Stabilization with a Specified External Gain for Linear MIMO Systems and Its Applications to Control of Networked Systems
Abstract
This paper studies a stabilization problem for linear MIMO systems subject to external perturbation that further requires the closedloop system render a specified gain from the external perturbation to the output. The problem arises from control of networked systems, in particular, robust output synchronization of heterogeneous linear MIMO multiagent systems via output feedback/communication. We propose a new approach that converts a class of MIMO systems into a normal form via repeated singular value decomposition and prove that a stabilization controller with a specified external gain can be explicitly constructed for the normal form. Two scenarios with static state feedback and dynamic output feedback are investigated. By integrating the reference model and internal model techniques, the robust output synchronization problem for MIMO multiagent systems is converted into a stabilization problem with a specified external gain and solved by the developed approach.
A]Lijun Zhu , B] Zhiyong Chen, C] Xi Chen, A] David J. Hill
Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong
School of Electrical Engineering and Computing, The University of Newcastle, Callaghan, NSW 2308, Australia
Engineering Research Center of Metallurgical Automation and Measurement Technology, Ministry of Education, Wuhan University of Science and Technology, Wuhan 430081 China
Key words: Stabilization, External gain, MIMO, Synchronization, Multiagent systems,
1 Introduction
Stabilization for linear MIMO systems is a well developed and widely used technique in modern control theory. The main focus of this paper is study a stabilization problem for linear MIMO systems subject to external perturbation that further requires the closedloop system render a specified gain from the external perturbation to the output. The tool to characterize the gain from the external perturbation to the output is the socalled inputtooutput stability (IOS) gain with the external perturbation regarded as an input to the closedloop system [1, 2].
It is known that when a linear system is stabilizable, a feedback controller that renders the asymptotic stability for the system free of external perturbation also makes the closedloop system IOS when external perturbation is taken into consideration. The IOS gain is determined by the closedloop system structure. However, it remains a challenging task when an inputtooutput stabilization further requires an arbitrarily specified external gain. That requires an inputtooutput stabilization controller to further achieve an arbitrarily specified external gain. Such a stabilization is interesting by itself when one is interested in managing the external influence to system output through the stabilization controller design. Also, the problem is well motivated from studying networked systems.
A relevant research topic in the literature is and (sub)optimal control. For example, Sontag discussed the close relation between inputtostate/output formulation and and (sub)optimal control in [3]. It is known that the solution of and control relies on the solution of Riccati equations for a linear system [4] and HamiltonJacobiIsaacs partial differential equations for nonlinear systems [5, 6, 7, 8]. What differs from inputtooutput stabilization here is that the or gain through admissible state feedback and output feedback control is greater or equal than a minimum which depends on the system structure and the solutions to Riccati equations. In other words, the impact of a perturbation in terms of or gain can be optimized/minimized by feedback control, but not made arbitrarily small. However, the problem studied in this paper requires an arbitrarily specified external gain through feedback controller design.
One sufficient condition for the solvability of or control is that the linear system is stabilizable and detectable and the solutions to the corresponding Riccati equations exist. The conditions needed for the stronger requirement on arbitrarily specified external gain are studied in paper. In particular, with some additional conditions, linear MIMO systems can be transformed to a socalled normal form for which the problem can be solved. It is noted that the same problem has been solved for nonlinear minimumphase systems in the previous paper [9] using the backstepping technique and the improved small gain theorem. But the systems in [9] are SISO and in the socalled lower triangular form. Therefore, the result cannot be applied in the present MIMO and nonlowertriangular systems.
An important motivation or application of the proposed technique of stabilization with a specified external gain is the robust output synchronization problem for heterogeneous multiagent systems through output feedback and output communication. Heterogeneity among agents is commonly encountered in realworld applications such as vehicle platoons in [10] and power systems in [11] where the dynamics of subsystems are not uniform. Synchronization of multiagent systems aims to achieve the agreement on agents’ outputs via local communication among agents despite the heterogeneity. For homogeneous multiagent systems, the synchronization pattern is naturally embedded in the homogeneous part of agents’ dynamics, but it is not explicit for heterogeneous systems. In [12], a necessary condition for synchronization of heterogeneous multiagent systems called internal model principle has been given as that there must exist a homogeneous kernel for each agent that embraces the synchronization pattern. The homogeneous kernel is either embedded in the original dynamics or explicitly constructed through the controller design according to the task.
