Robust Output Feedback Consensus for Networked Heterogeneous Nonlinear Negative-Imaginary Systems
aa r X i v : . [ ee ss . S Y ] J un Robust Output Feedback Consensus for Networked HeterogeneousNonlinear Negative-Imaginary Systems
Kanghong Shi † , Igor G. Vladimirov † , Ian R. Petersen † Abstract — This paper provides a control protocol for therobust output feedback consensus of networked heterogeneousnonlinear negative-imaginary (NI) systems. Heterogeneous non-linear output strictly negative-imaginary (OSNI) controllers areapplied in positive feedback according to the network topologyto achieve output feedback consensus. The main contribution ofthis paper is extending the previous studies of the robust out-put feedback consensus problem for networked heterogeneouslinear NI systems to nonlinear NI systems. Output feedbackconsensus is proved by investigating the internal stability ofthe closed-loop interconnection of the network of heterogeneousnonlinear NI plants and the network of heterogeneous nonlinearOSNI controllers through the network topology. The networkof heterogeneous nonlinear NI systems is proved to be alsoa nonlinear NI system, and the network of heterogeneousnonlinear OSNI systems is proved to be a nonlinear NI system.Under suitable conditions, the nonlinear OSNI controllers leadto the convergence of the outputs of all nonlinear NI plants to acommon limit trajectory, regardless of the system model of eachplant. Hence, the protocol is robust with respect to uncertaintyin the system models of the heterogeneous nonlinear NI plantsin the network. This paper also describes some typical first-order and second-order nonlinear OSNI systems that can beused as controllers for the robust output feedback consensus ofheterogeneous nonlinear NI plants.
Index Terms — nonlinear negative-imaginary systems, nonlin-ear output strictly negative-imaginary systems, heterogeneoussystems, consensus, robust control.
I. I
NTRODUCTION
Negative-imaginary (NI) systems theory was introducedby Lanzon and Petersen in 2008 [1]. NI systems theory hasattracted a lot of interest among control theory researchers(see [2]–[7], etc.). NI systems theory complements positive-real (PR) systems theory because it can be applied to systemswith a relative degree from zero to two, while PR systemstheory can only deal with systems with relative degree ofzero or one. Typical NI systems are mechanical systems withco-located force actuators and position sensors. Positive-position feedback control is often used for NI systems, whichcan be applied to flexible structures with highly resonantdynamics due to the robustness of NI systems with respectto uncertainty in system models and external disturbances. NIsystems theory has already achieved success in some fields,such as nano-positioning (see [8]–[10], etc.).
This work was supported by the Australian Research Council under grantDP190102105. † Research School of Electrical, Energy and Materi-als Engineering, College of Engineering and ComputerScience, Australian National University, Canberra, Acton,ACT 2601, Australia. [email protected] , [email protected] , [email protected] . NI systems theory was recently extended to nonlinearsystems [11]. A system is said to be nonlinear NI if it isdissipative with the supply rate w = u T ˙ y , where u and y arethe input and output of the system, respectively, and y onlydepends on the system state x . While the positive-feedbackinterconnection of a linear NI system and a linear strictlynegative-imaginary (SNI) system is asymptotically stableif their cascaded DC gain is less than unity, the positive-feedback interconnection of a nonlinear NI system and a so-called weak strictly nonlinear NI system is proved in [11] tobe also asymptotically stable under reasonable assumptions.A class of NI systems called output strictly negative-imaginary (OSNI) systems was introduced in [12] and [13]for linear systems and was recently extended to nonlinearsystems in [14]. A nonlinear system is said to be nonlinearOSNI if it is dissipative with the supply rate w ( u, ˙ y ) = u T ˙ y − ǫ | ˙ y | , where u , x and y are the input, state andoutput of the system, respectively. Also, y is only dependenton x . Here, ǫ > is an index that quantifies the level ofoutput strictness of the system. It is proved in [14] that theclosed-loop interconnection of a nonlinear NI system and anonlinear OSNI system is asymptotically stable under certainconditions.A robust cooperative control problem for networked het-erogeneous NI systems is investigated in [15] for linearsystems. A network of systems is said to have outputfeedback consensus if the outputs of all subsystems convergeto a common limit trajectory under the effect of the networkcommunication between subsystems. With certain conditionssatisfied, the outputs of heterogeneous linear NI systemsconnected according to an undirected connected graph canconverge to the same limit trajectory if edge-based linear SNIcontrollers are connected to the plants in positive feedbackaccording to the network topology.This paper extends the investigation of the output feedbackconsensus problem for networked heterogeneous linear NIsystems in [15] to nonlinear NI systems by using the resultsin [11] and [14]. Output feedback consensus of networkedheterogeneous nonlinear NI systems is proved by analysingthe stability of the closed-loop interconnection of a networkof heterogeneous nonlinear NI plants and a network ofheterogeneous nonlinear OSNI controllers. The main con-tribution of this paper is providing a control framework tosynchronise multiple heterogeneous nonlinear NI systemsunder certain conditions. The protocol is robust with respectto uncertainty in the system models of the heterogeneousnonlinear NI systems. This paper is also applicable to real-world control systems considering that differences in systemodels are inevitable in a network of real-world plants orcontrollers due to manufacturing uncertainties, even if theyare designed to be identical. This paper provides theoreticalsupport for the feasibility of real-world synchronisation prob-lems of nonlinear NI systems when uncertainties are takeninto consideration.Notation: The notation in this paper is standard. R and C denote the fields of real and complex numbers, respectively. R m × n and C m × n denote the spaces of real and complexmatrices of dimension m × n , respectively. A T denotes thetranspose of matrix A . u denotes a constant vector or scalar. I n is the n × n identity matrix. A ⊗ B denotes the Kroneckerproduct of matrices A and B . For a nonlinear system H withinput u and output y , y = H ( u ) describes its input-outputrelationship.Graph theory preliminaries: G = ( V , E ) , where V = { v , v , · · · , v n } and E = { e , e , · · · , e l } ⊆ V × V ,describes an undirected graph with n nodes and l edges. Thesymmetric adjacency matrix A = [ a ij ] ∈ R n × n is defined sothat a ii = 0 , and ∀ i = j , a ij = 1 if ( v i , v j ) ∈ E and a ij = 0 otherwise. A sequence of unrepeated edges in E that joins asequence of nodes in V defines a path. An undirected graphis connected if there is a path between every pair of nodes.Given an undirected graph G , a corresponding directed graphcan be obtained by defining a direction for each edge of G .The incidence matrix Q = [ q ev ] ∈ R l × n of a directed graphis defined so that the elements in Q are given by q ev := if v is the initial vertex of edge e, − if v is the terminal vertex of edge e, if v does not belong to edge e. In this paper, the initial and terminal vertices of an edge in adirected graph can both send information to each other. Foran undirected graph G , the choice of a corresponding directedgraph is not unique. However, the Laplacian matrix L n of G has the following relationship with the incidence matrix Q of any directed graph corresponding to G : L n = Q T Q .II. P RELIMINARIES
