#485)

* Avoid unnecessarily large realization for feedback of TransferFunction

* Fix and added more tests.

* Change to numpoly, denpoly

* switch u layout for lsim

* Update src/timeresp.jl

Co-authored-by: Fredrik Bagge Carlson <baggepinnen@gmail.com>

* More updates, one error in test_timeresp

* Fix tests

* Change to AbstractVecOrMat

* Catch CuArray in matrix conversion

* General zero vectors for x0 to support GPUs

* Update src/timeresp.jl

Co-authored-by: Mattias Fält <mfalt@users.noreply.github.com>

* Update src/timeresp.jl

Co-authored-by: Mattias Fält <mfalt@users.noreply.github.com>

* Move f outside lsim

* Remove GPU compatible x0, save for later

* Fix doctest

* add kwargs

* Remove variable and generalize type

Co-authored-by: Fredrik Bagge Carlson <baggepinnen@gmail.com>
Co-authored-by: Mattias Fält <mfalt@users.noreply.github.com>

* QOL improvements for plotting

* remove spurious getindex

* update docstring

* multiple Ms in nyquist

* bugfixes

* make rings appear in all subplots

* Minor fixes to gang-of-four functionality.

* Fixes to Nyquist plots.

* Updated the nyquistplot docstring

Co-authored-by: Fredrik Bagge Carlson <baggepinnen@gmail.com>

* updates to nyquistplot

* Update src/plotting.jl

Co-authored-by: olof3 <olof3@users.noreply.github.com>

* Update src/plotting.jl

Co-authored-by: olof3 <olof3@users.noreply.github.com>

Co-authored-by: olof3 <olof3@users.noreply.github.com>

* fix traces in rlocus

* Update src/pid_design.jl

Co-authored-by: Albin Heimerson <albin.heimerson@control.lth.se>

Co-authored-by: Albin Heimerson <albin.heimerson@control.lth.se>

until hessenberg is properly used

* allow balance when converting tf to ss

* use zeros(T) instead of fill(zero(T))

* Update src/types/StateSpace.jl

Co-authored-by: olof3 <olof3@users.noreply.github.com>

* Update src/types/conversion.jl

Co-authored-by: Albin Heimerson <albin.heimerson@control.lth.se>

Co-authored-by: olof3 <olof3@users.noreply.github.com>
Co-authored-by: Albin Heimerson <albin.heimerson@control.lth.se>

* delay error should be correct now

* warn limit

* add Tustin discretization

* no indexing after c2d

* implement f_prewarp in tustin and test vs. matlab

* fixe use wrong Ts

* f_prewarp -> w_prewarp

* w_prewarp in tests

* pi place and tests

* Fix test

* Add ending space in file

* Update numvec denvec

* Fixed error

* Update test/test_pid_design.jl

Co-authored-by: olof3 <olof3@users.noreply.github.com>

* Update test/test_pid_design.jl

Co-authored-by: olof3 <olof3@users.noreply.github.com>

* Update forms

* Fix test

* Fix test

* Update src/pid_design.jl

Co-authored-by: olof3 <olof3@users.noreply.github.com>

* Update src/pid_design.jl

Co-authored-by: olof3 <olof3@users.noreply.github.com>

* Update src/pid_design.jl

Co-authored-by: olof3 <olof3@users.noreply.github.com>

Co-authored-by: olof3 <olof3@users.noreply.github.com>

* Use nstates instead of nx in lsimplot

* up innovation_form and add noise_model

* keep innovation_form, add predictor

* export predictor

* add hats

* updates to `obsv`

All computing an arbitrary number of rows in the observability matrix and accept `AbstractStateSpace`

* Update src/matrix_comps.jl

Co-authored-by: olof3 <olof3@users.noreply.github.com>

Co-authored-by: olof3 <olof3@users.noreply.github.com>

* Add tests for place.

* Removed luenberger and exented place instead.

Co-authored-by: Fredrik Bagge Carlson <baggepinnen@gmail.com>

* allow AbstractStateSpace in several places

* maintain zpk type in c2d

* Fix type in typeconversion for c2d tf

* Fix type in c2d tf/zpk

* Remove assertion on tf/zpk SISO for c2d

* Test for c2d(zpk(..)..) unbroken

* Keep MIMO assertion c2d tf

* Add comment and fix test

* Fix test

* h to Ts

* Fix test

* remove assert

* split into methods

* Replace asserts

* add error types

* fix conversion for custom types

* special numeric_type for AbstractSS

* fix conversion from ss to tf without type

* omre abstract statespaces

* even more ASS

* remove simple feedback in favor of mor egeneral version

* propagate timeevol

* add block diagram to feedback docstring

* Updates to docstring

* fix docstring formatting

* delete redundancy in feedback docstring

Co-authored-by: olof3 <olof3@users.noreply.github.com>

* add controller function

* rename controller and predictor

* change formatting in dampreport

* update dampreport to use ±

* even better formatting

* up formatting for complex systems

* add additive identity element for statespace and TF

* rm zero from type. Test MIMO zero

* Fix struct_ctrb_obsv, closes #409

* Update src/simplification.jl

Co-authored-by: Fredrik Bagge Carlson <baggepinnen@gmail.com>

* add to docstring

Co-authored-by: olof3 <olof3@users.noreply.github.com>

* optional epsilon in dcgain

* Update analysis.jl

Would be good to have a regression test for `BigFloat` data which was the reason I found this bug. That seems slightly involved to fix though.

I assume that this would also have failed for `ComplexF64`, so eventually a test for that too would be good.

remove version checks

* write `struct_ctrb_states` in terms of `iszero` instead of `== 0`

The reason is that `iszero` always returns a bool, whereas `== 0` may return anything. The difference appears for symbolic variables where
```julia
julia> q0 == 0
q0(t) == 0

julia> iszero(q0)
false
```

* Update simplification.jl

* remove issmooth

* drop extra `]`

* Return a result structure from lsim

This is by design a non-breaking change to the lsim interface since the structure allows both getindex and destructuring to behave just like if lsim returned a tuple of arrays like before.  Indeed, the tests for lsim were not touched in this commit.

This commit also adds a plot recipe to the result structure. All plot recipes for lsimplot, stepplot, impulseplot have been replaced by the new recipe. This is a breaking change since the names of the previous plots no longer exist. A slight change is that the plots for a step response no longer show the text "step response", but I think that's an acceptable change, the user can supply any title they prefer themselves.

* remove automatic title

* introduce additional abstract result type

* do not destructure sys

The functionality was very buggy and poorly tested. A much improved version with proper tests are available in https://github.com/JuliaControl/RobustAndOptimalControl.jl/blob/master/src/lqg.jl