We present an approach for data-driven modeling and evolving control of unknown dynamic systems called State-Space Evolving Granular Control. The approach is based on elements of granular computing, discrete state-space systems, and online learning. First, the structure and parameters of a granular model is developed from a stream of state data. The model is formed by information granules comprising ifrst-order diference equations. Partial activation of granules gives global nonlinear approximation capability. The model is supplied with an algorithm that constantly updates the granules toward covering new data; however, keeping memory of previous patterns. A granular controller is derived from the granular model for parallel distributed compensation. Instead of diference equations, the content of a control granule is a gain matrix, which can be redesigned in real-time from the solution of a relaxed locally-valid linear matrix inequality derived from a Lyapunov function and bounded control-input conditions. We have shown asymptotic stabilization of a chaotic map assuming no previous knowledge about the source that produces the stream of data.
Evolving granular-computing models are structures with online learning, and summarizing and representational capabilities [1, 2, 3, 4]. An evolving granular model is equipped with an incremental algorithm that gradually builds its structure of interconnected elements. The elements are called information granules. Granules features, their location, size and geometry on search spaces, and the content they carry, may change at any time according: (i) to the spatio-temporal patterns noticed on a numerical, word, image, or relatively-finer granular-data stream; and (ii) to inter-granular relationships. From the dynamic aggregation of the local estimates – ofered by the content of the current set of active granules – emerging nonlinear and time-varying behaviors and outputs can be produced by the model, including those often needed to track concept drifts and shifts.
Evolving granular systems [1] is a general-purpose online-learning framework to construct classifiers, regressors, predictors, and controllers in which at least one of several possible https://sites.google.com/site/danfl7 (D. Leite) aspects of a problem may assume a granular value. In other words, non-pointwise uncertain characterization (e.g., interval, fuzzy, rough, statistical, and mixtures of uncertain objects) can be admitted to: original data instances, pre-processed or space-transformed instances; to model parameters, learning equations, covering regions, and so on [1, 3, 5]. Generalized constraints, in the sense of Zadeh’s general theory of uncertainty [6], are used to delimit granules. Data granulation, granular computation, and mapping granules of diferent domains are attractive features to manage data uncertainty, to reduce complexity, and to reason in a structured and more meaningful way from the point of view of human-centered systems [7, 8, 9].
Evolving granular systems are aligned with issues and needs of Big data processing, fully autonomous systems, and eXplainable Artificial Intelligence. The incremental evolution of the structure of a granular model can be performed by supervised [10, 11], unsupervised [12, 13], semi-supervised [14, 15], and weakly (inaccurately) supervised [16] algorithms. As the algorithms operate on an instance-per-instance basis, they are generally very fast compared to machine learning algorithms in general. Nonetheless, time granulation – aiming at reducing the sampling rate of fast data streams and/or synchronizing concurrent data streams that are provided at random time intervals – has also been discussed [1]. A time granule describes the data stream for a certain time period. The result of dynamic time granulation is a unique granule per segment, which can further be used as an input for spatial computation. Of concern to the rest of this paper is state-space evolving fuzzy granular control only.
Fuzzy control, as an instance of granular control, is a well-founded framework to deal with complex nonlinear systems [17, 18, 19]. If the equations that describe a dynamical system are known, an exact fuzzy representation can be obtained, e.g., by the Sector Nonlinearity method [19, 20]. Thus, the accurateness on the assumptions toward obtaining nonlinear equations is the sole aspect that restricts the performance of a fuzzy model-based controller applied to a physical system to be similar to the performance required from the controller in computer simulations. Data-stream-driven fuzzy control comes into play when the system equations are time-varying, uncertain, and/or partially known or unknown [21, 22, 23]. In particular, time varying systems imply a clear drawback on the use of ofline-designed model-based and model-free control methods in general, since some initial assumptions and some parameters inherent to the system change over time [21].
