—i _ i 1* __ __To make the design less sensitive to the perturbation A , fictitious noise will be added to the system at the uncertainty in the design process. Actually, fictitious noise is considered only in obtaining the optimal estimator gain, and is not present in the plant. The configuration shown in Figure l8 recalls the general A-P-K framework presented in Chapter ll and illustrates this technique (see Figure I9) Flctltloualiiolu0W YFigure 19. General A-P-K Framework with Fictitious Noise Added at the Uncertainty.in this case the assumption of fictitious white noise, in addition to any white noisethat may actually be present, enhances the robustness as the intensity of the fictitiousnoise tends to infinity asymptotically.43" “_i"‘ I — r

As the plant copes with the fictitious noise, the LQG controller becomes morerobust to gain and phase plant input, but the optimality for the original nominal stochasticmodel is no longer guaranteed. In the case of non-minimum phase plants,o -> ac will notlead to loop recovery. Thus, to recover open loop transfer functions with a seriescompensator. a plant must be minimum phase.The following example includes a servo in parallel with the vibrational modedesigned with LQG techniques. The sequence illustrates how the good robustnessproperties of an LQR controller (Gain Margin = I, Phase Margin 2 60°, 6 dB marginagainst gain reductions) can be recovered by injecting fictitious noise at the control input.Saberi, Ali, Chen, Ben M., and Peddapullaiah, Sannuti, Loop Transfer recoveryAnalysis and Design, Springer-Verlag, I993.

in the case of the Linear Quadratic Gaussian Controller Design for the flexiblemissile the uncertainties were not taken into account. As a result, Nominal Performancewas achieved but Robust Performance and Robust stability were not. In other words, theLQG Design becomes sensitive to parameter variation or uncertainties The accelerationtime response and the rsults of the u—analysis test are presented in Figure 24The performance of the LQG design in the presence of fictitious noise injected atone or two uncertainties was evaluated. Loop Transfer Recovery through injection offictitious noise at M, in the presence of an uncertainty at Z“ is applied to enhance the

robustness of the design. The effect of this technique on the time response and resulting|.t-analysis tests are illustrated in Figures 25 through 28.It can be seen in Figure 2S(a) that onoe an uncertainty at Z, is considered, the LQGcontroller is unable to track the desired reference input. In addition, robust performance,robust stability, and nominal performance requirements are not met (Figure 2$(b)).

As shown in Figure 26, time response is improved through the addition of fictitiousnoise at M,, and u-analysis results indicate a corresponding improvement in robuststability. Nominal performance and robust performance, however, are not met.As the variance of the fictitious noise added at M,, increases, the acceleration timeresponse of the system tends to track the desired reference input, and robust stability isachieved, as shown in Figures 27 and 28. Once again, neither robust performance nornominal performance are met.So far, robust stability is achieved with enough intensity of the fictitious noise butas a consequence of loosing nominal performance. it must be recognized that there existsa tradeoff between performance loss for the nominal model and robustness gain. Thesimultaneous achievement of robust performance, robust stability, and nominalperformance will require application of more sophisticated techniques, such as linearquadratic gaussian loop transfer recovery with formal synthesis and l-L optimal control,and is beyond the scope of the present work.