ELEC4631 Continuous-Time Control System Design
The Complete Exam Bible · T2 2026Continuous-Time Control System Design
ELEC4631 Continuous-Time Control System Design is a 6 UoC undergraduate course taught in person at UNSW Sydney in Term 2 2026. The official indicative workload is fifteen hours per week across a ten-week teaching pattern, and all assessment is individual. The mathematical progression is unusually coherent. Linear algebra and vector calculus establish eigenvalues, eigenvectors, algebraic and geometric multiplicity, quadratic forms and gradients. State-space modelling then packages a continuous-time plant as x-dot = Ax + Bu and y = Cx + Du. The state equation and matrix exponential turn eigenstructure into trajectories. Lyapunov theory asks what can be certified without solving every trajectory, first for nonlinear systems through a candidate V and its derivative, then for LTI systems through spectra and the Lyapunov equation. Controllability and observability identify which modes can be reached by input and reconstructed from output; those rank conditions become the permissions for state-feedback pole placement and Luenberger observer design. LQR replaces exact pole choice with a quadratic performance objective and a Riccati equation. The final course blocks introduce convex linear matrix inequalities, convert bilinear feedback conditions with X and Y substitutions, recover controller and observer gains, then address nonzero references and nonlinear Lyapunov feedback. The official 2026 schedule—not the topic location in an older paper—governs that order. The assessment map has a deliberate asymmetry. CLO1–3, covering state-space analysis, stability, controllability and observability, are mapped to both the midterm and final. CLO4, covering state feedback, observers, output feedback and tracking, and CLO5, covering convex optimisation, are mapped to the final but not the midterm. CLO6 is demonstrated in the laboratory. This means final preparation must preserve the early foundations while allocating new time to observers, LMIs and tracking. The current Quiz 1 instructions provide a strong signal about answer culture: analytical workings are required and MATLAB-only solutions receive no marks. The available older midterm reference solutions reinforce the structure without predicting current questions. They state the mathematical object, show the working, state the decisive condition and then name the conclusion—for example, eigenvalues followed by algebraic and geometric multiplicities before diagonalisability, or Lyapunov properties before the stability adjective. All worked examples in this guide use new numbers and independently written solutions. Administrative specificity is kept separate from mathematical specificity. The Course Outline confirms a one-hour Week 7 midterm that is in person, invigilated and closed book, and a two-hour written final that is centrally timetabled, in person, invigilated and closed book. It does not publish the course-specific final date, question count, calculator rule or topic mark split. A separate midterm information note says three questions, Lectures 1–8 and Tutorials 1–5, and a one-sided handwritten A4 note, but the same file prints a stale 2025 year string. Those details must therefore remain attributed and be confirmed on Moodle rather than silently merged with the Course Outline. No separate component hurdle is stated. The ordinary overall pass threshold applies, while current submission, special-consideration and live examination instructions remain authoritative on Moodle and the official timetable.
What ELEC4631 covers
ELEC4631 progresses from linear algebra and state-space models into the state equation and Lyapunov stability, then uses controllability and observability as permissions for state feedback and observers. The second half adds LQR, output feedback, convex LMI foundations and LMI synthesis before finishing with reference tracking and nonlinear Lyapunov feedback. The chapters follow the official T2 2026 sequence rather than the location of topics in older papers: early CLO1–3 material remains mapped to both examinations, while feedback and tracking in CLO4 and convex optimisation in CLO5 are mapped to the final but not the midterm.
How ELEC4631 is assessed
| Component | Weight | Format |
|---|---|---|
| Laboratory | 12% | Individual. The laboratory assesses computational application of modelling, analysis and control design and is the component mapped to CLO6. The exact task timing and submission instructions should be taken from the current Moodle activity. |
| Quizzes | 12% | Individual. The current Quiz 1 has three questions weighted 28, 37 and 35 raw quiz marks across eigenstructure, quadratic forms and the matrix exponential. It requires analytical workings and states that MATLAB-only solutions receive no marks. The available due line is 11 June 2026 at 5 pm; use the live Moodle activity for any update. |
| Midterm Exam | 20% | Individual; one hour in Week 7; in person, invigilated and closed book. The Course Outline says material covered thus far and that the exact date/time is announced. A separate information note says three questions, Lectures 1–8 and Tutorials 1–5, and one side of handwritten A4 notes, but carries a stale 2025 year string; confirm those attributed details on Moodle. |
| Final Examination | 56% | Individual; two-hour written examination, centrally timetabled, in person, invigilated and closed book. Any aspect of the course may be examined unless specifically indicated otherwise. The course-specific date, question count, calculator/stationery rules and per-topic mark split are not stated in the Course Outline material used here. |
From eigenstructure to a complete free-response conclusion
- step 1Compute det(lambda I - A) = (lambda + 1)^2. The only eigenvalue is -1 and its algebraic multiplicity is two.
