Gain-scheduled Leader-follower Tracking Control for Interconnected Parameter Varying SystemsAccepted for publication in International Journal of Robust and Nonlinear Control This work was supported by the Australian Research Council under Discovery Projects funding scheme (projects DP0987369 and DP120102152).

# Gain-scheduled Leader-follower Tracking Control for Interconnected Parameter Varying Systems

## Abstract

This paper considers the gain-scheduled leader-follower tracking control problem for a parameter varying complex interconnected system with directed communication topology and uncertain norm-bounded coupling between the agents. A gain-scheduled consensus-type control protocol is proposed and a sufficient condition is obtained which guarantees a suboptimal bound on the system tracking performance under this protocol. An interpolation technique is used to obtain a protocol schedule which is continuous in the scheduling parameter. The effectiveness of the proposed method is demonstrated using a simulation example.

Key words: Gain scheduling; leader-follower tracking control; interconnected parameter varying systems; interpolation technique

## 1 Introduction

In recent years, the topic of cooperative control has attracted much attention. The objective of the cooperative control problem is to propose distributed control laws to achieve a desired system behavior [1]. A related problem is that of synchronization of complex dynamical systems where all the components are controlled to exhibit a similar behavior by interconnecting them into a network [2, 3].

There are several approaches to the synchronization problem for complex systems consisting of many dynamic subsystems-agents. In the average consensus problem, which has received considerable attention in the last decade [4, 5], the objective is to synchronize all the agents to a common state. Another approach is to employ a suitable internal model for the system. For instance, it is shown in [6] that the existence of such an implicit internal model is necessary and sufficient for synchronization of a system of heterogeneous linear agents considered in that paper. However, the internal model approach does not directly address the overall system performance, since each system is controlled to follow its own internal model dynamics. In general, it may be difficult to determine an internal model that guarantees a good synchronization performance. Yet another approach is to designate one of the agents to serve as a leader, and design interconnections within the system so that the rest of the system follows the leader [7]; also, see [8] for a recent example. This idea leads to the leader-follower tracking problem, which is the main focus of this paper.

A common feature of many papers that consider the leader-follower problem is that the dynamics of agents are usually assumed to be dynamically decoupled [22, 23, 24, 25, 26, 27]. In many complex systems, however, interactions between subsystems are inevitable and must be taken into account [11]. Examples of systems with dynamical interactions between subsystems include spacecraft control systems and power systems [9]. For example, it was noted in [10] that weak uncertain couplings between generators is one of the reasons for dynamic instability in power systems and, therefore, they should not be neglected. This motivates us to consider the leader-follower tracking problem for interconnected systems. While we do not consider a specific application, our approach makes a step towards using consensus feedback for synchronization of such systems, compared to other contributions in the area of networked control systems which do not address the presence of interconnections.

In this paper, we are concerned with the leader-follower tracking problem for interconnected systems that depend on a time varying parameter. Analysis and control of linear parameter varying systems has attracted much attention in the last two decades due to their applications in flight control [12], turbofan engines and wind turbine systems [13, 14]. For example, an application of the distributed control approach to control and synchronization of wind generation systems modeled as parameter varying systems has been presented in [14]. Even though a discrete time model was considered in [14], parameter varying system modeling was motivated by the fact that the wind energy source driving wind turbines exhibits time-varying nature. In order to integrate a wind generation system into a power grid in a grid-friendly manner, the total power output of the wind turbines must be regulated to conform to a constant output. Each turbine then serves as a node and is controlled with respect to its power output by turning the blade pitch angle, and the value of the blade pitch angle is propagated through the communication network using the leader-follower algorithm. Thanks to this and many other potential applications, the theory of cooperative control for parameter varying systems has been gaining attention in recent years [15, 16].

