Linear quadratic control

In control engineering and systems and control theory, linear quadratic control or LQ control refers to controller design for a deterministic linear plant based on the minimization of a quadratic cost functional. The method is founded on the state space formalism and is a fundamental concept in linear systems and control theory.

There are two main versions of the method, depending on the setting of the control problem:

1. Discrete time linear quadratic control
2. Continuous time linear quadratic control

LQ control aims to find a control signal that minimizes a prescribed quadratic cost functional. In the so-called optimal regulator problem, this functional can be viewed as an abstraction of the "energy" of the overall control system and minimization of the functional corresponds to minimization of that energy.

Discrete time linear quadratic control

Plant model

In discrete time, the plant (the system to be controlled) is assumed to be linear with input $u_{k}$, state $x_{k}$ and output $y_{k}$, and evolves in discrete time k=0,1,... according to the following dynamics:

$x_{k+1}=Ax_{k}+Bu_{k};\,x_{0}=x$

$y_{k}=Cx_{k}+Du_{k},$

where $x,x_{k}\in \mathbb {R} ^{n}$, $u_{k}\in \mathbb {R} ^{m}$ and $y_{k}\in \mathbb {R} ^{p}$ for all $k\geq 0$ and $A,B,C,D$ are real matrices of the corresponding sizes (e.g., for consistency, $A$ should be of the dimension $n\times n$ while $B$ should be of dimension $n\times m$). Here $x$ is the initial state of the plant.

Cost function

For a finite integer $K>0$, called the control horizon or horizon, a real valued cost functional $J$ of the initial state x and the control sequence $u_{0},u_{1},\ldots ,u_{K-1}$ up to time $K-1$ is defined as follows:

$J^{K}(x,u_{0},u_{1},\ldots ,u_{K-1})=\sum _{k=0}^{K-1}(x_{k}^{T}Qx_{k}+u_{k}^{T}Ru_{k})+x_{K}^{T}\Gamma x_{K},\,\,(1)$

where $Q,\Gamma$ are symmetric matrices (of the corresponding sizes) satisfying $Q,\Gamma \geq 0$ and $R$ is a symmetric matrix (of the corresponding size) satisfying $R>0$. Note that each term on the right hand side of (1) are non-negative definite quadratic terms and may be interpreted as abstract "energy" terms (e.g., $x_{k}^{T}Qx_{k}$ as the "energy" of $x_{K}$). The horizon $K$ represents the terminal time of the control action. The final term $x_{K}^{T}\Gamma x_{K}$ is called the terminal cost and it penalizes the energy of the plant at the final state $x_{K}$ .

The LQ regulator problem in discrete time

The objective of LQ control is to solve the optimal regulator problem:

(Optimal regulator problem) For a given horizon K and initial state x, find a control sequence ${\tilde {u}}_{0},{\tilde {u}}_{1},\ldots ,{\tilde {u}}_{K-1}\in \mathbb {R} ^{m}$ that minimizes the cost functional $J^{K}$, that is,

$J^{K}(x,{\tilde {u}}_{0},{\tilde {u}}_{1},\ldots ,{\tilde {u}}_{K-1})\leq J^{K}(x,u_{0},u_{1},\ldots ,u_{K-1}),\,\,(2)$

over all possible control sequences $u_{0},u_{1},\ldots ,u_{K-1}\in \mathbb {R} ^{m}$.

The optimal regulator problem is a type of optimal control problem and the control sequence ${\tilde {u}}_{0},{\tilde {u}}_{1},\ldots ,{\tilde {u}}_{K-1}\in \mathbb {R} ^{m}$ is an optimal control sequence.

Related topics

Linear quadratic Gaussian control