L7: Lecture 7 LQG Design Linear Quadratic Gaussian regulator Control-estimation duality SRL for optimal estimator Example of LQG design for IO plant
LQG regulator L7:2 If the process and measurement noises are assumed Gaussian, then the estimate is statistically most liely, as well as producing the minimum squared error his assumption is not necessary, but has led to the name Linear Quadratic Gaussian LQG for a regulator designed with an optimal control law u = Kx and a Kalman estimator atlab provides the function lqgreg to form the feedbac regulator: w v» Hlqg = lqgregkest, K + G + Form the closed-loop system with feedbac r = u y Kˆ x Hlqg
Selection of covariance matrices Q and R L7:3 easurement noise R noise signals from the sensors are usually uncorrelated R is diagonal expected value of rms noise σ i from each sensor may be available from manufacturer s specifications R = diagσ 2 i rocess noise Q more difficult to specify rational values: often need trial-and-error with simulation studies unliely to have information on crosscorrelation Q is diagonal if there is a random process disturbance which can be approximated as white noise with a mean-square spectral density S i, then Q i S i /
L7:4 Duality between optimal control and estimation he time-varying optimal control equations are: [ ] [ ] Q R K R Φ + = Φ Φ + = + = [ ] w w + Φ = Φ + = + = Q R C L C R C C C his is a bacward recursion starting from N = Q, KN = he time-varying optimal estimation equations are: his is a forward recursion starting from { } ~ ~ E x x =
L7:5 Duality between optimal control and estimation [ ] [ ] Q R K R Φ + = Φ Φ + = + = [ ] w w + Φ = Φ + = + = Q R C L C R C C C Control: Estimation: R C Q R Q w w Φ Φ Estimation Control Apart from the different direction, the recursion relationships are identical, with these dualities:
Symmetric root locus for SISO system estimator Consider a system with: L7:6 a single process noise input w with rms value σ w a single sensed output y corrupted with sensor noise v with rms value σ v he uncertainty of the plant model relative to the sensor data could then be represented by ρ = Q/R = σ w /σ v 2 he control estimation duality has he transfer function from w to y is Y z N z = C zi Φ w =, say W z D z he estimator poles will hence be the stable solutions of the characteristic equation 2 w Q / R Q / R N z N z D z D z + ρ = = ρ
L7:7 Symmetric root locus for SISO system estimator If the process noise and control signal are additive, so that w = u, the control and estimation optimal root loci will be identical In a pole placement design, the control and estimator poles could then be selected from the same symmetric root locus In the estimator case, increasing the gain ρ = Q/R corresponds to putting more reliance on the sensor data larger estimator gains L w v u + lant + y K xˆ Estimator
Example: SRL pole-placement design of controller and estimator for flexible structure script flex_srlc.5.5 For controller, ρ =.67 dclz =.79±j.4,.73±j.4 K = [-.59.48.45.68] For estimator, ρ = 349 depz =.43±j.57,.39±j.6 L = [2.8 3.97 6.49 4.8] Symmetric root locus estimator poles 2-3 times faster than controller poles o: disp d - -2 Initial Condition Results L7:8 A xi s g a Im -.5 Amplitude o: disp y 2-2 - o: force u - control force» prev design -.5 -.5 - -.5.5.5 Real Axis 2 4 6 8 2 4 6 ime sec.
.5.5 Example: SRL pole-placement design of controller and estimator for flexible structure script flex_srlc For controller, ρ =.67 dclz =.79±j.4,.73±j.4 K = [-.59.48.45.68] For estimator, ρ = 2.5e5 depz = -.7±j.52,.3±j.3 L = [3.7 9.6 2. 8.2] Symmetric root locus estimator poles 4-5 times faster than controller poles o: disp d - Initial Condition Results L7:9 A xi s g a I -.5 - -.5 -.5 - -.5.5.5 Real Axis Amplitude o: force u o: disp y 5-5 5 5-5 control force even bigger! 2 4 6 8 2 4 6 ime sec.
