A near-optimal decentralized control approach for polynomial nonlinear interconnected systems

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

Decentralized control optimal control nonlinear systems Kronecker product stability analysis shifted Legendre polynomials

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 $[- 1, 1]$ , with a weight function $w (τ) = 1$ . The set of Legendre polynomials is obtained from the formula of Olinde-Rodrigues (Datta and Mohan, 1995)

ℓ_{n} (τ) = \frac{1}{2^{n} n!} \frac{d^{n} {(τ^{2} - 1)}^{n}}{d τ^{n}}

(1)

This gives

ℓ_{0} (τ) = 1, ℓ_{1} (τ) = τ, ℓ_{2} (τ) = \frac{3 τ^{2} - 1}{2}, \dots

(2)

These polynomials can also be obtained from the recursive relationship (Datta and Mohan, 1995)

(n + 1) ℓ_{n + 1} (τ) = (2 n + 1) τ ℓ_{n} (τ) - n ℓ_{n - 1} (τ)

(3)

with $ℓ_{0} (τ) = 1$ and $ℓ_{1} (τ) = τ$

SLPs

In order to obtain orthogonal Legendre polynomials over time interval $[0, t_{f}]$ , which is more useful in control synthesis, we perform the following change of variable

τ = \frac{2 t}{t_{f}} - 1 with 0 \leq t \leq t_{f}

(4)

The recursive relationship (3) becomes (Hwang and Chen, 1985)

(n + 1) s_{n + 1} (t) = (2 n + 1) (\frac{2 t}{t_{f}} - 1) s_{n} (t) - n s_{n - 1} (t)

(5)

where $s_{n} (t)$ denotes an elementary SLPs defined over $0 \leq t \leq t_{f}$ , $s_{0} (t) = 1$ and $s_{1} (t) = \frac{2 t}{t_{f}} - 1$ .

The principle of orthogonality of the SLPs is expressed by the following equation (Datta and Mohan, 1995)

\int_{0}^{t_{f}} s_{i} (t) s_{j} (t) dt = \frac{t_{f}}{2 i + 1} δ_{ij}

(6)

where $δ_{ij}$ is the Kronecker symbol.

The operational matrix of integration of SLPs

In the case of SLPs, the operational matrix of integration $P_{N}$ is defined as follows (Hwang and Chen, 1985)

\int_{0}^{t} S_{N} (τ) d τ ≅ P S_{N} (t)

(7)

where

\begin{matrix} P = \frac{t_{f}}{2} \\ (\begin{matrix} 1 & 1 & 0 & 0 & \dots & 0 & 0 & 0 \\ - \frac{1}{3} & 0 & \frac{1}{3} & 0 & \dots & 0 & 0 & 0 \\ 0 & - \frac{1}{5} & 0 & \frac{1}{5} & \dots & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & \dots & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & - \frac{1}{2 N - 3} & 0 & \frac{1}{2 N - 3} \\ 0 & 0 & 0 & 0 & \dots & 0 & - \frac{1}{2 N - 1} & 0 \end{matrix}) \end{matrix}

is an $(N \times N)$ constant matrix and $S_{N} (t) = [s_{0} (t), s_{1} (t), \dots, s_{N - 1} (t)]^{T}$ denotes a shifted Legendre basis of dimension N.

The integration of the cross product

The integration of the cross product of two SLPs vectors can be obtained as (Marzban and Razzaghi, 2004)

C_{p} = \int_{0}^{t_{f}} S_{N} (t) S_{N}^{T} (t) dt = t_{f} [\begin{matrix} 1 & 0 & \dots & 0 \\ 0 & \frac{1}{3} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & \frac{1}{2 M - 1} \end{matrix}]

(8)

$C_{p}$ is an $(N \times N)$ constant matrix.

Operational matrix of the Kronecker product

The product of two SLPs $s_{i} (t)$ and $s_{j} (t)$ can be expressed by Hwang and Chen (1985)

s_{i} (t) s_{j} (t) ≅ \sum_{k = 0}^{M - 1} ψ_{ijk} L_{k} (t)

(9)

with

ψ_{ijk} = \frac{2 k + 1}{t_{f}} \int_{0}^{t_{f}} s_{i} (t) s_{j} (t) s_{k} (t) dt

(10)

Then, we may write

s_{j} (t) S_{N} (t) = [\begin{matrix} ψ_{i 00} \\ ψ_{i 11} \\ ⋮ \\ ψ_{i (N - 1) (N - 1)} \end{matrix}] = K_{s}^{i} S_{N} (t)

(11)

where $K_{s}^{i}$ is an $N \times N$ constant matrix.

Then it becomes

S_{N} (t) \otimes S_{N} (t) = [\begin{matrix} K_{s}^{0} \\ K_{s}^{1} \\ ⋮ \\ K_{s}^{N - 1} \end{matrix}] = K_{S} S_{N} (t)

(12)

where ⊗ denotes the Kronecker product (Brewer, 1978). $K_{S} \in R^{N^{2} \times N}$ is the Kronecker product operational matrix of SLPs.