More specifically, as exposed in many works, e.g., [12, 13, 14, 15, 16], the synchronization controller of heterogeneous multiagent systems can be explicitly constructed in a twostep manner. The first step is the consensus of reference models that are constructed to be homogeneous for each agent and embed the homogeneous kernel. The second step is the regulation of each individual agent’s output to the output of its own reference model. In particular, when agent dynamics contain no uncertainties, the regulation can be achieved by the feedforward compensation control. This method was utilized for the synchronization of linear systems in [12] and nonlinear systems in [17]. However, the exact feedforward compensation becomes impossible if agent dynamics have uncertainties (see detailed discussion in [15]). In this case, researchers appeal to robust output regulation theory to handle uncertainties. It relies on a class of dynamic observers called internal model (in the context of robust output regulation theory) whose dynamics do not depend on the uncertainties but asymptotically generate the steadystate solution for states and/or control input. With the internal model, the regulation problem can be converted into a stabilization problem for which a feedback controller can be designed. This method has been used for the robust output synchronization for linear systems in [18] and nonlinear systems in [14, 16].
More specifically, synchronization controllers can be classified to rely on the state or output communication. For state communication, each agent is allowed to transmit its internal states to its neighbors, while for output communication, it becomes slightly more restrictive and only the output information is allowed to transmit. The freedom to choose what internal information to transmit over the network makes synchronization on state communication less complicated than that on output communication. Consequently, the aforementioned two actions, namely consensus for reference models and regulation of each agent to its reference, can be completely separated for state communication, which simplifies the controller design procedure [ChenXi2016]. For the output communication case, the work in [16] shows that these two actions are mutually perturbed by each other (hence further called perturbed consensus and perturbed regulation). In particular, the consensus action is perturbed by the regulation error and the regulation action is disturbed by the disagreement in consensus. The output synchronization framework via output communication developed in [16] suggests that synchronization is achieved if both perturbed consensus and perturbed regulation are solved as well as a small gain condition is satisfied.
The robust output synchronization of heterogeneous linear MIMO multiagent systems via output communication is yet to be investigated and is the focus of the second part of this paper. In particular, we need to propose a modified internal model and show that the robust output synchronization problem of MIMO multiagent systems can also be converted into perturbed consensus and perturbed regulation problems. Moreover, the perturbed regulation problem corresponds to the stabilization problem which can be solved using techniques developed in the first part of the paper
The rest of this paper is structured as follows. In Section 2, we will first introduce the problem of stabilization with a specified external gain and its motivation. The main results on stabilization with a specified external gain are given in Section 3. In particular, we propose a state feedback controller for the MIMO systems in the normal form as well as a dynamic output feedback controller for systems of a special structure. In Section 4, we formulate the robust output synchronization problem for linear heterogeneous MIMO multiagent systems and introduce the framework that converts the problem into a standard perturbed consensus problem and a perturbed regulation problem. Moreover, with the aid of modified internal model design, we show that the perturbed regulation problem is equivalent to the stabilization problem with a specified gain as studied in Section 2. The numerical simulation is conducted in Section 5 and the paper is concluded in Section 6.
2 Problem Formulation and Motivation
We consider a class of linear MIMO control systems typically represented by the following equations
(1) 
where is the state, the input, the output, and external perturbation. The matrices , , and have compatible dimensions. The external perturbation may represent external signals or influence of other subsystem to the system (1) in an interconnected setting.
A general linear controller takes a static form
(2) 
or a dynamic form
(3) 
that induces a closedloop system
(4) 
for or , respectively.
Without the external perturbation , the stabilization controller design for a linear system is well known in the literature. Moreover, for any stabilization controller that ensures a Hurwitz , the basic property of a linear system implies that the closedloop system (4) is automatically bounded input bounded output (BIBO) or inputtooutput stable (IOS), with as the input and the output. Also, the gain from to can be explicitly computed. However, it remains a difficult task to design a stabilization controller such that the closedloop system (4) has an arbitrarily specified gain from to .
More specifically, we will use a quadratic function to characterize the relationship between the input and output for the linear closedloop system (4).
Definition 2.1
The linear MIMO system (4) with a Hurwitz matrix is said to admit a quadratic IOSLyapunov function with if there exist positive constants and such that
(5) 
In particular, is called an IOS gain.
Remark 2.1
With the aforementioned setting, the main objective of this paper is rigorously formulated in the following problem.
The problem of stabilization with a specified external gain (stabilization) for the system (1) aims, for an arbitrarily specified , to find a controller (2) or (3) such that the system (4) has a Hurwitz matrix and an external gain .
