OPTIMAL SPARSE OUTPUT FEEDBACK FOR NETWORKED SYSTEMS WITH PARAMETRIC UNCERTAINTIES

. This paper investigates the design of block row/column-sparse distributed static output H 2 feedback control for interconnected systems with polytopic uncertainties. The proposed approach is applicable to the networked systems with publisher/subscriber communication topology. We added two ad- ditional terms into the optimisation index function to penalise the number of publishers and subscribers. To optimally select a subset of available publishers and/or subscribers in the network, we introduced both an explicit scheme and an iterative process to handle this problem. We demonstrated the eﬀectiveness by using a numerical example. The example showed that the simultaneous identiﬁcation of favourable networks topologies and design of controller strategy can be achieved by using the proposed method.

1. Introduction. The modern complex systems, e.g., water distribution system, transportation systems, and power distribution networks, are often treated as interconnected large-scale systems, for which the decentralized and distributed control schemes have been presented. The major constraint of the decentralised control is only the local state information is available to stabilize and regulate the subsystems. Thus there is no control network for decentralized control configuration. This can be effective only when the interconnections between the subsystems are not strong [14,20,15]. When the interconnections cannot be overlooked, the distributed control framework is often adopted. In this case, each individual subsystem can exploit local states and some of the other neighbor subsystems' states. Comparing with the decentralised configuration, the distributed solutions can guarantee the stability of large-scale network systems when interconnections are strong [19]. Meanwhile, compared to the centralised scheme, it is also less complex and has reduced computational load.
The configuration of the distributed controller network for the interconnected systems is often constrained because of several facts such as implementation-related issues and communication expenses. This limitation in distributed systems is called as information pattern. Different with traditional distributed control structure where all the related sub-controllers utilize the same information, the sub-controllers could acquire different information [16]. Because the fully distributed controller configuration is not always realistic, the design of distributed control systems with imposing a priori limitations on communication network configuration can be a possible solution. Otherwise, designing a control network with the minimum number of communication channels while satisfying a global performance requirements [12,1] should be considered as another choice for the distributed control systems. Actually, a trade off between the sparsity of the feedback gain matrices and the control performance should be explored [10,4,21,8,6].
Addressing network sparsification problem, in worst case, one has to test all possible structures, which means a comprehensive examination for numerous structures. This may result an exponential number of growth of the communication links, and it is essentially problematic and impractical to perform. In [18], it is clarified to avoid an exhaustive searching, a compromise choice is either in the searching strategy or in the selection of index function. Another option is to construct a multi-objective optimization of both controller structure and control law design by including secondary index functions into the main cost function, which represents a performance specification of the closed-loop system [10,17]. This secondary index functions can be constructed as the convex approximations of the original non-convex optimization of 0 -quasi-norms and can be applied to sparsify the distributed controller. The reweighted 1 (REL1) norm algorithms [10] can be further developed to find the optimal sparse feedback gain. In the REL1 algorithms, the entries of the weighting matrix are calculated at each individual step, which are inversely proportional to the strength of the elements of the gain in the preceding step. This approach has been frequently utilized to sparsify a key matrix at the element-wise level to minimise the number of communication channels in distributed control networks exploiting the so-called bilateral communication scheme [16]. Still, the current REL1 algorithms have disadvantages when the sparsity is defined at a group (e.g. block column or row) level, where the strength of sets of variables (block elements of feedback gain) should be considered. To be more specific, block column/row sparsified feedback gain has an application in the diffusion based networks [16], where a published information in the communication network is available to subscribers, the sub-controllers. Then, the objective is to minimise the number of subscribers/publishers in the system rather than that of bilateral communication links. This is obviously equivalent to seeking for feedback gains with maximum number of block columns/ rows with zero off-diagonal blocks. This paper presents a multi-objective optimisation problem by integrating two secondary index functions into the main index function to penalise the number of block columns/rows with non-zero off-diagonal blocks. Then, to cope with the underlying problem, an iterative process is developed, using the relaxed block column/row sparsity promoting penalty functions, to simultaneously penalise the number of subscribers and publishers in the control network. A similar framework has been used to develop an approach for optimal actuator/sensor selection in over-actuated/sensed systems [2,3].
The control scheme presented in this manuscript is static output feedback (SOF). Different with most of the approaches in the existing literature, it does not need the information of all the system states. Instead, it only utilizes the available sensor outputs. In addition, the overall networked system considered in this paper involving parametric uncertainties. Thus, this paper proposes a novel scheme for the design of H 2 -based sparse block column/row -wise SOF for systems with polytope uncertain. An obvious benefit of the proposed SOF is that it is capability to deal with the networked systems whose output matrix has parametric uncertainty; cf. [5].

