achapt 13

# achapt 13 - SENSITIVITY ANALYSES 13 SENSITIVITY ANALYSES...

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SENSITIVITY ANALYSES 13 SENSITIVITY ANALYSES The previous chapters hinted at the importance of the sensitivity of a system to its parameters. As mentioned in Section 4 of Chapter 3, in the absence of feedback, control forces do not really move a system in a highly predictable manner. The feedback acts as a continuously-acting correction of errors resulting from not knowing precisely the system’s parameters, the external disturbances, and even the control forces being applied. The question arises how one can study the sensitivity of a system to gain confidence that a control system design will stabilize a system as intended. Sensitivity analyses can be performed numerically or analytically. The numerical approach consists of repeatedly simulating the response of a system model while varying different parameters, like the system’s physical parameters, the control parameters, and the disturbances. This approach can be extremely effective. Another approach is to gain more insight into the nature of the errors. This is accomplished by analytical methods. The effect of the changes in the system’s parameters can be described by MAE 461: DYNAMICS AND CONTROLS

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SENSITIVITY ANALYSES relationships that express the changes in the closed-loop decay rates and closed-loop frequencies in terms of changes in the system’s parameters, that is, the physical parameters and the control gains. Before diving into these issues, there is the more general question of the mechanisms that produce instabilities. This section begins with a description of the most important principles that govern linear stability. These principles provide general conditions that guarantee stability of linear systems. Next, we show how a perturbation analysis can be performed to determine formulas for the changes in the closed-loop decay rates and the frequencies of oscillation in terms of changes in the system’s physical parameters. Then, we show how an approach called the root-locus method can be performed to determine the changes in the closed- loop decay rates and the frequencies of oscillation in terms of changes in control gains. 1. Principles of Linear Stability The equations of a dynamical system, as we know, are in general nonlinear. They can be written in the general form (13 – 1) F x x f x M + = ) , ( & & & where M is a symmetric mass matrix ( M = M T ), f is a nonlinear function of x and , and where F is a forcing function. For the purposes of determining the equilibrium position, we let F = 0 . At static equilibrium, x & MAE 461: DYNAMICS AND CONTROLS
SENSITIVITY ANALYSES (13 – 2) ) , ( 0 0 x f 0 = To linearize Eq. (13 – 1), we perform a Taylor series approximation of f about x 0 . Thus, (13 – 3) ) ( ) ( ) , ( ) , ( 0 0 0 x x f x x x f 0 x f x x f + + = & & & T T Substituting Eq. (13 – 3) into (13 – 1) yields the linearized equations (13 – 4) F Ax x B x M = + + & & & where T T = = x f B x f A & For purposes that will become apparent momentarily, let’s show that any matrix A can be

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## This note was uploaded on 09/17/2009 for the course MAE 469 taught by Professor Silverberg during the Fall '08 term at N.C. State.

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achapt 13 - SENSITIVITY ANALYSES 13 SENSITIVITY ANALYSES...

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