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)
Fal!2001 16.3120-5 ically use TS /r)s+1 where ro relative weight at low freq(< 1) roo relative weight at high freq ( -1/T approx freq at which relative uncertainty is 100% 10 1/τ 10 0 10 Freq(rad/sec) Figure 4: Typical input uncertainty weighting. Low error at low fre- quency and larger error at high frequency
Fall 2001 16.31 20—5 • Typically use τs + r0 E0(s) = (τ/r∞)s + 1 where — r0 relative weight at low freq (¿ 1) — r∞ relative weight at high freq (≥ 2) — 1/τ approx freq at which relative uncertainty is 100%. 101 10−2 10−2 10−1 100 101 102 r ∞ 1/τ r 0 100 |E 0| 10−1 Freq (rad/sec) Figure 4: Typical input uncertainty weighting. Low error at low frequency and larger error at high frequency
Fal!2001 16.3120-6 Note that Lp= LN(1+E)= Lp- LN= LN E . So we have that LpGw)-LNGu)= Lng)elu)I< LNga) Eo(jw) e At each frequency point, we must test if LpG)-Lwgju)<a is which is equivalent to saying that the actual Ltf is anywhere within a circle(radius a) centered at point LNju) Example: Consider a simple system with 0.18s+0.09 G(S)=+128+20wih()=0.5s+1 Weight Eo( 三10 0911/.511/21 10 Freq Figure 5: Uncertainty weighting
Fall 2001 16.31 20—6 • Note that Lp = LN(1 + E) ⇒ Lp − LN = LN E • So we have that |Lp(jω) − LN(jω)| = |LN(jω) E(jω)| ≤ |LN(jω) E0(jω)| • At each frequency point, we must test if |Lp(jω) − LN(jω)| < α is which is equivalent to saying that the actual LTF is anywhere within a circle (radius α) centered at point LN(jω). • Example: Consider a simple system with −8s + 64 0.18s + 0.09 G(s) = s2 + 12s + 20 with E0(s) = 0.5s + 1 Weight E0 (s) 100 10−1 10−2 .09*[1/.5 1]/[1/2 1] 10−2 10−1 100 101 102 103 Freq Magnitude Figure 5: Uncertainty weighting
Fal!2001 Possible Perturbations to the LTF: Multiplicative 16.3120-7 Real Figure 6: Nominal loop tf and possible multiplicative errors Possible Perturbations to the LTF: Multiplicative (1+jE Real Figure 7: Consider 4 possible multiplicative perturbations Lp(s)=LN(S)(1+E(s)) And can have E(s)=E0(s)E(s)=-E0(s) E(s=jEo(s E(s)=-jEo(s
Possible Perturbations to the LTF: Multiplicative Fall 2001 2 16.31 20—7 1 0 −1 −2 −3 −4 −2 −1 0 1 2 3 4 Real Figure 6: Nominal loop TF and possible multiplicative errors. Possible Perturbations to the LTF: Multiplicative 2 1 0 −1 −2 −3 −4 −2 −1 0 1 2 3 4 LN LN(1+E0 ) LN(1−E0 ) LN(1−jE0 ) LN(1+jE0 ) Real Figure 7: Consider 4 possible multiplicative perturbations. Lp(s) = LN(s)(1 + E(s)) And can have E(s) = E0(s) E(s) = −E0(s) E(s) = jE0(s) E(s) = −jE0(s) Imag Imag
Fal!2001 16.3120-8 Robust stability Tests From the nyquist plot, we developed a measure of the "closeness of the loop transfer function(LTF)to the critical point dlj) 1+ LNga) N Magnitude of nominal sensitivity transfer function S(s) Based on this result, the test for robust stability is whether 1TN)}=-(u 1+LNu) EoGjw ) Magnitude bound on the nominal complementary sensitiv ity transfer function T(s) Recall that S(s)+T(s,1 Proof: With d ju)=1+ LN(ja), criterion of interest for robust stability is whether the possible changes to the ltF Lpla)-) exceed the distance from the ltf to the critical point dlj)= 1+ Lnga) Because if it does, then it is possible that the modeling error could change the number of encirclements Actual system could be unstable
Fall 2001 16.31 20—8 Robust Stability Tests • From the Nyquist Plot, we developed a measure of the “closeness” of the loop transfer function (LTF) to the critical point: 1 1 = |d(jω)| |1 + LN(jω)| , |SN(jω)| — Magnitude of nominal sensitivity transfer function S(s). • Based on this result, the test for robust stability is whether: ¯ ¯ ¯ LN(jω) ¯ 1 ¯ ¯ |TN(jω)| = ¯ ¯ < ∀ω 1 + LN(jω) |E0(jω)| — Magnitude bound on the nominal complementary sensitivity transfer function T(s). — Recall that S(s) + T(s) , 1 • Proof: With d(jω)=1 + LN(jω), criterion of interest for robust stability is whether the possible changes to the LTF |Lp(jω) − LN(jω)| exceed the distance from the LTF to the critical point |d(jω)| = |1 + LN(jω)| — Because if it does, then it is possible that the modeling error could change the number of encirclements ⇒ Actual system could be unstable
Fal!2001 16.3120-9 By geometry, we need to test if LpG)- LNg)<d(w)= 1+ LNga)l a But Lp= Ln(1+E) LN=LN E e So we must test whether LN E <1+LN Vw LN E<1 +L Recall that T(s, l(s/(1+L(s)) TN(ju)E(ju)=|TNj)·|E(ju)≤|N(ju)·|Eo(ju) So the test for robust stability is to determine whether TN(ju)·|Eoi)<1u
Fall 2001 16.31 20—9 • By geometry, we need to test if: |Lp(jω) − LN(jω)| < |d(jω)| = |1 + LN(jω)| ∀ω • But Lp = LN(1 + E) ⇒ Lp − LN = LN E • So we must test whether |LN E| < |1 + LN| ∀ω or ¯ ¯ ¯ LN ¯ ¯ E¯ ¯ < 1 ¯1 + LN • Recall that T(s) , L(s)/(1 + L(s)) |TN(jω) E(jω)| = |TN(jω)| · |E(jω)| ≤ |TN(jω)|·|E0(jω)| • So the test for robust stability is to determine whether |TN(jω)|·|E0(jω)| < 1 ∀ω
