3.5.36 · D5Guidance, Navigation & Control (GNC)
Question bank — LQG — LQR + Kalman filter, separation principle
Quick symbol reminder, so nothing here is used before it means something:
- = the true state (what we want to control); = our estimate of it; = estimation error.
- = control input, built from the estimate, with regulator gain .
- = noisy measurement; = Kalman gain (how hard we correct toward ).
- = controller (regulator) closed loop; = estimator error dynamics.
- = process-noise covariance, = sensor-noise covariance.
- CARE = Continuous-time Algebraic Riccati Equation (the steady-state Riccati equation solved for the regulator).
True or false — justify
Separation means controller and estimator poles are the union of the two designs.
True. The joint matrix is block upper-triangular (see the red block in the figure), so its eigenvalues are exactly .
Separation means the process and sensor noise no longer cost you anything.
False. Separation decouples the gain design and pole placement, not the performance. The off-diagonal term injects estimation error into the state, so LQG cost is strictly larger than ideal full-state LQR cost.
LQG inherits LQR's guaranteed phase margin.
False. Doyle (1978): "LQG has no guaranteed margins." Inserting the Kalman filter can wipe out LQR's robustness; you use LQG-LTR Loop Transfer Recovery to try to recover it.
Making larger relative to produces a faster, more aggressive regulator.
True. Big penalizes state error heavily, so the optimizer accepts large control effort to kill error quickly, pushing poles further left.
Making sensor noise larger makes the Kalman gain larger.
False — it makes smaller. Algebraically , and shrinks as grows, so : a noisy sensor makes the innovation untrustworthy, and we lean on the model prediction instead.
The Kalman filter and the LQR are solved by two different kinds of equation.
False. Both solve an algebraic Riccati Equation; the filter one is the exact dual of the control one under .
If the true system is nonlinear, LQG is still exactly optimal.
False. The "L" in LQG is Linear; optimality is proven only for linear dynamics, Gaussian noise, and quadratic cost. On nonlinear plants it is at best a good local approximation.
For the same numbers, the scalar LQR gain and scalar Kalman gain come out identical.
True (when the data is dual). The CARE with is ; the filter Riccati with is the term-for-term dual — same equation, same positive root , so .
Spot the error
"The optimal control is : just feed the measurement back through the gain."
Wrong twice: carries sensor noise , and only sees a projection of the state, so unmeasured components are invisible. Correct is using the filtered estimate.
"Since the closed loop poles separate, the estimation error never reaches the state."
The zero block sits in the error's row, meaning can't corrupt — not that can't reach . The term in the state row shows error does drive the state.
"We plug into the running estimator, using the true ."
The whole point of estimation is that we don't have ; the implementable law is . Using would be full-state LQR, which is not the LQG problem.
"The CARE is ."
The quadratic term is subtracted: . That minus sign is what stabilizes the closed loop; a plus would give the wrong (anti-stabilizing) .
"Big makes the controller aggressive because we care more about control."
Backwards. Big penalizes control effort, so the optimizer uses gentle inputs — slower, fuel-saving, less aggressive.
" is the state; the filter estimates it."
is the error covariance, a fixed matrix from the filter Riccati equation. The estimated state is ; tells you how uncertain that estimate is.
Why questions
Why do we choose a quadratic cost rather than, say, the absolute value of error?
Squaring keeps every term positive, punishes big deviations disproportionately, and — decisively — makes the minimizing control linear in the state, which is what yields the clean feedback law .
Why do we guess for the cost-to-go before we've solved anything?
A quadratic guess is the natural match to a quadratic running cost and linear dynamics; substituting it into the HJB equation self-consistently reproduces a quadratic, so the guess is justified by working.
Why is the single block "the entire mathematical content of separation"?
A block-triangular matrix's eigenvalues are just its diagonal blocks' eigenvalues, so that zero forces controller and estimator spectra to split cleanly — no cross terms, no coupled optimization.
Why must (strictly positive), while may be only semidefinite?
appears in ; if were singular that inverse wouldn't exist. only weights states, so allowing zero penalty on some states is fine.
Why is the Kalman filter called the dual of LQR rather than just "similar"?
The error matrix has the same algebraic form as under the exact swap , so solving one Riccati equation solves the other by relabelling — a precise duality, not a loose analogy.
Why does inserting the Kalman filter risk destroying LQR's robustness?
The estimator adds its own dynamics and lag into the loop; the guaranteed LQR margins were proven for direct state feedback, and that proof no longer applies once replaces .
Edge cases
What if the system is not observable?
Then some state modes leave no fingerprint in , so no can drive their estimation error to zero — the filter Riccati equation has no stabilizing and LQG fails.
What if the system is not controllable (or not stabilizable)?
Then some unstable mode can't be moved by , so no stabilizes — the CARE lacks a stabilizing and the LQR half of LQG is ill-posed.
What if the sensor-noise covariance is singular (not invertible)?
The Kalman gain needs , so a singular makes the filter Riccati equation ill-posed — it says some measurement direction is perfectly noise-free, demanding infinite gain there. You either remove that channel (treat it as an exact algebraic constraint) or add a tiny regularizing floor to .
What if the process-noise covariance is singular (non-invertible)?
itself isn't inverted, so the filter Riccati equation isn't immediately broken; but a singular means some state directions get no excitation, and a unique stabilizing is only guaranteed when is stabilizable — otherwise unexcited unstable modes leave undefined, mirroring the controllability requirement on the LQR side.
What if process noise (perfect model, no gusts)?
The filter increasingly trusts prediction: small, the estimator relaxes toward open-loop propagation since there's little disturbance to chase.
What if sensor noise (a perfect sensor of )?
blows up to correct hard toward the flawless measurement; the estimator poles push far left, effectively reconstructing the observed states almost instantly.
What if and simultaneously?
No noise means no estimation error to worry about; and LQG collapses to plain full-state LQR, recovering its cost and its margins.
What if we use the Kalman estimate in a system that's stable but only detectable, not observable?
Detectability guarantees the unobservable modes are already stable, so the filter still yields a stabilizing estimator even though those modes can't be estimated precisely — LQG remains well-posed.