ashfordeOU Observer Design (gnc-autonomy/control/observer-design)
Use when the task is deterministic full-order state estimation for feedback control: the Luenberger observer gain, the placement of the observer error dynamics eigenvalues, the stability of those error dynamics, and the separation principle that joins them with a controller gain.
Domain quick reference
- Plant: the continuous-time linear time-invariant system x_dot = A x + B u with measured output y = C x, x in R^n. The full-order observer is x_hat_dot = A x_hat + B u + L (y - C x_hat) with the estimation error e = x - x_hat obeying e_dot = (A - L C) e.
- Observability: the observability matrix O = [C; C A; ...; C A^(n-1)] has full column rank n exactly when the pair (A, C) is observable; a rank-deficient O means some state direction never reaches the output and no observer can reconstruct it.
- Ackermann formula: with the desired characteristic polynomial phi(s) = prod_i (s - p_i) built from the observer poles p_i, the estimator gain is L = phi(A) O^{-1} e_n, where e_n is the last unit vector and O must be square (single measured output row). The error dynamics eigenvalues are then exactly the chosen p_i.
- Worked, double integrator: A = [[0, 1], [0, 0]] (position, velocity) with C = [[1, 0]] (position measured only). O = I2, so the pair is observable. Choosing observer poles -4 and -5 rad/s gives phi(s) = s^2 + 9s + 20 and L = [9, 20]; then A - L C = [[-9, 1], [-20, 0]] with characteristic polynomial s^2 + 9s + 20, so the error decays as e^(-4t) and e^(-5t).
- Complex poles: choosing -2 +/- 3j rad/s gives phi(s) = s^2 + 4s + 13 and L = [4, 13] on the same plant; the gain stays real because the poles form a conjugate pair.
- Worked, three states: A = [[-1, 0, 1], [0, -2, 0], [0, 1, -3]] with C = [[1, 0, 0]] and poles -10, -11, -12 rad/s gives L = [27, 720, 216] and error dynamics polynomial s^3 + 33 s^2 + 362 s + 1320 = (s+10)(s+11)(s+12), stable by the Routh array.
- Settling time: for the 2% band t_s = 4 / sigma, where sigma = min_i |Re(p_i)| is the distance of the slowest error pole from the imaginary axis. The worked double integrator gives t_s = 1.0 s; the complex pair -2 +/- 3j gives t_s = 2.0 s.
- Separation principle: with output feedback u = -K x_hat the closed loop is x_dot = (A - B K) x + B K e, e_dot = (A - L C) e, a block upper triangular system whose characteristic polynomial factors into the controller polynomial det(sI - (A - B K)) and the observer polynomial det(sI - (A - L C)). Controller and observer poles can be designed independently. Worked: K = [1, 1] on the double integrator gives controller polynomial s^2 + s + 1 and observer polynomial s^2 + 9s + 20, whose product s^4 + 10 s^3 + 30 s^2 + 29 s + 20 is the closed-loop polynomial.
- Rule of thumb: place observer poles 4 to 10 times faster than the controller poles (in real part) so the estimates settle before the controlled response, without amplifying measurement noise too much.
- Characteristic polynomial: computed by the Faddeev-LeVerrier algorithm from the matrix trace recursion; Hurwitz verdict by the Routh array, which needs every first-column entry strictly positive (marginal roots on the imaginary axis are not stable).
- Units: SI. Pole locations in rad/s, settling time in seconds.
Workflow
- Write the state-space model A, B, C of the plant with the state x, the input u, and the measured output y; the Ackermann gain needs a single output row, C is 1 x n.
- Compute observability_matrix(A, C) and confirm is_observable(A, C) returns True; a rank-deficient O means the observer task is ill-posed for this measurement set.
- Choose the observer poles p_i: strictly negative real parts, real values or conjugate pairs, typically 4 to 10 times faster than the controller poles.
- Compute the estimator gain with observer_gain_ackermann(A, C, poles); the function raises ValueError when the system is not observable, the pole count is wrong, a pole is unstable, or a non-conjugate complex pole set would give a complex gain.
- Verify the design with error_dynamics(A, C, L): the returned error matrix A - L C, its characteristic polynomial, and the Hurwitz stability verdict; the polynomial coefficients must match the desired phi(s).
- Size the convergence with settling_time(poles); 4 / sigma seconds to reach the 2% error band.
- When a controller gain K exists (for example from lqr-design or root-locus-design), run separation_closed_loop(A, B, C, K, L) and confirm factorizes is True, so the closed-loop polynomial is the product of the controller and observer polynomials.
- Implement the observer as flight software: x_hat_dot integration with the injected output error L (y - C x_hat), u = -K x_hat as the feedback path.
Pitfalls
- Routing stochastic estimation here: Kalman gain, innovation, process and measurement noise covariances, and the covariance recursion belong to gnc-autonomy/navigation/kalman-filter-design; observer design is deterministic pole placement with no noise statistics.
- Routing analysis here: controllability and observability verdicts, the state transition matrix, eigenvalue stability, and canonical forms belong to gnc-autonomy/control/state-space-analysis; this leaf synthesizes the estimator gain that the verdict enables, it does not repeat the rank tests as an end in themselves.
- Routing controller gain design here: the feedback gain K from quadratic cost belongs to gnc-autonomy/optimal-control/lqr-design, and pole-placement controller design belongs to gnc-autonomy/control/root-locus-design; this leaf designs the estimator side L, and u = -K x_hat combines both through the separation principle.
- Forgetting the sign convention: the gain enters the observer as +L (y - C x_hat) and the error dynamics are A - L C; a sign flip in either place moves the error poles into the right half plane.
- Designing unstable or marginal observer poles: the error dynamics must be strictly Hurwitz, poles on the imaginary axis never decay.
- Making the observer too fast: 100x faster poles amplify measurement noise through L; 4 to 10x the controller bandwidth is the practical band.
- Applying Ackermann to a multi-output measurement: the formula needs a square observability matrix, that is a single measured output row; multi-output estimation needs a different gain synthesis.
- Reading the Routh array wrong: all polynomial coefficients must be positive and every first-column entry strictly positive; a zero first-column entry means marginal or unstable roots, not a pass.
- Ignoring observability: placing poles on an unobservable pair is silently ineffective for the unobservable state direction, check rank(O) = n first.
- Confusing settling time with the time constant: t_s = 4 / sigma is the 2% band, the time constant is 1 / sigma.
Behavior contract (gate 3)
The observability matrix and rank verdict, the Ackermann estimator gain, the characteristic polynomial, the Hurwitz stability verdict, the separation principle factorization, and the settling time are exercised by the gate 3 contract test: scripts/test_observer_design.py against scripts/observer_design_logic.py (stdlib unittest, offline). Run: python3 scripts/test_observer_design.py
Compliance
- Standards referenced, not reproduced: ARP4754A frames development assurance for aircraft systems and DO-178C frames the flight software that hosts the observer implementation; the observer design equations are common control-theory knowledge, summary-only per standards-map.yaml, both reference-only: true.
- compliance: STANDARDS-REF, gated: false.