Description of the studied system class

We consider, in this study, a global nonlinear polynomial system $(S)$ consisting of M interconnected subsystems $(S_{i})$ described by the following state equations

(S_{i}) {\begin{matrix} {\overset{\cdot}{x}}_{i} (t) = f_{i} (x_{i} (t)) + g_{i} (x_{i} (t)) u_{i} (t) + h_{i} (x (t)) \\ y_{i} (t) = C_{i} x_{i} (t) \end{matrix}

(13)

where $x_{i} (t) \in R^{n_{i}}$ , $u_{i} (t) \in R^{m_{i}}$ denotes local states and control, respectively, for the i-th subsystem. $x (t) = [x_{1} (t)] x_{2} (t) \dots x_{M} (t)]^{T}$ , $u (t) = [u_{1} (t)] u_{2} (t) \dots u_{M} (t)]^{T}$ designate the global system states and control, respectively. Notice that

x (t) \in R^{n}, n = \sum_{i = 1}^{M} n_{i}

u (t) \in R^{m}, m = \sum_{i = 1}^{M} m_{i}

and

f_{i} (x_{i} (t)) = \sum_{k = 1}^{p_{i}} A_{ik} x_{i}^{[k]} (t)

$f_{i} (x_{i} (t))$ is called polynomial since $x^{[i]} (t)$ is the $i - th$ power of $x (t)$ in the sense of the Kronecker product

x^{[i]} (t) = \underset{i - times}{\underset{︸}{x (t) \otimes x (t) \otimes \dots \otimes x (t)}}, x^{[0]} (t) = 1

\begin{matrix} g_{i} (x_{i} (t)) = \sum_{s = 0}^{q_{i}} [\begin{matrix} G_{1 is} x_{i}^{[s]} (t) ⋮ & G_{2 is} x_{i}^{[s]} (t) ⋮ & \dots & ⋮ G_{m_{i} is} x_{i}^{[s]} (t) \end{matrix}] \\ = \sum_{s = 0}^{q_{i}} G_{is} (I_{m_{i}} \otimes x_{i}^{[s]} (t)) \end{matrix}

with

G_{is} = [\begin{matrix} G_{1 is} ⋮ & G_{2 is} ⋮ & \dots & ⋮ G_{m_{i} is} \end{matrix}]

and the nonlinear interconnection term is given by

h_{i} (x (t)) = \sum_{\binom{j = 1}{j \neq i}}^{M} \sum_{k = 1}^{p} \sum_{l = 1}^{k} H_{ij}^{kl} x_{i}^{[k - l]} (t) \otimes x_{j}^{[l]}

The overall interconnected system $(S)$ has the following state equation

\overset{\cdot}{x} (t) = F (x (t)) + G (x (t)) u

(14)

with

F (x (t)) = \sum_{j = 1}^{p} A_{j} x^{[j]} (t)

where matrix $A_{j}$ could be identified from matrices $A_{ik}$ and $H_{ij}^{kl}$ .

Also

G (x (t)) = \sum_{j = 0}^{q} G_{j} (I_{m} \otimes x^{[j]} (t))

where matrix $G_{j}$ could be identified from matrices $G_{is}$ .

Problem formulation

In this work, we propose to find the optimal subsystems state and control trajectories ( $x_{i}^{*} (t)$ , $u_{i}^{*} (t)$ ), based on shifted Legendre direct approach, by minimizing the performance index

J = \frac{1}{2} \sum_{i}^{M} \int_{t_{0}}^{t_{f}} [x_{i}^{T} (t) Q_{i} x_{i} (t) + u_{i}^{T} (t) R_{i} u_{i} (t)] dt

(15)

where $Q_{i} \in R^{n_{i} \times n_{i}}$ , $R_{i} \in R^{m_{i} \times m_{i}}$ , are symmetric positive matrices and M is the number of interconnected subsystems.

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

u_{i} (t) = \sum_{l = 1}^{r_{i}} K_{il} x_{i}^{[l]} (t)

(16)

that are the desired suboptimal controllers.

Remark 1: The global control could be written with respect to the following form

u (t) = \sum_{j = 1}^{r} K_{j} x^{[j]} (t)

(17)

We suggest choosing the nonlinear state feedback controller multiplicity such that

q + r = p

where

p = max_{i} (p_{i})

q = max_{i} (q_{i})

$p_{i}$ and $q_{i}$ are Kronecker multiplicity of $f (x_{i} (t))$ and $g (x_{i} (t))$ in (13).

It is worth noting that control gains in equations (17) and (16) are linked with the following relation

K_{j} = [\begin{matrix} K_{1 j} M_{1}^{[j]} \\ K_{2 j} M_{2}^{[j]} \\ ⋮ \\ K_{Mj} M_{M}^{[j]} \end{matrix}], \forall j = 1, \dots, r .

(18)

where matrices $M_{i} \in R^{n_{i} \times n}$ could be constructed as follows

\begin{matrix} M_{1} = (\begin{matrix} I_{n_{1}} & 0_{n_{1} \times n_{2}} & \dots & 0_{n_{1} \times n_{M}} \end{matrix}) \\ M_{2} = (\begin{matrix} 0_{n_{2} \times n_{1}} & I_{n_{2}} & 0_{n_{2} \times n_{3}} & \begin{matrix} \dots & 0_{n_{2} \times n_{M}} \end{matrix} \end{matrix}) \\ ⋮ \\ M_{M} = (\begin{matrix} 0_{n_{M} \times n_{1}} & \dots & 0_{n_{M} \times n_{M - 1}} & I_{n_{M}} \end{matrix}) \end{matrix}

where $I_{n_{i}}$ is the identity matrix of dimension $n_{i}$ and $0_{n_{i} \times n_{j}}$ is the zero matrix of dimension $(n_{i} \times n_{j})$ .

Notice that matrix $K_{1}$ with respect to (18) is $diag (K_{i 1})$ , $\forall i = 1, 2 \dots, M$ . that corresponds perfectly to the well known decentralized linear controller.

