State Feedback, and Optimal Control

LQR - Linear Quadratic Regulator

Given a continous-time LTI system in state space form, where D = 0

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

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

• Q and R are positive-definite weighting matrices

  Positive-Definite Matrix

  A symmetric matrix with all positive eigenvalues. NOTE: All symmetric matrices have only real eigenvalues

  For a matrix, $M$, to be positive-semidefinite the following has to hold

  Criterion

  $x^{T}Mx>0,\forall x\in {\mathbb{R}}^n\backslash \{0\}$ Given that $M$ is a $n\times n$ symmetric real matrix

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  Can be written generally as

  $$M=\left[\begin{array}{cc}{m}_1& m_2\\ m_2& m_3\end{array}\right]$$Link to original

• $Q,R>0$, $Q,R=Q^T,R^T$
• 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
• PROTIP: WRITE OUT THE EQUATION TO SEE DIRECTLY HOW EACH PART OF Q AND R AFFECTS EACH STATE

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

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 $u(t)$

Examples