Decentralized Control of Complex Systems: Lyapunov Function Approach

Abstract

In this contribution, we generalize existing methods for decentralized control design, providing a unified methodological framework that applies to linear continuous and discrete-time complex systems, as well as certain classes of nonlinear complex systems. Our approach leverages the direct connection between the stability properties of the overall complex system and those of its individual subsystems. By conducting the entire controller design process at the subsystem level, we circumvent the need for explicit interconnection values. Through numerical examples, we demonstrate that the proposed method ensures asymptotic stability of the full complex system within a specified region, and also guarantees stability of the isolated subsystems. In particular, we obtain quantifiable stability margins (e.g., ${\gamma }_c$ bounds) and closed-loop eigenvalue placements that verify the effectiveness of the design. These results highlight not only the method’s theoretical robustness, but also its practical significance in simplifying the design process, reducing computational overheads, and enhancing scalability for large and interconnected engineering systems.

1. Introduction

Today’s real-world systems are growing in scale and complexity as they strive to satisfy rising expectations in terms of both quantity and quality. Because of this, real processes have become complex or large-scale. Large-scale systems (LSSs) are complex, uncertain, and constrained by their information structure, making centralized control challenging to implement effectively.

Large-scale systems appear in various fields, such as industrial manufacturing, power systems, transportation networks, water distribution systems, and electronic circuits. For instance, power grids are inherently large-scale, requiring decentralized control to ensure stability and efficiency across geographically dispersed nodes. Similarly, in manufacturing systems, decentralized control is vital for coordinating robotic units or processes that operate independently while contributing to a shared goal. In electronic systems, decentralized control can be applied to modular circuits or communication networks, ensuring robust operation under constraints of limited communication or processing power. The applicability of decentralized control in these domains highlights its importance for optimizing performance, improving scalability, and enhancing fault tolerance.

LSSs utilize decentralized control strategies with information-constraint structures. Numerous methods for decentralized controller design have been developed for linear time-invariant large-scale systems, addressing both frequency and time domain frameworks. Within the frequency domain, prominent methods include the independent design approach [1], the sequential design technique [1], and the equivalent subsystem method [2]. Frequency-domain approaches also facilitate the analysis of low-frequency stability phenomena in complex interconnected systems. For instance, recent work has examined low-frequency stability in train–grid interactions [3] and multiple vehicle-traction network systems [4], demonstrating the practicality and importance of frequency-based analyses in decentralized control design. The first two methods are rather conservative and complex in terms of their respective solutions. The equivalent subsystem method provides a structured framework for decentralized controller design, guaranteeing the fulfillment of both necessary and sufficient stability conditions. Within the time domain, three main methodological categories have emerged: stability analysis and decentralized control design using the aggregation matrix approach [5], the vector Lyapunov function technique [6], and innovations in decentralized controller design for LSSs based on LMI-BMI methods, as outlined in [7]. However, these approaches typically necessitate a comprehensive model of the entire complex system for effective stability analysis and decentralized controller design.

The main idea of the proposed article can be summarized as follows. Let us assume that the complex system is stable. Interactions in a complex system, the parameters of the subsystems, and the decentralized controllers are configured to ensure the stability of the complex system. Let us find out what properties the parameters of subsystems and their decentralized regulators have in a case where a complex system is stable. This way, it is possible to obtain so-called ’isolated’ subsystems, for which a decentralized controller is designed so that the newly acquired properties of the subsystems with decentralized controllers are qualitatively better in terms of ensuring the stability of the complex system than in the previous case. For a decentralized controller designer, one of the ways to achieve more stable properties of a complex system is to use the aggregation matrix method approach [5]. In the proposed work, we methodically summarized the partial results published by our group in previous works and created a unified method for designing decentralized control without taking into account the interactions between subsystems for continuous, discrete, and nonlinear complex systems. If the complex system is unstable, then we modify the model of the complex system by adding a diagonal matrix so that the sum of both matrices creates a stable matrix. The designer must take this modification of the model into consideration.

Previous studies on decentralized control have categorized complex systems into two types. These studies demonstrate strong and weak interaction links. This work proposes that using system stability as a criterion offers a more effective basis for classifying complex systems. Such an approach opens new possibilities to design a better method for decentralized controller design. For both stable and unstable complex systems, designing decentralized controllers at the isolated subsystem level, without considering the specific interaction values, is considerably simpler compared to the approaches presented in prior studies.