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

\begin{matrix} x_{i} (t) \approx X_{i}^{T} S_{N} (t) \\ u_{i} (t) \approx U_{i}^{T} S_{N} (t) \end{matrix}

(19)

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

\begin{matrix} vec (x_{i} (t)) \approx (S_{N} {(t)}^{T} (t) \otimes I_{n_{i}}) vec (X_{i}^{T}) \\ vec (u_{i} (t)) \approx (S_{N} {(t)}^{T} (t) \otimes I_{m_{i}}) vec (U_{i}^{T}) \end{matrix}

(20)

where $I_{n_{i}}$ and $I_{m_{i}}$ are $n_{i} \times n_{i}$ and $m_{i} \times m_{i}$ identity matrices.

Moreover, at the initial time, $t_{0} = 0$ , the initial state could be written as follows

x_{i} (0) ≅ X_{i 0}^{T} S_{N} (t)

(21)

where $X_{i 0}^{T} = [x_{i 0} 0 \dots 0]$ is an N constant vector.

We notate $z_{i}$ the unknown parameters vector associated to the i-th subsystem. $z_{x_{i}} = vec (X_{i}^{T})$ and $z_{u_{i}} = vec (U_{i}^{T})$ are, respectively, the i-th subsystem state and control parameters, such that

z_{i} = [\begin{matrix} z_{x_{i}} \\ z_{u_{i}} \end{matrix}]

(22)

and

z_{xi 0} = vec (X_{i 0}^{T}) .

Problem reformulation using SLPs

Each integral term $\int_{0}^{t_{f}} x_{i}^{T} (t) Q_{i} x_{i} (t) + u_{i}^{T} (t) R_{i} u_{i} (t) dt$ in the criterion is approached as (Marzban and Razzaghi, 2004)

\int_{0}^{t_{f}} [z_{x_{i}}^{T} (S_{N} (t) \otimes I_{n}) Q_{i} (S_{N} (t)^{T} \otimes I_{n}) z_{x_{i}} + z_{u_{i}}^{T} (S_{N} (t) \otimes I_{m}) R_{i} (S_{N} (t)^{T} (t) \otimes I_{m}) z_{u_{i}}] dt

(23)

which is equivalent to

\int_{0}^{t_{f}} [z_{x_{i}}^{T} (S_{N} (t) S_{N} (t)^{T} \otimes Q_{i}) z_{x_{i}} + z_{u_{i}}^{T} (S_{N} (t) S_{N} (t)^{T} \otimes R_{i}) z_{u_{i}}] dt

(24)

Using the integral of the cross operational matrix C, it reduces to

z_{x_{i}}^{T} (C \otimes Q_{i}) z_{x_{i}} + z_{u_{i}}^{T} (C \otimes R_{i}) z_{u_{i}}

(25)

System path approximation

The expansion of the system state over the shifted Legendre basis requires the development of functions $f_{i} (x_{i} (t))$ , $g_{i} (x_{i} (t)) u_{i} (t)$ and $h_{i} (x (t))$ over that basis. Throughout the development, some useful lemmas are recalled.

Lemma 1: The development of the $i^{th}$ Kronecker power of the state vector over an shifted Legendre basis $S_{N} (t)$ gives (Bouafoura and Benhadj Braiek, 2019)

x^{[i]} (t) = X_{[i]}^{T} . S_{N} (t)

where

X_{[i]}^{T} = X^{T [i]} . κ_{[i]}

with

κ_{[i]} = (κ_{[i - 1]} \otimes I_{N}) . K_{S}, for i = 3, 4, \dots

and

κ_{[2]} = K_{S}, κ_{[1]} = κ_{[0]} = 1

We recall that $X^{T [i]}$ denotes the $i^{th}$ Kronecker power of the state coefficients $X^{T}$ , $K_{S}$ being the operational matrix of the Kronecker product.

Expansion of $f_{i} (x_{i} (t))$ over the shifted Legendre basis

$f_{i} (x_{i} (t))$ could be approached as follows

f_{i} (x_{i} (t)) \approx F_{i}^{T} . S_{N} (t)

(26)

where

F_{i}^{T} = \sum_{k = 1}^{p_{i}} A_{ik} X_{i}^{T [k]} κ_{[k]}

Expansion of $g_{i} (x_{i} (t)) u_{i} (t)$ over the shifted Legendre basis

Notice that the term of $g_{i} (x_{i} (t)) u_{i} (t)$ under the sum could be expanded over the hybrid basis as follows

\begin{matrix} (I_{m_{i}} \otimes {x_{i}}^{[s]} (t)) u_{i} (t) \approx ({U_{i}}^{T} \otimes {X_{i}}^{T [s]} κ_{[s]}) K_{S} S_{N} (t) \\ \approx ({U_{i}}^{T} \otimes {X_{i}}^{T [s]}) (I_{N} \otimes κ_{[s]}) K_{S} S_{N} (t) \end{matrix}, \forall s = 1, 2, \dots

(27)

Then it becomes

g (x (t, τ)) u (t) \approx G_{i}^{T} . S_{N} (t)

(28)

with

{G_{i}}^{T} = G_{i 0} {U_{i}}^{T} + \sum_{s = 1}^{q_{i}} G_{is} ({U_{i}}^{T} \otimes {X_{i}}^{T [s]}) (I_{N} \otimes κ_{[s]}) K_{S}

Expansion of $h_{i} (x (t))$ over the shifted Legendre basis

The term of $h_{i} (x (t))$ under the sums could developed over the hybrid basis as follows

{x_{i}}^{[k - l]} (t) \otimes {x_{j}}^{[l]} (t) \approx (X_{i}^{T [k - l]} \otimes X_{j}^{T [l]}) κ_{[k - l, l]} K_{S} . S_{N} (t)

(29)

where we notate $κ_{[k - l, l]} = κ_{[k - l]} \otimes κ_{[l]}$ .

