Distributed static output feedback control for interconnected systems in finite‐frequency domain

Funding information National Natural Science Foundation of China, Grant/Award Number: 61673218 Abstract This paper aims at designing a distributed static output feedback (DSOF) controller for the interconnected system in finite-frequency domain. Motivated by the fact that the signal transmitting process is usually affected by surrounding environments, the interconnection communication between subsystems are assumed to be non-ideal. By introducing some dilated multipliers into the extended condition that guarantees the well-posedness, stability and finite-frequency ∞ performance of non-ideally interconnected systems, a new equivalent condition is derived, which is more effective for parameterizing the DSOF controller. A two-stage approach is then developed for constructing a DSOF controller: design an initial distributed full information (DFI) controller first and then derive a desired DSOF from the DFI controller. Moreover, two iterative linear matrix inequality based algorithms are proposed to improve the solvability of the DFI controller and DSOF controller design problems. Finally, an example is given to show the effectiveness of the proposed two-stage DSOF controller design method.

, authors proposed a state-space description for the interconnected system and analyzed the well-posedness, stability, contractiveness and distributed control of such interconnected system. On this basis, considerable works on distributed control of interconnected systems were carried out, such as  2 distributed control [6], distributed finite-time control [7], distributed control of discrete-time interconnected systems [8][9][10] etc.
As a significant component of control theory, static output feedback (SOF) control problem has been extensively studied in recent years [11]. Compared with the dynamic output feedback controller, SOF controller has a simpler structure and is less expensive to be implemented. Moreover, it is shown in [12] that the dynamic output feedback control problem can be transformed into an SOF control problem. The SOF control problem is essentially a bilinear matrix inequality problem and generally considered to be NP-hard. In the last two decades, we have witnessed lots of research on SOF controller design for linear time-invariant systems through linear matrix inequality (LMI) based methods, to name a few, iterative LMI methods [13,14], direct LMI methods with extra constraints [15,16]. Recently, the two-stage approach has been shown to be a promising method for dealing with the SOF controller design problem [17][18][19][20]. The idea of the two-stage approach is finding a state feedback (SF) controller at the first stage and then explore an SOF controller from the obtained SF controller at the second stage. For interconnected systems, most efforts have been devoted to decentralized SOF controller design till now but few research is focused on distributed SOF (DSOF) controller design. Thus, in this paper, one of our motivations is to extend the two-stage approach for designing a DSOF controller for the interconnected system.
Moreover, it is notable that most efforts so far have been devoted to controller design for the interconnected system in the entire-frequency (EF). Actually, in some real applications, there is no need to consider the system properties and design specifications in the EF range, as many systems may just belong to certain frequency range and the energies of many signals concentrate only in some finite-frequency (FF) ranges [21]. The generalized Kalman-Yakubovich-Popov (GKYP) lemma establishes the equivalence between an infinite number of frequency domain inequalities characterizing the properties of a transfer function in a finite-frequency range and a numerically tractable linear matrix inequality for its state space realization, which enables one to deal with the restricted frequency-domain specifications [22][23][24][25][26][27][28]. Based on the GKYP lemma developed for interconnected systems, a great quantity of results on control synthesis with restricted frequency-domain specifications have been obtained [7,[29][30][31][32]. However, no research on frequencydomain constrained feedback controller design for interconnected systems has been reported, which is another motivation of our work.
In this paper, we are going to study the DSOF controller design problem for interconnected systems subject to restricted frequency-domain specifications. Based on the GKYP lemma, an efficient two-stage approach is developed for constructing an FF DSOF controller. Moreover, for improving the solvability of the developed two-stage approach, we look into the conditions in depth and propose iterative LMI algorithms for getting a desired initial distributed full information controller and exploring the DSOF controller. Finally, an example is given to illustrate the validity and effectiveness of the proposed methods. The contributions of this paper are threefold: (1) Compared with the existing results based on a common communication channel [6,10,31], the communication channels between subsystems we considered in this paper are affected by uncertainty and attenuation, which is more general and can reflect the reality. (2) By introducing some dilated multipliers, we derive a new equivalent condition for characterizing the performance of interconnected systems with a finite-frequency constraint, which is more effective for parameterizing the DSOF controller. (3) A two-stage approach is proposed, as a first attempt, to design a DSOF for the interconnected system subject to restricted frequency-domain specifications, which fills the blank of distributed static output feedback controller design for interconnected systems.
The remainder of this paper is organized as follows. The system description and some primaries are given in Section 2. Section 3 presents the two-stage approach of designing the DSOF controller for interconnected systems. Section 4 gives a way to construct a desired initial distributed full information controller. An example is included in Section 5 to demonstrate the validity and effectiveness of the derived results. Section 6 concludes this paper.
Notations. The sets of real numbers and nonnegative integers are denoted by ℝ and ℤ + , respectively. ℂ n denotes ndimensional complex space. ℂ m×n and ℍ n refer to the m × n complex matrix set and n × n Hermitian matrix set, respectively. For a matrix A ∈ ℂ m×n , A * denotes its complex conjugate transpose; represents a permutation matrix where the i-th block of row-partitioned matrix  i is an appropriately dimensioned identity matrix. The "⋆" in a matrix block denotes the transposition of its conjugate symmetric term.

