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dft, Feedback Control Theory, Doyle - Coggle Diagram
dft, Feedback Control Theory, Doyle
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Chapter 6, Design Constraints
Algebraic Constraints
The identity S + T = 1 always holds. This is an immediate consequence of the definitions of S and T. So in particular, |S(jω)| and |T(jω)| cannot both be less than 1/2 at the same frequency ω.
Robust performance ||W1S| + |W2T||∞ < 1 implies min{|W1(jω)|, |W2(jω)|} < 1, ∀ω
If p is a pole of L in Res ≥ 0 and z is a zero of L in the same half-plane, then S(p) = 0, S(z) = 1, T(p) = 1, T(z) = 0.
Analytic Constraints
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The Waterbed Effect
Theorem 1 Suppose that P has a zero at z with Rez > 0. Then there exist positive constants c1 and c2, depending only on ω1, ω2, and z, such that c1 logM1 + c2 logM2 ≥ log |Sap(z)^{−1}| ≥ 0.
The waterbed effect is amplified if the plant has a pole and a zero close together in the right half-plane.
The waterbed effect applies to non-minimum-phase plants only. In fact, a very good result in minimum-phase case can be proved (Section 10.1)
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Chapter 2, Norms for Signals and Systems
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G(s) is called proper if G(j∞) is finite, strictly proper if G(j∞) = 0
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Chapter 3, Basic Concepts
well-posedness
all closedloop transfer functions exist,
that is, all transfer functions from the exogenous inputs to all internal signals and the outputs of the summing junctions
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Internal Stability
If the closedloop transfer functions above are stable, then the feedback system is said to be internally stable.
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Performance
sensitivity function S := 1 / (1 + L), e = Sr
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Conclusion: Performance specs that involve e result in weights on S and performance specs on u result in weights on T.
Chapter 4, Uncertainty and Robustness
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Robust Stability
A controller C provides robust stability if it provides internal stability for every plant in P a set of uncertainty model.
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Robust Performance
Robust performance condition is a combination of robust stability condition and nominal performance condition (for all plant in P), i.e., |W2T|∞ < 1 and |W1~{S}|∞ < 1 (in multiplicative uncertainty model)
Theorem 2 A equivalent condition for robust performance is ||W1S| + |W2T||∞ < 1.
(Multiplicative uncertainty model)
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