Observer Design and Real-Time Estimation

Overview

• Lecture covers observers (part B of Lecture 6), focusing on real-time state estimation.
• Goal: implement full-state feedback when only outputs y(t) are measured.
• Key idea: design an observer that estimates states x̂(t) so estimation error e(t) → 0.
• Main result: existence and eigenvalue placement of observers equivalent to system observability.

Observability Vs Implementation

• States x are representation artifacts; measured signals are outputs y(t).
• C is typically not invertible (fewer outputs than states), so instantaneous inversion fails.
• Observability classically uses output history over a time interval to infer initial state.
• Practical need: real-time state estimates for feedback — this motivates dynamic observers.

Observer Block Diagram And Goals

• Inputs to observer: measured outputs y(t) and known control input u(t).
• Observer output: x̂(t) (estimate of the state).
• Desired properties:
  • Design observer so e(t) = x̂(t) − x(t) → 0 for all initial conditions.
  • When controller uses x̂(t) (u = K x̂), closed-loop system remains stable.
• Twin goals: (1) design stable observer; (2) ensure closed-loop stability with observer-based control.

General Linear Observer Structure

• Consider a linear observer with dynamics driven by y(t) and u(t).
• Most general linear finite-dimensional observer can be parameterized by several matrices (up to six).
• Practical design restricts observer order to n (same as plant state dimension) — less is hard/NP-hard.

Error Dynamics Derivation (General Case)

• Define estimation error e(t) = x̂(t) − x(t).
• Writing full dynamics with observer internal state leads to many coupling terms (y, u, z, u̇, etc.).
• Goal: choose observer structure to cancel extra terms and yield simple ė dynamics.

Luenberger Observer (Preferred Practical Form)

• Luenberger observer form:
  • x̂̇ = A x̂ + B u + L (y − C x̂)
  • output estimate ŷ = C x̂
• Interpretation:
  • Propagate state estimate with plant dynamics (A x̂ + B u).
  • Continuously correct estimate using innovation (y − ŷ) multiplied by observer gain L.
• This reduces the many parameters to a single design parameter L (the observer gain).

Simplified Error Dynamics With Luenberger Observer

• Error dynamics simplify to ė = (A + L C) e.
• Design task: choose L such that A + L C is Hurwitz (stable).
• Analogy with controller design where closed-loop A + B K must be Hurwitz.

Duality And Eigenvalue Assignment

• Observer design maps to controller synthesis of the dual system:
  • Consider (Aᵀ, Cᵀ) as the controllability pair for dual system.
  • Using results from controller design, eigenvalues of A + L C can be arbitrarily assigned iff system is observable.
• Detectability notion (briefly mentioned) refines existence for asymptotic estimation when full observability not present.

LMI Approach For Observer Design

• Stabilizability LMIs from controller design carry over to observability via transpose duality.
• Solve LMI for Aᵀ + Cᵀ K stabilization; then Lᵀ = K and L = Kᵀ gives observer gain.
• This produces Luenberger gains guaranteeing e(t) → 0 and allows eigenvalue placement.

Table: Observer Design Summary

Observer-Based Controller And Closed-Loop Stability

• Implement controller u = K x̂ using the state estimate.
• Aggregate closed-loop state uses [x; x̂] or alternatively [x; e].
• The aggregate system eigenvalues equal the union of eigenvalues of (A + B K) and (A + L C).
• Therefore stability achieved if both A + B K and A + L C are Hurwitz.
• Practical design guideline: choose observer eigenvalues (A + L C) significantly faster (more negative) than controller eigenvalues for robustness, though not strictly required.

Proof Outline For Closed-Loop Eigenstructure

• Construct closed-loop state matrix; perform similarity transformation to block upper-triangular form.
• Eigenvalues of block upper-triangular matrix are eigenvalues of diagonal blocks: A + B K and A + L C.
• Hence observer and controller eigenvalues determine closed-loop spectrum.

Using LMIs For Region Constraints

• Can impose D-stability (LMIs that constrain eigenvalues to specified complex-plane regions).
• Stack LMIs for controller (K) and observer (L) to ensure closed-loop eigenvalues lie in desired region.
• Applicable to specifications like settling time and percent overshoot.

Table: Design Workflow (Controller + Observer)

Key Terms And Definitions

• Observability: ability to infer initial state from output history.
• Luenberger Observer: dynamic estimator x̂̇ = A x̂ + B u + L (y − C x̂).
• Innovation (or residual): y − ŷ, used to correct state estimate.
• Detectability: weaker than observability; ensures estimation of unstable modes.
• Duality: mapping observer design to controller design on transposed system matrices.
• LMI (Linear Matrix Inequality): convex condition used to synthesize gains and enforce region constraints.

Action Items / Next Steps

• For given plant (A, B, C, D):
  • Check observability and controllability.
  • Use LMI methods or pole placement to compute K and L.
  • Prefer designing observer poles faster than controller poles (rule of thumb).
• Prepare to study discrete-time versions and summary (covered briefly at end of lecture).