- step 2A + I = [[0,1],[0,0]], so its null space is span{[1,0]^T}. The geometric multiplicity is one.
- step 3Because geometric multiplicity one is smaller than algebraic multiplicity two, A is defective and not diagonalisable.
- step 4Write A = -I + N with N = [[0,1],[0,0]] and N^2 = 0. Since the two terms commute, exp(At) = exp(-t)(I+tN) = exp(-t)[[1,t],[0,1]].
- step 5Check exp(A0)=I and differentiate at t=0 to recover A. The factor t comes from the size-two Jordan block.
- step 6Both modes contain a decaying exponential; t exp(-t) also tends to zero. A is Hurwitz, so the LTI origin is globally asymptotically stable despite the defective eigenstructure.
Key terms
- Algebraic multiplicity
- The repetition count of an eigenvalue as a root of the characteristic polynomial. It must be compared with geometric multiplicity before a diagonalisability claim.
- Geometric multiplicity
- The dimension of ker(A-lambda I), equivalently the number of independent eigenvectors for that eigenvalue. It is never larger than algebraic multiplicity.
- Hurwitz matrix
- A continuous-time matrix whose eigenvalues all have strictly negative real part. For x-dot=Ax this is equivalent to global asymptotic stability of the origin.
- Lyapunov function
- A positive function about an equilibrium whose derivative along trajectories has a sign that certifies stability. Domain, definiteness and invariance determine the strength of the conclusion.
- Controllability
- The property that the input can move the state through every state-space direction in finite time. For an n-state LTI pair, rank[B AB ... A^(n-1)B]=n.
- Observability
- The property that the output history distinguishes every initial state. For an n-state LTI pair, the stacked matrix [C; CA; ...; CA^(n-1)] must have rank n.
- Separation principle
- For a controllable and observable nominal LTI plant, state-feedback and observer gains can be designed independently; the combined eigenvalues are the union of those of A-BK and A-LC.
- Linear matrix inequality
- A symmetric matrix inequality whose matrix depends affinely on its decision variables. Affinity is the source of a convex feasible set.
- Riccati equation
- The differential or algebraic matrix equation that defines the quadratic value matrix in LQR. The infinite-horizon CARE requires selection of the stabilising solution.
- Reference prefilter
- A feedforward factor chosen so a stabilised nominal closed loop has the desired constant steady-state output. It does not by itself guarantee disturbance rejection or robustness.
ELEC4631 FAQ
What is actually on the ELEC4631 midterm?
The official Course Outline confirms a one-hour, Week 7, in-person, invigilated, closed-book midterm covering material taught thus far. A separate information note says three questions and identifies Lectures 1–8 and Tutorials 1–5, plus one side of handwritten A4 notes. That note also prints a stale 2025 year string, so use the content as attributed guidance and confirm the current date, coverage boundary and note rule on Moodle. The official teaching sequence through that point includes linear algebra, state space, Lyapunov stability, LTI stability, controllability, observability, transformations, pole placement and the beginning of LMI material. Do not let an older paper override the 2026 sequence, because the course itself warns that earlier papers can place topics differently after restructuring.
What is confirmed about the final examination?
It is worth 56% and the official 2026 Course Outline describes it as a two-hour written, centrally timetabled, in-person, invigilated, closed-book examination. It can examine any course aspect unless the course specifies otherwise. The final date, question count, topic mark split, calculator rule and stationery details are not stated in the material used here. Take the live date, time and venue from the official timetable and any equipment instruction from Moodle. The CLO map is useful for revision priority but not for predicting marks: CLO1–3 remain cumulative, while CLO4 feedback/tracking and CLO5 convex optimisation add final-but-not-midterm emphasis.
Can MATLAB replace the analytical derivation?
