The client that approached us for a solution is in the spray-painting business. The objects to spray-paint are of various kinds. In addition, various kinds of paint are used which require different drying rates. To enable a suitable drying condition for a specific object surface and paint type combination, the client will provide a specific looked-up drying temperature.
The client has a heat source at the center of the drying space and needs optimal guidance on how far away from the heat source each spray-painted object needs to be placed to comply with the looked-up drying temperature. Objects that need to be spray-painted are placed on a radio-controlled cart with a protected heat sensor. Once spray-painting is completed, a remote controller (containing the in-silico agent) are called upon to position the cart at the optimal distance from the heat source so that the drying paint are placed at the specified temperature.
We simulate an agent that can relocate the cart in a temperature gradient field. The agent aims to position the cart at a desired temperature relative to the heat source. Our simulation setup is an adaptation of the setup described by Buckley et al. (2017).
versioninfo() ##. Julia versionJulia Version 1.10.4
Commit 48d4fd48430 (2024-06-04 10:41 UTC)
Build Info:
Official https://julialang.org/ release
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: 12 × Intel(R) Core(TM) i7-8700B CPU @ 3.20GHz
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-15.0.7 (ORCJIT, skylake)
Threads: 1 default, 0 interactive, 1 GC (on 12 virtual cores)
Environment:
JULIA_NUM_THREADS = # import Pkg; Pkg.activate(".."); Pkg.instantiate();
import Pkg
# Pkg.add(Pkg.PackageSpec(;name="RxInfer", version="3.0.0"))
Pkg.add(Pkg.PackageSpec(;name="RxInfer", version="3.6.0"))
Pkg.add(Pkg.PackageSpec(;name="Plots"))
Pkg.add(Pkg.PackageSpec(;name="LaTeXStrings"))
## Pkg.resolve() #.
using RxInfer, Plots
using Random; Random.seed!(51233) # Set random seed
using LaTeXStrings
import RxInfer.ReactiveMP: getrecent, messageout Updating registry at `~/.julia/registries/General.toml`
Resolving package versions...
No Changes to `~/.julia/environments/v1.10/Project.toml`
No Changes to `~/.julia/environments/v1.10/Manifest.toml`
Precompiling project...
✓ Unitful
✓ Unitful → ConstructionBaseUnitfulExt
✓ UnitfulLatexify
✓ Plots
✓ Plots → FileIOExt
✓ Plots → UnitfulExt
6 dependencies successfully precompiled in 92 seconds. 260 already precompiled.
Resolving package versions...
No Changes to `~/.julia/environments/v1.10/Project.toml`
No Changes to `~/.julia/environments/v1.10/Manifest.toml`
Resolving package versions...
No Changes to `~/.julia/environments/v1.10/Project.toml`
No Changes to `~/.julia/environments/v1.10/Manifest.toml`Pkg.status()Status `~/.julia/environments/v1.10/Project.toml`
[b964fa9f] LaTeXStrings v1.3.1
[91a5bcdd] Plots v1.40.8
[86711068] RxInfer v3.6.0There is no pre-existing data to be analyzed.
There is no pre-existing data to be prepared.
The next figure (from Bert de Vries at Eindhoven Technical University) shows the interactions between the agent and the environment:
This section attempts to answer three important questions:
For this problem, we will only track the temperature.
Decisions will be in the form of agent-prescribed velocity actions.
The only source of uncertainty relating to the environment will be the noise in the measurements of the temperature.
The system-under-steer is the radio-controlled paint drying cart. The cart is subject to a temperature gradient field given by:
The temperature \(\cal T\) is a function of postion \(s\) and follows the profile
\[ \cal T(\tilde{s}) = \frac{\cal T_0}{\tilde{s}^2 + 1} \]
where
Please see the next code and chart.
## Environmental process parameters
## Temperature at the heat source (z=0)
_𝚃₀ = 100.0 ## [\]mttT[tab] 'math teletype', closest for now
## Actual temperature profile; this function is hidden from the agent
𝚃(s̃) = [_𝚃₀/(s̃[1]^2 + 1.0)] ## return a 1-element vector𝚃 (generic function with 1 method)_d = 0.0:0.01:6.0 ## distance range #.
