Constructor for Statespace Realizations

Any rational (\((m,n)\)-dimensional) matrix \(x(z)\) which has no pole at \(z=0\) may be represented as $$x(z) = C(z^{-1} I_s - A)^{-1} B + D$$ Here \(I_s\) denotes the \((s,s)\)-dimensional identity matrix and \(A,B,C,D\) are (real or complex valued) matrices of size \((s,s)\), \((s,n)\), \((m,s)\) and \((m,n)\) respectively. The integer \(s \geq 0\) is called the state dimension of the above statespace realization of \(x(z)\).

Usage

stsp(A, B, C, D)

Arguments

A

\((s,s)\) matrix (or a vector of length \((s^2)\))

B

\((s,n)\) matrix (or a vector of length \((sn)\))

C

\((m,s)\) matrix (or a vector of length \((ms)\))

D

\((m,n)\) matrix (or a vector of length \((mn)\))

Value

An object of class stsp.

Details

Internally statespace realizations are stored as an \((s+m, s+n)\)-dimensional matrix with an attribute order = c(m,n,s) and a class attribute c("stsp","ratm").

Any of the integers \(m,n,s\) may be zero.

If the arguments A,B,C are missing, then a statespace realization with statespace dimension \(s=0\) is constructed. In this case D must a be matrix or a scalar. If A,B,C are given (and compatible) and D is missing, then D = diag(x = 1, nrow = m, ncol = n) is used.

See also

• test_stsp generates random polynomials.

• checks: is.stsp, is.miniphase, is.stable and is.minimal.

• generic S3 methods: dim, str and print.

• arithmetics: Ops.ratm, matrix multiplication %r%, ...

• matrix operations: transpose t.stsp, Hermitean transpose Ht, bind, [.stsp, ...

• extract the system matrices \(A,B,C,D\) with $.stsp.

• statespace realizations related tools: state transfromation state_trafo, observability and controllability matrices obs_matrix and ctr_matrix, Grammian matrices grammians and balanced realizations balance..

• reflect_poles and reflect_zeroes may be used to reflect poles and zeroes of a rational matrix in statespace form by multiplication with allpass rational matrices.

• zeroes, pseries, zvalues, ...

Examples

### x(z) =  I
stsp(D = diag(3))
#> statespace realization [3,3] with s = 0 states
#>      u[1] u[2] u[3]
#> x[1]    1    0    0
#> x[2]    0    1    0
#> x[3]    0    0    1

### random (2 x 3) rational matrix in statespace form with state dimension s = 4
x = stsp(A = stats::rnorm(4*4), B = stats::rnorm(4*3), C = stats::rnorm(2*4))

is.stsp(x)
#> [1] TRUE
dim(x)
#> m n s 
#> 2 3 4 
str(x)
#> ( 2 x 3 ) statespace realization with s = 4 states
print(x, digits = 3)
#> statespace realization [2,3] with s = 4 states
#>        s[1]   s[2]   s[3]   s[4]   u[1]   u[2]   u[3]
#> s[1]  0.383  1.445 -0.910 -0.719 -0.306  0.934 -0.727
#> s[2]  0.476 -3.237 -0.510  1.410 -0.150  0.159 -1.070
#> s[3] -0.328 -0.239 -0.232 -0.111 -1.604 -0.028  0.623
#> s[4] -1.020 -0.225  0.982  1.249  0.016 -0.314 -1.199
#> x[1] -0.001  1.631 -0.415  0.111  1.000  0.000  0.000
#> x[2]  0.998  0.056 -0.437  1.105  0.000  1.000  0.000