QuantizedStateSystems.jl

txarray,xarray,tqarray,qarray = qss1(f,x₀,tspan,Δq,tuarray,uarray)

Solve the initial value problem (IVP) for a given first-order ODE with inputs using the method of hysteretically quantized state system (QSS) or order 1.

For a single first-order explicit ODEs (aka state equation) ẋ = f(x,u), with the initial value x₀ specified at an initial time t₀, and possibly a piecewise (control) input u given by a pair of vectors tuarray and uarrayof times and values, find the solution on the time span tspan=(t₀,t₁). The only parameter for the method is the quantum Δq, while the hysteresis band has the same width as the quantum.

Arguments

• f: function defining the right hand side of the differential equation.
• x₀: initial condition.
• tspan: time span.
• Δq: quantum.
• tuarray: array of times at which the the input changes to a new constant value.
• uarray: array of values of the piecewise constant input.

Outputs

• txarray: a vector of times when the value of x is computed, between these values it evolves linearly.
• xarray: a vector of values of x corresponding to the times.
• tqarray: a vector of times when the value of piecewise constant q changes.
• qarray: a vector of values of the piecewise constant quantized q.

Note that for a first-order system with no external events the x and q arrays are identical up to the initial values (x₀ before vs. after quantization).