_y⁰ = [𝚃([s̃ₖ])[1] for s̃ₖ in _d] ## observation range (noise-free) #.
_s̃₀ = [2.0] ## initial position #.
_x₊ = [4.0] ## target/goal temperature
plot(
_d, _y⁰, color="black", xlabel="Position", ylabel="Temperature", #.
label="Temperature")
scatter!(_s̃₀, [𝚃(_s̃₀)], label="Initial position temperature") #.The state at time \(t\) of the system-under-steer/environment (envir) will be given by \(\tilde{s}_t\) which will be the position of the cart relative to the heat souce.
The decision variables represent what we control.
The environment is steered by decisions/actions \(a_t\) that reflect the velocity of the cart. Decisions/actions are in the form of agent-prescribed velocity actions. An action suggests the adjustment of the radial velocity to either move away from the heat source or to move towards it. This velocity is limited to the interval \((-V^{max}, -V^{max})\). It is given by:
\[ \begin{align} V^a &= V^{max} ⋅ \mathrm{tanh}(a) \\ &= 0.05 ⋅ \mathrm{tanh}(a) \end{align} \] where \(a_t\) is the velocity action at time \(t\).
Since the velocity have limits, we use the \(\tanh(\cdot)\) function to limit the velocity action to the interval \((-V^{max}, -V^{max})\).
Let us visualize how the velocity action, \(V^a\), varies with the amount of action \(a\).
_Vᵐᵃˣ = 0.5
Vᵃ = (a::Real) -> _Vᵐᵃˣ*tanh(a)#13 (generic function with 1 method)_a = range(-10, 10, length=400) #.
_Vᵃ = [ _Vᵐᵃˣ*tanh(xs) for xs in _a ] #.
plot(
_a, _Vᵃ, #.
title="Limits on velocity actions",
label="Landscape",
color="black",
xlabel=L"Action, $a$",
ylabel=L"Action velocity, $V^a$",
legend=nothing,
ylimits=(-0.5, 0.5)
)There are no exogenous information variables for this problem.
The transition function captures the dynamics of the environment/system-under-steer/generative process:
\[ \begin{aligned} \tilde{s}_t &= R_{t}(\tilde{s}_{t-1}, Vᵃ_t) \\ &= \tilde{s}_{t-1} + Vᵃ_t \\ &= \tilde{s}_{t-1} + Vᵐᵃˣ\mathrm{tanh}(a_t) \end{aligned} \]
The environment generates outcomes as noisy observations of the current state with an observation noise variance
\[ \vartheta = 10^{-4} \]
The observation function can be represented by: \[ y_t \sim \cal N(\cal T(\tilde{s}_t), \vartheta) \]
_γ = 1e4 ## transition precision (system noise)
_ϑ = 1e-4 ## observation variance (observation noise)0.0001The objective function is such that the Bethe free energy is minimized. This aspect will be handled by the RxInfer Julia package.
Because states of the agent are unknown to the world, we wrap them in a comprehension that only returns functions for interacting with the agent. Internal beliefs cannot be directly observed, and interaction is only allowed through the Markov blanket of the agent (i.e., the sensors and actuators).
function create_envir(; 𝚃, s̃₀)
s̃ₜ₋₁ = s̃₀
s̃ₜ = s̃ₜ₋₁
yₜ = 𝚃(s̃ₜ)[1] + sqrt(_ϑ)*randn() ##Report noisy temperature at current position; maybe this is repeated to keep place in the records
execute = (aₜ::Float64) -> begin
s̃ₜ = s̃ₜ₋₁ + [Vᵃ(aₜ)] ##Compute next state #.
yₜ = 𝚃(s̃ₜ)[1] + sqrt(_ϑ)*randn() ##Report noisy temperature at current position; maybe this is repeated to keep place in the records
s̃ₜ₋₁ = s̃ₜ ##Reset state
end
observe = () -> begin
return [yₜ]
end
return (execute, observe)
endcreate_envir (generic function with 1 method)The only uncertainty we have for the environment is the noise associated with an observation as captured in the observation function above: \[ y_t \sim \cal N(\cal T(\tilde{s}_t), \vartheta) \]
The agent consists of:
According to the agent the state of the system-under-steer/environment/generative process will be \(s_t\), rather than \(s̃_t\), which is the distance of the agent from the heat source.
