Communication-Aware Dissipative Output Feedback Control

IEEE CONTROL SYSTEMS LETTERS , V OL. XX, NO . XX, XXXX 2017 1 Communication-A w are Dissipativ e Output F eedbac k Control Ingyu Jang, Gr aduate Student Member , IEEE , and Leila J . Bridgeman, Member , IEEE Abstract — Communication-aware control is essential to reduce costs and complexity in large-scale networks. This work proposes a method to design dissipativity-augmented output feedbac k controller s with reduced online commu- nication. The contributions of this w ork are three fold: a generalized well-posedness condition for the controller network, a conve x relaxation for the constraints that infer stability of a network from dissapativity of its agents, and a synthesis algorithm integrating the Network Dissipativity Theorm, alternating direction method of multiplier s, and iterative con vex overbounding. The proposed approach yields a sparsel y interconnected controller that is both rob ust and applicable to networks with heter og eneous non- linear agents. The efficiency of these methods is demon- strated on heterogeneous networks with uncer tain and un- stable agents, and is compared to standar d H ∞ control. Index T erms — Communication-A ware Contr ol, Rob ust Control, Networked System Contr ol, Heterogeneous Net- work, Nonlinear System Contr ol I . I N T R O D U C T I O N N ETWORKED control systems are increasingly prev alent in modern infrastructure, including smart grids, po wer plant netw orks, and swarm robotics. While centralized con- trol schemes often suf fer from scalability issues, such as prohibitiv e communication ov erhead, fully decentralized ap- proaches can degrade closed-loop performance. As networked systems grow in complexity , controller architectures must optimize the trade-off between performance and communi- cation efficienc y , specifically by promoting sparsity in agent interconnections [1]. Many methods balance controller archi- tecture against performance [2]–[12], but most framew orks are restricted to linear time-in variant (L TI) agents or employ full- state feedback, which is unrealistic in real physical situations. This paper addresses the synthesis of rob ust, communication- aware controllers for networks with heterogeneous nonlinear agents under partial state feedback. Imposing an ℓ 0 norm (cardinality) constraint can directly yield the optimal sparse controller , but this results in NP- hard problems [13]. T o circumv ent this computational chal- lenge, v arious sparsity-promoting methods hav e been dev el- oped, including ℓ 1 -norm relaxations [2]–[4], gradient-based algorithms [5], [6], methods minimizing perturbations from *This work is suppor ted by ONR Grant No. N00014-23-1-2043. Ingyu Jang (PhD Student), and Leila Br idgeman (Assistant Pro- fessor) are with the Depar tment of Mechanical Engineer ing and Ma- terial Science, Duke University , Durham, NC 27708 USA (email: ij40@duke.edu; ljb48@duke.edu , phone: 919-225-4215). a well-performing reference controller [7], [8], and alternat- ing direction methods of multipliers (ADMM) with sparsity penalties [9]–[11]. Howe ver , the majority of existing works are restricted to L TI plants or assume full-state feedback framew orks. Although some works [3], [8], [10] ha ve e xtended these methods to observer-based control with sparse matrix parameters, the resulting architectures still exchange full ob- server states between agents. The more realistic setting is one where controller communication is restricted to local output information, similar to the plant interconnections, rather than full internal states. Dissipativity [14], [15] offers a versatile framework for analyzing nonlinear dynamics by modeling systems as in- put–output operators rather than relying on internal state descriptions. This perspectiv e is especially powerful due to its compositional properties [16]. The Network Dissipativity Theorem (NDT) [17], [18] leverages this modularity to certify robust network stability using only open-loop dissipati vity characteristics of individual agents. By decoupling agent- lev el dynamics from the network topology , NDT is uniquely suited for networked control with heterogeneous, nonlinear agents, as seen in [19], [20], which applied NDT to the design of centralized or decentralized controllers for large- scale networks. It also offers a natural extension to distributed optimization paradigms [21], which was lev eraged in [22]. Building upon [12], which addressed dissipativity-based communication-aware control under full-state feedback, this paper considers the more realistic scenario of networked dynamics output-feedback control, in which controllers com- municate only their filtered output information with others. Our objective is to identify the optimal input-output (IO) com- munication links between local controllers while minimizing a global H ∞ -norm performance objective, by integrating NDT, ADMM [21], and iterativ e con vex ov erbounding (ICO) [23]. The primary contrib utions of this work are threefold. First, we establish a generalized well-posedness condition for the controller network, ensuring a reasonable global control law as an extension of classical feedback well-posedness. Second, we derive a conv ex relaxation for the global H ∞ -norm con- straint within a networked dynamics output-feedback frame- work, enabling efficient computation. Third, we propose a computationally tractable synthesis algorithm that combines NDT, ADMM, and ICO to solve the sparse controller design problem. This approach yields a sparsely interconnected con- troller that is robust and applicable to networks of nonlinear and heterogeneous agents. Owing to the modularity of the NDT framew ork, the proposed approach yields a sparsely 2 IEEE CONTROL SYSTEMS LETTERS , V OL. XX, NO . XX, XXXX 2017 interconnected controller that is both robust and applicable to networks with heterogeneous nonlinear agents. Furthermore, this modular structure ensures that the synthesis problem is readily extendable to distrib uted optimization paradigms. I I . P R E L I M I N A R I E S A. Notation The sets of real and natural numbers up to 𝑛 are denoted by R and N 𝑛 , respectiv ely . The set of real 𝑛 × 𝑚 matrices is R 𝑛 × 𝑚 . If ( A ) 𝑖 𝑗 ∈ R 𝑛 𝑖 × 𝑚 𝑗 and A ∈ R  𝑁 𝑖 = 1 𝑛 𝑖 ×  𝑀 𝑗 = 1 𝑚 𝑗 , then ( A ) 𝑖 𝑗 is said to be a “block” of A , and A is said to be in R 𝑁 × 𝑀 block-wise. The set of 𝑛 × 𝑛 symmetric matrices is S 𝑛 . The notation A ≺ 0 and A ⪯ 0 indicates that A is negativ e definite and negativ e semi-definite, respectively . For brevity , He ( A ) = A + A 𝑇 and asterisks, ∗ , denote duplicate blocks in symmetric matrices. T 1 0 ( A ) is the 1st order T aylor expansion of the matrix variable A from its initial point A 0 , meaning T 1 0 ( A ) = A 0 + 𝛿 A . The set of square integrable functions is L 2 . The Frobenius norm and L 2 norm are denoted by ∥ · ∥ 𝐹 and ∥ · ∥ 2 , respectiv ely . The truncation of a function y ( 𝑡 ) at 𝑇 is denoted by y 𝑇 ( 𝑡 ) , where y 𝑇 ( 𝑡 ) = y ( 𝑡 ) if 𝑡 ≤ 𝑇 , and y 𝑇 ( 𝑡 ) = 0 oth- erwise. If ∥ y 𝑇 ∥ 2 2 = ⟨ y 𝑇 , y 𝑇 ⟩ =  ∞ 0 y 𝑇 𝑇 ( 𝑡 ) y 𝑇 ( 𝑡 ) 𝑑 𝑡 < ∞ for all 𝑇 ≥ 0 , then y ∈ L 2 𝑒 , where L 2 𝑒 is the extended L 2 space. B. QSR -Dissipativity of Large-Scale Systems In this paper , controllers are synthesized based on the QSR - dissipativity of each agent, defined as follo ws. Definition 1 ( 𝑄 𝑆 𝑅 -Dissipativity , [18]): Let Q ∈ S 𝑙 , R ∈ S 𝑚 , and S ∈ R 𝑙 × 𝑚 . The operator 𝒢 : L 𝑚 2 𝑒 ↦→ L 𝑙 2 𝑒 is QSR -dissipativ e if there exists 𝛽 ∈ R such that for all u ∈ L 𝑚 2 𝑒 and 𝑇 ≥ 0  𝑇 0  𝒢 ( u ( 𝑡 ) ) u ( 𝑡 )  𝑇  Q S ∗ R   𝒢 ( u ( 𝑡 ) ) u ( 𝑡 )  𝑑 𝑡 ≥ 𝛽 . (1) Theorem 1 relates dissipativity to IO stability , defined next. Definition 2 (IO or L 2 -stability , [24]): An operator 𝒢 : X 𝑚 𝑒 ↦→ X 𝑙 𝑒 is IO-stable, if for any u ∈ X 𝑚 and all x 0 where