10 - RobustStability10 filled

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online