The main contribution of this paper concerns the leader-follower control of parameter varying interconnected systems. A related problem, from the synchronization viewpoint, has been studied in [18, 19]. In the first reference, the synchronization problem is solved for heterogeneous systems which depend on the parameters in an affine fashion, under the assumption that the network of LPV agents allows for an internal model for synchronization while the agents are decoupled from each other. That is, the agents interact over the control protocol only. In [19], while the linear dependency on the parameter is not required, the leader is assumed to be given and be completely decoupled from the agents. In contrast with these references, we assume the leader to be chosen from the group of parameter varying agents; it is interconnected to the rest of the network, and the linear dependency on the parameter is not required.

The fact that the leader is interconnected with the followers makes it difficult to apply regular centralized tracking techniques to the problem under consideration. Many tracking techniques assume that the reference trajectory is generated by an exosystem or is a priori known, and is independent of the followers. This is not the case in this paper. Alternatively, a centralized tracking controller can be obtained by solving a stabilization problem for a large-scale system comprised of subsystems describing dynamics of individual tracking errors. This can be done by applying one of the existing centralized gain-scheduling robust control techniques, e.g., using the technique from [21]. However, in general the centralized controller obtained this way will not have a desired information structure. Indeed, a controller obtained in such a way will not generally guarantee that the information from subsystems that are not observed by a node is not required for feedback. One way to enforce such an information constraint is to employ a block diagonal Lyapunov function. This is the solution approach undertaken in this paper and is its main difference from solutions which could be obtained in a centralized setup.

Different from many papers that study synchronization of decoupled systems (including the above mentioned references [18, 19], also see [2, 3]), in the case of coupled systems, it is essential to distinguish the network representing the existing interactions between the subsystems (including interactions with the leader) from the network which realizes communication and control. This leads us to consider a two-network structure in this paper. The rationale for this is twofold. Firstly, our aim is to construct synchronization protocols for all agents excluding the leader, as we wish to avoid perturbing the leader dynamics (other than through an unavoidable physical coupling). Hence, the communication graph of the network must be different from the interconnection graph. The second reason is that the interconnection graph describes dynamical couplings between subsystems and thus may be different from the communication and control graph.

Our main result is a sufficient condition for the design of a gain-scheduled leader-follower control protocol for parameter varying multi-agent systems with directed control network topology and linear uncertain couplings subject to norm-bounded constraints. The condition involves checking feasibility of parameterized linear matrix inequalities (LMIs) at several operating points of the system. These LMIs serve as the basis for the design of a continuous (in the scheduling parameter) control protocol for coupled parameter varying systems, by interpolating consensus control protocols computed for those operating points; cf. [29, 19, 21]. Interpolation allows us to mitigate detrimental effects of transients arising when the system traverses from one operating condition to another.

The remainder of the paper proceeds as follows. In Section 2, we formulate the leader follower control problem for parameter varying coupled multi-agent systems and give some preliminaries. The main results are given in Section 3. Section 4 gives an example which illustrates the theory presented in the paper. Finally, the conclusions are given in Section 5.

## 2 Problem Formulation and Preliminaries

### 2.1 Interconnection and communication graphs

As stated in the introduction, in this paper we draw a distinction between the communication topology of the system used for control and the topology of interactions between subsystems. The example of the two-graph structure is shown in Fig. 1, where the edges of the interconnection graph are indicated by the solid lines and the edges of the communication graph are shown by the dashed lines. Both coupling and communication topologies are described in terms of directed graphs defined on the common node set . Without loss of generality, node 0 will be assigned to be the leader of the network, while the nodes from the set will represent the followers. The coupling graph and the communication graph will be denoted as and , respectively.

The edge sets of both graphs are subsets of the set and consist of pairs of nodes. The pair in each edge set denotes the directed edge which originates at node and ends at node . Edge in the edge set of the directed coupling graph , denoted , describes the fact that node is influenced by node through a directed interaction between nodes and .

Also, denotes an edge set of the communication graph , consisting of ordered pairs of nodes. Each such edge indicates the information flow between the nodes, that is, if and only if node obtains information from node , which it can use for control.

