lqr

Linear-Quadratic Regulator (LQR) design

Description

example

[K,S,P] = lqr(sys,Q,R,N) calculates the optimal gain matrix K, the solution S of the associated algebraic Riccati equation, and the closed-loop poles P for the continuous-time or discrete-time state-space model sys. Q and R are the weight matrices for states and inputs, respectively. The cross term matrix N is set to zero when omitted.

example

[K,S,P] = lqr(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 continuous-time state-space matrices A and B.

Examples

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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 θ. The control input u is the horizontal force on the cart.

[x˙x¨θ˙θ¨]=[01000-0.13000010-0.5300][xx˙θθ˙]+[0205]uy=[10000010][xx˙θθ˙]+[00]u

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

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

Since the outputs are x and θ, 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 δ and the output is the aircraft pitch angle θ.

[α˙q˙θ˙]=[-0.31356.70-0.0139-0.4260056.70][αqθ]+[0.2320.02030][δ]y=[001][αqθ]+[0][δ]

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

  • Rise time less than 2 seconds

  • Settling time less than 10 seconds

  • Steady-state error less than 2%

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 5 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.

Input Arguments

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Dynamic system model, specified as an ss model object.

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

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

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:

Qi,i=1maximum acceptable value of (errorstates)2, i{1,2,...,n}Q=[Q1,1000Q2,200000Qn,n]

Here, n is the number of states.

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:

Rj,j=1maximum acceptable value of (errorinputs)2, j{1,2,...,m}R=[R1,1000R2,200000Rm,m]

Here, m is the number of inputs.

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

Output Arguments

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

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.

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

Limitations

The input data must satisfy the following conditions:

  • The pair A and B must be stabilizable.

  • [Q,N;N',R] must be nonnegative definite.

  • R>0 and QNR1NT0.

  • (QNR1NT,ABR1NT) has no unobservable mode on the imaginary axis (or unit circle in discrete time).

Tips

  • lqr supports descriptor models with nonsingular E. The output S of lqr is the solution of the algebraic Riccati equation for the equivalent explicit state-space model:

    dxdt=E1Ax+E1Bu

Algorithms

For continuous-time systems, lqr computes the state-feedback control u=Kx that minimizes the quadratic cost function

J(u)=0(xTQx+uTRu+2xTNu)dt

subject to the system dynamics x˙=Ax+Bu.

In addition to the state-feedback gain K, lqr returns the solution S of the associated algebraic Riccati equation

ATXE+ETXA+ETXGXE-(ETXB+S)R-1(BTXE+ST)+Q = 0

and the closed-loop poles P = eig(ABK). The gain matrix K is derived from S using

K=R1(BTS+NT).

For discrete-time systems, lqr computes the state-feedback control un=Kxn that minimizes

J=n=0{xTQx+uTRu+2xTNu}

subject to the system dynamics xn+1=Axn+Bun.

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

Introduced before R2006a