WOLFRAM SYSTEM MODELER

# discreteRiccatiIterative

Newton's method with exact line search for solving discrete algebraic Riccati equation # Wolfram Language

In:= `SystemModel["Modelica.Math.Matrices.Utilities.discreteRiccatiIterative"]`
Out:= # Information

This information is part of the Modelica Standard Library maintained by the Modelica Association.

#### Syntax

```           X = Matrices.Utilities.discreteRiccatiIterative(A, B, R, Q, X0);
(X, r) = Matrices.Utilities.discreteRiccatiIterative(A, B, R, Q, X0, maxSteps, eps);
```

#### Description

This function provides a Newton-like method for solving discrete-time algebraic Riccati equations. It uses Exact Line Search to improve the sometimes erratic convergence of Newton's method. Exact line search in this case means, that at each iteration `i` a Newton step `delta_i`

```  X_i+1 = X_i + delta_i
```

is taken in the direction to minimize the Frobenius norm of the residual

```  r = || A'X_i+1*A - X_i+1 - A'X_i+1*G_i*X_i+1*A + Q ||
```

with

```                       -1
G_i = B*(R + B'*X_i*B) *B'
```

Output `r` is the norm of the residual of the last iteration.

The inputs "maxSteps" and "eps" specify the termination of the iteration. The iteration is terminated if either maxSteps iteration steps have been performed or the relative change delta_i/X_i became smaller than eps.

With an appropriate initial value X0 a sufficiently accurate solution might be reach with a few iteration steps. Although a Lyapunov equation of order `n` (n is the order of the Riccati equation) is to be solved at each iteration step, the algorithm might be faster than a direct method like Matrices.discreteRiccati, since direct methods have to solve the 2*n-order Hamiltonian system equation. The algorithm is taken from  and .

#### References

```   Benner, P., Byers, R.
An Exact Line Search Method for Solving Generalized Continuous-Time Algebraic Riccati Equations
IEEE Transactions On Automatic Control, Vol. 43, No. 1, pp. 101-107, 1998.
 Datta, B.N.
Numerical Methods for Linear Control Systems
```

#### Example

```     A  = [0.9970,    0.0000,    0.0000,    0.0000;
1.0000,    0.0000,    0.0000,    0.0000;
0.0000,    1.0000,    0.0000,    0.0000;
0.0000,    0.0000,    1.0000,    0.0000];

B  = [0.0150;
0.0000;
0.0000;
0.0000];

R = [0.2500];

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

X0=identity(4);

(X,r) = Matrices.Utilities.discreteRiccatiIterative(A, B, R, Q, X0);

//  X = [30.625, 0.0, 0.0, 0.0;
0.0,   1.0, 0.0, 0.0;
0.0,   0.0, 1.0, 0.0;
0.0,   0.0, 0.0, 1.0];

// r =   3.10862446895044E-015
```

Matrices.Utilities.continuousRiccatiIterative
Matrices.discreteRiccati

# Syntax

(X, r) = discreteRiccatiIterative(A, B, R, Q, X0, maxSteps, eps)

# Inputs (7)

A Type: Real[:,size(A, 1)] Description: Matrix A of discrete Riccati equation Type: Real[size(A, 1),:] Description: Matrix B of discrete Riccati equation Default Value: identity(size(B, 2)) Type: Real[size(B, 2),size(B, 2)] Description: Matrix R of discrete Riccati equation Default Value: identity(size(A, 1)) Type: Real[size(A, 1),size(A, 2)] Description: Matrix Q of discrete Riccati equation Default Value: identity(size(A, 1)) Type: Real[size(A, 1),size(A, 2)] Description: Initial approximate solution discrete Riccati equation Default Value: 10 Type: Integer Description: Maximal number of iteration steps Default Value: Matrices.frobeniusNorm(A) * 1e-9 Type: Real Description: Tolerance for stop criterion

# Outputs (2)

X Type: Real[size(X0, 1),size(X0, 2)] Type: Real

# Revisions

• 2010/04/30 by Marcus Baur, DLR-RM