Adaptive and evolving control, as well as model-free and model-based control, should be diferentiated. Evolving control is a step toward a higher level of autonomy compared to adaptive control and classical parameter identification methods [ 24, 25]. The evolving aspect accounts for unbounded amounts of data, changing concepts, and structural adaptation. Evolving control does not depend upon prior structural knowledge. In model-free control, the controller generally receives error-related variables (after comparing sensor data and reference values) and outputs control signals. In model-based control, besides accurate control signals, the accompanying model may carry meaningful information about the underlying process and, therefore, allow analysis of closed-loop system properties and the design of a controller oriented to attain certain performance requirements. The possibility of analyzing stability, performance, and robustness has given rise to a keen interest in model-based control [21, 26, 27].
This paper presents a state-space variety of evolving granular model, which we call SS-EGM, and an evolving granular control approach that we call SS-EGC. SS-EGC is designed based on the SS-EGM aiming at parallel distributed compensation in a sense that closed-loop Lyapunov stability and bounded control inputs are guaranteed [19, 21]. While stability is a fundamental concern in control design, bounded inputs may avoid actuator saturation. SS-EGM observes the dynamical system from a state data stream. The model provides one-step state estimates. Its structure gives support to control design. Model granules and their contents may change at any time step. In this case, the corresponding granules of the controller have their gains redesigned from the solution of a relaxed linear matrix inequality (LMI). Eficient LMI parser and solver, Yalmip’18 [28], and Mosek’20 [29], are utilized so that new gains for the changed control granules are secured quickly in online environment.
The efectiveness of the SS-EGM as a one-step predictor, and that of SS-EGC on regularizing a chaotic system, known as the Henon map [30], is demonstrated. Chaotic systems are nonlinear systems whose trajectories are sensitive to the initial conditions, parameter variations, and perturbation. The stabilization of unknown chaotic systems is challenging since slightly inaccurate parameter estimates may trigger undesired closed-loop behavior [31]. Although trajectories of chaotic systems are bounded, this is not the case when the system is influenced by control, that is, the solution may escape in finite time [ 32]. Broadly speaking, we have achieved stabilization of unknown, nonlinear and nonstationary systems using no a priori information and fully-autonomous online granular learning. We visualize applications on secure communication, such as in smart IoT and synchronization of cyber-physical systems; suppression of interference and artifacts in cardiac, electroencephalogram, and speech signals; and control of switched and time-varying systems in general, such as mobile robots and unmanned aerial vehicles, to mention some.
Consider a discrete, nonlinear and time-varying system,
x( + 1) = (x(), u(), )
y( + 1) = ℎ (x(), ),
in which x() ∈ , u() ∈ , and y() ∈ are the state, input and output vectors at time step
. The maps and ℎ describe the relations (
Since is unknown, the control law, , can only be designed based on ′ – a model of . An
evolving granular model ′ constructed from a stream of state data is considered. As is time
varying in general, the model ′ must be provided with mechanisms for learning from stream
data x() , = 1, ... . In other words, ′ must be evolved over time to track the behavior of . It
is expected that is stabilized by the control u() if ′ is a relatively accurate approximation of
, and is an appropriately designed control law.
(
A solution x∗() of (
lim (x() − x∗()) = 0.
→∞
(
A discrete state-space fuzzy variety of evolving granular model is described. SS-EGM learning emphasizes data coverage and memory of the past. The model aims to assist online LMI-based control design for parallel distributed compensation.
