Distributed adaptive control for H∞ tracking of uncertain interconnected dynamical systems

A distributed adaptive control is proposed for H∞$$ {H}_{\infty } $$ tracking of an interconnected dynamical system in the presence of L2$$ {L}_2 $$ disturbances and system/interconnection uncertainties. A reference model which achieves a robust tracking against L2$$ {L}_2 $$ disturbances is introduced by using H∞$$ {H}_{\infty } $$ control with transients. Then a distributed adaptive control law is developed for an uncertain interconnected dynamical system, where it employs the specified reference model. It is shown that the boundedness of the error dynamics behaviors as well as zero tracking error in the steady state is guaranteed by the proposed distributed adaptive control in the presence of disturbances and uncertainties. An explicit error bound related to H∞$$ {H}_{\infty } $$ tracking is also established.


INTRODUCTION
An adaptive control system automatically compensates for variations in system dynamics by adjusting the controller characteristics so that the overall system performance is maintained at a desirable level. 1 In fact, adaptive control has been extensively studied for various complex systems such as interconnected and/or large-scale systems under different perspectives. For example, an adaptive control with performance enforcement is proposed for a class of uncertain dynamical systems that consist of actuated and unactuated portions that are physically interconnected to each other. 2 A decentralized adaptive control of interconnected systems is developed for stabilization and tracking via several approaches. 3,4 Considering a wide array of applications of interconnected systems, a major research area is the development of distributed control architectures such that the subsystems perform given tasks through local interactions. For instance, a distributed adaptive control approach is proposed for uncertain multi-agent systems with coupled dynamics. [5][6][7] A distributed adaptive control law is proposed for large-scale systems with unknown interconnection parameters 8 and for large-scale modular systems interconnected physically 9 via graph theoretic approach.
Performance guarantee is also an issue in distributed adaptive control. In fact, a work 9 provided an error bound of tracking based on a Lyapunov solution of a given reference model. This technique has further been investigated in the context of several type of uncertain systems in References 5-8.
When we introduce a performance index for realizing a desirable tracking, it is natural to explore a performance guarantee with respect to the index. However, there are few available results in this context. For example, a paper 7 considers a distributed adaptive control for large-scale systems, where boundedness of the internal signals is investigated, while any performance guarantee related to nominal tracking is not provided. In this regard, the authors of the present paper evaluate the performance degradation caused by adaptation in terms of the performance index which is used for designing the reference model as LQ optimal tracking. 10 Here, an appropriate selection of control input which incorporates the evaluation of performance degradation is needed so that the tracking error is minimized for the nominal system and an explicit error bound of the optimal tracking is obtained. This paper follows this line of research. 10 We focus on a class of interconnected dynamical systems that is characterized by sets of uncertain dynamics with an unknown physical interconnection between these dynamics. Specifically, we propose a distributed adaptive control for realizing a robust tracking of uncertain dynamical systems. To this end, employing H ∞ control with transients, [11][12][13] we introduce a reference model which achieves an H ∞ tracking in the presence of L 2 disturbances. Then we develop a distributed adaptive control law for uncertain dynamical systems, where the law utilizes the specified reference model. We show that the proposed distributed adaptive control guarantees the boundedness of the error dynamics behaviors in the presence of disturbances and uncertainties, where it achieves zero tracking error in the steady state as well. Furthermore, we establish an explicit error bound of tracking, which enables us to evaluate a transient performance of the control system. Numerical examples illustrate that the results developed in this paper are useful.
A preliminary version of this paper was presented at a conference, 14 where an adaptive H ∞ tracking of a single plant is considered for a step-type reference signal. On the other hand, the present paper deals with a distributed adaptive H ∞ tracking of an interconnected system for a general reference signal, which clarifies a possible performance guarantee for a type of adaptive control law.
This paper is organized as follows. We state notations and definitions in Section 2. We describe the statement of problem in Section 3, and select the desired reference model for H ∞ tracking in Section 4. We discuss the distributed adaptive control scheme for uncertain systems with interconnections in Section 5, and evaluate the performance degradation to the original H ∞ tracking in Section 6. In Section 7, the effectiveness of the proposed scheme is demonstrated to show results of the theoretical findings. Some concluding remarks are discussed in Section 8.