The stabilization problem has an independent interest when one is interested in managing the influence of an external signal to system output through the stabilization controller design. Also, the problem is well motivated from studying networked systems. One simple motivating example is given below. A complete application of stabilization can be found in Section 4 in effectively solving the robust output synchronization problem for heterogeneous MIMO multiagent systems.
Example 2.1
Consider the interconnection of two subsystems illustrated in Fig. 1. The first subsystem is a network of linear MIMO control systems described by (1), that is,
(6) 
where , and are the state, input and output, respectively and is the composite output of . The second subsystem is given by
(7) 
where is the state and the output. The subsystems and are coupled through their respective outputs and . Suppose admits a quadratic IOSLyapunov function and an external gain . Then, the stabilization for each with the external gain guarantees stability of the overall system by using Proposition 2.1 stated below.
For , it is a typical interconnection of two subsystems studied in many references such as [19, 20]. The present scenario with derives from the robust output synchronization studied in, e.g., [16]. The two interconnected subsystems and correspond to the dynamics for perturbed consensus and perturbed regulation problems to be elaborated in Section 4.
The following proposition shows how stabilization for each guarantees the stability of the overall system in Example 2.1.
Proposition 2.1
Proof: Let the quadratic IOSLyapunov function for be that satisfies
(8) 
for some satisfying . Let the state of the system composed of and its stabilization controller be . Then, solvability of the stabilization problem for means the existence of a quadratic IOSLyapunov function that satisfies
(9) 
for some satisfying .
Let and
whose derivative along the trajectory of the overall closedloop system is
As , we can pick
that is,
Applying LasalleYoshizawa Theorem ([21]) leads to the state of the closedloop has the following asymptotical property
(10) 
which further implies
(11) 
The proof is thus completed.
3 Main Results: Stabilization
The first main result is to explicitly find a state feedback controller (2) to solve the stabilization problem for the system (1). For this purpose, we introduce a state transformation based on the singular value decomposition of the input matrix that puts the system into a class of normal form.
3.1 A Normal Form
Let us recursively define a sequence of matrix pairs , in the following algorithm. In the algorithm, denote the rank of by , i.e., For convenience of notation, let .

Let and set the initial values

If has a full row rank or , exit.

One has
and the singular value decomposition (SVD) of is
(12) where is a diagonal matrix bearing all nonzero singular values on the diagonal entries and and are unitary matrices.
Since does not have a full row rank and (otherwise exit at step (ii)), one has and hence the following decomposition
Calculate
(13) 
Let and go to step (ii).
Then, the next lemma shows a useful result from the above algorithm under a controllability assumption. The proof is given in the Appendix.
Assumption 3.1
The pair of matrices is controllable.
Lemma 3.1
Throughout the paper, we call in Lemma 3.1 the number of SVD steps. With the finite number , we define the orthogonal matrix
(14) 
where . Accordingly, we introduce the coordinate transformation
(15) 
with the dimension of compatible with .
We need one more assumption to obtain the inputtooutput stabilization normal form for the system (1).
Assumption 3.2
The matrices satisfy for .
Lemma 3.2
Proof: The proof is given in Appendix.
Now, we can define the following matrices recursively,
and, for ,
Also, define
Then, the system (1) with the state transformation (15) under Assumptions 3.1 and 3.2 can be converted to the following form, with ,
(17) 
Remark 3.1
3.2 State Feedback Control
In this subsection, we will show how a static state controller can be designed for the stabilization problem of the system (17) and hence (1).
Theorem 3.1
Proof: For the normal form transformation given in the previous subsection, it suffices to solve the stabilization problem of the system (17). For this purpose, we introduce a recursive state transformation to (17) as follows,
(19) 
with .
With the matrices properly selected, the system (17) with can be put in the form
for some ’s. It is easy to see that, by properly selecting ’s, there exists a quadratic Laypunov function such that
(20) 
for some and . Therefore, the stabilization problem is solved by the controller
for some matrix depending on .
To explicitly calculate the matrices , we only consider the case with . The calculation can be extended for the general case with using the backstepping technique. Specifically, for , let
(21) 
As a result, the system (17) with becomes
for and . We choose and . Let be the quadratic IOSLyapunov function. Using the fact
a simple calculation shows
for some . Thus, the stabilization problem is solved.
3.3 Extension to a Class of Output Feedback Control Systems
In this subsection, we will extend the method in the previous subsection to construct the output feedback controller to solve the stabilization problem for a particular linear MIMO system described as follows
(22) 
where and are the augmented state. The matrix has a full column rank with being the pseudoinverse of , i.e., . This specific linear MIMO system structure arises from the robust output synchronization problem to be studied in Section 4. The problem is studied under the following additional assumption.