The structure of this paper is outlined as follows. In the second section, we obtain the original decentralized control design method which should be applied to continuous, discrete-time linear systems and with minor manipulation of the proposed design method to nonlinear systems. The third section validates the proposed method through the design of decentralized controllers for both nonlinear and linear complex systems. Finally, the Conclusion summarizes the obtained results.

2. An Original Procedure for Decentralized Controller Development

In this section, methodically summarizing the results of our previous research, a new design procedure for decentralized controller design is proposed for linear continuous, discrete-time and specific nonlinear systems. The novel design methodology introduced in this paper will be illustrated using a linear LSS model in the following form:

$$\dot{x}=AxA\in R^{n\times n}$$

(1)

Let us denote

$$\dot{x}=A_{v}x=(A+vI)x=(A_d+vI+A_o)x$$

where v is a real number, and $A_d$ and $A_o$ are the diagonal and off-diagonal parts of the complex system. The system described by (1) consists of interconnected subsystems structured as follows:

$${\dot{x}}_j=(A_{jv}+v{I}_j)x_j+\sum _{i=1}^m{A}_{sj}{x}_i,j,s=1,2,\dots ,m,j\ne i$$

(2)

where m denotes the overall count of subsystems, $x_j\in {\mathbb{R}}^{n_j}$, and $A_{vj}$ and $A_{sj}$ are the state matrix of the j-th subsystem, the constant matrix of the j-th subsystem and the interaction matrices of the corresponding dimensions, respectively. The next lemma holds for the stability of complex system (1).

Lemma 1.

There exists a real value of v such that complex system $A_v$ is asymptotically stable in the sense of Lyapunov stability theory.

Proof. 

The Lyapunov function for complex system $A_v$ is defined as $V_v\left(x\right)=x^T{P}_{v}x$. Its time derivative is calculated as

$$\dot{V}\left(x\right)=x^T(A_v^T{P}_v+P_v{A}_v)x$$

The asymptotic stability of the complex system is ensured if the following Lyapunov inequality holds:

$$A_v^T{P}_v+P_v{A}_v={(A_d+vI+A_o)}^T{P}_v+P_v(A_d+vI+A_o)<0$$

The solution of the above inequality gives the positive definite matrix $P_v$ and the value of the auxiliary variable v. Note that if $v<0$, the complex system is unstable, and if $v>0$, it is stable. The obtained real value of v, as the solution of the Lyapunov matrix inequality, determines the value by which it is necessary to shift the eigenvalues of $A_d$ to reach the stability boundary of the complex system. This means that the complex system is on the stability boundary when ${\lambda }_n+v\le 0$ holds for the maximal real part of the eigenvalues of matrix $A_v$. Assume that the off-diagonal elements of matrix $A_o$ are constant and its parameters do not change when the decentralized controller is in operation. In this case, one can say that matrix $(A_d+vI)$ determines the stability boundary of the complex system because the off-diagonal matrix $A_o$ is a fixed part of a complex system, which cannot be influenced by any decentralized controller. The maximal value of subsystem eigenvalues is calculated as $MaxSubsEigen=\{{\gamma }_{1n_1},\dots {\gamma }_{m{n}_m}\}$, and the maximal eigenvalue $MAXEIG{A}_d=max\left(MAXSUBSEigen\right)$ To obtain the stability boundary of a complex system, one needs to move in the complex plane $MAXEIG{A}_d$ by v to the right if $v>0$, or to the left if $v<0$. The above principle assists the designer in selecting an appropriate performance quality function for the design of a decentralized controller for each subsystem. Let the eigenvalues of complex system A be $eig\left(A\right)={\lambda }_1,\dots ,{\lambda }_n$. From (1), the eigenvalues of matrix $A_v$ are calculated as $eig\left(A_v\right)={\lambda }_1+v,\dots ,{\lambda }_n+v$, that is, there exist such a value of v that all real part eigenvalues of matrix $A_v$ are negative and the complex system is asymptotically stable, that is,

$${\lambda }_k+v<0,k=1,2,\dots ,n$$

which proves the lemma. □

The second step results in the following practical method for decentralized controller design. Let the complex system be asymptotically stable. The eigenvalues of the j-th subsystem without a controller are given as ${\gamma }_{jk},k=1,2,\cdots ,n_j,j=1,2,\cdots ,m$. For each j-th subsystem, where $j=1,2,\cdots ,m$, the decentralized controller must be designed to ensure that the real parts of the closed-loop subsystem eigenvalues are less than those of the corresponding open-loop eigenvalues.

