LinSys; Linear Quadratic Regulator - LQR

The LQR problem consists of finding $u(t)$ that makes the following equation as small as possible

$$\begin{array}{r}{J}_{LQR}:={\int }_0^{\infty }y(t{)}^{T}Qy(t)+u(t{)}^{T}Ru(t)dt\end{array}$$

And then find the corresponding $K$ (feedback).

LQR seeks to find a controller that minimizes both energies using the weighting matrices Q and R to establish a trade-off between the control signal and output energy.

• Q - How fast the system reacts
• R - How much energy can we allow in the system

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Properties of $Q$ and $R$:

1. Q and R are positive-definite weighting matrices (Positive-Definite Matrix) → They are equal themselves transposed - $Q,R>0$, $Q,R=Q^T,R^T$

• The term ${\int }_0^{\infty }y(t{)}^{T}Qy(t)dt$ is a measure of the output energy
• The term ${\int }_0^{\infty }u(t{)}^{T}Ru(t)dt$ is a measure of the control signal energy
• WRITE OUT THE EQUATION TO SEE DIRECTLY HOW EACH PART OF Q AND R AFFECTS EACH STATE

How to calculate LQR?

The “trick” is to rewrite the cost functional on the special form:

$$J=H(x(\cdot ),u(\cdot ))+{\int }_0^{\infty }\Lambda (x(t),u(t))dt$$

• Where the $H$ term is not affected by the input, directly or indirectly
• and the $\Lambda$ term has an obvious minimum in terms of $u(t)$