NOTATIONS AND DEFINITIONS
In this paper, R denotes the set of real numbers, R n denotes the set of n-dimensional real column vectors, and R n×m denotes the set of n × m real matrices. In addition, we write P T for the transpose of a real matrix P, R −1 for the inverse of a matrix R, rank S for the rank of a matrix S, and trU for the trace of a square matrix U. Furthermore, we define that is, ||f || denotes the L 2 norm of a vector-valued function f .

PROBLEM FORMULATION
Let us consider an interconnected system consisting of N uncertain dynamical subsystems with uncertain interconnection. The topology of the interconnection is expressed by a graph  = (, ), where  = {1, 2, … , N} is the set of the nodes each of which corresponds to a subsystem, and  ⊆  ×  is the set of the edges which represents the interaction among the subsystems. The set of neighborhood of the i-th subsystem is denoted by  i = {j ∈ |(i, j) ∈ }, where the i-th subsystem is affected by the j-th subsystem through uncertain interconnection if j ∈  i . Here we assume that  is known and time-invariant.
With the graph , we describe the dynamics of the i-th subsystem aṡ where x i (t) ∈ R n i is the state, u i (t) ∈ R m i is the control input restricted to the class of admissible controls consisting of measurable functions, y i (t) ∈ R m i is the controlled output. Throughout the paper, the subscripts i and j correspond to the i-th and the j-th subsystems, respectively, that is, i ∈  and j ∈  i ⊂ . In addition, d i (t) ∈ R p i is an unknown disturbance.
Here we assume that d i ∈ L ∞ ∩ L 2 andḋ i ∈ L ∞ . In other words, we assume that d i (t) satisfies 15 The matrices A i ∈ R n i ×n i , B i ∈ R n i ×m i , C i ∈ R m i ×n i , and E i ∈ R n i ×p i represent the nominal known part of the subsystem, where the pair (A i , B i ) is controllable and the pair (C i , A i ) is observable. On the other hand, the matrix Λ i ∈ R m i ×m i and the vector-valued function Δ i,j ∶ R n j → R m i represents the uncertain part of the subsystem as well as the uncertain interaction among the subsystems. That is, Δ i,i (x i (t)) expresses the uncertain dynamics of the i-th subsystem itself, while Δ i,j (x j (t)) (j ≠ i) expresses the uncertain influence from the j-th subsystem to the i-th subsystem.
Here we introduce the following assumption for Λ i and Δ i,j (x j (t)).

Assumption 1.
The control effectiveness Λ i of the i-th subsystem is an unknown symmetric and positive definite matrix. The state dependent uncertainty Δ i,j (x j ) of the i-th subsystem is linearly parameterized as for all j ∈  i ∪ {i}, where F i,j ∈ R m i ×s i,j is an unknown weight matrix and i,j ∶ R n j → R s i,j is a given basis function.
For this system (1), we define its nominal system aṡ That is, when In this paper, we consider a reference signal r i (t) ∈ R m i generated bẏ where x ri (t) ∈ R r i is the state of the reference signal generator, the eigenvalues of A ri are on the imaginary axis and all distinct from one another, and the pair (C ri , A ri ) is observable. That is, we can deal with a sinusoidal reference signal as well as a step-type reference signal, while both of them are bounded. We define the initial time t = 0 at the time when the reference signal is applied. The initial state x ri0 of (4) is arbitrary. It is known that the controlled output y i (t) of the nominal subsystem (3) can follow any reference signal r i (t) of (4) in the steady state if rank for all eigenvalues ri of A ri . We assume this condition for all subsystems (1). The objective of this paper is to construct a robust distributed adaptive control law of the form such that the output y i (t) of the given system (1) asymptotically tracks the reference signal r i (t) of (4) in the presence of the L 2 disturbance d i (t) and the system uncertainty described by Λ i and Δ i,j (x i ) satisfying Assumption 1. Here, x  i (t) denotes the set of x j (t) with j ∈  i , and thus the control law (6) utilizes the knowledge of the i-th subsystem itself and its surrounding neighbors only. In this sense, we call the form (6) a distributed adaptive control law. Then we provide a reference model via an H ∞ type robust tracking control for the nominal system (3) and derive a robust distributed adaptive tracking control law for the interconnected uncertain system (1) with a performance guarantee related to an H ∞ type measure. Here, the performance measure is the induced norm of the tracking error over all possible disturbances and initial states for each subsystem.

