Topic #20 16.31 Feedback Control Robustness Analysis ? Model Uncertainty ? Robust Stability (RS) tests ? RS visualizations Copyright 2001 by Jonathan How. 1 Fall 2001 16.31 20—1 Model Uncertainty ? Prior analysis assumed a perfect model. What if the model is in- correct ? actual system dynamics G A (s) are in one of the sets — Multiplicative model G p (s)= G N (s)(1 + E(s)) — Additive model G p (s)= G N (s)+ E(s) where 1. G N (s) is the nominal dynamics (known) 2. E(s)isthe modeling error — not known directly, but bound E 0 (s) known (assumed stable) where |E(jω)| ≤ |E 0 (jω)| ?ω ? If E 0 (jω) small, our confidence in the model is high ? nominal model is a good representation of the actual dynamics ? If E 0 (jω)large, our confidence in the model is low ? nominal model is not a good representation of the actual dynamics G N 10 0 10 ?1 10 ?2 10 ?3 10 ?4 10 ?5 10 ?6 10 ?1 10 0 10 1 10 2 multiplicative uncertainty Freq (rad/sec) |G| Figure 1: Typical system TF with multiplicative uncertainty Fall 2001 16.31 20—2 ? Simple example: Assume we know that the actual dynamics are ω 2 n G A (s)= s 2 (s 2 +2ζω n s + ω 2 n ) but we take the nominal model to be G N =1/s 2 . ? Can explicitly calculate the error E(s), and it is shown in the plot. ? Can also calculate an LTI overbound E 0 (s) of the error. Since E(s) is not normally known, it is the bound E 0 (s)thatisusedinour analysis tests. 10 3 10 2 10 1 10 0 |G| G N E=G A /G N ?1 E 0 G A G A G N E E 0 10 ?1 10 ?2 10 ?3 10 ?4 10 ?1 10 0 10 1 Freq (rad/sec) Figure 2: Various TF’s for the example system Fall 2001 16.31 20—3 10 4 10 2 10 0 10 ?2 10 ?4 10 ?6 G N Possible G’s given E 0 G A G N 10 ?1 10 0 10 1 Freq (rad/sec) Figure 3: G N with one partial bound. Can add many others to develop the overall bound that would completely include G A . ? Usually E 0 (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 G p (s) ? Robust Stability |G| Fall 2001 16.31 20—4 Unstructured Uncertainty Model ? 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. G p (s)= G N (s)(1 + E(s)) — E(s) is any stable TF that satisfies the magnitude bound |E(jω)| ≤ |E 0 (jω)| ?ω u E G - - - ? y ? Called an unstructured modeling error and/or uncertainty. — With a controller G c (s), we have that G p G c = G N G c (1 + E) ? L p = L N (1 + E) — Which is a set of possible perturbed loop transfer functions. ? Can use |E 0 (jω)| to accentuate the model uncertainty in certain frequency ranges (percentage error) Fall 2001 16.31 20—5 ? Typically use τ s + r 0 E 0 (s)= (τ /r ∞ )s +1 where — r 0 relative weight at low freq (? 1) — r ∞ relative weight at high freq (≥ 2) — 1/τ approx freq at which relative uncertainty is 100%. 10 1 10 ?2 10 ?2 10 ?1 10 0 10 1 10 2 r ∞ 1/τ r 0 10 0 |E 0 | 10 ?1 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—6 ? Note that L p = L N (1 + E) ? L p ? L N = L N E ? So we have that |L p (jω) ? L N (jω)| = |L N (jω) E(jω)| ≤ |L N (jω) E 0 (jω)| ? At each frequency point, we must test if |L p (jω) ? L N (jω)| < α is which is equivalent to saying that the actual LTF is anywhere within a circle (radius α) centered at point L N (jω). ? Example: Consider a simple system with ?8s +64 0.18s +0.09 G(s)= s 2 +12s +20 with E 0 (s)= 0.5s +1 Weight E 0 (s) 10 0 10 ?1 10 ?2 .09*[1/.5 1]/[1/2 1] 10 ?2 10 ?1 10 0 10 1 10 2 10 3 Freq Magnitude Figure 5: Uncertainty weighting. 