Assumption 3.3
The pair of matrices is detectable.
Theorem 3.2
(Dynamic Output Feedback) Consider the linear MIMO system (22) under Assumptions 3.1, 3.2 and 3.3. Assume is Hurwitz and is controllable. Let the stabilization of the system (1) of the same be solved by a state feedback controller . Then, the stabilization of the system (22) is solved by the following output feedback controller
(23) 
where and is selected such that is Hurwitz.
Proof: Since is detectable and is controllable, one has that is also detectable (see Theorem 6.23 in [22]). The state transformation and lead to
For the dynamics, we can regard and as the external perturbation. Since the state feedback controller solves the stabilization problem of the linear MIMO system described by matrices , according to the proof of Theorem 3.1 (see (20)), there exists a quadratic IOSLyapunov function , whose derivative satisfies
(24) 
for some and .
Since and are Hurwitz, there exists positive definite matrices and such that and . Let and . Then, the derivative of along the dynamics is
for some . The derivative of along the dynamics is . Let where and . Then,
Thus, the stabilization of the system (22) is solved with .
4 Robust Output Synchronization of MASs
In this section, we will apply the controller design method developed in Section 3 to solve the robust output synchronization problem for a class of linear MIMO uncertain heterogeneous MASs whose dynamics are described by
(25) 
where , , are state, input and output of agent , respectively and an uncertain parameter vector. Without loss of generality, we assume as the nominal value of and has a full column rank. The output synchronization problem is to find a distributed control strategy for each agent (25) such that outputs of all agents synchronize to an agreed trajectory, regardless of the uncertainties .
The group of MAS (25) is said to achieve robust output synchronization if there exist neighborhoods of , , on which the output trajectories of all agents satisfy
The synchronization pattern for the output trajectories, denoted by , is typically governed by the dynamics
(26) 
where , , and are two prescribed matrices. The autonomous system (26) represents a general class of patterns including constant and/or harmonic series up to a certain order. The definition of output synchronization, as introduced in, e.g., [16], is revisited as follows.
In this paper, we assume that each agent can only receive relative outputs from its neighbors with the specified weights, denoted by
(27) 
where is the weight on the information transmitted between agent and and is the set of neighboring agents from which agent can receive the relative output . We consider a directed graph to represent the communication topology where the set of nodes denotes agents and the set of edges represents the information flow. The weighted adjacency matrix of a graph is with and , more specifically, for and for . Denote as the Laplacian of the graph, where and . Throughout the section, the graph is assumed to have a spanning tree, that is, there exists a node to which all other nodes can be linked via a directed path.
For each agent , a distributed controller that uses the relative output network communication and the output is designed such that the group of closedloop agents achieves the robust output synchronization. Denote and when no confusion is caused. Some assumptions are needed for the problem.
Assumption 4.1
The pair is detectable.
Assumption 4.2
For , the pair is controllable, the pair is detectable, and for where is the number of SVD steps of .
Assumption 4.3
For any ,
where denotes the spectrum of .
Remark 4.1
Assumption 4.1 is common for output synchronization of linear homogeneous multiagent systems (see [18, 23]). As shown in [16], it ensures that perturbed consensus of the reference models can be achieved. Assumption 4.2 follows the assumptions in the previous section as stabilization will be shown as a required step in the present output synchronization problem. By Theorem 1.9 in [22], Assumption 4.3 is required such that there exists a unique solution pair for the following regulator equations, for all ,
(28) 
The pair can be used to define the steadystate state and input to (25) as the the output synchronization is achieved in the pattern (26).
As introduced in [16], the robust output synchronization problem of uncertain multiagent systems can be solved in a framework by addressing two coupled problems, namely perturbed consensus and perturbed regulations problems. The perturbed consensus problem is standard and it has been solved by introducing the homogeneous reference model for each agent as follows
(29) 
Define for as the local regulation error. Let and . The reference model (30) can be rewritten as
(30) 
It was shown that the consensus of reference models is perturbed by the regulation error . By properly selecting matrices , , and , in particular, with a Hurwitz , such that consensus of (30) with is achieved. Let be the full state that represents disagreements among reference models. In particular, there exists a quadratic IOSLyapunov function such that
for some positive constant and . As be a part of and is Hurwitz, there exists a quadratic IOSLyapunov function with such that
(31) 
for some positive constant and . Denote .
The next socalled perturbed regulation problem aims to design such that for the MIMO agent dynamics repeated as follows
(32) 
Note that is governed by the first equation of (30), i.e.,
(33) 
that is regarded as an exosystem with perturbation .