Improving performance of low order robust controllers for parametric uncertain systems

Abstract

The classical control problem deals with the design of closed-loop systems that are stable with fixed controller structure, the P, PI and PID type controllers. In addition to stability, reference tracking, noise attenuation and disturbance rejection can be addressed during the controller design. One of the popular design methods for reference tracking is the dominant pole placement method. The time domain specifications are adjusted based on a second-order polynomial and additional poles out of the dominant region, the region of the poles of the second-order polynomial, are added to the polynomial. The polynomial is then equated to the characteristic polynomial of the closed-loop transfer function. In the end, it is desired to end up in a dominant pole pair to dominate the behavior of the closed-loop system. For the dominant pole placement method, a certain settling time and overshoot value are chosen according to the application and based on these a complex pole pair or in other words the second-order polynomial is defined. For closed-loop systems with zero overshoot, the formulae used during the dominant pole placement method lose their validity, since the damping ratio is greater than or equal to one. The damping ratio for zero overshoot is an inequality that implies that there are multiple solutions. The damping ratio inequality defines the family of critically and over-damped all-pole systems. The settling time of such systems is investigated in the literature, however, most are only for analysis purposes. To be able to synthesize controllers, it is necessary to be able to choose a settling time value and convert this information into some pole locations on the s-domain. Since most of the work uses iterative models, a new method for this "inverse" relation between settling time and poles is proposed. The model proposed is developed for over-damped systems up to order three, but it has been mentioned that for higher-order systems the precision of the model loses significance. Several examples of controller synthesis are provided. A different approach is the characteristic ratio assignment method, which uses the coefficients of the polynomial to calculate the ratios instead of using the poles directly. The characteristic ratios are chosen based on filters with known characteristics so that the designed closed-loop system has the same properties. A common choice is the Butterworth filter for low overshoot characteristics. Hence, the problem encountered during the dominant pole placement is solved, but at the cost of losing the ability to fix the structure of the controller. Since, the CRA method designs controllers equal to the order of the plant, resulting in a closed-loop system twice the order of the plant. This is not desirable considering the practical world. Another loss is the lack of the integrator in the controller in the feedforward path, where because of it the reference tracking is less robust. The CRA method is based on the maximally-flatness property of a system, which uses the frequency domain Bode gain to come up with low overshoot step responses. This work combines the aforementioned design methods, by choosing classical controllers and the maximally-flatness property of the CRA method. The closed-loop transfer function is calculated with the chosen classical controller and the gain of the closed-loop transfer function is computed in the frequency domain. The magnitude square, also called the power gain, of the closed-loop system is obtained and a direct low-pass inequality is stated, which results in inequalities that for low-degree systems resemble the characteristic ratios. These inequalities then are used to ensure low or even zero overshoot. Due to the difficulty of assessing the overshoot of high-order systems, the resulting overshoot is called low or zero. From numerical studies, it is observed that most of the designs prohibit zero overshoot. It has been shown that using the inequalities obtained by setting the power gain of a transfer function less than or equal to one and setting the infinity norm of a transfer function equal to one are strongly connected. Thanks to this connection, the term "low or zero overshoot" for first and second-order transfer functions is calculated. It has been proposed that inequalities arising by setting the power gain to less than or equal to one or setting the infinity norm equal to one can be used to enforce low or zero overshoot. Based on this a design approach is given where the controller is a PI or PI-PD controller, the plant is a FOPDT or SOPDT plant and based on the closed-loop the inequalities are added to design such a controller. Since the FOPDT and SOPDT plants include delay in their models, the Pad\'e approximation method is used during the design steps. The Pad\'e approximation method substitutes the nonlinear delay term with a set of zeros in the right-half-plane and poles in the left-half-plane. The design conditions are obtained as inequalities and introducing interval-type uncertain parameters with lower and upper bounds results in a robust design problem that can be simplified by checking specific corner points instead of the whole uncertain family. A theorem is stated for this reduction, which simplifies the robust design problem, significantly. The optimization problems for robust PI controller design for FOPDT and robust PI-PD controller design for SOPDT systems are stated. Some numerical case studies are given to back up the optimization problems. The unique aspect of this work is summed up as follows; \begin{itemize} \item inequalities from polynomial coefficients are produced based on maximally-flatness properties, which is more powerful than CRA since it takes zeros into account \item a general formula is worked out for the inequalities \item nominal low-order controllers are designed with characteristic-ratio-like inequalities \item the structure of the proposed inequalities is exploited and a reduction theorem is proposed for plants with uncertainty \item based on the proposed theorem robust controller design problem is stated \item a settling time model for critically and over-damped systems is proposed \item it has been shown that the proposed settling time model can be used to determine the locations of the poles with prefixed settling time \end{itemize} During this work a settling time model for over-damped systems is proposed which can be used to determine the settling time by plugging in the poles and to determine the locations of the poles by choosing a prefixed settling time. It has been shown that it is possible to design controllers with the proposed settling time. It has been observed that using the maximally-flatness approach in order to get low or zero overshoot design or setting the H infinity norm of the closed-loop equal to one produce the same conditions. Using these conditions it is possible to get low or zero overshoot. Due to the structural advantage of the proposed inequalities a theorem that reduces the complexity of evaluating and ensuring that the inequalities hold for every point in the uncertainty space is proposed. Both nominal and robust PI and PI-PD controller design for FOPDT and SOPDT systems is stated.