Questions: State Feedback Control and Pole Placement

Question 1

A 3rd-order system has one uncontrollable mode. An engineer applies state feedback u = −Kx and attempts to move all three poles to stable locations in the left half-plane. What will actually happen?

• A. All three poles can be placed arbitrarily if the gain K is chosen large enough
• B. The two controllable modes can be placed anywhere, but the uncontrollable mode remains fixed at its open-loop location
• C. The feedback will stabilize all modes because u = −Kx affects the full state vector x
• D. Ackermann's formula will find a valid K regardless of controllability

Answer: (B)

The pole placement theorem is precise: arbitrary pole assignment is possible if and only if the system is controllable. An uncontrollable mode is completely disconnected from the input — no choice of K can affect it, because the control signal never reaches that mode. The gain K acts on all elements of x, but if a mode of the dynamics does not respond to u, the feedback has no leverage there. Ackermann's formula will fail (the controllability matrix will be rank-deficient), and the uncontrollable pole stays put.

Question 2

An engineer designs state feedback poles very far into the left half-plane to achieve extremely fast settling. What practical concern does this raise?

• A. The system will lose controllability as the poles move farther left
• B. Poles in the far left half-plane produce imaginary eigenvalues of (A − BK)
• C. Large feedback gains amplify sensor noise and may saturate actuators, causing real behavior to diverge from the linear model
• D. The observer design that follows will be unable to estimate the state quickly enough

Answer: (C)

Placing poles far to the left requires large entries in K, which means the control signal u = −Kx becomes very sensitive to the state — including measurement noise in each state variable. High-gain feedback amplifies noise, potentially driving actuators into saturation where the linear model no longer applies. The design trade-off is fundamental: faster response costs proportionally more control effort and increases sensitivity to noise and model uncertainty. Option D is a concern but is separate from the pole placement design step itself.

Question 3

In state feedback design, the closed-loop poles are the eigenvalues of the matrix (A − BK).

• T. True
• F. False

Answer: True

This follows directly from the closed-loop dynamics. With u = −Kx substituted into ẋ = Ax + Bu, we get ẋ = Ax + B(−Kx) = (A − BK)x. The eigenvalues of (A − BK) are the closed-loop poles — the values that determine stability and transient response. By choosing K to shape (A − BK), the designer moves those eigenvalues to desired locations. This is the mathematical core of state feedback design.

Question 4

Because u = −Kx feeds back the full state vector x, state feedback automatically handles situations where some states cannot be directly measured with sensors.

• T. True
• F. False

Answer: False

This is a common and important misconception. State feedback requires access to the complete state vector x, but in real systems many states are not directly measurable — for example, you might measure position but not velocity, or angular displacement but not angular rate. State feedback assumes x is available; when it is not, a state observer (Luenberger observer or Kalman filter) must be designed to estimate x from available measurements. This is precisely why observer design is the next topic in the sequence: the mathematical elegance of u = −Kx depends on having x, which in practice must often be reconstructed.

Question 5

Why does controllability determine whether state feedback can place closed-loop poles at arbitrary locations, and what happens physically when a mode is uncontrollable?

Model answer: Controllability means every mode of the system dynamics can be influenced by the input u. An uncontrollable mode has a direction in state space that the input cannot reach — the control signal effectively has zero leverage on that mode's behavior. When you apply u = −Kx, the feedback modifies the closed-loop matrix to (A − BK), but an uncontrollable mode corresponds to a direction where BK has no effect, so the eigenvalue associated with that mode is unchanged regardless of K. Physically, it means some aspect of the system's natural dynamics is completely decoupled from the actuator.

The controllability matrix C = [B, AB, A²B, ...] spans exactly the subspace reachable by control inputs. If this matrix is rank-deficient, there are modes the input cannot excite. These modes evolve on their own according to the open-loop dynamics — they are invisible to the feedback law. This is why controllability is a prerequisite: pole placement only works in the controllable subspace, and an uncontrollable unstable mode cannot be stabilized by state feedback alone.