REFERENCE MODEL SELECTION
In this section, we select a suitable reference model for our adaptive control. To this end, we employ an H ∞ type control [11][12][13] in order to design a robust tracking control law for the nominal system (3) to the reference signal (4). We first introduce a variation system for tracking control. Under the assumption (5), there exist a unique state x si (t) and a unique control input u si (t) described by for which the controlled output y i (t) is identical to the reference signal r i (t), 16 where L xi and L ui are defined by We denote the variations of x i (t) and u i (t) from x si (t) and u si (t) bỹ and the tracking error of the controlled output y i (t) by Using these notations, the variation system is defined bẏx In this way, we can recast the original tracking problem as a stabilization problem of the variation system. That is, if a feedback control lawũ stabilizes the variation system given by (10), it turns out that lim t→∞xi (t) = 0, that is, lim t→∞ y i (t) = r i (t).
In order to select a suitable control law (11) for tracking in the presence of L 2 disturbance d i (t), we employ a robust control design which is called H ∞ control with transients. 11 That is, in this paper, we utilize a performance specification where i ∈ R is a specified positive number. The supremum is taken over allx i0 ∈ R n i and d i ∈ L 2 which satisfy ‖d i ‖ 2 + x T i0 R ixi0 ≠ 0. Then, we see that there exists a state feedback (11) which stabilizes (10) and achieves (12) if and only if there exist X i = X T i ∈ R n i ×n i and G i ∈ R m i ×n i which satisfy the linear matrix inequalities where such a state feedback gain K i of (11) is obtained by This is a direct consequence of the existing work. 13 Throughout this paper, we assume that such an X i and a G i exist for a given i > 0. As a matter of fact, the performance (12) is guaranteed by the feedback gain (14) as follows. When we define and use (14), we can rewrite (13) as where we see that A i + B i K i is Hurwitz, that is, the resultant closed-loop system is stable. The inequality (16) together with (10) and (11) implies that d dt for anyx i (t) ≠ 0. Integrating this inequality from 0 to ∞, with (17), we have where we use lim t→∞x T i (t)P ixi (t) = 0 which is guaranteed by the closed-loop stability. The above inequality shows that the performance specification (12) holds.
Using (7) and (9), we rewrite the control law (11) as for the nominal system (3), where K i is given by (14) and H i is represented as That is, the H ∞ tracking control law for the nominal system (3) is composed of a feedback from x i (t) and a feedforward from x ri (t). The resultant control system with (3) and (19) is described bẏ Since it achieves the performance specification (12), it is suitable as a reference model for adaptive control. However, the unknown d i (t) should be excluded in the reference model. Also, the initial states of the given system and the reference model may be different. Thus, in this paper, we employ the systeṁ as the reference model for adaptive control, where x mi (t) ∈ R n i is the state, K i is given by (14) based on the LMIs (13), and H i is given by (20) with this K i . Then, we develop a model reference adaptive control and derive its performance guarantee.

DISTRIBUTED ADAPTIVE CONTROL SCHEME
Let us go back to the uncertain dynamical system (1). Referring to the selected reference model (22), we rewrite each subsystem (1) aṡx where we define (6). In fact, from Assumption 1, the signal i (t) must be linearly parameterized by using an unknown weight W i ∈ R m i ×q i and the corresponding basis function i ∶ R n i +n  i +r i → R q i which contains x i (t), x ri (t), and i,j (x j (t)) (j ∈  i ∪ {i}), where n i = ∑ j∈ i n j and q i ≤ n i + r i + ∑ j∈ i ∪{i} s i,j . Then we introduce a distributed adaptive feedback control law where we define the update rule of the adaptive control gainŴ i (t) ∈ R m i ×q i aṡŴ Notice that x mi (t) of (25) is generated by (22). The learning rate i is any positive real number, the performance specification i for the nominal system is the one in (12), and P i is a symmetric and positive definite matrix defined by (15), where X i is a solution of the linear matrix inequalities (13).
We see that the control law (24) is distributed indeed. In fact, the basis i (x(t), x  i (t), x ri (t)) is distributed and x mi (t) of (22) is given by x ri (t). Thus, if the interconnection of the overall system (1) is sparse, we enjoy a sparse structure of the control, where the control law (24) utilizes the knowledge of the i-th subsystem itself and its surrounding neighbors only. See also the numerical example in Section 7 for further details. Now, let us define the errors from the ideal case as where x mi (t) and y mi (t) are defined in (22). With (22), (23), (24), and (25), we havė which describes the error dynamics from the reference model (22). The next theorem presents the result of this section.

