ELEC4632 Chap.5 Optimal, Nonlinear and Predictive Control
Optimal, Nonlinear and Predictive Control
The advanced-design block moves from nonlinear discrete-time systems and singular-point stability to linearisation, Lyapunov functions, dynamic programming, linear-quadratic control and model-predictive control. This chapter focuses on the reasoning sequence, the assumptions behind each method and the distinctions an analytical examination answer should preserve.
Use the numerical plant data supplied with the current lecture, tutorial or assessment task for calculation.
What this chapter covers
- 01
Equilibrium verification for nonlinear sampled systems
- 02
Local linearisation in deviation coordinates
- 03
Lyapunov candidates and one-step change
- 04
Bellman optimality and backward recursion
- 05
Linear-quadratic trade-offs between state and input
- 06
Model-predictive control over a receding horizon
- 07
Constraints, computation time and implementation feasibility
Worked example · free
Plan a local nonlinear stability analysis
- +1Verify that the candidate satisfies the equilibrium equation under its operating input.
- +1Define deviation variables and form the local linear approximation around the equilibrium.
- +1Inspect the local discrete-time eigenvalues and label the conclusion as local.
- +1If a Lyapunov candidate is supplied, test its positivity and one-step change over the claimed region.
Key terms
- Equilibrium Point
- An equilibrium point reproduces itself under the system update and its associated operating input.
- Linearisation
- Linearisation is a first-order approximation of nonlinear dynamics near a selected operating point.
- Value Function
- A value function records the optimal remaining cost or return from a state and stage.
- Quadratic Objective
- A quadratic objective penalises state deviation and control effort using weighted squared terms.
- Receding Horizon
- A receding horizon repeatedly shifts forward after the first action of each finite plan is applied.
Optimal, Nonlinear and Predictive Control FAQ
What does local stability mean for a nonlinear system?
A local result describes behaviour near the analysed equilibrium, where the linear approximation or Lyapunov conditions are valid. It does not automatically describe trajectories that begin far away, because neglected nonlinear terms may become important.
How does dynamic programming avoid a greedy decision?
It evaluates each current action using both its immediate contribution and the optimal value of the next state over the remaining horizon. This preserves the future consequence of stored energy or another state carried between stages.
Why can MPC be harder to implement than LQ control?
MPC solves a constrained finite-horizon problem repeatedly, so model size, horizon and solver timing must fit inside the sample interval. A precomputed LQ gain is lighter online, although its assumptions and unconstrained behaviour still require verification.
What must be checked before linearising a nonlinear model?
Verify that the candidate operating point reproduces itself under the stated input. Then define deviations and form the local Jacobian model. If the point is not an equilibrium, the resulting eigenvalue discussion is not a stability analysis of that operating condition.
Why does terminal information matter in dynamic programming?
A terminal cost, constraint or required final state changes the value of earlier actions during backward recursion. Omitting it can make a locally attractive action appear optimal even when it leaves an unacceptable state at the end of the horizon.
What makes an optimisation controller feasible in real time?
The model, horizon and constraints must define a solvable problem whose computation finishes inside the sample interval. Validation should report solver timing and constraint behaviour as well as the output trajectory, because a late solution cannot control the current sample.
Exam move
Use four headings for every advanced method: model, objective, constraints and claim. For nonlinear stability, verify the equilibrium before linearising and label the result local. For dynamic programming, identify state, action, transition, stage cost and terminal condition. For LQ and MPC, compare horizon, online computation and constraint handling.
Take numerical weights, horizons and plant data from the current Moodle materials assigned to your class. Build a one-page comparison with rows for linearisation, Lyapunov analysis, LQ control, dynamic programming and MPC. For each, record the question it answers, the mathematical object it needs, the scope of its conclusion and the implementation burden it creates.
Work one symbolic nonlinear analysis from equilibrium verification through the deviation model and unit-circle test; the point is the order, not invented coefficients. For Bellman reasoning, narrate a two-stage decision backwards so immediate reward and remaining value are clearly separated. For LQ, explain qualitatively what happens when state or input penalties increase.
For MPC, rehearse the operational loop: estimate, optimise, apply the first move, measure and resolve. Attach a timing question to every optimisation design because sample-time feasibility is part of the controller. Finish comparisons with bounded language. A method is suitable for the stated model and objective under its assumptions; it is not automatically superior in every plant.
Name the simulation, constraint and computation evidence that would support implementation. Add one implementation line to every method card. For linearisation, identify the operating region; for Lyapunov analysis, name the region in which signs are established; for LQ, inspect actuator demand; for MPC, compare solve time with the sample interval; and for dynamic programming, state the terminal condition.
This turns abstract definitions into decision tools. When revising, explain why changing one assumption could change the recommended method rather than memorising a fixed ranking.
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