10 - RobustStability10 filled

10 - RobustStability10 filled - ME 575 Handouts Robust...

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Unformatted text preview: ME 575 Handouts Robust Stability Nyquist Diagram Nominal: 2.5 Lo ( s ) = 2 1.5 Actual: 1 L ( s) = 0.5 Imaginary Axis 3 (s + 1)3 0 3 e−0.5s 3 (s + 1) -0.5 -1 -1.5 -2 -2.5 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 Real Axis ME575 Session 10 – Robust Stability School of Mechanical Engineering Purdue University Robust Stability Robust Stability Theorem Robust Consider a plant with a nominal TF of Go(s) and a true TF of G(s). Assume that Consider (s) G(s). a controller C(s) has been designed to achieve nominal internal stability (i.e., C(s) no unstable pole/zero cancellation and Lo(s)= Go(s)C(s) is stable). Also (s)= (s)C(s) assume that Lo(s)= Go(s)C(s) and L(s)= G(s)C(s) have the same number of (s)= (s)C(s) L(s)= G(s)C(s) unstable poles. Then, a sufficient condition for stability of the actual feedback the loop obtained by applying the controller to the true plant is that that To ( jω) GΔ ( jω) = where L o ( jω ) 1 + L o ( jω) GΔ ( j ω ) < 1 GΔ ( jω) is the frequency response of the multiplicative modeling error. ME575 Session 10 – Robust Stability School of Mechanical Engineering Purdue University 1 ME 575 Handouts Robust Stability Example • Problem Formulation Nominal System 3 Lo ( s ) = (s + 1)3 Actual System 3 L ( s) = e− τs 3 (s + 1) MME GΔ ( s ) = e−τs − 1 • Find exact value of time delay that leads to instability. ME575 Session 10 – Robust Stability School of Mechanical Engineering Purdue University Bode Plots of Nominal System Magnitude (dB) Bode Diagram Gm = 8.5206 dB (at 1.7322 rad/sec), Pm = 41.691 deg (at 1.0393 rad/sec) 20 0 -20 -40 -60 0 Phase (deg) -45 -90 -135 -180 -225 -270 -1 10 10 0 10 1 Frequency (rad/sec) ME575 Session 10 – Robust Stability School of Mechanical Engineering Purdue University 2 ME 575 Handouts Robust Stability Example (cont.) • Estimate the critical time delay using Robust Stability. ME575 Session 10 – Robust Stability School of Mechanical Engineering Purdue University Plots of |To(jω)||GΔ(jω)| Robust Stability 1.4 τ = 0.7 τ = 0.56 1.2 τ = 0.5 1 Magnitude 0.8 0.6 0.4 0.2 0 10 -1 10 0 10 1 Frequency (rad/sec) ME575 Session 10 – Robust Stability School of Mechanical Engineering Purdue University 3 ME 575 Handouts Robust Stability Example • Comment on any differences in the two values for time delay. ME575 Session 10 – Robust Stability School of Mechanical Engineering Purdue University Actual and Nominal Sensitivity • Actual Achieved Sensitivity Functions S(s) = So (s)SΔ (s) T(s) = To (s) (1 + GΔ (s)) SΔ (s) Si (s) = Sio (s) (1 + GΔ (s)) SΔ (s) where Su (s) = Suo (s)SΔ (s) SΔ (s) = 1 , Error Sensitivity 1 + To (s)GΔ (s) GΔ (s) = G(s) − Go (s) , Go (s) ME575 Session 10 – Robust Stability MME School of Mechanical Engineering Purdue University 4 ME 575 Handouts Performance Robustness • To ensure that achieved performance is close to nominal performance, we need S Δ ( jω) ≈ 1 ME575 Session 10 – Robust Stability School of Mechanical Engineering Purdue University 5 ...
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