Here, we recall the definitions of nonlinear negative-imaginary systems and nonlinear output strictly negative-imaginary systems.Consider the following general nonlinear system: ˙ x ( t ) = f ( x ( t ) , u ( t )); (1) y ( t ) = h ( x ( t )) (2)where f : R p × R m → R p is a Lipschitz continuous functionand h : R p → R m is a class C function. Definition 1: [14] The system (1), (2) is said to be anonlinear negative-imaginary system if there exists a positivedefinite storage function V : R p → R of class C such that ˙ V ( x ( t )) ≤ u ( t ) T ˙ y ( t ) (3)for all t ≥ . Definition 2: [14] The system (1), (2) is said to be anonlinear output strictly negative-imaginary system if there exists a positive definite storage function V : R p → R ofclass C and a constant ǫ > such that ˙ V ( x ( t )) ≤ u ( t ) T ˙ y ( t ) − ǫ | ˙ y ( t ) | (4)for all t ≥ . The index ǫ quantifies the level of outputstrictness of the system.III. R OBUST OUTPUT FEEDBACK CONSENSUS
PSfrag replacements H p u p y p u p y p u pn y pn H p H p H pn ...... ... Fig. 1. System H p : a nonlinear system consisting of n independent andheterogeneous nonlinear systems H pi ( i = 1 , , · · · , n ) , with independentinputs and outputs combined as the input and output of the networked system H p . Consider n heterogeneous nonlinear systems H pi ( i =1 , , · · · , n ) described as ˙ x pi ( t ) = f pi ( x pi ( t ) , u pi ( t )); (5) y pi ( t ) = h pi ( x pi ( t )) (6)where f pi : R p × R m → R p is a Lipschitz continuousfunction and h pi : R p → R m is a class C function. Theyoperate independently in parallel and each of them has itsown input u pi ∈ R m and output y pi ∈ R m , ( i = 1 , , · · · , n ),which is shown in Fig. 1. The subscript “ p ” indicates thatthis system will play the role of a plant in what follows. Wecombine the inputs and outputs respectively as the vectors U p = u p u p ... u pn ∈ R nm × , and Y p = y p y p ... y pn ∈ R nm × . Lemma 1:
If the H pi are nonlinear NI systems for all i =1 , , ..., n , then H p is also a nonlinear NI system. Proof:
According to Definition 1, each nonlinear NIsystem H pi ( i = 1 , , · · · , n ) must have a correspond-ing positive definite storage function V pi ( x pi ) such that ˙ V pi ( x pi ) ≤ u Tpi ˙ y pi , where x pi is the state of the system H pi . We define the storage function for the system H p as V p = P ni =1 V pi ( x pi ) , which is positive definite. Then ˙ V p = n X i =1 ˙ V pi ( x pi ) ≤ n X i =1 u Tpi ˙ y pi = U Tp ˙ Y p , (7)which implies the NI inequality (3). Therefore, H p is anonlinear NI system.Now we give a definition of output feedback consensusfor a network of systems as shown in Fig. 1. efinition 3: A distributed output feedback control lawachieves output feedback consensus for a network of systemsif y pi ( t ) − y ss ( t ) → as t → + ∞ , ∀ i ∈ , , · · · , n . Here, y ss ( t ) is the limit trajectory.Consider a series of heterogeneous nonlinear OSNI sys-tems H ck ( k = 1 , , · · · , l ) applied as controllers corre-sponding to the edges in the network. The OSNI controllershave the following state-space models: ˙ x ck ( t ) = f ck ( x ck ( t ) , u ck ( t )); (8) y ck ( t ) = h ck ( x ck ( t )) , (9)where f ck : R q × R m → R q is a Lipschitz continuousfunction and h ck : R q → R m is a class C function. Theyoperate independently in parallel and each of them has itsown input u ck ∈ R m and output y ck ∈ R m , ( k = 1 , , · · · , l ),which is shown in Fig. 2. We combine the inputs and outputsrespectively as the vectors U c = u c u c ... u cl ∈ R lm × , and Y c = y c y c ... y cl ∈ R lm × . PSfrag replacements H c u c y c u c y c u cl y cl H c H c H cl ...... ... Fig. 2. System H c : a nonlinear system consisting of l independent andheterogeneous nonlinear systems H ck ( k = 1 , , · · · , l ) , with independentinputs and outputs combined as the input and output of the networked system H c . Let us consider the networked controllers connected ac-cording to the graph network topology ˆ H c as shown inFig. 3, where Q is the incidence matrix of a directedgraph that represents the communication links between theheterogeneous nonlinear NI plants.PSfrag replacements H c H c H c H cl U c Y c . . . ˆ H c ˆ U c ˆ Y c Q ⊗ I m Q T ⊗ I m Fig. 3. Heterogeneous nonlinear OSNI controllers connected according tothe directed graph network topology.
