=Paper=
{{Paper
|id=Vol-3380/paper2
|storemode=property
|title=State-Space Evolving Granular Control of Unknown Dynamic Systems
|pdfUrl=https://ceur-ws.org/Vol-3380/paper2.pdf
|volume=Vol-3380
|authors=Daniel Leite
|dblpUrl=https://dblp.org/rec/conf/wcci/Leite22
}}
==State-Space Evolving Granular Control of Unknown Dynamic Systems==
https://ceur-ws.org/Vol-3380/paper2.pdf
State-Space Evolving Granular Control
of Unknown Dynamic Systems
Daniel Leite1
1
Adolfo Ibanez University (UAI), 2640 Diag. Las Torres, Santiago, Chile
Abstract
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
first-order difference 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 difference 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.
Keywords
Evolving Control, Granular Computing, Dynamic Systems, Chaos
1. Introduction
1.1. Evolving Granular Systems
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 – offered 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
OLUD 2022: First Workshop on Online Learning from Uncertain Data Streams, July 18, 2022, Padua, Italy.
Envelope-Open daniel.furtado@uai.cl (D. Leite)
GLOBE https://sites.google.com/site/danfl7 (D. Leite)
Orcid 0000-0003-3135-2933 (D. Leite)
© 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073
CEUR Workshop Proceedings (CEUR-WS.org)
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 different 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 offline-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
differentiated. 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). Efficient 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 effectiveness 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 non-
linear 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 commu-
nication, 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.
1.2. Problem definition
Consider a discrete, nonlinear and time-varying system,
x(𝑘 + 1) = 𝑓 (x(𝑘), u(𝑘), 𝑘)
y(𝑘 + 1) = ℎ (x(𝑘), 𝑘) , (1)
in which x(𝑘) ∈ 𝑅𝑛 , u(𝑘) ∈ 𝑅𝑚 , and y(𝑘) ∈ 𝑅𝑝 are the state, input and output vectors at time step
𝑘. The maps 𝑓 and ℎ describe the relations (1). Consider all states accessible. Then, y(𝑘) = x(𝑘),
and the output equation is omitted. The goal is to design a state feedback control law
u(𝑘) = 𝑔 (x(𝑘), 𝑘) , (2)
such that a fixed point of the unknown map 𝑓, in particular the origin, is stabilized.
Since 𝑓 is unknown, the control law, 𝑔, can only be designed based on 𝑓 ′ – a model of 𝑓. An
evolving granular model 𝑓 ′ constructed from a stream of state data is considered. As 𝑓 is time
varying in general, the model 𝑓 ′ must be provided with mechanisms for learning from stream
data x(𝑘), 𝑘 = 1, .... In other words, 𝑓 ′ must be evolved over time to track the behavior of 𝑓. It
is expected that 𝑓 is stabilized by the control u(𝑘) if 𝑓 ′ is a relatively accurate approximation of
𝑓, and 𝑔 is an appropriately designed control law.
A solution x∗ (𝑘) of (1), with initial condition x∗ (0), is called chaotic if it is Lyapunov unsta-
ble and all the solutions starting from some neighborhood of x∗ (0) are bounded on (−∞, ∞).
Stabilizing x∗ (𝑘) means to drive the states x(𝑘) to x∗ (𝑘), i.e.,
lim (x(𝑘) − x∗ (𝑘)) = 0. (3)
𝑘→∞
Figure 1 shows the closed-loop system with the state-space evolving granular model, SS-EGM,
playing the role of 𝑓 ′ , and its associated online learning algorithm; and the evolving granular
controller, SS-EGC, playing the role of 𝑔, and its associated online LMI-based design method. In
̃ + 1) is a one-step estimate of the states; and 𝑧 −1 is a unit delay. The unknown
the figure, x(𝑘
nonlinear time-varying system, 𝑓, is a general term that extends to phenomena, processes,
machines, robots, and a variety of virtual and real-world systems that can be controlled. We
access, learn, and change the dynamics of 𝑓 from a stream of state data.
Figure 1: State-space evolving granular control system
2. State-Space Evolving Granular Model
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.