Problem Formulation.
2.1. Problem statement. The following model describes a linear-time-invariant (LTI) large-scale networked system with h subsystems: where ξ i ∈ R ni , u i ∈ R mi , z i ∈ R qi , and y i ∈ R pi are the system state, control input, performance output, and output of the i-th subsystem, respectively. The matrices in (1) are all constant with appropriate dimensions. A ij , i, j = 1, · · · , h, j = i is the interactions between the subsystems (A i , i = 1, · · · , h), i.e., A ij = 0 if the subsystem j does not directly influence the subsystem i. It is also assumed, without loss of generality, that m i ≤ q i ≤ n i , and rank(B 2,i ) = m i . w i (t) ∈ R mw i denotes the external disturbance. Define and where A ii = 0 and C z,ii = 0. Based on (1), (2) and (3), the total system can be described as the following state space equations:

Interactions among subsystems
Communications among subsystems u1 u2 uh where N is the number of vertices. Suppose that there exists a static gain F y such that A l + B 2,l F y C l is stable.
In the existing literatures, the frameworks of distributed control design for networked systems are built based on bilateral communication scheme, where the subsystems have two-direction communications. In this case, searching for a sparse feedback gain is equivalent to using less communication channels in the control. However, these frameworks have nothing to do with the so-called publisher/subscriber communication scheme; see Fig. 1. See [16], the diffusion based networks (e.g., the factory instrumentation protocol, such as EN 50170 and IEC 61158/IEC 61784 standards) is considerably different from the bilateral one. In this communication strategy, a published information is available to the sub-controllers that are subscriber in the communication network. In this study, the main target is the optimization of the number of publishers and/or subscribers in the system. This is comparable to investigating for feedback gains with maximum number of block columns/rows with zero off-diagonal blocks. The major aim of this study is highlighted by the following problems: Problem 1. Given the state space representation in Equation (4) of a networked system, choose a subset of available publishers/subscribers and simultaneously find a distributed controller. This controller employs only the available sparse information while minimising the degradation of an optimisation index (e.g., H 2 norm of the closed-loop transfer function from w to z) in contrast to the circumstance where all the system information are exploited.
To solve Problem 1, firstly build a structure for the design of a controller utilizing a priori specified subset of system information. This structure can be applied to deal with different control network configurations. (4); build a distributed controller employing a priori specified subset of system information to minimise an optimisation metric, e.g., H 2 norm of the closed-loop transfer function from w to z.

Problem 2. Consider a networked system depicted as in Equation
Because the systems in (1) or (4) have the parametric uncertainties, it is required to develop a strategy for the design of sparsely distributed controllers for networked systems. This strategy employs publisher/subscriber communication topology to cope with the parametric uncertainties. To this end, we investigate the structured static output feedback synthesis with a H 2 performance requirement. We present some useful definitions as follows.
Definition 2.1. We call a matrix whose elements are either 0 or 1 as a structure matrix. Let ∆ = [∆ ij ] m×n be a block matrix with ∆ ij ∈ R ai×bj , then the structure matrix of ∆ is obtained as  [11] Given that two matrices U and V are of column dimension m and a matrix Z m×m is symmetric, there exists an unstructured matrix X that satisfies iff where N U and N V are matrices whose columns form a basis of the null spaces of matrices U and V.
[2] Assume a closed loop system is described by A cl , B cl , C cl . Then, the closed loop system is stable and ||C cl (sI − A cl ) −1 B cl || 2 2 < γ, if and only if, there exist matrices P > 0 and Z > 0 such that trace(Z) < 1.
Lemma 2.7. Assume a closed loop system is described by A cl , B cl , C cl . Then, the closed loop system is stable and ||C cl (sI − A cl ) −1 B cl || 2 2 < γ, if and only if, there exist a scale η > 0, matrices X > 0, Z > 0, and V , such that trace(Z) < 1.
Proof. We define the matrices X , Z, U, N U , V, and N V of Lemma 2.5 as follows: It can easily be checked that U T X V + V T X T U + Z < 0 is equivalent to (11), by letting X = P , N T U ZN U < 0 in (7) is equivalent to (8), and N T V ZN V < 0 in (7) is equivalent to the following trivial inequality Then, based on Lemma 2.5 and Lemma 2.6, this lemma can be proved.
Lemma 2.8. The following three statements, involving X > 0, Z > 0, a general matrix variable V are equivalent.
Proof. In Lemma 2.7, if let η = 1, A cl = A + B 2 F, B cl = B 1 , C cl = C z + D z F, based on Lemmas 2.7 and 2.6 and define Y = FX, this lemma can be proved.
It should be emphasized here; as V + V T > 0, V is nonsingular and thus if the LMI (19) is feasible, the state feedback would be derived as F = Y V −1 . It is also important to note that as post-multiplication retains the row-sparse structure, if Y is row-sparse, the corresponding feedback gain F = Y V −1 will trivially be row-sparse. Moreover, the specific LMI characterisation in (19) enables us to use different Lyapunov matrices for each of the related LMI constraints in the problem. The reason is the product terms between the matrix A and the Lyapunov matrix in the LMI (19) disappeared. In such a situation, the feedback gain can be achieved independent of the Lyapunov matrix. This feature has an important implication in the controller design of systems with parametric uncertainties.
The H 2 problem by supposing the controller as u(t) = Fx(t) for the system (4) with parametric uncertainty (5), can be embedded in the following optimisation: subject to (19), (20) and (21).
Also from the item iii) of Lemma 2.8, for each vertex l, the following inequalities can used:  where X l > 0, Z l > 0, Y l and V l are variables. However, addressing the optimisation problem in (22) with the above inequalities (for l = 1, · · · , N ) constraints by exploiting different Y l and V l cannot get a unique state feedback control F. These inequalities in the next subsection will be used to design a sparse row/column-wise SOF for networked systems with parametric uncertain by using the proposed two specific matrix variable transformations.

