dlqr

Linear-quadratic (LQ) state-feedback regulator for discrete-time state-space system

Syntax

[K,S,P] = dlqr(A,B,Q,R,N)

Description

[K,S,P] = dlqr(A,B,Q,R,N) calculates the optimal gain matrix K, the solution S of the associated algebraic Riccati equation, and the closed-loop poles P using the discrete-time state-space matrices A and B. This function is only valid for discrete-time models. For continuous-time models, use lqr.

Input Arguments

A — State matrix

n-by-n matrix

State matrix, specified as an n-by-n matrix, where n is the number of states.

B — Input-to-state matrix

n-by-m matrix

Input-to-state matrix, specified as an n-by-m input-to-state matrix, where m is the number of inputs.

Q — State-cost weighted matrix

matrix

State-cost weighted matrix, specified as an n-by-n matrix, where n is the number of states. You can use Bryson's rule to set the initial values of Q given by:

$$\begin{array}{l}{Q}_{i,i}=\frac{1}{\text{maximum acceptable value of }{\left(erro{r}_{states}\right)}^2},\text{ i}\in \left\{1,2,\mathrm{...},n\right\}\\ Q=\left[\begin{array}{cccc}{Q}_{1,1}& 0& \cdots & 0\\ 0& Q_{2,2}& \cdots & 0\\ 0& 0& \ddots & ⋮\\ 0& 0& \cdots & Q_{n,n}\end{array}\right]\end{array}$$

Here, n is the number of states.

R — Input-cost weighted matrix

scalar | matrix

Input-cost weighted matrix, specified as a scalar or a matrix of the same size as D'D. Here, D is the feed-through state-space matrix. You can use Bryson's rule to set the initial values of R given by:

$$\begin{array}{l}{R}_{j,j}=\frac{1}{\text{maximum acceptable value of }{\left(erro{r}_{inputs}\right)}^2},\text{ j}\in \left\{1,2,\mathrm{...},m\right\}\\ R=\left[\begin{array}{cccc}{R}_{1,1}& 0& \cdots & 0\\ 0& R_{2,2}& \cdots & 0\\ 0& 0& \ddots & ⋮\\ 0& 0& \cdots & R_{m,m}\end{array}\right]\end{array}$$

Here, m is the number of inputs.

N — Optional cross term matrix

0 (default) | matrix

Optional cross term matrix, specified as a matrix. If N is not specified, then lqr sets N to 0 by default.

Output Arguments

K — Optimal gain

row vector

Optimal gain of the closed-loop system, returned as a row vector of size n, where n is the number of states.

S — Solution of the associated algebraic Riccati equation

matrix

Solution of the associated algebraic Riccati equation, returned as an n-by-n matrix, where n is the number of states. In other words, S is the same dimension as state-space matrix A. For more information, see idare.

P — Poles of the closed-loop system

column vector

Poles of the closed-loop system, returned as a column vector of size n, where n is the number of states.

Algorithms

dlqr computes the optimal gain matrix K such that the state-feedback law $$u\left[n\right]=-Kx\left[n\right]$$ minimizes the quadratic cost function

$$J(u)={\displaystyle \sum _{n=1}^{\infty }(x{\left[n\right]}^{T}Qx\left[n\right]+u{\left[n\right]}^{T}Ru\left[n\right]+2x{\left[n\right]}^{T}Nu}\left[n\right])$$

for the discrete-time state-space model $$x\left[n+1\right]=Ax\left[n\right]+Bu\left[n\right]$$.

In addition to the state-feedback gain K, dlqr returns the infinite horizon solution S of the associated discrete-time Riccati equation

$$A^{T}SA-S-(A^{T}SB+N){(B^{T}SB+R)}^{-1}(B^{T}SA+N^T)+Q=0$$

and the closed-loop eigenvalues $$P=eig\left(A-BK\right)$$. The gain matrix K is derived from S using

$$K={(B^{T}SB+R)}^{-1}(B^{T}SA+N^T)$$

In all cases, when you omit the cross term matrix N, dlqr sets N to 0.

Version History

Introduced before R2006a