observers.LinearObserver
observers.LinearObserver(system, L)Linear state observer with constant gain.
Implements the observer: d x̂/dt = f(x̂, u) + L(y - h(x̂)) [Continuous-time] x̂[k+1] = f(x̂[k], u[k]) + L(y[k+1] - h(f(x̂[k], u[k]))) [Discrete-time]
where: - x̂ is the state estimate - L is the observer gain matrix - y is the measurement - h(x̂) is the predicted measurement
Attributes: system: SymbolicDynamicalSystem or GenericDiscreteTimeSystem L: Observer gain matrix (nx, ny) x_hat: Current state estimate (nx,)
Example - Continuous-Time Observer: >>> # Design Kalman gain >>> system = SymbolicPendulum(m=1.0, l=0.5) >>> Q_process = np.diag([0.001, 0.01]) >>> R_measurement = np.array([[0.1]]) >>> L = system.kalman_gain(Q_process, R_measurement) >>> >>> # Create observer >>> observer = LinearObserver(system, L) >>> observer.reset(x0=torch.zeros(2)) >>> >>> # Observer loop >>> dt = 0.01 >>> for t in range(num_steps): >>> observer.update(u[t], y_measured[t], dt) >>> x_estimate = observer.x_hat
Example - Discrete-Time Observer: >>> # Create discrete system >>> system_ct = SymbolicQuadrotor2D() >>> system_dt = GenericDiscreteTimeSystem(system_ct, dt=0.01) >>> >>> # Design discrete Kalman gain >>> L = system_dt.discrete_kalman_gain(Q_process, R_measurement) >>> >>> # Create observer >>> observer = LinearObserver(system_dt, L) >>> >>> # Observer loop (no dt needed for discrete) >>> for k in range(num_steps): >>> observer.update(u[k], y[k], dt=None) # dt ignored for discrete >>> x_estimate = observer.x_hat
Example - Output Feedback Control: >>> # Design LQG controller >>> K, L = system.lqg_control(Q_lqr, R_lqr, Q_process, R_measurement) >>> >>> # Create controller and observer >>> controller = LinearController(K, system.x_equilibrium, system.u_equilibrium) >>> observer = LinearObserver(system, L) >>> >>> # Closed-loop with output feedback >>> x_true = x0 >>> observer.reset(x0=system.x_equilibrium) # Start from equilibrium guess >>> >>> for t in range(num_steps): >>> # Measure (with noise) >>> y = system.h(x_true) + torch.randn(system.ny) * 0.1 >>> >>> # Update observer >>> observer.update(u, y, dt) >>> >>> # Compute control based on estimate >>> u = controller(observer.x_hat) >>> >>> # Update true system >>> x_true = system(x_true, u) if discrete else integrate(x_true, u, dt)
Notes: - Gain L is constant and designed at equilibrium - For nonlinear systems, only accurate near equilibrium - No covariance tracking - Can be used with both continuous and discrete systems - Lower computational cost than EKF
See Also: kalman_gain: Design L for continuous systems discrete_kalman_gain: Design L for discrete systems ExtendedKalmanFilter: Adaptive nonlinear observer LinearController: State feedback controller lqg_control: Combined controller and observer design
Methods
| Name | Description |
|---|---|
| reset | Reset observer to initial state estimate. |
| to | Move observer to specified device (CPU/GPU). |
| update | Update observer state estimate. |
reset
observers.LinearObserver.reset(x0=None)Reset observer to initial state estimate.
Args: x0: Initial state estimate (nx,). Uses equilibrium if None.
Example: >>> # Reset to known initial condition >>> observer.reset(x0=torch.tensor([0.1, 0.0])) >>> >>> # Reset to equilibrium >>> observer.reset() # Uses system.x_equilibrium
Notes: - Called automatically in init() - Unlike EKF, no covariance to reset - Start observer from best available estimate
to
observers.LinearObserver.to(device)Move observer to specified device (CPU/GPU).
Args: device: torch.device or string (‘cpu’, ‘cuda’)
Returns: Self for chaining
update
observers.LinearObserver.update(u, y, dt)Update observer state estimate.
Continuous-time: d x̂/dt = f(x̂, u) + L(y - h(x̂)) x̂_new ≈ x̂_old + dt * [f(x̂, u) + L(y - h(x̂))]
Discrete-time: x̂_pred = f(x̂, u) x̂_new = x̂_pred + L(y - h(x̂_pred))
Args: u: Control input (nu,) y: Measurement (ny,) dt: Time step (used for continuous systems, ignored for discrete)
Example: >>> u = torch.tensor([1.0]) >>> y_measured = torch.tensor([0.15]) # Noisy angle measurement >>> observer.update(u, y_measured, dt=0.01) >>> print(f”Estimate: {observer.x_hat}“)
Notes: - For continuous systems: uses Euler integration - For discrete systems: dt parameter is ignored - Innovation = y - h(x̂_pred) is the measurement residual - Large innovation suggests either bad estimate or bad measurement - Gain L determines how much to trust innovation vs model