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Getting Started with Moving Horizon Estimation

James B. Rawlings

Department of Chemical and Biological EngineeringUniversity of Wisconsin–Madison

Insitut für Systemtheorie und RegelungstechnikUniversität StuttgartStuttgart, Germany

June 20, 2011

Stuttgart – June 2011 MPC short course 1 / 25

Outline

1 Least squares estimation for linear, unconstrained systems

2 Moving horizon estimation

3 Convergence of the state estimator



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State Estimation

In most applications, the variables that are conveniently or

economically measurable (y ) are a small subset of the variablesrequired to model the system (x ).

The measurement is corrupted with sensor noise and the stateevolution is corrupted with process noise.

Determining a good state estimate for use in the regulator in the face of anoisy and incomplete output measurement is a challenging task.


State estimation

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0 00 01 11 1 0 00 00 01 11 11 1

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Measurement

MH EstimateMPC control

Forecast

t time

Reconcile the past Forecast the future

sensorsy

actuatorsu

minx 0,w (t )

0−T

|y − g (x , u )|2R + |ẋ − f (x , u )|2Q dt

ẋ = f (x , u ) + w (process noise)

y = g (x , u ) + v (measurement noise)



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Least Squares Estimation

Consider the state estimation problem as a deterministic optimizationproblem, given:

a time horizon with measurements y (k ), k = 0, 1, . . . , T

the prior information to be our best initial guess of the initial state

x (0), denoted x (0)weighting matrices P −(0), Q , and R for the initial state, processdisturbance, and measurement disturbance

The objective function is

V T (x(T )) = 1

2

|x (0) − x (0)|2(P −(0))−1 +

T −1k =0

|x (k + 1) − Ax (k )|2Q −1 +

T k =0

|y (k ) − Cx (k )|2R −1 (1)in which x(T ) := {x (0), x (1), . . . , x (T )}.


Forward Dynamic Programming (DP)

Using forward DP, we can decompose and solve recursively the leastsquares state estimation problem:

minx(T )

V T (x(T )) (2)

First we combine the prior and the measurement y (0) into the quadraticfunction V 0(x (0)) as shown in the following equation

minx (T ),...,x (1)

arrival cost V −1 (x (1)) minx (0)

1

2

|x (0) − x (0)|2(P −(0))−1 + |y (0) − Cx (0)|

2R −1

combine V 0(x (0))

+ |x (1) − Ax (0)|2Q −1

+

|y (1) − Cx (1)|2R −1

+ |x (2) − Ax (1)|2Q −1

+ · · · +

|x (T ) − Ax (T − 1)|2Q −1

+ |y (T ) − Cx (T )|2R −1



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Forward DP

Then we optimize over the first state, x (0). This produces the arrival costfor the first stage, V −1 (x (1)), which we will show is also quadratic

V −1 (x (1)) = 1

2 x (1) − x̂ −(1)

2

(P −(1))−1

Next we combine the arrival cost of the first stage with the nextmeasurement y (1) to obtain V 1(x (1))

minx (T ),...,x (2)

arrival cost V −2 (x (2)) minx (1)

1

2

x (1) − x̂ −(1)2(P −(1))−1

+ |y (1) − Cx (1)|2R −1

combine V 1(x (1))

+ |x (2) − Ax (1)|2Q −1

+

|y (2) − Cx (2)|2R −1 + |x (3) − Ax (2)|

2Q −1 + · · · +

|x (T ) − Ax (T − 1)|2Q −1

+ |y (T ) − Cx (T )|2R −1

(3)


Combine Prior and Measurement

Combining the prior and measurement defines V 0

V 0(x (0)) = 1

2

|x (0) − x (0)|2(P −(0))−1

prior

+ |y (0) − Cx (0)|2R −1 measurement

(4)

We can combine these two terms into a single quadratic function

V 0(x (0)) = (1/2) (x (0) − x (0) − v )H̃ −1(x (0) − x (0) − v ) + constant

in which

v = P −(0)C (CP −(0)C + R )−1C R −1 (y (0) − C x (0))

H̃ = P −(0) − P −(0)C (CP −(0)C + R )−1CP −(0)

and we set the constant term to zero because it does not depend on x (1).