Then, it becomes

h_{i} (x (t)) \approx H_{i}^{T} . S_{N} (t)

(30)

with

{H_{i}}^{T} = \sum_{\binom{j = 1}{j \neq i}}^{M} \sum_{k = 1}^{p_{i}} \sum_{l = 1}^{k} H_{ij}^{kl} ({X_{i}}^{T [k - l]} \otimes {X_{j}}^{T [l]}) κ_{[k - l, l]} K_{S}

The integration of the system equation by introducing the operational matrix of integration P with respect to notations (30) and (26) gives

X_{i}^{T} - X_{i 0}^{T} = F_{i}^{T} P + G_{i}^{T} P + H_{i}^{T} P

(31)

Our objective is to express the constraint (31) in terms of decision variables $z_{x}$ and $z_{u}$ , to this end we apply the vec operator to (31). That allows us to state

\begin{matrix} z_{x_{i}} - z_{xi 0} = vec (\sum_{k = 1}^{p_{i}} A_{ik} X_{i}^{T [k]} κ_{[k]} P + G_{i 0} {U_{i}}^{T} P \\ + \sum_{s = 1}^{q_{i}} G_{is} ({U_{i}}^{T} \otimes {X_{i}}^{T [s]}) (I_{N} \otimes κ_{[s]}) K_{S} P \\ + \sum_{\binom{j = 1}{j \neq i}}^{M} \sum_{k = 1}^{p_{i}} \sum_{l = 1}^{k} H_{ij}^{kl} ({X_{i}}^{T [k - l]} \otimes {X_{j}}^{T [l]}) κ_{[k - l, l]} K_{S} P) \end{matrix}

(32)

Using the linearity property of the vec operator, it comes out as

\begin{matrix} z_{x_{i}} - z_{xi 0} = \sum_{k = 1}^{p_{i}} vec (A_{ik} X_{i}^{T [k]} κ_{[k]} P) + vec (G_{i 0} {U_{i}}^{T} P) \\ + \sum_{s = 1}^{q_{i}} vec (G_{is} ({U_{i}}^{T} \otimes {X_{i}}^{T [s]}) (I_{N} \otimes κ_{[s]}) K_{S} P) \\ + \sum_{\binom{j = 1}{j \neq i}}^{M} \sum_{k = 1}^{p_{i}} \sum_{l = 1}^{k} vec (H_{ij}^{kl} ({X_{i}}^{T [k - l]} \otimes {X_{j}}^{T [l]}) κ_{[k - l, l]} K_{S} P)) \end{matrix}

(33)

which is equivalent to

\begin{matrix} z_{x_{i}} - z_{xi 0} = \sum_{k = 1}^{p_{i}} (P^{T} {κ_{[k]}}^{T} \otimes A_{ik}) vec (X_{i}^{T [k]}) + (P^{T} \otimes G_{i 0}) vec ({U_{i}}^{T}) \\ + \sum_{s = 1}^{q_{i}} (P^{T} {K_{S}}^{T} (I_{N} \otimes {κ_{[s]}}^{T}) \otimes G_{is}) vec ({U_{i}}^{T} \otimes {X_{i}}^{T [s]}) \\ + \sum_{\binom{j = 1}{j \neq i}}^{M} \sum_{k = 1}^{p_{i}} \sum_{l = 1}^{k} ((P^{T} {K_{S}}^{T} {κ_{[k - l, l]}}^{T}) \otimes H_{ij}^{kl}) vec ({X_{i}}^{T [k - l]} \otimes {X_{j}}^{T [l]}) \end{matrix}

(34)

The major concern here is to express $vec (X_{i}^{T [k]})$ in terms of decision variable $z_{x_{i}}$ . This is possible by applying the following result (Bouafoura and Benhadj Braiek, 2019).

Lemma 2:

\begin{matrix} vec (X_{i}^{T [k]}) = Π_{(n_{i}^{k - 1}, N^{k - 1})} (mat (z_{x_{i}})) . Π_{(n_{i}^{k - 2}, N^{k - 2})} (mat (z_{x_{i}})) \dots \\ Π_{(n_{i}, N)} (mat (z_{x_{i}})) . z_{x_{i}} \\ = Ψ_{k} (z_{x_{i}}) \end{matrix}

for $k = 2, 3, \dots$ , where the matrix $Π_{(., .)}$ is defined in the Appendix.

Similarly

\begin{matrix} vec (U_{i}^{T} \otimes X_{i}^{T [s]}) = Π_{(n_{i}^{s}, N^{s})} (mat (z_{u_{i}})) . vec (X_{i}^{T [s]}) \\ = Π_{(n_{i}^{s}, N^{s})} (mat (z_{u_{i}})) Ψ_{s} (z_{x_{i}}) \end{matrix}

(35)

and

\begin{matrix} vec (X_{i}^{T [k - l]} \otimes X_{j}^{T [l]}) = Π_{(n^{l}, N^{l})} (mat (vec (X_{i}^{T [k - l]})) Ψ_{l} (z_{x_{j}}) \\ = Π_{(n^{l}, N^{l})} (mat (Ψ_{k - l} (z_{x_{i}})) Ψ_{l} (z_{x_{j}}) \end{matrix}

(36)

