Linear-Quadratic Regulator (LQR) design - MATLAB lqr (2024)

pendulumModelCart.mat contains the state-space model of an inverted pendulum on a cart where the outputs are the cart displacement x and the pendulum angle $\theta$. The control input u is the horizontal force on the cart.

$$\begin{array}{l}\left[\begin{array}{c}\underset{}{\overset{\dot{}}{x}}\\ \underset{}{\overset{¨}{x}}\\ \underset{}{\overset{\dot{}}{\theta }}\\ \underset{}{\overset{¨}{\theta }}\end{array}\right]=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& -0.1& 3& 0\\ 0& 0& 0& 1\\ 0& -0.5& 30& 0\end{array}\right]\left[\begin{array}{c}x\\ \underset{}{\overset{\dot{}}{x}}\\ \theta \\ \underset{}{\overset{\dot{}}{\theta }}\end{array}\right]+\left[\begin{array}{c}0\\ 2\\ 0\\ 5\end{array}\right]u\\ y=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\end{array}\right]\left[\begin{array}{c}x\\ \underset{}{\overset{\dot{}}{x}}\\ \theta \\ \underset{}{\overset{\dot{}}{\theta }}\end{array}\right]+\left[\begin{array}{c}0\\ 0\end{array}\right]u\end{array}$$

First, load the state-space model sys to the workspace.

load('pendulumCartModel.mat','sys')

Since the outputs are x and $\theta$, and there is only one input, use Bryson's rule to determine Q and R.

Q = [1,0,0,0;... 0,0,0,0;... 0,0,1,0;... 0,0,0,0];R = 1;

Find the gain matrix K using lqr. Since N is not specified, lqr sets N to 0.

[K,S,P] = lqr(sys,Q,R)

K = 1×4 -1.0000 -1.7559 16.9145 3.2274

S = 4×4 1.5346 1.2127 -3.2274 -0.6851 1.2127 1.5321 -4.5626 -0.9640 -3.2274 -4.5626 26.5487 5.2079 -0.6851 -0.9640 5.2079 1.0311

P = 4×1 complex -0.8684 + 0.8523i -0.8684 - 0.8523i -5.4941 + 0.4564i -5.4941 - 0.4564i

Although Bryson's rule usually provides satisfactory results, it is often just the starting point of a trial-and-error iterative design procedure to tune your closed-loop system response based on the design requirements.

aircraftPitchModel.mat contains the state-space matrices of an aircraft where the input is the elevator deflection angle $\delta$ and the output is the aircraft pitch angle $\theta$.

$$\begin{array}{l}\left[\begin{array}{c}\underset{}{\overset{\dot{}}{\alpha }}\\ \underset{}{\overset{\dot{}}{q}}\\ \underset{}{\overset{\dot{}}{\theta }}\end{array}\right]=\left[\begin{array}{ccc}-0.313& 56.7& 0\\ -0.0139& -0.426& 0\\ 0& 56.7& 0\end{array}\right]\left[\begin{array}{c}\alpha \\ q\\ \theta \end{array}\right]+\left[\begin{array}{c}0.232\\ 0.0203\\ 0\end{array}\right]\left[\delta \right]\\ y=\left[\begin{array}{ccc}0& 0& 1\end{array}\right]\left[\begin{array}{c}\alpha \\ q\\ \theta \end{array}\right]+\left[0\right]\left[\delta \right]\end{array}$$

For a step reference of 0.2 radians, consider the following design criteria:

Load the model data to the workspace.

load('aircraftPitchModel.mat')

Define the state-cost weighted matrix Q and the control weighted matrix R. Generally, you can use Bryson's Rule to define your initial weighted matrices Q and R. For this example, consider the output vector C along with a scaling factor of 2 for matrix Q and choose R as 1. R is a scalar since the system has only one input.

R = 1

R = 1

Q1 = 2*C'*C

Q1 = 3×3 0 0 0 0 0 0 0 0 2

Compute the gain matrix using lqr.

[K1,S1,P1] = lqr(A,B,Q1,R);

Check the closed-loop step response with the generated gain matrix K1.

sys1 = ss(A-B*K1,B,C,D);step(sys1)

Since this response does not meet the design goals, increase the scaling factor to 25, compute the gain matrix K2, and check the closed-loop step response for gain matrix K2.

Q2 = 25*C'*C

Q2 = 3×3 0 0 0 0 0 0 0 0 25

[K2,S2,P2] = lqr(A,B,Q2,R);sys2 = ss(A-B*K2,B,C,D);step(sys2)

In the closed-loop step response plot, the rise time, settling time, and steady-state error meet the design goals.

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 icare and 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.