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Combine Prior and Measurement

By defining

P (0) = P −(0) − P −(0)C (CP −(0)C + R )−1CP −(0)

L(0) = P −

(0)C

(CP −

(0)C

+ R )−1

C

R −1

and with the state estimate x̂ (0) as follows

x̂ (0) = x (0) + v

x̂ (0) = x (0) + L(0)(y (0) − C x (0))

We have derived the following compact expression for the function V 0:

V 0(x (0)) = (1/2) |x (0) − x̂ (0)|2P (0)−1


State Evolution and Arrival Cost

Now we add the next term and denote the sum as V (·):

V (x (0), x (1)) = V 0(x (0)) + (1/2) |x (1) − Ax (0)|2Q −1

V (x (0), x (1)) = 1

2

|x (0) − x̂ (0)|2P (0)−1 + |x (1) − Ax (0)|

2Q −1

We can add the two quadratics to obtain

V (x (0), x (1)) = (1/2) |x (0) − v |2H −1 + d in which

v = x̂ (0) + P (0)A

AP (0)A + Q −1

(x (1) − Ax̂ (0))

d = (1/2)

|v − x̂ (0)|2P (0)−1 + |x (1) − Av |2Q −1



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State Evolution and Arrival Cost

We define the arrival cost to be the result of this optimization

V −1 (x (1)) = minx (0)

V (x (0), x (1))

Substituting v into the expression for d and simplifying gives

V −1 (x (1)) = (1/2) |x (1) − Ax̂ (0)|2(P −(1))−1

in whichP −(1) = AP (0)A + Q

We define x̂ −(1) = Ax̂ (0) and express the arrival cost compactly as

V −1 (x (1)) = (1/2) x (1) − x̂ −(1)2(P −(1))−1


Combine Arrival Cost and Measurement

Now combine the arrival cost and measurement for the next stage:

V 1(x (1)) = V −

1 (x (1)) prior

+ (1/2) |(y (1) − Cx (1))|2R −1 measurement

V 1(x (1)) = 1

2

x (1) − x̂ −(1)2(P −(1))−1

+ |y (1) − Cx (1)|2R −1

This equation is exactly the form as (4) of the previous step.By simply changing the variable names, we have that

P (1) = P −(1) − P −(1)C (CP −(1)C + R )−1CP −(1)

L(1) = P −(1)C (CP −(1)C + R )−1

x̂ (1) = x̂ −(1) + L(1)(y (1) − C ̂x −(1))



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Combine Arrival Cost and Measurement

The cost function V 1 is defined as

V 1(x (1)) = (1/2)(x (1) − x̂ (1))P (1)−1(x (1) − x̂ (1))

in which

x̂ −(1) = Ax̂ (0)

P −(1) = AP (0)A + Q


Recursion and Termination

The recursion can be summarized by two steps:

1 Adding the measurement at time k produces

P (k ) = P −(k ) − P −(k )C (CP −(k )C + R )−1CP −(k )

L(k ) = P −(k )C (CP −(k )C + R )−1

x̂ (k ) = x̂ −(k ) + L(k )(y (k ) − C x̂ −(k ))

2 Propagating the model to time k + 1 produces

x̂ −(k + 1) = Ax̂ (k )

P −(k + 1) = AP (k )A + Q

The arrival cost, V −k , and arrival cost plus measurement, V k , for eachstage are given by

V −k (x (k )) = (1/2)x (k ) − x̂ −(k )

(P −(k ))−1

V k (x (k )) = (1/2) |x (k ) − x̂ (k )|(P (k ))−1



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Moving Horizon Estimation

For least squares method, we must optimize all the states in thetrajectory x(T ) simultaneously to obtain the state estimates.

When using nonlinear models or considering constraints on theestimates, this optimization problem becomes computationallyintractable as T increases.

We cannot solve recursively the least squares problem.

Moving horizon estimation (MHE) removes this difficulty byconsidering only the most recent N measurements and finds only themost recent N values of the state trajectory.