Finally, the system path constraint could be implemented using the following equation

\begin{matrix} z_{x_{i}} - z_{xi 0} = \sum_{k = 1}^{p_{i}} (P^{T} κ_{[k]}^{T} \otimes A_{ik}) Ψ_{i} (z_{x_{i}}) + (P^{T} \otimes G_{i 0}) z_{u_{i}} \\ + \sum_{s = 1}^{q_{i}} (P^{T} {K_{L}}^{T} (I_{N} \otimes {κ_{[s]}}^{T}) \otimes G_{is}) Π_{({n_{i}}^{s}, N^{s})} (mat (z_{u_{i}})) Ψ_{s} (z_{x_{i}}) \\ + \sum_{\binom{j = 1}{j \neq i}}^{M} \sum_{k = 1}^{p_{i}} \sum_{l = 1}^{k} ((P^{T} {K_{L}}^{T} {κ_{[k - l, l]}}^{T}) \otimes H_{ij}^{kl}) Π_{({n_{i}}^{l}, N^{l})} (mat (Ψ_{k - l} (z_{x_{i}}))) Ψ_{l} (z_{x_{j}}) \end{matrix}

(37)

Now, it could be noticed that the system path constraint is expressed properly in terms of unknown parameters $z_{x_{i}}$ and $z_{u_{i}}$ .

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 $z_{x_{i}}$ and $z_{u_{i}}$ that minimizes the overall criterion

J = \sum_{i = 1}^{M} J_{i}

(38)

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

z_{i}^{*} = [\begin{matrix} z_{x_{i}}^{*} \\ z_{u_{i}}^{*} \end{matrix}]

(39)

the optimal state and control vectors coefficients of each subsystem.

The idea is to find control matrices $K_{ij}$ such that the optimal vector (39) verify the control equation (17).

Expanding equation (17) over the shifted Legendre basis yields

U_{i}^{T} = \sum_{j = 1}^{r} K_{ij} X_{i}^{T [j]} κ_{[j]}

(40)

substituting the control and state coefficients with their optimal values and applying the vec operator gives

z_{u_{i}}^{*} = \sum_{j = 1}^{r} α_{ij} vec (K_{ij})

(41)

with α_{ij} = (κ_{[j]}^{T} (mat (z_{x_{i}}^{*}))^{[j] T} \otimes I_{m_{i}}) .

Finding control parameters could be then reduced to solving, in the least square sense, the following problems

C_{i} . θ_{i} = D

(42)

where i = 1 \dots M, and

C_{i} = [α_{i 1} ⋮ α_{i 2} ⋮ \dots ⋮ α_{ir}], D = z_{u_{i}}^{*} and θ_{i} = [\begin{matrix} vec (K_{i 1}) \\ vec (K_{i 2}) \\ ⋮ \\ vec (K_{ir}) \end{matrix}]

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.

Theorem 1: The overall system (14) controlled by the state feedback (17) is stable if all eigenvalues of matrix $A = A_{1} + G_{0} K_{1}$ have a strictly negative real part and if

‖ x_{0} ‖ \leq ε_{0},

(43)

where $ε_{0} > 0$ , $λ > 0$ and $ω < 0$ are given scalars satisfying

‖ e^{At} ‖ < λ e^{ω t} \forall t \geq 0

(44)

and

ε_{0} = \frac{ε_{1}}{λ}

(45)

with $ε_{1}$ is the unique positive solution of the following equation

λ \sum_{i = 2}^{p} ‖ A_{i} + \sum_{k = 2}^{r} G_{0} K_{k} + \sum_{j = 1}^{q} \sum_{k = 1}^{r} G_{j} (K_{k} \otimes I_{n^{j}}) ‖ ε^{i - 1} + ω = 0

(46)

such that $i = j + k$ .

Proof 1: The second term of system (14) with control (17) could be written as follows

G (x (t)) u (t) = \sum_{j = 0}^{q} \sum_{k = 1}^{r} G_{j} (K_{k} \otimes I_{n^{j}}) x^{[j + k]} (t)

(47)

Thus, the closed-loop system becomes

\begin{matrix} \overset{\cdot}{x} (t) = \sum_{i = 1}^{p} A_{i} x^{[i]} (t) + \sum_{j = 0}^{q} \sum_{k = 1}^{r} G_{j} (K_{k} \otimes I_{n^{j}}) x^{[j + k]} (t) \\ = (A_{1} + G_{0} K_{1}) x (t) + \sum_{i = 2}^{p} A_{i} x^{[i]} (t) + \sum_{k = 2}^{r} G_{0} K_{k} x^{[k]} (t) \\ + \sum_{j = 1}^{q} \sum_{k = 1}^{r} G_{j} (K_{k} \otimes I_{n^{j}}) x^{[j + k]} (t) \end{matrix}

(48)

The latter state equation could be rewritten as follows

\overset{\cdot}{x} (t) = (A_{1} + G_{0} K_{1}) x (t) + \sum_{i = 2}^{p} Δ_{i} (K_{k}) x^{[i]} (t)

(49)

with

Δ_{i} (K_{k}) = A_{i} + \sum_{k = 2}^{r} G_{0} K_{k} + \sum_{j = 1}^{q} \sum_{k = 1}^{r} G_{j} (K_{k} \otimes I_{n^{j}})

such that $i = j + k$ .

