ELEC4631 Continuous-Time Control System Design

How ELEC4631 is assessed

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.

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.

Worked example · free

From eigenstructure to a complete free-response conclusion

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

Questions students ask in each ELEC4631 chapter

Is the final really closed book and two hours?

Yes. Unlike many course builds where those details are absent, the official 2026 Course Outline explicitly states a two-hour written final that is centrally timetabled, in person, invigilated and closed book. It does not state the course-specific date, question count, per-topic marks or calculator rule. Those live details belong to the official timetable and Moodle. Chapter 01 · How This Course Is Marked: Lab 12, Quizzes 12, Midterm 20 and Final 56

Is every repeated eigenvalue defective?

No. Repetition is algebraic multiplicity. A repeated eigenvalue is defective only when its eigenspace has smaller dimension. A scalar multiple of I has a repeated eigenvalue and a full eigenvector basis; a Jordan block has fewer independent eigenvectors. Chapter 02 · State-Space Models: Eigenstructure, Realisations and Similarity

Can I exponentiate A entry by entry?

Only when A is diagonal in the coordinates being used. In general, matrix powers mix coordinates and the exponential is defined by the matrix power series or an equivalent spectral/resolvent method. Entrywise exponentiation usually fails Phi(0)=I or Phi-dot=A Phi. Chapter 03 · Solving the State Equation: Matrix Exponential, Modes and Free Response

Does V-dot<=0 prove asymptotic stability?

Not by itself. It usually proves stability under the other direct-method conditions. To establish convergence with a semidefinite derivative, analyse the largest invariant subset of the zero-derivative set, often through LaSalle. If that invariant subset contains only the equilibrium, asymptotic stability can follow. Chapter 04 · Lyapunov Stability I: Direct Method, Energy Functions and Invariant Sets

Are imaginary-axis eigenvalues always unstable?

No. With no RHP eigenvalues, semisimple imaginary-axis modes can yield Lyapunov stability but not asymptotic decay. A nontrivial Jordan block on the axis creates polynomial growth and instability. State multiplicity conditions rather than using the location alone. Chapter 05 · Lyapunov Stability II: The Lyapunov Equation and Linear Stability Tests

Why does the controllability matrix stop at A^(n-1)B?

Cayley–Hamilton expresses higher powers of A as combinations of I through A^(n-1), so later propagated input directions add no new span. With multiple inputs the matrix is wide, but the decisive comparison remains rank with n. Chapter 06 · Controllability, Observability, Minimality and Pole-Zero Cancellation

Why must controllability be shown first?

An uncontrollable mode cannot be moved by any state-feedback gain. Ackermann’s inverse also assumes a full-rank square single-input controllability matrix. Stating rank n is the mathematical permission, not ceremonial setup. Chapter 07 · State Feedback and Pole Placement: Ackermann and Canonical Form

How is LQR different from pole placement?

Pole placement starts with exact desired eigenvalues. LQR starts with Q and R and chooses the gain minimising the resulting cost; closed-loop poles are consequences. LQR does not promise specified poles unless weights happen to produce them. Chapter 08 · Optimal Control and LQR: Riccati Equations and Cost Weighting

Why does the input disappear from error dynamics?

Plant and observer use the same known Bu term and the same direct feedthrough in the predicted output, so subtraction cancels them. If the observer model or D term is inconsistent, rederive the error equation; cancellation is not automatic. Chapter 09 · Observer Design, Separation Principle and Output Feedback

How can I tell whether a matrix condition is an LMI?

Box every unknown. Each may appear only affinely: fixed offset plus fixed matrices multiplied by scalar decisions, or equivalent linear matrix operations. Products between unknowns, decision inverses and squared decisions break affinity. Repeat the audit after any substitution. Chapter 10 · Linear Matrix Inequalities: Convexity, Feasibility and Schur Complements

Why is the original Lyapunov condition a BMI?

Expanding it produces K^TB^TP and PBK. When P and K are both decisions, their products are bilinear. The stability theorem is correct, but its direct joint design model is nonconvex. Chapter 11 · Controller and Observer Synthesis Using LMIs

Why does stabilising K not automatically track r?

With u=-Kx and no feedforward, the stable closed-loop equilibrium is normally x=0 and hence y=0. A nonzero command must be embedded through a compatible steady state, prefilter, integral state or another reference model. Chapter 12 · Reference Tracking and Nonlinear Lyapunov Feedback

What should I revise for the midterm?

The Course Outline says material covered thus far; the attributed information note says Lectures 1–8 and Tutorials 1–5. Build skills in eigenstructure, quadratic forms, state-space models, the state equation, Lyapunov direct method, LTI stability, controllability, observability, transformations and pole placement. Tag old questions by the current schedule because restructuring can move topics. Chapter 13 · Midterm and Final Playbook: How ELEC4631 Asks and Marks Control Questions

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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