Neural Stochastic Control: Exponential Stabilizer and Asymptotic Stabilizer
Anonymous Author(s)
AffiliationAddress email
Abstract
Control problems are always challenging since they arise from the real-worldsystems where stochasticity and randomness are of ubiquitous presence.
Thisnaturally and urgently calls for developing efficient neural control policies forstabilizing not only the deterministic equations but the stochastic systems as well.
Here, in order to meet this paramount call, we propose two types of controllers, viz.
,the exponential stabilizer (ES) based on the stochastic Lyapunov theory and theasymptotic stabilizer (AS) based on the stochastic asymptotic stability theory.
The
ES can render the controlled systems exponentially convergent but it requires a longcomputational time; conversely, the AS makes the training much faster but it canonly assure the asymptotic (not the exponential) attractiveness of the control targets.
These two stochastic controllers thus are complementary in applications.
Wealso investigate rigorously the linear controller and the proposed neural stochasticcontrollers in both convergence time and energy cost and numerically compare themin these two indexes.
More significantly, we use several representative physicalsystems to illustrate the usefulness of the proposed controllers in stabilization ofdynamical systems.
Introduction
[Figure]
Exponential Stabilizer
Asympyotic Stabilizer
ICNN or Quadratic form
FNN
Slow
Fast
High
Low
C o m p u t a t i o n a l c o s t
L o n g t
i m e
Controlledorbit
Orbit
FNN =
( , ( ), ( ), , ) =
( , ( ), ( ), ) e t a r e c n e g r e v n o C ( ) =
( ) + ( ) ( )
= ( ) + ( ( ) + ( ))
Initial state Equilibrium
Original system
Controlled system 1 x 2 x 3 x 1 d x − d x . . .
∗ ∗ ∗
(FNN).
In the field of controlling dynamical systems,one of the major missions is to find efficientcontrol policies for stabilizing ordinary differential equations (ODEs) to targeted equilibriums.
The policies for stabilizing linear or polynomialdynamical systems have been fully developedusing the standard Lyapunov stability theory,e.g., the linear quadratic regulator (LQR) Khalil(2002) and the sum-of-squares (SOS) polynomials through the semi-definite planning (SDP)
Parrilo (2000).
As for stabilizing more generaland nonlinear dynamical systems, linearizationtechnique around the targeted states is often utilized and thus the existing control policies areeffective in the vicinity of the targeted states(Sastry & Isidori, 1989) but likely lose efficacyin the region far away from those states.
Moreover, in real applications, the explicit forms of the con- trolled nonlinear systems are often partially or completely unknown, so it is very difficult to directlydesign controllers only using the Lyapunov stability theory.
To overcome these difficulties, designingthe controllers via training neural networks (NNs) become one of the mainstream approaches in thecommunity of cybernetics (Polycarpou, 1996).
Recent outstanding developments using NNs includeenlarging the safe region (Richards et al., 2018), learning the stable dynamics (Takeishi & Kawahara,2021), and constructing the Lyapunov function and the control function simultaneously (Chang et al.,2019).
In Kolter & Manek (2019) a projected NN has been constructed to directely learn a stabledynamical system and fits the observed time series data well, but it did not focus on learning a controlpolicy to stabilize the original dynamics.
All these existing developments are formulated only fordeterministic systems but inapplicable directly to the dynamical systems described by stochasticdifferential equations (SDEs), requiring us to include the stochasticity appropriately into the use ofneural controls to different types of dynamical systems.
The stability theory for stochastic systems has been systematically developed in the past severaldecades.
Representative contributions in the literature include the Lyapunov-like stability theoryfor SDEs Mao (2007), stabilization of unstable states in ODEs only using noise perturbations Mao(1994b), and the stability induced by randomly switching structures Guo et al.
Generally, forany SDEs governed by d x = f ( x )d t + g ( x )d B t , control policies as u
= ( u f , u g ) are introduced,which transforms the original equations into the controlled system d x = [ f ( x )+ u f ( x )]
d t +[ g ( x )+u g ( x )]
d B t .