Let us note $A = A_{1} + G_{0} K_{1}$ , the solution of (49) is given by

x (t) = e^{At} x_{0} + \int_{0}^{t} e^{A (t - σ)} \sum_{i = 2}^{p} Δ_{i} (K_{k}) x^{[i]} (σ) d σ

(50)

Since, all eigenvalues of matrix A have negative real part, then there exist two reals $λ > 0$ and $ω < 0$ such that equation (44) holds. Hence, $x (t)$ could be bounded as

‖ x (t) ‖ \leq λ e^{ω t} ‖ x_{0} ‖ + λ e^{ω t} \sum_{i = 2}^{p} ‖ Δ_{i} (K_{k}) ‖ \int_{0}^{t} e^{- ω σ} ‖ x^{[i]} (σ) ‖ d σ

(51)

we recall that

‖ x^{[i]} (t) ‖ = ‖ x (t) ‖^{i}

Then, we can state

‖ x (t) ‖ \leq λ e^{ω t} ‖ x_{0} ‖ + λ e^{ω t} \sum_{i = 2}^{p} ‖ Δ_{i} (K_{k}) ‖ \int_{0}^{t} e^{- ω σ} ‖ x (σ) ‖^{i} d σ

(52)

Let assume that

‖ x (t) ‖ \leq ε

(53)

Then

‖ x (t) ‖^{i} \leq ε^{i - 1} ‖ x (t) ‖

and inequality (52) becomes

‖ x (t) ‖ e^{- ω t} \leq λ ‖ x_{0} ‖ + λ \sum_{i = 2}^{p} ‖ Δ_{i} (K_{k}) ‖ ε^{i - 1} \int_{0}^{t} e^{- ω σ} ‖ x (σ) ‖ d σ

(54)

By Grönwall-Bellman lemma (see Appendix), one may have

‖ x (t) ‖ e^{- ω t} \leq λ ‖ x_{0} ‖ e^{\int_{0}^{t} λ \sum_{i = 2}^{p} ‖ Δ_{i} (K_{k}) ‖ ε^{i - 1} ds}

(55)

Then

‖ x (t) ‖ \leq λ ‖ x_{0} ‖ e^{(λ \sum_{i = 2}^{p} ‖ Δ_{i} (K_{k}) ‖ ε^{i - 1} + ω) t}

(56)

It is then clear that the system state would be bounded if

λ \sum_{i = 2}^{p} ‖ Δ_{i} (K_{k}) ‖ ε^{i - 1} + ω < 0

(57)

and the state would satisfy then

‖ x (t) ‖ < ε = λ ‖ x (0) ‖

(58)

Condition (57) is satisfied for $ε < ε_{1}$ , where $ε_{1}$ the unique positive solution of $λ \sum_{i = 2}^{p} ‖ Δ_{i} (K_{k}) ‖ ε^{i - 1} + ω = 0$ .

Hence, the initial state should verify

‖ x (0) ‖ < ε_{0} = \frac{ε_{1}}{λ}

(59)

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.

The tripled inverted pendulums subject to control inputs can be described by (Yoo et al., 2009)

{\begin{matrix} {\overset{\cdot\cdot}{θ}}_{1} = & \frac{g}{l} \sin θ_{1} + u_{1} + \frac{k_{1} a^{2}}{m_{1} l^{2}} (\sin θ_{2} \cos θ_{2} - \sin θ_{1} \cos θ_{1}) \\ {\overset{\cdot\cdot}{θ}}_{2} = & \frac{g}{l} \sin θ_{2} + u_{2} + \frac{k_{1} a^{2}}{m_{2} l^{2}} (\sin θ_{1} \cos θ_{1} - \sin θ_{2} \cos θ_{2}) \\ + \frac{k_{2} a^{2}}{m_{2} l^{2}} (\sin θ_{3} \cos θ_{3} - \sin θ_{2} \cos θ_{2}) \\ {\overset{\cdot\cdot}{θ}}_{3} = & \frac{g}{l} \sin θ_{3} + u_{3} + \frac{k_{2} a^{2}}{m_{3} l^{2}} (\sin θ_{2} \cos θ_{2} - \sin θ_{3} \cos θ_{3}) \end{matrix}

(60)

where $θ_{i}$ is the angle of the i-th pendulum, g is the acceleration due to gravity ( $g = 9.8 kgm / s 2$ ), $m_{i}$ is the mass of the i-th rod. $m_{1} = 0.2 kg$ , $m_{2} = 0.4 kg$ and $m_{3} = 0.3 kg$ , l is the length of each rod ( $l = 0.9 m$ ), a is the distance from the pivot to the centre of gravity of the rod ( $a = 0.3 m$ ), $k_{1}$ and $k_{2}$ are the spring constants ( $k 1 = 1 N / m$ and $k 2 = 1.4 N / m$ ) and $u_{i}$ are the control inputs, where $i = 1, 2, 3$ .

We consider that the angles . $θ_{i}, i = 1, 2, 3$ remain in the interval $[- π / 4, π / 4]$ . Then, it would be possible to consider the following third-order polynomial approximations of the nonlinear trigonometric functions, such that

\begin{matrix} \sin θ_{i} \approx θ_{i} - \frac{θ_{i}^{3}}{6} \\ \begin{matrix} \sin θ_{i} \cos θ_{i} \approx (θ_{i} - \frac{θ_{i}^{3}}{6}) (1 - \frac{θ_{i}^{2}}{2}) \\ \approx θ_{i} (1 - \frac{2}{3} θ_{i}^{2}) \end{matrix} \end{matrix}

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

{\overset{\cdot}{x}}_{1} (t) = A_{11} x_{1} (t) + A_{13} x_{1}^{[3]} (t) + G_{10} u (t) + H_{12}^{11} x_{2} (t) + H_{12}^{33} x_{2}^{[3]} (t)