For the system ˆ H c , we have the following lemma. Lemma 2:
If the H ck are nonlinear OSNI systems for all k = 1 , , · · · , l , then the system ˆ H c is a nonlinear NI system. Proof:
Let ˆ u ci denote the i -th ( i = 1 , , · · · , n ) m × vector in the input ˆ U c and let ˆ y ci denote the i -th m × vectorin the output ˆ Y c of the system ˆ H c . Let u ck denote the k -th( k = 1 , , · · · , l ) m × vector in the input U c and let y ck denote the k -th m × vector in the the output Y c of thesystem H c . Using the properties of the incidence matrix Q ,the following equations are obtained: u ck = n X i =1 q ki ˆ u ci , ˆ y ci = l X k =1 q ki y ck . (10)For every nonlinear OSNI system H ck , we have a positivedefinite storage function V ck ( x ck ) and a constant index ǫ k > such that ˙ V ck ( x ck ) ≤ u Tck ˙ y ck − ǫ k | ˙ y ck | , (11)where ǫ k is the level of output strictness of the system H ck .From (10) and (11), we obtain ˙ V ck ( x ck ) ≤ n X i =1 q ki ˆ u Tci ˙ y ck − ǫ k | ˙ y ck | , (12)For the system ˆ H c , we define its storage function ˆ V c as thesum of the storage functions of all the networked controllers: ˆ V c := l X k =1 V ck ( x ck ) > . The time derivative of ˆ V c satisfies the following inequalitydue to (10) and (12): ˙ˆ V c = l X k =1 ˙ V ck ( x ck ) ≤ l X k =1 n X i =1 q ki ˆ u Tci ˙ y ck − l X k =1 ǫ k | ˙ y ck | = n X i =1 ˆ u Tci l X k =1 q ki ˙ y ck − l X k =1 ǫ k | ˙ y ck | = n X i =1 ˆ u Tci ˙ˆ y ci − l X k =1 ǫ k | ˙ y ck | = ˆ U Tc ˙ˆ Y c − l X k =1 ǫ k | ˙ y ck | . (13)Hence, the system ˆ H c satisfies the definition of a nonlinearNI system. In addition, the term P lk =1 ǫ k | ˙ y ck | represents anon-negative output dissipation that comes from all of thecontrollers. This completes the proof.We assume that the following conditions are satisfied.a) For each individual nonlinear OSNI controller H ck ( k =1 , , · · · , n ) with input u ck ( t ) , state x ck ( t ) and output y ck ( t ) described by the state-space model (8), (9), suppose:Assumption I: Over any time interval [ t a , t b ] where t b >t a , y ck ( t ) remains constant if and only if x ck ( t ) remainsconstant; i.e., ˙ y ck ( t ) ≡ ⇐⇒ ˙ x ck ( t ) ≡ .Assumption II: Over any time interval [ t a , t b ] where t b >t a , x ck ( t ) remains constant only if u ck ( t ) remains constant;i.e., ˙ x ck ( t ) ≡ ⇒ ˙ u ck ( t ) ≡ .ssumption III: In the single-input single-output (SISO)case, if the system H ck is in steady state; i.e., u ck ( t ) ≡ ¯ u ck , x ck ( t ) ≡ ¯ x ck and y ck ( t ) ≡ ¯ y ck , then ¯ y ck > ⇐⇒ ¯ u ck > , ¯ y ck = 0 ⇐⇒ ¯ u ck = 0 and ¯ y ck < ⇐⇒ ¯ u ck < .PSfrag replacements H p ¯ U p Y p ( t ) ˆ H c ˆ U c ( t ) ¯ˆ Y c Fig. 4. Open-loop interconnection of the networked nonlinear NI plants H p and the networked nonlinear OSNI controllers ˆ H c . b) For the open-loop interconnection of the systems H p and ˆ H c shown in Fig. 4, suppose:Assumption IV: Given any constant input U p ( t ) ≡ ¯ U p for the system H p , we obtain a corresponding output Y p ( t ) ,which is not necessarily constant. Given Y p ( t ) as input ˆ U c ( t ) to the system ˆ H c , if the corresponding output of the system ˆ H c is a constant ˆ Y c ( t ) ≡ ¯ˆ Y c , then there exists a constant γ ∈ (0 , such that ¯ U p and ¯ˆ Y c satisfy ¯ U Tp ¯ˆ Y c ≤ γ (cid:12)(cid:12) ¯ U p (cid:12)(cid:12) . (14)Now consider the closed-loop interconnection of the net-worked plants shown in Fig. 1 and the networked controllersshown in Fig. 3 in positive feedback, which is depicted inFig. 5. In this paper, the robust output consensus of hetero-geneous nonlinear NI plants is achieved by constructing acontrol system with the block diagram shown in Fig. 5 andchoosing suitable controllers that satisfy certain conditions.The connections between the plants and controllers can bebetter visualised from the undirected graph, as shown in theexample in Fig. 6.PSfrag replacements H p H c H p H p H pn H c H c H cl U p Y p U c Y c . . .. . . ˆ H c ˆ U c ˆ Y c Q T ⊗ I m Q ⊗ I m Fig. 5. Positive feedback interconnection of heterogeneous nonlinearNI plants and nonlinear OSNI controllers according to the directed graphnetwork topology.