In the last step, a suitable method for decentralized controller design must be chosen to meet the conditions identified during the previous step.

The advantages of the design method proposed in this paper compared to the design method from the references using the aggregation matrix approach are given below. The stability of the isolated subsystems

$${\dot{x}}_j=A_j{x}_j,j=1,2,\dots ,m$$

(3)

can be determined using the Lyapunov function $v_j={\left(x_j^T{P}_j{x}_j\right)}^{{\displaystyle \frac{1}{2}}}$. Based on the findings in [8], the j-th subsystem is deemed stable if the following inequalities are fulfilled:

$${\alpha }_{j2}\left|\right|x_j\left|\right|\ge v_j\ge {\alpha }_{j1}\left|\right|x_j\left|\right|$$

(4)

$${\dot{v}}_j\le -{\alpha }_{j3}\left|\right|x_j\left|\right|\left|\right|grad{v}_j\left|\right|\le {\alpha }_{j4}$$

(5)

where

$${\alpha }_{j1}={\lambda }_m^{{\displaystyle \frac{1}{2}}}\left(P_j\right){\alpha }_{j2}={\lambda }_M^{{\displaystyle \frac{1}{2}}}\left(P_j\right)$$

(6)

$${\alpha }_{j3}=0.5{\displaystyle \frac{{\lambda }_m\left(G_j\right)}{{\lambda }_M^{0.5}\left(P_j\right)}}{\alpha }_{j4}=0.5{\displaystyle \frac{{\lambda }_M\left(P_j\right)}{{\lambda }_m^{0.5}\left(P_j\right)}}$$

(7)

where ${\lambda }_m$ and ${\lambda }_M$ denote the smallest and largest eigenvalues of the respective matrices, while $G_j$ represents a positive definite matrix that satisfies the Lyapunov equation $A_j^T{P}_j+P_j{A}_j=-G_j$. It follows that if inequalities (4) and (5) are satisfied, the j-th subsystem is asymptotically stable. According to [5,8], the stability criterion for LSSs can be represented using the following aggregated matrix formulation:

$$\dot{v}\le Wv,W=\left\{w_{ij}\right\}$$

(8)

where $v=[v_1\dots v_m]$ represents the vector Lyapunov function, as described in [6]:

$$w_{ij}=-{\alpha }_{j2}^{-1}{\alpha }_{j3}j=i,$$

(9)

$$w_{ij}={\eta }_{ij}{\alpha }_{j1}^{-1}{\alpha }_{j4}j\ne i$$

(10)

where ${\eta }_{ij}={\lambda }_M^{{\displaystyle \frac{1}{2}}}\left(A_{ij}^T{A}_{ij}\right)$.

The LSS achieves asymptotic stability provided that the aggregation matrix (8) is also asymptotically stable. Equation (9) clearly shows that the parameters ${\alpha }_{j2}$ and ${\alpha }_{j3}$ depend on the j-th decentralized controller. In this approach, the designer must select the controller parameters to ensure the stability of the Metzler matrix W. Ensuring the stability of subsystems (4) and (5), along with the aggregated model (8), collectively establishes connective stability for the entire system, as stated in [5].

3. Decentralized Controller Design: Examples

3.1. Designing Decentralized Controllers for Nonlinear Complex Systems

We consider a nonlinear system characterized by the state x, control input u and output y expressed in the following form:

$$\begin{array}{c}{\dot{x}}_1=a_{11}{x}_1+d_{11}{x}_1^2+a_{12}{x}_2+\dots a_{1n}{x}_n+b_{1m}{u}_1\\ \dots \\ {\dot{x}}_n=a_{nn}{x}_n+d_{nn}{x}_n^2+a_{n1}{x}_1+\dots b_{nm}{u}_m\end{array}$$

After some manipulation, we obtain

$$\dot{x}=Ax+D\left(x\right)x+Bu,D\left(x\right)=diag\{d_{ii}{x}_i,i=1,2,\dots ,n\}$$

(11)

where

$$A={\left\{a_{ij}\right\}}_{n\times n},B={\left\{b_{i,j}\right\}}_{n\times m},y=Cx$$

where $D\left(x\right)$ is a nonlinear function matrix that the controller designer must construct according to the actual nonlinear structure of the plant. Assume that $\left|\right|D\left(x\right)\left|\right|$ approaches zero as x approaches zero. Additionally, we consider the matrices B and C to have a decentralized structure. If the output matrix lacks a decentralized structure, the approach outlined in [9] is applied. For additional results on the design of decentralized controllers for nonlinear systems, see the references [10,11,12,13,14,15,16] and others. In the above papers, the primary limitation associated with decentralized controller design is the requirement to use the full complex model in the decentralized controller design procedure. Note that in the approach proposed in this paper, all decentralized controller designs are performed at the isolated subsystem level. From the first step of the decentralized design procedure, we obtain the following stability condition for a complex system.