X is any semi-inner product space and X 𝑒 is its extension, there exists a constant 𝜅 > 0 and a function 𝛽 ( x 0 ) such that ∥ ( 𝒢 ( u ) ) 𝑇 ∥ X ≤ 𝜅 ∥ u 𝑇 ∥ X + 𝛽 ( x 0 ) (2) where ∥ · ∥ X is the induced norm of the innerproduct space. If the space X is L 2 , then IO stability is called L 2 stability . Theor em 1: The operator is L 2 stable if and only if it is 𝑄 𝑆 𝑅 -dissipativ e with Q ≺ 0 . NDT, stated next, shows how agent-lev el dissipativity ex- tends to the network lev el, thereby guaranteeing the L 2 - stability of the networked system. Theor em 2 (NDT, [25]): Consider 𝑁 Q 𝑖 S 𝑖 R 𝑖 dissipativ e operators, 𝒢 𝑖 : L 𝑚 𝑖 2 𝑒 ↦→ L 𝑙 𝑖 2 𝑒 , interconnected by matrices, ( H ) 𝑖 𝑗 : L 𝑙 𝑗 2 𝑒 ↦→ L 𝑚 𝑖 2 𝑒 as y 𝑖 = 𝒢 𝑖 u 𝑖 , u 𝑖 = e 𝑖 +  𝑗 ∈ N 𝑁 ( H ) 𝑖 𝑗 y 𝑗 , y = 𝒢 e , u = e + Hy , (3) where u = col ( u 𝑖 ) 𝑖 ∈ N 𝑁 , y = col ( y 𝑖 ) 𝑖 ∈ N 𝑁 , e = col ( e 𝑖 ) 𝑖 ∈ N 𝑁 , and 𝒢 = diag ( 𝒢 𝑖 ) 𝑖 ∈ N 𝑁 . Then, 𝒢 : L 𝑚 2 𝑒 ↦→ L 𝑙 2 𝑒 is L 2 stable if Q + SH + H 𝑇 S 𝑇 + H 𝑇 RH ≺ 0 (4) with Q = diag ( Q 𝑖 ) 𝑖 ∈ N 𝑁 , and S and R defined analogously . C . ICO Optimal control synthesis problems frequently inv olve non- con vex bilinear matrix inequalities (BMIs) of the form Q + He ( XNY ) ≺ 0 , (5) where N ∈ R 𝑝 × 𝑞 is fixed, and Q ∈ S 𝑛 , X ∈ R 𝑛 × 𝑝 . and Y ∈ R 𝑞 × 𝑛 are design variables. T o handle the general NP-hardness of (5), con vex conserv atishm can be introduced via Theorem 3. Theor em 3 ( [26]): Consider the matrices Q ∈ S 𝑛 , N ∈ R 𝑝 × 𝑞 , X ∈ R 𝑛 × 𝑝 . and Y ∈ R 𝑞 × 𝑛 , where Q , X , and Y are design vari- ables. The BMI condition Q + He ( XNY ) ≺ 0 is implied by  Q XN + Y 𝑇 G 𝑇 N 𝑇 X 𝑇 + G Y − He ( G )  ≺ 0 (6) for any G ∈ R 𝑞 × 𝑞 satisfying He ( G ) ≻ 0 . The conservati ve effect of (6) can be mitigated by iterativ ely updating a feasible point, ( X 𝑖 , Y 𝑖 ) , satisfying (5).W ith the update, ( X 𝑖 + 1 , Y 𝑖 + 1 ) = ( X 𝑖 + 𝛿 X , Y 𝑖 + 𝛿 Y ) , 𝛿 X and 𝛿 Y serve as the decision v ariables. The tightening of (6) relative to (5) then lies in proportion to these perturbations. As detailed in [23], this iterativ e scheme reduces the conservatism inherent in Theorem 3. Each optimization problem remains feasible since 𝛿 X = 0 and 𝛿 Y = 0 yield the initial feasible point. Remark 1: In this paper , I is used as G , but any G satisfying He ( G ) ≻ 0 can be used for (6). I I I . S P A R S I T Y - P R O M O T I N G D I S S I P A T I V I T Y - A U G M E N T E D C O N T R O L Consider a multi-agent networked system 𝒢 consisting of 𝑁 heterogeneous agents 𝒢 𝑖 interconnected through H . The ov erall network dynamics are described by 𝒢 𝑖 : ¤ x 𝑖 = 𝑓 𝑖 ( x 𝑖 , u 𝑖 ) , y 𝑖 = ℎ 𝑖 ( x 𝑖 ) 𝒢 : ¤ x = 𝑓 ( x , e ) , z = ℎ ( x ) , u = e + Hy , z = y (7) where x 𝑖 ∈ L 𝑛 𝑖 2 𝑒 , u 𝑖 ∈ L 𝑚 𝑖 2 𝑒 , and y 𝑖 ∈ L 𝑙 𝑖 2 𝑒 are the states, inputs, and outputs of the 𝑖 th agent, respectiv ely . u , e , and y are stacked v ectors defined in Theorem 2, where e represents the exogenous input to the global network. The global network output is denoted by z , which is equiv alent to y . A. Well-P osedness of Controller Interconnection Consider a network of agents and local dynamic output feedback controllers, 𝒞 𝑖 , that communicate through their out- put measurements. This paper aims to jointly design local con- trollers for each agent and a sparse network topology , so that the resulting global controller 𝒞 stabilizes and regulates the network 𝒢 . Lemma 1 provides conditions for well-posedness, which ensures the existence and uniqueness of the closed-loop solution. Definition 3 (Chapter 5.2 [27]): An interconnected system is said to be well-posed if all interconnected transfer matrices are well-defined and proper . Lemma 1: Consider 𝑁 L TI systems with minimal state- space realizations 𝒞 𝑖 : ¤  x 𝑖 =  A 𝑖  x 𝑖 +  B 𝑖  u 𝑖 ,  y 𝑖 =  C 𝑖  x 𝑖 +  D 𝑖  u 𝑖 . Con- struct the global L TI system 𝒞 through the interconnections  u =  H 𝑦  e + H  y and  z =  H  𝑦  y , where  e and  z are the input