Linear Quadratic Gaussian (LQG)

Topic #19 16.31 Feedback Control Systems • Stengel Chapter 6 • Question: how well do the large gain and phase margins discussed for LQR map over to DOFB using LQR and LQE (called LQG)? Fall 2010 16.30/31 19–2 Linear Quadratic Gaussian (LQG) • When we use the combination of an optimal estimator (not discussed in this course) and an optimal regulator to design the controller, the compensator is called Linear Quadratic Gaussian (LQG) • Special case of the controllers that can be designed using the sep­ aration principle. • 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 mechanics - the entire process is fast and automated. • Designer can focus on the “performance” related issues, being con­ fident that the LQG design will produce a controller that stabilizes the system. � Selecting values of Rzz , Ruu and relative sizes of Rww & Rvv • This sounds great – so what is the catch?? • 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., the compensator might work very well if the plant gain α = 1, but be unstable if α = 0.9 or α = 1.1. • LQG is also prone to plant–pole/compensator–zero cancelation, which tends to be sensitive to modeling errors. • J. Doyle, ”Guaranteed Margins for LQG Regulators”, IEEE Transac­ tions on Automatic Control, Vol. 23, No. 4, pp. 756-757, 1978. November 5, 2010 Fall 2010 16.30/31 19–3 • 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, different definitions of what makes a controller good, but one important criterion is whether there is a reasonable chance that it would work on the real system as well as it does in Matlab. ⇒ Robustness. • The controller must be able to tolerate some modeling error, be­ cause our models in Matlab are typically inaccurate. � Linearized model � Some parameters poorly known � Ignores some higher frequency dynamics • Need to develop tools that will give us some insight on how well a controller can tolerate modeling errors. November 5, 2010 Fall 2010 16.30/31 19–4 LQG Robustness Example • Cart with an inverted pendulum on top. m, l θ L F M x Image by MIT OpenCourseWare. • Force actuator and angle sensor • Can develop the nonlinear equations for large angle motions • Linearize for small θ (I + ml2)θ¨ − mglθ = mLẍ (M + m)ẍ + bẋ − mLθ¨ = F � �� � � � 2 2 2 (I + ml )s − mgL −mLs θ(s) 0 = 2 2 −mLs (M + m)s + bs x(s) F which gives θ(s) mLs2 = F [(I + ml2)s2 − mgL][(M + m)s2 + bs] − (mLs2)2 x(s) (I + ml2)s2 − mgL = F [(I + ml2)s2 − mgL][(M + m)s2 + bs] − (mLs2)2 • Set M = 0.5, m = 0.2, b = 0.1, I = 0.006, L = 0.3 to get: −1.818s2 + 44.55 x = 4 F s + 0.1818s3 − 31.18s2 − 4.45s which has poles at s = ±5.6, s = 0, and s = −0.14 and plant zeros at ±5. November 5, 2010 Fall 2010 16.30/31 19–5 • Define � q= θ x � � , x= q q̇ � Then with y = x ẋ = Ax + Buu y = Cy x • Very simple LQG design - main result is fairly independent of the choice of the weighting matrices. • The resulting compensator is unstable (+23!!) • This is somewhat expected. (why?) 5 Mag 10 0 10 G Gc −5 10 −2 10 −1 10 0 10 Freq (rad/sec) 1 2 10 10 Phase (deg) 200 150 G Gc 100 50 0 −2 10 −1 10 0 10 Freq (rad/sec) Fig. 1: Plant and Controller November 5, 2010 1 10 2 10 Fall 2010 16.30/31 19–6 2 10 Mag Loop L 0 10 −2 10 −2 10 −1 10 0 10 Freq (rad/sec) 1 10 2 10 Phase (deg) 0 −100 −200 −300 −2 10 −1 10 0 10 Freq (rad/sec) 1 10 2 10 Bode Diagram Gm = −0.0712 dB (at 5.28 rad/sec) , Pm = −0.355 deg (at 3.58 rad/sec) 30 Magnitude (dB) 20 10 0 −10 −20 Phase (deg) −30 180 90 0 −90 −180 −2 10 −1 10 0 10 Frequency (rad/sec) Fig. 2: Loop and Margins November 5, 2010 1 10 2 10 Fall 2010 16.30/31 19–7 Root Locus 10 10 0.64 0.76 8 0.5 0.34 0.16 8 6 6 0.86 Imaginary Axis 4 4 0.94 2 2 0.985 0 −2 0.985 −4 2 0.94 4 −6 0.86 −8 0.76 0.64 −10 −10 −8 6 8 0.5 −6 0.34 −4 0.16 −2 100 Real Axis 2 4 6 8 10 Root Locus 2 0.64 0.76 1.5 0.86 1 Imaginary Axis 0.94 0.5 0.5 0.34 2 0.16 1.75 1.5 1.25 1 0.75 0.5 0.985 0.25 0.985 0.25 0 −0.5 0.94 −1 0.86 −1.5 0.76 0.64 −2 −2 −1.5 0.5 0.75 1 1.25 1.5 1.75 0.5 0.34 0.16 2 −1 −0.5 0 Real Axis 0.5 1 1.5 2 Fig. 3: Root Locus with frozen compensator dynamics. Shows sensitivity to overall gain – symbols are a gain of [0.995:.0001:1.005]. November 5, 2010 Fall 2010 16.30/31 19–8 • Looking at both the Loop TF plots and the root locus, it is clear this system is stable with a gain of 1, but • Unstable for a gain of 1 ± � 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 margins your controller really has. • Question: what analysis tools should we use? November 5, 2010 Fall 2010 16.30/31 19–9 Frequency Domain Tests • Frequency domain stability tests provide further insights on the sta­ bility margins. • Recall that the Nyquist Stability Theorem provides a binary mea­ sure of stability, or not. • But already discussed that we can use “closeness” of L(s) to the critical point as a measure of “closeness” to changing the number of encirclements. • Closeness translates to high sensitivity which corresponds to LN (jω) being very close to the critical point. • Ideally you would want the sensitivity to be low. Same as saying that you want L(jω) to be far from the critical point. • 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 can destabilize the system. November 5, 2010 Fall 2010 16.30/31 19–10 Nichols: Unstable Open−loop System 1 Nichols: Unstable Open−loop System 10 1.05 1 0.99 1.01 1.04 1.03 1.02 Mag Mag 1.01 0 10 1 0.99 0.98 0.97 0.96 −1 10 −260 −240 −220 −200 −180 −160 Phase (deg) −140 −120 −100 0.95 −180.5 −180 −179.5 Phase (deg) −179 −178.5 Fig. 4: Nichols Plot (|L((jω))| vs. arg L((jω))) for the cart example which clearly shows sensitivity to overall gain and/or phase lag. Sensitivity Plot 3 10 |S| |L| 2 10 1 Mag 10 0 10 −1 10 −2 10 −2 10 −1 10 0 10 Freq (rad/sec) 1 10 2 10 Fig. 5: Sensitivity plot of the cart problem. Difficulty in this example is that the open-loop system is unstable, so L(jω) must encircle the critical point ⇒ hard for L(jω) to get too far away from the critical point. November 5, 2010 Fall 2010 16.30/31 19–11 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 different. • Basic analysis tool is the Sensitivity 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” approach. • 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. November 5, 2010