2.1. Basic framework
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(𝑘), x(𝑘 − 1), ..., x(𝑘 − 𝑑); control inputs u(𝑘), u(𝑘 − 1), ..., u(𝑘 − 𝑣); and other external variables
can be considered attributes. Nonetheless, we use descriptors of granules of the form
𝑅𝑖 : IF 𝑥1 (𝑘) is ℳ1𝑖 AND ... AND 𝑥Ψ (𝑘) is ℳΨ𝑖
THEN x𝑖 (𝑘 + 1) = 𝐴𝑖 x(𝑘) + 𝐵𝑖 u(𝑘)
in which x(𝑘) = [𝑥1 (𝑘) ... 𝑥𝜓 (𝑘) ... 𝑥Ψ (𝑘)]𝑇 , and u(𝑘) = [𝑢1 (𝑘) ... 𝑢𝜙 (𝑘) ... 𝑢Φ (𝑘)]𝑇 . A variable
amount of rules 𝑅𝑖 , 𝑖 = 1, ..., 𝑐, describes smooth granules boundaries and the granules content.
In evolving modeling, 𝐴𝑖 is a matrix of appropriate dimension with variable coefficients; and
ℳ𝜓𝑖 , 𝜓 = 1, ..., Ψ, are normal (height 1) trapezoidal membership functions defined by strictly
increasing parameters, ℳ𝜓𝑖 = (𝑙𝜓𝑖 , 𝜆𝜓𝑖 , Λ𝑖𝜓 , 𝐿𝑖𝜓 ), which are updated in light of changes of the system
states reflected on the data stream. We assume 𝐵𝑖 = [1 0 ... 0]𝑇 common to all rules, i.e., the
control input is applied to 𝑥1 , without loss of generality. Superscript 𝑖 on the left-hand side of
the consequent equation refers to a local one-step estimate.
Affine terms are added to consequent matrices and state and control vectors as follows,
1 0 0 1 𝑢
𝐴̃𝑖 = [ 𝑖 𝑖 ̃ =[ 0 ],
] , 𝐵̃ 𝑖 = [ 𝑖 ] , x̃ = [ ] , u (4)
a0 𝐴 𝐵 x u
in which a𝑖0 = [𝑎10
𝑖 ... 𝑎 𝑖 ... 𝑎 𝑖 ]𝑇 ; and 𝑢 is the controller offset term, which is obtained
𝜓0 Ψ0 0
straightforwardly from the design method (Section 3). Granule descriptors are rewritten as
𝑅𝑖 : IF 𝑥1 (𝑘) is ℳ1𝑖 AND ... AND 𝑥Ψ (𝑘) is ℳΨ𝑖
THEN x̃ 𝑖 (𝑘 + 1) = 𝐴̃𝑖 x(𝑘)
̃ + 𝐵̃ 𝑖 u
̃(𝑘)
We omit ‘(𝑘)’ from the time-varying membership functions ℳ𝜓𝑖 and system matrices 𝐴𝑖 .
The overall state estimate of the granular model is given by
𝑐
̃ + 1) = ∑ 𝜇 𝑖𝑟 x̃ 𝑖 (𝑘 + 1),
x(𝑘 (5)
𝑖=1
in which 𝜇 𝑖𝑟 is the rescaled activation degree of the 𝑖-th rule,
𝑐
𝜇𝑖
𝜇 𝑖𝑟 = 𝑐 , so that 𝜇 𝑖𝑟 ≥ 0 and ∑ 𝜇 𝑖𝑟 = 1. (6)
∑ 𝜇𝑖 𝑖=1
𝑖=1
Rules activation degrees 𝜇 𝑖 , 𝑖 = 1, ..., 𝑐, can be determined by any T-norm aggregation [33],
i.e., 𝜇 𝑖 = 𝑇 (𝜇1𝑖 , ..., 𝜇Ψ𝑖 ), in which 𝜇𝜓𝑖 is the membership degree of 𝑥𝜓 (𝑘) in ℳ𝜓𝑖 . Nonetheless, we
obtain 𝜇 𝑖 from a similarity measure as follows.