SYSTEM DESCRIPTION AND PROBLEM FORMULATION
Consider the interconnected system G which is composed of L heterogeneous subsystems and the subsystems are interconnected over a communication network. The topology of such interconnected system can be modeled by an undirected graph  = (,  ). The vertex set  of the graph is defined as  = {G i , i = 1, … , L}, where G i represents i-th subsystem and has the following dynamics: where t ∈ ℝ + ; x i (t ) ∈ ℝ m i , z i (t ) ∈ ℝ q i , d i (t ) ∈ ℝ p i , y i (t ) ∈ ℝ r i and u i (t ) ∈ ℝ s i denote the state, performance output, external disturbance, measured output and control input of subsystem G i , respectively; v i (t ), w i (t ) ∈ ℝ n i are the input signal from the neighboring subsystems and output signal to neighbors, respectively. The set of undirected edges is defined by  := {(G i , G j ), i, j = 1, … , L} and represents the interconnection between subsystems. The weight of edge (G i , G j ) is an integer n i j . The situation that the subsystem G i and G j are not interconnected can also be captured by allowing n i j = 0. It is assumed that vector v i (t ) and w i (t ) can be subdivided into where v i j (t ) and w i j (t ) are the internal input and output transferring between subsystem G i and G j . Moreover, consider the interconnection relationships between subsystems to be } denotes a class of systems which capture the nature of the interconnection between G i and G j , such as being lossy, affected by possible uncertainties or attacked by adversaries in the transmission. Notably, the situation that the interconnection is ideal, that is, Δ i j = I , is included in the this interconnection relationship. Figure 1 gives an intuitive view of the interconnected system with L = 7 subsystems. In this paper, we are expected to design a distributed static output feedback controller K with sub-controller K i being where v k i (t ), w k i (t ) ∈ ℝ n i are the input signal from the neighboring sub-controllers and output signal to neighbors, respectively. The interconnection relationship between sub-controllers follows with the one between subsystems, that is, v k i j (t ) = Δ ji w k ji (t ). In Figure 2, we have depicted the distributed control structure of the interconnected system for a clear illustration. Substituting the distributed controller into the interconnected system leads to the closed-loop systemG with subsys-temG i being and the system matrices are shown as follows: Then the transfer function matrix of closed-loop systemG can be compactly given as: := diag{I m , T * diag{P * , P * }T }, ∑L i=1 n i and permutation matrices P, T , respectively, satisfying Finally, the finite-frequency distributed static output feedback controller design problem can be expressed as follows: 1. The closed-loop systemG with subsystemG i in (4) is wellposed and asymptotically stable; 2. The transfer function matrix G ( j ) of closed-loop system G satisfies where is a positive scalar and Ω := { ∈ ℝ : l ≤ ≤ h , l , h ∈ ℝ + } represents a finite-frequency region.
Remark 1. The construction of the interconnected system and interconnection relation may not always be well-defined as the signals satisfying the interconnection may not exist or be unique. Thus we require that the closed-loop system is well-posed, which means that the system is physically realizable.
Remark 2. In this paper, the finite-frequency region Ω in (7) is called middle frequency range, which also includes the low frequency range. The frequency range here can be substituted with high frequency, that is, Ω = { ∈ ℝ : | | ≥ h , h ∈ ℝ + }, or even the entire-frequency, and the corresponding analysis results are similar to those in following sections. Therefore, the FF controller design problem is a generalization of the  ∞ control problem [5]. For more discussion on the frequency range, the readers can refer to [22,31].
Before closing this section, we present the following lemma for continuous-time non-ideally interconnected system G , which can be reviewed as an extension of Theorem 2 in [31].

