Abstract
This article tackles the decentralized near-optimal control problem for the class of nonlinear polynomial interconnected system based on a shifted Legendre polynomials direct approach. The proposed method converts the interconnected optimal control problems into a nonlinear programming one with multiple constraints. In light of the formulated NLP optimization, state and control coefficients are used to design a nonlinear decentralized state feedback controller. Overall closed-loop system stability sufficient conditions are investigated with the help of Grönwall lemma. The triple inverted pendulum case is considered for simulation. Satisfactory results are obtained in both open-loop and closed-loop schemes with comparison to collocation and state-dependent Riccati equation techniques.
Keywords
Introduction
Nowadays, the large-scale interconnected system is a hot topic in researching and a dominant class of systems that involve many practical applications such as electrical power systems (Abidi and Elloumi, 2020; Pham et al., 2017; Tlili, 2018), smart grids networks (Lo and Ansari, 2013), aircrafts (Hou et al., 2017; Xu et al., 2013), management (Miranbeigi et al., 2014), and many others. Thus, the control of those applications needs suitable control approaches. Maybe the most forward and economical way to govern that kind of systems is the decentralized control strategy. The key idea is to make each subsystem perfectly regulated using only its local state variables, with strong engagement to guaranty the global stability of the overall system.
In the past few decades, several important research results on large-scale linear systems have been already published (Lefebvre et al., 1980; Mahmoud et al., 1985). These results, even though with a major theoretical contribution, encounter some limitations in practice since most of the real systems are inherently nonlinear; that is why there has been an increasing interest in developing control schemes for interconnected systems with linearities in their dynamics and/or interconnections (Attia et al., 2014; Dhbaibi et al., 2009; Feydi et al., 2019; Jamshidi and Wang, 1984; Rtibi et al., 2018; Tlili, 2017; Yoo et al., 2009).
Possible modeling of nonlinear interconnected systems is to consider polynomial expansion, in the sense of the Kronecker power, of every nonlinear term appearing in the global system. Hence, with an appropriate truncation, a compact model is obtained that can describe correctly the dynamical behavior of a large set of processes such as electrical machines, robot manipulators arms and various mechanical processes in general (Belhaouane and Benhadj Braiek, 2011; Belhaouane et al., 2010). The decentralized control paradigm has been applied to such a system class. Elloumi and Benhadj Braiek (2012) solved the stabilization problem of polynomial interconnected systems with odd Kronecker power multiplicity. Meanwhile, Rtibi et al. (2018) address the problem of robust stabilization of uncertain interconnected polynomial systems. Solutions in these works are obtained by solving a system of decoupled Ricatti equations. Even though interconnection independent obtained controllers are proved to stabilize the overall system in both studies, neglecting interconnections terms in controller design phase may alter the efficiency of the global system especially when optimizing a performance index.
It is worth noting that reaching optimality in a nonlinear decentralized optimal control problem needs more computational efforts. For example, in the successive approximation approach (Abidi and Elloumi, 2020), instead of directly solving the nonlinear large-scale two-point boundary value problem (TPBVP), derived from the maximum principle, a sequence of nonhomogeneous linear time-varying TPBVPs is solved iteratively. Nevertheless, solving time-varying equations is much more difficult than solving time-invariant ones. That is why researchers are generally more concerned with suboptimal controllers (Lavaei et al., 2007; Miller and Davison, 2014; Ranganathan and Miller, 2018).
In this paper, we propose to design a near-optimal decentralized nonlinear state feedback control for the class of nonlinear polynomial systems. The controller design relies on a tailored direct spectral method (Jaddu, 2002) where shifted Legendre polynomials (SLPs) (Iben Warrad et al., 2015; Perng, 1986) with some important properties are exploited to transform the complex nonlinear optimal problem into a nonlinear programming (NLP) one with multiple equality constraints. Indeed, the important operational properties of the used orthogonal basis, such as the Kronecker product (Bouafoura and Benhadj Braiek, 2019; Iben Warrad et al., 2018) and the integration operational matrices (Hwang and Chen, 1985; Hwang and Shih, 1986; Datta and Mohan, 1995), reduce the differential equations with nonlinearities into easier-to-process algebraic equations (Bichiou et al., 2018) feeding the NLP solver. Closed-loop system with proposed suboptimal control stability is investigated and sufficient conditions are given. Open-loop and closed-loop results are compared by handling the decentralized control problem of three interconnected inverted pendulums.