Let the expansion region of ℳ𝜓𝑖 be
𝐸𝜓𝑖 = [𝐿𝑖𝜓 − 𝜌, 𝑙𝜓𝑖 + 𝜌], (7)
in which 𝜌 is the maximum width that ℳ𝜓𝑖 , ∀𝜓 , 𝑖, is allowed to expand to enclose a datum 𝑥𝜓 .
In other words, 𝐿𝑖𝜓 − 𝑙𝜓𝑖 ≤ 𝜌, ∀𝜓 , 𝑖, at any time step 𝑘.
Define ℳ 𝑖 = [ℳ1𝑖 ... ℳΨ𝑖 ]𝑇 as a Ψ-dimensional information granule within the state space of
a dynamical system. ℳ 𝑖 is obtained from the cylindrical extension of its components across the
state space. E𝑖 = [𝐸1𝑖 ... 𝐸Ψ𝑖 ]𝑇 is the expansion region of the granule ℳ 𝑖 . Geometrically, E𝑖 is a
hyperbox enclosing the support of ℳ 𝑖 , that is, supp(ℳ 𝑖 ) ⊆ E𝑖 .
Given a numerical instance x at instant 𝑘, the activation degree of the 𝑖-th granule is 𝜇 𝑖 =
𝑆(x, ℳ 𝑖 ) if x ∈ E𝑖 – being 𝑆(.) a similarity measure; otherwise 𝜇 𝑖 = 0. We use
Ψ
1
𝑆(x, ℳ 𝑖 ) = 1 − ∑ (|𝑥 − 𝑙 𝑖 | + 2|𝑥𝜓 − 𝜆𝜓𝑖 | + 2|𝑥𝜓 − Λ𝑖𝜓 | + |𝑥𝜓 − 𝐿𝑖𝜓 |). (8)
6Ψ 𝜓 =1 𝜓 𝜓
The value of 𝑆 is equal to 1 – indicating maximum activation – if the trapezoids (𝑙𝜓𝑖 , 𝜆𝜓𝑖 , Λ𝑖𝜓 , 𝐿𝑖𝜓 ),
∀𝜓, are degenerated in singletons, and match x. The similarity reduces linearly as x withdraws
from ℳ 𝑖 in any dimension. In particular, the summatory function in Eq. (8) determines a
Hamming-like distance [11, 34].
In online environment no granule exists a priori. Model granules are created and updated to
cover the state space. The amount of granules, 𝑐, increases by a unit if 𝜇 𝑖 = 0, ∀𝑖. In this case,
𝜇 𝑐+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.
2.2. Online learning
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 time-
varying nonlinear dynamical systems and avoids time-consuming batch training – common to
conventional offline and some online window-based learning methods.
Expansion regions, E𝑖 , (7), are essential to the decision on whether or not a new instance
belongs to a granule ℳ 𝑖 . Different values of the hyperparameter 𝜌 ∈ (0, 1) produce different
granular representations of the same dynamical system. The lower the value of 𝜌, the larger the
amount of descriptors, and the greater the details the model can seize. However, if 𝜌 approaches
0, a granule is created for each non-coincident instance, and the model becomes excessively
complex. In contrast, if 𝜌 is equal to 1, a single granule covers all instances in the unit hypercube
(rescaled data), which is insufficient to nonlinear modeling and control.
A granule is created if one or more entries of x(𝑘) do not belong to the expansion regions E𝑖 of
ℳ 𝑖 , ∀𝑖, 𝑖 = 1, ..., 𝑐. The new granule ℳ 𝑐+1 is built from trapezoidal fuzzy sets ℳ𝜓𝑐+1 , 𝜓 = 1, ..., Ψ,
whose parameters match x(𝑘),
ℳ𝜓𝑐+1 = (𝑙𝜓 , 𝜆𝜓 , Λ𝜓 , 𝐿𝜓 )𝑐+1 = (𝑥𝜓 , 𝑥𝜓 , 𝑥𝜓 , 𝑥𝜓 ). (9)
This is a bottom-up procedure since a granule starts as a point, and tends to enlarge shortly
afterwards. The opposite – top-down procedure – is also possible, that is, a granule can be
initialized as large as possible based on 𝜌, and shrink gradually.