The adjacency matrix of the directed interconnection graph is denoted as , its -th entry is 1 if and only if . The adjacency matrix of the directed communication graph is defined in the same manner. Since according to a standard convention we assume that both the coupling graph and communication graph have no self-loops, the diagonal entries of and are all equal to zero.

To distinguish between the leader and the rest of the network, we define subgraphs , of the graphs , defined on the node set , with the edge sets and , respectively. Selecting the subgraphs and induces the partition of the matrices , ,

 Aφ=[0¯ddAφ0],Ac=[00gAc0],

where , , and , are the adjacency matrices of the subgraphs and , respectively. The zero row of the matrix reflects our assumption that the leader node does not receive the state information from other nodes of the network. However, the corresponding row of may be nonzero since the leader node may be physically coupled with some of the followers.

In this paper we are concerned with the case where only some of the followers receive the state information directly from the leader. We refer to such nodes as pinned nodes; the corresponding entries of the adjacency matrix , , and if node is not pinned. The matrix is referred to as a pinning matrix.

The Laplacian matrix of the subgraph is defined as , where is the in-degree matrix of , i.e., the diagonal matrix, whose diagonal elements are the in-degrees of the corresponding nodes of the graph , for , where are the elements of the th row of the matrix .

Finally, we give the definition and the notation for neighborhoods in the above graphs.

###### Definition 1

is a communication graph, each node represents a subsystem in the interconnected system and edge means that node obtain information from node .

###### Definition 2

is an interconnection graph, each node represents a subsystem in the interconnected system and edge indicates that subsystem is influenced by subsystem through a directed interaction between subsystems and .

Node is called a neighbor of node in the graph (or , , , respectively) if ( or , respectively). The sets of neighbors of node in the graphs and are denoted as , and , respectively. The sets of neighbors of node in the subgraphs and are denoted as and , respectively.

We conclude this subsection by specifying our standing requirements on the communication topology of the network which are assumed to hold throughout the paper.

###### Assumption 1

The communication subgraph contains a spanning tree, whose root node is the pinned node, i.e., .

###### Remark 1

Assumption 1 ensures that the information flows from the leader to its pinned followers, as well as between other followers [25, 28].

### 2.2 Notation

Throughout the paper, the following notation will be used.

and are a real Euclidean -dimensional vector space and a space of real matrices.

denotes an interval . The symbols , , , , etc., will represent various points within the interval , and , will denote sets of points within which will be defined later in the paper. Also, is a scalar function, defined on and taking values in the interval . This function will describe a time-varying scheduling parameter of the multi-agent system under consideration.

Unless stated otherwise, the notations , , etc., will refer to matrices of appropriate dimension parameterized by .

For , denotes the diagonal matrix with the entries of as its diagonal elements.

denotes the Kronnecker product of two matrices.

and will denote the largest and the smallest eigenvalues of a real symmetric matrix.

is the identity matrix, and is the vector whose all entries are equal to . When the dimension is clear from the context, the subscript will be suppressed, and will represent the identity matrix of an appropriate dimension.

For two symmetric matrices and of the same dimensions, , () if and only if is positive semidefinite (positive definite).

Consider the Laplacian matrix of the subgraph , . According to [28], under Assumption 1, the matrix is nonsingular (also see [27]), and all the entries of the vector are positive. Also, the matrix is positive definite, where . The following two constants will be used in the proofs of our results:

 σ = 12λmin(Θ−1(Lc0+G)+(Lc0+G)′Θ−1), ^λ = λmax((Lc0+G)′Θ−2(Lc0+G)). (1)

### 2.3 Problem Formulation

The system under consideration consists of parameter-varying dynamical agents, coupled with their neighbors; the topology of interconnections is captured by the directed graph . Dynamics of the th agent are described by the linear equation

 ˙xi=A(ρ(t))xi+B1ui+B2∑j∈Nφiφij(t,xj−xi),i=0,…,N, (2)

where is the state of agent , is the control input and is the time-varying parameter, which is available to all agents. It is assumed that is a continuous function. Also in equation (2), and are real matrices of appropriate dimensions, and is the composition of a continuous matrix-valued function and the function defined above.