Appropriate forms of control policies are able to steer the controlled system to theequilibriums that are unstable in the original SDEs.
Traditional control methods focus on designingdeterministic control u f and regard noise as negative part.
Innovatively, we treat noise as a beneficialpart and design stochastic control u g to achieve the stabilization.
In this article, we articulate two frameworks of neural stochastic control which can complement eachother in terms of convergence rate and the computational time of training NNs.
Additionally, weanalytically investigate the convergence time and the energy cost for the classic linear control and theproposed neural stochastic control and numerically compare them.
We further extend our frameworksto model-free case with existing data reconstruction methods.
The major contributions of this articleare multi-folded, including:
• designing two frameworks of neural stochastic control, viz., the ES and the AS, and presenting their advantages in the stochastic control,• providing theoretical estimation for ES/AS and classic linear control in terms of convergencetime and the energy cost,• computing the convergence time and the energy cost of particular stochastic neural control,• demonstrating the efficacy of the proposed stochastic neural control in important control
problems arising from representative physical systems, and we make our code available at
https://anonymous.4open.science/r/Neural-Stochastic-Control-DE3F/ .
1.1 Related Works
Lyapunov Method in Machine
Learning The recent work Chang et al.
(2019) proposed an
NN framework of learning the Lyapunov function and the linear control function simultaneouslyfor stabilizing ODEs.
In comparison, we select several specific types of NNs which have typicalproperties of the Lyapunov function.
For instance, we use the input convex neural network (ICNN)
Amos et al.
(2017), constructing a positively definite convex function as a neural Lyapunov function(Kolter & Manek, 2019; Takeishi & Kawahara, 2021),
and we construct the NN in a quadratic form(Richards et al., 2018; Gallieri et al., 2019) for linear or sublinear systems where the SDP method isoften used to find the SOS-type Lyapunov function (Henrion & Garulli, 2005; Jarvis-Wloszek et al.
,2003; Parrilo, 2000).
Stochastic Stability Theory of SDEs Stochastic stability theory for SDEs have been systematically and fruitfully achieved in the past several decades (Kushner, 1967; Arnold, 2007; Mao,1991, 1994a).
The positive effects of stochasticity have also been cultivated in control fields (Mao et al., 2002;
Deng et al., 2008; Caraballo et al., 2003; Appleby et al., 2008; Mao et al., 2007).
These,therefore, motivate us to develop only neural stochastic control to stabilize different sorts of dynamicalsystems in this article.
More stochastic stability theory for different kinds of systems are included in
Appleby et al.
; Appleby (2003); Caraballo & Robinson (2004); Wang & Zhu (2017).
Preliminaries
To begin with, we consider the SDE which is written in a general form as:
[Equation]
where F : R d → R d is the drift function, and G : R d → R d × r is the diffusion function with R d × r ,a space of d × r matrices with real entries, and B t ∈ R r , a r -dimensional
( r -D) Brownian motion.
Without loss of generality, we set F ( 0 ) = 0 and G ( 0 ) =
so that x 0 = 0 is a zero solution Eq.
Notations.
Denote by ∥ · ∥ the L 2 -norm for any given vector in R d .
Denote by | · | the absolute valueof a scalar number or the modulus length of a complex number number.
For A = ( a ij ) , a matrix ofdimension d × r , denote by ∥ A ∥ 2F = (cid:80) di =1 (cid:80) rj
=1 a 2 ij the Frobenius norm.
Assumption 2.1 (Locally Lipschitzian Continuity)
For every integer n ≥ 1 , there is a number
K n > 0 such that
[Equation]
for any x , y ∈ R d with ∥ x ∥ ∨ ∥ y ∥ ≤ n .
Definition 2.1 (Derivative Operator) Define the differential operator L associated with Eq.
(1) by
[Equation]
Definition 2.2 (Exponential Stability)
The zero solution of Eq.
(1) is said to be almost surely exponentially stable, if lim sup t →∞ 1 t log ∥ x ( t ; x 0 )
∥ < 0 a.s. for all x 0 ∈ R d .
Here and throughout,a.s. stands for the abbreviation of almost surely.