Subsequently, when x(𝑘 + 1) is available, a supervised learning step is given by considering
the input-output pair, (x(𝑘), x(𝑘 + 1)), and the Recursive Least Squares method [21, 24]. Thus,
1 0 0
𝐴̃𝑐+1 = [ ] and 𝐵 ̃ 𝑐+1 = [ ], (10)
a𝑐+1
0 𝐴𝑐+1 𝐵𝑐+1
with offset coefficients initialized as
𝑥𝜓 (𝑘 + 1)
𝑎𝜓𝑐+1
0 = , 𝜓 = 1, ..., Ψ. (11)
𝑥𝜓 (𝑘)
𝐵𝑐+1 = [1 0 ... 0]𝑇 is constant.
∗ ∗ ∗
Updating a granule ℳ 𝑖 consists in enlarging the support [𝑙𝜓𝑖 , 𝐿𝑖𝜓 ] and updating the core
∗ ∗
[𝜆𝜓𝑖 , Λ𝑖𝜓 ] of its components. Among all granules ℳ 𝑖 that can be expanded to include an x(𝑘),
∗
that with the highest similarity, ℳ 𝑖 , in which
𝑖∗ = 𝑎𝑟𝑔 max (𝑆(x, ℳ 𝑖 )) (12)
𝑖=1,...,𝑐
is chosen. 𝑆(.) is obtained from (8). Granular coverage of the space of states and model memory
to support control design are emphasized by: (i) avoiding typical granule deleting and merging
procedures [35, 36]; (ii) keeping the model granularity 𝜌 constant; and (iii) considering the
following updating relations, which keep granules centers static:
∗ ∗ ∗
If 𝑥𝜓 (𝑘) ∈ [𝐿𝑖𝜓 − 𝜌, 𝑙𝜓𝑖 ] then 𝑙𝜓𝑖 (𝑘 + 1) = 𝑥𝜓 (𝑘) (13)
∗ ∗ ∗
If 𝑥𝜓 (𝑘) ∈ [𝐿𝑖𝜓 , 𝑙𝜓𝑖 + 𝜌] then 𝐿𝑖𝜓 (𝑘 + 1) = 𝑥𝜓 (𝑘) (14)
∗ ∗ ∗ ∗
Otherwise, 𝑙𝜓𝑖 (𝑘 + 1) = 𝑙𝜓𝑖 (𝑘) and 𝐿𝑖𝜓 (𝑘 + 1) = 𝐿𝑖𝜓 (𝑘) (15)
for 𝜓 = 1, ..., Ψ. Core parameters are updated from
∗ ∗
∗ ∗
𝑙𝜓𝑖 (𝑘 + 1) + 𝐿𝑖𝜓 (𝑘 + 1)
𝑖 𝑖
𝜆𝜓 (𝑘 + 1) = Λ𝜓 (𝑘 + 1) = . (16)
2
∗
The learning method creates a new granule ℳ 𝑐+1 or adapts the parameters of ℳ 𝑖 , accordingly.
∗
The Recursive Least Squares method updates 𝐴̃𝑖 [21, 24].
3. Control Design
From the granular state-space model, SS-EGM, we formulate conditions for closed-loop Lya-
punov 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.
3.1. Closed-loop system
We consider parallel distributed compensation, i.e., evolving granular model and controller
share the same attributes. A control rule is given by
𝑅𝑖 : IF 𝑥1 (𝑘) is ℳ1𝑖 AND ... AND 𝑥Ψ (𝑘) is ℳΨ𝑖
THEN u ̃𝑖 (𝑘 + 1) = 𝐾 𝑖 x(𝑘)
̃
in which x(𝑘)
̃ = [1 𝑥1 (𝑘) ... 𝑥Ψ (𝑘)]𝑇 and u
̃(𝑘) = [𝑢0 (𝑘) 𝑢1 (𝑘) ... 𝑢Φ (𝑘)]𝑇 . 𝐾 𝑖 ∈ ℜ(Φ+1×Ψ) is a gain
matrix – with offset 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 𝑖 on the left-hand side of the consequent state feedback law means a local control
input. The overall control signal, effectively sent to actuators, is
𝑐
u ̃𝑖 (𝑘 + 1),
̃(𝑘 + 1) = ∑ 𝜇 𝑖𝑟 u (17)
𝑖=1
in which 𝜇 𝑖𝑟 is the rescaled activation degree (6).