and output of 𝒞 , respecti vely ,  u = col (  u 𝑖 ) 𝑖 ∈ N 𝑁 , and  y ,  x ,  e ,  z are defined analogously . Then, 𝒞 is well-posed if and only if I −  D 𝑑 H is in vertible, where  D 𝑑 = diag (  D 𝑖 ) 𝑖 ∈ N 𝑁 . A UTHOR et al. : PREP ARA TION OF P APERS FOR TEXTSCIEEE CONTROL SYSTEMS LETTERS (NO VEMBER 2021) 3 (a) Closed-loop representation (b) Decomposed representation Fig. 1. T wo representations of a multi-agent system and its controller . Pr oof: 𝒞 can be modeled as a closed-loop system with an external input, with gains  H 𝑦 and  H  𝑦 on the input and output, respectiv ely . The plant is a L TI system with (  A 𝑑 ,  B 𝑑 ,  C 𝑑 ,  D 𝑑 ) , where  A 𝑑 ,  B 𝑑 and  C 𝑑 are defined analogously to  D 𝑑 , and the feedback gain is H . Then, the closed-loop realization of 𝒞 with input  e and output  z is giv en by  A =  A 𝑑 +  B 𝑑 H ( I −  D 𝑑 H ) − 1  C 𝑑 ,  B = (  B 𝑑 +  B 𝑑 H ( I −  D 𝑑 H ) − 1  D 𝑑 )  H 𝑦 ,  C =  H  𝑦 ( I −  D 𝑑 H ) − 1  C 𝑑 ,  D =  H  𝑦 ( I −  D 𝑑 H ) − 1  D 𝑑  H 𝑦 . Since the plant has a state-space realization, the well- posedness of 𝒞 is equiv alent to the existence of ( I −  D 𝑑 H ) − 1 [27, Chapter 5.2]. T o ensure well-posedness, each local controller 𝒞 𝑖 and the resulting global controller 𝒞 are giv en by 𝒞 𝑖 : ¤  x 𝑖 =  A 𝑖  x 𝑖 +  B 𝑖  u 𝑖 ,  y 𝑖 =  C 𝑖  x 𝑖 +  D 𝑖  u 𝑖 , 𝒞 : ¤  x =  A  x +  B  e ,  z =  C  x +  D  e ,  u =  H 𝑦  e ,  z =  H  𝑦  y , (8) where  x 𝑖 ∈ L 𝑛 𝑖 2 𝑒 ,  u 𝑖 ∈ L 𝑙 𝑖 2 𝑒 ,  y 𝑖 ∈ L 𝑚 𝑖 2 𝑒 ,  A ,  B ,  C , and  D follows the setup in Lemma 1. 𝒞 is always well-posed since H = 0 . B. Closed-loop Networked System Consider the multi-agent networked system 𝒢 : e → z , defined in (7), interconnected in feedback with the networked con- troller 𝒞 :  e →  z , defined in (8). Let n and  n be exogenous disturbances affecting the plant and controller inputs, respec- tiv ely , so that e = n +  z and  e =  n + z , as illustrated in Figure 1a. This configuration can be equiv alently viewed as a network interconnection among agents, 𝒢 𝑖 , and their respectiv e local controllers, 𝒞 𝑖 . The resulting global interconnection is  u  u  =  n + Hy +  H  𝑦  y  H 𝑦 ( z +  n )  =  n  H 𝑦  n  + H  y  y  , H =  H  H  𝑦  H 𝑦 0  , (9) where we used z = y . In the global interconnection matrix H , ( H ) 𝑖 𝑖 = 0 to preclude local self-feedback. This closed-loop architecture is illustrated in Figure 1b. The objectiv e is to synthesize sparse structures for  H 𝑦 and  H  𝑦 within H so that the controller 𝒞 stabilizes the plant 𝒢 in the presence of n and  n , while improving the network performance. C . Dual-Model Synthesis IO stability is closely linked to the QSR -dissipati vity prop- erties of the agents and controllers. If each agent is Q 𝑖 S 𝑖 R 𝑖 - dissipativ e, stability of the system can be guaranteed by enforcing  Q 𝑖  S 𝑖  R 𝑖 -dissipativity on the corresponding local con- trollers and applying NDT with H . Howe ver , according to (9), the external input to the system is ( n ,  H 𝑦  n ) , not ( n ,  n ) , which differs from the standard configuration of NDT. Ne vertheless, NDT remains applicable, because premultiplication by a time- in variant matrix preserves L 2 -integrability of signals. In this work, the objecti ve of the controller is to attenuate the impact of n and  n on the plant while guaranteeing IO stability of the network via NDT, and this attenuation can be performed using a nominal linearized plant, 𝒢 𝑙 𝑡 𝑖 , described by 𝒢 𝑙 𝑡 𝑖 𝑖 : ¤ x 𝑖 = A 𝑖 x 𝑖 + B 𝑖 u 𝑖 , y 𝑖 = C 𝑖 x 𝑖 , 𝒢 𝑙 𝑡 𝑖 : ¤ x = ( A 𝑑 + B 𝑑 HC 𝑑 ) x + B 𝑑 e , z = C 𝑑 x , u = e + Hy , z = y , (10) and a global controller 𝒞 in (8). The parameters of the resulting closed loop of 𝒢 𝑙 𝑡 𝑖 and 𝒞 are A 𝑐𝑙 = A +  B  K  C , B 𝑐𝑙 = B +  B  K  H , C 𝑐𝑙 = C +  H  K  C , D 𝑐𝑙 =  H  K  H , (11) where the auxiliary system matrices are defined as A =  A 𝑑 + B 𝑑 HC 𝑑 0 0 0  , B =  B 𝑑 0 0 0  , C =  C 𝑑 0 0 0  ,  K =   A 𝑑  B 𝑑  C 𝑑  D 𝑑  ,  B =  0 B 𝑑  H  𝑦 I 0  ,  C =  0 I  H 𝑦 C 𝑑 0  ,  H =  0 0 0  H  𝑦  ,  H =  0 0 0  H 𝑦  . This attenuation process is formulated as minimizing the H ∞ - norm bound 𝜈 ≥ 0 subject to the existence of Y ≻ 0 satisfying       Y A 𝑐𝑙 + A 𝑇 𝑐𝑙 Y YB 𝑐𝑙 C 𝑇 𝑐𝑙 B 𝑇 𝑐𝑙 Y − 𝜈 I D 𝑇 𝑐𝑙 C 𝑐𝑙 D 𝑐𝑙 − 𝜈 I       ≺ 0 . (12) D . Sparsity Promoting Controller Synthesis Including a penalty term 𝑔 ( H ) in the objecti ve function, such as the ℓ 1 norm, weighted ℓ 1 norm, sum-of-logs, and the cardinality function of H [9], promotes sparsity in the network topology H . Accordingly , the sparsity-promoting dissipati vity augmented controller is obtained by solving arg min  K , F ,  F , H 𝐽 (  K ) + 𝛾 𝑔 ( H ) (13a) s.t.       Y A 𝑐𝑙 + A 𝑇 𝑐𝑙 Y YB 𝑐𝑙 C 𝑇 𝑐𝑙 B 𝑇 𝑐𝑙 Y − 𝜈 I D 𝑇 𝑐𝑙 C 𝑐𝑙 D 𝑐𝑙 − 𝜈 I       ≺ 0 , (13b) F 𝑖 ∈ { F 𝑖 | Constraint for dissipativity holds } , ∀ 𝑖 ∈ N 𝑁 (13c) −   C 𝑇 𝑖  Q 𝑖  C 𝑖  C 𝑇 𝑖  Q 𝑖  C 𝑖  D 𝑇 𝑖  Q 𝑖  C 𝑖  R 𝑖 +  D 𝑇 𝑖  Q 𝑖  D 𝑖  + He  P 𝑖 0 0 − S 𝑇 𝑖   K 𝑖  ⪯ 0 , (13d) Q + He ( SH ) + H 𝑇 RH ≺ 0 , (13e) where  K 𝑖 =   A 𝑖  B 𝑖  C 𝑖  D 𝑖  , F 𝑖 = { Q 𝑖 , S 𝑖 , R 𝑖 } ,  F 𝑖 = {  Q 𝑖 ,  S 𝑖 ,  R 𝑖 } , Q = diag ( diag ( Q 𝑖 ) 𝑖 ∈ N 𝑁 , diag (  Q 𝑖 ) 𝑖 ∈ N 𝑁 ) , S and R are defined analogously , F =  𝑁 𝑖 = 1 F 𝑖 , and  F =  𝑁 𝑖 = 1  F 𝑖 . Simultaneously enforcing (13b) and (13e) together constitutes the dual-model synthesis approach explained in Section III-C. E. Advantage of NDT The primary adv antage of NDT lies in its modular frame- work, enabling the stabilization of a global network using only open-loop characteristics of agents and their local con- trollers. By decoupling agent-lev el dynamics from the network topology , any heterogeneous agents can be integrated into (13) once they satisfy dissipati vity conditions, which can be formulated into (13c). In addition, various dissipati vity char- acterizations can be employed to formulate (13c) as an linear matrix inequality (LMI). For instance, [28, Equation 3.2], [29, Theorem 3.1], or [30, Theorem 4] provide suitable formula- tions for agents modeled as L TI systems, L TI systems with 4 IEEE CONTROL SYSTEMS LETTERS , V OL. XX, NO . XX, XXXX 2017 input/output/state delays, or input-affine nonlinear systems, respectiv ely . If dissipativity parameters of nonlinear agents are established a priori, e.g. ( Q 𝑝 𝑖 , S 𝑝 𝑖 , R 𝑝 𝑖 ) , these predefined parameters may be scaled by a design v ariable 𝜆 𝑖 ≥ 0 , yielding ( Q 𝑖 , S 𝑖 , R 𝑖 ) = 𝜆 𝑖 ( Q 𝑝 𝑖 , S 𝑝 𝑖 , R 𝑝 𝑖 ) , and (13c) can be omitted. I V . A L G O R I T H M The difficulty of (13) lies in its non-con vex penalty and the constraints (13b), (13d), and (13e). This section details an approach to solve (13) by reformulating (13b), (13d), and (13e) using Theorem 3 and computing a feasible sparse controller based on ADMM. A. Conv ex Ov erbounding of BMI Constraints This section derives conv ex reformulations of (13b), (13d), and (13e) based on Theorem 3 using giv en initial points. First, Corollary 1 provides LMI implying (13b). Cor ollary 1: Gi ven  K 0 ,  B 0 ,  C 0 ,  H 0 ,  K , and Y 0 , if there exist 𝛿  K , 𝛿  B , 𝛿  C , 𝛿  H , 𝛿  H , 𝛿 Y , and 𝜈 > 0 such that Y 0 + 𝛿 Y ≻ 0 and  Φ 11 ∗ Φ 21 Φ 22  ≺ 0 , (14) where Φ 11 =       He ( T 1 0 ( Y ( A +  B  K  C ) ) ) ∗ ∗ ( T 1 0 ( Y ( B +  B  K  H ) ) ) 𝑇 − 𝜈 I 0 T 1 0 ( C +  H  K  C ) T 1 0 (  H  K  H ) − 𝜈 I       , Φ 21 =            ( Y 0  B 0 𝛿  K ) 𝑇 + 𝛿  C 0 ( Ξ 𝑙 𝛿 Π 𝑘 ) 𝑇 ( Y 0  B 0 𝛿  K ) 𝑇 𝛿  H (  H 0 𝛿  K ) 𝑇 ( 𝛿 Y 0 𝛿  B ) 𝑇 + 𝛿  K  C 0 +  K 0 𝛿  C 𝛿  K  H 0 +  K 0 𝛿  H 0 𝛿  K  C 0 +  K 0 𝛿  C 𝛿  K  H 0 +  K 0 𝛿  H 𝛿  H 𝑇 L 𝑇 1 + 𝛿 Y L 𝑇 2 0            , Φ 22 =           − 2 I 0 ∗ ∗ ∗ 0 − 2 I ∗ ∗ ∗ 𝛿  K 𝛿  K − 2 I 0 ∗ 𝛿  K 𝛿  K 0 − 2 I 0 (  B 0 𝛿  K ) 𝑇 (  B 0 𝛿  K ) 𝑇 ( 𝛿  B ) 𝑇 0 − 2 I           , L 1 = 𝛿  B  K 0  C 0 +  B 0 𝛿  K  C 0 +  B 0  K 0 𝛿  C , and L 2 = 𝛿  B  K 0  H 0 +  B 0 𝛿  K  H 0 +  B 0  K 0 𝛿  H , then  K 0 + 𝛿  K ,  B 0 + 𝛿  B ,  C 0 + 𝛿  C ,  H 0 + 𝛿  H ,  H 0 + 𝛿  H , and Y 0 + 𝛿 Y are feasible points