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 L N L N (1+E 0 ) L N (1?E 0 ) L N (1?jE 0 ) L N (1+jE 0 ) Real Figure 7: Consider 4 possible multiplicative perturbations. L p (s)= L N (s)(1 + E(s)) And can have E(s)= E 0 (s) E(s)= ?E 0 (s) E(s)= jE 0 (s) E(s)= ?jE 0 (s) Imag Imag 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+ L N (jω)| , |S N (jω)| — Magnitude of nominal sensitivity transfer function S(s). ? Basedonthisresult, the testfor robust stability is whether: ˉ ˉ ˉ L N (jω) ˉ 1 ˉ ˉ |T N (jω)| = ˉ ˉ < ?ω 1+ L N (jω) |E 0 (jω)| — Magnitude bound on the nominal complementary sensitiv- ity transfer function T (s). — Recall that S(s)+ T (s) , 1 ? Proof: With d(jω)=1+ L N (jω), criterion of interest for robust stability is whether the possible changes to the LTF |L p (jω) ? L N (jω)| exceed the distance from the LTF to the critical point |d(jω)| = |1+ L N (jω)| — 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—9 ? By geometry, we need to test if: |L p (jω) ? L N (jω)| < |d(jω)| = |1+ L N (jω)| ?ω ? But L p = L N (1 + E) ? L p ? L N = L N E ? So we must test whether |L N E| < |1+ L N | ?ω or ˉ ˉ ˉ L N ˉ ˉ ˉ E ˉ < 1 ˉ 1+ L N ? Recall that T (s) , L(s)/(1 + L(s)) |T N (jω) E(jω)| = |T N (jω)|·|E(jω)| ≤ |T N (jω)|·|E 0 (jω)| ? So the test for robust stability is to determine whether |T N (jω)|·|E 0 (jω)| < 1 ?ω Fall 2001 16.31 20—10 Visualization of Robustness Tests ? Stability robustness test with multiplicative uncertainty given by: |T N (jω)| < 1 |E 0 (jω)| ?ω ? Consider typical case of a system with poorly known high frequency dynamics, so that — |E 0 (jω)|? 1 ? ω < ω l — |E 0 (jω)|à 1 ? ω > ω h 10 2 10 1 10 0 Good Perf |T N | 1/|E 0 | Robustness Boundary |.| 10 ?1 10 ?2 10 ?3 10 ?2 10 ?1 10 0 10 1 10 2 Freq (rad/sec) Figure 8: Visualization of the robustness test. ? Bottom line: With high frequency uncertainty in the system dy- namics, we must limit the bandwidth of the nominal system control if we want to achieve robust stability. Fall 2001 16.31 20—11 Summary ? Robust Stability Analysis — Use G N (s)todesign G c (s) — Develop bound for uncertainty model E 0 (s)(stable,min phase) — Check that |T N (jω)| < 1/|E 0 (jω)| ?ω ? State space tools for testing this condition are imperative. Can use the bounded gain theorem to determine if max |T N (jω)E 0 (jω)| < 1 ω ? Robust Stability Synthesis — Explicitly design the controller G c (s) to ensure that |T N (jω)| < 1/|E 0 (jω)| ?ω — Harder, but can do this using H ∞ techniques. ? Primary di?erence between additive and multiplicative uncertainties is at high frequency. Additive approach still allows large errors, but the multiplicative errors are washed out by the roll-o? in G. ? Potential problem with this approach is that the test only considers the magnitude of theerror. All phases areallowed, sinceweonly restrict |E| < |E 0 |. — Actual error could be very large, but with a phase that takes it away from the critical point. — Tests exist to add the phase information, but these are harder to compute.