Having model and controller 𝑐 rules, (5) and (17) combined yield the closed-loop system
𝑐 𝑐
̃ + 1) = ∑ ∑ 𝜇 𝑖𝑟 𝜇 𝑗𝑟 𝐺 𝑖𝑗 x(𝑘),
x(𝑘 ̃ (18)
𝑖=1 𝑗=1
in which 𝐺 𝑖𝑗 ∶= 𝐴̃𝑖 + 𝐵̃ 𝑖 𝐾 𝑗 , or, equivalently,
𝑐 𝑐
𝑖𝑟 2 𝑖𝑖 𝐺 𝑖𝑗 + 𝐺 𝑗𝑖
̃ + 1) = ∑(𝜇 ) 𝐺 x(𝑘)
x(𝑘 ̃ + 2 ∑ 𝜇 𝑖𝑟 𝜇 𝑗𝑟 ( ) x(𝑘).
̃ (19)
𝑖=1 𝑖<𝑗 2
If the unforced system is stable, 𝐾 𝑖 ∀𝑖 may be used to improve the transient response. Unstable
systems require suitable 𝐾 𝑖 ’s for stabilization primarily. An issue in online environment is that
granules ℳ 𝑖 , system matrices 𝐴𝑖 , and the number of rules 𝑐, are time-varying. Local 𝐾 𝑖 ’s should
be reviewed after changes of the corresponding model granules.
3.2. Lyapunov stability
A fuzzy Lyapunov function is a fuzzy combination of quadratic functions of the system states,
𝑐
𝑉 (x) = ∑ 𝜇 𝑖𝑟 x𝑇 𝑃 𝑖 x, (20)
𝑖=1
in which 𝑃 𝑖 > 0, ∀𝑖 (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
𝑋 𝑖 = (𝑃 𝑖 )−1 and matrices 𝑄 𝑖 and 𝑍 𝑖 , 𝑖 = 1, ..., 𝑐, such that
𝑋 𝑖 − (𝑍 𝑗 )𝑇 − 𝑍 𝑗 (𝑍 𝑗 )𝑇(𝐴𝑖 )𝑇+(𝑄 𝑗 )𝑇(𝐵𝑖 )𝑇
[ ]<0 (21)
𝐴𝑖 𝑍 𝑗 + 𝐵𝑖 𝑄 𝑗 − 𝑋𝑘
holds true for all combinations of 𝑖, 𝑗, 𝑘 = 1, ..., 𝑐. See [37] for a proof. If a feasible solution for
(21) is found, the controller gains are
𝐾 𝑗 = 𝑄 𝑗 (𝑍 𝑗 )−1 , 𝑗 = 1, ..., 𝑐. (22)
The number of granules affects the complexity of the LMI analysis. Finding a Lyapunov
function for a large ‘𝑐’ may be difficult [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, 𝔵, for an instance x(𝑘) is equal to the number of
terms that make the rescaled activation degree 𝜇 𝑖𝑟 (x(𝑘)) > 0, 𝑖 = 1, ..., 𝑐.
Additionally, we reduce the number of concatenated rows in (21) by making matrices 𝑍 𝑗 ∀𝑗
and 𝑋 𝑘 ∀𝑘 equal to 𝑋 𝑖 ∀𝑖, with 𝑋 𝑖 symmetric, to obtain a theorem.
Theorem: The closed-loop system (18) is asymptotically stable if there are positive definite
matrices 𝑋 𝑖 = (𝑃 𝑖 )−1 , and matrices 𝑄 𝑗 ; 𝑖, 𝑗 = 1, ..., 𝔵, such that
−𝑋 𝑖 𝑋 𝑖 (𝐴𝑖 )𝑇 +(𝑄 𝑗 )𝑇 (𝐵𝑖 )𝑇
[ ]<0 (23)
𝐴𝑖 𝑋 𝑖 + 𝐵𝑖 𝑄 𝑗 −𝑋 𝑖
are satisfied for all combinations of 𝑖, 𝑗 = 1, ..., 𝔵 – being 𝔵 the number of active granules for an
instance x(𝑘).