This article is organized as follows. In Section 2, SLPs operational properties are recalled. Section 3 is reserved for the description of the studied system class. Section 4 exposes the control objectives. In Section 5, the main developments concerning the control synthesis are carried. Section 6 enounces sufficient conditions for the closed-loop stability analysis. Section 7 is dedicated to the application of the proposed approach to the triple interconnected inverted pendulums. Finally, conclusions and future works are drawn in Section 8.
Preliminaries
Legendre polynomials
The Legendre polynomials are orthogonal on the interval
This gives
These polynomials can also be obtained from the recursive relationship (Datta and Mohan, 1995)
with
SLPs
In order to obtain orthogonal Legendre polynomials over time interval
The recursive relationship (3) becomes (Hwang and Chen, 1985)
where
The principle of orthogonality of the SLPs is expressed by the following equation (Datta and Mohan, 1995)
where
The operational matrix of integration of SLPs
In the case of SLPs, the operational matrix of integration
where
is an
The integration of the cross product
The integration of the cross product of two SLPs vectors can be obtained as (Marzban and Razzaghi, 2004)
Operational matrix of the Kronecker product
The product of two SLPs
with
Then, we may write
where
Then it becomes
where ⊗ denotes the Kronecker product (Brewer, 1978).
Description of the studied system class
We consider, in this study, a global nonlinear polynomial system
where
and
with
and the nonlinear interconnection term is given by
The overall interconnected system
with
where matrix
Also
where matrix
Problem formulation
In this work, we propose to find the optimal subsystems state and control trajectories (
where
More specifically, at first, our objective is to determine control and state coefficients expanded over the SLPs basis by solving a formulated nonlinear constrained optimization problem. That would characterize an open-loop solution to the optimal control problem of the considered interconnected polynomial systems.
In the sequel, we suggest to identify, based on above results, the gains of the following nonlinear decentralized feedbacks
that are the desired suboptimal controllers.
We suggest choosing the nonlinear state feedback controller multiplicity such that
where
It is worth noting that control gains in equations (17) and (16) are linked with the following relation
where matrices
where
Notice that matrix
Control synthesis
Open loop framework
The main purpose of the study is to transform the decentralized optimal control problem under dynamic constraints to a NLP problem. To this end, each subsystem state and control variables are approximated by a finite length of unknown parameters as follows
where X and U are unknown state and control parameters, respectively. Applying the vec operator (see Appendix) and related Kronecker product property (Brewer, 1978) yields
where
Moreover, at the initial time,
where
We notate
and
Problem reformulation using SLPs
Each integral term
which is equivalent to
Using the integral of the cross operational matrix C, it reduces to
System path approximation
The expansion of the system state over the shifted Legendre basis requires the development of functions
where
with
and
We recall that
Expansion of
over the shifted Legendre basis
where
Expansion of
over the shifted Legendre basis
Notice that the term of
Then it becomes
with
Expansion of
over the shifted Legendre basis
The term of
where we notate
Then, it becomes
with
The integration of the system equation by introducing the operational matrix of integration P with respect to notations (30) and (26) gives
Our objective is to express the constraint (31) in terms of decision variables
Using the linearity property of the vec operator, it comes out as
which is equivalent to
The major concern here is to express
for
Similarly
and
Finally, the system path constraint could be implemented using the following equation
Now, it could be noticed that the system path constraint is expressed properly in terms of unknown parameters
The NLP problem
The optimal control problem has been approximated by a NLP problem and is given by the following: Find the optimal vector z of the unknown parameters
subject to M constraints of type (37).
The mathematical programming problem can be solved by using available NLP solvers.
Closed loop framework
Once the optimal open loop results are obtained by solving the NLP problem (38)–(37), we notate
the optimal state and control vectors coefficients of each subsystem.
