PeriodicMatrices.jl

Handling of periodic time-varying matrices

Compatibility

Julia 1.10 and higher.

About

PeriodicMatrices.jl provides the basic tools to handle periodic time-varying matrices. The time dependence can be either continuous or discrete.

A continuous-time periodic matrix can be specified in the following forms:

• periodic matrix function
• harmonic matrix series
• periodic matrix time series with uniform time grid
• periodic matrix time series with non-uniform time grid
• periodic symbolic matrix
• Fourier matrix series approximation

A discrete-time periodic matrix can be specified in the following forms:

• periodic matrix time series with time-varying dimensions with uniform time grid
• periodic matrix time series with time-varying dimensions with non-uniform time grid
• periodic matrix time series with constant dimensions with uniform time grid
• periodic matrix time series with constant dimensions with non-uniform time grid

For a periodic matrix A(t) of period T it is not assumed that T is the minimum value which satisfies the periodicity condition A(t) = A(t+T) for all values of t. To describe matrices having multiple periods, a subperiod Tsub := T/n can be defined, such that A(t) = A(t+Tsub), for all t. This allows a substantial memory saving for some classes of periodic representations.

The provided classes of periodic representation extend the classes used in the Periodic Systems Toolbox for Matlab (see [1]).

Several operations on periodic matrices are implemented, such as, inversion, transposing, norms, derivative/shifting, trace. All operations with two periodic matrices such as addition/substraction, multiplication, horizontal/vertical concatenation, block-diagonal appending, allow different, but commensurate, periods/subperiods.

Functions are provided to compute the characteristic multipliers and characteristic exponents of periodic matrices, using methods based on the periodic Schur decomposition of matrix products or structure exploitung fast algorithms. These functions are instrumental to apply Floquet theory to study the properties of solutions of various classes of differential equations (Mathieu, Hill, Meissner) and the stability of linear periodic systems (see PeriodicSystems package).

Example: Floquet-analysis of differential equations with periodic parameters

A frequently encountered periodic differential equation is of second order, expressible as

$$\ddot{x} + 2\zeta\dot{x} + (a - 2q\psi(t))x = 0 ,$$

where $ψ(t) = ψ(t+T)$ is a periodic function of period $T$. The parameter $a$ represents a constant portion of the coefficient of $x$ and $q$ accounts for the magnitude of the time variation. A positive damping coefficient $\zeta$ is frequently used in practical applications.
The above equation for $\zeta = 0$ is generally known as the Hill-equation and the form in which it is expressed is that most widely encountered in applications. When $\psi(t) = cos 2t$ , the equation becomes the Mathieu equation. If $\phi(t)$ is a rectangular function, the corresponding particular form is known as the Meissner equation. If $\zeta > 0$, we speak of the lossy variants of these equations.

The above equation can be equivalently expressed as a first order system of differential equations, by defining

$$y(t) = \left[ \begin{array}{c} x(t)\\ \dot{x}(t) \end{array} \right]$$

to recast the second order differential equation into the form

$$ \dot{y}(t) = A(t)y(t)$$

where $A(t)$ is a $2\times 2$ matrix of constant or periodically varying coefficients of the form

$$A(t) = \left[ \begin{array}{cc} 0 & 1\\\ -a+2q\psi(t) & -2\zeta \end{array} \right]$$

The state transition matrix $\Phi(t,0)$ over the time interval $(0,t)$ satisfies the differential equation

$$\dot{\Phi}(t,0) = A(t)\Phi(t,0), \,\, \Phi(0,0) = I$$

and the monodromy matrix $\Psi := \Phi(T,0)$, i.e., the state transition matrix over one full period.

The Floquet-theory based stability analysis of the above equations addresses the determination of characteristic multipliers $\lambda_i$ as the eigenvalues of the monodromy matrix or alternatively the characteristic exponents $\mu_i$ related to the characteristic multipliers as

$$ \lambda_i = exp(\mu_iT) .$$

The solution $x(t)$ is stable if it remains bounded as time goes to infinity. For stability it is sufficient that $Re(\mu_i) < 0 , \forall i$, which is the same as $|\lambda_i| < 1 , \forall i$. Such a solution will also be stable if in addition one $Re(\mu_i) = 0$ or one $|\lambda_i| = 1$.

In what follows we illustrate how to perform the stability analysis for the three types of equations based on the characteristic multipliers/exponents.

The lossless Meissner equation with a rectangular waveform coefficient

This is Example of Fig 3.1 in [3]. Assume the period is $T = \pi$ and let $\tau = T/3$ be the switching time. We consider the periodic function $\psi(t) = 1$ if $t \in [0,\tau)$ and $\psi(t) = -1$ if $t \in [\tau,\pi)$. We can describe the periodic matrix $A(t)$ as a PeriodicSwitchingMatrix with two components corresponding to the two constant values of $\psi(t)$ and switching times at $t = 0$ and $t = \tau$. The following code can be used for stability analysis purposes:

using PeriodicMatrices

# set parameters
a = 1; q = .1; T = pi
ts = [0; τ; T] 

ψ(t,ts) = isodd(findfirst(mod(t,T) .< ts)) ? -1 : 1

# setup of A(t)
A = PeriodicSwitchingMatrix([[0. 1.; -a+2*q*ψ(t,ts) 0] for t in ts[1:end-1]], ts[1:end-1], T)
ce = psceig(A)  

# stability test
all(real(ce) .< 0)

The computed characteristic exponents are:

julia> ce 
2-element Vector{ComplexF64}:
 0.041379744661220644 + 1.0im
  -0.0413797446612206 + 1.0im

and therefore the solutions are unstable. The computations can be easily extended to several switching points in $\tau$ as well.

For a lossy Meissner equation, the computations reveal stability:

# setup of A(t)
ζ = 0.2
A = PeriodicSwitchingMatrix([[0. 1.; -a+2*q*ψ(t,ts) -2ζ] for t in ts[1:end-1]], ts[1:end-1], T)
ce = psceig(A)  

# stability test
all(real(ce) .< 0)

and

julia> ce
2-element Vector{ComplexF64}:
 -0.14798307933724503 + 1.0im
   -0.252016920662755 + 1.0im

References

[1] A. Varga. A Periodic Systems Toolbox for Matlab. Proc. of IFAC 2005 World Congress, Prague, Czech Republic, 2005.

[2] S. Bittanti and P. Colaneri. Periodic Systems - Filtering and Control, Springer Verlag, 2009.

[3] J. A. Richards. Analysis of Periodically Time-Varying Systems, Springer Verlag, 1983.