μ-Synthesis FO-PID for Twin Rotor Aerodynamic System

Abstract

μ-synthesis is a NP-hard optimization problem based on the generalized Robust Control framework which manages to find a controller which fulfills both robust stability and robust performance. In order to solve such problems, nonsmooth optimization techniques are employed to find nearly-optimal parameters values. However, the free parameters available for tuning must be involved only in classical arithmetic operations, which leads to a problem for the fractional-order operator or for its integer-order approximation, exponential operations being involved. The main goal of the current article consists of presenting a possibility to integrate a fixed-structure multiple-input-multiple-output (MIMO) fractional-order proportional-integral-derivative (FO-PID) controller in the μ-synthesis optimization problem. The solution consists in a possibility to find a set of tunable parameters isomorphic with the fractional-order such that the coefficients involved in the approximation of the fractional element, along with the formulation of a fixed-structure mixed-sensitivity loop shaping μ-synthesis control problem. The proposed design procedure is applied to a twin rotor aerodynamic system (TRAS) using both MATLAB numerical simulation and practical experiments on laboratory scale equipment. Moreover, a comparison with the unstructured μ-synthesis is performed, highlighting the advantages of the proposed solution: simpler form and guaranteed robust stability and performance.

Sampling rate selection for multi‐loop cascade control systems in an optimal manner

“…This subsection concerns with two classical closed-loop structures: single-loop control structure and cascade control structure. There are multiple control design approaches available in the literature for linear and time-invariant systems, starting from classical PID, linear quadratic regulator (LQR), and leading to the robust fractional-order PID (FO-PID) [33,34]. Robust synthesis methods allows the design of both structured or unstructured regulators alike [35,36].…”

“…based on the functional  Ω ( ) from (33), the following quasioptimal sampling time optimization problem:…”

“…The two experiments' functional weights are presented in Table 1, where it can be seen that, for high fidelity, an optimal solution becomes  ⋆ 1 = (0.012[s], 1), while by weighting the implementation penalty functionals more, an optimal solution becomes  ⋆ 2 = (0.0419[s], 3). Figures 5 and 6 illustrate the global extremum points for the approximated cost functionals as in (33), with the contour plots denoting on the (X , Y , Z )-axes the variables (log 10 T , −20 log 10  Ω ( )), which provide an intuitive view of the dynamics of the functional based on the input  . The figures also illustrate the highly non-convex nature of the problem and the different influences the weighting parameters c 1,8 have in the resulting contour plots.…”

“…A pair of corresponding PI regulators have been designed using the root locus methodology, while the time delays have been considered from first-order Padé approximants: rameters for the two conducted experiments are presented in Table 2, where it can be seen that, for high fidelity emphasis, an optimal solution becomes  ⋆ 1 = (2.887[s], 2), while by weighting for lower implementation requirements, an obtained optimal solution is  ⋆ 2 = (13.219[s], 4). Figures 8 and 9 portray the global optimum points for the approximated functionals from (33), where the contours denote on the (X , Y , Z )-axes the variables (log 10 T , k, −20 log 10  Ω ( )), with the same purpose of obtaining higher readability as a function of  =(T , k). The figures also illustrate the highly non-convex nature of the optimization problem and the influences the weighting parameters c 1,8 have in the resulting plots.…”

Reduced-Order Approximation of Fractional-Order Controllers by keeping Robust Stability and Robust Performance

“…However, fixed-structure controller synthesis can be performed if the controller can be described using only arithmetical expressions in terms of tunable parameters, which is a limitation for a large number of available approximations, including ORA. In order to benefit from this method, the authors proposed a solution based on Taylor approximations in [9], which was further developed in [10] and improved in [11], where the fractional-order element approximated using ORA has been successfully integrated into the RCF using an isomorphic expression of the tunable fractional-order.…”