Topic #18 16.31 Feedback Control Closed-loop system analysis Robustness State-space — eigenvalue analysis Frequency domain — Nyquist theorem. Sensitivity Copyright 2001 by Jonathan How. 1 Fall 2001 16.31 18—1 Combined Estimators and Regulators When we use the combination of an optimal estimator and an opti- mal regulator to design the controller, the compensator is called Linear Quadratic Gaussian (LQG) — Special case of the controllers that can be designed using the separation principle. The great news about an LQG design is that stability of the closed- loop system is guaranteed. — The designer is freed from having to perform any detailed me- chanics - the entire process is fast and can be automated. Now the designer just focuses on: — How to specify the state cost function (i.e. selecting z = C z x) and what value of r to use. — Determine how the process and sensor noise enter into the system and what their relative sizes are (i.e. select R w & R v ) So the designer can focus on the “performance” related issues, be- ing confident that the LQG design will produce a controller that stabilizes the system. This sounds great — so what is the catch?? Fall 2001 16.31 18—2 The remaining issue is that sometimes the controllers designed using these state-space tools are very sensitive to errors in the knowledge of the model. — i.e., Might work very well if the plant gain α = 1, but be unstable if it is α =0.9orα =1.1. — LQG is also prone to plant—pole/compensator—zero cancellation, which tends to be sensitive to modeling errors. The good news is that the state-space techniques will give you a controller very easily. — You should use the time saved to verify that the one you designed is a “good” controller. There are, of course, di?erent definitions of what makes a controller good, but one important criterion is whether there is a reason- able chance that it would work on the real system as well as it do es in Matlab. ? Ro bu s t ne s s. — Thecontrollermustbeabletotoleratesomemodelingerror, because our models in Matlab are typically inaccurate. 3 Linearized model 3 Some parameters poorly known 3 Ignores some higher frequency dynamics Need to develop tools that will give us some insight on how well a controller can tolerate modeling errors. Fall 2001 16.31 18—3 Example Consider the “cart on a stick” system, with the dynamics as given in the notes on the web. Define q = ? θ x ? , x = ? q ˙q ? Then with y = x ˙x = Ax+ Bu y = Cx For the parameters given in the notes, the system has an unstable pole at +5.6andoneats = 0. There are plant zeros at ±5. The target locations for the poles were determined using the SRL for both the regulator and estimator. — Assumes that the process noise enters through the actuators B w ≡B, which is a useful approximation. — Regulator and estimator have the same SRL. — Choose the process/sensor ratio to be r/10 so that the estimator poles are faster than the regulator ones. The resulting compensator is unstable (+16!!) — But this was expected. (why?) Fall 2001 16.31 18—4 ?8 ?6 ?4 ?2 0 2 4 6 8 ?10 ?8 ?6 ?4 ?2 0 2 4 6 8 10 Real Axis Imag Axis Symmetric root locus Figure 1: SRL for the regulator and estimator. 