Consider the following candidate Lyapunov function proposed for system (11):

$$V\left(x\right)=x^{T}Px,P>0$$

(12)

Let us assume that the system matrix A is asymptotically stable. The first time derivative of (12) with respect to the linear component of (11) yields the following equation:

$$\dot{V}\left(x\right)=x^T(A^{T}P+PA)x=-x^{T}Qx,Q>0$$

(13)

To establish the stability conditions for matrix A, we propose modifying A in the following manner:

$$\dot{V}=x^T({(A+\alpha I)}^{T}P+P(A+\alpha I))x\le 0$$

(14)

To solve the Lyapunov matrix Equation (14) for P and $\alpha$, it can be observed that the system (11) is unstable when $\alpha <0$. Let $A_c=A+\alpha I$. The derivative of the candidate Lyapunov function with respect to (11) and $A_c$ is given as follows:

$${\dot{V}}_c=x^T(A_c^{T}P+P{A}_c)x+x^{T}Qx+x^T(D^T\left(x\right)P+PD\left(x\right))x\le 0$$

(15)

From (15) we obtain the following assertion: Given any $\gamma >0$, it is possible to find $r>0$ such that for $\left|\right|D\left(x\right)\left|\right|<\gamma$ and $\left|\right|x\left|\right|<r$, we obtain the following inequality:

$${\dot{V}}_c\le [-{\lambda }_{min}\left(Q\right)+{2\gamma \left|\right|P\left|\right|\left]\right|\left|x\right||}^2,\left|\right|x\left|\right|<r$$

(16)

Select $\gamma \le {\displaystyle \frac{{\lambda }_{min}\left(Q\right)}{2\left|\right|P\left|\right|}}$. From Equation (16), it follows that $x\in [0,\gamma ]$ exists such that ${\dot{V}}_c<0$, which ensures the asymptotic stability of the nonlinear system (11). The matrix $A_c$ can be partitioned as follows:

$$A_c=\left[\begin{array}{cc}{A}_{11}& \dots .A_{1m}\\ A_{m1}& \dots A_{mm}\end{array}\right]$$

(17)

Drawing from the results discussed earlier, the theorem below can be formulated:

Theorem 1.

Assume that matrix $A_c$ is asymptotically stable. The subsystem parameters, in the absence of decentralized controllers, ensure the stability of the complex system, and the maximal subsystem eigenvalues under this condition are denoted by α, which guarantees the stability of the complex system. To maintain the stability of the complex system, the decentralized controllers for the subsystems must be designed such that the following inequality is satisfied for the maximal value of the subsystem closed-loop eigenvalues β:

$$\beta \le \alpha$$

(18)

Consider complex system (11) with the following parameters:

$$A=\left[\begin{array}{ccc}-0.2& 0.05& 0.07\\ 0.12& -035& 0.16\\ 0.1& 0.12& -0.41\end{array}\right]$$

$$B=\left[\begin{array}{ccc}1& 0& 0\\ 0.2& 0.3& 0\\ 0& 0& 1\end{array}\right]C=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\end{array}\right]$$

From (14), we calculate $\alpha =0.1024,\gamma =0.0016$, indicating that the complex system is asymptotically stable within the region $x\in [0,\gamma ]$, and $A_c=A+\alpha I$ lies on the stability boundary. Since the complex system is stable, we form two isolated subsystems as follows:

$${\dot{x}}_i=A_{ii}{x}_i+B_i{u}_i{y}_i=C_i{x}_{i}i=1,2$$

where

$$A_{11}=\left[\begin{array}{cc}-0.2& 0.05\\ 0.12& -0.35\end{array}\right]A_{22}=-0.41$$

$$B_1=\left[\begin{array}{c}1\\ 0.2\end{array}\right]B_2=1$$

$$C_1=\left[10\right]C_2=1$$

The objective is to develop two PI controllers for the subsystems, ensuring asymptotic stability for both the individual subsystems and the entire complex system.

