Simulation

Simulation#

Simulation involves evaluating a trajectory of the system starting from an initial state \(x_0\).

See also

Execution#

Simulation of an inclusion problem \(0 \in \sum_{i=1}^s F_i(\beta^*)\) with \(\beta^* \in \R^d\) is conducted by the alg_sim object.

The arguments to alg_sim are the system (sys), and the dimensionality of \(\beta\) in simulation (d). The output of an algorithm execution for \(T\) time steps is obtained through the sim() command,

1simulator = alg_sim(sys, d);
2sim_result = simulator.sim(T);

By default, algorithm execution will occur with \(x_0=0\), and \(w_p = 0\). The sampler field of alg_sim allows for random generation and external signals. The attributes of sampler are:

Field

Description

x0

Initial Condition

wp

Performance Input

param0

Initial value of problem-dependent parameters

param

Subsequent value of problem-dependent parameters

The output of alg_sim.sim(T) is an alg_sim_out object. The fields of alg_sim_out include

xn

state of network

z

input to operators

w

output from operators

xi

state of controller

zp

performance output

wp

performance input

k

time index

y

output to controller

u

input from controller

f

function value

res_w

optimality error \(\norm{\sum_{i=1}^s w^i_k}_2\)

res_z

consensus error \(\norm{z^i_k - z^i_{\text{average}, k}}_2\)

mode

subsystem for switched systems

param

problem-dependent parameters

Each field of alg_sim_out is a numeric or cell array with last dimension indexed by \(k \in \{1, \ldots, T\}\).

Plotting#

The alg_plotter class accepts a result from simulation, and can then plot any signal stored in alg_sim_out. A plot of the signals (\(w\), \(z\), \(u\), \(y\)) is accomplished by performing

1plt = alg_plotter(sim_result);
2fig = plt.plot({"w", "z", "u", "y"});

The figure number can be set by an optional second argument to plot()

1plt = alg_plotter(sim_result);
2fig1 = plt.plot({"w", "z", "res_w", "res_z"}, 100); %figure number 100
3fig2 = plt.plot({"xi", "u", "y"}, 101); %figure number 101

Helper functions of alg_plotter include

Method

Description

Plotted Signals

plot_6()

States, oracles, and convergence

(xn, w, res_w, xc, z, res_z)

plot_6f()

States, oracles, convergence, function values

(x, w, res_w, f, z, res_z)

plot_4()

oracles, convergence

(w, res_w, z, res_z)

See also

Plotting contains a full list of helper functions and more details.

Details of Execution#

The System sys executed by first interconnecting the Network and Controller, and then interconnecting the possibly nonlinear operator \(F\). This is mathematically described by

\[\begin{split}\begin{align*} \text{Operator}: & & w_k & \in F(z_k), \\ \text{Algorithm}: & & \mat{c}{x_{k+1} \hl z_k \\ z_{p k}} &= \mat{c|cc}{\Acl & \Bcl_z & \Bcl_{z_p} \hl \Ccl_z & \Dcl_{zw} & \Dcl_{z w_p} \\ \Ccl_{z_p} & \Dcl_{z_p w} & \Dcl_{z_p w_p}} \mat{c}{x_k \hl w_k \\ w_{p k}}, \end{align*}\end{split}\]

Well-posedness requires that the map \(H: = (F^{-1} - \Dcl{zw})^{-1}\) is globally defined and continuous.

The closed loop state \(x\) is the concatenation \(x = [x^N, x^c]\).

alg_sim execution using sim() requires well-posedness and a block-triangular information structure (closed loop \(\Dcl\) matrix).

Under these conditions, the System can be partitioned as

\[\begin{align*} \text{Operator}: & & w_k & \in F(z_k), \\ \text{Algorithm}: & & \mat{c}{x_{k+1} \hl z_k^1 \\ z_k^2 \\ \vdots \\ z_k^s \\ z_{p k}} &= \mat{c|cccc:c}{\Acl & \Bcl_{z,1} & \Bcl_{z,2} & \cdots & \Bcl_{z,s} & \Bcl_{z_p} \hl \Ccl_{z 1} & \Dcl_{zw,11} & 0 & \cdots & 0 & \Dcl_{z w_p, 1} \\ \Ccl_{z 1} & \Dcl_{zw,21} & \Dcl_{zw,22} & \cdots & 0 & \Dcl_{z w_p, 2} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ \Ccl_{z 1} & \Dcl_{zw, s1} & \Dcl_{zws2} & \cdots & \Dcl_{zw,ss} & \Dcl_{z w_p s} \\ \Ccl_{z_p} & \Dcl_{z_p w,1} & \Dcl_{z_p w, 2} & \cdots & \Dcl_{z_p w, s} & \Dcl_{z_p w_p}} \mat{c}{x_k \hl w_k^1 \\ w_k^2 \\ \vdots \\ w_k^s \\ w_{p k}}, \end{align*}\]

The iterative loop for algorithm simulation is to evaluate the equations for each \(k \in \N\)

\[\begin{split}\begin{align} w^i_k &= (F_i^{-1} - \Dcl_{zw, ii})^{-1} (\Ccl_i x_k + \textstyle \sum_{j=1}^{i-1} \Dcl_{zw, ij} w^j_k), & & \forall i \in 1, \ldots, s,\\ x_{k+1} &= \Acl x_k + \textstyle \sum_{i=1}^s \Bcl_{wi} w^i_k + \Bcl_{w_p} w_{pk}. \\ z_{p, k+1} &= \Ccl_{z_p} x_k + \textstyle \sum_{i=1}^s \Dcl_{z_p w i} w^i_k + \Dcl_{z_p w_p} w_{pk}. \end{align}\end{split}\]

The operator \(H_i: = (F_i^{-1} - \Dcl_{zw, ii})^{-1}\) can be evaluated using the op_sim methods

Evaluation

Used Method

Condition

Operation \(z_i \mapsto H_i z_i\)

Explicit

fw()

\(\Dcl_{zw, ii} = 0\)

\(z \mapsto F_i(z_i)\),

Implicit

bw()

\(\Dcl_{zw,ii}\) is invertible

\(z_i \mapsto \Dcl_{zw, ii}^{-1} [z - (I - \Dcl_{zw, ii} F_i)^{-1}(z_i)]\)