Then, the following Lyapunov stability theorem will be used in the establishment of our main results.
Theorem 2.2 Mao (2007) Suppose that Assumptions 2.1 holds.
Suppose further that there exist afunction V ∈ C 2 ( R d ; R + ) with V ( 0 ) = 0 , constants p > 0 , c 1 > 0 , c 2 ∈ R and c 3 ≥ 0 , such that (i)c 1 ∥ x ∥ p ≤ V ( x ) , (ii) L V ( x ) ≤ c 2 V ( x ) , and (iii) |∇ V ⊤ ( x ) G ( x ) | 2 ≥ c 3 V 2 ( x ) for all x ̸ = 0 andt ≥ 0 .
Then,
[Equation]
In particular, if c 3 − 2 c 2 > 0 , the zero solution of Eq.
(1) is exponentially stable almost surely.
The following asymptotic theorem also will be used in the establishment of our main results.
Theorem 2.3 Appleby et al.
Suppose that Assumptions 2.1 holds.
Suppose furthermin ∥ x ∥ =
M ∥ x ⊤ G ( x ) ∥ > 0 for any M > 0 and there exists a number α ∈ (0 , 1) such that
[Equation]
Then, the unique and global solution of Eq.
(1) satisfies lim t →∞ x ( t, x 0 )
=
a.s. , and we call thisproperty as asymptotic attractiveness.
Designing Stable Stochastic Controller
Here, we assume that the zero solution of the following SDE:
[Equation]
is unstable.
Note that, for any nontrivial targeted equilibrium x ∗ , a direct transformation y = x
− x ∗can make the zero solution as the equilibrium of the transformed system.
Thus, our mission is tostabilize the zero solution only.
As such, we are to use the NNs to design the control u : R d → R d × rwith u ( 0 )
=
and apply it to Eq.
(4) as
[Equation]
Since u is integrated with d B t in the controlled system (5), we regard it as a stochastic controller.
Inwhat follows, two frameworks of neural stochastic control, the exponential stabilizer (ES) and theasymptotic stabilizer (AS), are articulated, respectively, in Sections 3.1 and 3.2.
All these controlpolicies are intuitively depicted in Figure 1.
3.1 Exponential Stabilizer
Once we find the Lyapunov function V and the neural controller u , making the controlled system (5)meet all the conditions assumed in Theorem 2.2, the equilibrium 0 can be exponentially stabilized.
To this end, we first provide two different types of functions for constructing V , which actually couldbe complementary in applications.
Then, we design the explicit forms of control function and lossfunction.
ICNN V Function.
We use the ICNN (Amos et al., 2017) to represent the candidate Lyapunovfunction V .
This guarantees V as a convex function with respect to the input x .
In order to furthermake V as a true Lyapunov function, we use the following form:
[Equation]
6) as introduced in Manek & Kolter (2020).
Here, W i , b i are real-valued weights, U i are positiveweights, σ i are convex, monotonically non-decreasing activation functions in the i -th layer, ε isa small positive constant, and F is a continuously differentiable and invertible function.
In ourframework, we require V ∈ C 2 ( R d ; R + ) according to Definition 2.1; however, each activationfunction σ i ≡ σ in Manek & Kolter (2020) is C 1 only.
Thus, we modify the original function as:
[Figure]
[Equation]
which not only approximates the typical ReLU activation but also becomes continuously differentiableto the second order (see Figure 2).
Quadratic V Function.
For any x ∈ R d , let V θ ∈ R
d be a multilayered feedforward NN of theinput x and with tanh( · ) as the activation functions, where θ is the parameter vector.
To meet thecondition used in Definition 2.1, we cannot use the ReLU , a non-smooth function, as the activationfunction.
Hence, we use the candidate Lyapunov function as:
[Equation]
8) which was introduced in Gallieri et al.
Here, ε is a small positive constant.
Control Function.
We introduce a multi-layer feedforward NN (FNN), denoted by NN ( x ) ∈ R r ,to design the controller u .
Since we require u ( 0 ) = 0 , we set u ( x ) ≜ NN ( x ) − NN ( 0 ) oru ( x ) ≜ diag( x ) NN ( x ) with r = d .
