This paper investigates the H∞ controller design method for a class of singular networked control systems (SNCS) based on the singular plant. In view of the network-induced delay less than or equal to a sampling period, finite external disturbance, clock-driven sensors, event-driven controller and actuators as well as impulse behavior and structural instability of singular plants, the H∞ controller design method of SNCS with state feed- back way and dynamic output feedback way is investigated respectively by means of the linear matrix inequality method. The existence condition of H∞ control law, the solving approaches of H∞ controller parameters and disturbance attenuation degree are presented. Finally, a simulation example is given to illustrate the effectiveness and feasibility of the presented method.
Networked control system (NCS) is a distributed realtime feedback control system where the system node situates different geographical position exchange data and control signal with controller via communication network [
It should be pointed out that, most of the results in the existing literature are focused on linear normal system, while the study of singular networked control system (SNCS) based on singular system has not been addressed intensively. Since the dynamics of singular system is quite different from normal linear/nonlinear system, and has many characteristics such as pulse characteristics, no causality, no solution, no uniqueness, structure instability, etc. [
In this paper, we aim to investigate the stabilization and controller design method for a class of SNCS subject to the double characteristics of singular systems and NCS. In this work, network-induced delay, limited input disturbance, impulse behaviour are taken into simultaneous consideration. The control method of SNCS with state feedback way and dynamic output feedback way is investigated respectively by means of the linear matrix inequality method. The existence condition of control law, the solving approaches of controller parameters and disturbance attenuation degree in different feedback way are presented. Finally, a simulation example is given to illustrate the effectiveness and feasibility of the proposed method.
The SNCS based on singular plant is shownin
In this paper, it is assumed that sensors are driven by clock, controller and actuators are driven by event, the measuresensorssample the state value or output value of the plant with period, the measured value are transmitted to the remote controller via network after A/D conversion and packaging; controller respond immediately to calculate control law and transmit to actuator node after receiving the information from sensors, and actuator node work immediately to implementadjustment job after receiving the control signal from the controller.
As the system is shown in
employed by the communication network. The delay maybe is constant, random, limited, even Markov chain feature. In order to enhance the system performance, in a general way, we make as far as possible it constant. Furthermore, there exists single packet and multiple packet transmission, data packet loss, network connection interrupt and channel interference etc. All of these problems will make the structure characteristics of closeloop systems change, and influence the stability and control performance of the SNCS.
In this paper, the considered singular plant is shown in equation (1):
where, , and
are state vector, control input vector, output vector and expectation output vector, respectively., and are constant matrix, is singular matrix, i.e.; is finite external disturbance, are corresponding dimension constant matrix.
Throughout this paper, the following assumptions are made:
1) The singular plant is regular and impulse free, which is achieved by adjustingthe part structure and componentconfiguration of plant, such that one of the following holds:
a)
b)
2) The network-induced delay of closed-loop system is less than or equal to a sampling period, i.e., and the sample period is constant, which is achieved bychoosing suitable communication protocol of control network and designing part device of the system.
3) The network communication is single packet transmission, and there is no packet loss.
4) The external input disturbance of the plant is finite energy, i.e. the close-loop transfer function from to satisfies, is a scalar.
According to condition (1), when the singular plant is regular and impulse free, there are always two nonsingular matrices, such that
,
,
Let, equation (1) can be equivalent transformed as:
When the network-induced delay, control input is piecewise continuous in a sampling period, the discrete-time model of equation (2) in a sampling period can be shown as equation (3):
where
,
,
,
.
The state feedback controller model is shown in equation (4).
Combine equation (3) with equation (4), the following closed-loop system is obtained
Let augmented state vectortherefore, the close-loop model of state feedback SNCS is as follows:
When the state variables are not measurable, or partial state is measurable, we will put to use the following dynamic output feedback controller:
Combine equation (3) and equation (6), the following can be obtained:
Let augmented state vector
, then, the close-loop model of dynamic output feedback SNCS is shown in equation (7):
Clearly, whether put to use state feedback or output feedback, the close-loop system model of SNCS is a linear normal system depending on time delay. When is constant quantity, the close-loop system model of SNCS is a linear time-invariant system, when changes with time, the close-loop system model of SNCS is a time-varyingsystem.