For the third step of the design procedure, we use the simple decentralized control design method. This means that the derivative of the Lyapunov function in conjunction with the decentralized controller must be negative definite. The following gains of the decentralized controller and subsystem eigenvalues are obtained:

$$Eigclosed1=-4.09,-1.0959,-0.3557$$

The gains are $kp=-5,ki=-4.5$, and for the second subsystem, one obtains

$$Eigclosed2=-4.3834,-1.0266$$

The controller gains are $kp=-5,ki=-4.5$. For the complex controlled system, the obtained ${\gamma }_c=0.3173$. These results clearly indicate that the complex system achieves asymptotic stability within the region $x\in [0,{\gamma }_c]$. This shows that the obtained results of the eigenvalues of the closed-loop system and the open-loop one satisfy the stability condition of the complex system. This is a very interesting advantage of the proposed design method.

3.2. Decentralized Controller Design for Linear Complex System

This example has been borrowed from [17]. The example consists of two subsystems, a stable one and an unstable one, along with a complex system. The authors state that the complex system with a decentralized controller could be stabilized with state feedback. In this paper, we assume that the first states of both subsystems will be equal to the output feedback. The task involves designing a PI controller for the first subsystem and a combination of a PID controller and state feedback for the second subsystem, guaranteeing both the stability and performance of individual subsystems as well as the entire complex system. The composition and parameters of the linear complex system are detailed below. The first subsystem’s interaction, input, and output matrices are defined as follows:

$${\dot{x}}_i=A_i{x}_i+B_i{u}_i+L_{ij}{x}_{j}i=1,2i\ne j$$

(19)

where

$$A_1=\left[\begin{array}{cc}0& 1\\ -1& -1\end{array}\right]B_1=\left[\begin{array}{c}0\\ 1\end{array}\right]C_1=\left[10\right]$$

$$A_2=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]B_2=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]C_2=\left[100\right]$$

$$L_{12}=\left[\begin{array}{ccc}0& 0& 0\\ 1& 1& 1\end{array}\right]L_{21}=\left[\begin{array}{cc}0& 0\\ 0& 0\\ 1& 1\end{array}\right]$$

Based on the first step, the stability of the uncontrolled complex system must be evaluated first. The complex system is given as follows:

$$Aco=\left[\begin{array}{cc}{A}_1& L_{12}\\ L_{21}& A_2\end{array}\right]$$

and its stability is calculated as follows:

$${(Aco+\alpha \ast I)}^{T}P+P(Aco+\alpha \ast I)\le 0$$

(20)

$P>0$ is the Lyapunov matrix and $\alpha$ is the auxiliary variable. The solution of (20) gives $\alpha =-1.618$. For the obtained $\alpha$, we can conclude that the complex system is unstable, with the maximal value of the eigenvalue to the matrix $Aco$, ${\gamma }_{max}=1.618$. If we choose $alph{a}_c<\alpha$, the complex system $A_{cr}=Aco+{\alpha }_c\ast I$ will be asymptotically stable. In this case (stable complex system) we could choose to design decentralized controllers using the model given by (20) with the condition $L_{12}=L_{21}=0$. To ensure the stability of the complex system, it is necessary that the designed decentralized controllers shift the maximal eigenvalue of the subsystems to the left by ${\alpha }_c$.

We are now prepared to proceed with the design of the decentralized controller. We will adopt the method of $\dot{V}<0$, meaning that the Lyapunov function’s first derivative for subsystems with a decentralized controller must be negative definite. Decentralized controllers need to be designed for the two subsystems: a PI controller for the first subsystem and a PID-plus-state controller for the second one. The following control algorithm will be used.

The second subsystem is as follows:

$${\dot{x}}_n=\left[\begin{array}{cc}A+B{k}_{p}C+BKD& B{k}_i{r}_i\\ C& 0\end{array}\right]x_n+\left[\begin{array}{cc}BkC& 0\\ 0& 0\end{array}\right]{\dot{x}}_n$$

where $KD=k_{d1}C2+k_{d2}C3,{\dot{r}}_i=Cx$

$$C2=\left[010\right],C3=\left[001\right]$$

$k_{di};i=2,3$ gains for state feedback. The PI controller parameters and the corresponding closed-loop eigenvalues are