Here, diag( x ) is a diagonal matrix with its i -th diagonalelement as x i .
Remark 3.1 As reported in Chang et al.
(2019), a single-layer NN without the bias constants in itsarguments, which degenerates as linear control, could sufficiently take effect in the stabilization ofmany deterministic systems.
However, this is NOT always the case for achieving the stabilization of highly nonlinear systems or even SDEs.
The following proposition with Figure 3 provides anexample, where neither the classic linear controller nor the stochastic linear controller can stabilizethe unstable equilibrium in a particular SDE.
The proof of this proposition is included in Appendix
A.1.
Proposition 3.2 Consider the following 1 -D SDE:
[Equation]
) with a zero solution x ∗ = 0 .
Then, for u ( x ) = kx with any k and x 0 ̸ = 0 , x ∗ = 0 is neitherexponentially stable nor of globally asymptotic attractiveness almost surely.
For u ( x ) = 2 x 2 , x ∗ = 0is of globally asymptotic attractiveness.
For u ( x ) ≡ 0 , the deterministic system cannot be stabilizedby any classic linear controller.
[Figure]
Loss Function.
When the learning procedureupdates the parameters in the NNs such thatthe constructed V and u with the coefficientfunctions, f and g u ≜ g + u , in the controlledsystem (5) meet all the conditions assumed in
Theorem 2.2, the exponential stability of thecontrolled system is assured.
Thus, we demanda suitable loss function to evaluate the likelihood that those conditions are satisfied.
First, from
Theorem 2.2, it follows that V ( x ) ≥ ε ∥ x ∥ 2 for all x ∈ R d .
Thus, Conditions (ii)-(iii) together withc 3 − 2 c 2 > 0 in Theorem 2.2 equivalently become
[Equation]
These conditions further imply that
[Equation]
With these reduced conditions, we design the ES loss function for the controlled system (5) as follows.
Definition 3.1 ( ES loss) Consider a candidate Lyapunov function V and a controller u for thecontrolled system (5).
Then, the ES loss is defined as
[Equation]
where the state variable x obeys the distribution µ .
In practice, we consider the following empiricalloss function:
[Equation]
where { x i } Ni =1 are sampled from the distribution µ = µ (Ω) and Ω is some closed domain in R d .
For convenience, we summarize the developed framework in Algorithm 1.
Here, b is a hyperparameter that can be adjusted as required by solving a specific problem.
Remark 3.
In Section 5, we show numerically that the conditions reduced in (11) are sufficientlyeffective for designing the ES loss.
Actually, it is not necessary to design the loss function using theconditions in (10) .
Now, for controlling any nonlinear ODEs or SDEs, we design the ES according to Algorithm 1.
Assuch, using the ES framework can not only stabilize those unstable equilibriums (constant states)of the given systems, but also can stabilize those unstable oscillators, e.g., the limit cycle.
Thisis because the solution corresponding to the oscillator can be regarded as a zero solution of thecontrolled system after appropriate transformations are implemented.
Another point needs attention.
During the construction of V in (6), ε ∥ x ∥ 2 , the L 2 -regularization, isused to guarantee the positive definiteness of V .
However, often in the application of the Lyapunovstability theory, the form of ∥ x ∥ 2 is not always a suitable candidate for the Lyapunov function.
Itmay restrict the generalizability of using our framework, so it needs necessary adjustments.
Thefollowing example illustrates this point.
Example 3.4 Consider a 2 -D SDE as follows:
[Equation]
In Appendix A.2, the zero solution of this system is validated to be exponentially stable almost surely;however, k ∥ x ∥ 2 for any k ∈ R cannot be a useful auxiliary function to identify the exact stability ofthe zero solution.
To be candid, using the current framework takes a longer time for training and constructing the neural
Lyapunov function.
In the next subsection, we thus establish an alternative control framework thatcan reduce the training time.
3.2 Asymptotic Stabilizer
Here, in light of Theorem 2.3, we are to establish the second framework, the AS, for stabilizingthe unstable equilibrium of system (5).