According to the agent the action on the environment at time \(t\) will be represented by \(u_t\), also known as the control state of the agent.
We start by specifying a probabilistic model for the agent that describes the agent’s internal beliefs over the external dynamics of the environment. Assuming the current time is \(t\) and \(t=1\),
The generative model is defined as follows:
\[\begin{aligned} p'(x,s,\theta_A,\theta_B,u) = \underbrace{p(s_{t-1})}_{\substack{ \text{Initial} \\ \text{state} \\ \text{prior}}} \underbrace{p(\theta_A)}_{\substack{ \text{Parameter} \\ \theta_{A} \\ \text{prior}}} \underbrace{p(\theta_B)}_{\substack{ \text{Parameter} \\ \theta_{B} \\ \text{prior}}} \prod_{k=t}^{t+T} \underbrace{p(x_k \mid s_k,\theta_A)}_{\substack{ \text{Observation} \\ \text{model}}} \, \underbrace{p(s_k \mid s_{k-1},\theta_B,u_k}_{\substack{\text{Transition} \\ \text{model}}} \, \underbrace{p(u_k)}_{\substack{ \text{Control} \\ \text{prior}}} \ \nonumber \end{aligned}\]
The generative model includes future time steps.
Omitting parameters \(\theta_A\) and \(\theta_B\),
The generative model is defined as follows:
\[\begin{aligned} p'(x,s,u) = \underbrace{p(s_{t-1})}_{\substack{ \text{Initial} \\ \text{state} \\ \text{prior}}} \prod_{k=t}^{t+T} \underbrace{p(x_k \mid s_k)}_{\substack{ \text{Observation} \\ \text{model}}} \, \underbrace{p(s_k \mid s_{k-1},u_k}_{\substack{\text{Transition} \\ \text{model}}} \, \underbrace{p(u_k)}_{\substack { \text{Control} \\ \text{prior}}} \ \nonumber \end{aligned}\]
The generative model includes future time steps.
To infer goal-driven (i.e. purposeful) behavior, we add prior beliefs \(p^+(x)\) about desired future observations. This leads to an extended agent model:
\[\begin{aligned} p(x,s,u) &= \frac{p'(x,s,u) p^+(x)}{\int_x p'(x,s,u)p^+(x) dx} \\ &\propto \underbrace{p(s_{t-1})\prod_{k=t}^{t+T} p(x_k \mid s_k)p(s_k \mid s_{k-1},u_k)p(u_k)}_{\text{original generative model}} \, \underbrace{p^+(x_k)}_{\substack { \text{extension} \\ \text{Goal} \\ \text{prior}}} \\ &\propto \underbrace{p(s_{t-1})}_{\substack{ \text{Initial} \\ \text{state} \\ \text{prior}}} \prod_{k=t}^{t+T} \underbrace{p(x_k \mid s_k)}_{\substack{ \text{Observation} \\ \text{model}}} \, \underbrace{p(s_k \mid s_{k-1},u_k}_{\substack{\text{Transition} \\ \text{model}}} \, \underbrace{p(u_k)}_{\substack { \text{Control} \\ \text{prior}}} \, \underbrace{p^+(x_k)}_{\substack { \text{Goal} \\ \text{prior}}} \nonumber \end{aligned}\]
The factors are defined as:
\[ \begin{align} p(x_k \mid s_k) &= \mathcal{N}(x_k \mid -s_k, \vartheta) \\ &= \mathcal{N}(x_k \mid -s_k, 10^{-2}) \end{align} \] where \(x_k\) denotes observations of the agent after interacting with the environment. Note that we hampered the observation model. Instead of the actual temperature profile, we specify that the observed temperature decreases with position linearly.
\[ \begin{aligned} p(s_k \mid s_{k-1},u_k) &= \mathcal{N}(s_k \mid s_{k-1} + u_k, ϑ) \\ p(s_{t-1}) &= \mathcal{N}(s_{t-1} \mid m_{t-1},\,V_{t-1}) \end{aligned} \]
The current state is a linear function of the previous state and action. We have endowed the agent with an accurate model of the system dynamics.