The functions ,

 φij(t,x)=Δij(t)Cijx,x∈Rn,

describe how agent influences the dynamics of agent . Here and are respectively constant and time-varying matrices. According to the above model, we focus on linear time-varying interactions between the subsystems whose strengths depend on the relative state of subsystem with respect to subsystem , . Also, we will assume that the coefficients are uncertain, and satisfy the constraint

 Δ′ij(t)Δij(t)≤I,∀t∈[0,∞). (3)

That is, we consider a class of norm-bounded uncertain interactions, which will be denoted by .

Since we have designated agent 0 to be the leader, and the rest of the agents to be controlled are to follow the leader, then according to (2) dynamics of the th follower are described by the equation

 ˙xi= Missing or unrecognized delimiter for \Big (4)

while the leader dynamics are given by the equation

 ˙x0=A(ρ(t))x0+B2∑k:¯dk=1φ0k(t,xk−x0). (5)

Unlike agents , , the leader is not controlled, i.e., . On the contrary, all other agents will be controlled to track node 0.

###### Remark 2

In this paper, no a priori stability assumptions are made about the matrices . This is in contrast to, for example, the synchronization problem considered in [22], where the dynamics of the leader were required to be stable or marginally stable. In the sequel, however, certain conditions, in an LMI form, will be imposed on the coefficients of the leader and followers dynamics. While we formally make no special provisions regarding these coefficients, except for the feasibility of these LMI conditions, for such LMIs to be feasible one can reasonably expect the overall system dynamics to have certain collective stabilizability and detectability properties; see Remark 3 below.

Define tracking performance associated with the control input as

 J(u) =N∑i=1∫∞0((x0−xi)′Q(x0−xi)+u′iRui)dt, (6)

where and are given weighting matrices. In this paper we are concerned with the following problem.

###### Problem 1

For each follower , find gain schedule functions so that the control protocols of the form

 ui=−Ki(ρ(t))⎧⎨⎩∑j∈Sci(xj−xi)+gi(x0−xi)⎫⎬⎭, (7)

ensure a bounded worst-case tracking performance,

 supΞJ(u)

where denotes a constant which can depend on and .

## 3 The Main Results

In this section, we first revisit the leader follower tracking control problem for multi-agent systems with fixed parameters considered in [17] to obtain a sufficient condition for the design of a control protocol for a more general class of systems involving coupling between the leader and the agents. This sufficient condition will then be applied to develop an interpolation technique to obtain a continuous gain scheduling tracking control protocol for the system (2). The interpolation technique is the main result of this paper.

### 3.1 Leader follower control for fixed parameter systems

We now revisit the results of [17]. This revision is prompted by a more general structure of the system (2) which allows for physical connections between the leader and the followers. As a result, the form of the control protocol is somewhat different here in that the resulting controller gains depend on .

Consider a fixed-parameter version of the system (2) described by the equation

 ˙xi= A(ρ)xi+βξi(t,xi)+B1ui+B2∑j∈Nφiφij(t,xj−xi),i=0,…,N, (9)

where is fixed, and is a positive constant. Compared to (2), the system (9) includes an additional uncertainty element , which satisfies the following constraint

 ∥ξ0(t,x)−ξi(t,y)∥2≤∥x−y∥2,∀x,y∈Rn. (10)

In the sequel, we will show that when the difference is sufficiently small, the system (2) can be represented as the system (9) subject to (10).

We now derive a distributed protocol of the form (7) under which the fixed-parameter uncertain system (9) satisfies the performance requirement (8).

Define the leader tracking error vectors as , . Dynamics of the variable satisfy the equation

 ˙ei= A(ρ)ei−B1ui+β~ξi(t)−B2∑k:¯dk=1φ0k(t,ek) −B2∑j∈Sφi(φij(t,ei)−φij(t,ej))−B2diφi0(t,ei), (11)

where . It follows from (10) that for all .

