> For the complete documentation index, see [llms.txt](https://nelson-9.gitbook.io/nelson/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://nelson-9.gitbook.io/nelson/markdown/en/control_system/lqr.md).

# lqr

Linear-Quadratic Regulator (LQR) design.

## 📝 Syntax

* \[K, S, P] = lqr(sys, Q, R, N)
* \[K, S, P] = lqr(A, B, Q, R, N)

## 📥 Input argument

* sys - LTI model
* Q - State-cost weighted matrix
* R - Input-cost weighted matrix
* N - Optional cross term matrix: 0 by default.
* A - State matrix: n x n matrix.
* B - Input-to-state matrix: n x m matrix.

## 📤 Output argument

* K - Optimal gain: row vector.
* S - Solution of the Algebraic Riccati Equation.
* p - Poles of the closed-loop system: column vector.

## 📄 Description

In the context of continuous-time state-space matrices A and B, the command \[K, S, P] = lqr(A, B, Q, R, N) computes the optimal gain matrix K, the solution S to the associated algebraic Riccati equation, and the closed-loop poles P.

This syntax is applicable exclusively to continuous-time models.

When applied to a continuous-time or discrete-time state-space model represented by sys, the command \[K, S, P] = lqr(sys, Q, R, N) computes the optimal gain matrix K, the solution S to the associated algebraic Riccati equation, and the closed-loop poles P.

The weight matrices Q and R govern the importance of states and inputs, and the cross term matrix N is zero by default when not specified.

## 💡 Example

```matlab
A = [-0.313 56.7 0; -0.0139 -0.426 0; 0 56.7 0];
B = [0.232; 0.0203; 0];
C = [0 0 1];
D = 1;
Ts = 1.2;
sys1 = ss(A, B, C, D, Ts);
sys2 = ss(A, B, C, D);

P = 2;
Q = P * C' * C;
R = 2;
[K1, S1, e1] = lqr(sys1, Q, R)
[K2, S2, e2] = lqr(sys2, Q, R)

```

## 🔗 See also

[care](/nelson/markdown/en/control_system/care.md), [dare](/nelson/markdown/en/control_system/dare.md), [lqe](/nelson/markdown/en/control_system/lqe.md).

## 🕔 History

| Version | 📄 Description  |
| ------- | --------------- |
| 1.0.0   | initial version |