$$k_{p1}=-4.999;k_{i1}=-4.5;Eig1=\{-2.999;-2.366;-0.639\}$$

The following parameters are obtained for the PID+state controller of the second subsystem:

$$k_{p2}=-7.5;k_{i2}=-7;k_{d2}=-2.5;k_{d2}=-7.5;k_{d3}=-7.5$$

The eigenvalues of the closed-loop second subsystem are

$$Eig2=-5.6653;-0.1061\pm 0.8662i;-1.6226$$

The closed-loop complex system is given as follows

$$Acom=\left[\begin{array}{cccccc}-4.9998& 1-4.5& 0& 0& 0& 0\\ -1& -1& 0& 1& 1& 1& 0\\ 1& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 1& 1& 0& -7.5& -11.5& -7.5& -7\\ 0& 0& 0& 1& 0& 0& 0\end{array}\right]$$

and the eigenvalues of the closed-loop complex system are

$$Eigcom=\{-5.57;-4.1997;-1.4732\pm 0.9232i;-0.612;-0.0858\pm 0.8488i\}$$

A analysis of the eigenvalues demonstrates that the closed-loop complex system employing the proposed two decentralized controllers is asymptotically stable.

In the above examples, the controller parameters were calculated using YALMIP [18], a MATLAB-based optimization modeling framework. Rather than providing closed-form analytic solutions, YALMIP formulates the optimization problem symbolically and then utilizes various compatible numerical solvers to iteratively search for optimal solutions. This approach simplifies the setup of complex optimization problems, ensures flexibility in selecting and changing solvers, and facilitates rapid prototyping, making it easier to refine and improve controller designs.

4. Conclusions

This work introduces a unified methodological framework that generalizes decentralized control design methods for continuous, discrete, and certain classes of nonlinear complex systems. By leveraging Lyapunov-based techniques, the proposed approach ensures system-wide stability without requiring direct consideration of the interaction magnitudes between subsystems. Instead of classifying systems based on the strength of interaction links (strong or weak), the focus shifts to whether the overall system is stable or unstable. This reframing significantly simplifies the controller design process.

The key practical advantage of this methodology lies in its ability to reduce design complexity and computational overheads. Dividing the complex system into manageable linear subsystems streamlines stability analysis and controller design tasks, making it more accessible and scalable for large, interconnected systems. As a result, engineers can independently tune subsystems for improved stability and performance, without becoming entangled in intricate interconnection details. In practical terms, this leads to controller solutions that are easier to implement and maintain and more adaptable to a wide range of engineering domains.

The method’s versatility is further highlighted by its applicability to linear and nonlinear systems, enabling the use of Proportional–Integral (PI), Proportional–Integral–Derivative (PID), or more sophisticated controllers. By ensuring asymptotic stability within a specified region, this decentralized control design provides higher stability margins, allowing the overall system to withstand greater disturbances while maintaining the desired performance. The method’s real-world applicability is demonstrated through practical examples, showcasing how the approach can be employed in fields such as robotics, industrial automation, and other dynamic engineering environments to achieve reliable, high-performance control.

In summary, the contributions and benefits of this work are as follows:

• This paper proposes a novel perspective on decentralized control for complex systems, focusing on overall system stability rather than interaction strength.
• The decentralized control design procedure is simpler and more efficient, directly ensuring that each subsystem’s stability enhancements translate into improved overall system performance.
• The controller design is carried out at the subsystem level without explicit interaction values, allowing for straightforward integration of modern control theory techniques.
• Regional pole placement is emphasized for designing decentralized controllers suitable for both stable and unstable systems.
• Conditions involving Bilinear Matrix Inequalities (BMIs) can be transformed into Linear Matrix Inequalities (LMIs) using established methods, reducing computational complexity.
• Open questions regarding the stability conditions of complex systems, including optimal parameter choices for unstable subsystems, are identified for future research.

In addition to the benefits outlined above, we plan to extend the proposed decentralized control method through the application of neuro-evolution techniques. Neuro-evolution, which combines neural networks [19,20] and evolutionary algorithms, offers a promising avenue for optimizing controller parameters and enhancing adaptability to varying system dynamics. Integrating an Artificial Neural Network (ANN) with two or three outputs, respectively, for PI or PID controllers offers a dynamic approach to control design allowing for the real-time adaptation of control parameters based on system feedback. By employing neuro-evolution techniques to tune the ANN, the controller can continuously adjust its outputs, ensuring optimal performance and stability under varying operating conditions.