For node of the subgraph , introduce matrices , , where are the elements of the neighborhood set , and are the nodes with the property ; and are, respectively, the in-degree and the out-degree of node in the graph . Also, let , where , are the constants defined in (1).

In order to formulate the extension of Theorem 1 in [17], with each node , , we associate a collection of positive constants , , , , (only for those nodes for which ), and (only for those nodes for which ). Also, let be an matrix. Using these constants and the matrix, for each node introduce a matrix defined depending on , as follows.

Case 1. and . For each such node , define the matrix as

 Πi=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣ZiY¯Q1/2Y^C′iY¯C′iYYC′i0YC′0i¯Q1/2Y−I00000^CiY0−Φi0000¯CiY00−Ωi000Y000−Λi00Ci0Y0000−1νi0I0C0iY00000−1Nμ0iI⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦, (12)

where

 Φi= diag[1νijI,j∈Sφi],Ωi=diag[1μjiI,j:i∈Sφj],Λi=1πiI, Zi= A(ρ)Y+YA(ρ)′−B1R−1B′1+1πiβ2I +(∑j∈Sφi(1νij+1μij)+1νi0+∑k:¯dk=11μ0k)B2B′2. (13)

Case 2. and . For each such node , the constant is not defined. Accordingly, we define the matrix by removing the second last column and row from the matrix in (12):

 Πi=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣ZiY¯Q1/2Y^C′iY¯C′iYYC′0i¯Q1/2Y−I0000^CiY0Φi000¯CiY00Ωi00Y000Λi0C0iY0000−1Nμ0iI⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦, (14)

where the matrix is modified to be

 Zi=A(ρ)Y+YA(ρ)′−B1R−1B′1+1πiβ2I+(∑j∈Sφi(1νij+1μij)+∑k:¯dk=11μ0k)B2B′2. (15)

Case 3. and . For each such node the constant is not defined, hence the corresponding matrix will be defined by removing the last column and row from the matrix in (12):

 Πi=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣ZiY¯Q1/2Y^C′iY¯C′iYYC′i0¯Q1/2Y−I0000^CiY0−Φi000¯CiY00−Ωi00Y000−Λi0Ci0Y0000−1νi0I⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦, (16)

The matrix is the same as in (13).

Case 4. and . In this case, both and are not defined, and the corresponding matrix is defined by removing two last columns and rows from the matrix in (12):

 Πi=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣ZiY¯Q1/2Y^C′iY¯C′iY¯Q1/2Y−I000^CiY0−Φi00¯CiY00−Ωi0Y000−Λi⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦, (17)

The matrix is the same as in (15).

The following theorem is an extension of Theorem 1 in [17].

###### Theorem 1

Under Assumption 1, let a matrix , , constants , , , , , and constants (for those with ), (for those with ) exist such that the following LMIs are satisfied simultaneously

 Πi<0,i=1,…,N. (18)

Then the control protocol (7) with

 Ki(ρ)=−(ϑiσ)−1R−1B′1Y−1 (19)

solves the leader follower tracking control problem for the fixed parameter system (9). Furthermore, this protocol guarantees the following performance bound

 supΞJ(u)≤^λσ2N∑i=1(x0(0)−xi(0))′Y−1(x0(0)−xi(0)) (20)

for all uncertainties for which (10) holds.

###### Remark 3

The feasibility of the LMIs (18) in Theorem 1 is a sufficient condition for the existence of a control scheme to guarantee the performance bound (20). The LMIs are numerically tractable and can be solved using the existing software. Additionally, using the standard tools of the control theory, such as the Strict Bounded Real Lemma and the associated Riccati equation, the feasibility of the LMIs (18) can be related to collective stabilizability and/or detectability properties of the coefficients of the interconnected system (9). Since this system consists of identical agents, one can expect that each agent must necessarily have corresponding stabilizability and detectability properties for these collective properties to hold; cf. [20].

