idare

Implicit solver for discrete-time algebraic Riccati equations

Syntax

[X,K,L] = idare(A,B,Q,R,S,E)

[X,K,L,info] = idare(___)

[___] = idare(___,'noscaling')

[___] = idare(___,'anti')

Description

[X,K,L] = idare(A,B,Q,R,S,E) computes the unique stabilizing solution X, state-feedback gain K, and the closed-loop eigenvalues L of the following discrete-time algebraic Riccati equation.

$$A^{T}XA-E^{T}XE-\left(A^{T}XB+S\right){\left(B^{T}XB+R\right)}^{-1}{\left(A^{T}XB+S\right)}^T+Q=0$$

The stabilizing solution X puts all the eigenvalues L inside the unit disk.

Algebraic Riccati equations play a key role in LQR/LQG control, H2- and H-infinity control, Kalman filtering, and spectral or co-prime factorizations.

[X,K,L,info] = idare(___) also returns a structure info which contains additional information about the solution to the discrete-time algebraic Riccati equation.

[___] = idare(___,'noscaling') turns off the built-in scaling and sets all entries of the scaling vectors info.Sx and info.Sr to 1. Turning off scaling speeds up computation but can be detrimental to accuracy when A,B,Q,R,S,E are poorly scaled.

[___] = idare(___,'anti') computes the anti-stabilizing solution X that puts all eigenvalues L outside the unit disk.

Examples

Solve Discrete-Time Algebraic Riccati Equation

For this example, solve the discrete-time algebraic Riccati equation considering the following set of matrices:

$$A=\left[\begin{array}{cc}-0.9& -3\\ 0.7& 0.1\end{array}\right]B=\left[\begin{array}{c}1\\ 1\end{array}\right]Q=\left[\begin{array}{cc}1& 0\\ 0& 3\end{array}\right]R=0.1.$$

Find the stabilizing solution using idare to solve for the above matrices with default values for S and E.

A = [-0.9,-0.3;0.7,0.1];
B = [1;1];
Q = [1,0;0,3];
R = 0.1;
[X,K,L,info] = idare(A,B,Q,R,[],[])

X = 2×2

    4.7687    0.9438
    0.9438    3.2369

K = 1×2

   -0.2216   -0.1297

L = 2×1

   -0.4460
   -0.0027

info = struct with fields:
        Sx: [2x1 double]
        Sr: 1
         U: [2x2 double]
         V: [2x2 double]
         W: [0.0232 0.0428]
    Report: 0

Here, X is the unique stabilizing solution, K contains the state-feedback gain, L contains the closed-loop eigenvalues, while info contains additional information about the solution.

Anti-Stabilizing Solution of Discrete-Time Algebraic Riccati Equation

For this example, solve the discrete-time algebraic Riccati equation considering the following set of matrices:

$$A=\left[\begin{array}{cc}-0.9& -3\\ 0.7& 0.1\end{array}\right]B=\left[\begin{array}{c}1\\ 1\end{array}\right]Q=\left[\begin{array}{cc}1& 0\\ 0& 3\end{array}\right]R=0.1.$$

Find the anti-stabilizing solution using the 'anti' option to solve for the above matrices with default values for S and E.

A = [-0.9,-0.3;0.7,0.1];
B = [1;1];
Q = [1,0;0,3];
R = 0.1;
[X,K,L] = idare(A,B,Q,R,[],[],'anti')

X = 2×2

   -0.5423    0.4996
    0.4996   -0.5569

K = 1×2

 -118.0177  490.9023

L = 2×1

 -371.4426
   -2.2420

Here, X is the unique anti-stabilizing solution, K contains the state-feedback gain, and L contains the closed-loop eigenvalues.

Input Arguments

A,B,Q,R,S,E — Input matrices

matrices

Input matrices, specified as matrices.

The matrices Q and R must be Hermitian. A square matrix is Hermitian if it is equal to its complex conjugate transpose, that is, $$a_{i,j}={\overline{a}}_{j,i}$$.

For more information on Hermitian matrices, see ishermitian.

Matrix E must be non-singular.

When matrices R, S and E are omitted or set to [], idare uses the following default values:

• R = I

• S = 0

• E = I

If the inputs Q and R are scalar-valued, idare interprets them as multiples of the identity matrix.

'noscaling' — Option to turn off built-in scaling

'noscaling'

Option to turn off built-in scaling, specified as 'noscaling'. When you turn off the built-in scaling, idare sets all entries in the scaling vectors info.Sx and info.Sr to 1. Turning off scaling speeds up computation but can be detrimental to accuracy when A,B,Q,R,S,E are poorly scaled.