The simplest form of MHE is the following least squares problem

minxN (T )

V̂ T (xN (T )) (5)

in which the objective function is

V̂ T (xN (T )) = 1

2

T −1k =T −N

|x (k + 1) − Ax (k )|2Q −1 +

T k =T −N

|y (k ) − Cx (k )|2R −1

(6)

The measurements are yN (T ) = {y (T − N ), . . . , y (T )}.

The states to be estimated are xN (T ) = {x (T − N ), . . . , x (T )}.

Assume T ≥ N − 1 to ignore the initial period in which theestimation window fills with measurements.



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0000000000000000000000000000011111111111111111111111111111

00000000000000001111111111111111

T T − N 0

x (T )

moving horizon

full information

x (T − N )

y (T ) y (T − N )


MHE in Terms of Least Squares

From our previous DP recursion, the full least squares problem is also of form

V T (xN (T )) = V −T −N (x (T − N ))+

1

2 T −1

k =T −N |x (k + 1) − Ax (k )|2Q −1 +

T

k =T −N |y (k ) − Cx (k )|2R −1

in which V −T −N (·) is the arrival cost at time T − N .It is clear that the simplest form of MHE is equivalent to setting up a fullleast squares problem but then setting the arrival cost function V −T −N (·) tozero.



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Convergence of Estimator

Given an initial estimate error, and zero state and measurement noises,

does the state estimate converge to the state as time increases and more

measurements become available?

If the answer is yes, we say the estimates converge, or the estimatorconverges.

As with the regulator, optimality of an estimator does not ensure itsstability.

Now we go back to the Kalman filtering or full least squares problem...

Recall that this estimator optimizes over the entire state trajectoryx(T ) := {x (0), . . . , x (T )} based on all measurements y(T ) := {y (0),

. . . , y (T )}.


Convergence of Estimator Cost

Lemma 1 (Convergence of estimator cost)

Given noise-free measurements y(T ) =

Cx (0), CAx (0), . . . , CAT x (0)

,

the optimal estimator cost V 0T (y(T )) converges as T → ∞.

The optimal estimator cost converges regardless of system observability.But if we want the optimal estimate to converge to the state, the systemneeds to be restricted further as in the following lemma.



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Lemma 2 (Estimator convergence)For (A, C ) observable, Q , R > 0, and noise-free measurements y(T ) =

Cx (0), CAx (0), . . . , CAT x (0)

, the optimal linear state estimate

converges to the state

x̂ (T ) → x (T ) as T → ∞



The simplest form of MHE, which discounts prior data completely, isalso a convergent estimator, as discussed in Exercise 1.28.

The estimator convergence result in Lemma 2 is the simplest toestablish.

As in the case of the LQ regulator, however, we can enlarge the classof systems and weighting matrices (variances) for which estimator

convergence is guaranteed: The system restriction can be weakened from observability to

detectability , which is discussed in Exercises 1.31 and 1.32. The restriction on the process disturbance weight (variance) Q can be

weakened from Q > 0 to Q ≥ 0 and (A, Q ) stabilizable , which isdiscussed in Exercise 1.33.

The restriction R > 0 remains to ensure uniqueness of the estimator.



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Recommended exercises

MHE Convergence. Exercise 1.28.1

Detectability and semi-definite state noise. Exercise 1.31, 1.32, 1.33.

Least squares and recursive least squares. Exercises 1.41, 1.48.

Arrival cost. Exercises 1.51, 1.52, and 1.53.

1Rawlings and Mayne (2009, Chapter 1). Downloadable fromwww.che.wisc.edu/~jbraw/mpc.


Further Reading

J. B. Rawlings and D. Q. Mayne. Model Predictive Control: Theory and Design. Nob Hill Publishing, Madison, WI, 2009. 576 pages, ISBN

978-0-9759377-0-9.


http://www.che.wisc.edu/~jbraw/mpchttp://www.che.wisc.edu/~jbraw/mpchttp://www.che.wisc.edu/~jbraw/mpchttp://www.che.wisc.edu/~jbraw/mpchttp://www.che.wisc.edu/~jbraw/mpc


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Acknowledgments

The author is indebted to Luo Ji of the University of Wisconsin andGanzhou Wang of the Universität Stuttgart for their help in organizing thematerial and preparing the overheads.


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07 mpc jbr stanfd