The idea is to find control matrices
Expanding equation (17) over the shifted Legendre basis yields
substituting the control and state coefficients with their optimal values and applying the vec operator gives
Finding control parameters could be then reduced to solving, in the least square sense, the following problems
Closed loop system stability analysis
We consider the proposed global state feedback control (17). The following theorem gives sufficient conditions for state feedback stabilization of the overall considered nonlinear system.
where
and
with
such that
Thus, the closed-loop system becomes
The latter state equation could be rewritten as follows
with
such that
Let us note
Since, all eigenvalues of matrix A have negative real part, then there exist two reals
we recall that
Then, we can state
Let assume that
Then
and inequality (52) becomes
By Grönwall-Bellman lemma (see Appendix), one may have
Then
It is then clear that the system state would be bounded if
and the state would satisfy then
Condition (57) is satisfied for
Hence, the initial state should verify
Application to the three inverted interconnected pendulums control
In this section, we will illustrate the performance of the near-optimal direct approach, discussed in the previous section, on a system composed of three interconnected inverted pendulums, as shown in Figure 1.

Triple inverted pendulums.
The tripled inverted pendulums subject to control inputs can be described by (Yoo et al., 2009)
where
We consider that the angles .
To design the decentralized optimal controller, the system of equation (60) can be seen as an interconnection of three subsystems.
The first subsystem is given by
with
The second subsystem is given by
with
The third subsystem is given by
with
where
While the global system (60) with the same polynomial truncation could be written under the form (14)
and matrix
where matrices
Open loop optimization
Simulation of this example is led by considering
The optimization problem (37)–(38) is solved by the mean of the routine fmincon of MATLAB Software.
The curves of Figure 2 show the evolution of the state variables of the interconnected system and the corresponding control signal evolution, with the following initial conditions:

Optimal states and control
Obtained results are compared on this same figure to the results of a 1000-mesh-point collocation technique implemented in the bvp5c routine of MATLAB. It is then clear that the proposed SLPs approach is valid and systems optimal coefficients could be now used for a nonlinear state feedback design.
Closed loop synthesis
According to (16), we propose to control each pendulum separately as follows
By solving the posed least square problems (42), it is possible to determine the decentralized control parameters.
The first subsystem controller has the following gains
The second subsystem controller has the following gains
The third subsystem controller has the following gains
The curves of Figure 3 show the evolution of the state variables of the interconnected system and the corresponding decentralized control signals evolution. Results of the state-dependent Riccati equation approach (Feydi et al., 2019) are depicted in the same figure. It can be seen that both decentralized approaches can counteract the effect of nonlinear interconnections. It is also shown that the proposed approach results are very close to those of the SDRE method with a good enough accuracy compared with the optimal solution.

Suboptimal states and control
Moreover, obtained gains verify stability conditions enounced in Theorem 1, with
Thus, such decentralized controllers ensure the global stability of the overall system (14) (Benhadj Braiek, 1995). Finding this controller is very difficult or even impossible. However, a compromising controller in this example could be determined by enlarging the stability region by solving the following optimization problem
s.t.
where
Solving the above optimization problem lead to
Obtained gains verify stability conditions enounced in theorem 1, with
Nonetheless, it is worth noting that our work has the merit to derive an explicit nonlinear state feedback controller that is more suitable for implementation. This is in contrast to the feedback controller in Feydi et al. (2019), where the state-dependent matrices expressions solution of the differential Riccati equations are not determined, but obtained numerically with an iterative technique. This could be a major limitation in practice unless other steps are necessary for implementation, the complexity of which increases especially in a development environment other than MATLAB.
Conclusion
In this paper, a computational direct approach based on SLPs is proposed for solving the nonlinear decentralized optimal control problem. The studied interconnected systems are described by polynomial models in the sense of Kronecker power. A near-optimal polynomial state feedback controller is determined. The stability of the global closed-loop system could be checked with given sufficient conditions.
The three interconnected pendulums benchmark is considered for simulation. As expected, these simulations illustrate that the SLPs-based solution is effectively optimal by referring to the collocation technique. It follows that designed decentralized feedback controllers can steer all subsystems states to their optimal trajectories despite the effect of inherent nonlinear interconnections.
In future works, we expect to solve the decentralized nonlinear optimal tracking control problem for interconnected polynomial systems with state and input delays under bounded control inputs.
Footnotes
Appendix
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.
