \end{array}$$ where - $\exp({\color{darkblack}x}) = (e^{ {\color{darkblack}x}_1}, e^{ {\color{darkblack}x}_2}, \cdots, e^{ {\color{darkblack}x}_n})$ - ${\color{blue}a_j} = ({\color{blue}\alpha}_{1,j}, \cdots, {\color{blue}\alpha}_{n,j})$ - ${\color{blue}b_j} = \log({\color{orange}c_j})$ --- ### 🏗️ Geometric Programming Workflow .mermaid[ <pre> graph LR A(("Design Variables")) --> B[Posynomial Constraints] B --> C[GP Solver] C --> D[Optimal Solution] style A fill:#c8e6c9,stroke:#2e7d32,stroke-width:3px style B fill:#fff3e0,stroke:#e65100,stroke-width:3px style C fill:#e3f2fd,stroke:#1565c0,stroke-width:3px style D fill:#f3e5f5,stroke:#7b1fa2,stroke-width:3px </pre> ] --- class: nord-light, middle, center ## 🛡️ Robust Geometric Programming --- ### 🛠️ Robust GP .pull-left[ - GP in the convex form can be solved efficiently by interior-point methods 📈. - In robust version, coefficients ${\color{orange}c_s}$ are functions of ${\color{coral}q}$ 🔄. - 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? 🔦🧐 ] --- count: false class: nord-dark, middle, center .pull-left[ ## Q&A 🎤 ] .pull-right[  ]