Chapter 5 Estimator Design - Pole placement Motivation Control law - - PDF document

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Chapter 5 Estimator Design - Pole placement Motivation

• Control law design with state feedback needs all the

state variables.

• Not all state variables are available since the cost of

the required sensors may be prohibitive, or it may be physically impossible to measure all the state variables. Estimator: reconstruct all the state variables x of a system from measurements. The reconstructed state variables ˆ x are then used for control.

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Motivation example

Example Boeing 747 aircraft control In the Boeing 747 aircraft model, there are 4 states : the side-slip angle β, the yaw rate r, the roll rate p and the roll angle φ. Only the yaw rate r is measured with a gyroscope, while all other states are not measured. Challenge : Can we estimate the states β, p and φ from the measurement of r such that the state feedback control law can be implemented? What are the effects of the estimated states on the final closed loop system?

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Full Order Estimators (Observers) Open loop estimator

Estimator: Reconstruct a full order model of the plant dynamics: ˙ ˆ x = Aˆ x + Bu. ˆ x is the estimate of the state x. The dynamics of this estimator: Define the error in the estimate: ˜ x

∆

= x − ˆ x Then the dynamics of this error system are given by (sub- tracting the estimate from the state): ˙ ˜ x = A˜ x, ˜ x(0) = x(0) − ˆ x(0).

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˙ ˆ x = Aˆ x + Bu ˙ x = Ax + Bu u Process Estimator x C ˆ x y

Disadvantage of the open loop estimator:

• The error is converging to zero at the same rate as the

natural dynamics of A (if A is stable). There is no way to influence the rate at which the state estimate converges to the true state.

• The error will NOT converge if A is unstable.
• If the model dynamics (A, B, C, D) are different from

the system dynamics (due to inaccurate modeling for instance), in general, ˜ x will not converge to 0.

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Estimator with feedback

Estimator: Insert the feedback signal of the difference between the mea- sured and estimated outputs (y−Cˆ x) to correct the model: ˙ ˆ x = Aˆ x + Bu + L(y − Cˆ x). where L is a constant matrix in Rn×p. Dynamics of the estimator: The error dynamics are given as follows (by subtracting the estimate from the state): ˙ ˜ x = (A − LC)˜ x, ˜ x(0) = x(0) − ˆ x(0).

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˙ x = Ax + Bu Process ˙ ˆ x = Aˆ x + Bu + L˜ y Estimator C C − + y ˜ y u x ˆ x ˆ y

Properties:

• By choosing the gain matrix L, the error dynamics of

A − LC can be stable and much faster than the open loop dynamics A.

• In case of inaccurate A and/or C, the error dynamics

can still be stabilized and be faster ⇒ Robustness.

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Estimator Design - Pole Placement

Choose the gain matrix L such that the poles of the estimator are in desired positions. Let the desired locations be given by s = s1, s2, . . . , sn then the desired estimator characteristic equation is αe(s)

∆

= (s − s1)(s − s2) . . . (s − sn)

Direct method

Determine L by comparing coefficients of the two polyno- mials on both sides of the following equation : det (sI − (A − LC)) = (s − s1)(s − s2) . . . (s − sn)

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Duality of estimation and control designs

The problem to find L such that det (sI − (A − LC)) = (s − s1)(s − s2) . . . (s − sn) is equivalent to the problem to find K = LT such that det

• sI − (AT − CTK)
• = (s − s1)(s − s2) . . . (s − sn)

which is the problem for state feedback control design. Further, the controllability matrix C of (A, B) is the trans- pose of the observability matrix O of (AT, BT). Thus we have the following duality relations: Control Estimation A AT B CT K LT C OT

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Ackermann’s formula for SISO

L = αe(A)O−1       . . . 1       where O is the observability matrix.

Sylvester equation for MIMO:

Design procedure:

• Pick an arbitrary matrix G ∈ Rp×n.
• Solve the Sylvester equation for X:

ATX − XΛ = CTG Λ =         α1 β1 −β1 α1 ... λ1 ...         , which has eigenvalues: α1 ±jβ1, . . . , λ1, . . . . which are the desired poles of the estimator.

• Obtain the static feedback gain L = (GX−1)T.

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Estimator Pole Selection

The dynamics of the state estimation error are governed by the eigenvalues of A − LC. Obviously, all eigenvalues are chosen to be stable. Concerning the estimator pole se- lection there is a fundamental trade-off between sensitivity

• f the estimation error to sensor noise and speed of con-
• vergence. The smaller Re{eig(A − LC)}, the faster the

estimation error goes to zero. However, for small eigenval- ues rapid changes in the signals will also introduce rapid changes in the estimation error. The estimated state will change nervously. In the next chapter we will describe an

• ptimal solution for this dilemma which is the Kalman fil-

ter. Consider a control system with process noise w and sensor noise v: ˙ x = Ax + Bu + B1w, y = Cx + v then the estimator error will be : ˙ ˜ x = (A − LC)˜ x + B1w − Lv

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Rules of thumb:

• Trade-off between a fast dynamic response (of the esti-

mator) and a good sensor noise reduction. ⇒ “large” L or “small ” L.