Define 1: Given a positive constant, for the state feedback case, if close-loop system (5) is asymptotically stable under zero initial condition,external disturbance and expected output satisfy norm constraint condition, then,
singular plant (1) realizes second best state feedback control, the system disturbance attenuation degree is defined as, the corresponding state control law is defined as second best state feedback control law; further optimization make minimum, in this case, the state feedback control law is defined as best state feedback control law.
Define 2: Given a positive constant, for the dynamic output feedback case, if close-loop system (7) is asymptotically stable, and when zero initial state
, external disturbance and expectation output satisfy norm constraint condition, then singular plant (1) realizes
second best dynamic output feedback control, the system disturbance attenuation degrees is defined as, the corresponding dynamic output feedback control law is defined as second best dynamic output feedback control law; further optimization make minimum, in this case, the dynamic output feedback control law is defined as best dynamic output feedback control law.
Lemma1: [
if and only if there is a scale, such that
Theorem 1: Without regard to the external disturbance, under the control of state feedback controller (4), if there exist positive definite matrices, such that
where
,
,
then state feedback SNCS (5) is asymptotically stable.
Proof: Choose positive definite matrices and, define a Lyapunov function as follows:
.
then the forward differential of along trajectory of closed-loop system (5) is as follows:
where,
By Lyapunov stability theory, if,then system (5) is asymptotically stable, so,the asymptotically stability condition is as follows:
By Schur complement, equation (9) can be transformed as:
Multiplying on the left side and the right side of equation (10), it is derived that
Let, then equation (11) is equivalent to equation (8), the proof is completed.
Theorem 2: For singular plant (1), under the control of state feedback controller (4), for given disturbance attenuation degree, if there exist symmetric positive definite matrices, such that
where
,
,
then singular plant (1) canrealize second best state feedback control.
Proof: The external disturbance is taken into accountaccording to definition 1, to make hold, Let, choosepositive definite matrices, and define a Lyapunovfunction.
For close-loop system (5), when meet theorem 1, the system is asymptotically stable, in the zero initial conditions, for, it is derived as
where,
,
,
,
By Schur complement, Equation (13) can be transformed as
Similarly, further transform, equation (12) can be derived, the proof is completed.
Theorem 3: For singular plant (1), under the control of state feedback controller (4), if there exist symmetric positive definite matrices, matrices, scalars and compatible dimension unit matrix, such that
then the disturbance attenuation degree, second best state feedback controller is as following:
Proof: For singular plant (1), if second best state feedback control law exists, then theorem 2 is true.
Spread out, then equation (12) can be expressed as
Equation (17) can be written as:
From lemma 1 and Schur complement, equation (18) can be transformed as
where. further transform, equation (19) is derived that
Multiplying on the left side and the right side of equation (20), and Let
, and then Let, ,
, , equations (15) and (16) are derived, the disturbance attenuation degree. The proof is completed.
Theorem 4: For singular plant (1), if the following optimization problem has feasible solutions:
the minimum disturbance attenuation degree is,
best state feedback controller is
By means of feasibility problem Solver “feasp” and optimization problem Solver “mincx” of MATLAB LMI tool-box, if the feasible solutions of theorem 3 and theorem 4 exist, second best state feedback controller, best state feedback controller as well as corresponding disturbance attenuation degree are obtained.
Theorem 5: when the external disturbance is not taken into account, under the control of dynamic output feedback controller, if there exist positive definite matrices, such that
where, , , , then dynamic output feedback SNCS
(7) is asymptotically stable.