10 ?2 10 ?1 10 0 10 1 10 2 10 ?4 10 ?2 10 0 10 2 10 4 Freq (rad/sec) Mag Plant G Compensator Gc 10 ?2 10 ?1 10 0 10 1 10 2 0 50 100 150 200 Freq (rad/sec) Phase (deg) Plant G Compensator Gc Figure 2: Plant and Controller Fall 2001 16.31 18—5 10 ?2 10 ?1 10 0 10 1 10 2 10 ?2 10 ?1 10 0 10 1 Freq (rad/sec) Mag Loop L 10 ?2 10 ?1 10 0 10 1 10 2 ?300 ?250 ?200 ?150 ?100 Freq (rad/sec) Phase (deg) Figure 3: Loop and Margins Looking at both the Loop plots and the root locus, this system is stable with a gain of 1, but — Unstable for a gain of 1±2 and/or a slight change in the system phase (possibly due to some unmodeled delays) — Very limited chance that this would work on the real system. Of course, this is an extreme example and not all systems are like this, but you must analyze to determine what robustness mar- gins your controller really has. Question: what analysis tools should we use? Fall 2001 16.31 18—6 ?10 ?8 ?6 ?4 ?2 0 2 4 6 8 10 ?10 ?8 ?6 ?4 ?2 0 2 4 6 8 10 Real Axis Imag Axis Figure 4: Root Locus with frozen compensator dynamics. Shows sen- sitivity to overall gain — symbols are a gain of [0.995:.0001:1.005]. ?2 ?1.5 ?1 ?0.5 0 0.5 1 1.5 2 ?2 ?1.5 ?1 ?0.5 0 0.5 1 1.5 2 Real Axis Imag Axis Fall 2001 16.31 18—7 Analysis Tools to Use? Eigenvalues give a definite answer on the stability (or not) of the closed-loop system. — Problem is that it is very hard to predict where the closed-loop poles will go as a function of errors in the plant model. Considerthecasewerethemodelofthesystemis ˙x = A 0 x + Bu — Controller also based on A 0 ,sonominal closed-loop dynamics: ? A 0 ?BK LC A 0 ?BK?LC ? ? ? A 0 ?BK BK 0 A 0 ?LC ? Butwhatiftheactual system has dynamics ˙x =(A 0 +?A)x + Bu — Then perturbed closed-loop system dynamics are: ? A 0 +?A ?BK LC A 0 ?BK?LC ? ? ? A 0 +?A?BK BK ?AA 0 ?LC ? Transformed ˉ A cl not upper-block triangular, so perturbed closed- loop eigenvalues are NOT the union of regulator & estimator poles. — Can find the closed-loop poles for a specific ?A, but — Hard to predict change in location of closed-loop poles for a range of possible modeling errors. Fall 2001 16.31 18—8 Frequency Domain Tests Frequency domain stability tests provide further insights on the “stability margins”. Recall from the Nyquist Stability Theorem: — P = # poles of L(s)=G(s)G c (s)intheRHP — Z = # closed-loop poles in the RHP — N = # clockwise encirclements of the Nyquist Diagram about the critical point -1. Can show that Z = N + P (see notes on the web). So for the closed-loop system to be stable, need Z , 0 ? N = ?P If the loop transfer function L(s) has P poles in the RHP s-plane (and lim s→∞ L(s) is a constant), then for closed-loop stability, the locus of L(jω)forω ∈ (?∞,∞) must encircle the critical point (-1,0) P times in the counterclockwise direction [Ogata 528]. — This provides a binary measure of stability, or not. Fall 2001 16.31 18—9 Can use “closeness” of L(s) to the critical point as a measure of “closeness”tochangingthenumberofencirclements. — Premise is that the system is stable for the nominal system ? has the right number of encirclements. Goal of the robustness test is to see if the possible perturbations to our system model (due to modeling errors) can change the number of encirclements In this case, say that the perturbations candestabilizethe system. ?260 ?240 ?220 ?200 ?180 ?160 ?140 ?120 ?100 10 ?1 10 0 10 1 Nichols: Unstable Open?loop System Mag Phase (deg) ?180.5 ?180 ?179.5 ?179 ?178.5 0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 Nichols: Unstable Open?loop System Mag Phase (deg) 1 0.99 1.01 Figure 5: Nichols Plot for the cart example which clearly shows the sensitivity to the overall gain and/or phase lag. Fall 2001 16.31 18—10 ?1.5 ?1 ?0.5 0 0.5 1 1.5 ?1.5 ?1 ?0.5 0 0.5 1 1.5 Imag Part Real Part stable OL L N (jω) L A (jω) ω 1 ω 2 Figure 6: Plot of Loop TF L N (jω)=G N (jω)G c (jω) and perturbation (ω 1 →ω 2 ) that changes the number of encirclements. Model error in frequency range ω 1 ≤ω ≤ω 2 causes a