• Trade-off between process noise reduction and sensor

noise reduction. ⇒ “large” L or “small ” L. Consider the following first order system ˙ x = −x + u y = x + v where v is sensor noise. Now compare 2 estimators : A−LC = −10 and A−LC = −2. There is no input, but the initial state is 1. The dynamics for A − LC = −10 are faster but more subject to noise :

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −0.2 0.2 0.4 0.6 0.8 1 1.2 time (s) − state evolution −− estimated state, A−LC=−10 .. estimated state, A−LC=−2

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Examples of estimator design

Example Boeing 747 aircraft control - Estimator design with pole placement (Ackermann’s method) We have designed a state feedback control law, and the desired closed-loop poles are at −0.0051, −0.468, −1.106, −9.89, −0.279 ± 0.628i. For a fast response of the estimator, we make the poles of the estimator 5 times faster than the desired closed-loop poles : −0.0255, −2.34 , −5.53, −49.45, −1.395 ± 3.14i. The characteristic polynomial of the estimator

αe(s) = s6 + 60.14s5 + 575.4s4 + 2453s3 + 6595s2 + 7721s + 192.6

αe(A) =            −1.13e + 6 −4.48e + 4 −5.15e + 3 −4.49e + 3 6.01e + 2 2.24e + 2 −5.38e + 5 3.27e + 3 −4.06e + 3 1.91e + 2 1.50e + 2 1.81e + 5 −8.17e + 3 1.55e + 4 −3.62e + 3 −6.12e + 2 −1.72e + 4 −1.45e + 4 8.58e + 3 4.54e + 3 −6.54e + 1 5.38e + 4 3.17e + 3 3.84e + 3 −3.14e + 1 5.48e + 1 −1.73e + 3            ESAT–SCD–SISTA

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The observability matrix equals

O =            1 0 −3.33e − 1 −4.75e + 0 5.98e − 1 −4.48e − 1 −3.18e − 2 1.11e − 1 4.96e + 1 −2.04e − 1 −4.46e − 1 7.70e − 2 2.48e − 2 −3.70e − 2 −4.94e + 2 −4.90e − 1 2.50e − 1 −1.32e − 2 −8.49e − 3 1.23e − 2 4.94e + 3 2.17e − 1 4.66e − 1 −4.96e − 2 −2.03e − 2 −4.12e − 3 −4.94e + 4 4.18e − 1 −2.95e − 1 5.31e − 3 9.01e − 3 1.37e − 3           

Then the feedback gain for the estimator is L =            2.5047e + 01 −2.0517e + 03 −5.1935e + 03 −2.4851e + 04 −4.0914e + 04 −1.5728e + 04           

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Example Tape drive control - Estimator design with pole placement (Sylvester equation) The desired estimator poles are again set to be 5 times faster than the desired closed-loop poles: −2.2550±4.6850i, −4.7350±2.9050i, −5.8000, −5.8000. Take an arbitrary matrix G:

G =

• −0.508 −0.248 −0.445 −0.209 −1.064 1.133

0.885 −0.726 −0.613 0.562 0.352 0.150

• Solve the Sylvester equation for X:

ATX − XΛe = CTG Λe is a block diagonal matrix with the eigenvalues equal to the desired estimator poles: Λc =            −2.225 4.685 −4.685 −2.225 −4.735 2.905 −2.905 −4.735 −5.8 −5.8            .

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X =

           −4.0150e − 02 −7.4418e − 02 −3.5794e − 02 −2.4713e − 02 −1.0413e − 01 9.2375e − 02 −6.1041e − 02 3.7229e − 03 2.4053e − 02 −3.0478e − 02 2.4528e − 02 −3.8791e − 02 4.0795e − 02 −3.4260e − 02 −5.2180e − 02 3.4518e − 02 −7.9299e − 02 1.0297e − 01 2.1710e − 02 1.2860e − 02 1.5984e − 03 9.5250e − 03 4.1517e − 02 −3.1544e − 02 2.9984e − 03 8.9685e − 03 −4.3541e − 05 6.1539e − 03 −3.8326e − 03 6.0611e − 03 1.0521e − 03 −3.7573e − 03 −1.1269e − 03 −1.0362e − 03 −6.4870e − 03 4.9287e − 03           

Now obtain the estimator feedback gain L = X−TGT: L =            5.38e + 01 −4.21e + 01 2.04e + 02 −1.56e + 02 3.90e + 01 −7.92e + 00 4.00e + 02 −3.08e + 02 6.51e + 02 −5.82e + 02 1.77e + 03 −1.50e + 03           

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Matlab Functions

• bsv

poly real polyvalm acker lyap place

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