--- name: math-thumb-control-theory description: "Apply Chapter 25 (Control Theory: Performance, Robustness, and Implementation Reality) of Mathematical Rules of Thumb to solve, check, or teach problems. Use it to translate implementation constraints into a delay and sampling budget before tuning a controller." --- # Control Theory: Performance, Robustness, and Implementation Reality Use this chapter to help the reader make a checked mathematical decision. All 10 numbered rules are available in [the chapter source](references/chapter.md). [The workbook](references/notebook.md) contains a lab, exercises, solutions, and the full rule checklist. [The local rule index](references/rules.json) supplies discovery metadata. ## Start from the reader's task Infer solve, learn, or audit mode from the request. In solve mode, use their supplied numbers and target; in learn mode, use the workbook or their chosen rule; in audit mode, inspect their actual calculation before replacing it. Gather only missing information that changes the choice: Plant and loop structure; bandwidth/crossover conventions; delays; sample rate; actuator limits; noise; robustness and time-response targets. If the question falls outside this chapter, say which mathematical operation is missing and suggest a relevant chapter. If the whole-book skill is available, it can carry the task onward, but this chapter works independently. ## Select and apply a rule - **Is the requirement stated in time units?** Translate it into a bandwidth interval, then verify the full step response rather than trusting one constant. - **Is the persistent disturbance approximately constant?** Add integral structure only if the loop can remain stable and the actuator has enough authority. - **Will a reduced model guide tuning?** Compare pole decay, residues, zeros, and full-versus-reduced responses in every important channel. - **Is the loop ordinary SISO with a clear crossover?** Use phase and gain margins as starting screens, then inspect peak sensitivity. - **Are there multiple crossovers, unstable poles, or MIMO coupling?** Escalate from classical margins to Nyquist, disk, or structured robustness analysis. - **What latency is unavoidable?** Convert every component to phase at crossover before raising bandwidth. - **Will the design run digitally?** Include sample, hold, computation, jitter, filtering, and quantization in the implemented model. - **Can the actuator saturate or the derivative see noise?** Add anti-windup and derivative filtering before nonlinear simulation or hardware tests. Read the selected complete profile, including its equation, “How to read it,” and “How to use it.” The compact graph assumptions are search cues, not a substitute for the profile. Preserve the numbered citation and role. **Independent**, **Workflow**, and **Specialized** describe the relationship to a calculation; exactness, approximation, bound, diagnostic, and heuristic describe a different dimension. Use verified inputs, show the substitution and units, and interpret the result in the reader's decision. Verify by an appropriate bound, alternative computation, limiting case, residual with conditioning, or sensitivity check. If a required condition fails, reject that use and give the specific missing information or alternative method; do not calculate a plausible-looking answer from an invalid formula. **Essential boundary:** State Hz versus rad/s and identify the actual crossover. Classical margins and sampling ratios are design screens, not permission to deploy a controller without model and implementation checks. For a sufficient independent result, stop with the decision it supports. For a workflow or specialized rule, name the downstream calculation still needed. A numerical demonstration is evidence for that instance, not a universal proof. ## Teach and check understanding Use the [workbook](references/notebook.md) for guided practice. Start with 25.2.1, 25.3.1, 25.3.2 when the reader wants a starting exercise. Ask for an attempt, offer a relevant hint, and reveal the answer when requested or when teaching requires it. Do not force a quiz when the reader asked for a worked solution. Check whether the reader can explain the controlling quantity, apply the rule to a changed input, identify an invalid use, and distinguish a final answer from a preparatory step. Track only demonstrated work. Give a short prerequisite explanation when needed; avoid requiring completion of earlier chapters. ## Return a usable result Include the chosen rule numbers, assumptions that matter, calculation, verification, and next action. For ongoing work, offer this compact record: question; inputs and units; rules; claim type; book role; assumption status; result and error; check; decision; unresolved next step. Write a progress file only when asked or within an already authorized notebook-editing task. The source chapter is a fixed book snapshot. Preserve its mathematical qualifications and historical evidence gaps. Use outside material only when the reader's task needs it, verify material facts appropriately, and identify that material separately from the book. ## Illustrated exploration Open [the browser reader](assets/reader.html) or [the saved illustrated notebook](assets/notebook.ipynb). [Equation cards](references/equations.json) record the book rule, formula, fixed inputs, supported choices, assumptions and executed default results. - **C25-D01: Connect a real step response with bandwidth**: rule 25.1.1. - **C25-D02: Budget latency at an assumed crossover**: rule 25.3.1. - **C25-D03: Simulate integral windup and a remedy**: rule 25.3.3. - **C25-D04: Remove steady error with integral action**: rule 25.1.2. - **C25-D05: See what phase margin looks like in a step response**: rule 25.2.1. Use a saved illustration only when its conditions fit. Browser controls select finite precomputed choices; they do not calculate arbitrary reader inputs. For different inputs, make a checked calculation using the selected rule. Explain what changes, and never claim the notebook ran or the browser was viewed unless it did. Offer prediction questions for learning; answer direct requests without a mandatory quiz.