Coordinated Control for a Group of Interconnected Pairwise Subsystems

Abstract

The implementation of pairwise decomposition is discussed on an interconnected system with uncertainties. Under the concept of system inclusion, two systems with the same expanded system achieved by the same expand transformation are considered as approximations. It is proven that a coordinated controller can be found to stabilize both the two systems. This controller is contracted from the coordinated controller of expanded system, with each pairwise subsystem having information structure constraint taken into consideration. At last, this controller design process is applied on a four-area power system treated as a group of subsystems with information structure constraints.

1. Introduction

Complex systems in real world are usually composed of a large group of interconnected subsystems. The interconnections among the subsystems are commonly presented in dynamics, and not only their weight values but also their connections with others keep evolving from time to time. Decentralized control is an ideal control strategy to handle the structural perturbations. Inclusion principle [1–3] is widely used as a general mathematical framework of decomposition, for example, automatic generation control (AGC) for a four-area power system [4–8], formation control of unmanned aerial vehicles [9], and structural vibration control of tall buildings under seismic excitations [10–12].

Particularly, pairwise decomposition provided in [4–8] can take full use of interconnections in the system, by treating each pair of subsystems with information structure constraint as a basic connected unit. Based on the inclusion principle framework, the system will be expanded into a much bigger space in a recurrent reverse order, so that the system is completely decomposed. Then a pairwise coordinated controller for the expanded system will be constructed by achieving coordinated consensus of each pairwise subsystem in parallel. After properly compensated, the controller can be contracted into the original space to fix the original system.

However, to apply pairwise decomposition methodology, an explicitly defined overall system model in particular superposition form is needed, and this condition may not always be satisfied due to system complexity. The work of this paper is to present an implementation approach of pairwise decomposition for interconnected system with state uncertainties. As the basis of system expansion and contraction, adequate knowledge of interconnection structure between pairwise subsystem is necessary, and this is the presumption to apply pairwise decomposition in this paper. For the system whose model is uncertain, the inclusion principle can not achieve its expansion exactly. But under the circumstance that the interconnection structure of system is available, an approximate expanded system can be constructed instead. Motivated by the idea that the whole system could achieve high performance only if each part could be consistent, an expanded system can be constructed, which comprises all pairwise subsystems with information structure constraints of the original system, and this expanded system is treated as an approximate expansion of the origin. According to the inclusion conditions, the expanded system can be contracted to the original space. A contraction dual to the expansion can always be found, so that the contracted system and original system are approximate in state dynamic. In this way, the coordinated controller that can stabilize the contracted system is also suitable for the original system. Similar to the system level, the coordinated controller of contracted system is also established by contracting from that of expanded system properly. In fact, this contracted controller can be used directly on the original system. As long as the dynamic of original system is adequately included in the expanded system, the contracted controller can be used as a suboptimal controller of the origin. The approximation of this paper mainly represents how good the expanded system would include the original system state. However, it is difficult to describe the approximation without a comparison of control performances. Considering the uncertainties of system state, a static state feedback controller at each subsystem is designed to robustly stabilize the system dynamics. This control design process mainly depends on the decentralized system form; it can suit a group of systems which can only use local information, for example, the multiagent system. Moreover, just the same as the ordinary pairwise decomposition, this process is also able to deal with the information structure constraints variation.

The organization of this paper is as follows. In the next section, preliminaries of permuted inclusion principle and system contraction are provided. The main result is presented in Section 2, where the approximate expansion under the concept of system inclusion is discussed, as well as the controller design procedure. In Section 3, a simulation example is provided to illustrate the proposed method on a group of subsystems with information structure constraints.

2. Preliminaries

The controller design process provided in this paper mainly relies on the permuted inclusion principle [6, 7] and system contraction [7, 13] that are presented in the following.

2.1. Permuted Inclusion Principle

Suppose that system 𝐒 is a group of interconnected subsystems and each subsystem 𝐒_𝑖 is connected to every other counterparts. Then system 𝐒 can be decomposed into the expanded space of 𝑁(𝑁−1)/2 pairwise subsystems with a pair recurrent reverse order subscripts as follows: 𝐒_𝑖𝑗∶𝐒₁₂,𝐒₂₃,𝐒₁₃,𝐒₃₄,𝐒₂₄,𝐒₁₄,…,𝐒_{(𝑁−1)𝑁},𝐒_{(𝑁−2)𝑁},…,𝐒_2𝑁,𝐒_1𝑁. (1)

Notice that the pairwise subsystems are arranged by a reverse order of subscript 𝑗, and this unnatural order enables the last one or some subsystems of the sequence to disconnect from, or connect to on the contrary, the overall system without impact on the remaining orders. It is convenient for representing the system information structure constraints variations.

