For the technology of mechanical elastic energy storage utilizing spiral torsion springs as the energy storage media presented previously, a global multivariable control algorithm based on nonlinear internal model principle under multiclass external disturbances is proposed. The nonlinear external disturbances with nonharmonic periodic characteristics are generated by multiclass nonlinear external systems. New equations of nonlinear internal model are designed to estimate the multiclass external disturbances. On the basis of constructing the control law of nominal system, a state feedback controller is designed to guarantee the closed-loop system globally uniformly bounded, and a Lyapunov function is constructed to theoretically prove the global uniform boundedness of the multivariable closed-loop system signals. The simulation results verify the correctness and effectiveness of the presented algorithm.
Energy storage technologies have a great practical significance for the solution of new energy interconnecting to the grid, peak regulation, frequency modulation, and stability control [
One of the core questions in control field is to guarantee asymptotical stability of unforced close-loop system, implement the asymptotical tracking of system output for given trajectories, and reject exogenous disturbances [
Another central issue in control field is to extend the established control theory to more complex and generalized circumstances. In [
The mainly theoretical contribution of the paper is to propose a global multivariable disturbance control method to reject multiclass nonlinear external nonharmonic disturbances generated by multiclass nonlinear exosystems for general multivariable nonlinear system, and the presented control algorithm is employed to regulate an actual nonlinear system. The validity and effectiveness of the proposed method are testified by the simulation results.
The organization of the paper is as follows. It starts with an introduction of the research status of disturbance rejection and points out the significance of rejection of nonharmonic external disturbances in Section
For the convenience of understanding and reutilization, the model of the whole system for PMSM based MEES system proposed in [
The model of PMSM based MEES system.
In the process of energy storage, PMSM runs in the state of electric motor. For PMSM, assume that the inductance of
The ratio of gear case is assumed to be
Suppose the outer end of STS to be fixed in spring box as V type, and in terms of the national standard
Due to the large mass and high inertia of MEES system, the working rotation velocities for the main shaft of energy storage box and the rotor of PMSM are both assumed to be invariable. Hence, the relationship between the angular velocity
Equation (
For a given STS, (
The mathematical model for the whole system of PMSM based MEES can be obtained by combining differential equation (
Consider the multivariable nonlinear system with a standard affine form under multiclass disturbances:
If the nonlinear disturbance inputs are ignored, for system (
The essence of solving stability problems for a multivariable input system is to convert these problems into the stability problems of multiple single-input systems [
For system (
The trajectory of the vector field for the nonlinear exosystem (
The functions, which meet Assumption
There exists a smooth function
The problem to be solved in the paper can be described as in the following definition.
For any given compact subset
In the paper, internal model principle (IMC) is utilized to reject the multiclass disturbances. Disturbances rejection by IMC belongs to indirect suppression algorithm. Hence, appropriate internal model equations should be firstly established to estimate the inputting nonlinear disturbances. Because the exosystem discussed in the paper is nonlinear, the internal model equations established should also be nonlinear. Therefore, Assumption
For the nonlinear exosystem (
Consequently, the multiclass nonlinear internal model equations can be designed as follows:
Define an auxiliary error
In terms of the nonlinear internal models shown in (
Construct a Lyapunov function as follows:
Derivative of Lyapunov function
In terms of Assumption
Applying permanent establishment inequality
Substitute (
Choose appropriate
By above knowable, all the variables are bounded. Combine with the application of the invariant set theorem, it can be obtained that
There exist positive definite matrices
Theorem
The verification of the proposed algorithm in the paper is performed by means of numerical simulation in a 0.018 kWh/1.1 kW PMSM based MEES system. The specific parameters of the MEES system are shown as follows: the rating torque of PMSM
Considering the multiclass nonlinear disturbances, the mathematical model for the whole system of PMSM based MEES system is converted into the form of (
If the nonlinear disturbances
For the sake of convenience, the inputs of nonlinear external disturbances
Suppose that
Assume that
Let
Supposing that
Choose
Therefore, Assumption
The above analysis has verified that MEES system (
The numerical simulations are conducted in Matlab environment. The whole simulation time is set as 60 s with the sampling interval 0.001 s; let the initial condition of the simulation be
Nonlinear disturbance inputs
System control inputs
System states
In light of the strong coupling and nonlinear characteristics of PMSM based MEES system, a global multiclass nonharmonic disturbances rejection method for general multivariable nonlinear system under multiclasses nonlinear exosystems is proposed in the paper. For multiclass nonlinear external disturbances with different periodic bounded nonharmonic characteristics, different nonlinear internal model equations are designed. Based on design of control law for nominal system, a state feedback controller for original system is presented and a Lyapunov function is established to theoretically testify the global boundedness of all signals in multivariable close-loop system. The simulation results show that the multiclass different nonlinear disturbance inputs are all completely rejected and the close-loop system can track the reference signals promptly. Consequently, high accuracy servo control for PMSM based MEES system is realized.
In addition to PMSM based MEES system, many other practical engineering systems, including turbine motor, generator, power flexible manipulator, and communication circuit, are frequently affected by the nonharmonic disturbances generated by the external nonlinear exosystems. As the most famous and typical nonlinear circuit, Van der Pol circuit researched in the paper will excite nonharmonic disturbances and make the system mentioned above produce nonharmonic forced oscillation. The algorithm presented in the paper can eliminate the harmful oscillation and improve the stability for these practical systems.
The critical points of the output regulation problem under nonharmonic disturbances are to model the nonlinear exosystems and propose reasonable algorithm to stabilize the closed-loop system. In the future, the proposed algorithm in the paper can be able to be extended to uncertainly multivariable systems and unknown external signals; correspondingly, the innovative control technologies should be researched to cope with the more complex and generalized circumstances.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (Grant no. 51077053), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant no. 20120036130001), the Fundamental Research Funds for the Central Universities of China (Grant no. 2014MS93), and the Independent Research Funds of State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources of China (Grant no. 201209).