\[ \begin{aligned} p(u_t) &= \prod_{k=t}^{t+T} \mathcal{N}(u_k \mid 0,\,\Xi) \\ &= \prod_{k=t}^{t+T} \mathcal{N}(u_k \mid 0,\,ξ ⋅ \mathbf{I}) \\ &= \prod_{k=t}^{t+T} \mathcal{N}(u_k \mid 0,\,0.5) \\ &= \mathcal{N}(u_t \mid m_u,\,V_u)\\ \end{aligned} \]
This represents the control priors.
\[ \begin{aligned} p^+(x_t) &= \prod_{k=t}^{t+(T-1)} \mathcal{N}(x_k \mid 0,\,\sigma^{huge}) \cdot \mathcal{N}(x_T \mid x_+, \, \Sigma)\\ &= \prod_{k=t}^{t+(T-1)} \mathcal{N}(x_k \mid 0,\,\sigma^{huge}) \cdot \mathcal{N}(x_T \mid x_+, \, \sigma ⋅ \mathbf I)\\ &= \prod_{k=t}^{t+(T-1)} \mathcal{N}(x_k \mid 0,\,10^{12}) \cdot \mathcal{N}(x_T \mid 4.0, \, 10^{-4} ⋅ \mathbf I)\\ &= \mathcal{N}(x_t \mid m_x,\,V_x) \\ \end{aligned} \]
This represents the target/goal priors and encodes a belief about a preferred signal strength \(x₊=4.0\).
Setting \(t=1\), \[ p(s_0) = \mathcal{N}(s_0 \mid 0, 10^{12}) \]
This means we set a vague prior for the initial state.
The code in the next block defines the agent’s internal beliefs over the external dynamics and its probabilistic model of the environment, which correspond accurately by directly using the functions defined above. We use the @model macro from RxInfer to define the probabilistic model and the meta block to define approximation methods for the nonlinear state-transition functions.
In the model specification we include the beliefs over its future states (up to T steps ahead), in addition to the current state of the agent:
## @model function thermostat_model(; T, 𝚃)
@model function thermostat_model(mᵤ, Vᵤ, mₓ, Vₓ, mₛ₍ₜ₋₁₎, Vₛ₍ₜ₋₁₎, T, 𝚃)
## Transition function
g = (sₜ₋₁::AbstractVector) -> begin
sₜ = similar(sₜ₋₁) ## Next state
sₜ = 𝚃(sₜ₋₁)
return sₜ
end
Γ = _γ*diageye(1) ## Transition precision
𝚯 = _ϑ*diageye(1) ## Observation variance
sₜ₋₁ ~ MvNormal(mean=mₛ₍ₜ₋₁₎, cov=Vₛ₍ₜ₋₁₎)
sₖ₋₁ = sₜ₋₁
local s
for k in 1:T
## Control
u[k] ~ MvNormal(mean=mᵤ[k], cov=Vᵤ[k])
hIuI[k] ~ MvNormal(mean=sₖ₋₁ + u[k], precision=Γ)
## State transition
s[k] ~ g(hIuI[k]) where { meta=DeltaMeta(method=Unscented(alpha=1.9)) }
## Likelihood of future observations
x[k] ~ MvNormal(mean=s[k], cov=𝚯)
## Target/Goal prior
x[k] ~ MvNormal(mean=mₓ[k], cov=Vₓ[k])
sₖ₋₁ = s[k]
end
return (s, )
endNext, we define the agent and the time-stepping procedure.
