Recent advances on micro-control for near-critical complex systems

Di Zhao·Yi Dong

Abstract In this paper，we review some existing control methodologies for complex systems with particular emphasis on those that are near critical.Due to the shortage of the classical control theory in handling complex systems，the reviewed control methods are mainly associated with machine learning techniques，game-theoretical approaches，and sparse control strategies.Additionally，several interesting and promising directions for future research are also proposed.

Keywords Complex systems·Near criticality·Micro-control·Machine learning·Game theory·Sparse control

1 Introduction

Near-critical systems are a class of complex systems that are“close”to criticality and possess potentials to be induced or manipulated to become critical[1，2].Such systems demonstrate advantages in multi-disciplinary application scenarios.For example，when a biological predator-prey system is near critical，the escape probability of the prey is the highest and the whole system possesses a highly dynamical and flexible structure [3]. When a real-time computing network is near critical，it can perform complex computations on time series[4].Actually，dynamical systems that are capable of fulfilling complex computational tasks often operate near the edge of chaos.To fully take advantage of near criticality，how to estimate and control of near-critical complex systems becomes a pressing issue.

Classical control theory，which mainly includes linear systems theory，sampled-data control theory，stochastic control theory， nonlinear systems theory and so on， concentrates on various topics such as stability of dynamical systems，time-domain and frequency-domain analysis，synthesis and compensation of controllers. However， most of them are inapplicable to the control of complex systems， mainly due to the following reasons. The classical control theory is model based， whereas in most case， we cannot obtain a mathematical model of a complex system， let alone an accurate one， since such a complex system is mostly featured with highly nonlinear dynamics，emergent properties，self-adaption and self-organization.For example，the classical frequency-domain methods，such as frequency response analysis and root locus methods，are inapplicable without a model. The control for complex systems is usually considered on a case by case basis and realized by inputting noise[5] and micro-control by constructing weak feedback loop[6]in some scenerios，as opposed to the model-based control theory for general linear or nonlinear dynamical systems.

Intelligent control sheds new light on the complex system analysis and provides an effective control methodology[7]， including fuzzy logic， neural networks， evolutionary computation， machine learning technique， and swarming intelligence. In particular， it is promising in studying the near-critical phenomenon of complex systems. Both from scientific and practical considerations， the estimation and measurement of the near-critical state and the control of near-critical complex systems such as biological systems，computing systems， social systems， etc.， is of paramount importance in academic and industrial societies.

In this paper， we review some existing methods for the control of near-critical complex systems. First， by utilizing machine learning techniques， we estimate the near-critical state， based on which the set of near-critical parameters are further manipulated via a game-based theory. We then investigate the micro-control strategy based on temporal and spatial sparse constraints，rendering control performance of complex systems satisfactory. Finally， possible research directions about the micro-control of complex systems are pointed out.

2 Near-critical measurements and micro-control for complex systems based on machine learning

The difficulties for estimating and measuring the near-critical state of a complex system can be described as follows.First of all，different definitions of critical system have been used in the literature， and second， the identification of the conditions under which evolutionary dynamics favour critical systems is still a fundamental open question. Third， the importance of the environment on system criticality and of the system openness are often overlooked[8].With the rapid development of artificial intelligence and computing capability， machine learning techniques have been adopted to study the complex problems，and thus the related research on estimation of near criticality and micro-control of complex systems has attracted a lot of attention in the areas of artificial intelligence.

Machine learning based methods can be used to estimate near-critical parameters of complex systems.It is extremely difficult or even impossible in practice to construct mathematical models for carbon emission systems， rail traffic control systems and power transmission systems， etc. Fortunately，machine learning based methods are advantageous in dealing with systems without accurate mathematical models.Take the unmanned driving system for example.Tami et al. [9] considered an approach based on machine learning to identify the near-critical state of the dynamical situation and to enable autonomous driving system to make an appropriate decision for safe driving. Lu et al. [10] proposed a data-driven framework to identify the critical components for a fault detection system. The criticality of a component with respect to the safe/failed state of the detection system is assessed with a feature selection technique based on random forest classification. It is certainly an interesting topic to further estimate the near-critical parameters of unmodeled dynamics via data-based machine learning methods.