The nodes p i ( i = 1 , · · · , in this example) represent theheterogeneous nonlinear NI plants, while the heterogeneousnonlinear OSNI controllers c k ( k = 1 , · · · , in this example)correspond to the edges. Given any directed graph corre-sponding to the graph in Fig. 6 with the incidence matrix Q , each edge will have a direction. Then the correspondingconnection between the plants and the controller is as shownin Fig. 7. The controller takes the difference between theoutputs of the plants as its input and feeds back its outputPSfrag replacements p p p p p c c c c c Fig. 6. An example of the networked connection of plants and controllers. to the plants with a positive or negative sign correspondingto the edge direction. Each plant takes the sum of all theoutputs of the controllers connected to it with correct signsas its input.PSfrag replacements p p c ++ + −− d = 0 d = 0 y p y p y c = H c ( y p − y p ) Fig. 7. Detailed block diagram corresponding to a pair of nodes connectedby an edge.
For simplicity, we consider SISO systems (with m = 1 )in the following theorem. Theorem 1:
Consider an undirected connected graph G that models the communication links for a network ofheterogeneous nonlinear NI systems H pi ( i = 1 , , · · · , n ) as shown in Fig. 1, and any directed graph corresponding to G with the incidence matrix Q . Also, consider the hetero-geneous nonlinear OSNI control laws H ck ( k = 1 , , · · · , l ) for all the edges. Suppose Assumptions I-IV are satisfied andthe storage function, defined as W := V p + ˆ V c − Y Tp ˆ Y c , is positive definite, where V p and ˆ V c are positive definitestorage functions that satisfy (7) for the system H p and (13)for the system ˆ H c , respectively. Here, Y p and ˆ Y c are outputsof the systems H p and ˆ H c , respectively. Then the robustoutput feedback consensus can be achieved via the protocol U p = ( Q T ⊗ I m ) H c (( Q ⊗ I m ) Y p ) , or equivalently, u pi = l X k =1 q ki H ck n X j =1 q kj y pj , for each plant i , as shown in Fig. 5, where P nj =1 q kj y pj represents the difference between the outputs of the plantsconnected by the edge e k . Proof:
We apply the Lyapunov’s direct method and takethe time derivative of the storage function W . According to7) and (13), we have ˙ W = ˙ V p + ˙ˆ V c − ˙ Y Tp ˆ Y c − Y Tp ˙ˆ Y c = ˙ V p + ˙ˆ V c − U Tp ˙ Y p − ˆ U Tc ˙ˆ Y c ≤ − l X k =1 ǫ k | ˙ y ck | ≤ . (15)Hence, the closed-loop system is at least Lyapunov stable.Now we apply LaSalle’s invariance principle. According to(15), ˙ˆ W can remain zero only if P lk =1 ǫ k | ˙ y ck | remains zero,which means ˙ y ck ( t ) remains zero for all k = 1 , , · · · , l .According to Assumptions I and II, for the system H ck , ˙ y ck ( t ) ≡ implies ˙ x ck ( t ) ≡ , which holds only if ˙ u ck ( t ) ≡ . In other words, for all k = 1 , , · · · , l , the controllers H ck are in steady-state; i.e., u ck ( t ) ≡ ¯ u ck , x ck ( t ) ≡ ¯ x ck and y ck ( t ) ≡ ¯ y ck . Consider the setting in Fig. 5, in which U c ( t ) , Y c ( t ) and ˆ Y c ( t ) are all constant vectors; i.e., U c ( t ) ≡ ¯ U c , Y c ( t ) ≡ ¯ Y c and ˆ Y c ( t ) ≡ ¯ˆ Y c . Therefore, U p ( t ) ≡ ¯ U p also remains constant. ˆ U c ( t ) and Y p ( t ) are not necessarilyconstant. According to the closed-loop setting that ¯ U p ≡ ¯ˆ Y c ,the inequality (14) implies ¯ U Tp ¯ˆ Y c = (cid:12)(cid:12) ¯ U p (cid:12)(cid:12) ≤ γ (cid:12)(cid:12) ¯ U p (cid:12)(cid:12) . This condition can only hold if ¯ U p = 0 , which implies ¯ˆ Y c =0 . We will now show that since