function create_agent(; T=20, 𝚃, x₊, s₀, ξ=0.5, σ=1e-4)
## Set control priors
Ξ = fill(ξ, 1, 1) ##Control prior variance
mᵤ = Vector{Float64}[ [0.0] for k=1:T ] ##Set control priors
Vᵤ = Matrix{Float64}[ Ξ for k=1:T ]
## Set target/goal priors
Σ = σ*diageye(1) ##Target/Goal prior variance
mₓ = [zeros(1) for k=1:T] ##mean for x [vector]
mₓ[end] = x₊ ##Set prior mean to reach goal at t=T
Vₓ = [huge*diageye(1) for k=1:T] ##Variance for x [matrix]
Vₓ[end] = Σ ##Set prior variance to reach goal at t=T
## Set initial brain state prior
mₛ₍ₜ₋₁₎ = s₀
Vₛ₍ₜ₋₁₎ = tiny*diageye(1)
## Vₛ₍ₜ₋₁₎ = huge*diageye(1) ##in writeup
## Set current inference results
result = nothing
## Bayesian inference by message passing
## The `infer` function is the heart of the agent
## It calls the `RxInfer.infer` function to perform Bayesian inference by message passing
compute = (υₜ::Float64, ŷₜ::Vector{Float64}) -> begin ##.align with mountain car
mᵤ[1] = [υₜ] ## Register action with the generative model
Vᵤ[1] = fill(tiny, 1, 1) ## Clamp control prior to performed action
mₓ[1] = ŷₜ ## Register observation with the generative model
Vₓ[1] = tiny*diageye(1) ## Clamp target/goal prior to observation
result = infer(
model=thermostat_model(T=T, 𝚃=𝚃),
data=Dict(
:mᵤ => mᵤ,
:Vᵤ => Vᵤ,
:mₓ => mₓ,
:Vₓ => Vₓ,
:mₛ₍ₜ₋₁₎ => mₛ₍ₜ₋₁₎,
:Vₛ₍ₜ₋₁₎ => Vₛ₍ₜ₋₁₎))
end
## The `act` function returns the inferred best possible action
act = () -> begin
if result !== nothing
return mode(result.posteriors[:u][2])[1]
else
return 0.0 ## Without inference result we return some 'random' action
end
end
## The `future` function returns the inferred future states
future = () -> begin
if result !== nothing
return getindex.(mode.(result.posteriors[:s]), 1)
else
return zeros(T)
end
end
## The `slide` function modifies the `(mₛ₍ₜ₋₁₎, Vₛ₍ₜ₋₁₎` for the next step
## and shifts (or slides) the array of future goals `(mₓ, Vₓ)`
## and inferred actions `(mᵤ, Vᵤ)`
slide = () -> begin
model = RxInfer.getmodel(result.model)
(s, ) = RxInfer.getreturnval(model)
varref = RxInfer.getvarref(model, s)
var = RxInfer.getvariable(varref)
slide_msg_idx = 3 ##This index is model dependent
(mₛ₍ₜ₋₁₎, Vₛ₍ₜ₋₁₎) = mean_cov(getrecent(messageout(var[2], slide_msg_idx)))
mᵤ = circshift(mᵤ, -1)
mᵤ[end] = [0.0]
Vᵤ = circshift(Vᵤ, -1)
Vᵤ[end] = Ξ
mₓ = circshift(mₓ, -1)
mₓ[end] = x₊ ##x_target
Vₓ = circshift(Vₓ, -1)
Vₓ[end] = Σ
end
return (act, future, compute, slide)
endcreate_agent (generic function with 1 method)In this simulation we are going to perform a naive action policy. In this case, with limited engine power, the agent should not be able to achieve its goal:
_Nⁿᵃⁱᵛᵉ = 100 ## Total simulation time
_πⁿᵃⁱᵛᵉ = 0.5*ones(_Nⁿᵃⁱᵛᵉ) ## Naive policy for right full-power only
(execute_naive, observe_naive) = create_envir(; ## Let there be a world
𝚃=𝚃,
s̃₀=_s̃₀
);
_yⁿᵃⁱᵛᵉ = Vector{Vector{Float64}}(undef, _Nⁿᵃⁱᵛᵉ)
