Allow place to handle multi input systems

[albheim]

Currently place only handles SISO systems and it might be nice to try to implement something similar as matlab has where they find a robust set of poles for multi input systems according to Robust Pole Assignment in Linear State Feedback. Have not managed to understand the algorithm but could try to take a look at it if it is of interest.

[baggepinnen]

In anticipation of a better method, the following does an okay job

function generic_place(A, B::AbstractMatrix, desired_poles; max_tries = 20, λ=1e-8)
    function loss(K)
        e = eigvals(A-B*K)
        mean(abs2.(e-desired_poles)) + λ*mean(abs2, K)
    end    
    desired_poles = sort(complex.(desired_poles), by=LinearAlgebra.eigsortby)
    nu = size(B,2)    
    local res
    for i = 1:max_tries
        if nu > 1
            v = randn(nu)
            B0 = B*v
            K0 = place(A,B0, desired_poles)
            K = v*K0
        else
            K = place(A, B, desired_poles) .+ 0.1 .* randn.()
        end
        res = Optim.optimize(
            loss,
            K,
            BFGS(),
            Optim.Options(
                store_trace       = false,
                show_trace        = false,
                show_every        = 1,
                iterations        = 10000,
                allow_f_increases = false,
                time_limit        = 100,
                x_tol             = 1e-10,
                f_tol             = 1e-10,
                g_tol             = 1e-12,
                f_calls_limit     = 0,
                g_calls_limit     = 0,
                ),
                )
        (res.f_converged || res.g_converged || res.x_converged) && break
    end
    real.(res.minimizer), res
end

nx = 10
A = randn(nx,nx) - 2I
B = randn(nx,3)

desired_poles = sort(randn(nx) .- 4)


PK = place(A,B,desired_poles)
OK, res = generic_place(A,B,desired_poles, λ=1e-8)

@test PK ≈ OK rtol=1e-2

Of course, you may add additional penalties to the loss function, encouraging use of potential redundancy to reduce noise influence etc. And it should probably be initialized using the trick to project B along a random direction and calling place
EDIT: I made that update

[baggepinnen]

Closed by #854