Theorem 1. Consider the uncertain interconnected dynamical system described by (1) subject to Assumption 1. Consider, in addition, the reference model given by (22) and the distributed adaptive controller given by (24) and (25)
. Then, all of the solutions (x ei (t), W ei (t)) (i = 1, 2, … , N) given by (26) and (27)  Proof. Consider a candidate of Lyapunov function where i and P i are taken from (25) and Λ i of (1) satisfies Assumption 1, which means that Differentiating this candidate along the trajectories of (26) and (27), we havė for anyx ei (t) ≠ 0, which partially follows completing the square used in (18). Integrating this inequality from 0 to any T > 0, we have Since d i ∈ L 2 , the right-hand side of the inequality is bounded. Thus, V i (x ei (T), W ei (T)) is bounded for any T > 0, which implies that the solution (x ei (t), W ei (t)) is bounded. Also, the boundedness of the right-hand side of (28) implies that Furthermore, sincėy ei (t) = C iẋei (t) anḋx ei (t) of (26) is represented by the signals , and x ri (t), we can see that In fact, x ei (t) and W ei (t) are bounded as we have proved above, while d i ∈ L ∞ is bounded and x ri (t) of (4) is bounded. The boundedness of x i (t) and x  i (t) follows the boundedness of x ei (t) (i = 1, 2, · · · , N) and x mi (t) (i = 1, 2, · · · , N) of (22). With (29), (30), and a version of the Barbȃlat lemma, 15 we conclude that all of the tracking errors y ei (t) (i = 1, 2, … , N) satisfy lim t→∞ y ei (t) = 0.
This completes the proof of the theorem. ▪ Theorem 1 establishes boundedness of the error dynamics behaviors via the proposed distributed adaptive control in the presence of disturbances and uncertainties. The theorem also shows that zero steady state tracking error is achieved by this control.

PERFORMANCE EVALUATION
Since the proposed adaptive control given by (24) and (25) employs an H ∞ control (22) for the nominal system (3) as the reference model, one of our interests is to evaluate the performance degradation due to introduction of adaptive mechanism.
In this regard, let us recall the performance specification (12) for the nominal system (3). It says that holds for any d i ∈ L 2 , x i0 ∈ R n i , and x si0 ∈ R n i . On the other hand, when we use the adaptive control of (22), (24), and (25), we can evaluate the tracking error ||r i − y i || of the uncertain systems (1) similarly as follows.

Theorem 2. Consider the uncertain interconnected dynamical system described by (1) subject to Assumption 1. Consider, in addition, the reference model given by (22) and the distributed adaptive controller given by (24) and (25). Then, the tracking error is bounded as
Proof. Since (31) holds for (21), we have for the reference model (22) which does not contain d i ∈ L 2 . Regarding the adaptive control, we have established (28) for any T > 0, which means that Thus we have where we use (17). These evaluations (33) and (34) together with the triangle inequality gives the tracking error bound of the theorem. ▪ This bound given by Theorem 2 shows a transient performance of the proposed distributed H ∞ adaptive control. It says that L 2 type gain from the disturbance and the initial values to the tracking error of the distributed adaptive control is bounded by i as that of the corresponding nominal closed-loop system (21) is. In addition, if we use large learning rate i , the transient performance becomes better, which will be confirmed in the numerical example below.

