reduce allocations and improve performance in sminreal (#652)

* reduce allocations and improve performance in sminreal

* add pointers between minreal and sminreal docstrings

Author: baggepinnen

src/simplification.jl

@@ -13,10 +13,11 @@ trunc_zero!(A) = A[abs.(A) .< 10eps(maximum(abs, A))] .= 0
 trunc_zero!(sys.A); trunc_zero!(sys.B); trunc_zero!(sys.C)
 sminreal(sys)
 ```
+See also [`minreal`](@ref)
 """
 function sminreal(sys::StateSpace)
     A, B, C, inds = struct_ctrb_obsv(sys)
-    return StateSpace(A, B, C, sys.D, sys.timeevol)
+    return basetype(sys)(A, B, C, sys.D, sys.timeevol)
 end
 
 # Determine the structurally controllable and observable realization for the system
@@ -39,10 +40,16 @@ Compute a bit-vector, expressing whether a state of the pair (A, B) is
 structurally controllable based on the location of zeros in the matrices. 
 """
 function struct_ctrb_states(A::AbstractVecOrMat, B::AbstractVecOrMat)
-    bitA = .!iszero.(A)
+    size(A,1) > typemax(UInt16) && error("Maximum size of A exceeded. If you encounter this error, please open an issue. This limit is not fundamental and excists for performance reasons only.")
+    # UInt16 can only store up to 65535, so if A is completely dense and of size larger than 65535, the computations below might overflow. This is exceedingly unlikely though.
+    bitA = UInt16.(.!iszero.(A)) # Convert to Int because mutiplying with a bit matrix is slow
     x = vec(any(B .!= 0, dims=2)) # index vector indicating states that have been affected by input
-    for i = 1:size(A, 1) # apply A nx times, similar to controllability matrix
-        x = x .| .!iszero.(bitA * x)
+    xi = bitA * x
+    xi2 = similar(xi)
+    @. xi = (xi != false) | !iszero(x)
+    for i = 2:size(A, 1) # apply A nx times, similar to controllability matrix
+        mul!(xi2, bitA, xi)
+        @. xi = (xi2 != false) | !iszero(xi)
     end
-    x
-end
+    xi .!= false # Convert back to BitVector
+end

src/types/StateSpace.jl

@@ -389,6 +389,8 @@ end
 Minimal realisation algorithm from P. Van Dooreen, The generalized eigenstructure problem in linear system theory, IEEE Transactions on Automatic Control
 
 For information about the options, see `?ControlSystems.MatrixPencils.lsminreal`
+
+See also [`sminreal`](@ref), which is both numerically exact and substantially faster than `minreal`, but with a much more limited potential in removing non-minimal dynamics.
 """
 function minreal(sys::T, tol=nothing; fast=false, atol=0.0, kwargs...) where T <: AbstractStateSpace
     A,B,C,D = ssdata(sys)