Proof: Let, , , , , then equation(7)
can be written as
When the external disturbance of system is not taken into account, choose positive definite matrices and define a Lyapunov as follows:
Then the forward differential of along trajectory of close-loop system (7) is as follows:
where,
,
,
By Lyapunov stability theory, the asymptotically stability condition of system (7) is as follows:
By Schur complement, the above equation (24) can be transformed as:
Multiplying on the left side and the right side of equation (25), and Let, equation (25) is equivalent to equation
(23), the proof is completed.
Theorem 6: For plant (1), under the control of dynamic output feedback controller (6), for given disturbance attenuation degree, if there are symmetric positive definite matrices, such that
then singular plant (1) realizes second best dynamic output feedback control.
Proof: The external disturbance is taken into account, by definition 2, to make the following equation exist:
, we Let
, choose positive definite matrices, define a Lyapunov function as follows:
For dynamic output feedback close-loop system (7), when satisfies theorem 5, the system is asymptotically stable, in the zero initial conditions, for, we have:
Let, , , we have:,
, ,
, ,
, ,
Further transform, inequality (27) can be derived, the proof is completed.
Theorem 7: For singular plant (1), under the control of dynamic output feedback controller (6), if there exist symmetric positive definite matrices, matrices, scalars and compatible dimension unit matrix, such that
where, , then the disturbance attenuation degree, second best dynamic output feedback control law is as follows:
Proof: if plant (1) can realize second best dynamic output feedback control, then theorem 6 exists. Spreading out of inequality (26), inequality (26) can be written as
From lemma 1, inequality (30) can be transformed as
Multiplying on the left side and the right side of the above inequality, and Let
,
, ,
, we can obtain inequality (28). By calculating the feasible solutions of inequality (28), we can get controller parameters and equation (29), thereforethe proof is completed.
Theorem 8: For dynamic output feedback SNCS (7), if the feasible solutions following optimization problem (31) exist:
The minimum disturbance attenuation degree
, best dynamic output feedback
control law :
To illustrate the effectiveness of proposed method, we focus on state feedback control way. A typical singular plant model with external disturbance is as follows:
The sampling period, the network-induced delay.
Choose nonsingular matrices as follows:
,
The above singular plant model can be transformed as
Its discrete model parameter is as follows:
, ,
, , ,
, , ,
Choose the controllerby LMI tool-box of MATLAB to solve the feasible solutions of theorem 1, it is shown that the system is asymptotically stable. When initial state, the system state response trajectory of external Sine disturbance is as blue solid line shown in
By control, use theorem 3 to solve its feasible solutions as follows:
,
,
Therefore the disturbance attenuation degree
; the second state feedback
controller is
Under the same conditions, the system state response trajectory is as black dotted line shown in
By LMI tool-box of MATLAB to find the optimized solutions of theorem 4, the obtained corresponding solutions are as follows:
.
Therefore the minimum disturbance attenuation
, thebest state feedback
control law is as follows:
.
After putting optimal into effect, the system state response trajectory is as dotdashline shown in
Before and after optimization control, the system expectation output is as blue solid line and black dotted line shown respectively in
The system simulation shows that the disturbance attenuation degree can decrease to 0.0591 from 35.9133 after optimization control, and the anti-interference performance is enhanced markedly. As a result, the system stability performance has been improved.
In this paper, when focused on network communication
characteristics and singular plant characteristics simultaneously, the optimal control problems for a class of SNCS are addressed with both state feedback case and dynamical output feedback case. The network communication characteristics include the network-induced delay less than or equal to a sampling, limited input disturbance, clock-driven sensors as well as eventdriven controller and actuators. The singular plant characteristics include impulse behavior, structuralinstability and something like that. This paper presents respectively the optimal control method, the existence condition of control law and the solving method of control law and disturbance attenuation degree. The simulation results show that optimal control of SNCS makes the disturbance attenuation degrees decrease obviously and makes the anti-interference performance enhance obviously. Therefore, the analytical method and the results are valid and feasible.
This project was supported by the National Natural Science Foundation of China (61074029, 61104093) and the Science Research Program of Liaoning Province (2011216007).