Time-varying controller synthesis for structures subjected to parametric periodic loadings

Linear and nonlinear time-varying controller synthesis for systems represented by nonlinear differential equations with periodic coefficients is addressed. A recently developed technique, based on the Liapunov-Floquet (L-F) theorem, is employed so that time-varying control gains can be obtained via time-invariant techniques. Further, a simple time-varying pole-placement approach for the design of linear control has also been devised for linear time-periodic systems. The robustness of the above control designs under structured perturbations of the nominal system matrices is studied. In many cases, the linear control design alone may not meet the desired performance specifications of the nonlinear periodic systems due to the time-varying nature of the problem. Therefore, to improve the controlled response of the nonlinear system, a nonlinear time-varying controller is designed and incorporated. In this paper, three control design methodologies are compared. In the first two methods, linear and nonlinear controllers are used to stabilize and improve the response specifications, respectively. The linear control designs are based on the L-F transformation approach and the time-varying poleplacement approach whereas, the nonlinear controller is obtained using the Liapunov's Direct Method. In the third approach, a purely nonlinear controller is employed to stabilize the system, even though the linear part is unstable. The responses obtained through the above three approaches are compared and the advantages and disadvantages of the methods are discussed. Noticeably, the combination of linear and nonlinear controllers based on L-F transformation approach has been found to have better performance and robustness characteristics than the other approaches. The gains obtained via purely nonlinear control is found to be large and sensitive to initial conditions of the system and therefore, does not represent a robust control design. Introduction Periodically time-varying systems appear naturally in various branches of science and engineering. Numerous practical applications can be found in the areas of (i) structures subjected to periodic loading1, (ii) helicopter rotor blades in forward flight2, (iii) asymmetric rotorbearing systems3, (iv) robots performing repetitive tasks4, (v) quantum mechanics5 and (vi) electrical circuits5. The equations of motion for these systems, in general, have time-varying periodic coefficients and are nonlinear. The control problems associated with linear periodic systems alone are quite challenging due to their time-varying nature. One of the principal reasons that could be attributed to this fact is that the time-varying eigenvalues of the periodic matrix does not determine the stability of the systems and one must resort to Floquet analysis. Hence it is difficult to apply classical as well as modern control techniques to these problems as opposed to the case of time-invariant systems. Methods for control synthesis of linear time-varying systems have been reported in the past by several authors. Invariably, these methods are based on transforming the original system into a suitable canonical form so that some of the special properties of the canonical system can be utilized for controller design^^.^.^^. However, such transformations, if they exist, are not unique and are tedious to implement, especially for higher dimensional systems. Calico and Wiese17 discussed the active control problems associated with time-periodic systems where an iterative procedure was suggested to achieve control via the pole placement techniques. However, the gains thus obtained were only a subset of possible gain selections and did not represent the most general possibilities. shammas has reported conditions for robust finite-energy input-output stability of nonlinear plants with nonlinear time-varying perturbations or linear time-varying plants subject to linear time-varying perturbations. A modified version of the small-gain *Ph.D. Candidate, Student Member AlAA tprofessor, Senior Member A I M Copyright c 19% by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved 1 theorem has been employed in this work. Nevertheless, when the perturbations are not small, the applicability of such an approximation is still not clear. Calico et aL9 developed a method to design fixed-gain controllers for time-periodic systems. The approach was quite complicated and required two levels of iteration in calculating the feedback gains. In this regard, Joseph1', Sinha and ~ o s e p h l ~ have recently developed a new strategy in designing controllers for linear periodic systems through an application of the Liapunov-Floquet (L-F) transformation matrix which permits the construction of an equivalent time-invariant problem. Then, the wellknown time-invariant techniques can be employed to design the various controllers. Finally, the time-varying gains for the periodic systems are obtained easily due to the invertible nature of the L-F transformation matrix. However, the robustness characteristics of all the designs stated above, especially when they are implemented in periodically time-varying nonlinear systems, are not known. In this paper, new methods to design time-varying controllers for nonlinear time-periodic systems are presented. Both linear andlor nonlinear controller synthesis are designed and their robustness characteristics addressed. The linear controller designs are based on either the L-F transformation matrix approach or the simple time-varying pole placement approach. Further, to improve the closed-loop system performance, nonlinear controllers are incorporated via the Liapunov's Direct Method. Three design methodologies are presented. In the first two methods, linear as well as nonlinear controllers are used to stabilize and improve the response specifications. In the third approach, a purely nonlinear controller is employed to stabilize the system, even though the linear part is unstable. In the following, the mathematical preliminaries necessary for the development of time-varying control designs are briefly discussed. Mathematical Preliminaries Stability of Nonlinear Periodic Systems Consider the control problem of a periodically timevarying nonlinear system represented by with A(t) = A( t+n ,B( t ) = B(t+n,f(x, t ) = f (x , t+n where x is an n x 1 state vector, u = uL + uN is an m x 1 control vector, A(t) is a n x n system matrix, B(t) is a n xm control matrix and f(x,t) is the nonlinear function associated with the system which satisfies the condition f(0,t) = 0 . uL and uN are the linear and nonlinear control vectors, respectively. Here, the aim is to select the appropriate control vector u such that the system is asymptotically stable and meets the system performance specifications. In the following, the control and stability of equation (1) is discussed using the Liapunov's Direct method. Let the linear control function be selected as