If a feasible solution to (23) is found, then the control gains assigned to the active control
granules are redefined as
𝐾 𝑖 = 𝑄 𝑗 𝑃 𝑖 , 𝑖, 𝑗 = 1, ..., 𝔵. (24)
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 𝑖 = 1, ..., 𝔵; and (18) is stable using 𝐾 𝑖 (24). The proof of the Theorem follows
analogously to that in the appendix of [21]. Efficient LMI parser and solver, Yalmip’18 [28], and
Mosek’20 [29], are available.
3.3. Bounded input
Being the state x(𝑘) known, the following result applies for a bounded control input [19, 21].
Result: Given positive definite matrices 𝑋 𝑖 , and matrices 𝑄 𝑗 = 𝐾 𝑖 𝑋 𝑖 , 𝑖 = 1, ..., 𝔵, as in (24). The
constraint ||u(𝑘 + 1)||2 ≤ 𝜁 is enforced if
1 x(𝑘)𝑇 𝑋 𝑖 (𝑄 𝑗 )𝑇
[ ] > 0 and [ ]>0 (25)
x(𝑘) 𝑋 𝑖 𝑄 𝑗 𝜁 2𝐼
hold true for 𝑖, 𝑗 = 1, ..., 𝔵. See [19] for a proof. We replaced the initial state x(0) in (25) by the
current state x(𝑘) for online design. Parameter 𝜁 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.
4. Control of Chaotic Map: Result and Discussion
The effectiveness 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.
4.1. Henon map
The nonlinear equations of the Henon map are:
𝑥1 (𝑘 + 1) = 𝑥2 (𝑘) + 1 − 𝛼𝑥1 (𝑘)𝑥1 (𝑘)
𝑥2 (𝑘 + 1) = 𝛽𝑥1 (𝑘). (26)
The phase portrait for 𝛼 = 1.4, 𝛽 = 0.3, and initial state x(0) = [1 0]𝑇 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(𝑘) to the origin x∗ = [0 0],
lim𝑘→∞ (x(𝑘) − x∗ ) = 0, we add a control input, 𝑢(𝑘), to the first relation. Thus,
𝑥1 (𝑘 + 1) = 𝑥2 (𝑘) + 1 − 1.4𝑥1 (𝑘)𝑥1 (𝑘) + 𝑢(𝑘)
𝑥2 (𝑘 + 1) = 0.3𝑥1 (𝑘). (27)
The online model-based granular controller is designed to opportunely inject values 𝑢(𝑘) to
lead the states x(𝑘) to the origin x∗ . The Henon system as in (27) is completely controllable
and observable. The states are considered measurable.
Figure 2: Phase plane of the Henon chaotic map (26)
4.2. Online modeling and one-step estimation
Consider the controller, SS-EGC, off for 𝑘𝑓 time steps. Then, 𝑢(𝑘) = 0, 𝑘 = 1, ..., 𝑘𝑓 . 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
𝑘𝑓 𝑛
1 2
RMSE = ∑ ∑ (𝑥𝑗 (𝑘 + 1) − 𝑥̃𝑗 (𝑘 + 1)) , (28)
𝑘𝑓 𝑘=1
√ 𝑗=1
in which x(𝑘 + 1) is the actual value given by (27); and x(𝑘
̃ + 1), given by (5), is the SS-EGM
estimate. Table 1 shows the one-step prediction results for 𝑘𝑓 = 500, and different maximum
widths, 𝜌, allowed for the model granules.
Table 1
One-step SS-EGM prediction based on different model granularities
Granularity (𝜌) Final Structure (# granules) RMSE
0.1 60 0.0305
0.2 32 0.0317
0.3 20 0.0359
0.4 16 0.0442
0.5 10 0.0445
0.6 9 0.0478
0.7 7 0.0487
0.8 5 0.0573
We notice from Table 1 that the number of resulting granules and, therefore, the number of
SS-EGM parameters, reduces with the increasing of 𝜌, 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-off between accuracy and model compactness is observed.
