Amir Nassirharand

Tracking and decoupling of multivariable non-linear systems

This paper presents a new systematic procedure for decoupling and command tracking of multivariable, non-linear, and unstable systems. The design methodology is based on stabilisation of the multivariable system followed by generating its describing function models; two algebraic procedures for decoupling and tracking are used. Finally, the design must be verified by a non-linear simulation to make sure that approximations made during design are valid. The major outcome of this research is that controllers of a general class of non-linear systems may be designed taking into account the amplitude dependency features of that system. This amplitude dependency is the most important characteristic of a non-linear system. This is the first paper in the area of simultaneous decoupling and signal tracking of multivariable non-linear systems that is based on the application of the describing function technique and two other algebraic techniques. In fact, this is the first work in the area of decoupling of general ...

Controller design for nonlinear multivariable systems

A new method of controller design for use with nonlinear multivariable systems with discontinuous nonlinear terms is developed. The procedure is based on generating the sinusoidal-input describing function models of the plant followed by application of a nonlinear optimization technique. The procedure is applied to bank-angle control of a highly nonlinear unmanned aerial vehicle (UAV), and the results are compared with three other controllers that were previously designed for the same system. It is concluded that the present controller out-performs a linear controller, and it competes with a nonlinear controller.

Nonlinear Lead-Lag Controller Synthesis for Unstable Nonlinear Systems

AbstractA new systematic procedure for synthesis of nonlinear lead-lag controllers for highly nonlinear unstable plants is developed. The procedure is to stabilize the unstable plant followed by generation of sinusoidal-input describing function (SIDF) models at various operating regimes of interest. The unstable plant frequency domain behavior is extracted, and a set of linear lead-lag controllers are designed that would force the open-loop behavior of the system (comprised of the lead-lag controller and the nonlinear plant) and mimic that of desired. Finally, a describing function inversion is used, and the nonlinear functions describing the values of the lead-lag coefficients as a function of the error signal are obtained. Experimental results are presented for verification, and bounded-input bounded-output stability is demonstrated by successful generation of the describing function models of the closed-loop system comprised of the synthesized nonlinear lead-lag controller and the nonlinear plant.

Frequency Domain Modeling

Mathematical modeling, in general, is one of those steps that a control engineer usually assumes that has been done by others. In today’s complicated industrialized world, mathematical models are complex, and they are usually implemented in terms of complicated and massive computer codes. Such computer models are given to a control engineer so he or she would design a robust closed-loop feedback system. One reliable approach to deal with such complicated models without any restrictions on nonlinearity type, arrangement, and the number of nonlinear terms, system order, and the number of input and/or the outputs is the frequency domain modeling that is developed in this chapter.

New Methodology for Control of Nonlinear Unstable Multivariable Systems

AbstractIn this paper, a new methodology for control of nonlinear unstable multivariable systems is developed. The method involves stabilization of the nonlinear plant followed by generation of sinusoidal input describing function (SIDF) models of the stabilized closed-loop system. With the known SIDF model of the closed-loop system and the known stabilizing controller, the model of the unstable nonlinear multivariable system is extracted; then a controller for the model is designed. Finally, the design is verified by simulation. Stability is demonstrated by successful generation of SIDF models of the designed closed-loop system for various amplitudes of excitation. The method is applied to a two-input/two-output (TITO) model of a nonlinear unstable robotic arm, and the results are compared with an alternative approach.

Design of Dual-Range Linear Multivariable Controllers for Nonlinear Systems with Application to UAV Control

AbstractIn this research, the previous design procedure for single-variable dual-range linear controllers for highly nonlinear systems is extended to the multivariable case. The procedure is based on generating the sinusoidal input describing function models of the plant followed by selection of two of these models for optimization of the dual-range linear controller. A single multivariable controller is designed by optimization to achieve a stable closed-loop system that satisfies a set of user-defined performance measures. Finally, the design is verified by digital simulation of the nonlinear plant and linear controller. Stability is demonstrated by successful generation of the sinusoidal input describing function models of the final closed-loop system. The procedure developed and the associated software are applied to bank-angle control of an unmanned aerial vehicle (UAV) with a discontinuous nonlinear term. It is shown that performance of a dual-range linear controller is superior to that of both a si...

Nonlinear Proportional and Rate Feedback Controller Design Synthesis with Experimental Verification

AbstractA new procedure for synthesis of nonlinear proportional and nonlinear rate-feedback controllers for use with unstable nonlinear systems with application to a direct drive inverted pendulum is presented. The approach is to stabilize the nonlinear system followed by generation of the corresponding describing function models at various operating regimes of interest. With the known stabilizing controller and the stabilized frequency domain models, the frequency domain models of the unstable plant are algebraically extracted. A computer-aided design technique is used, and a set of proportional plus rate feedback controllers for the set of obtained open-loop frequency domain models is designed. The table of rate feedback gains as a function of rate feedback block input amplitudes is treated as the describing function model of the unknown nonlinear rate feedback gain; describing function inversion is used to obtain the nonlinear rate feedback gain. One linear proportional controller at an arbitrary opera...

A transformed descriptor approach to stabilizability of constrained parallel splitting systems

This paper presents the application of the asymptotically positive realness constraint (APRC) in the stabilization for interconnected systems with a descriptor system approach. Handling parallel redundant subsystems that have unknown splitting ratios is an essential part of this development. We have introduced the parallel masking technique in previous work whereby the APRC and the dissipation inequalities are applied to different hierarchies of subsystem interconnections. In this work, the dissipativity criterion for parallel splitting systems is derived from a transformed descriptor system with the employment of a specially structured storage function. The pre-heating an desilication unit operation in an alumina refinery is simulated to illustrate the theoretical results.

Experimental Study: Unstable SISO Systems

In this chapter, a new nonlinear PID controller synthesis approach using describing function inversion technique for unstable systems that a mathematical model may not be available is outlined; the approach is applied to a highly nonlinear inverted pendulum experimental setup. The procedure involves stabilization of the unstable system followed by generation of the describing function models of the stabilized closed-loop system. Then, the corresponding unstable open-loop frequency domain models at various operating regimes are extracted. A controller at nominal conditions is designed, followed by obtaining the corresponding desired open-loop frequency domain model. A set of controllers that force the open-loop behavior of the system mimic that of desired at various operating regimes is designed by optimization. Finally, the controller gains are inverted using a describing function inversion technique followed by experimental verification. The nonlinear PID design is compared with two other alternative designs. The experimental results indicate that the proposed approach is a viable and effective nonlinear controller synthesis technique for use with unstable nonlinear systems.

Single-Range Controller Design

In many cases, it is observed that open-loop frequency domain models of a nonlinear plant at various operating regimes of interest are almost identical. In such cases, a single-range controller most likely suffices to achieve the objectives of control. An algorithm for design of single-range linear controllers for both the single-variable and multivariable nonlinear systems is presented. Example problems are presented to demonstrate the procedure. For additional example problems, see Appendix C.