This framework only makes the equilibrium asymptoticallyattractive almost surely.
Its control function is designed in the same way as the one used in the ESframework, whereas the loss function is differently designed.
Definition 3.2 ( AS loss) Utilization of the notations used in Definition 3.1, the loss function for thecontrolled system (5) with the controller u is defined as:
[Equation]
Akin to Definition 3.1, we set the empirical loss function as:
[Equation]
Here, α is an adjustable parameter, which is related to the convergence time and the energy cost usingthe controller u .
We show in Appendix A.8 the influence of selecting different α .
For convenience,we summarize the AS framework in Algorithm 2.
Understanding AS Loss
We utilize the formula ∥ x ∥ 2 (2 ⟨ x , F ( x ) ⟩ + ∥ G ( x ) ∥ 2F ) − (2 −α )
∥ x ⊤ G ( x ) ∥ 2 ≜ q ( x ) in (3) to construct the AS loss.
Here we explain this term in more detail.
The q ( x ) is derived from the chain derivative of ∥ x
∥ α = ( ∥ x ∥ 2 ) α/ 2 according to It ˆo ’s formula,we then have d ∥ x ∥ α = α/ 2 ∥ x
∥ α − 4 q ( x )d t + α ∥ x ∥ α − 2 ∥ x ⊤ G ( x ) ∥
d B t .
The drift term is less thanzero when q ( x ) ≤ 0 , which can guarantee the bounded existence and asymptotic stability of ∥ x
∥ αfrom the semi-martingale convergence theorem (Liptser & Shiryayev, 2012) .
Computational Complexity The AS framework is computationally efficient because we onlyneed the tensor operation in the training process.
The computational complexity of this procedure is
O ( mn ) for m data on n -D dynamics.
The training for ES framework is not as efficient as AS.
Themajor reason is that we should compute the Hessian matrix of V to get L V and the computationalcomplexity of this operator is O ( mn 2 ) .
Convergence Time and Energy Cost
The convergence time and the energy cost are the crucial factors to measure the quality of a controller(Yan et al., 2012;
Li et al., 2017; Sun et al., 2017).
In this section, we provide a comparative study between the traditional stochastic linear control and the ES/AS, the above-articulated neural stochasticcontrol.
To this end, we first present a theorem on the estimations of the convergence time and the energy costfor the stochastic linear control on a general SDE.
Theorem 4.1 Consider the SDE with a stochastic linear controller as:
[Equation]
14) where ⟨ x , f ( x ) ⟩ ≤ L ∥ x
∥ 2 and u ( x ) =
k x with | k | > √ 2 L .
Then, for ϵ < ∥ x 0 ∥ , we have
[Equation]
where, for a sufficiently small ϵ > 0 , we denote the stopping time by τ ϵ ≜ inf { t > 0 :
∥ x ( t ) ∥
= ϵ }and denote the energy cost by
[Equation]
The proof of this theorem is provided in Appendix A.3.1.
We further consider the case for NN controller u ( x ) .
In general, the u ( x ) is Lipschitz continuouswith Lipshcitz constant k u under a suitable activation function such as ReLU in NN (Fazlyab et al.
,2019; Pauli et al., 2021).
Then we have the following upper bound estimation of the convergencetime and the energy cost for ES and AS, whose proofs are provided in Appendix A.3.2, A.3.3.
Theorem 4.2 ( Estimation for ES) For ES stabilizer u ( x ) in (14) with ⟨ x , f ( x ) ⟩ ≤ L ∥ x ∥ 2 , ε <
∥ x 0 ∥ , under the same notations and conditions in Theorem 2.2 with c 3 − 2 c 2 > 0 , we have
[Equation]
Theorem 4.3 ( Estimation for AS) For (14) with ⟨ x , f ( x ) ⟩ ≤ L ∥ x ∥ 2 , ε <
∥ x 0 ∥ , underthe same notations and conditions in Theorem 2.3, if the left term in (3) further satisfiesmax ∥ x ∥≥ ε ∥ x ∥ α − 4 ( ∥ x ∥ 2 (2 ⟨ x , f ( x ) ⟩ + ∥ u ( x ) ∥ 2F ) − (2 − α )
∥ x ⊤ u ( x ) ∥ 2 )
= − δ ε < 0 , then for
NN controller u ( x ) with Lipschizt constant k u , we have
[Equation]
Based on the theoretical results for ES and AS, we can further analyze the effects of hyperparametersb, α and neural network structures on the convergence time and energy cost in the control process.