Figure 3 shows how the state space is granulated for 𝜌 = {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 difference 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.
Figure 3: SS-EGM partitions of the Henon map using 60, 32, 20, and 10 granules
As an example, the parameters of an SS-EGM descriptor, using 𝜌 = 0.3 at 𝑘 = 500, are:
Granule 1:
ℳ11 = (0.9382, 1.1437, 1.1437, 1.3491)
ℳ21 = (−0.1194, 0.0046, 0.0046, 0.1285)
1 0 0
𝐴1 = [ −0.3484 −0.6542 0.7188 ] (29)
16.0058 0.4261 0.0800
Vector 𝐵1 = [0 1 0]𝑇 , which is common to all descriptors.
Figure 4: SS-EGM structural evolution for different model granularities 𝜌
4.3. Closed-loop stabilization
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, 𝑡𝑠 , as the number of time steps from the application of the control
input, 𝑢, until all states, x, enter and remain within a 2% band around the origin, x∗ . The
range of values of the system (26) are 𝑥1 ∈ [−1.2836, 1.2727] and 𝑥2 ∈ [−0.3851, 0.3818]. Thus,
𝑥1 (𝑘) ≤ 𝑥1(2%) = 0.0256, and 𝑥2 (𝑘) ≤ 𝑥2(2%) = 0.0077, ∀𝑘, after 𝑢 ≠ 0, determines 𝑡𝑠 .
Let the energy 𝐸x of the states x, for 𝑢 ≠ 0, be
𝑛
𝐸x = ⟨x(𝑘), x(𝑘)⟩ = ∑ ∑ |𝑥𝑗 (𝑘)|2 . (30)
𝑗=1 𝑘𝑢≠0
Particularly, 𝑛 = 2 is the number of states; and 𝐸x is a measure of the energy of the closed-loop
system during the stabilization transient. We turned the controller on at 𝑘 = 500, and off
at 𝑘 = 600. The lower the values of 𝑡𝑠 and 𝐸x , the better – as this implies that the chaotic
behavior was dissipated faster. Table 2 summarizes the SS-EGC closed-loop results for different
granularities, 𝜌, and upper bounds on the control input, 𝜁.
We notice from Table 2 that an intermediate granularity, 𝜌 = 0.3, and a wider operation range
for inputs, 𝜁 = 1.5, provide the best closed-loop performance. As expected, larger values of 𝜌
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
Table 2
SS-EGC results on the stabilization of the Henon chaos
Granularity 𝜌 Upper bound 𝜁 Steps for settling [𝑘] Energy 𝐸x
0.1 1.5 21 1.9574
0.2 1.5 23 1.9629
0.3 1.5 17 1.5953
0.4 1.5 85 1.9152
0.5 1.5 39 1.9131
0.6 1.5 61 2.7607
0.1 1.1 53 2.1995
0.2 1.1 53 2.1264
0.3 1.1 77 2.0304
0.4 1.1 46 2.4073
0.5 1.1 – 2.5735
0.6 1.1 – 3.8334
Figure 5: SS-EGC regularization of the Henon chaos. Example using 𝜌 = 0.3 and 𝜁 = 1.5. The controller
is designed and updated from 𝑘 = 500 to 𝑘 = 600 based on the SS-EGM
states still spiral on a band larger than 2% around x∗ after the 100 time steps with 𝑢 ≠ 0. To
illustrate the regularization of the chaotic behavior, Figure 5 shows the states convergence when
𝑢 is enabled at 𝑘 = 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 off at 𝑘 = 600.
Figure 6: Phane plane perspective of the result in Figure 5 emphasizing the convergence of the Henon
states toward the origin (0, 0) in a spiral-shaped trajectory. Control is applied at 𝑘 = 500. The blue
arrows show the deviation of the states from the usual unforced trajectory. The system backs to the
chaotic orbit at 𝑘 = 600 when the evolving controller is turned off (red arrows)
5. Conclusion
We described an evolving state-space variation of granular method to model and control un-
known 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.
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