for t = 1:_Nⁿᵃⁱᵛᵉ
execute_naive(_πⁿᵃⁱᵛᵉ[t]) ## Execute environmental process
_yⁿᵃⁱᵛᵉ[t] = observe_naive() ## Observe external states
end_yⁿᵃⁱᵛᵉ100-element Vector{Vector{Float64}}:
[16.71681084161624]
[14.162811043613832]
[12.130010375269212]
[10.477036308444694]
[9.129242440313945]
[8.014134634991786]
[7.11502449829092]
[6.335537417093641]
[5.668086254476602]
[5.09459637411509]
⋮
[0.18373948326761114]
[0.18838046098425149]
[0.18520102232614685]
[0.17734073536314304]
[0.16833059836318884]
[0.17676855961853477]
[0.16856284746341046]
[0.12573434370159794]
[0.1461178658812947]_pa=plot(
map(x -> x[1], _πⁿᵃⁱᵛᵉ), label="actions",
xlabel="t", ylabel="velocity",
)
_py=plot(
map(x -> x[1], _yⁿᵃⁱᵛᵉ), label="observations",
color="red", xlabel="t", ylabel="temperature",
)
_py=plot!(
[0, _Nⁿᵃⁱᵛᵉ], [_x₊[1], _x₊[1]],
label="target/goal", color="green", xlabel="t", ylabel="temperature")
plot(_pa, _py, layout=@layout([ a; b ]))In the active inference approach we are going to create an agent that models the environment around itself as well as the best possible actions in a probabilistic manner. That should help agent to understand that the brute-force approach is not the most efficient one and hopefully to realise that a little bit of swing is necessary to achieve its goal.
### Simulation parameters
## Total simulation time
_Nᵃⁱ = 100
## Lookahead time horizon
_Tᵃⁱ = 20
## Initial position
_s₀ = [2.0]
## Control prior variance value
# _Εᵛᵃˡ = .5
_ξ = 0.5
## Target prior variance value
_σ = 1e-4
## Target/Goal signal strength
_x₊ = [4.0]1-element Vector{Float64}:
4.0# ## OVERRIDES
## Total simulation time
# # _Nᵃⁱ = 50 #100
# # _Nᵃⁱ = 100
# _Nᵃⁱ = 200
# # _Nᵃⁱ = 500
# # _Nᵃⁱ = 1000
## Lookahead time horizon
# # _Tᵃⁱ = 20#200 #100 #50 #20
# _Tᵃⁱ = 50
# # _Tᵃⁱ = 100
# ## Initial position
# # _s₀ = [2.0]#[2.0]
# # _s₀ = [1.0]
# ## Control prior variance value
# _ξ = 0.01
# _ξ = 0.1
# _ξ = 1.0
# _ξ = 10.0
# _ξ = 100.0
# _ξ = 1000.0
# _ξ = 10000.0
# _ξ = 1e12
## Target prior variance value
# _σ = 1e-4
# ## Target/Goal temperature
# # _x₊ = [25.0]#[5.0]#[4.0] #[25.0]#[4.0]
# # _x₊ = [50.0]
# # _x₊ = [20.0]
# _x₊ = [15.0](execute_ai, observe_ai) = create_envir(;
𝚃=𝚃, #.
s̃₀=_s₀
)
(act_ai, future_ai, compute_ai, slide_ai) = create_agent(;
T =_Tᵃⁱ,
𝚃=𝚃,
x₊=_x₊,
s₀=_s₀,
ξ = _ξ,
σ = _σ
)
## Step through experimental protocol
_as = Vector{Float64}(undef, _Nᵃⁱ) ## Actions
_fs = Vector{Vector{Float64}}(undef, _Nᵃⁱ) ## Predicted future
_ys = Vector{Vector{Float64}}(undef, _Nᵃⁱ) ## Observations #.
for t = 1:_Nᵃⁱ
## 1. Act-Execute-Observe: #.execute() & observe() from create_envir()
_as[t] = act_ai() ## Invoke an action from the agent
_fs[t] = future_ai() ## Fetch the predicted future states
execute_ai(_as[t]) ## The action influences hidden external states
_ys[t] = observe_ai() ## Observe the current environmental outcome (update p) #.