the controllers H ck arein steady-state for all k = 1 , , · · · , n , ¯ˆ Y c = 0 implies ¯ y ck = 0 for all k = 1 , , · · · , l , according to AssumptionIII. We will show this by contradiction. Indeed, suppose ∃ k ∈ { , , · · · , n } such that ¯ y ck = 0 , then we have ¯ u ck = 0 , according to Assumption III. This implies ∃ i, j ∈{ , , · · · , n } such that ¯ y pi = ¯ y pj . Consider the plants whoseoutput is equal to the maximum of all the plant outputs. Sincethe graph is connected and the outputs of all the plants arenot the same, there must be at least one plant p r with themaximum output ¯ y pr = max { ¯ y p , ¯ y p , · · · , ¯ y pn } connectedwith a plant p s with the output ¯ y ps < ¯ y pr by an edge e w .Then, we have ¯ u cw = ¯ y pr − ¯ y ps > if v r is the initial vertexof e w and ¯ u cw = ¯ y ps − ¯ y pr < if v r is the terminal vertex of e w . According to Assumption III, ¯ u cw > ⇐⇒ ¯ y cw > and ¯ u cw < ⇐⇒ ¯ y cw < . According to the distributedcontrol protocol, ¯ u pr = ¯ˆ y cr is the sum of all ¯ y cµ > if v r is the initial vertex of e µ minus the sum of all ¯ y cµ < if v r is the terminal vertex of e µ . Therefore, ¯ u pr = ¯ˆ y cr ispositive and this contradicts the condition ¯ˆ Y c = 0 . Thus, wecan conclude that ¯ y ck = 0 for k = 1 , , · · · , l .According to Assumption III, ¯ y ck = 0 implies ¯ u ck = 0 ,which implies y pi ( t ) ≡ y pj ( t ) for all ( v i , v j ) ∈ E . Thismeans the output consensus is achieved for all the heteroge-neous nonlinear NI plants. Otherwise, ˙ W cannot remain atzero and W will keep decreasing until ˙ W ≡ or the statesof all the plants p i ( i = 1 , , · · · , n ) converge to zero, whichalso implies the output consensus. This completes the proof. Remark 1:
The protocol in Theorem 1 is robust withrespect to uncertainty in system models for heterogeneousnonlinear NI plants connected in a network. Indeed, theoutput consensus can always be achieved with the protocolin Theorem 1, regardless of the system models for the het-erogeneous nonlinear NI systems connected in the network.We now provide some typical first-order and second-orderdynamical systems as possible choices for nonlinear OSNIcontrollers.
Lemma 3:
Consider a first-order system with the state-space model: ˙ x ( t ) = ρ ( x ( t )) + αu ( t ); y ( t ) = x ( t ) where x ( t ) , u ( t ) and y ( t ) are scalar functions of time, ρ : R → R is a Lipschitz continuous function and α > is aconstant. If the function V , given by V ( x ) = − α R x ρ ( z ) dz is positive definite, then the system is nonlinear OSNI withlevel of strictness ǫ ∈ (0 , α ] and with V being a storagefunction. Proof:
Let us define D ( x ) = ˙ V ( x ) − (cid:0) u ˙ y − ǫ ˙ y (cid:1) . Weprove in the following that D ( x ) ≤ for ǫ ∈ (0 , α ] . D ( x ) = ˙ V ( x ) − (cid:0) u ˙ y − ǫ ˙ y (cid:1) = ∂V ( x ) ∂x ˙ x − u ˙ x + ǫ ˙ x = ˙ x (cid:20) ∂V ( x ) ∂x − u + ǫ ˙ x (cid:21) = ( ρ ( x ) + αu ) (cid:20) − α ρ ( x ) − u + ǫ ( ρ ( x ) + αu ) (cid:21) = (cid:18) ǫ − α (cid:19) ( ρ ( x ) + αu ) ≤ when ǫ ∈ (cid:0) , α (cid:3) . Therefore, this system is a nonlinear OSNIsystem according to Definition 2. Lemma 4:
Consider a second order system with the fol-lowing state-space model: (cid:20) ˙ x ( t )˙ x ( t ) (cid:21) = (cid:20) x ( t ) η ( x ( t )) − βx ( t ) + αu ( t ) (cid:21) ; y ( t ) = x ( t ) where x ( t ) , x ( t ) , u ( t ) and y ( t ) are scalar functions oftime, η : R → R is a Lipschitz continuous function and α > and β > are constants. If the function V , givenby V ( x , x ) = − α R x η ( z ) dz + α x is positive definite,then the system is nonlinear OSNI with level of strictness ǫ ∈ (0 , βα ] and with V being a storage function. Proof:
Define D ( x , x ) = ˙ V ( x , x ) − (cid:0) u ˙ y − ǫ ˙ y (cid:1) .e prove in the following that D ( x , x ) ≤ for ǫ ∈ (0 , βα ] . D ( x , x ) = ˙ V ( x , x ) − (cid:0) u ˙ y − ǫ ˙ y (cid:1) = ∂V ( x , x ) ∂x ˙ x + ∂V ( x , x ) ∂x ˙ x − u ˙ x + ǫ ˙ x = − α η ( x ) x + 1 α x [ η ( x ) − βx + αu ] − ux + ǫx = (cid:18) ǫ − βα (cid:19) x ≤ when ǫ ∈ (cid:16) , βα i . Hence, this system is a nonlinear OSNIsystem according to Definition 2.IV. E XAMPLE
This section illustrates the robust output feedback consen-sus protocol described in Theorem 1 with an example ofnetworked heterogeneous pendulum systems.Consider three pendulum systems connected by an undi-rected connected graph G as shown in Fig. 8. The LaplacianPSfrag replacements e e Fig. 8. An undirected and connected graph consisting of three nodes. matrix of graph G is L = − − − − . To obtain adirected graph corresponding to G , we can arbitrarily decidethe direction of each edge. If we decide the directions ofthe edges as e = ( v , v ) and e = ( v , v ) , then theincidence matrix of the directed graph corresponding to G is Q = (cid:20) − − (cid:21) .These pendulum systems have the following state-spacemodel: (cid:20) ˙ x ˙ x (cid:21) = (cid:20) x ml ( − κx − mgl sin x + u ) (cid:21) ; y = x where m is the mass of each bob, l is the length of each rod, κ is the spring constant of a torsional spring installed in eachpivot and g ≈ . m/s is the gravitational acceleration. m , l and κ are different for the three heterogeneous networkedpendulums. For each pendulum system, the system state x is the counterclockwise angular displacement from thevertically downward position and x is the system angularvelocity. The system input u is an external torsional force inthe counterclockwise direction, and y is the system output.The system is a nonlinear NI system with the storagefunction V ( x , x ) = κx + ml x + mgl (1 − cos x ) .According to Lemma 3, we choose the following nonlinearOSNI system as the control law corresponding to each edge: ˙ x c = − βx c − φx c + αu c ; y c = x c where β > , φ > and α > are constants. The nonlinearOSNI property of this system can be proved with the storagefunction V c ( x c ) = β α x c + φ α x c .Suppose the pendulums have the following parameters:pendulum 1: m = 1 kg, l = 0 . m and κ = 3 N m/rad ; pendulum 2: m = 1 . kg, l = 0 . m and κ = 5 N m/rad ; pendulum 3: m = 0 . kg, l = 0 . m and κ = 6 N m/rad.
The parameters for the controllers are chosen to be:controller 1: β = 10 , φ = 15 and α = 20; controller 2: β = 20 , φ = 5 and α = 30 . According to Theorem 1, we control the pendulums withthe distributed control law: u p = H c ( y p − y p ) , u p = − H c ( y p − y p ) + H c ( y p − y p ) and u p = − H c ( y p − y p ) , respectively. Here, H ck ( · ) represents the output ofcontroller c k . It can be verified that Assumptions I-IV aresatisfied and the storage function of the entire networkedsystem is positive definite. As shown in Fig. 9, the pendulumsystems approach the same limit trajectory under the effectof the heterogeneous nonlinear OSNI controllers.PSfrag replacements O u t pu t Time (s)Output Feedback Consensus of Pendulums
Pendulum 1Pendulum 2Pendulum 3 -1-0.5000.51 5 10 15 20 25 30
Fig. 9. Robust output feedback consensus for a network of heterogeneouspendulum systems with heterogeneous nonlinear OSNI controllers applied.
