Abstract: Based on differential geometry theory, a nonlinear affine model of an inductor current continuous (CCM) Buck converter is established using the pulse waveform integration method. Under the condition of satisfying exact feedback linearization, the nonlinear coordinate transformation matrix and state feedback expression are derived using the exact feedback linearization method, resulting in a linearized model of the Buck converter. Taking the linearized Bruno canonical form as the new controlled object, a controller is designed using linear quadratic optimal control. Simulation experiments show good control performance, and the influence of weighting matrices Q and R on system performance indicators is analyzed. This also demonstrates that exact feedback linearization can achieve optimal control for piecewise linear systems like the Buck converter, possessing general theoretical and practical significance.
Keywords: precise feedback linearized optimal control, Buck switching converter
Chinese Library Classification Number: TM461 Document Identification Code: A
0 Introduction
In the past two decades, differential geometry and differential game theory have been combined with the design problem of nonlinear control systems[1], providing a feasible way to solve complex nonlinear systems. The greatest advantage of differential geometry is that it transforms the study of the submanifolds of the differential manifold into the study of the subdistributions of the tangent space, such as the correspondence between the Nanda submanifold and the controllable Lie algebra[2]. Therefore, differential geometry greatly facilitates the structural analysis, decomposition and structure-related control design of nonlinear systems, enabling differential geometry to be directly applied to the linearization, decoupling, zero dynamic systems and feedback stabilization of nonlinear systems, thus providing the possibility of using differential geometry to realize nonlinear control of DC/DC switching converters. However, due to the nonlinear discontinuous operation of DC/DC switching converters, differential geometry cannot be directly applied and must be analyzed and applied conditionally based on the model of the DC/DC switching converter[3].
This paper takes the most conventional current-continuous Buck converter as the research object. Based on the feedback linearization method of differential geometry, the original nonlinear system is equivalent to a fully controllable linear system. Then, a linear quadratic optimal controller is designed to achieve optimal control of the switching converter. Simulation results show that the control strategy has good dynamic performance and robustness.
1. Mathematical Model of CCMBuck Converter
Since the DC/DC switching converter switches periodically between multiple linear systems, it is a typical piecewise linear system. In order to model the converter in a unified manner, the pulse model integration method is introduced [4], and the pulse function waveform is defined as shown in Figure 1.
The working principle diagram of the PWMBuck switching converter is shown in Figure 2, where Figure 2(a) is the circuit schematic, Figure 2(b) shows the switching transistor in the on state, and Figure 2(c) shows the switching transistor in the off state.
2. Buck converter precise feedback linearization
2.1 Determine the degree of relation
The Li derivative is calculated based on the exact feedback linearization theory in reference [5]:
4. LQR Controller Design
The state equation of a given linear time-invariant system is:
Therefore, we get K=1 and K=1.4142.
The state feedback control law of this system is:
When the Buck converter circuit operates in steady state, the average value of the load voltage[7] is
This also leads to a decrease in the average load voltage. Therefore, increasing Q to obtain a smaller rise time comes at the cost of sacrificing the duty cycle.
6. Conclusion
This paper introduces differential geometry theory into power electronic control systems and applies linear quadratic optimal control based on precise feedback linearization to improve the control characteristics of power electronic converters. Based on a unified modeling of the CCMBuck converter using the impulse model integral method, this paper discusses the conditions for achieving precise linearization of the converter and presents its feedback rate and control expressions. Simulation experiments demonstrate excellent control performance, and the influence of the weighting matrices Q and R on the system performance indicators is analyzed.
References
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[2] Cheng Daizhan. Geometric Theory of Nonlinear Systems [M]. Beijing: Science Press, 1982.
[3] Lu Qiang, Sun Yuanzhang. Nonlinear control of power systems [M]. Beijing: Science Press, 1993.
[4] Sato Y, kataoka. State feedback control of current type PWM ac-to-dc converters [J]. IEEE Trans. on Ind. Applicant, 1993, (99):1090-1097.
[5] Hu Yueming. Theory and Application of Nonlinear Control Systems [M]. Beijing: National Defense Industry Press, 2005.
[6] Liu Bao. Modern Control Theory [M]. Beijing: China Machine Press, 2007.
[7] Wang Zhaoan and Huang Jun (eds.). Power Electronics Technology [M]. Beijing: China Machine Press, 2000.