No. The clearest current-course instruction appears in Quiz 1: analytical workings are required and MATLAB-only solutions receive no marks. Use MATLAB or another tool after a derivation to check eigenvalues, ranks, Riccati residuals, LMI margins and simulations. In a written response, show the determinant, null space, derivative, rank matrix, desired polynomial, Riccati equation or variable substitution that produces the result. A software check is valuable precisely because it is independent; it becomes weak when it replaces the argument the question is assessing.
Why does the guide include LMIs and tracking instead of Gramians?
The official 2026 schedule explicitly includes introduction to LMIs in Week 7, controller and observer synthesis using LMIs in Week 8, then reference tracking and nonlinear Lyapunov feedback in Week 9. The available Lecture 11–14 material reinforces that late-course sequence. Controllability and observability are taught through rank tests, duality, minimality and cancellation. Gramians and a full Kalman decomposition are legitimate control theory, but the captured course material does not support giving them a large chapter at the expense of the actual LMI and tracking blocks.
How should older midterm solutions be used?
Use them for mathematical answer shape, not question prediction. The available reference solutions make several habits visible: state eigenvalues and both multiplicities before diagonalisability; identify the symmetric quadratic-form matrix before its eigenvalue signs; list positivity, derivative sign, radial unboundedness or invariant-set facts before naming a stability class. Rework the method with fresh matrices. Do not infer the 2026 paper’s numbers, marks, ordering or coverage from the old document, and remember the course warning that restructuring can move topics between assessments.
Can Sia help me study ELEC4631?
Yes—outside assessments and within the course’s integrity rules. Ask for a fresh matrix that tests a specific skill, show your own derivation and request the first incorrect line, or ask for contrasting examples such as a repeated eigenvalue that is diagonalisable versus defective, a semidefinite Lyapunov derivative that needs LaSalle, or a bilinear inequality that becomes an LMI after substitution. Sia should teach and check reasoning, not complete graded work. The final is categorised as no assistance for generative AI, and the current quiz demands the student’s own analytical working.
Is there a hurdle?
No separate component hurdle is stated in the official Course Outline used for this guide. The ordinary overall course pass threshold applies, subject to UNSW rules. This is not permission to ignore a component: the final carries 56%, early CLO1–3 remain cumulative, and every assessment is individual. Confirm any updated course rule on Moodle. For late work, the Course Outline states that work more than five days, or 120 hours, late is not accepted and receives zero; component instructions and approved special consideration still govern specific cases.
How to study for the exam
Treat ELEC4631 as one chain of permissions and certificates rather than thirteen unrelated formula lists. In the first pass, build a one-page map: eigenstructure determines modes and the shape of exp(At); the state equation combines transported initial condition and input convolution; Lyapunov functions certify behaviour without solving trajectories; controllability permits K design; observability permits L design; separation combines the two; LQR selects K from a cost; LMIs search for certificates and gains under affine matrix constraints; tracking adds a compatible nonzero equilibrium; nonlinear feedback shapes V-dot. In the second pass, practise one algorithm from a blank page in the course’s visible answer order. For eigenstructure, write the characteristic equation, eigenvalues, algebraic and geometric multiplicities, null spaces and conclusion. For Lyapunov analysis, locate the equilibrium, classify V, compute V-dot, inspect the zero-derivative invariant set and state local or global scope. For reachability and visibility, build the matrices with correct orientation, compare rank with n and identify the design consequence. For pole placement and observers, state permission, expand the target polynomial, calculate the gain under a declared sign convention and verify the closed-loop polynomial. For LQR, identify horizon, Q and R, solve the RDE or CARE, select the stabilising P, recover K and check A-BK. For LMIs, box every decision variable, prove affinity, retain strictness and side conditions, recover the physical gain and verify the original dynamics. Use computation as a second route: compare eigenvalue sum and product with trace and determinant, test Phi(0)=I and its derivative, inspect Lyapunov and Riccati residuals, and check LMI eigenvalue margins. Weekly, attempt tutorial problems before viewing solutions and repair the first incorrect line afterwards. For the final, use the CLO map rather than an invented topic split: keep short mixed drills for CLO1–3 and longer connected blocks for observers, output feedback, convex synthesis, tracking and nonlinear feedback. Because the exam is closed book, practise reconstructing formulas from definitions. Confirm the final date, time, venue and equipment rules on the official timetable and Moodle, and keep all AI assistance outside assessments that prohibit it.
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