(61)

with

\begin{matrix} A_{11} = [\begin{matrix} 0 & 1 \\ \frac{g}{l} - \frac{k_{1} a^{2}}{m_{1} l^{2}} & 0 \end{matrix}], G_{10} = [\begin{matrix} 0 \\ \frac{1}{m_{1} l^{2}} \end{matrix}], \\ A_{13} = [\begin{matrix} 0 & 0 & \dots & 0 \\ - \frac{g}{6 l} + \frac{2 k_{1} a^{2}}{3 m_{1} l^{2}} & 0 & \dots & 0 \end{matrix}] \in R^{2 \times 8}, \\ H_{12}^{11} = [\begin{matrix} 0 & 0 \\ \frac{k_{1} a^{2}}{m_{1} l^{2}} & 0 \end{matrix}], \\ H_{12}^{33} = [\begin{matrix} 0 & 0 & \dots & 0 \\ - \frac{2 k_{1} a^{2}}{3 m_{1} l^{2}} & 0 & \dots & 0 \end{matrix}] \in R^{2 \times 8} \end{matrix}

The second subsystem is given by

\begin{matrix} {\overset{\cdot}{x}}_{2} (t) = A_{21} x_{2} (t) + A_{23} x_{2}^{[3]} (t) + G_{20} u (t) + H_{21}^{11} x_{1} (t) + H_{21}^{33} x_{1}^{[3]} (t) \\ + H_{23}^{11} x_{3} (t) + H_{23}^{33} x_{3}^{[3]} (t) \end{matrix}

(62)

with

\begin{matrix} A_{21} = [\begin{matrix} 0 & 1 \\ \frac{g}{l} - \frac{(k_{1} + k_{2}) a^{2}}{m_{2} l^{2}} & 0 \end{matrix}], G_{20} = [\begin{matrix} 0 \\ \frac{1}{m_{2} l^{2}} \end{matrix}], \\ A_{23} = [\begin{matrix} 0 & 0 & \dots & 0 \\ - \frac{g}{6 l} + \frac{2 (k_{1} + k_{2}) a^{2}}{3 m_{2} l^{2}} & 0 & \dots & 0 \end{matrix}] \in R^{2 \times 8} . \\ H_{21}^{11} = [\begin{matrix} 0 & 0 \\ \frac{k_{1} a^{2}}{m_{2} l^{2}} & 0 \end{matrix}] . \\ H_{21}^{33} = [\begin{matrix} 0 & 0 & \dots & 0 \\ - \frac{2 k_{1} a^{2}}{3 m_{2} l^{2}} & 0 & \dots & 0 \end{matrix}] \in R^{2 \times 8}, H_{23}^{11} = [\begin{matrix} 0 & 0 \\ \frac{k_{2} a^{2}}{m_{2} l^{2}} & 0 \end{matrix}], \\ H_{23}^{33} = [\begin{matrix} 0 & 0 & \dots & 0 \\ - \frac{2 k_{2} a^{2}}{3 m_{2} l^{2}} & 0 & \dots & 0 \end{matrix}] \in R^{2 \times 8} \end{matrix}

The third subsystem is given by

{\overset{\cdot}{x}}_{3} (t) = A_{31} x_{3} (t) + A_{33} x_{3}^{[3]} (t) + G_{30} u (t) + H_{32}^{11} x_{2} (t) + H_{32}^{33} x_{2}^{[3]} (t)

(63)

with

A_{31} = [\begin{matrix} 0 & 1 \\ \frac{g}{l} - \frac{k_{2} a^{2}}{m_{3} l^{2}} & 0 \end{matrix}], G_{30} = [\begin{matrix} 0 \\ \frac{1}{m_{3} l^{2}} \end{matrix}],

A_{33} = [\begin{matrix} 0 & 0 & \dots & 0 \\ \frac{g}{6 l} + \frac{2 k_{2} a^{2}}{3 m_{3} l^{2}} & 0 & \dots & 0 \end{matrix}] \in R^{2 \times 8}, H_{12}^{11} = [\begin{matrix} 0 & 0 \\ \frac{k_{2} a^{2}}{m_{3} l^{2}} & 0 \end{matrix}],

H_{12}^{33} = [\begin{matrix} 0 & 0 & \dots & 0 \\ - \frac{2 k_{2} a^{2}}{3 m_{3} l^{2}} & 0 & \dots & 0 \end{matrix}] \in R^{2 \times 8}

where $x_{i} (t) = [θ_{i} {\overset{\cdot}{θ}}_{i}]^{T}$ for $i = 1, 2, 3$ .

While the global system (60) with the same polynomial truncation could be written under the form (14)

A_{1} = [\begin{matrix} A_{11} & H_{12}^{11} & 0_{2 \times 2} \\ H_{21}^{11} & A_{21} & H_{23}^{11} \\ 0_{2 \times 2} & H_{32}^{11} & A_{31} \end{matrix}],

and matrix $A_{3}$ is given by

A_{3} = [\begin{matrix} A_{13} M_{1}^{[3]} + H_{12}^{33} M_{2}^{[3]} \\ A_{23} M_{2}^{[3]} + H_{21}^{33} M_{1}^{[3]} + H_{23}^{33} M_{3}^{[3]} \\ A_{33} M_{3}^{[3]} + H_{32}^{33} M_{2}^{[3]} \end{matrix}]

where matrices $M_{i}$ are firstly defined in (18), and

G_{0} = diag (G_{i 0})

Open loop optimization

Simulation of this example is led by considering $t_{f} = 2$ , the shifted Legendre basis dimension $N = 13$ , weighting matrices $Q_{i} = [\begin{matrix} 500 & 0 \\ 0 & 10 \end{matrix}]$ and $R_{i} = 10$ , for all $i = 1, 2, 3$ .

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: $x_{11} = 0.5$ , $x_{12} = - 0.5$ , $x_{21} = - 0.5$ , $x_{22} = 0.7$ , $x_{31} = 0.5$ and $x_{32} = 0.7$ .

