NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...
NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ... NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...
Figure 1.1: TORA System One of the challenges of evaluating the non-linear control design techniques that are being developed is to nd a reasonable method of gauging their performance. There exists a large variation in the complexity, behavior, and dynamics within the set of non-linear systems, and a given control design might work very well on one speci c type of system while providing poor performance on another. Thus the need for a benchmark non-linear problem { a mathematical system that contains signi cant non-linearities, has a tractable mathematical model, and is fairly easy to implement in hardware. In recent years, the translational oscillator rotational actuator (TORA) [1] system (also referred to as the Non-linear Benchmark Problem or NLBP) has been proposed as just such a benchmark system. In this system a mass is restricted to a linear oscillatory motion by ideal springs, and the system is actuated by arotational proof mass. The coupling between the rotational and linear motion provides the non- linearity. Many di erent researchers have used this system as a testbed to evaluate various non-linear control strategies, so this system was selected as a way to eval- uate and compare the performance of the successive Galerkin technique with other robust, non-linear control laws as well as with standard optimal linearized methods. Speci cally, six unique control laws are derived and implemented on a hardware im- plementation of the TORA system. A non-linear Galerkin approximation to the H 2 2
problem and a Galerkin approximation to the non-linear H1 problem are studied and compared with the results of a linearized optimal control, a linearized H1 control, a passivity-based control, and a control law based on integrator backstepping. These control strategies are chosen because they are current topics of research, and most are proposed as control laws that will be provide robust performance. The speci c hardware system, upon which the tests were performed, is de- scribed in Chapter 2 along with a derivation of the equations of motion. Chapter 3 presents an overview of the control strategies tested, and it explains the details of the implementations of each algorithm. Chapter 3 also explains the successive Galerkin approximation technique and shows how it is implemented on the hardware system. Chapter 4 explains the mechanics of the testbed and the methods of comparing per- formance, and a comparison of the six controls in simulation is presented. The actual results of the tests as performed in hardware are compared and discussed in Chap- ter 5, and an analysis of the robustness of the control laws is presented in Chapter 6. Finally Chapter 7 sums up the results of the experiment the conclusions of the research are presented along with recommendations for future research. 1.2 Contributions There is often a wide gap between mathematical systems and the physical plants these systems attempt to model. The TORA system is proposed in [2] as a benchmark problem so that nonlinear control algorithms can be compared on a common system { a system that can be easily implemented in hardware. While many control designs perform well in the idealized environment of mathematics, the true test of a nonlinear algorithm must lie in its regulation of a physical plant. The SGA technique is applied to the nonlinear benchmark problem, and it performs better than other well known nonlinear techniques when applied to a physical implementation of this TORA system. This result is the main contribution of this thesis: it o ers experimental evidence that the SGA method produces excellent results when applied to real-life problems and systems. This thesis also justi es further research into understanding, clarifying, and improving the SGA technique. 3
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Figure 1.1: TORA System<br />
One of the challenges of evaluating the non-linear control design techniques<br />
that are being developed is to nd a reasonable method of gauging their performance.<br />
There exists a large variation in the complexity, behavior, and dynamics within the<br />
set of non-linear systems, and a given control design might work very well on one<br />
speci c type of system while providing poor performance on another. Thus the need<br />
for a benchmark non-linear problem { a mathematical system that contains signi cant<br />
non-linearities, has a tractable mathematical model, and is fairly easy to implement<br />
in hardware.<br />
In recent years, the translational oscillator rotational actuator (TORA) [1]<br />
system (also referred to as the Non-linear Benchmark Problem or NLBP) has been<br />
proposed as just such a benchmark system. In this system a mass is restricted to a<br />
linear oscillatory motion by ideal springs, and the system is actuated by arotational<br />
proof mass. The coupling between the rotational and linear motion provides the non-<br />
linearity. Many di erent researchers have used this system as a testbed to evaluate<br />
various non-linear control strategies, so this system was selected as a way to eval-<br />
uate and compare the performance of the successive Galerkin technique with other<br />
robust, non-linear control laws as well as with standard optimal linearized methods.<br />
Speci cally, six unique control laws are derived and implemented on a hardware im-<br />
plementation of the TORA system. A non-linear Galerkin approximation to the H 2<br />
2
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