We give an online learning algorithm to build a granular model with state-space consequent from a stream of data generated by an unknown dynamical system. In general, the original system equations are unknown, nonlinear and time-varying. A finite number of past states x(), x( − 1), ..., x( − ) ; control inputs u(), u( − 1), ..., u( − ) ; and other external variables can be considered attributes. Nonetheless, we use descriptors of granules of the form
THEN x ( + 1) = x() + u() in which x() = [ 1() ... () ... Ψ()] , and u() = [ 1() ... () ... Φ()] . A variable amount of rules , = 1, ..., , describes smooth granules boundaries and the granules content. In evolving modeling, is a matrix of appropriate dimension with variable coeficients; and ℳ , = 1, ..., Ψ , are normal (height 1) trapezoidal membership functions defined by strictly increasing parameters, ℳ = ( , , Λ ,
), which are updated in light of changes of the system states reflected on the data stream. We assume = [1 0 ... 0 ] common to all rules, i.e., the control input is applied to 1, without loss of generality. Superscript on the left-hand side of the consequent equation refers to a local one-step estimate.
Afine terms are added to consequent matrices and state and control vectors as follows, 1
0 ̃ = [ 0 ], ̃= [ ], x̃= [ a 0 x ], ũ = [ u0 ], 1 in which a0 = [ 10 ... 0 ... Ψ0] ; and 0 is the controller ofset term, which is obtained straightforwardly from the design method (Section 3). Granule descriptors are rewritten as
THEN x ̃( + 1) = ̃ x̃() + ̃ũ() We omit ‘() ’ from the time-varying membership functions ℳ
The overall state estimate of the granular model is given by and system matrices .
(
x̃( + 1), =
∑ =1 , so that ≥ 0 and ∑
= 1. =1 Let the expansion region of ℳ
be obtain from a similarity measure as follows.
Rules activation degrees , = 1, ..., , can be determined by any T-norm aggregation [33], i.e., = ( 1, ..., Ψ), in which is the membership degree of () in ℳ . Nonetheless, we = [ − , + ], in which is the maximum width that ℳ
− ≤ , ∀ , , at any time step . hyperbox enclosing the support of ℳ , that is, supp(ℳ ) ⊆ E .
ℳ
Define a dynamical system. ℳ is obtained from the cylindrical extension of its components across the = [ℳ1 ... ℳΨ] as a Ψ-dimensional information granule within the state space of state space. E = [ 1 ... Ψ] is the expansion region of the granule ℳ . Geometrically, E is a
Given a numerical instance x at instant , the activation degree of the -th granule is = ( x, ℳ ) if x ∈ E – being (.) a similarity measure; otherwise = 0. We use , ∀ , , is allowed to expand to enclose a datum . ( x, ℳ ) = 1 − 1 6Ψ =1 Ψ ∑(| − | + 2| − | + 2| − Λ | + | − |).
The value of is equal to 1 – indicating maximum activation – if the trapezoids ( ,
∀ , are degenerated in singletons, and match x. The similarity reduces linearly as x withdraws
from ℳ
in any dimension. In particular, the summatory function in Eq. (
In online environment no granule exists a priori. Model granules are created and updated to cover the state space. The amount of granules, , increases by a unit if = 0, ∀ . In this case, +1 = 1, i.e., the new granule matches the instance responsible for its own creation. A granular structure and a granular input-output map evolve without human in the loop.
whose parameters match x() , An SS-EGM is generally built from scratch. The model retains new patterns if changes or new behaviors arise in the flowing data. We describe a learning method that deals with timevarying nonlinear dynamical systems and avoids time-consuming batch training – common to conventional ofline and some online window-based learning methods.
Expansion regions, E , (
A granule is created if one or more entries of x() do not belong to the expansion regions E of
ℳ , ∀ , = 1, ..., . The new granule ℳ+1 is built from trapezoidal fuzzy sets ℳ+1 , = 1, ..., Ψ ,
(
(
Subsequently, when x( + 1) is available, a supervised learning step is given by considering the input-output pair, (x(), x( + 1)), and the Recursive Least Squares method [21, 24]. Thus, ̃+1 = [ a+1 1 0
0 +1 ] and ̃+1 = [ +1 ], 0 with ofset coeficients initialized as 0 +1 = ( + 1) ()
, = 1, ..., Ψ. ∗, in which ∗ = =1,..., max (( x, ℳ )) +1 = [1 0 ... 0 ] is constant.