of (12). Moreover , (14) is always feasible if  K 0 ,  B 0 ,  C 0 ,  H 0 ,  H 0 , and Y 0 are feasible for (12). Pr oof: The proof follows by applying the o verbounding condition in (6) of Theorem 3 sequentially with G = I . First, use  K =  K 0 + 𝛿  K and  C =  C 0 + 𝛿  C . Next, use  H =  H 0 + 𝛿  H . Then use  B =  B 0 + 𝛿  B and  H =  H 0 + 𝛿  H . Finally , use Y = Y 0 + 𝛿 Y . The LMIs in Corollaries 2 and 3, introduced in [12], [20], imply Equations 13d and 13e, respectively . Cor ollary 2: [20] Giv en  A 0 ,  B 0 ,  C 0 ,  D 0 ,  Q 0 ,  S 0 ,  R 0 , and  P 0 , suppose there exist 𝛿  A , 𝛿  B , 𝛿  C , 𝛿  D , 𝛿  Q , 𝛿  S , 𝛿  R , and 𝛿 Y such that Y 0 + 𝛿 Y ≻ 0              T 1 0  −   C 𝑇  Q  C  C 𝑇  Q  C  D 𝑇  Q  C  R +  D 𝑇  Q  D  + He  P 0 0 − S 𝑇   K   ∗ ∗ ∗  𝛿  P 0 0 − 𝛿  S  + 𝛿  K − 2 I 0 0 − 1 2  Q 0  𝛿  C 𝛿  D  0 − 2 I ∗  𝛿  C − 𝛿  Q  C 0 𝛿  D − 𝛿  Q  D 0  0 − 1 2 𝛿  Q − 2 I              ⪯ 0 , (15) Then  A 0 + 𝛿  A ,  B 0 + 𝛿  B ,  C 0 + 𝛿  C ,  D 0 + 𝛿  D ,  Q 0 + 𝛿  Q ,  S 0 + 𝛿  S ,  R 0 + 𝛿  R , and  P 0 + 𝛿  P are feasible points of (13d). Moreover , if  A 0 ,  B 0 ,  C 0 ,  D 0 ,  Q 0 ,  S 0 ,  R 0 , and  P 0 are feasible for (13d), (15) is alw ays feasible. Cor ollary 3: [12] Giv en Q 0 , S 0 , R 0 , and H 0 , suppose there exist 𝛿 Q , 𝛿 S , 𝛿 R , and 𝛿 H such that          T 1 0 ( Q + He ( SH ) + H 𝑇 RH ) ∗ ∗ ∗ 𝛿 S 𝑇 + 𝛿 H − 2 I 0 0 1 2 R 0 𝛿 H + 𝛿 H 0 − 2 I ∗ 𝛿 RH 0 + 𝛿 H 0 1 2 𝛿 R − 2 I          ≺ 0 , (16) Then Q 0 + 𝛿 Q , S 0 + 𝛿 S , R 0 + 𝛿 R , and H 0 + 𝛿 H are feasible points of (13e). Moreover , if Q 0 , S 0 , R 0 , and H 0 are feasible for (13e), (16) is always feasible. Applying Corollaries 1 to 3 yields the conv ex problem arg min 𝛿  K , 𝛿 F , 𝛿  F , 𝛿 H 𝐽 (  K ) , (17a) s.t. N ( 𝛿  K , 𝛿 F , 𝛿  F , 𝛿 H ) ⪯ 0 , (17b) to construct the centralized controller for given initial feasible points, where N ( 𝛿  K , 𝛿 F , 𝛿  F , 𝛿 H ) composes: (14) to ensure that H ∞ norm of the network; agent dissipativity requirements encoded as LMIs; (15) to impose controller dissipativity; and (16) to ensure that the network is stable via NDT. B. Initialization Solving (17) requires a set of initial feasible points, {  A 0 𝑖 ,  B 0 𝑖 ,  C 0 𝑖 ,  D 0 𝑖 } for 𝑖 ∈ N 𝑁 and { Q 0 , S 0 , R 0 , H 0 } . If all agents are open-loop stable, the technique in [20, Section 6] pro vides this. Otherwise, local controllers 𝒞 𝑖 can be designed via standard synthesis procedures, such as PID, Linear Quadratic Gaussian (LQG), or H ∞ design, while assuming a decentralized struc- ture by fixing  H 𝑦 = I and  H  𝑦 = I . If the initial design fails to satisfy the dissipativity/stability constraints, the feasibility test is repeated with increased controller gains until a v alid feasible point is achiev ed. The iterativ e relaxation approach of [23] can also be used. After feasible local controllers are obtained, solving (17) yields an initial centralized feasible controller with dense initial interconnection matrices  H 0 𝑦 and  H 0  𝑦 . C . Sparsity Promotion Once the initial centralized feasible points are obtained, the sparse topology can be determined by arg min 𝛿  K , 𝛿 F , 𝛿  F , 𝛿 H 𝐽 ( 𝛿  K ) + 𝛾 𝑔 ( H 0 + 𝛿 H ) , (18a) s.t. N ( 𝛿  K , 𝛿 F , 𝛿  F , 𝛿 H ) ⪯ 0 (18b) where the penalty function 𝑔 ( H 0 + 𝛿 H ) promotes sparsity in the global controller . Following [12, Section IV C.], we consider the weighted ℓ 1 , 𝑔 1 ( H ) , and cardinality penalty , 𝑔 0 ( H ) , 𝑔 1 ( H 0 + 𝛿 H ) =  𝑖 , 𝑗 ∈ N 2 𝑁 min  ∥ ( H 0 + 𝛿 H ) 𝑖 𝑗 ∥ − 1 𝐹 , 𝜖 − 1 𝑙  ∥ ( H 0 + 𝛿 H ) 𝑖 𝑗 ∥ 𝐹 , (19) 𝑔 0 ( H 0 + 𝛿 H ) =  𝑖 , 𝑗 ∈ N 2 𝑁 card  ∥ ( H 0 + 𝛿 H ) 𝑖 𝑗 ∥ 𝐹  . (20) The problem is con ve x with (19), but not with (20). The ADMM iteration from [12, Section IV C.] can be employed A UTHOR et al. : PREP ARA TION OF P APERS FOR TEXTSCIEEE CONTROL SYSTEMS LETTERS (NO VEMBER 2021) 5 with (20) as 𝛿 H 𝑟 + 1 = arg min 𝛿  K , 𝛿 F , 𝛿  F , 𝛿 H 𝐽 ( 𝛿  K ) + 𝜌 2 ∥ H 0 + 𝛿 H − Z 𝑟 + 𝚲 𝑟 ∥ 2 𝐹 (21a) s.t. N ( 𝛿  K , 𝛿 F , 𝛿  F , 𝛿 H ) , Z 𝑟 + 1 = arg min Z 𝛾 𝑔 ( Z ) + 𝜌 2 ∥ H 0 + 𝛿 H 𝑟 + 1 − Z + 𝚲 𝑟 ∥ 2 𝐹 , (21b) 𝚲 𝑟 + 1 = 𝚲 𝑟 + ( H 0 + 𝛿 H 𝑟 + 1 − Z 𝑟 + 1 ) , (21c) where Z is the clone of H 0 + 𝛿 H , 𝚲 is the dual variable, 𝑟 is the iteration index of ADMM, and 𝜌 > 0 is the augmented Lagrangian parameter . The block component of the unique solution to (21b), ( Z 𝑟 + 1 ) 𝑖 𝑗 , is ( V ) 𝑖 𝑗 if ∥ ( V ) 𝑖 𝑗 ∥ 𝐹 >  2 𝛾 / 𝜌 and 0 otherwise, where V = H 0 + 𝛿 H 𝑟 + 1 + 𝚲 𝑟 [9]. The stopping criteria of ADMM are 𝑟 𝑝 = ∥ H 0 + 𝛿 H 𝑟 − Z 𝑟 ∥ 𝐹 ∥ Z 𝑟 ∥ 𝐹 ≤ 𝜖 𝑝 and 𝑟 𝑑 = ∥ Z 𝑟 − Z 𝑟 − 1 ∥ 𝐹 ∥ Z 𝑟 ∥ 𝐹 ≤ 𝜖 𝑑 . After solving (18), the local controllers and networks are up- dated to  A 1 𝑖 =  A 0 𝑖 + 𝛿  A ★ 𝑖 ,  B 1 𝑖 =  B 0 𝑖 + 𝛿  B ★ 𝑖 ,  C 1 𝑖 =  C 0 𝑖 + 𝛿  C ★ 𝑖 ,  D 1 𝑖 =  D 0 𝑖 + 𝛿  D ★ 𝑖 for all 𝑖 ∈ N 𝑁 , and H 1 = H 0 + 𝛿 H ★ , where 𝛿  A ★ 𝑖 , 𝛿  B ★ 𝑖 , 𝛿  C ★ 𝑖 , 𝛿  D ★ 𝑖 for all 𝑖 ∈ N 𝑁 , and 𝛿 H ★ are the optimal perturbation obtained from the solution. The matrices Q 1 , S 1 , and R 1 are updated accordingly . The subspace H is then defined as the set of block matrices that share the same sparsity pattern as H 1 . D . Structured ICO Once the sparse structure H is identified, we consider arg min 𝛿  K , 𝛿 F , 𝛿  F , 𝛿 H 𝑁  𝑖 = 1 𝐽 𝑖 ( 𝛿  K 𝑖 ) , (22a) s.t. N ( 𝛿  K , 𝛿 F , 𝛿  F , 𝛿 H )⪯ 0 , H 𝑘 + 𝛿 H ∈ H , (22b) and apply ICO to (22) to determine the optimal controller parameters  K ★ , using the feasible points {  A 1 𝑖 ,  B 1 𝑖 ,  C 1 𝑖 ,  D 1 𝑖 } for all 𝑖 ∈ N 𝑁 , and { Q 1 , S 1 , R 1 , H 1 } obtained from the process in Section IV -C. ICO is applied to (22) instead of (18) since the cardinality penalty’ s discontinuity at zero inv alidates the con vergence guarantee of ICO [31]. The ov erall procedure of Section IV is summarized in Algorithm 1. E. Conv ergence of Algor ithm 1 Algorithm 1 combines two stages, sparsity promotion and structured ICO. The ICO stage is recursively feasible and con- ver ges to a local optimum, as each iteration is initialized with the previous feasible point and guarantees a non-increasing cost. In contrast, the con ver gence of the sparsity-promotion stage depends on the penalty function: 𝑔 1 ensures conv ergence because the problem is formulated as a semidefinite program (SDP), but 𝑔 0 is noncon ve x and therefore lacks con vergence guarantees. Howe ver , as noted in [9], ADMM con ver ges regardless with a suf ficiently large 𝜌 . V . N U M E R I C A L E X A M P L E Sparse controllers are synthesized using Algorithm 1 for a networked system with polytopic uncertainty . W e consider 𝑁 = 10 agents randomly distributed over a 2 × 2 grid. In a modified version of [32], the individual agents’ dynamics deviate randomly from their nominal models, ¤ x 𝑖 = A 𝑛 𝑖 x 𝑖 +  0 1   e 𝑖 +  𝑗 ≠ 𝑖 𝑒 − 𝛼 ( 𝑖 − 𝑗 ) y 𝑗  , y 𝑖 =  1 1  x 𝑖 , Algorithm 1 Sparsity-Promoting Controller Synthesis Require: 𝑔 ,  K 0 𝑖 for 𝑖 ∈ N 𝑁 , Q 0 , S 0 , R 0 , H 0 , 𝜖 𝑝 , 𝜖 𝑑 , 𝜖 𝑙 , 𝜖 Ensure:  K 𝑖 for 𝑖 ∈ N 𝑁 , H 1: if 𝑔 ( H 0 + 𝛿 H ) follo ws (19) then find 𝛿 H ★ by solving (18) 2: else if 𝑔 ( H 0 + 𝛿 H ) follows (20) then 3: while 𝑟 𝑝 > 𝜖 𝑝 or 𝑟 𝑑 > 𝜖 𝑑 do 4: Find 𝛿 H 𝑘 + 1 by solving (21a) 5: Find Z 𝑘 + 1 by solving (21b) 6: Find 𝚲 𝑘 + 1 by solving (21c) 7: end while 8: end if 9: Set H 1 = H 0 + 𝛿 H ★ , and update feasible points,  K 1 𝑖 for 𝑖 ∈ N 𝑁 , Q 1 S 1 , and R 1 analogous to H 1 . 10: Define the structured subspace H 11: while 𝐽 𝑖 ( 𝛿  K 𝑘 𝑖 ) − 𝐽 𝑖 ( 𝛿  K 𝑘 + 1 𝑖 ) 𝐽 ( 𝛿  K 𝑘 + 1 𝑖 ) ≰ 𝜖 for all 𝑖 ∈ N 𝑁 do 12: Solve (22) using  K 𝑘 𝑖 for 𝑖 ∈ N 𝑁 , Q 𝑘 , S 𝑘 , R 𝑘 , and H 𝑘 13: Set H 𝑘 + 1 = H 𝑘 + 𝛿 H ★ , and other feasible points,  K 𝑘 + 1 𝑖 , for 𝑖 ∈ N 𝑁 , Y 𝑘 + 1 , Q 𝑘 + 1 , S 𝑘 + 1 , and R 𝑘 + 1 analogous to H 1 14: end while 20 40 60 80 100 120 140 160 180 200 17 18 19 20 21 22 23 Nonzero blocks in  H 𝑦 and  H  𝑦 𝐽 (  K ) W eighted ℓ 1 Cardinality Centralized Decentralized [20] 5% threshold (a) Different sparsity le vels 0 20 40 60 80 100 120 140 160 180 200 16 17 18 19 20 21 22 23 Sample numbers 𝐽 (  K ) H ∞ -control W eighted ℓ 1 Cardinality (b) Different uncertain agents Fig. 2. H ∞ -norm of resulting sparse controllers; In Figure 2b, the number of nonzero bloc ks in e H 𝒚 and e H b 𝒚 is 100 and 102 for weighted ℓ 1 norm and cardinality , respectively . where A 𝑛 𝑖 =  1 1 1 2  for 𝑖 ∈ N 5 and  − 2 1 1 − 3  otherwise, 𝛼 = 0 . 