The expected pairwise subsystems order is established by both row and column permutation matrices, which are composed of a series of basic permutation matrices representing a special case of nonsingular transformations. Assume that 𝐼_𝑘 is a subidentity matrix corresponding to the subsystem 𝐒_𝑘, as provided in [6, 7], 𝑃=^⟵Π^⟵𝑁−2_𝑖=1Π^{⟵𝑁−𝑖−1}_𝑗=1Π₍₎^{𝑁(𝑁−𝑗)−𝑖(𝑗+1)}_{𝑘=1+𝑖𝑖−1}𝑝_𝑘(𝑘+1)𝑃⁻¹=⟶Π^𝑁−2_𝑖=1⟶Π^{𝑁−𝑖−1}_𝑗=1⟶Π^{𝑁(𝑁−𝑗)−𝑖(𝑗+1)}_{k()=1+𝑖𝑖−1}𝑝^𝑇_𝑘()𝑘+1, (2) where the signs “←” and “→” indicate right and left directional multiplying operations, and 𝑝_𝑘(𝑘+1)=blockdiag𝐼₁,…,𝐼_𝑘−1,0𝐼_𝑘𝐼_𝑘+10,𝐼_𝑘+2𝑁,𝑝,…,𝐼_{𝑘()−1𝑘+1}=𝑝^𝑇_𝑘()𝑘+1=blockdiag𝐼₁,…,𝐼_𝑘−1,0𝐼_𝑘+1𝐼_𝑘0,𝐼_𝑘+2𝑁,…,𝐼 (3) are the basic permutation matrices for the 𝑘 and 𝑘+1 groups of adjacent columns and rows, respectively. The 𝑁 in (3) indicates the number of subsystems in the expanded system, and here 𝑁=𝑁(𝑁−1). The literature [8] provides an alternative matrix position-based form to construct this permutation matrix 𝑃 more simply. Use 𝑃(𝑚,𝑛)_𝑏 to notate the block position in 𝑃 of subidentity matrices 𝐼_𝑖 and 𝐼_𝑗 corresponding to pairwise subsystem 𝐒_𝑖𝑗 in 𝑃; then it comes 𝑃𝑁𝑖−𝑗+1,𝑗𝑗−1−2(𝑖−1)−1_𝑏=𝐼_𝑖,𝑃𝑁𝑗−1−𝑖+1,𝑗𝑗−1−2(𝑖−1)_𝑏=𝐼_𝑗,𝑖,𝑗=1,2,…,𝑘,…,𝑁,𝑖≠𝑗. (4)

Example 1. Consider an expansion for system 𝐒 with full network structure and 𝑁=3; its pairwise subsystems can be ordered as 𝐒_𝑖𝑗∶𝐒₁₂,𝐒₂₃,𝐒₁₃. (5)
According to (4), the block positions of 𝐼_𝑖 and 𝐼_𝑗 to 𝐒_𝑖𝑗 can be obtained as 𝐒₁₂∶𝑃(2,1)_𝑏=𝐼₁,𝑃(3,2)_𝑏=𝐼₂,𝐒₂₃∶𝑃(4,3)_𝑏=𝐼₂,𝑃(5,4)_𝑏=𝐼₃,𝐒₁₃∶𝑃(1,5)_𝑏=𝐼₁,𝑃(6,6)_𝑏=𝐼₃. (6) that is to say 𝑃=0000𝐼₁0𝐼₁000000𝐼₂000000𝐼₂000000𝐼₃0000000𝐼₃. (7) It is equivalent to the result of (2) when 𝑁=3.

Consider an interconnected system 𝐒 in compacted form and its expanded system 𝐒_𝑃 as follows: 𝐒∶𝑥̇=𝐴𝑥+𝐵𝑢,𝑦=𝐶𝐒𝑥,_𝑃∶̇𝑥_𝑃=𝐴_𝑃𝑥_𝑃+𝐵_𝑃𝑢_𝑃,𝑦_𝑃=𝐶_𝑃𝑥_𝑃, (8) where 𝑥(𝑡)∈𝐑^𝑛, 𝑢(𝑡)∈𝐑^𝑚, and 𝑦(𝑡)∈𝐑^𝑙 are the state, input, and output vectors of the system 𝐒; 𝑥_𝑃(𝑡)∈𝐑𝑛, 𝑢_𝑃(𝑡)∈𝐑𝑚, and 𝑦_𝑃𝑙̃(𝑡)∈𝐑 are those of system 𝐒_𝑃. It is supposed that 𝑛≤𝑛, 𝑚≤𝑚, and 𝑙≤𝑙̃.

For the input-state-output inclusion principle mentioned in [14, 15], a definition of the permuted inclusion principle is given.

Definition 2. The system 𝐒_𝑃 includes the system 𝐒, or 𝐒_𝑃⊃𝐒, if there exists a quadruplet of full rank matrices {𝑉_𝑃,𝑈_𝑃,𝑄_𝑃,𝑆_𝑃} satisfying 𝑈_𝑃𝑉_𝑃=𝐼_𝑛, such that for any 𝑥₀∈𝐑^𝑛 and any 𝑢∈𝐑^𝑚, the conditions 𝑥_𝑃0=𝑉_𝑃𝑥₀ and 𝑢=𝑄_𝑃𝑢_𝑃 imply 𝑥(𝑡;𝑡₀,𝑥₀,𝑢)=𝑈_𝑃𝑥_𝑃(𝑡;𝑡₀,𝑥_𝑃0,𝑢_𝑃) and 𝑦[𝑥(𝑡)]=𝑆_𝑃𝑦_𝑃[𝑥_𝑃(𝑡)] for all 𝑡≥𝑡₀.

Call the system 𝐒 a contraction of system 𝐒_𝑃. It is supported by the inclusion principle that all information about the behavior of 𝐒 is included in 𝐒_𝑃, such as stability and optimality. One of the necessary and sufficient conditions for the inclusion is restriction, the following theorem considers the restriction type (d) ([2, 3, 7]).