Updating a granule ℳ
∗]of its components. Among all granules ℳ that can be expanded to include an x() ,
that with the highest similarity, ℳ
∗ consists in enlarging the support [ ∗,
∗] and updating the core
is chosen. (.) is obtained from (
If () ∈ [ ∗ − , ∗] then ∗( + 1) = ()
If () ∈ [ ∗ , ∗ + ] then ∗( + 1) = ()
Otherwise, ∗( + 1) = ∗() and ∗( + 1) = ∗()
∗( + 1) = Λ ∗( + 1) = ∗( + 1) + ∗( + 1) 2 .
The Recursive Least Squares method updates ̃ ∗ [21, 24].
The learning method creates a new granule ℳ+1 or adapts the parameters of ℳ ∗, accordingly.
From the granular state-space model, SS-EGM, we formulate conditions for closed-loop
Lyapunov stability and bounded control inputs as an LMI feasibility problem. The gain matrices of
the granular controller, SS-EGC, are derived automatically from the solution of a relaxed LMI
solved only for active granules in a time step. Some SS-EGC local gains are quickly redesigned
when their corresponding SS-EGM granules change.
(
Having model and controller rules, (
ũ ( + 1), x̃( + 1) = ∑ ∑ x̃(),
We consider parallel distributed compensation, i.e., evolving granular model and controller share the same attributes. A control rule is given by
THEN ũ ( + 1) = x̃() in which x̃() = [1 1() ... Ψ()] and ũ() = [ 0() 1() ... Φ()] . ∈ ℜ(Φ+1×Ψ) is a gain matrix – with ofset terms in the first column – to be determined in order to make the closed loop system asymptotically stable and/or to drive the states faster and smoother to a reference. Superscript on the left-hand side of the consequent state feedback law means a local control input. The overall control signal, efectively sent to actuators, is (17) (18) (19) (20) =1 =1 =1 =1 < ( x) = ∑ x x, in which > 0, ∀ (positive definiteness). A stabilization result for the closed-loop system (18) based on (20) is as follows [37].
Result: The system (18) is asymptotically stable if there exist positive definite matrices = ( )−1 and matrices and , = 1, ..., , such that in which ∶= ̃ + ̃ , or, equivalently, =1 x̃( + 1) = ∑( )2 x̃() + 2 ∑ ( + 2 ) x̃().
If the unforced system is stable, ∀ may be used to improve the transient response. Unstable systems require suitable ’s for stabilization primarily. An issue in online environment is that granules ℳ , system matrices , and the number of rules , are time-varying. Local ’s should be reviewed after changes of the corresponding model granules.
A fuzzy Lyapunov function is a fuzzy combination of quadratic functions of the system states, = ( )−1, = 1, ..., .
The number of granules afects the complexity of the LMI analysis. Finding a Lyapunov function for a large ‘ ’ may be dificult [ 19, 21, 26]. In view of the features of the SS-EGM learning method, namely, inactive granules do not change, then, recalculation of gains (22) is needed only for the active granules at a time step. Therefore, the number of rows and columns of (21) can be greatly reduced by considering active granules only.
Definition : The number of active granules, , for an instance x() is equal to the number of terms that make the rescaled activation degree (x()) > 0, = 1, ..., .
Theorem: The closed-loop system (18) is asymptotically stable if there are positive definite matrices = ( )−1, and matrices ; , = 1, ..., , such that [
− + ( ) +( ) ( ) − are satisfied for all combinations of , = 1, ..., – being the number of active granules for an instance x() .
If a feasible solution to (23) is found, then the control gains assigned to the active control granules are redefined as [ − ( ) − + ( ) ( ) +( ) ( ) − holds true for all combinations of , , = 1, ..., . See [37] for a proof. If a feasible solution for (21) is found, the controller gains are
= , , = 1, ..., .
