Topic #16 16.31 Feedback Control State-Space Systems Open-loop Estimators Closed-loop Estimators Observer Theory (no noise) — Luenberger IEEE TAC Vol 16, No. 6, pp. 596—602, December 1971. Estimation Theory (with noise) — Kalman Copyright 2001 by Jonathan How. 1 Fall 2001 16.31 16—1 Estimators/Observers Problem: So far we have assumed that we have full access to the state x(t) when we designed our controllers. — Most often all of this information is not available. Usually can only feedback information that is developed from the sensors measurements. — Could try “output feedback” u = Kx ? u = ? Ky — Same as the proportional feedback we looked at at the beginning of the root locus work. — This type of control is very di?cult to design in general. Alternative approach: Develop a replica of the dynamic sys- tem that provides an “estimate” of the system states based on the measured output of the system. New plan: 1. Develop estimate of x(t)thatwillbecalled?x(t). 2. Then switch from u = ?Kx(t)tou = ?K?x(t). Two key questions: — How do we find ?x(t)? — Will this new plan work? Fall 2001 16.31 16—2 Estimation Schemes Assume that the system model is of the form: ˙x = Ax + Bu , x(0) unknown y = Cx where 1. A, B,andC are known. 2. u(t)isknown 3. Measurable outputs are y(t)fromC 6= I Goal: Develop a dynamic system whose state ?x(t)=x(t) for all time t ≥ 0. Two primary approaches: — Open-loop. — Closed-loop. Open-loop Estimator Given that we know the plant matrices and the inputs, we can just perform a simulation that runs in parallel with the system ˙ ?x(t)=A?x + Bu(t) Then ?x(t) ≡ x(t) ? t provided that ?x(0) = x(0) Major Problem: We do not know x(0) Fall 2001 16.31 16—3 Analysis of this case: ˙x(t)=Ax + Bu(t) ˙ ?x(t)=A?x + Bu(t) Define the estimation error ?x(t)=x(t)? ?x(t). Now want ?x(t)=0? t. (But is this realistic?) Subtract to get: d dt (x? ?x)=A(x? ?x) ? ˙ ?x(t)=A?x which has the solution ?x(t)=e At ?x(0) — Gives the estimation error in terms of the initial error. Fall 2001 16.31 16—4 Does this guarantee that ?x =0? t? Or even that ?x → 0ast →∞ ? (which is a more realistic goal). — Response is fine if ?x(0)=0.Butwhatif?x(0) 6=0? If A stable, then ?x → 0ast →∞ ,butthedynamicsoftheestima- tion error are completely determined by the open-loop dynamics of the system (eigenvalues of A). — Could be very slow. — No obvious way to modify the estimation error dynamics. Open-loopestimationdoesnotseemtobeaverygoodidea. Closed-loop Estimator An obvious way to fix this problem is to use the additional informa- tion available: — How well does the estimated output match the measured output? Compare: y = Cx with ?y = C?x — Then form ?y = y ? ?y ≡ C?x Fall 2001 16.31 16—5 Approach: Feedback ?y to improve our estimate of the state. Basic form of the estimator is: ˙ ?x(t)=A?x(t)+Bu(t)+L?y(t) ?y(t)=C?x(t) where L is the user selectable gain matrix. Analysis: ˙ ?x = ˙x? ˙ ?x =[Ax + Bu]?[A?x + Bu+ L(y ? ?y)] = A(x? ?x)?L(Cx?C?x)=A?x?LC?x =(A?LC)?x So the closed-loop estimation error dynamics are now ˙ ?x =(A?LC)?x with solution ?x(t)=e (A?LC)t ?x(0) Bottom line: Can select the gain L to attempt to improve the convergence of the estimation error (and/or speed it up). — But now must worry about observability of the system model. Fall 2001 16.31 16—6 Note the similarity: — Regulator Problem: pick K for A?BK 3 Choose K ∈R 1×n (SISO) such that the closed-loop poles det(sI ?A + BK)=Φ c (s) are in the desired locations. — Estimator