There are some interesting phenomena such as the monotonicity of T ε along α for AS change withthe relative relationships of ∥ x 0 ∥ and ε , this inspires us to select suitable α according to the specificproblems.
We leave more discussions in Appendix A.3.4.
Now, we numerically compare the performances of the linear controller u ( x ) =
kx and the ASon the convergence time and the energy cost of the controls applied to system (14) with specificconfigurations (see Figure 4).
We numerically find that u ( x ) =
kx can efficiently stabilize theequilibrium for k > k c = 5 . 6 .
Without loss of generality, we fix k = 6 . 0 , and compare thecorresponding performances.
As clearly shown in Figure 4, the AS outperforms u ( x ) = kx fromboth perspectives, the convergence speed and the energy cost.
In the simulations, the energy cost
E ( τ ϵ , T ϵ ) defined above is computed in a finite-time duration as E ( τ ϵ , T ∧ T ϵ ) , where T < ∞ isselected to be appropriately large.
We leave more results of the comparison study in Appendix A.5.5.
[Figure]
Experiments
In this section
[Figure]
we demonstrate the efficacy of the above-articulated frameworks of stochastic neuralcontrol, the ES and the AS, on several representative physical systems.
We also compare thesetwo frameworks, highlighting their advantages and weaknesses.
The detailed configurations forthese experiments are included in Appendix A.5.
Additional illustrative experiments are included in
Appendix A.
[Figure]
5.1 Harmonic Linear Oscillator
[Figure]
First, consider the harmonic linear oscillator¨ y + 2 β ˙
y + w 2 y = 0 , where w is the natural frequency and β > 0 is the damping coefficient representing the strength of the external force on thevibrator (Dekker, 1981).
Although this system isexponentially stable, the system with stochasticperturbations ¨ y +(2 β + ξ 2 ) ˙
y
+( w 2 + ξ 1 ) y
= 0 becomes unstable even if E [ ξ 1 ( t )]
=
E [ ξ 2 ( t )] = 0(Arnold
et al., 1983).
Now, we apply the nonlinear ES(+ICNN), the linear ES(+Quadratic),and the nonlinear AS, respectively, to stabilizing this unstable dynamics (A.5.1) with w 2 = 1 ,β = 0 .
5 , ζ 1 = − 3 , and ζ 2 = 2 . 15 , the resultsare shown in Figure 5.
Indeed, we find that the two nonlinear stochastic neural controls are morerobust than the linear control, and that the ES(+ICNN), rather than the AS, makes the controlledsystem more stable.
[Table]
In Table 1 we compute the training time (Tt)used for the loss function converging to 0 , thenumber of iterations (Ni), the distance (Di) between the trajectory and the targeted equilibriumat time T = 4 , and the convergence time (Ct)when the distance between the trajectory and thetargeted equilibrium is less than 0 .
The results are obtained through averaging the corresponding quantities produced by 20 randomly-sampledtrajectories and the detailed training configurations are shown in Appendix A.5.1.
[Table]
[Figure]
We further provide a comparison study of our newlyproposed ES(+ICNN) with HDSCLF in (Sarkar et al.,2020), BALSA in (Fan et al., 2020) and classic LQRcontroller in controlling the harmonic linear oscillator.
Both HDSCLF and BALSA are based on the Quadratic
Program(QP), and they seek the control policy dynamically for each state in the control process.
By contrast, our proposed learning control policy is directlyused in the control process.
Hence, our method is more efficient in the practical control problems.
The results are shown in Figure.
6 (Please see more details in Appendix A.5.1).
As can be seen thatour learning control methods outperforms all others.
5.2 Inverted Pendulum
[Figure]
and we compare the energy cost between linear control u ( x ) =
kx
and our Asymptotic Stabilizer learning control in Figure 4.
