Research On Controller Parameterization For Bifurcation Polynomial Systems Based On Symbolic Computation

Bifurcation polynomial system is a widespread dynamic system, and the control of which is an important research direction in the control of nonlinear systems. Aiming at the Hopf bifurcation problem in bifurcation polynomial system, this paper proposes a design method of the controller parameterization, which is used to design a controller for this kind of system to realize the trajectory stabilization of the system state at the equilibrium point. Compared to the existing research results, the controller designed by this paper is composed by the vector of equation of original system, which is in simple form. At the meanwhile, the controller keeps the position of the original system equilibrium point unchanged, and can achieve stabilization and control at all equilibrium points of the system. In the process of parameterized controller design approximate calculation and experience formula are typically used, and the obtained constraints are of strong limitation. From these constraints, only a limited number of solutions will be obtained generally; as a result, in this paper the constraints of controller are computed and simplified by symbolic computation technology. The obtained constraints are composed of equations and inequalities. Direct solution of control parameters from these constraints is tedious and difficult, and in this paper the parameters of controller are solved by the algebraic geometry method. The problem of equality and multiple inequality solving is transformed into problem of region division and search in the space of controller parameters, so as to realize the full solution of the constraint condition. The main works of this paper are as follows:(1) In this paper, the general expression of the controller with parameters is first defined, and the construction method of the controller parameterization is proposed. Then, according to the Hurwitz criterion, the constraints of the controller parameters are derived and the validity of the controller is proved.(2) In this paper, when solving the controller parameters from constraints, the problem of solving complex semi algebraic set is transformed into the problem of region division in controller parameter space, and the system controller parameter region which satisfies the constraints of the system is searched out. For the controlled system, whose controller parameters in this region, can be realized the Hopf bifurcation or stable control near the equilibrium. In this paper, the R?ssler system, which has two equilibrium points, is used as an example to illustrate the design process of the controller, and the effects of different controller parameters at each equilibrium point are analyzed.(3) In this paper, the proposed controller parameterization method is applied to the bifurcation polynomial system, which has a switch on bifurcation parameter, and the system is realized to stable control. Due to the sensitivity of the system state to the bifurcation parameters change, the designed controller usually needs to be changed or even to be re-designed when the bifurcation parameter is switched. In the design process of controller, this article deals the system with switched bifurcation parameter as two separate subsystems to be designed. Using controller parameterization design method and taking into account the effects of constraints of each equilibrium points when bifurcation parameter switching, the range of controller parameters satisfying all constraints is solved by the method of algebraic geometry. In this way, the system can be stable controlled and need not to change the controller. Simulation results show the effectiveness of controller parameterization method in this article for the bifurcation polynomial system with switched bifurcation parameters.(4) Based on the controller parameterization method of this paper, the robust stability controller design method is proposed to bifurcation polynomial system. Because of the complexity of nonlinear system structure, it is difficult to solve exactly the bifurcation parameter values. The system state characteristics are very sensitive to changes of bifurcation parameters, and the dynamic characteristics of the system may be affected by mall changes in the bifurcation parameters. So it is expected that the controller has some adaptive ability for the bifurcation parameter change near bifurcation point, namely, which has some robustness characteristics, while design the controller. Taking Lorenz system as an example to illustrate the controller derivation and design process, and then Van der Pol oscillator system is used as an example to the engineering application.(5) The constraints of the controller parameters increase exponentially with the number of equilibrium points of the system. To solve this problem, in this paper the calculation process of the proposed controller parameterization design method of bifurcation polynomial system is optimized to simplify the constraints of the system and reduce the number of constraints. The optimization of the controller parameters is studied. An optimization algorithm is proposed for the system state adjusting time. After getting the parameter range of the controller, the optimization algorithm can be used to improve the adjusting time of the system state.(6) The controller parameterization design method is used on a brushless DC motor system model for an engineering application. Added the controller, the chaotic states of original system are stabilized to the equilibrium point, and the simultaneous stabilization of system states can be realized at three equilibrium points. When the equivalent load is fully loaded and there is a sinusoidal disturbance, the controller can also stabilize the system at three equilibrium points simultaneously and the trajectories of system states are convergent.