Theorem 3. The system 𝐒 is a typical restriction of the system 𝐒_𝑃, if there is a triplet of full rank matrices {𝑉_𝑃,𝑄_𝑃,𝑆_𝑃} such that 𝑉_𝑃𝐴=𝐴_𝑃𝑉_𝑃,𝑉_𝑃𝐵𝑄_𝑃=𝐵_𝑃,𝐶=𝑆_𝑃𝐶_𝑃𝑉_𝑃. (9)

Proof. The proof follows directly from the results in [6, 7, 14, 15].
The systems 𝐒 and 𝐒_𝑃 are related by 𝐴_𝑃=𝑉_𝑃𝐴𝑈_𝑃+𝑀^𝑃_𝐴,𝐵_𝑃=𝑉_𝑃𝐵𝑄_𝑃+𝑀^𝑃_𝐵,𝐶_𝑃=𝑇_𝑃𝐶𝑈_𝑃+𝑀^𝑃_𝐶, (10) where 𝑀^𝑃_𝐴, 𝑀^𝑃_𝐵, and 𝑀^𝑃_𝐶 are complementary matrices with proper dimensions. See [6, 7] for details.

2.2. System Contraction

One of the difficulties in applying system contraction by inclusion principle is that the conditions may be too restrictive, and a complete contraction from the given expanded system 𝐒_𝑃 to system 𝐒 will not always exist. It is indicated by the restriction conditions of (9) that system 𝐒_𝑃 completely includes 𝐒 if and only if it is uncontrollable. A natural way to resolve this problem is to introduce an incomplete contraction as an approximation. Split the permuted state matrix 𝐴_𝑃 into two parts as 𝐴_𝑃=𝐴_𝑅𝑃+𝑀_𝐴𝑃, (11) where 𝐴_𝑅𝑃 is the part that can be contracted as (8) implies, 𝑀_𝐴𝑃 is a complementary matrix with proper dimension standing for the remnant after contraction from the expanded space. System 𝐒 is a reduced-order model of system 𝐒_𝑃, according to the restriction conditions in (9) and (10), and take the state matrix for example, this incomplete system contraction requires that 𝑉_𝑃𝐴=𝐴_𝑃−𝑀_𝐴𝑃𝑉_𝑃. (12) There are arbitrary choices of the expanding transformation matrix 𝑉_𝑃. Since this paper is based on the pairwise decomposition methodology, 𝑉_𝑃 is chosen as the same form of that in [6, 7], which will be presented in next section. Anyway, when 𝑉_𝑃 is confirmed according to the inclusion condition, here goes 𝑀_𝐴𝑃𝑉_𝑃=𝐴_𝑃𝑉_𝑃−𝑉_𝑃𝐴. (13) To satisfy the restriction condition, there must be 𝐴=𝑉^𝑇_𝑃𝑉_𝑃⁻¹𝑉^𝑇_𝑃𝐴_𝑃𝑉_𝑃 (14) so that ‖𝑀_𝐴𝑃𝑉_𝑃‖ will be minimum, and this results in the minimal norm solution 𝑀_𝐴𝑃=𝐼−𝑉_𝑃𝑉^𝑇_𝑃𝑉_𝑃⁻¹𝑉^𝑇_𝑃𝐴_𝑃𝑉_𝑃𝑉^𝑇_𝑃𝑉_𝑃⁻¹𝑉^𝑇_𝑃. (15)

3. Pairwise Decomposition for a Group of Interconnected Subsystems

Assume that system 𝐒 is composed of a group of interconnected subsystems as the coordinated control target, 𝐒=𝐒_𝑖∶̇𝐒𝑥=𝐴𝑥+𝐵𝑢,𝑦=𝐶𝑥,_𝑖∶𝑥̇_𝑖=𝐴_𝑖𝑖𝑥_𝑖+𝐵_𝑖𝑖𝑢_𝑖+^𝑁_{𝑗=1𝑗≠𝑖}𝑒_𝑖𝑗(𝑡)𝐴_𝑖𝑗𝑥_𝑗,𝑦_𝑖=𝐶_𝑖𝑖𝑥_𝑖,𝑖=1,2,…,𝑁, (16) where 𝑥_𝑖(𝑡)∈𝐑^𝑛𝑖, 𝑢_𝑖(𝑡)∈𝐑^𝑚𝑖, and 𝑦_𝑖(𝑡)∈𝐑^𝑙𝑖 are the state, input, and output vectors of 𝐒_𝑖 at time 𝑡∈𝐑 system matrices 𝐴, 𝐵, and 𝐶 are compacted forms of 𝐴_𝑖𝑗, 𝐵_𝑖𝑖, and 𝐶_𝑖𝑖 in proper dimensions, respectively. Notations 𝑥_𝑖(𝑡;𝑡₀,𝑥_𝑖0,𝑢_𝑖) and 𝑦_𝑖[𝑥_𝑖(𝑡)] denote unique solutions of 𝐒_𝑖 for the initial time 𝑡₀, the initial state vector 𝑥_𝑖0, and a fixed control input 𝑢_𝑖, 𝑒_𝑖𝑗(𝑡) is element of the interconnection matrix 𝐸=(𝑒_𝑖𝑗)∈𝐑^𝑁×𝑁. Under the concept of pairwise decomposition, if at least one of interconnections 𝑒_𝑖𝑗(𝑡)≠0 or 𝑒_𝑗𝑖(𝑡)≠0, then it appears that subsystems 𝐒_𝑖 and 𝐒_𝑗 are connected. Call 𝐒_𝑖𝑗∶𝑥̇_𝑖=𝐴_𝑖𝑖𝑥_𝑖+𝑒_𝑖𝑗(𝑡)𝐴_𝑖𝑗𝑥_𝑗+𝐵_𝑖𝑖𝑢_𝑖,𝑦_𝑖=𝐶_𝑖𝑖𝑥_𝑖𝑥̇_𝑗=𝐴_𝑗𝑗𝑥_𝑗+𝑒_𝑗𝑖(𝑡)𝐴_𝑗𝑖𝑥_𝑖+𝐵_𝑗𝑗𝑢_𝑗,𝑦_𝑗=𝐶_𝑗𝑗𝑥_𝑗𝑖,𝑗=1,2,…,𝑁,𝑖≠𝑗 (17) a pairwise subsystem with basic interconnection.

