LQR Optimal Control Input for Q=I, R=1 | AI Research

LQR Optimal Control Input for $Q=I$ , $R=1$

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Nov 3, 2026

In the continuous-time Linear Quadratic Regulator (LQR), you minimize the cost functional

J=\int_{0}^{\infty}\left(x(t)^{T}Qx(t)+u(t)^{T}Ru(t)\right)dt

subject to linear dynamics. For LQR, the optimal control is a state-feedback law of the form

u(t)=-Kx(t),

where $K$ is computed from the solution to the (algebraic/differential) Riccati equation. With $Q=I$ and $R=1$ , the structure of the optimal input remains the same: it is still proportional to $-x(t)$ (only the gain value changes).

Therefore, among the choices, the correct option is (i) $-Kx$ .

A useful way to connect the options to LQR is to note that:

$u$ depends on the state via $x$ (state-feedback), not on $Qx$ directly and not on $Ru$ (that would be circular).
$-K$ alone would be a constant control, but LQR gives a gain times the state vector, i.e., $-Kx$ .

LQR: Optimal Control Law u = -Kx

Choosing the Correct Control Input Form

1
Step 1
For standard continuous-time LQR, the optimal input is state feedback $u(t)=-Kx(t)$ , where $K$ comes from solving the Riccati equation.
2
Step 2
Setting $Q=I$ and $R=1$ changes the numerical value of $K$ (the gain), but not the functional dependence on $x(t)$ .
3
Step 3
Options like $-K$ (constant input), $Qx$ (uses the wrong weighting directly), or $Ru$ (not a closed-form control law) do not match the LQR state-feedback form.
4
Step 4
The only option with the LQR-required structure $u=-Kx$ is (i) $-Kx$ .

Pro Tip

In LQR, $Q$ and $R$ mainly determine the gain $K$ via the Riccati solution; the controller form stays $u=-Kx$ for state-feedback LQR.

Common Exam Pitfall

Don't confuse the cost term $x^TQx$ with the control law. LQR does not produce $u=Qx$ ; it produces $u=-Kx$ .

Matching Options to LQR Structure

Qualitative check: which option has the correct state-feedback dependence on $x$ .

How LQR Produces the Control Law

Define cost

$J=\int (x^TQx+u^TRu)dt$ "

Solve Riccati

Find $P$ (or $P(t)$ ) from Riccati equation"

Compute gain

$K=R^{-1}B^TP$ (gain depends on $Q,R$ )"

Apply state feedback

Use $u(t)=-Kx(t)$ "

Quick FAQ: Why it’s $-Kx$

Knowledge Check

Question 1 of 3

Q1Single choice

For continuous-time LQR with cost $J=\int (x^TQx+u^TRu)dt$ , the optimal control input has which form?

$u(t)=-Kx(t)$

$u(t)=-K$

$u(t)=Qx(t)$

$u(t)=Ru(t)$

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LQR Optimal Control Input for Q=IQ=IQ=I, R=1R=1R=1