change in the number of encirclements of the critical point (?1,0) — Nominal closed-loop system stable L N (s)=G N (s)G c (s) — Actual closed-loop system unstable L A (s)=G A (s)G c (s) Bottom line: Large model errors when L N ≈?1 are very dan- gerous. Fall 2001 16.31 18—11 Frequency Domain Test ?1.5 ?1 ?0.5 0 0.5 1 ?1 ?0.5 0 0.5 1 stable OL Real Part |L N (jω)| |d(jω)|Imag Part Figure 7: Geometric interpretation from Nyquist Plot of Loop TF. |d(jω)| measures distance of nominal Nyquist locus to critical point. By vector addition gives ?1+d(jω)=L N (jω) ? d(jω)=1+L N (jω) Actually more convenient to plot 1 |d(jω)| = 1 |1+L N (jω)| , |S(jω)| the magnitude of the sensitivity transfer function S(s). Fall 2001 16.31 18—12 So high sensitivity corresponds to L N (jω)beingvery close to the critical point. 10 ?2 10 ?1 10 0 10 1 10 2 10 ?2 10 ?1 10 0 10 1 10 2 10 3 Sensitivity Plot Freq (rad/sec) Mag |S| |L| Figure 8: Sensitivity plot of the cart problem. Ideally you would want the sensitivity to be much lower than this. — Same as saying that you want L(jω) to be far from the critical point. — Di?culty in this example is that the open-loop system is unsta- ble, so L(jω) must encircle the critical point ? hard for L(jω) to get too far away from the critical point. Fall 2001 16.31 18—13 Figure 9: Sensitivity for Example 1 G(s)= 8·14·20 (s+8)(s+14)(s+20) with a low bandwidth controller ?260 ?240 ?220 ?200 ?180 ?160 ?140 ?120 ?100 10 ?1 10 0 10 1 Nichols: Stable Open?loop System Mag Phase (deg) ?195 ?190 ?185 ?180 ?175 ?170 ?165 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Nichols: Stable Open?loop System Mag Phase (deg) 1 0.95 1.05 10 ?1 10 0 10 1 10 2 10 3 10 ?2 10 ?1 10 0 10 1 10 2 Sensitivity Plot Freq (rad/sec) Mag |S| |L| Fall 2001 16.31 18—14 Figure 10: Sensitivity for Example 3 G(s)= 8·14·20 (s?8)(s?14)(s?20) ?260 ?240 ?220 ?200 ?180 ?160 ?140 ?120 ?100 10 ?1 10 0 10 1 Nichols: Unstable Open?loop System Mag Phase (deg) ?195 ?190 ?185 ?180 ?175 ?170 ?165 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Nichols: Unstable Open?loop System Mag Phase (deg) 1 0.95 1.05 10 ?1 10 0 10 1 10 2 10 3 10 ?2 10 ?1 10 0 10 1 10 2 Sensitivity Plot Freq (rad/sec) Mag |S| |L| Fall 2001 16.31 18—15 Figure 11: Sensitivity for Example 1 G(s)= 8·14·20 (s+8)(s+14)(s+20) with a high bandwidth controller ?260 ?240 ?220 ?200 ?180 ?160 ?140 ?120 ?100 10 ?1 10 0 10 1 Nichols: Stable Open?loop System Mag Phase (deg) ?195 ?190 ?185 ?180 ?175 ?170 ?165 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Nichols: Stable Open?loop System Mag Phase (deg) 1 0.95 1.05 10 ?1 10 0 10 1 10 2 10 3 10 ?2 10 ?1 10 0 10 1 10 2 Sensitivity Plot Freq (rad/sec) Mag |S| |L| Shows that as the controller bandwidth increases, can expect L(jω) to get much closer to the critical point. “Push—pop” Fall 2001 16.31 18—16 Summary LQG gives you a great way to design a controller for the nominal system. But there are no guarantees about the stability/performance if the actual system is slightly di?erent. — BasicanalysistoolistheSensitivity Plot No obvious ways to tailor the specification of the LQG controller to improve any lack of robustness — Apart from the obvious “lower the controller bandwidth” ap- proach. — And sometimes you need the bandwidth just to stabilize the system. Very hard to include additional robustness constraints into LQG — See my Ph.D. thesis in 1992. Other tools have been developed that allow you to directly shape the sensitivity plot |S(jω)| — Called H ∞ and μ Good news: Lack of robustness is something you should look for, but it is not always an issue.