2.3.
Sparse row/column-wise H 2 SOF. Based on the discussions given previously, we specify the requirements of Problem 2 in the following problem.
Problem 3. Given a networked system with the state space representation in Equation (4) involving the parametric uncertainties, design a sparse row/column-wise SOF such that it ensures the H 2 performances, i.e. T wz 2 2 < γ, while S(F y ) ⊆ Γ, where Γ is a priori specified sparse row/column wise structure matrix and F y is the SOF.
The SOF problem can be considered as a constrained state feedback problem; i.e. a state feedback (say F) which satisfies the additional constraint F = F y C [13]. Effective schemes to address a similar non-convex optimisation problem for the design of an H ∞ SOF and mixed H 2 /H ∞ SOF are proposed in [13,5]. In this paper, unlike [13,5], the output matrix C belongs to the polytope (5). We now introduce specific LMI decision variables transformations as where V Θ ∈ R (n−p)×(n−p) and V Ω ∈ R p×p are symmetric matrices, and Y Ω ∈ R m×p . Besides, Θ l = null(C l ) ∈ R n×(n−p) and Ω l ∈ R n×p is any matrix that satisfies C l Ω l = I. In general form, Ω l can be considered as Ω l = C † l + Θ l Φ l , where Φ l ∈ R (n−p)×p is a given matrix and C † l = C T l (C l C T l ) −1 . As seen, the matrix transformation proposed in (26) is essentially different from the ones proposed in [13,5]. Now by letting the variables V l and Y l be (26), the SOF gain can be obtained through the following lemma.
As it is guaranteed that V l is invertible, C l V l is of rank p, and thereby V Ω should have rank p. Then, as V Ω is a full rank square matrix, it is invertible as well. As a result, Now the sparse row/column-wise H 2 SOF problem, by exploiting LMI approach, can be set as the following optimisation problem: However, this choice requires solving the optimisation problem in (22) subject to LMIs in (23)-(25) in advance to find V l associated with each vertex l. Clearly, if no solution can be attained by solving (22), the SOF design problem would not be feasible and no further action is required to be taken for the output feedback problem.
3. Identifying Favourable Sparse Row/Column-Wise Structures. The previous section developed a framework for the design of an H 2 -based SOF while constraining the structure of the feedback gain. In this section, we aim to address the objective mentioned in Problem 1; i.e. seeking for an optimal subset of available publishers/subscribers in the networked system while the H 2 -norm degradation of the closed-loop system is minimised relative to the fully distributed topology, or equivalently, finding favourable sparse block row/column-wise SOF gains. This problem can also be seen as searching for the redundant publishers and subscribers of the networked system. We indeed aim to construct a multi-objective optimisation program, where the block row/column sparsity of the SOF gain is directly incorporated into the index function. This is encapsulated in the following problem.
subject to the constraints in (27), where Y Ω is a full decision matrix; i.e. S(Y Ω ) = Γ = 1 m×p , the off-row-0 (offcol-0) is a quasi-norm that counts the number of non-zero off-diagonal block rows (columns) of Y Ω , and , is a weighting matrix that implies the emphasis on the off-diagonal block row-sparsity (column-sparsity) of Y Ω , and thus the SOF F y . For example, a larger ψ s,i (ψ p,j ) will lead to not employing i-th subscriber (j-th publisher) in the control system.
Obviously, an intractable combinatorial search is required to address the optimisation problem above, hence the computation time would grow faster than polynomial, as the order of the networked system system grows [17]. A number of convex approximation of quasi-zero-norms are proposed yet, such as 1 -norm or weighted 1 -norm [7]. In addition, the paper [7] proposes the reweighted 1 (REL1) minimisation method which is nothing but an iterative program that solves a sequence of weighted minimisation problems, in which at each iteration the weights are updated based on the previous iteration's solution. The REL1 algorithm has recently been used by a number of researchers (e.g., see [9], [21]) for the design of sparse controllers for the distributed systems. Nevertheless, the developed REL1 schemes in these references do not promote row/column-sparsity of the feedback gain which is required in (28). Here, we need to develop a novel method in which the variable selection should amount to the selection of the important groups of variables (block rows and/or columns), rather than important individual variables (elements in the feedback gain).