Comparison of state estimates L7: low ρ L = [2.8 3.97 6.49 4.8] high ρ L = [3.7 9.6 2. 8.2] Response of states and predictive estimates to x = 4 5 Response of states and predictive estimates to x = 4 2-2 x =d x hat x 2 =ddot x 2 hat 2 x =d x hat 5-5 - x 2 =ddot x 2 hat -4 5 ime s -5 5 ime s -2 5 ime s -5 5 ime s 4 5 5 x 3 =y x 3 hat 5 x 4 =ydot x 4 hat 2 x 3 =y x 3 hat x 4 =ydot x 4 hat -5-5 -2-5 - 5 ime s - 5 ime s -4 5 ime s - 5 ime s State estimates converge more rapidly, but initial errors larger larger initial control force
LQG design of IO regulator for flexible structure script flex_lqg w u + lant + v L7: y = [d y] K xˆ Estimator Controller design: weight displacements, not velocities Q c =diagq,, Q 22, scalar input R c =» K = dlqrhi, Gam, Qc, Rc Kalman estimator design: assume % rms sensor noise on d and y R e = diag. 2,. 2 trial-and-error values for process noise Q e
LQG design of IO regulator for flexible structure script flex_lqg L7:2 Kalman estimator design: lant model must have process noise input:» = sshi, [Gam Gam], C, [D D],» [Kest, L] = alman, Qe, Re Form regulator:» Hreg = lqgregkest, K Close loop:» Gcl = feedbacgpd, Hreg, + u + w lant + v y = [d y] K xˆ Estimator
Simulate model sim_flex_lqg L7:3 4 up Sensor noise v w rocess noise Sum Gp:2, Continuous plant model Sum 2 ys yp Hreg Discrete regulator 3 u
L7:4 Q =.67, Q22 = Qe =.5, Re = diag. 2,. 2 As per actual disturbances
L7:5 Actual noise corresponds with estimator design Sensor rms noise ten times estimator design value
Some further considerations L7:6 ole placement control design For arbitrary placement of controller poles, the plant A, B or Φ, must be controllable For arbitrary placement of estimator poles, the plant A, C or Φ, C must be observable Optimal control design he state-weighting matrix Q must be positive semidefinite x Qx he control-weighting matrix R must be positive definite u Ru > he plant A, B or Φ, must at least be stabilisable i.e., any unstable modes are controllable he plant A, C or Φ, C must at least be detectable i.e., any unstable modes are observable
Robustness L7:7 For an LQ regulator with a diagonal R, the closedloop system will have a gain margin of gain-reduction margin of.5 phase margin 6º in each plant input control channel Given the duality between optimal control and estimation, the Kalman filter has similar robust properties However, the LQG combination can have arbitrarily poor stability margins! Increasing the speed of the estimator dynamics may reduce stability margins
Loop transfer recovery L7:8 It is possible to recover the robustness of the LQ regulator, for minimum phase systems, by a procedure called loop transfer recovery LR his involves cancellation of the plant zeros by some of the filter poles; the remaining poles may become arbitrarily fast LR concept: suppose an LQG regulator is designed for realistic process and sensor noise values now add increasing amounts of fictitious process noise adjustment to estimator design in coping with added noise, LQG controller becomes more robust to gain and phase changes at plant input however, it is no longer optimal for actual noise levels LR can be targeted say to vicinity of gain crossover frequency by frequency shaping of fictitious noise
LR procedure + G For the LQR, u = Kx, so the loop gain is HG = K si A B H We have seen that this gives desirable robustness properties phase and gain margins L7:9 w v r = u + y For the LQG regulator, u = Kxˆ, so that the loop gain is HG = K si A + BK + LC L C si A [ ] B It can be shown that this approaches the ideal LQR loop gain if L = ρb, as ρ. Glad & Ljung, ch. 9 One way of achieving this is to design the Kalman filter assuming w = process noise is additive with control input, and setting Q = αr. As α increases large process noise, the optimal estimation solution approaches L = αb. he advantage of this approach is that the estimator is guaranteed stable for any α.
Case studies L7:2 Hot steel rolling mill atlab Control Systems oolbox User s Guide On-line pdf documentation milldemo.m agnetic tape drive Franlin, owell & Worman 3rd edn, sec. 9.5.4