Topic #20 16.31 Feedback Control Robustness Analysis · Model Uncertainty Robust Stability(rs)tests ● RS visua| izations Copyright 2001 by Jonathan How
Topic #20 16.31 Feedback Control Robustness Analysis • Model Uncertainty • Robust Stability (RS) tests • RS visualizations Copyright 2001 by Jonathan How. 1
Fal!2001 16.3120-1 Model Uncertain Prior analysis assumed a perfect model. What if the model is in correct= actual system dynamics GA(s)are in one of the sets Multiplicative model G,(s=GN(s(1+E(s)) Additive model Gp(S)=GN(S)+E(s) where 1. GN(s)is the nominal dynamics(known 2. E(s) is the modeling error- not known directly, but bound Eo(s) known(assumed stable) where E(ju)≤|Eo(ju) If Eo(jw) small, our confidence in the model is high = nominal model is a good representation of the actual dynamics If Eo(jw)large, our confidence in the model is low = nominal mode is not a good representation of the actual dynamics Figure 1: Typical system TF with multiplicative uncertainty
Fall 2001 16.31 20—1 Model Uncertainty • Prior analysis assumed a perfect model. What if the model is incorrect ⇒ actual system dynamics GA(s) are in one of the sets — Multiplicative model Gp(s) = GN(s)(1 + E(s)) — Additive model Gp(s) = GN(s) + E(s) where 1. GN(s) is the nominal dynamics (known) 2. E(s) is the modeling error — not known directly, but bound E0(s) known (assumed stable) where |E(jω)| ≤ |E0(jω)| ∀ω • If E0(jω) small, our confidence in the model is high ⇒ nominal model is a good representation of the actual dynamics • If E0(jω) large, our confidence in the model is low ⇒ nominal model is not a good representation of the actual dynamics G 100 N 10−1 10−2 10−3 10−4 10−5 10−6 10−1 100 101 102 multiplicative uncertainty Freq (rad/sec) |G| Figure 1: Typical system TF with multiplicative uncertainty
Fal!2001 16.3120-2 Simple example: Assume we know that the actual dynamics are GA(s +2Cwns+ but we take the nominal model to be gn= 1 Can explicitly calculate the error E(s), and it is shown in the plot Can also calculate an LTI overbound Eo(s) of the error. Since E(s) is not normally known, it is the bound eo(s that is used in our Lysis tests 10 10 E=G,G.-1 10 G N 10 10 10 10 Freq(rad/sec) Figure 2: Various TF's for the example system
Fall 2001 16.31 20—2 • Simple example: Assume we know that the actual dynamics are ω2 n GA(s) = s2(s2 + 2ζωns + ω2 n) but we take the nominal model to be GN = 1/s2. • Can explicitly calculate the error E(s), and it is shown in the plot. • Can also calculate an LTI overbound E0(s) of the error. Since E(s) is not normally known, it is the bound E0(s) that is used in our analysis tests. 103 102 101 100 |G| GN E=GA/GN−1 E0 GA GA GN E E0 10−1 10−2 10−3 10−4 10−1 100 101 Freq (rad/sec) Figure 2: Various TF’s for the example system
Fal!2001 16.3120-3 10 Possible G's given Ep 10 10 10 Freq(rad/sec) Figure 3: GN with one partial bound Can add many others to develop the overall bound that would completely include ga Usually EoGjw)not known, so we would have to develop it from our approximate knowledge of the system dynamics Want to demonstrate that the system is stable for any possible perturbed dynamics in the set Gp(s)= Robust Stability
Fall 2001 16.31 20—3 104 102 100 10−2 10−4 10−6 GN Possible G’s given E0 GA GN 10−1 100 101 Freq (rad/sec) Figure 3: GN with one partial bound. Can add many others to develop the overall bound that would completely include GA. • Usually E0(jω) not known, so we would have to develop it from our approximate knowledge of the system dynamics. • Want to demonstrate that the system is stable for any possible perturbed dynamics in the set Gp(s) ⇒ Robust Stability |G|
Fal!2001 Unstructured Uncertainty Mode/ 0.3120-4 Standard error model lumps all errors in the system into the actu- ator dynamics Could just as easily use the sensor dynamics, and for MImo systems, we typically use both 1(s)=GN(s)(1+E(s) E(s) is any stable TF that satisfies the magnitude bound E(ju)≤|Eo(ju) E G Called an unstructured modeling error and or uncertainty With a controller Gc(s), we have that GPGc= gnGc(l+e)= Lp= ln(1+e) Which is a set of possible perturbed loop transfer functions Can use Eo( ja) to accentuate the model uncertainty in certain frequency ranges(percentage error
Fall 2001 16.31 20—4 Unstructured Uncertainty Model • Standard error model lumps all errors in the system into the actuator dynamics. — Could just as easily use the sensor dynamics, and for MIMO systems, we typically use both. Gp(s) = GN(s)(1 + E(s)) — E(s) is any stable TF that satisfies the magnitude bound |E(jω)| ≤ |E0(jω)| ∀ω u E G - - - ? y • Called an unstructured modeling error and/or uncertainty. — With a controller Gc(s), we have that GpGc = GNGc(1 + E) ⇒ Lp = LN(1 + E) — Which is a set of possible perturbed loop transfer functions. • Can use |E0(jω)| to accentuate the model uncertainty in certain frequency ranges (percentage error)