We put more details and experiments in the appendix 0 2 4 6 8 10 k 051015 l og ( 1 + x ) 0 .
5 Time − 1000100200 x
Linearcontrol 0 0 . 075 0 .
15 Time − 2002040 x
Learnedcontrol (a) (b) (
5. Experiments
In this section we consider some problems with actual physical background that are not all direct equation stability problems, but can be transformed into equivalent forms of finding the stable control of the corresponding equations, and some recent articles have partially solved these problems.
So it can be used as a benchmark to compare with our method with respect to convergence speed and computation time.
We refer to Appendix C for implementation details and to Appendix D for additional experiments.
5.1. Harmonic Linear Oscillator
First, we consider harmonic linear oscillator ¨ y + 2 β ˙
y
+ w 2 y = 0 , where w is the natural frequency of oscillator, β > 0 is the damping coefficient representing the strength of external force on the vibrator.
Although the original system is exponentially stable, the perturbation system with random damping and random restoring force ¨ y + (2 β + ξ 2 ( t )) ˙
y
+ ( w 2 + ξ 1 ( t ))
y = 0 , (27) becomes unstable even if E ξ 1 ( t ) =
E ξ 2 ( t ) =
(Arnold et al., 1983).
Now we use our algorithms to stabilize this unstable random dynamics for w 2 = 1 ,β = 0 .
5 ,k 1 = − 3 ,k 2 = 2 .
and we plot the results in Figure  2 20 10trajectory 0 25000
10x1x2 (a) original 0. trajectory 0 25000 50000 75000 0.0 0.2 0.
x1x2 (b) controlled
[Equation]
(28) where m is the pendulum mass, g is the gravitational acceleration, l is the pole length, and β is the friction coefficient.
This system has two state variables x
= ( θ, ˙ θ ) , the angle θ from the upright equilibrium x = 0 and angular velocity ˙ θ .
[Figure]
We hope that the inverted pendulum can remain upside down, but this position corresponds to the unstable equilibrium point x = 0 of the system.
We illustrate this phenomenon in Figure 6.
We use the Asymptotic Stabilizer in
Algorithm 2 to stabilize the system (28) and show the results in Figure 6.
[Figure]
Controlling the inverted pendulum is one of theclassic nonlinear control problems (Anderson,1989; Huang & Huang, 2000), which is governed by ml
2 ¨ θ = mgl sin θ − β ˙ θ.
Here, mis the mass of the pendulum, g is the gravitational acceleration, l is the pole length, andβ is the friction coefficient.
Mathematically,the pendulum can be written as a system withtwo state variables: θ , the angle deviating fromthe vertical position x = 0 , and ˙ θ , the angular velocity.
Denote the 2 -D state variable byx =
( θ, ˙ θ ) .
Then, we apply the AS articulatedin Algorithm 2 to steer the system to the equilibrium x = 0 (sustaining the pendulum inverted).
The schematic diagram of the inverted pendulum system and successful control results are shownin Figure 7.
The sampled trajectories are initiated from different angles and angular velocities [i.e.,( θ (0) , ˙ θ (0))
=
( − 5 , 5) , ( − 1 , 3) , (3 , − 4) ].
5.3 Stuart-Landau Equations
In this subsection, we show that our frameworks are beneficial to realizing the control and thesynchronization of complex networks.
To this end, we consider the single Stuart–Landau oscillatorwhich is governed by the following complex-valued ODE:
[Equation]
This equation is a paradigmatic model undergoing the so-called Andronov bifurcation (Kuznetsov,2013): Stability of the equilibrium changes and the limit cycle emerges as the parameter passes somecritical value.
In what follows, we consider two cases based on system (15).
[Figure]
n t r o l
W i t
h
o u t c
o
n t r o l
[Figure]
Case 1 We set β = − 25 , γ =
[Figure]
and µ = 1 , so that system (15) has a stable equilibrium ρ = 0and an unstable limit cycle ρ = 5 .
Here, Z = x + i y
= ρe i θ .