1823 , x 𝑖 ∈ R 2 , and e 𝑖 ∈ R . The first 5 agents are nominally unstable, while the remaining 5 are stable. From local dynam- ics, H has blocks ( H ) 𝑖 𝑖 = 0 and ( H ) 𝑖 𝑗 = 𝑒 − 𝛼 ( 𝑖 − 𝑗 ) for 𝑖 ≠ 𝑗 . The actual dynamics A 𝑖 of each agent are uniformly dis- tributed within ± 4% of their nominal values A 𝑛 𝑖 , so each agent’ s dynamics can be modeled as an L TI system with polytopic uncertainty , represented by a polyhedron with 16 vertices, where each vertex corresponds to the maximum or minimum of one parameter in A 𝑖 . T o enforce agent dissipativ- ity , [12, Lemma 2] is used in (13c), with a small adjustment to generalize from y = x in [12] to y = Cx here. After computing the local LQG controller with Q 𝑛 = 100 I 2 and R 𝑛 = 1 , an initial local feasible controller is found by scaling the transfer function of the LQG controller by 10 − 3 and adding feedforward gain − 10 for all 10 agents. The parameters are 𝜖 = 𝜖 𝑝 = 𝜖 𝑑 = 𝜖 𝑙 = 10 − 3 , and 𝜌 = 1000 . The weighting factor 𝛾 is varied ov er 20 logarithmically spaced points on the inv erv al [ 2 × 10 − 5 , 1 . 5 ] for the weighted ℓ 1 penalty , and [ 1 × 10 − 6 , 5 ] for the cardinality penalty . Figure 2a sho ws the H ∞ norm of the resulting controllers 6 IEEE CONTROL SYSTEMS LETTERS , V OL. XX, NO . XX, XXXX 2017 T ABLE I B E S T A N D W O R ST H ∞ - N O R M O F F I G U R E 2 B H ∞ -control W eighted ℓ 1 Cardinality Best 16.7350 17.1313 17.1111 W orst 22.3155 17.2547 17.2325 for differ ent sparsity lev els. Since the parameter 𝜌 is suffi- ciently large, the sparsity-promotion method using cardinality successfully con ver ges to a sparse controller structure. The H ∞ norm obtained using a decentralized controller network, which is equiv alent to the approach in [20], is 22 . 66 , whereas the centralized controller achieves 17 . 09 . Starting from the fully decentralized setup, both methods rapidly reduce the H ∞ norm and reach a value within 5% of the gap between decentralized and centralized case, which is 17 . 37 . Before reaching the 5% threshold, the cardinality-based method achieves better performance than the weighted ℓ 1 -norm for the same sparsity lev els. After reaching the 5% threshold, the two methods exhibit similar performance. The H ∞ norms of resulting closed-loop networks are ev aluated ov er 200 randomly generated true dynamics of each agent. For the comparison, a standard H ∞ control ap- proach is used to compute the optimal H ∞ controller for the nominal agent dynamics. As shown in Figure 2b, the H ∞ norm obtained using the H ∞ controller exhibits significant fluctuations, whereas the norms obtained using the proposed approaches remain nearly constant while requiring half as many communication links. This behavior arises because the dissipativity constraint accounts for all admissible realizations of the true agent dynamics. Consequently , T able I sho ws that the worst-case H ∞ norm achiev ed by the proposed approach is significantly smaller than that obtained by the standard optimal H ∞ controller . V I . C O N C L U S I O N This paper presented the approaches of synthesizing dissipativity-based dynamic output feedback controllers and a sparse communication network simultaneously using NDT. This extends [12] from state to output feedback. W e first construct a condition to ensure the global controller is well- posed. Under this well-posed condition, the optimization prob- lem with NDT and a sparsity penalty was solved by mixing ADMM and ICO. 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