The gains related to inactive granules are kept as computed in previous time steps.
The feasibility problem (23) is dynamic and convex. Finding a solution means that (20) is Lyapunov for = 1, ..., ; and (18) is stable using (24). The proof of the Theorem follows analogously to that in the appendix of [21]. Eficient LMI parser and solver, Yalmip’18 [ 28], and Mosek’20 [29], are available. (21) (22) (23) (24)
Being the state x() known, the following result applies for a bounded control input [19, 21].
Result: Given positive definite matrices , and matrices = , = 1, ..., , as in (24). The constraint ||u( + 1)|| 2 ≤ is enforced if [ x(1) x()
] > 0 and [ ( ) 2 hold true for , = 1, ..., . See [19] for a proof. We replaced the initial state x(0)in (25) by the current state x() for online design. Parameter bounds the maximum input, thus keeping it within the operation range of actuators. The equations (25) are appended to (23) for a stable granular controller satisfying input constraints.
The efectiveness of the SS-EGM and associated SS-EGC is assessed through a deterministic chaotic system, the Henon map [30]. We use the Henon equations to generate a data stream. State-space granular modeling and control are performed on the fly.
The nonlinear equations of the Henon map are: 1( + 1) = 2() + 1 − 1() 1() 2( + 1) = 1(). (25) (26) (27) The phase portrait for = 1.4 , = 0.3 , and initial state x(0) = [1 0 ]is shown in Figure 2. In this case, [0.6314 0.1894]and [−1.1314 − 0.3394]are the fixed points of the map. In general, if the orbit or trajectory spreads over the phase plane, we have a stochastic process. As we see a deterministic orbit, then, we have chaos. In fact, the Henon map is a model of the Poincaré section of the Lorenz system. Notice that the orbit of the unforced system settles into an irregular oscillation – confined in a fractal set – which never repeats exactly.
Aiming at stabilizing (26), i.e., driving the chaotic orbit x() to the origin x∗ = [0 0 ], lim→∞ (x() − x∗) = 0, we add a control input, () , to the first relation. Thus, 1( + 1) = 2() + 1 − 1.4 1() 1() + () 2( + 1) = 0.3 1().
The online model-based granular controller is designed to opportunely inject values () to lead the states x() to the origin x∗. The Henon system as in (27) is completely controllable and observable. The states are considered measurable.
Consider the controller, SS-EGC, of for time steps. Then, () = 0, = 1, ..., . We want to assess the accuracy of the model, SS-EGM, in one-step prediction. The model is built from scratch, with no knowledge about the dynamic system that produces the data stream.
Let the root mean square error be 1 ∑ =1 √=1
RMSE =
∑ ( ( + 1) − ̃( + 1) ) ,
2
(28)
in which x( + 1) is the actual value given by (27); and x̃( + 1) , given by (
Granularity ( )
Final Structure (# granules) RMSE 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 60 32 20 16 10 9 7
We notice from Table 1 that the number of resulting granules and, therefore, the number of SS-EGM parameters, reduces with the increasing of , which facilitates further control design. However, the SS-EGM accuracy reduces, and, in this case, the controller is designed for a worse picture of the original system, which may lead worse-than-expected transient responses and instability. A trade-of between accuracy and model compactness is observed.
Figure 3 shows how the state space is granulated for = {0.1, 0.2, 0.3, 0.5} , yielding more detailed or coarser fuzzy partitions. Notice in the last of the graphs how wider supports of fuzzy granules (larger boxes) start to embrace data on both upper and lower parts of the Henon moon so that locally-valid estimates given by linear diference equations start to get confused with respect to a positive or negative slope. As a consequence, some red dots (one-step prediction) are noticed a bit farther from the black dots (actual values), specially in the middle, more curvy, part of the moon. Figure 4 shows the SS-EGM structural evolution. The smaller the support of the fuzzy granules, the more accurate the local model, but the more complex the LMI to be solved for further controller adaptation.