With the near-critical parameters estimated， the microcontrol can be further fulfilled by machine learning approaches. Such control methods are becoming more and more popular in the control design of linear/nonlinear systems.For instance， Lewis and Vamvoudakis [11] solved a linear quadratic regulation problem with unknown parameters based on the reinforcement learning methods. For a complex system， in most cases， it is neither controllable nor observable，and thus the design of traditional control strategies is formidable even with the model known. However，under special circumstances，micro-control strategies based on machine learning have achieved some success.Tami et al.[9]predicted the relation between control input and system state to ensure robust decision-making of autonomous vehicles with deep learning method.For the Hegselmann-Krause(HK)model for opinion dynamics，Su et al.[5]showed that noise could serve as the “micro-control input”， and below the critical value， the opinions of the noisy HK dynamics would tend to achieve “consensus” in finite time. It is expected to take advantage of more machine learning methods to realize micro-control of complex systems in the near future.

3 Parameter regulation for near-critical complex systems based on game theory

A critical system often behaves tremendously differently in variousmacroscopicaspects，suchasattractiveregions，bifurcation patterns and universality， due to a slight variation in parameters[1，2，12].When a nonlinear system is near critical，whether an attractor exists or not and how the bifurcation phenomena appear sensitively depend on the variation of parameters.For an uncontrollable and unobservable complex system，it is almost unlikely to drive it away from a chaotic state by classical control methods.

In certain scenarios of economics[13]，biology[14]and so on， it is promising to transform the chaotic state to the stable state by parameter regulation.Duopolistic Stackelberg game is a typical complex system in economics.Such a game in [15] was built based on a cost function that is nonlinear and depends on the quantities produced by firms and the announced plan products.The stability of the Nash equilibrium was investigated under various parameter setting.The critical parameters under which chaotic phenomena appear were identified by numerical simulation， and proper regulation on the identified parameters drives the system from chaos to stability.

Regarding the parameter sensitivity of near-critical complex systems，an appealing research direction is to utilize the game theory in achieving the optimization. To be specific，a multiple game strategy set may be built up to implement the most appropriate approach for optimizing the parameters and eventually driving the complex system towards desired steady state.

4 Micro-control for complex systems based on temporal and spatial sparse constraints

The classical control theory can be interpreted as methods of“strong control”since it usually requires a high capability of perception and actuation to implement the control.The complex systems including nonlinear dynamical networks and chaotic systems display limited actuation ability，making the“strong control”methods inapplicable to execute the control.In contrast with the“strong control”，the micro-control methods aim at reducing the deployment of actuators and lowering the actuating power while achieving the efficiency.In the scenario of a weakly actuated system， one of the most critical problems is how to optimize the deployment of actuators and the allocation of actuating energy to achieve micro-control.Based on the current sparse control theory，an optimal control strategy involving temporal and spatial sparse structure can significantly reduce the total deploying number of actuators and the cost of actuating energy by optimizing the allocation of actuating resources， and eventually facilitate the microcontrol.

The theory of sparse control has been intensively investigated for the linear and nonlinear dynamics. For linear dynamical models， various sparse control strategies have been developed. Ikeda and Kashima [16] characterized the solution set for optimal sparse control of general infinitedimensional linear systems and described the controllability.For complex nonlinear dynamical systems， methods for sparse identification and control have been developed for a lot of applications. Stadler [17] solved an elliptical control problem withL1norm cost，serving as a heuristic method to induce sparse control structure.Caponigro et al.[18]studied sparse control methods for Cucker-Smale nonlinear network model and proves the optimality under prescribed conditions.Kaiser et al.[19]proposed a sparse identification scheme for model predictive control for nonlinear systems.

To control complex systems with inadequate actuation capability，it is interesting to explore sparse control strategies from both the temporal and spatial perspectives.The temporal sparsification can be realized by asynchronous sampling on the multiple actuation signals，while the spatial sparsification can be achieved by solving optimal control problems with sparse structure constraints.A combination of the temporal and spatial sparsification schemes will eventually result in a systematic way for micro-control of complex systems.

5 Conclusions

In this paper， we have reviewed methodologies for the micro-control of near-critical complex systems， including techniques based on the machine learning，game theory and sparse control.Both the theoretical developments and practical implementations are still at early stages and have become increasingly popular due to multi-disciplinary demands on complex systems science.In the near future，we expect that vast applications in the engineering and military will require more accurate measurement of the near-critical state，and the associated micro-control will certainly become a more and more appealing research topic.