Remark 2.
In the existing literature, a scalar is used to weight the sparsity of the feedback gain; cf. [10], in an extended objective function, with the value of this scalar determining the emphasise on the sparsity of the feedback gain. However, in real cases, there may be prior information available about the control network. For example, some communication links can be infeasible or unattractive due to the high implementation costs. In this case, to assist the optimisation-based program proposed for the sparity pattern recognition, it would be advisable to incorporate this a priori knowledge into the optimisation problem by using different scalars for weighting different off-diagonal block rows (columns). This can be implemented simply by specification of diagonal matrices (Ψ s and Ψ p ) of separate weights corresponding to individual off-diagonal block rows (columns).
Let us now recast the objective function of the optimisation problem (28) as follows Problem 5. Given a system with the state space representation in Equation (1), and γ > 0 in the following optimisation program: subject to the constraints in (27), excluding the structural constraint on Y Ω . Here, f s (·) (f p (·)) denotes the relaxed off-diagonal block-row-sparsity (block-column-sparsity) promoting function.
The following subsection proposes candidates for f s (·) and f p (·).
3.1. REL1 for row sparsity promoting penalty function. A convex alternative for the non-convex off-row-0 quasi-norm, can be the following function where · F denotes the Frobenius norm, and similarly is a convex approximation for the non-convex off-col-0 quasi-norm in (29). Moreover, the update rule for W s,i can be considered as where k denotes the current iteration and we use 0 < 1 to provide stability and to ensure that a zero valued off-diagonal block row in Y Ω does not strictly prevent a non-zero value at the next step. The weighting matrix will be formed as As seen, the weights are updated without considering the Frobenius norm of the block diagonal entries of the SOF gain, because local measurements should not raise the communication cost. It is also worth noting that the obtained relaxed sparsity promoting function do not promote sparsity within the blocks, but at the level of block rows. Similarly, W p,j can be updated as Additionally, we form the weighting matrix as W p = [w p,ij ] h×h , where Remark 3. It is worth mentioning that minimising the number of subscribers and publishers or either of them have the possibility of promoting a completely decentralised control structure which is equal to a control network with no subscriber and publisher.

3.2.
An algorithm for solving Problem 5. Now, define the matrix = [τ ij ] h×h with The optimisation problem in (29), by letting f s (Y Ω ) and f p (Y Ω ) as (30) and (31), respectively, is equivalent to subject to the constraints in (28), where • denotes the Hadamard product (entry-wise product). For addressing the above convex problem and identifying a sparse row/column-wise SOF, the following algorithm is proposed.
3) Update W k s,i and W k p,j using the update rules in (32) and (34), respectively, and form W s = [w s,ij ] h×h and W p = [w p,ij ] h×h as in (33) and (35), respectively. 5) If F y − F k y ≤ α go to Step 6, else F k y = F y , k = k + 1 and return to Step 2. 6) Let the unnecessary block rows and columns of F y be zero and return Γ = S(F y ).
Eventually, in order to find the H 2 structured SOF with the identified Γ , we turn to the minimisation problem in (27). 4. Numerical examples. Consider a decentralised interconnected system, presented in [16], that consists of four subsystems: We additionally assume that there exists an uncertainty in the entry (4, 6) of the output matrix C by up to ±50% of its nominal value. As seen this interconnected system is fully coupled and open-loop unstable. Solving the convex problem in (27), by letting Γ = 1 m×p and assuming a block diagonal structure for V l , l = 1, 2 as , with V Ω,i ∈ R, are symmetric matrices, and Φ l = 0, results in a true H 2 -norm of 3.7689 (with nominal C). We now exploit Algorithm 1 with α = 0.01 and = 0.001. By increasing Ψ s and Ψ p from zero, the number of block rows and columns with nonzero off-diagonal blocks in the SOF gain decreases; see Fig. 2. Once the sparsity structures of controllers are identified for different Ψ s and Ψ p , the resulting patterns are used to solve (36), by identified Γ, in order to obtain the H 2 block row/columnsparse structured controllers.