As an example, the parameters of an SS-EGM descriptor, using = 0.3 at = 500 , are: Granule 1: ℳ11 = (0.9382, 1.1437, 1.1437, 1.3491) ℳ21 = (−0.1194, 0.0046, 0.0046, 0.1285) Vector 1 = [0 1 0 ], which is common to all descriptors. (29)
The granular controller, SS-EGC, is designed based on the model SS-EGM. Nonetheless, the output of the SS-EGC is applied to the original Henon system (27). Thus, x is the actual system state, i.e., we want to control a completely unknown chaotic system on the fly.
Define the settling time, , as the number of time steps from the application of the control input, , until all states, x, enter and remain within a 2% band around the origin, x∗. The range of values of the system (26) are 1 ∈ [−1.2836, 1.2727]and 2 ∈ [−0.3851, 0.3818 ]. Thus, 1() ≤ 1(2%) = 0.0256, and 2() ≤ 2(2%) = 0.0077, ∀ , after ≠ 0 , determines . x = ⟨x(), x()⟩ = ∑
∑ | ()| 2. =1 ≠0 (30) Particularly, = 2 is the number of states; and x is a measure of the energy of the closed-loop system during the stabilization transient. We turned the controller on at = 500 , and of at = 600 . The lower the values of and x, the better – as this implies that the chaotic behavior was dissipated faster. Table 2 summarizes the SS-EGC closed-loop results for diferent granularities, , and upper bounds on the control input, .
We notice from Table 2 that an intermediate granularity, = 0.3 , and a wider operation range for inputs, = 1.5 , provide the best closed-loop performance. As expected, larger values of cause higher modeling error, such that the controller is designed based on a less accurate model. Therefore, in cases such as for the granularities 0.5 and 0.6, despite converging evidence, the states still spiral on a band larger than 2% around x∗ after the 100 time steps with ≠ 0 . To illustrate the regularization of the chaotic behavior, Figure 5 shows the states convergence when is enabled at = 500 . Figure 6 highlights the asymptotic convergence to the origin (0, 0)from the phase plane perspective. Notice that the Henon states backtrack to their chaotic orbit when the control is turned of at = 600 .
We described an evolving state-space variation of granular method to model and control unknown nonlinear dynamical systems. The model SS-EGM and the controller SS-EGC have their structures developed from scratch and their parameters updated over time. The online learning method is particularly devoted to cover the data space and keep memory of past patterns. Control gains are derived from a relaxed LMI feasibility problem and guarantee closed-loop Lyapunov stability and bounded control inputs from parallel distributed compensation. Bounded inputs avoid actuator saturation.
We have shown asymptotic stabilization of the Henon chaos as an example. The results remark a relatively low RMSE in one-step SS-EGM prediction, 0.0359; and a settling time of 17 steps using the original Henon equations in the loop and an intermediate granularity, 0.3. The Henon equations are assumed unknown for any modeling and control design purpose. SS-EGM notices the dynamic system by means of the data stream only. In the future we envision a diversity of applications, such as on secure communication, artifact suppression in physiological signals, and control of nonlinear time-varying systems in general, such as mobile robots and unmanned aerial vehicles, to mention some. We should also address trajectory following control and LMI-optimal design issues. A higher level of autonomy, structural adaptability, human-level interpretability, and the possibility to deal with heterogeneous data streams have been achieved by means of evolving granular computing. Uncertain data modeling based on evolving ellipsoidal fuzzy information granules, IEEE Transactions on Fuzzy Systems 28 (2020) 2427–2436. [11] D. Leite, R. Ballini, P. Costa, F. Gomide, Evolving fuzzy granular modeling from nonstationary fuzzy data streams, Evolving Systems 3 (2012) 65–79. [12] C. Aguiar, D. Leite, Unsupervised fuzzy eix: Evolving internal-external fuzzy clustering, in:
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