The time-varying parameters 𝑒_𝑖𝑗(𝑡) and 𝑒_𝑗𝑖(𝑡) in (17) can describe the dynamic weight values between the connected subsystems 𝐒_𝑖 and 𝐒_𝑗. They represent the information structure constraints of the interconnected system and play a very important role in the system dynamic. In the literature [16], a fundamental interconnection (adjacency) matrix 𝐸=(𝑒_𝑖𝑗)∈𝐑^𝑁×𝑁 is defined in order to describe the normal structure of a given system graph. This notation can also be used here to indicate whether there is information structure constraint between a subsystem pair, by the rule 𝑒_𝑖𝑗=1,𝑒_𝑖𝑗(𝑡)≠0𝑒_𝑖𝑖=0.0,𝑒_𝑖𝑗(𝑡)=0, (18)

When 𝑒_𝑖𝑗=1, it indicates that there is interconnection from subsystem 𝐒_𝑖 to 𝐒_𝑗, and 𝑒_𝑖𝑗=0 indicates not. This binary interconnection matrix 𝐸 will be used later in the inclusion principle framework. If one of 𝑒_𝑖𝑗 and 𝑒_𝑗𝑖 is equal to 0, the pairwise subsystem 𝐒_𝑖𝑗 is half connected; the original information structures of 𝐒_𝑖𝑗 would be changed by using the coordinated control mentioned earlier. In this case, the sequential LQ optimization provided in [3, 17] can be consulted to keep the information structure of 𝐒_𝑖𝑗. Moreover, note that 𝑒_𝑖𝑗 and 𝑒_𝑗𝑖 will both be valued 0 under some circumstances, which means that the pairwise subsystem 𝐒_𝑖𝑗 will be disjointed. The disconnected modes have been discussed in [6, 7]. Particularly, when 𝑒_𝑖𝑗 and 𝑒_𝑗𝑖 evolute in a dynamical way and enforce 𝐒_𝑖𝑗 to disjoint and then joint again, the discussion is provided in [8].

Theoretically, any existing control technique can be applied to the coordinated control of this pairwise subsystem 𝐒_𝑖𝑗. Take pairwise subsystem 𝐒_𝑖𝑗 as this compact form 𝐒_𝑖𝑗∶𝑥̇_𝑖𝑗=𝐴_𝐷𝑖𝑗𝑥_𝑖𝑗+𝐵_𝐷𝑖𝑗𝑢_𝑖𝑗,𝑦_𝑖𝑗=𝐶_𝐷𝑖𝑗𝑥_𝑖𝑗, (19) with 𝑥_𝑖𝑗=[𝑥_𝑖,𝑥_𝑗]^𝑇, 𝑢_𝑖𝑗=[𝑢_𝑖,𝑢_𝑗]^𝑇, and 𝑦_𝑖𝑗=[𝑦_𝑖,𝑦_𝑗]^𝑇, and 𝐴_𝐷𝑖𝑗=𝐴_𝑖𝑖𝑒_𝑖𝑗(𝑡)𝐴_𝑖𝑗𝑒_𝑗𝑖(𝑡)𝐴_𝑗𝑖𝐴_𝑗𝑗,𝐵_𝐷𝑖𝑗=𝐵_𝑖𝑖00𝐵_𝑗𝑗,𝐶_𝐷𝑖𝑗=𝐶_𝑖𝑖00𝐶_𝑗𝑗. (20) Call 𝐂_𝑖𝑗∶𝑢_𝑖𝑗=−𝐾_𝐷𝑖𝑗𝑥_𝑖𝑗 (21) the basic coordinated controller, if it can stabilize the closed loop pairwise subsystem 𝐒^𝐶_𝑖𝑗∶𝑥̇_𝑖𝑗=𝐴_𝐷𝑖𝑗−𝐵_𝐷𝑖𝑗𝐾_𝐷𝑖𝑗𝑥_𝑖𝑗,𝑦_𝑖𝑗=𝐶_𝐷𝑖𝑗𝑥_𝑖𝑗. (22) For every pair of subsystems 𝐒_𝑖𝑗 with information structure constraints 𝑖=𝑗−𝑘, 𝑗=2,3,…,𝑁, 𝑘=1,2,…,𝑗−1, their basic coordinated controllers can be constructed in this way.