Lemma 1.
Consider the closed-looped systemG and let scalars l , h ∈ ℝ be given. The closed-looped systemG is well-posed, stable and satisfies the finite-frequency specification (7) if there exist symmetric matrices and (Z 11 Proof. The proof of Lemma 1 is close to the one of Theorem 2 in [31] and thus is omitted here. □ Remark 3. Lemma 1 is the continuous-time version of the result given in [31] for discrete-time interconnected systems. In addition, the difference between the proof of Lemma 1 and the one of Theorem 2 in [31] lies in that (Z 11 i ) C and ( 11 i ) C here should be positive definite as the interconnection relation between systems are non-ideal.

CONSTRUCTION OF DISTRIBUTED STATIC OUTPUT FEEDBACK CONTROLLERS
In this section, we will develop a sufficient condition, which is equivalent to the one in Lemma 1, for assuring the wellposedness, stability and finite-frequency  ∞ performance of the closed-loop system firstly. Then based on this condition, we will present a construction method of a DSOF controller which guarantees the required specifications.

Multiplier-based relaxation
In order to develop a construction method of a DSOF controller, we first introduce some matrices and then the conditions (8) and (9) in Lemma 1 and be dilated into the following matrix inequalities, respectively: where Next, based on the dilated conditions in (10) and (11), we now give conditions which are equivalent to (8) and (9) in Lemma 1.

Theorem 1.
Consider the closed-loop systemG with transfer function matrix G ( j ) given in (6). Let scalars l , h ∈ ℝ be given. There exist matrices such that the conditions (8) and (9) in Lemma 1 are satisfied if and only if there exist symmetric matrices } , } , Proof. (⇐): Pre-multiplying and post-multiplying the matrix inequality (12) by Υ ⟂ * i and Υ ⟂ i , and pre-multiplying and postmultiplying the matrix inequality (12) by Υ s i ⟂ * and Υ s i ⟂ , respectively, we can get Note that the null spaces of Υ i and Υ s i can be, respectively, selected as Then we can find that the condition (16) and (17) are exactly the conditions (8) and (9) in Lemma 1. Thus the sufficiency has been proved. (⇒): Definẽ .
Then it can be found that Thus the inequalities (8) and (9) can be equivalently represented as According to Finsler's lemma [33], we have that (18) and (19) are, respectively, equivalent to Then it is easy to see that there must exist a sufficiently large scalar such that the following inequalities hold: Note that the above inequalities can be, respectively, rewritten as: Thus by assigning the matrix variables in (12) and (13) as follows: we can finally see that (20) and (21) are exactly the same as (12) and (13), respectively. Thus the proof is completed. □ Remark 4. The product terms in Lemma 1 that the unknown controller parameter lies between two system matrices are the main difficulties when solving the DSOF controller synthesis problem. In Theorem 1, we have separated these product terms for computing the DSOF controller.

Computation of the DSOF controller
Based on Theorem 1, we will provide the following theorem for parameterizing a desired DSOF controller.

Theorem 2. Consider the interconnected system G and let scalars
l , h ∈ ℝ be given. There exist a DSOF controller such that the closed-loop systemG is well-posed, asymptotically stable and meets the finite-frequency specification (7) if there exist symmetric matri- for all i = 1, … , L, where

Moreover, if the previous conditions are satisfied, a realization of the DSOF controller can be obtain by
Proof. Firstly, since [R i lk ] 2×2 > 0 can be derived from (22) (22) and (23) are satisfied, the conditions in (12) and (13) are also satisfied by setting the gain of the DSOF controller as (25). Thus according to Theorem 1, the DSOF controller with its gain given in (25) can guarantee the wellposedness, stability and the finite-frequency specification in (7). The proof is completed. □ Note that the conditions in (22) and (23) are not LMIs, as there still exists the multiplication between unknown matrices R i lk and K i pq . Thus if the matrices K i pq can be specified a priori, the conditions in (22) and (23) will turn into LMIs with respect to other variables. Fortunately, it is not difficult to find that the matrices K i pq can be interpreted as the system matrices of a distributed full information controller which can guarantee the well-posedness, stability and the finitefrequency specification in (7) of the interconnected system G .
In the following, we will assume that matrices K i pq are known and give an algorithm (Algorithm 1) for designing a DSOF controller.
Remark 5. It can be observed from Algorithm 1 that Algorithm 1 gives a procedure of computing a desired DSOF controller when there exists a priori known distributed full information controller. Thus, it is the main part of the second stage. In next section, we will focus on constructing and optimizing a distributed full information controller given in (26) for Algorithm 1, in other words, the first stage.