## 2. Infer:
compute_ai(_as[t], _ys[t]) ## Infer beliefs from current model state (update q)
## 3. Slide:
slide_ai() ## Prepare for next iteration
end_as100-element Vector{Float64}:
0.0
-6.343207000354588e-12
-6.300452608128645e-12
-6.325567301730797e-12
-6.32821323632722e-12
-6.2999967981599965e-12
-6.32114312180386e-12
-6.301798495771919e-12
-6.3000011465939514e-12
-6.328374396370409e-12
⋮
-0.003942522330673979
-0.01943646046556582
0.0038905952643777204
0.02400063974579091
-0.00826943373310313
-0.010698347037095936
0.005816027556693469
0.013641346868345587
-0.0053588741909303935_fs100-element Vector{Vector{Float64}}:
[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
[19.98732212295191, 0.2506316816377108, 78.2434752339421, 0.024188291168147837, 82.12774910911314, 0.023343250705071573, 82.12968241977089, 0.023344835490577497, 82.1296742163674, 0.02334484547066802, 82.12967417433826, 0.02334484550648787, 82.12967385800283, 0.02334484585602179, 82.12902950374514, 0.023345557931323712, 80.81592674266264, 0.024796666594149613, -2595.101616631002, 2.9819507015123]
[20.012033919899157, 0.2500123051075404, 78.26164934260241, 0.024183478944838355, 82.12776174705, 0.023343255901790226, 82.12968238984718, 0.023344835531708123, 82.12967421619298, 0.023344845470820063, 82.12967417448836, 0.023344845506321828, 82.12967416511802, 0.023344845516629097, 82.12965536098811, 0.02334486629705572, 82.09133520704768, 0.023387213864201162, 4.000286103257123, 4.000077822209044]
[19.996940396244486, 0.25039033624168633, 78.25056091547398, 0.024186414040750052, 82.12775404036796, 0.023343252728926473, 82.1296824081125, 0.02334483550660844, 82.12967421614421, 0.0233448454708988, 82.12967385812027, 0.023344845855939968, 82.12902945105341, 0.023345557989553442, 80.81581936353287, 0.024796785258727912, -2595.3204400968066, 2.982192523124645, 4.000301064749503, 4.000077822472365]
[19.995717590896437, 0.2504210000770538, 78.24966093480106, 0.02418665239557995, 82.12775341431059, 0.023343252471713243, 82.12968240959393, 0.02334483550457186, 82.12967421630357, 0.0233448454707247, 82.12967416526138, 0.023344845516518695, 82.12965536098852, 0.02334486629705542, 82.09133520703394, 0.02338721386421635, 4.000286075247082, 4.000339273281338, 4.000339302230682, 4.000077823145877]
[20.010872666050012, 0.25004135935223804, 78.2607975653654, 0.02418370430375841, 82.12776115549383, 0.023343255657811535, 82.12968239109593, 0.02334483552995106, 82.12967389982823, 0.02334484582043638, 82.12902945119639, 0.023345557989443336, 80.8158193635329, 0.024796785258727985, -2595.3204400975956, 2.982192523125517, 4.000301036739258, 4.000339273544424, 4.000339302230687, 4.000077823145877]
[19.998324719865813, 0.2503556288722998, 78.25157947438923, 0.024186144304839383, 82.1277547488123, 0.023343253020085154, 82.12968240643134, 0.023344835508918315, 82.12967420705772, 0.023344845480937956, 82.12965536113182, 0.023344866296945047, 82.09133520703435, 0.023387213864216043, 4.0002860752470815, 4.0003392732843395, 4.000339274220319, 4.000339274217335, 4.0003393022306994, 4.000077823145877]
[20.013175547842916, 0.24998374683549687, 78.26248650755218, 0.024183257469094804, 82.12776232823045, 0.023343256141795478, 82.12968207209448, 0.023344835883226608, 82.12902949289074, 0.02334555795394937, 80.815819363689, 0.024796785258592614, -2595.3204400976274, 2.982192523125491, 4.000301036739258, 4.000339273547424, 4.000339274220324, 4.000339274217335, 4.0003393022306994, 4.000077823145877]
[20.011193708763845, 0.2500333264469247, 78.26103307176228, 0.024183641992904688, 82.12776131905511, 0.023343255725269553, 82.12968238163107, 0.023344835540514905, 82.12965540283759, 0.023344866261443286, 82.09133520717766, 0.023387213864105205, 4.0002860752470815, 4.0003392732843395, 4.000339274220319, 4.0003392742203365, 4.000339274220337, 4.000339274217335, 4.0003393022306994, 4.000077823145877]