Proof: Using the Schur complement, each LMI (18) can be transformed into the following Riccati inequality:

 A(ρ)Y+YA(ρ)′−B1R−1B′1+(∑j∈Sφi(1νij+1μij)+∑k:¯dk=11μ0k)B2B′2+1πiβ2I +Y(¯Q+∑j∈SφiνijC′ijCij+∑j:i∈SφjμjiC′jiCji+πiI)Y +(1νi0B2B′2+νi0YC′i0Ci0Y) (this term is present only if di=1) +Nμ0iYC′0iC0iY<0 (21) (this term is present only if ¯di=1).

Note that the last and the second last lines in the Riccati inequality (3.1) are present only for those nodes for which and/or , respectively.

After pre- and post-multiplying (3.1) by , and then using the expression (19) for in the resulting inequality, we obtain

 Y−1(A(ρ)+σϑiB1Ki(ρ))+(A(ρ)+σϑiB1Ki(ρ))′Y−1+∑j∈SφiνijC′ijCij+∑j:i∈SφjμjiC′jiCji +Y−1((∑j∈Sφi(1νij+1μij)+∑k:¯dk=11μ0k)B2B′2+1πiβ2I)Y−1+Y−1B1R−1B′1Y−1+¯Q+πiI +(1νi0Y−1B2B′2Y−1+νi0C′i0Ci0) (this term is present only if di=1) +Nμ0iC′0iC0i<0 (22) (this term is present only if ¯di=1).

Define and consider the following Lyapunov function candidate for the interconnected system consisting of the subsystems (3.1):

 V(e)=N∑i=1e′iY−1ei. (23)

Then

 dV(e)dt= N∑i=12e′iY−1(A(ρ)ei+B1Ki(ρ)(∑j∈Sci(ei−ej)+giei)) −2N∑i=1∑k:¯dk=1e′iY−1B2φ0k(t,ek)+2N∑i=1e′iY−1β~ξi(t,ei)−2N∑i=1die′iY−1B2φi0(t,ei) −2N∑i=1∑j∈Sφie′iY−1B2φij(t,ei)+2N∑i=1∑j∈Sφie′iY−1B2φij(t,ej). (24)

Note the following inequality:

 N∑i=12e′iY−1B1Ki(ρ)(∑j∈Sci(ei−ej)+giei) =−2e′(Θ(Lc0+G)⊗(Y−1B1(σR)−1B′1Y−1))e =−y′((Θ(Lc0+G)+(Lc0+G)′Θ)⊗Ip)y ≤−2σy′(IN⊗Ip)y =−2σe′(IN⊗Y−1B1(σR)−1B′1Y−1)e =−2N∑i=1e′iY−1B1R−1B′1Y−1ei, (25)

where .

From (3.1) and (3.1), one has

 dV(e)dt≤ N∑i=12e′iY−1(A(ρ)+σϑiB1Ki(ρ))ei+2N∑i=1e′iY−1β~ξi(t,ei) −2N∑i=1∑j∈Sφie′iY−1B2φij(t,ei)−2N∑i=1∑k:¯dk=1e′iY−1B2φ0k(t,ek) −2N∑i=1die′iY−1B2φi0(t,ei)+2N∑i=1∑j∈Sφie′iY−1B2φij(t,ej). (26)

Using the Riccati inequality (3.1), it follows from (3.1) that

 dV(e)dt≤ −N∑i=1e′i(Y−1B1R−1B′1Y−1+¯Q+∑j∈SφiνijC′ijCij+∑j:i∈SφjμjiC′jiCji +πiI+Y−1((∑j∈Sφi(1νij+1μij)+∑k:¯dk=11μ0k)B2B′2+1πiβ2I)Y−1)ei +N∑i:di=1e′i(1νi0Y−1B2B′2Y−1+νi0C′i0Ci0)ei+NN∑i:¯di=1e′iμ0iC′0iC0iei +2N∑i=1e′iY−1β~ξi(t,ei)−2N∑i=1∑j∈Sφie′iY−1B2φij(t,ei)−2N∑i=1die′iY−1B2φi0(t,