Problem: pick L for A?LC 3 Choose L ∈R n×1 (SISO) such that the closed-loop poles det(sI ?A + LC)=Φ o (s) are in the desired locations. Theseproblemsareobviouslyverysimilar—infacttheyarecalled dual problems. Fall 2001 16.31 16—7 Estimation Gain Selection For regulation, were concerned with controllability of (A,B) For a controllable system we can place the eigenvalues of A?BK arbitrarily. For estimation, were concerned with observability of pair (A,C). For a observable system we can place the eigenvalues of A?LC arbitrarily. Test using the observability matrix: rank M o , rank ? ? ? ? ? ? ? C CA CA 2 . . . CA n?1 ? ? ? ? ? ? ? = n The procedure for selecting L is very similar to that used for the regulator design process. Write the system model in observer canonical form ? ? ˙x 1 ˙x 2 ˙x 3 ? ? = ? ? ?a 1 1 0 ?a 2 0 1 ?a 3 00 ? ? ? ? x 1 x 2 x 3 ? ? + ? ? b 1 b 2 b 3 ? ? u y = ￡ 1 00 ¤ ? ? x 1 x 2 x 3 ? ? Fall 2001 16.31 16—8 Now very simple to form A?LC = ? ? ?a 1 1 0 ?a 2 0 1 ?a 3 00 ? ? ? ? ? l 1 l 2 l 3 ? ? ￡ 1 00 ¤ = ? ? ?a 1 ?l 1 1 0 ?a 2 ?l 2 0 1 ?a 3 ?l 3 00 ? ? — The closed-loop poles of the estimator are at the roots of det(sI ?A + LC)=s 3 +(a 1 + l 1 )s 2 +(a 2 + l 2 )s +(a 3 + l 3 )=0 So we have the freedom to place the closed-loop poles as desired. — Task greatly simplified by the selection of the state-space model used for the design/analysis. Fall 2001 16.31 16—9 Another approach: — Note that the poles of (A?LC)and(A?LC) T are identical. — Also we have that (A?LC) T = A T ?C T L T — So designing L T for this transposed system looks like a standard regulator problem (A?BK)where A ? A T B ? C T K ? L T So we can use K e = acker(A T ,C T ,P) ,L≡ K T e Note that the estimator equivalent of Ackermann’s formula is that L = Φ e (s)M ?1 o ? ? ? ? ? 0 . . . 0 1 ? ? ? ? ? Fall 2001 16.31 16—10 Estimators Example Simple system A = ? ?11.5 1 ?2 ? ,B= ? 1 0 ? ,x(0) = ? ?0.5 ?1 ? C = ￡ 1 0 ¤ ,D=0 — Assume that the initial conditions are not well known. — System stable, but λ max (A)=?0.18 — Test observability: rank ? C CA ? = rank ? 1 0 ?11.5 ? Use open and closed-loop estimators — Since the initial conditions are not well known, use ?x(0) = ? 0 0 ? Open-loop estimator: ˙ ?x = A?x + Bu ?y = C?x Closed-loop estimator: ˙ ?x = A?x + Bu+ L?y = A?x + Bu+ L(y ? ?y) =(A?LC)?x + Bu+ Ly ?y = C?x — Which is a dynamic system with poles given by λ i (A ? LC) and which takes the measured plant outputs as an input and generates an estimate of x. Fall 2001 16.31 16—11 Typically simulate both systems together for simplicity Open-loop case: ˙x = Ax + Bu y = Cx ˙ ?x = A?x + Bu ?y = C?x ? ? ˙x ˙ ?x ? = ? A 0 0 A ?? x ?x ? + ? B B ? u, ? x(0) ?x(0) ? = ? ? ? ? ? ?0.5 ?1 0 0 ? ? ? ? ? ? y ?y ? = ? C 0 0 C ?? x ?x ? Closed-loop case: ˙x = Ax + Bu ˙ ?x =(A?LC)?x + Bu+ LCx ? ? ˙x ˙ ?x ? = ? A 0 LC A?LC ?? x ?x ? + ? B B ? u Example uses a strong u(t)toshakethingsup Fall 2001 16.31 16—12 Figure 1: Open-loop estimator. Estimation error converges to zero, but very slowly. 0 0.5 1 1.5 2 2.5 3 3.5 4 ?1 ?0.5 0 0.5 1 Open loop estimator states time x1 x2 0 0.5 1 1.5 2 2.5 3 3.5 4 ?1 ?0.5 0 0.5 1 time estimation error Figure 2: Closed-loop estimator. Convergence looks much better. 