As the fundamental idea of pairwise decomposition, a given system should be expanded following the recurrent reverse order first, so that a coordinated controller can be designed to stabilize all of the pairwise subsystems and then contracted to the original space. However, restricted by the mathematical framework of inclusion principle, it is difficult to expand the system with uncertainties in its dynamics. Consider the procedure of pairwise decomposition, the original states can be almost included in the block-diagonal expanded system which is composed of state functions of all pairwise subsystems, 𝐒_𝑃𝐷=𝐒₁₂,𝐒₂₃,𝐒₁₃,𝐒₃₄,𝐒₂₄,𝐒₁₄,…,𝐒_2𝑁,𝐒_1𝑁∶̇𝑥_𝑃=𝐴_𝑃𝐷𝑥_𝑃+𝐵_𝑃𝐷𝑢_𝑃,𝑦_𝑃=𝐶_𝑃𝐷𝑥_𝑃,𝐴_𝑃𝐷=blockdiag𝐴_𝐷𝑖𝑗,𝐵_𝑃𝐷=blockdiag𝐵_𝐷𝑖𝑗,𝐶_𝑃𝐷=blockdiag𝐶_𝐷𝑖𝑗. (23) This block-diagonal system is a reasonable approximate expansion of the origin. To achieve this form, the interconnection structure of system 𝐒 should be available, and this is also the restriction in using inclusion principle. The interconnection structure is supposed to be given by the fundamental interconnection matrix 𝐸=(𝑒_𝑖𝑗). By expanding the original space of system 𝐒 into a bigger space of system 𝐒_𝑃𝐷 in recurrent reverse order, take the state matrix as an example, the transformation matrices of pairwise decomposition can be selected as 𝑉=blockdiag𝐼_𝑛1𝐼_𝑛1⋯^{∑𝑁𝑘=1𝑒1𝑘}𝐼_𝑛1,…,𝐼_𝑛𝑁𝐼_𝑛𝑁⋯^{∑𝑁𝑘=1𝑒𝑁𝑘}𝐼_𝑛𝑁^𝑇,1𝑈=blockdiag∑^𝑁_𝑘=1𝑒_1𝑘^{∑𝑁𝑘=1𝑒1𝑘}𝐼_𝑛1⋯𝐼_𝑛11,…,∑^𝑁_𝑘=1𝑒_𝑁𝑘^{∑𝑁𝑘=1𝑒𝑁𝑘}𝐼_𝑛𝑁⋯𝐼_𝑛𝑁, (24)𝑅, 𝑄, 𝑇, and 𝑆 have the same structure as their counterparts, respectively. Notice that there are arbitrary choices of these transformation matrices, and their forms are bound up with the inclusion form. Since the structure of expanded system 𝐒_𝑃𝐷 is confirmed, then the transformation matrices are also fixed, just as (24).

Consider the permuted inclusion principle; the transformation matrices will be permuted as 𝑉_𝑃=𝑃_𝐴⁻¹𝑉,𝑈_𝑃=𝑈𝑃_𝐴,𝑅_𝑃=𝑃_𝐵⁻¹𝑅,𝑄_𝑃=𝑄𝑃_𝐵,𝑇_𝑃=𝑃_𝐶⁻¹𝑇,𝑆_𝑃=𝑆𝑃_𝐶. (25) Therefore the state matrices of systems 𝐒 and 𝐒_𝑃𝐷 are related by (12), and the relationship of the state, input, and output vectors can be obtained by Definition 2 as 𝑥=𝑈_𝑃𝑥_𝑃,𝑢=𝑄_𝑃𝑢_𝑃,𝑦=𝑆_𝑃𝑦_𝑃. (26)

At the same time, a virtual system 𝐒 can be constructed as another contraction of system 𝐒_𝑃𝐷. The state dynamic of system 𝐒 is in a certain form, and 𝐒 is raised as an estimation of the original system 𝐒. According to the contraction condition, it is possible to use the transformation matrix 𝑉_𝑃 such that system 𝐒_𝑃𝐷 can be contracted to 𝐒 by (12) after an appropriate compensation. This process will also lead to the same relationship as (26). In this way, systems 𝐒 and 𝐒 may share the same state, input, and output vectors, since they have the same expanded system 𝐒_𝑃𝐷 which is calculated by the same transformation matrices. It can be concluded that systems 𝐒 and 𝐒 represent a pair of systems with approximate dynamics, and the bias between them is mainly reflected in compensation 𝑀_𝐴𝑃 of the contraction procedure.

Suppose that the expanded system 𝐒_𝑃𝐷 comprises every state function of pairwise subsystem 𝐒_𝑖𝑗 in system 𝐒, and the pairwise subsystems are arranged in the recurrent reverse order as (23). Each 𝐒_𝑖𝑗 is stabilized by the basic coordinated controller (21); then the coordinated controller for 𝐒_𝑃𝐷 can be constructed in a block-diagonal form as 𝐂_𝑃𝐷=𝐂_𝑖𝑗∶𝑢_𝑃=−𝐾_𝐷𝑥_𝑃=−blockdiag𝐾_𝐷𝑖𝑗𝑥_𝑃. (27) It is clear that a redundant control set is established with all pairwise controllers, which contains all necessary coordinated information 𝐾_𝐷𝑖𝑗 for both system 𝐒 and 𝐒. When the structure form of the estimator 𝐒 is determined, the coordinated controller of system 𝐒 can be obtained by contracting 𝐾_𝐷 together with a proper compensator 𝑀_𝐾𝑃. The contraction is checked by the following theorem.

