Frequency-domain controller design by linear programming

In this thesis, a new framework to design controllers in the frequency domain is proposed. The method is based on the shaping of the open-loop transfer function in the Nyquist diagram. A line representing a lower approximation for the crossover frequency and a line representing a new linear robustness margin guaranteeing lower bounds for the classical robustness margins are defined and used as constraints. A linear programming approach is proposed to tune fixed-order linearly parameterized controllers for stable single-input single-output linear time-invariant plants. Two optimization problems are proposed and solved by linear programming. In the first one, the new robustness margin is maximized given a lower approximation of the crossover frequency, whereas in the second one, the closed-loop performance in terms of load disturbance rejection, output disturbance rejection and tracking is maximized subject to constraints on the new robustness margin. The method can directly consider multi-model systems. Moreover, this new framework can be used directly with frequency-domain data. Thus, it can also consider systems with frequency-domain uncertainties. Using the same framework, an extension of the method is proposed to tune fixed-order linearly parameterized gain-scheduled controllers for stable single-input single-output linear parameter varying plants. This method directly computes a linear parameter varying controller from a linear parameter varying model or from a set of frequency-domain data in different operating points and no interpolation is needed. In terms of closed-loop performance, this approach leads to extremely good results. However, the global stability cannot be guaranteed for fast parameter variations and should be analyzed a posteriori. Nevertheless, for certain classes of switched systems and linear parameter varying systems, it is also possible to guarantee the stability within the design framework. This can be accomplished by adding constraints based on the phase difference of the characteristic polynomials of the closed-loop systems. This frequency-domain methodology has been tested on numerous simulation examples and implemented experimentally on a high-precision double-axis positioning system. The results show the effectiveness and simplicity of the proposed methodologies.

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