V. C
ONCLUSION
This paper provides a protocol for the output feedbackconsensus problem of heterogeneous nonlinear NI systems.For a network of heterogeneous nonlinear NI systems con-nected by an undirected and connected graph, heterogeneousedge-based nonlinear OSNI controllers can be applied inpositive feedback through a network topology leading toconvergence of the outputs of the nonlinear NI plants to acommon limit trajectory if certain conditions are satisfied.This protocol is robust with respect to uncertainty in thesystem models of the nonlinear NI plants and the nonlinearOSNI controllers so that any network of heterogeneousnonlinear NI systems can be synchronised with suitablenonlinear OSNI controllers that are not necessarily identical.Some typical first-order and second-order nonlinear systemsare also provided as possible choices for nonlinear OSNIcontrollers.
EFERENCES[1] A. Lanzon and I. R. Petersen, “Stability robustness of a feedback inter-connection of systems with negative imaginary frequency response,”
IEEE Transactions on Automatic Control , vol. 53, no. 4, pp. 1042–1046, 2008.[2] I. R. Petersen and A. Lanzon, “Feedback control of negative-imaginarysystems,”
IEEE Control Systems Magazine , vol. 30, no. 5, pp. 54–72,2010.[3] J. Xiong, I. R. Petersen, and A. Lanzon, “A negative imaginary lemmaand the stability of interconnections of linear negative imaginarysystems,”
IEEE Transactions on Automatic Control , vol. 55, no. 10,pp. 2342–2347, 2010.[4] M. A. Mabrok, A. G. Kallapur, I. R. Petersen, and A. Lanzon,“Generalizing negative imaginary systems theory to include freebody dynamics: Control of highly resonant structures with free bodymotion,”
IEEE Transactions on Automatic Control , vol. 59, no. 10,pp. 2692–2707, 2014.[5] J. Xiong, I. R. Petersen, and A. Lanzon, “On lossless negativeimaginary systems,” in . IEEE,2009, pp. 824–829.[6] Z. Song, A. Lanzon, S. Patra, and I. R. Petersen, “A negative-imaginary lemma without minimality assumptions and robust state-feedback synthesis for uncertain negative-imaginary systems,”
Systems& Control Letters , vol. 61, no. 12, pp. 1269–1276, 2012.[7] M. Mabrok, A. G. Kallapur, I. R. Petersen, and A. Lanzon, “Spectralconditions for the negative imaginary property of transfer functionmatrices,”
IFAC proceedings volumes , vol. 44, no. 1, pp. 1302–1306,2011.[8] S. K. Das, H. R. Pota, and I. R. Petersen, “Resonant controllerdesign for a piezoelectric tube scanner: A mixed negative-imaginaryand small-gain approach,”
IEEE Transactions on Control SystemsTechnology , vol. 22, no. 5, pp. 1899–1906, 2014.[9] ——, “A MIMO double resonant controller design for nanoposition-ers,”
IEEE Transactions on Nanotechnology , vol. 14, no. 2, pp. 224–237, 2014.[10] ——, “Multivariable negative-imaginary controller design for damp-ing and cross coupling reduction of nanopositioners: a referencemodel matching approach,”
IEEE/ASME Transactions on Mechatron-ics , vol. 20, no. 6, pp. 3123–3134, 2015.[11] A. G. Ghallab, M. A. Mabrok, and I. R. Petersen, “Extending neg-ative imaginary systems theory to nonlinear systems,” in . IEEE, 2018, pp. 2348–2353.[12] P. Bhowmick and S. Patra, “On LTI output strictly negative-imaginarysystems,”
Systems & Control Letters , vol. 100, pp. 32–42, 2017.[13] P. Bhowmick and A. Lanzon, “Output strictly negative imaginarysystems and its connections to dissipativity theory,” in . IEEE, 2019, pp.6754–6759.[14] K. Shi, I. G. Vladimirov, and I. R. Petersen, “Robust output feed-back consensus for networked identical nonlinear negative-imaginarysystems,”
To appear in the 24th International Symposium on Mathe-matical Theory of Networks and Systems , 2020.[15] J. Wang, A. Lanzon, and I. R. Petersen, “Robust cooperative controlof multiple heterogeneous negative-imaginary systems,”