Theorem 4. For the systems mentioned above, system 𝐒_𝑃𝐷 is the expansion for both systems 𝐒 and 𝐒. The state feedback controller 𝐶∶𝑢=−𝐾𝑥 can stabilize the closed loop system of 𝐒, if the controller 𝐶 of system 𝐒 can be contracted from 𝐶_𝑃, and it satisfies 𝐾=𝐾=𝑄_𝑃𝐾_𝐷+𝑀_𝐾𝑃𝑉_𝑃. (28)

Proof. Since system 𝐒 is a contraction system 𝐒_𝑃𝐷, supported by the contraction condition (12), the state function of system 𝐒_𝑃𝐷 is rewritten as ̇𝑥_𝑃=𝐴_𝑃𝐷−𝑀_𝐴𝑃𝑥_𝑃+𝐵_𝑃𝐷𝑢_𝑃, (29) and it apparently indicates the controller form of (5). According to the inclusion principle, here goes 𝑥=𝑈_𝑃𝑥_𝑃,𝑢=𝑄_𝑃𝑢_𝑃. (30) moreover, the approximation between systems 𝐒 and 𝐒 may indicate that 𝑥=𝑥 and 𝑢=𝑢, so that the controllers of systems 𝐒 and 𝐒_𝑃𝐷 are related as 𝑢=−𝐾𝑥,𝑢_𝑃=−𝐾_𝐷+𝑀_𝐾𝑃𝑥_𝑃. (31)
Notice that 𝑈_𝑃𝑉_𝑃=𝐼_𝑛, then (30) and (31) will conclude that 𝐾=𝑄_𝑃(𝐾_𝐷+𝑀_𝐾𝑃)𝑉_𝑃.

Remark 5. The literature [6] provides a sufficient condition of connective stability. But since far more information might be accumulated in the largest singular values of subsystem matrices, the criterion of connective stability might be somewhat conservative.

The virtual system 𝐒 is used as the estimator of system 𝐒, and it may have many possible forms. This diversity will mainly impact on the controller design process in determining the compensator 𝑀_𝐾𝑃. One of the most challenging problems in controller design for multiagent systems is the estimation of information structures among agents. The further research of this paper on implementation of pairwise decomposition in systems with dynamic information structure constraints, as well as the estimation of the interconnection structure, is undergoing. This issue is based on inclusion principle for time-varying system ([18]) and method in dealing with the structure perturbation under the concept of pairwise decomposition ([8]). However, in a particular case when the state function of each subsystem satisfies the linear superposition principle, there is a way to determine the structure of 𝑀_𝐾𝑃 much more easily.

Consider the mathematical framework of permuted inclusion principle; this position information can be concluded with (8) by using the block row-order of sub-identity matrices. Consider that system 𝐒 is in full network structure, the row-order of a particular pairwise subsystem 𝐒_𝑖𝑗 is 𝑐_𝑖=𝑗𝑗−1−2(𝑖−1)−1,𝑐_𝑗=𝑗𝑗−1−2(𝑖−1). (32) Besides, the row-order of every pairwise subsystem can also be concluded in this way as 𝑐_𝑖𝑘𝑖=𝑘_𝑖𝑘_𝑖−1−2(𝑖−1)−1,𝑖<𝑘_𝑖𝑖(𝑖−1)−2𝑘_𝑖−1,𝑖>𝑘_𝑖𝑐_𝑗𝑘𝑗=𝑘_𝑗𝑘_𝑗−1−2𝑗−1−1,𝑗<𝑘_𝑗𝑗𝑗−1−2𝑘_𝑗−1,𝑗>𝑘_𝑗𝑖,𝑗=1,2,…,𝑁,𝑘_𝑖,𝑘_𝑗=1,2,…,𝑁,𝑖≠𝑗,𝑘_𝑖,𝑘_𝑗≠𝑖,𝑗. (33) so that the complementary matrix 𝑀_𝐾𝑃 can be constructed by the following lemma.

Lemma 6. Suppose that system 𝐒 is in full network structure; the row-order of subsystems in each pairwise subsystem is concluded as (32) and (33), and 𝑀_𝐾𝑃 complements the information structure constraints bias between system 𝐒 and system 𝐒. Then 𝑀_𝐾𝑃 can be presented by the information structure of corresponding pairwise subsystem 𝐒_𝑖𝑗 as 𝑀_𝐾𝑃𝑐_𝑖𝑘𝑖,𝑐_𝑗𝑏=𝐾_𝑖𝑗,𝑀_𝐾𝑃𝑐_𝑗𝑘𝑗,𝑐_𝑖𝑏=𝐾_𝑗𝑖𝑖,𝑗=1,2,…,𝑁,𝑘_𝑖,𝑘_𝑗=1,2,…,𝑁,𝑖≠𝑗,𝑘_𝑖,𝑘_𝑗≠𝑖,𝑗. (34)

This matrix structure-based lemma is convenient for calculation, especially for real-time control in practice.

Example 7. Also consider system 𝐒 with 𝑁=3 subsystems, and its recurrent reverse order is presented as (5). According to (32) and (33), the calculation is proceeded as 𝐒₁₂∶𝑐_𝑖=1,𝑐_𝑗𝑘=2,_𝑖=𝑘_𝑗=3,𝑐_𝑖𝑘𝑖=5,𝑐_𝑗𝑘𝑗⟹𝑀=3,_𝐾𝑃(5,2)_𝑏=𝐾₁₂,𝑀_𝐾𝑃(3,1)_𝑏=𝐾₂₁𝐒₂₃∶𝑐_𝑖=3,𝑐_𝑗𝑘=4,_𝑖=𝑘_𝑗=1,𝑐_𝑖𝑘𝑖=2,𝑐_𝑗𝑘𝑗⟹𝑀=6,_𝐾𝑃(2,4)_𝑏=𝐾₂₃,𝑀_𝐾𝑃(6,3)_𝑏=𝐾₃₂𝐒₁₃∶𝑐_𝑖=5,𝑐_𝑗𝑘=6,_𝑖=𝑘_𝑗=2,𝑐_𝑖𝑘𝑖=1,𝑐_𝑗𝑘𝑗⟹𝑀=4,_𝐾𝑃(1,6)_𝑏=𝐾₁₃,𝑀_𝐾𝑃(4,5)_𝑏=𝐾₃₁. (35) So that the matrix 𝑀_𝐾𝑃 can be constructed by Lemma 6: 𝑀_𝐾𝑃=00000𝐾₁₃000𝐾₂₃𝐾00₂₁000000000𝐾₃₁00𝐾₁₂000000𝐾₃₂000. (36)

