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MIT2_017JF09_ch12

MIT2_017JF09_ch12 - 12 CONTROL SYSTEMS LOOPSHAPING 12 12.1...

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12 CONTROL SYSTEMS – LOOPSHAPING 94 12 CONTROL SYSTEMS – LOOPSHAPING 12.1 Introduction This section formalizes the notion of loopshaping for linear control system design. The loopshaping approach is inherently two-fold. First, we shape the open-loop transfer function (or matrix) P ( s ) C ( s ), to meet performance and robustness specifications. Once this is done, then the compensator must be computed, from from knowing the nominal product P ( s ) C ( s ), and the nominal plant P ( s ). Most of the analysis here is given for single-input, single-output systems, but the link to multivariable control is not too difficult. In particular, absolute values of transfer functions are replaced with the maximum singular values of transfer matrices. Design based on singular values is the idea of L 2 -control, or linear quadratic Gaussian (LQG) control and the loop transfer recovery (LTR). 12.2 Roots of Stability – Nyquist Criterion We consider the SISO feedback system with reference trajectory r ( s ) and plant output y ( s ), as given previously. The tracking error signal is defined as e ( s ) = r ( s ) - y ( s ), thus forming the negative feedback loop. The sensitivity function is written as e ( s ) S ( s ) = 1 = r ( s ) , 1 + P ( s ) C ( s ) where P ( s ) represents the plant transfer function, and C ( s ) the compensator. The closed- loop characteristic equation , whose roots are the poles of the closed-loop system, is 1 + P ( s ) C ( s ) = 0, equivalent to P ( s ) C ( s ) + P ( s ) C ( s ) = 0, where the underline and overline denote the denominator and numerator, respectively. The Nyquist criterion allows us to assess the stability properties of a feedback system based on P ( s ) C ( s ) only . This method for design involves plotting the complex loci of P ( s ) C ( s ) for the range s = , ω = [ -∞ , ]. Remarkably, there is no explicit calculation of the closed-loop poles, and in this sense the design approach is quite different from the root-locus method (see Ogata, also the rlocus() command in MATLAB). 12.2.1 Mapping Theorem To give some understanding of the Nyquist plot, we begin by imposing a reasonable assump- tion from the outset: The number of poles in P ( s ) C ( s ) exceeds the number of zeros. It is a reasonable constraint because otherwise the loop transfer function could pass signals with infinitely high frequency. In the case of a PID controller (two zeros) and a second-order zero-less plant, this constraint can be easily met by adding a high-frequency rolloff to the compensator, the equivalent of low-pass filtering the error signal.
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12 CONTROL SYSTEMS LOOPSHAPING 95 Now let F ( s ) = 1 + P ( s ) C ( s ) (the denominator of S ( s )). The heart of the Nyquist analysis is the mapping theorem, which answers the following question: How do paths in the complex s -plane map into paths in the complex F -plane? We limit ourselves to closed, clockwise (CW) paths in the s -plane, and the powerful result of the mapping theorem is Every zero of F ( s ) that is enclosed by a path in the s -plane generates exactly one CW encirclement of the origin in the F ( s ) -plane. Conversely, every pole of F ( s ) that is enclosed by a path in the s -plane generates exactly one CCW encirclement of the
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