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

u_{i} (t) = \sum_{l = 1}^{3} K_{il} x^{[l]} (t), for i = 1, 2, 3 .

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

K_{11} = [\begin{matrix} - 12.7 & - 2.56 \end{matrix}]

K_{12} = [\begin{matrix} 1.85 & - 2.3 & - 2.3 & - 0.62 \end{matrix}]

\begin{matrix} K_{13} = [\begin{matrix} 0.29 & - 1.12 & - 0.89 & - 0.65 & - 0.88 & - 0.65 & - 2.68 & - 0.54 \end{matrix}] \end{matrix}

The second subsystem controller has the following gains

\begin{matrix} K_{21} = [\begin{matrix} - 18.45 & - 4.09 \end{matrix}] \\ K_{22} = [\begin{matrix} - 6.27 & 2.68 & 2.68 & 0.39 \end{matrix}] \\ K_{23} = [\begin{matrix} 1.25 & - 1.25 & - 1.32 & - 2.06 & - 1.32 & - 2.06 & - 1.74 & - 0.73 \end{matrix}] \end{matrix}

The third subsystem controller has the following gains

\begin{matrix} K_{31} = [\begin{matrix} - 15.88 & - 1.92 \end{matrix}] \\ K_{32} = [\begin{matrix} 9.25 & - 2.5 & - 2.4 & 2.98 \end{matrix}] \\ K_{33} = [\begin{matrix} 2.84 & 2.1 & 1.96 & - 0.71 & 1.95 & - 0.71 & - 1.55 & 1.53 \end{matrix}] \end{matrix}

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.

Moreover, obtained gains verify stability conditions enounced in Theorem 1, with $λ = 5.8$ , $ω = 2$ and $ε_{0} = 10^{- 3}$ . That means obtained gains guarantee the global system asymptotic stability in a ball $B (O, ε_{0})$ centered at the origin but with a very reduced radius. While simulations show that the proposed decentralized controller is stabilizing the global system in a wider region to almost cover the polynomial model validity domain ( $| θ_{i} | < π / 4$ ). In fact, Theorem 1 gives only sufficient conditions for local exponential stability of the polynomial system, which is aimed at tracking optimal trajectories minimizing the quadratic cost (15). Hence, a tradeoff between tracking and stabilization according to Theorem 1 should be done.

Remark 2: Notice that the best stabilizing optimal controller should verify (41) and ensures that all eigenvalues of matrix $A = A_{1} + G_{0} K_{1}$ are negative and for $i \geq 2$

Δ_{j} (K_{k}) = 0 \forall k = 1, \dots, r

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

min_{K_{ij}} \sum_{i = 1}^{M} ‖ z_{u_{i}}^{*} - \sum_{j = 1}^{r} (κ_{[j]}^{T} {(mat (z_{x_{i}}^{*}))}^{[j] T} vec (K_{ij}) ‖

s.t.

eigenvalues (A_{1} + G_{0} K_{1}) < 0

‖ G_{0} K_{2} ‖ < ε_{1}

‖ A_{3} + G_{0} K_{3} ‖ < ε_{2}

where $ε_{1}$ and $ε_{2}$ are some positive scalars to be defined. Accordingly, we suggest the determine the following controllers gains

u_{i} (t) = K_{i 1} x (t) + K_{i 3} x^{[3]} (t), for i = 1, 2, 3 .

Solving the above optimization problem lead to

\begin{matrix} K_{11} = [- 11.14 - 2.7] \\ K_{13} = [0.13 - 0.32 - 0.32 0.24 - 0.32 0.24 0.24 0.27] \\ K_{21} = [- 12.92 - 3.25] \\ K_{23} = [0.17 - 0.17 - 0.17 0.27 - 0.17 0.27 0.27 - 0.02] \\ K_{31} = [- 10.19 - 2.5] \end{matrix}

K_{33} = [0.11 - 0.03 - 0.03 0.09 - 0.03 0.09 0.09 - 0.005]

Obtained gains verify stability conditions enounced in theorem 1, with $λ = 3.5$ , $ω = 1.7$ and a significantly larger stability domain defined by $ε_{0} = 0.15$ . These gains are also almost valid in the polynomial system validity region with slightly accuracy degradation regarding optimality with respect to imposed quadratic criterion. Further stability domain enlargement could be investigated by applying appropriate techniques (Bedoui et al., 2020) extending Theorem 1.

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.

ORCID iD

Mohamed Karim Bouafoura

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A near-optimal decentralized control approach for polynomial nonlinear interconnected systems

Abstract

Keywords

Introduction

Preliminaries

Legendre polynomials

SLPs

The operational matrix of integration of SLPs

The integration of the cross product

Operational matrix of the Kronecker product

Description of the studied system class

Problem formulation

Control synthesis

Open loop framework

Problem reformulation using SLPs

System path approximation

Expansion of f i ( x i ( t ) ) over the shifted Legendre basis

Expansion of g i ( x i ( t ) ) u i ( t ) over the shifted Legendre basis

Expansion of h i ( x ( t ) ) over the shifted Legendre basis

The NLP problem

Closed loop framework

Closed loop system stability analysis

Application to the three inverted interconnected pendulums control

Open loop optimization

Closed loop synthesis

Conclusion

Footnotes

Appendix

Declaration of conflicting interests

Funding

ORCID iD

References

Expansion of $f_{i} (x_{i} (t))$ over the shifted Legendre basis

Expansion of $g_{i} (x_{i} (t)) u_{i} (t)$ over the shifted Legendre basis

Expansion of $h_{i} (x (t))$ over the shifted Legendre basis