CONSTRUCTION OF INITIAL DISTRIBUTED FULL INFORMATION CONTROLLERS
Since not all choices of [K i pq ] 2×4 can lead to a desired DSOF controller through Algorithm 1, it is significant to find a suitable [K i pq ] 2×4 which has more possibility to produce a desired DSOF controller. Thus in this section, we will focus on constructing and optimizing a initial distributed full information controller (26) for Algorithm 1.

Computation of a initial controller
First, we have the following result on the existence of a distributed full information controller which can guarantee the well-posedness, stability and the finite-frequency specification in (7) of the closed-loop system. Algorithm 1 Design of the DSOF controller 1. Find matrices K i pq , p = 1, 2, q = 1, … , 4, i = 1, … , L such that the distributed full information controller in (26) can guarantee the well-posedness, stability and the finite-frequency specification in (7) of the interconnected system G . Let be a specified tolerance and set = 1.

2-1. Solve the following optimization problem for
If ( ) 1 < 0, then the controller with its gain given by (25) If ( ) 2 < 0, then the controller with its gain given by (25)

Theorem 3. Consider the interconnected system G and let scalars
l , h ∈ ℝ be given. There exist a distributed full information controller such that the resulting closed-loop system is well-posed, stable and satisfies the finite-frequency specification (7) if there exist symmetric matri- for all i = 1, … , L, wherē } , , ] ,

Proof.
First, pre-multiply and post-multiply inequality and its transpose, respectively, and we can get where Then, pre-multiply and post-multiply inequality (29) by [(Ξ 1 i ) * I ]T i and its transpose, respectively, we have Analogously, pre-multiply and post-multiply inequality (28) by and its transpose, respectively, we have Note that, as the controller is replaced by the distributed full information controller in (26) (29), (30) and Lemma 1, the closed-loop system resulting from the distributed full information controller in (26) is well-posed, stable and satisfies the finite-frequency specification (7). The proof is completed. □

Optimization of the initial controller
In Theorem 3, we have constructed a distributed full information controller as the initial controller for Algorithm 1. However, here comes the question: Is the constructed distributed full information controller a suitable choice for generating a desired DSOF controller? Taking a deep sight into conditions (22) and (23) in Theorem 2, we can find that a distributed full information controller with the following form: could be an ideal initial controller. In the following, we will present an algorithm to generate a new initial controller based on the constructed distributed full information controller in Theorem 3. First, we give the following theorem which will be used to update the initial controller.
Theorem 4. Given a distributed full information controller produced by Theorem 3, there exist symmetric matrices X i (22) and (23)  Proof. Suppose that a distributed full information controller with system matrices [K i pq ] 2×4 is derived from Theorem 3. According to the proof of Theorem 3, there exist  i T ,  i T ,  i and  i such that (29) is satisfied for all i = 1, … , L. Then there exist a sufficiently large scalar such that inequality (32) is satisfied.
Note that the terms in (22) are replaced as stated above. Hence, the condition in (32) is that in (22) by assigning: Analogously, it can be proved that there exist matrices such that (23) is also satisfied. Therefore, the distributed full information controller produced by Theorem 3 guarantees that there exist appropriate matrices such that (22) and (23) are satisfied. The proof of this theorem is completed. □ The basic thought of Algorithm 1 is to produce a DSOF controller with a known distributed full information controller. With some modifications, Algorithm 1 can also be used to update the initial distributed full information controller. Theorem 4 shows that, if the distributed full information controller generated from Theorem 3 is used as the initial controller, the optimization problem in Step 2-1 of Algorithm 1 with L i 11 C i Ty ,  2×4 . Moreover, noting that we are expected to produce a controller with its form given in (31), which is equivalent to Algorithm 2 Optimization of the initial controller 1-1. Construct an initial distributed full information controller [(K i pq ) (1) ] 2×4 according to Theorem 3. Let be a specified tolerance and set = 1.
we add the following constraints to Step 2-1 and Step 2-2 to optimize [ i pq ] 2×4 and [K i pq ] 2×4 , respectively, Then, we have Algorithm 2 for optimizing the initial controller. Algorithm 2 and Algorithm 1 together provides a way to construct a DSOF controller such that the resulting closed-loop systemG is well-posed, stable and satisfies the finite-frequency specification (7).