NUMERICAL EXAMPLE
In this section, we demonstrate the proposed distributed adaptive control through a numerical example. Let us consider a mass-spring-damper system having N masses in one line shown in Figure 1. Each mass m i (1 < i < N) is possibly connected with its neighbors m i−1 and m i+1 by springs k i−1,i , k i,i+1 and dampers c i−1,1 , c i,i+1 , that is, where q i (t) ∈ R is the position of the mass m i to be controlled, andq i (t) andq i (t) are its velocity and acceleration, respectively. The force u i (t) ∈ R is applied to the mass m i as the control input, while we assume the unknown disturbance force d i (t) ∈ R to m i satisfies d i ∈ L ∞ ∩ L 2 andḋ i ∈ L ∞ . We also assume that all physical parameters m i > 0, k i−1,i ≥ 0, k i,i+1 ≥ 0, c i−1,i ≥ 0, and c i,i+1 ≥ 0 are unknown. The cases i = 1 and i = N having one neighbor are not explicitly stated in this section, though it is clear that they can be described in a similar way. The mass-spring-damper system (36) stated above is consistent with the system description (1). In fact, we can rewrite (36) aṡx where we define ] , That is, all unknown parameters are included in Λ i and Δ i,j (x j (t)). Apparently, is observable, and the uncertainties Λ i and Δ i,j (x j (t)) satisfy Assumption 1. That is, the mass-spring-damper system (36) can be represented as (1), where we define  i = {i − 1, i + 1}.
F I G U R E 1 Interconnected mass-spring-damper system For this system, we consider a sinusoidal reference signal such as sin i t, that is, we define the coefficient matrices of (4) as where i > 0. We see that (C ri , A ri ) is observable. Also, the rank condition (5) is satisfied for the eigenvalues of A ri . Actually, we have the solutions of (8) as In the following, we set 2 i = 0.1 for all i ∈ . With = 1 and R i = 3I, we solved the LMIs (13). Then we obtained the H ∞ tracking gains (14) and (20) as which determines the reference model (22). We also have P i of (15) as which is used in the update rule (25) of the adaptive control gain. Referring to the above K i and H i , we rewrite the system (37) as the form of (23), that is, , It should be noted that its uncertainty is described as That is, due to the sparse structure of the system (36) of Figure 1, the basis i (x i−1 (t), x i (t), x i+1 (t), x ri (t)) contains only the states of its neighbors, that is, x i−1 (t) and x i+1 (t). Since the adaptive control law (24) and the update rule of the adaptive gain (25) have the form these become in fact a distributed control law thanks to the sparse structure of the basis i (x i−1 (t), x i (t), x i+1 (t), x ri (t)). Now, let us consider a numerical simulation. We investigate the case N = 3, that is,  = {1, 2, 3}, where we set the unknown and uncertain parameters as m 1 = m 2 = m 3 = 3, k 1,2 = k 2,3 = 2, and c 1,2 = c 2,3 = 1. We selected the initial states ] T for all i ∈ . We chose the disturbance as d i (t) = e −t∕10 sin t, that is, d i ∈ L ∞ ∩ L 2 andḋ i ∈ L ∞ for all i ∈ . Figures 2-7 show the tracking responses and the gain behaviors of the proposed distributed adaptive control for subsystems 1, 2, and 3, respectively, where we set the learning rates as 1 = 2 = 3 = 5. On the other hand, Figures 8-13 show the tracking responses and the gain behaviors of the proposed adaptive control, where we use 1 = 2 = 3 = 1. Note that y 1 (t), y 2 (t), y 3 (t) are indicated as solid lines and y m1 (t), y m2 (t), y m3 (t) are indicated as dashed lines in Figures 2,4,6,8,10, and 12. The elements of the adaptive gainsŴ 1 (t),Ŵ 2 (t), andŴ 3 (t) for subsystems 1, 2 and 3 are indicated as solid lines in Figures 3,5,7,9,11, and 13. In Figures 2-7, all signals are bounded and y i (t) tends to y mi (t) (i ∈ ) as t tends to infinity, which is consistent with Theorem 1. Furthermore, comparing Figures 2,4, and 6 with Figures 8,10, and 12, we see that a larger learning rate i gives a better performance, which is consistent with Theorem 2.

CONCLUSION
In this paper, we have proposed a distributed adaptive control for H ∞ tracking of uncertain interconnected dynamical system that asymptotically tracks the output of a reference model in the presence of system/interconnection uncertainties. We have employed an H ∞ type control [11][12][13] for a suitable reference model selection. Then we have developed a distributed adaptive control law for uncertain dynamical systems with L 2 disturbance. We have proved that the proposed distributed adaptive control guarantees the boundedness of the error dynamics behaviors in the presence of disturbances and uncertainties, where it achieves zero tracking error in the steady state as well. Furthermore, we have established an explicit error bound of tracking, which enables us to evaluate a transient performance of the control system. The numerical examples have shown that the results developed in this paper are useful.

ACKNOWLEDGMENTS
This work was supported by JSPS KAKENHI grant number JP20K04547.