0 0.5 1 1.5 2 2.5 3 3.5 4 ?1 ?0.5 0 0.5 1 Closed?loop estimator states time x1 x2 0 0.5 1 1.5 2 2.5 3 3.5 4 ?1 ?0.5 0 0.5 1 time estimation error Fall 2001 16.31 16—13 WheretoputtheEstimatorPoles? Location heuristics for poles still apply — use Bessel, ITAE, ... — Maindi?erence: probablywanttomaketheestimatorfaster than you intend to make the regulator — should enhance the control, which is based on ?x(t). — ROT: Factor of 2—3 in the time constant ζω n associated with the regulator poles. Note: When designing a regulator, were concerned with “band- width” of the control getting too high ? often results in control commands that saturate the actuators and/or change rapidly. Di?erent concerns for the estimator: — Loop closed inside computer, so saturation not a problem. — However, the measurements y are often “noisy”, and we need to be careful how we use them to develop our state estimates. ? High bandwidth estimators tend to accentuate the e?ect of sensing noise in the estimate. — State estimates tend to “track” the measurements, which are fluctuating randomly due to the noise. ? Low bandwidth estimators have lower gains and tend to rely more heavily on the plant model — Essentially an open-loop estimator — tends to ignore the mea- surements and just uses the plant model. Fall 2001 16.31 16—14 Can also develop an optimal estimator for this type of system. — Which is apparently what Kalman did one evening in 1958 while taking the train from Princeton to Baltimore... — Balances e?ect of the various types of random noise in the system on the estimator: ˙x = Ax + Bu+ B w w y = Cx+ v where: 3 w is called “process noise” — models the uncertainty in the system model. 3 v is called “sensor noise” — models the uncertainty in the measurements. A symmetric root locus exists for the optimal estimator. — Define G yw (s)=C(sI ?A) ?1 B w ≡ N(s)/D(s) — SRL for the closed-loop poles λ i (A?LC) of the estimator which are the LHP roots of: D(s)D(?s)± R w R v N(s)N(?s)=0 where R w and R v are, in some sense, associated with the sizes of the process/sensor noise (spectral density). — Pick sign to ensure that there are no poles on the jω-axis. Fall 2001 16.31 16—15 Relative size of the noises determine where the poles will be located. — Similar to role of control cost in LQR problem. As R w /R v → 0, the n polesgotothe 1. LHP poles of the system 2. Reflection of the RHP poles of the system about the jω-axis. — The “relatively noisy” sensor case ? Closed-loop estimator essentially reverts back to the open-loop case (but must be stable). — Low bandwidth estimator. As R w /R v →∞ ,then poles go to 1. LHP zeros (and reflections of the RHP zeros) of G yw (s). 2. ∞ along the Butterworth patterns — same as regulator case — The “relatively clean” sensor case ? Closed-loop estimator poles go to very high band- width to take full advantage o the information in y. — High bandwidth estimator. If you know R w and R v , then use them in the SRL, but more often than not we just use them as “tuning” parameters to develop low → high bandwidth estimators. — Typically fix R w and tune estimator bandwidth using R v Fall 2001 16.31 16—16 Final Thoughts Note that the feedback gain L in the estimator only stabilizes the estimation error. — If the system is unstable, then the state estimates will also go to ∞ , with zero error from the actual states. Estimation is an important concept of its own. — Not always just “part of the control system” — Critical issue for guidance and navigation system More complete discussion requires that we study stochastic pro- cesses and optimization theory. Estimation is all about which do you trust more: your measurements or your model.