4. Automatic Generation Control (AGC) for a Four-Area Power System

A four-area power system is shown in Figure 1; assume that areas 1, 2, and 3 contain three reheat turbine type thermal units and area 4 contains a hydro unit, respectively. Each pairwise subsystem is interconnected by tie line indicated by solid lines, and its information structure constraint is indicated by dotted ellipse. Details of the system description can be found in [19, 20]. References [4–8] implement the pairwise decomposition methodology in the procedure of coordinated control to this four-area power system AGC. As a counterpart, the controller design procedure of the new pairwise decomposition modality in this paper is presented here. Consider the system dynamic bias between system 𝐒 and system 𝐒 taken as approximation; each pairwise subsystem is robustly stabilized in terms of linear matrix inequalities (LMI) ([21, 22]).

Figure 1 

Schematic diagram of the four-area power system.

Suppose that the system graph is undirected and 𝑒_𝑖𝑗(𝑡)=𝑒_𝑗𝑖(𝑡)=1 for description convenience. The pairwise subsystem model is provided in (17) as 𝐒_𝑖𝑗∶𝑥̇_𝑖𝑥̇_𝑗=𝐴_𝑖𝑖𝐴_𝑖𝑗𝐴_𝑗𝑖𝐴_𝑗𝑗𝑥_𝑖𝑥_𝑗+𝑤_𝑖𝑤_𝑗+𝐵_𝑖𝑖00𝐵_𝑗𝑗𝑢_𝑖𝑢_𝑗+𝐺_𝑖𝑖00𝐺_𝑗𝑗𝜉_𝑖𝜉_𝑗,𝑦_𝑖𝑦_𝑗=𝐶_𝑖𝑖00𝐶_𝑗𝑗𝑥_𝑖𝑥_𝑗𝑖,𝑗=1,2,…,𝑁,𝑖≠𝑗, (37) where 𝑥_𝑖𝑗=[𝑥^𝑇_𝑖,𝑥^𝑇_𝑗]^𝑇, 𝑢_𝑖𝑗=[𝑢^𝑇_𝑖,𝑢^𝑇_𝑗]^𝑇, 𝜉_𝑖𝑗=[𝜉^𝑇_𝑖,𝜉^𝑇_𝑗]^𝑇, and 𝑦_𝑖𝑗=[𝑦^𝑇_𝑖,𝑦^𝑇_𝑗]^𝑇 are the state, control input, disturbance, and output vectors, respectively. 𝑤_𝑖𝑗=[𝑤^𝑇_𝑖,𝑤^𝑇_𝑗]^𝑇 is the unmodeled or uncertain state dynamic. 𝐴_𝐷𝑖𝑗=(𝐴_𝑖𝑗), 𝐵_𝐷𝑖𝑗=(𝐵_𝑖𝑖), 𝐺_𝐷𝑖𝑗=(𝐺_𝑖𝑖), and 𝐶_𝐷𝑖𝑗=(𝐶_𝑖𝑖) are system matrices with proper dimensions, respectively. The numerical values of the system matrices are given by 𝐴₁₁=,𝐴−0.056000−60−0.1−1.011.110000−3.333.3300−2.0800−550−0.2550000−0.061.3300000₁₂=0_5×10_5×5−0.4440_1×5,𝐴₁₃=0_4×10_4×40_4×100_1×40.06−0.4440_1×40,𝐴₁₄=𝐴₂₁=𝐴₃₂=𝐴₁₃,𝐴₂₃=A₃₁=𝐴₄₁=𝐴₁₂,𝐴₄₄=−0.07696.15000−6.158.78×10⁻⁴−22.2−0.1982.11×10⁻³0−4.39×10⁻⁴0−0.10.09891.05×10⁻³0−8.56×10⁻³,𝐵00−0.02050.02050−0.02550000−0.060.44400000₁₁=𝐵₂₂=𝐵₃₃=000500^𝑇,𝐵₄₄=0000.020500^𝑇,𝐺₁₁=𝐺₂₂=𝐺₃₃=−600000^𝑇,𝐺₄₄=−6.1500000^𝑇,𝐶₁₁=𝐶₂₂=𝐶₃₃=𝐶₄₄=𝐼₆. (38)𝐴₂₂ and 𝐴₃₃ are the same as 𝐴₁₁ except that 𝐴₂₂(6,1)=𝐴₃₃(6,1)=0.888, respectively.