Now, the AS steers the dynamics tothe unstable limit cycle, as successfully shown in Figure 8.
The trajectories (the left column) andthe phase orbits (the right column) of system (15), initiated from 30 randomly-selected initial states,without control (the upper panels) and with control of the AS (the lower panels).
The initial valuesinside (resp., outside) the limit cycle
= 5 are indicated
Case 2 Next, we consider the synchronization problem of the coupled Stuart-Landau equations.
Successful deterministic methods have been systematically developed for realizing synchronization, including the adaptive control with time delay Selivanov et al.
(2012) and the open-loop temporalnetwork controller Zhang & Strogatz (2021).
These methods majorly depend on the techniqueof linearization in the vicinity of the synchronization manifold.
Here, we show how to apply ourframework to achieving the synchronization in the coupled system.
We set the corresponding
Laplace matrix L =
( L jk ) n × n as (cid:80) nk =
1 L jk = 0 , which guarantees the synchronization manifoldis an invariant manifold of the coupled system (Pecora & Carroll, 1998).
Specifically, we select asn = 20 , σ = 0 .
01 , c 1 = − 1 . 8 , c 2 = 4 , and L jk =
δ jk − 1 n , where δ jk is the Kronecker function.
Then, we apply the AS to this system and realize the stabilization of the synchronous manifold, asshown in Figure 9.
In addition to the above controlled systems, we include the other illustrative examples, the controlledreservoir computing and the controlled Lorenz system, in Appendix A.6.
5.4 Stabilizing Cell Fate Dynamics through Data-Driven Pinning Controller
[Figure]
Indeed, our frameworks can be extended to themodel-free version via a combination with existing data reconstruction method.
It should benoted that the proposed methods only depend onthe samples of vector field in the learning stageinstead of the explicit formula of systems (see
Algorithm 1,2)
, implying that our frameworksinherently include the model-free situation.
To be concrete, we show that our framework can combinewith Neural ODEs (NODEs) (Chen et al., 2018) to learn the control policy from time series data forthe Cell Fate system (Sun et al., 2017; Laslo et al., 2006), which describes the interaction betweentwo suppressors during cellular differentiation for neutrophil and macrophage cell fate choices.
Thesystem ˙ x = f ( x ) , x = ( x 1 , ..., x 6 ) has three steady states: P 1 , 2 , 3 , where P 2 , 3 correspond to differentcell fates and are stable and P 1 represents a critical expression level connecting the two fates and isunstable.
The network structure of this 6 -D system is a treemap, where one root node x 1 can stabilizeitself under the original dynamic.
Hence, we choose root node x 2 with maximum our degree andadd pinning control on it to stabilize the system to unstable state P 1 .
The original trajectory thatconverges to P 2 (left) and the controlled trajectory that converges to P 1 (right) are shown in Figure 10.
The original trajectory is used to train the NODE to reconstruct the vector field
ˆ f , then we use thesample of ˆ f as training data to learn our stochastic pinning control.
We provide experimental detailsin Appendix A.5.6.
The results in this section do not conflict with those in the Appendix A.9 becausethe 3 -link pendulum does not has the similar treemap structure and self-stabilizing nodes as the cellfate system.
Conclusion and Future Works
In this article, we have proposed two frameworks of neural stochastic control for stabilizing differenttypes of dynamical systems, including the SDEs.
We have shown that the neural stochastic controloutperforms the classic stochastic linear control in both the convergence time and the energy costfor typical systems.
More importantly, using several representative physical systems, we havedemonstrated the advantages of our frameworks and showed part of their weaknesses possiblyemergent in real applications.
Also, we present some limitations of the proposed frameworks in
Appendix A.9.
Moreover, we suggest several directions for further investigations: (i) acceleration of the trainingprocess of the ES, (ii) the basin stability of the neural stochastic control (Menck et al., 2013), (iii)the trade-off between the deterministic controller u g using the NNs and the stochastic controller u fusing the NNs, (iv) the safe learning in Robotic control with small disturbances (Berkenkamp et al.,2017), and (v) the design of the purely data-driven stochastic neural control. Checklist For all authors...
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