Discrete-Time Luenberger Observers

Overview

Lecture 6 Part 3: brief introduction to discrete-time observers.
Goal: implement a state-feedback controller using estimated state x̂(k) when full state unavailable.
Focus on one-step Luenberger observers, mention two-step observers and implementation/ sampling issues.
Conclude with separation principle, eigenvalue placement, LMI summary, and a short numerical observability/controllability example.

Plant in discrete time with output y(k) = C x(k) + D u(k).
One-step Luenberger observer: propagate estimate, apply input effect, and add correction based on output error.
Observer update structure: x̂(k+1) = A x̂(k) + B u(k) + L (y(k) - Ĉ x̂(k)).
ŷ(k) denotes predicted output based on current state estimate.

Estimation error e(k) = x(k) - x̂(k).
Closed-loop error dynamics (one-step) reduce to e(k+1) = (A + L C) e(k).
Choose observer gain L so that A + L C is stable (eigenvalues inside unit circle).
Discrete-time analogue of continuous-time ė = (A + L C) e.

Augmented closed-loop (plant + observer-based controller) can be block-formed.
Eigenvalues of the augmented matrix equal union of eigenvalues of (A + B K) and (A + L C).
Design K (controller) and L (observer) independently via eigenvalue placement (e.g., Ackermann or place).
Discrete-time stability criterion: eigenvalues must lie inside the unit circle.

Systems naturally evolve in continuous time; discretization occurs due to sampling and controller update intervals.
Sampling of outputs and controller updates do not occur simultaneously in practice.
One-step observer uses current sampled output y(k) and current control u(k).
Two-step observers account for timing offsets; use predicted state estimates to align measurements and updates.
Two-step observer error dynamics often become e(k+1) = (A + L C A) e(k) (C replaced by C A in effect), similar in design complexity to one-step case.

There exist sets of LMIs for controller and observer design in both continuous and discrete time.
Multiple formulations exist: direct LMI forms, null-space formulations, and alternative equivalent LMIs.
Specific discrete-time and continuous-time LMIs were shown (from Duan and You).
Instructor notes some formulations are more complicated (e.g., null-space form).

Simple system: single input affecting state 2; outputs are states 1 and 3.
Controllability reasoning:
- Input directly affects state 2.
- State 2 affects state 1 directly → can control state 1 via state 2.
- State 3 affected by state 1 → can control state 3 indirectly via two-step mixing.
- Conclusion: system is controllable.
Observability reasoning:
- Outputs measure states 1 and 3 directly.
- State 2 dynamics depend on state 1 (which is measured), allowing indirect observation of state 2.
- Conclusion: system is observable.
Students encouraged to analyze a jet aircraft model similarly for controllability/observability.

Luenberger Observer: observer that propagates a state estimate and corrects via output prediction error.
One-Step Observer: observer using current sample and update within a single discrete step.
Two-Step Observer: observer that compensates for sampling/update timing by predicting state between steps.
Separation Principle: controller and observer can be designed independently; closed-loop eigenvalues are union of controller and observer eigenvalues.
Stability (Discrete-Time): system stable if all eigenvalues lie inside the unit circle.

Practice designing L such that A + L C has eigenvalues inside the unit circle.
Practice eigenvalue placement for controller K so that A + B K meets discrete-time specifications.
Apply LMI formulations to simple design problems (both continuous and discrete versions).
Analyze controllability and observability of provided aircraft model example using similar reasoning.