'anti' — Option to compute the anti-stabilizing solution

'anti'

Option to compute the anti-stabilizing solution, specified as 'anti'. When you enable this option, idare computes the anti-stabilizing solution X that puts all eigenvalues of (A-B*K,E) outside the unit disk.

The unique stabilizing and anti-stabilizing are both needed to know the complete phase portrait of the Riccati differential equations.

Output Arguments

X — Unique solution to the Riccati equation

matrix

Unique solution to the discrete-time algebraic Riccati equation, returned as a matrix.

By default, X is the stabilizing solution of the discrete-time algebraic Riccati equation. When the 'anti' option is used, X is the anti-stabilizing solution.

idare returns [] for X when there is no finite stabilizing solution.

idare returns [] when pencil is singular, that is, [B;S;R] is rank deficient. (since R2024a)

K — State-feedback gain

matrix

State-feedback gain, returned as a matrix.

The state-feedback gain K is computed as:

$$K={\left(B^{T}XB+R\right)}^{-1}\left(B^{T}XA+S^T\right).$$

idare returns [] for K when there is no finite stabilizing solution.

idare returns [] when pencil is singular, that is, [B;S;R] is rank deficient. (since R2024a)

L — Closed-loop eigenvalues

matrix

Closed-loop eigenvalues, returned as a matrix.

The closed-loop eigenvalues L is computed as:

$$L=eig\left(A-BK,E\right).$$

idare returns [] for X and K when there is no finite stabilizing solution. L is non-empty even when X and K are empty matrices.

idare returns L = NaN(n,1) when pencil is singular, that is, [B;S;R] is rank deficient. (since R2024a)

info — Information about the unique solution

structure

Information about the unique solution, returned as a structure with the following fields:

• Sx — Vector of values used to scale the states.

• Sr — Vector of values used to scale the R matrix.

• U, V and W — Vectors of values representing the basis of the stable invariant subspace of the associated scaled matrix pencil. For more information, see Algorithms.

• Report — A scalar with one of the following values:

  • 0 — The unique solution is accurate.

  • 1 — The solution accuracy is poor.

  • 2 — The solution is not finite.

  • 3 — No solution found since the Symplectic spectrum, denoted by [L;1./L], has eigenvalues on the unit circle.

  • 4 — Pencil is singular ([B;S;R] is rank deficient) (since R2024a)

Limitations

• (A-zE,B) must be stabilizable, E and R must be invertible, and [B;S;R] have full column rank for a finite stabilizing solution X to exist and be finite. While these conditions are not sufficient in general, they become sufficient when the following conditions are met:

  • $$\left[\begin{array}{cc}\begin{array}{c}Q\\ S^T\end{array}& \begin{array}{c}S\\ R\end{array}\end{array}\right]\ge 0$$

  • $$\left[\begin{array}{cc}A-B{R}^{-1}{S}^T& Q-S{R}^{-1}{S}^T\end{array}\right]$$ is detectible

Algorithms

Basis of the invariant subspace

idare works with the following pencil, and computes a basis [U;V;W] of the invariant subspace associated with the stable or anti-stable finite eigenvalues of this pencil.

$$M-zN=\left[\begin{array}{ccc}A& 0& B\\ -Q& E^T& -S\\ S^T& 0& R\end{array}\right]-z\left[\begin{array}{ccc}E& 0& 0\\ 0& A^T& 0\\ 0& -B^T& 0\end{array}\right]$$

The data is automatically scaled to reduce the sensitivity of eigenvalues near the unit circle and increase separation between the stable and anti-stable invariant subspaces.

Relationship between the solution, the state-feedback gain, and the scaling vectors

The solution X and state-feedback gain K are related to the scaling vectors, and U,V,W by the following set of equations:

$$\begin{array}{l}X={\text{ D}}_{x}V{U}^{-1}{\text{ D}}_x{E}^{-1}\text{,}\\ K=-D_r{\text{ WU}}^{-1}{\text{ D}}_x,\end{array}$$

where,

$$\begin{array}{l}{D}_x=\mathrm{diag}\left(S_x\right),\\ D_r=\mathrm{diag}\left(S_r\right).\end{array}$$

Version History

Introduced in R2019a

R2024a: Improved singularity detection

The function now returns X = [], K = [], L = NaN(n,1), and info.Report = 4 when pencil is singular, that is, [B;S;R] is rank deficient.