A PRACTICAL EXAMPLE
In this section, we will apply the proposed method to the control of a vehicle platoon which simulates a spring-massdamper system. Consider the situation that the transmissions of the position and velocity information between vehicles are not ideal. The system dynamics can be given as follows: where e i is the error between equilibrium position and realtime position of i-th vehicle; k i , c i , m i are the stiffness coefficient, damping constant, mass of vehicle, respectively; u i is the power provided by vehicle engine and is the control signal; y i is the measured output; 1 i, j ∈ (0, 1) is used to describe the non-ideal transmission of the position and velocity information between vehicles i and j . Moreover, as in horizontal direction, the body is sensitive to acceleration in the frequency range 1-2 Hz, we define the controlled output as z i =ë i . Our goal is to improve the driving comfort by reducingë i in the frequency range 1-2 Hz. Denoting x 1 i (t ) := e i , x 2 i (t ) :=̇e i and taking the noises during the movement of vehicle into consideration, the above vehicle platoon system can be concerted into the an interconnected system with subsystems given in (36) at the top of next page, and the interconnection relationship follows v i, j = j,i w j,i .
In this example, we consider there are L = 5 vehicles and i, j = 0.9 for all i, j = 1, … , 5. The masses, stiffness coefficients and damping constants are randomly chosen as follows: In addition to the finite-frequency specification in the frequency range Ω = [1, 2] × 2 , that is, ‖G ( j )‖ Ω ∞ < , we also include another constraint ‖G ( j )‖ ∞ < ∞ to avoid a bad disturbance attenuation performance in the entire-frequency range. We let = 0.07 and ∞ = 0.11.
Firstly, according to Theorem 3, we find a distributed full information controller as and other gain matrices are approximately zero matrices. Computing the i in (33) with [K i pq ] 2×4 shown in (37), we have ‖ 1 ‖ = 8.08, ‖ 2 ‖ = 11.29, ‖ 3 ‖ = 10.61, ‖ 4 ‖ = 10.21 and ‖ 5 ‖ = 7.36, which indicate that the distributed full information controller in (37) may not be a good initial controller for Algorithm 1. Now, based on the above controller, we run Algorithm 2 with = 0.01 and get a new distributed full information controller as and other gain matrices are approximately zero matrices. Computing the i in (33) with [K i pq ] 2×4 in (38), we have ‖ 1 ‖ = 1.18 × 10 −6 , ‖ 2 ‖ = 1.06 × 10 −6 , ‖ 3 ‖ = 2.18 × 10 −6 , ‖ 4 ‖ = 2.12 × 10 −6 and ‖ 5 ‖ = 1.77 × 10 −6 , which are much less than those obtained with [K i pq ] 2×4 shown in (37), that is, the distributed full information controller in (38) produced by Algorithm 2 has the excepted form presented in (31) and is more suitable than the controller in (37) to be the initial controller for Algorithm 1. Then, using this distributed full information controller in (38) as the initial controller for Algorithm 1, a desired DSOF controller is produced as and other gain matrices are approximately zero matrices. In order to show the advantage of controller design in finitefrequency range, we also design a conventional  ∞ DSOF which satisfies ‖G ( j )‖ ∞ < 0.11. The obtained  ∞ DSOF is produced by Algorithm 1 and Algorithm 2 as and other gain matrices are approximately zero matrices. In Figure 3, we have shown the maximum singular values of open-loop system, the closed-loop system with finitefrequency controller in (39) and the closed-loop system with entire-frequency controller in (40). From Figure 3, we can see that the closed-loop system with the finite-frequency DSOF controller (39) has a least gain over frequency range 1-2 Hz, which implies that the finite-frequency DSOF controller (39) can provide a better drive comfort. Moreover, in order to further illustrate the effectiveness of the finite-frequency DSOF controller (39), we assume that the first and third vehicles suffer the following disturbance signals: with A 1 = 0.3 and A 3 = 0.5. The frequency of the above disturbance is 1.25 Hz, which is covered by 1-2 Hz. We can see from Figure 4 that the horizontal acceleration of closed-loop system with finite-frequency DSOF controller is the smallest