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We consider a continuous linear time invariant system with ellipsoidal parametric uncertainty structured into subsystems. Since the design of a local controller uses only information on a subsystem and its neighbours, we combine the plug and play idea and robust distributed control to propose one distributed control strategy for linear system with ellipsoidal parametric uncertainty. Firstly for linear system with ellipsoidal parametric uncertainty, a necessary and sufficient condition for robust state feedback control is proposed by means of linear matrix inequality. If this necessary and sufficient condition is satisfied, this robust state feedback gain matrix can be easily derived to guarantee robust stability and prescribed closed loop performance. Secondly the plug and play idea is introduced in the design process. Finally by one example of aircraft flutter model parameter identification, the efficiency of the proposed control strategy can be easily realized.

Modern engineering system offers many examples of man-made systems characterized by a large numbers of states and inputs. One concept about large-scale systems is always thought as the result of many subsystems interacting through the coupling of physical variables or the transmission of information over one communication network. Complexity of these large-scale systems brings some challenges such as the application of simulation, analysis, and control design algorithms.

Consider the problem of control design algorithms for large-scale systems; two common used algorithms are centralized control scheme and distributed control scheme. In centralized control scheme, all control variables are computed by a single regulator. Therefore all subsystems transmit their outputs to the central controller in order to design the control inputs which will be sent back to actuators collocated with subsystems. But a centralized control scheme for the large-scale systems will suffer from many problems such as (1) computational complexity, a centralized regulator needs a considerable amount of computing power and memory in order to compute control inputs within a sampling interval, (2) communication network, a centralized control requires a star-like topology of the communication network that impacts the cost of the control system, and (3) reliability, a failure in a single subsystem or in a link could comprise the proper functioning of the overall controlled large-scale system. In order to remedy the above drawbacks, each subsystem in distributed control scheme is equipped with a local controller that receives the outputs from the corresponding subsystem and computes the control inputs. It means that each subsystem is equipped with a local controller, but controllers can transmit and receive quantities from other subsystems. Therefore if these pieces of information are properly used, the goal of stabilizing the closed loop large-scale system and guaranteeing prescribed level of performance can be achieved.

Now one new idea of distributed control scheme is proposed to design the local controller in large-scale systems in reference [

Using above descriptions and advantages of plug and play distributed control, in this short note, we apply this new plug and play idea into the robust control scheme to propose one new method (plug and play robust distributed control method). The goal of the robust control scheme is to guarantee that one

Consider a continuous time linear invariant system.

We treat

For the convergence of next analysis, we need the following two assumptions to express that subsystems

Matrix

The physical meaning of assumption 1 is that each subsystem is coupled through state variables only. In reference [

The matrix pairs

Under these two assumptions, if one stable distributed state feedback controller

In this section, we derive one linear matrix inequality condition in one state feedback

Assume the following equality holds

There exists one state feedback controller

There exists a symmetric matrix

From the results in [

Taking one quadratic function as that,

Theorem

Now by using that linear matrix inequality (

Consider the parameterized system structure (

There exists one state feedback controller

There exists a symmetric matrix

Considering system (

As linear matrix inequality (

In this section, we study the plug and play idea and the redesign of new controllers when subsystems are added to or removed from system (

(1) As a starting point, consider a system composed by subsystems

Consider the plugging of subsystem

(2) When subsystem

(3) Generally after combing all above descriptions, one plug and play robust distributed control algorithm is formulated as follows.

Partition the original continuous time linear invariant system as (

Compute matrices and other variables such as

Use linear matrix inequality tools from MATLAB 2009 to find one symmetric matrix

Find one appropriate state feedback controller

Merge the plug and play idea during the above process to verify all

Construct the original control input as

In this simulation part, we give one aircraft flutter model parameter identification example to confirm the efficiency of our plug and play robust distributed control strategy.

In the simulation example, we use one flutter mathematical model from [

Bode plots compared between the true system and the identified system.

In this short paper, we propose one plug and play robust distributed state feedback control for one kind of continuous linear invariant systems with ellipsoidal parametric uncertain set. After partitioning the original system into some subsystems, we derive one necessary and sufficient condition under which the robust distributed controller exists by using the linear matrix inequality tool for each subsystem. Then we apply the new plug and play idea into the robust control scheme to propose one method-plug and play robust distributed control method. But all the derivations are based on the assumption that the unknown parameter is known in one ellipsoidal set. So when any knowledge of unknown parameter in the state space matrix form is not known a priori, we should apply the input-output data to design one system identification experiment and obtain its parameter estimation. Then the next topic is how to apply the system identification concept into our control strategy.

The authors declare that there are no competing interests regarding the publication of this paper.