The quadratic constraints are imposed on the state uncertainty 𝑤_𝑖𝑗 as follows: 𝑤^𝑇_𝑖𝑗𝑤_𝑖𝑗≤𝛼²_𝑖𝑗𝑥^𝑇_𝑖𝑗𝑊^𝑇_𝑖𝑗𝑊_𝑖𝑗𝑥_𝑖𝑗, (39) where 𝛼_𝑖𝑗 is a positive number to be maximized and 𝑊_𝑖𝑗 is a full rank constant matrix; it can be set as an identity matrix in case that the information of uncertainty is unavailable. Pairwise subsystem (37) is robustly stabilizable with arbitrarily large basic coordinated controller degree 𝛼 by the basic coordinated controller (21), a LMI problem that can be obtained as minimize𝛾_𝑖𝑗subjectto𝑌_𝑖𝑗𝐴>0_𝐷𝑖𝑗𝑌_𝑖𝑗+𝑌_𝑖𝑗𝐴^𝑇_𝐷𝑖𝑗+𝐵_𝐷𝑖𝑗𝐿_𝑖𝑗+𝐿^𝑇_𝑖𝑗𝐵^𝑇_𝐷𝑖𝑗𝐴_𝐷𝑖𝑗𝑌_𝑖𝑗𝑊^𝑇_𝑖𝑗𝐴^𝑇_𝐷𝑖𝑗𝑊−𝐼0_𝑇𝑖𝑗𝑌_𝑖𝑗0−𝛾_𝑖𝑗𝐼<0, (40) where 𝐿_𝑖𝑗=𝐾_𝐷𝑖𝑗𝑌_𝑖𝑗 and 𝛾_𝑖𝑗=1/𝛼²_𝑖𝑗. Further details of this robust control procedure can be found in [21, 22].

According to the permuted inclusion principle, the expanded system 𝐒_𝑃 is supposed to contain those particular pairwise subsystems listed as following: 𝐒_𝑃=𝐒₁₂,𝐒₂₃,𝐒₁₃,𝐒₁₄. (41) Then compose the coordinated controller of system 𝐒_𝑃 by arranging the basic coordinated controllers of pairwise subsystems 𝐒₁₂, 𝐒₂₃, 𝐒₁₃, and 𝐒₁₄ in this exact recurrent reverse order, 𝐾_𝐷=blockdiag𝐾_𝐷12,𝐾_𝐷23,𝐾_𝐷13,𝐾_𝐷14𝐾=blockdiag¹¹𝐾₁₂𝐾₂₁𝐾₂₂,𝐾₂₂𝐾₂₃𝐾₃₂𝐾₃₃,𝐾₁₁𝐾₁₃𝐾₃₁𝐾₃₃,𝐾₁₁𝐾₁₄𝐾₄₁𝐾₄₄. (42)

Choose the transformation matrices by (24) 𝑉=blockdiag𝐼^𝐴₁𝐼^𝐴₁𝐼^𝐴₁^𝑇,𝐼^𝐴₂𝐼^𝐴₂^𝑇,𝐼^𝐴₃𝐼^𝐴₃^𝑇,𝐼^𝐴₄1𝑄=blockdiag3𝐼^𝐵₁𝐼^𝐵₁𝐼^𝐵₁,12𝐼^𝐵₂𝐼^𝐵₂,12𝐼^𝐵₃𝐼^𝐵₃,𝐼^𝐵₄, (43) and the permutation matrix 𝑃_𝐴 can be constructed by (2); 𝑃_𝐵 is in the same structure as 𝑃_𝐴, 𝑃_𝐴=000000𝐼^𝐴₁00000𝐼^𝐴₁𝐼000_𝐴100000000𝐼^𝐴₂00000000𝐼^𝐴₂00000000𝐼^𝐴₃000000000𝐼^𝐴₃000000000𝐼^𝐴₄, (44) where the dimensions of 𝐼^𝐴_𝑘 and 𝐼^𝐵_𝑘 are determined by the system state and control input vector, respectively. In this simulation example, 𝐼^𝐴_𝑘=𝐼_6×6, 𝐼^𝐵_𝑘=1.

Use (34) to construct the complementary matrix 𝑀_𝐾𝑃 as follows: 𝑀_𝐾𝑃=00000𝐾₁₃0𝐾₁₄000𝐾₂₃𝐾0000₂₁00000000000𝐾₃₁0000𝐾₁₂00000𝐾₁₄00𝐾₃₂000000𝐾₁₂000𝐾₁₃0000000000. (45) Finally, the coordinated controller of 𝐒 can be contracted as 𝐾=𝐾=𝑄_𝑃𝐾_𝐷+𝑀_𝐾𝑃𝑉_𝑃=𝐾₁₁𝐾₁₂𝐾₁₃𝐾₁₄𝐾₂₁𝐾₂₂𝐾₂₃0𝐾₃₁𝐾₃₂𝐾₃₃0𝐾₄₁00𝐾₄₄. (46)

Figures 2 and 3 illustrate the frequency and tie-line power perturbations of the group of subsystems. The respond curves are very similar to those of [4–7].

(a)

(b)

(c)

(d)

Figure 2 

Deviations of frequency among the power system.

(a)

(b)

(c)

(d)

Figure 3 

Deviations of tie-line power among the power system.

5. Conclusion

This paper presents a theoretical study of the pairwise decomposition, which can be seen as a reverse modality of this methodology. The proposed approach is able to coordinated the interconnected system with uncertainties, and it can achieve high quality control performance as well. Moreover, this process is convenient for a group of interconnected subsystems without a superposition-form overall system model, which is in the case that only local information is available. Further research is ongoing, and one task is to determine the structure of system 𝐒 as an estimator of the original system 𝐒. For this purpose, an update law which can fit the features of pairwise decomposition is needed as well as a calculation framework to deal with the structure perturbations effectively enough. The proposed approach can also motivate the application of pairwise decomposition to a nonlinear time-variant system.

Acknowledgment

This research reported herein was supported by the NSF of China under Grant no. 60874017.