[20.0076159566416, 0.25012286834120884, 78.25840758909462, 0.02418433672562668, 82.12775917901536, 0.02334325532290504, 82.12903767266343, 0.02334554800614642, 80.81581941001784, 0.024796785214489424, -2595.320440108609, 2.982192523115809, 4.000301036739257, 4.000339273547424, 4.000339274220323, 4.0003392742203365, 4.000339274220337, 4.000339274217335, 4.0003393022306994, 4.000077823145877]
⋮
[5.053281084686124, 4.000260505097566, 4.000339272840953, 4.000339274220312, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.000339274220337, 4.000339274217335, 4.0003393022306994, 4.000077823145877]
[5.070055050752981, 4.0002566621889155, 4.000339272773542, 4.000339274220312, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.000339274220337, 4.000339274217335, 4.0003393022306994, 4.000077823145877]
[5.0447298649948635, 4.000262418251974, 4.000339272874507, 4.000339274220312, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.000339274220337, 4.000339274217335, 4.0003393022306994, 4.000077823145877]
[5.022551490857471, 4.000267238128127, 4.000339272959048, 4.0003392742203125, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.000339274220337, 4.000339274217335, 4.0003393022306994, 4.000077823145877]
[5.05798405647362, 4.0002594397405655, 4.000339272822264, 4.0003392742203125, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.000339274220337, 4.000339274217335, 4.0003393022306994, 4.000077823145877]
[5.060617784836198, 4.000258839039103, 4.000339272811728, 4.0003392742203125, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.000339274220337, 4.000339274217335, 4.0003393022306994, 4.000077823145877]
[5.042620472987896, 4.000262885456829, 4.000339272882704, 4.0003392742203125, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.000339274220337, 4.000339274217335, 4.0003393022306994, 4.000077823145877]
[5.034017055851266, 4.000264771803537, 4.000339272915792, 4.0003392742203125, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.000339274220337, 4.000339274217335, 4.0003393022306994, 4.000077823145877]
[5.054822132291112, 4.000260157037838, 4.000339272834847, 4.0003392742203125, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.0003392742203365, 4.000339274220337, 4.000339274217335, 4.0003393022306994, 4.000077823145877]_ys100-element Vector{Vector{Float64}}:
[19.987269174689366]
[20.003012515936202]
[19.987828421880508]
[19.986659749224486]
[20.001821062932194]
[19.98921710129548]
[20.00411487425469]
[20.00207937166385]
[19.99850832336812]
[19.999713285209054]
⋮
[5.034437623873382]
[5.00696127942573]
[4.985408793409926]
[5.023876107109116]
[5.024288234007355]
[5.005348328839933]
[4.997513181834071]
[5.019801749717599]
[5.011007211431238]## _pa = plot(map(x -> x[1], _πⁿᵃⁱᵛᵉ), label="action", xlabel="t", ylabel="force")
## _py = plot(map(x -> x[1], _yⁿᵃⁱᵛᵉ), label="obsevation", color="red", xlabel="t", ylabel="temperature")
## _py = plot!([0,100], [_x₊[1], _x₊[1]], label="goal", color="green", xlabel="t", ylabel="temperature")
## plot(_pa, _py, layout=@layout([ a; b ]))
_pa = plot(
0.5*tanh.(_as), label="actions", xlabel="t", ylabel="velocity",
)
_py = plot(
map(x -> x[1], _ys), label="observations",
color="red", xlabel="t", ylabel="temperature",
)
_py = plot!(
[0, _Nᵃⁱ], [_x₊[1], _x₊[1]],
label="target/goal", color="green", xlabel="t", ylabel="temperature")
plot(_pa, _py, layout=@layout([ a; b ]))_s₀, 𝚃(_s₀), _x₊([2.0], [20.0], [4.0])Because \(s_0=[2.0]\), the temperature starts at \(\cal T([2.0]) = 20.0\). However, the target temperature is \(x_+=[4.0]\). The agent then issues actions in the form of velocity adjustments until the target temperature is reached.