representation, in which a state variable represents the number of individuals in the corresponding class.

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Systems and Control Theory Lecture Notes

Laura Giarr´ e

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Lab Session 3: Kalman

The evolution of a cattle breeding can be described by a population dynamic model in input-state-output

representation, in which a state variable represents the number of individuals in the corresponding class.

We consider a model with the following classes:

x 1 : young females; x 4 : young males x 2 : adult females; x 5 : adult males x 3 : old females; x 6 : old males

If x _i (k) represents the number of heads in the i-th class

during the year k, the population evolution is described by the

equations x(k + 1) = Ax(k) + u(k) + w(k)

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with

A =

⎡

⎢ ⎢

⎣

a ₁ a ₂ a ₃ 0 0 0

b ₁ 0 0 0 0 0

0 b ₂ b ₃ 0 0 0 c ₁ c ₂ c ₃ 0 0 0

0 0 0 d ₁ 0 0

0 0 0 0 d ₂ d ₃

⎤

⎥ ⎥

⎦

the coeﬃcients a _i and c _i , i = 1, 2, 3, are the birthrates of female and male heads, respectively, from cows belonging to the class x _i ;

the coeﬃcients b _i and d _i , i = 1, 2, 3, are the surviving rates of female heads of the class x _i and of male heads of the class x _i + 3, respectively;

u _i (k) is the buying and selling balance of heads in the i-th class, during the year k;

w _i (k) represents the discrepancy between the true evolution of

the i-th class and the nominal one predicted by the model.

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Assume that we have data from an annual census of the cattle breeding, reporting the total number of female and male

heads over a period of 101 years

y ₁ (k) = x ₁ (k) + x ₂ (k) + x ₃ (k) + v ₁ (k) y ₂ (k) = x 4 (k) + x 5 (k) + x 6 (k) + v 2 (k) (k = 0, 1, ..., 100)

where the variables v _j (k), j = 1, 2, represent the errors made

in the census.

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Exercise A

Assume that w _i (k), i = 1, . . . , 6 and v j (k), j = 1, 2, can be modeled as realizations of discrete-time stochastic white

processes, independent with each other, with zero mean and such that

the variance of processes w _i (k) is equal to σ _w ² ;

the variance of processes v _j (k) is equal to σ _v ²

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Design and implement a Kalman ﬁlter providing an estimate of the population in each class, by using the input data u(t) and the measurements y(t).

Plot the obtained estimates and compare them with the true evolution of each class.

Compute the state estimation error

x(k)ˆx(k) ² and

compare it with the corresponding expected value provided by the Kalman ﬁlter (the square root of the trace of the

covariance matrix P.)

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Exercise B

By using the Matlab function dlqe, design and implement the asymptotic Kalman ﬁlter relative to the same problem of

exercise A.

Compare the estimates provided by the asymptotic Kalman ﬁlter to those of the time-varying Kalman ﬁlter used in

exercise A.

Discuss the role of the Kalman ﬁlter initialization.

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Exercise C

Consider the same evolution of the state variables as in

exercise A (with the same input signal u(t)), but assume that a diﬀerent set of measurements y(t) is available. The

measurement errors aﬀecting these new data have a standard deviation equal to σ _v

_j

(k) = 3000.95 ^k ; j = 1, 2.

Apply the Kalman ﬁlter again, using these new measurements.

Compare the estimates with those provided by a Kalman ﬁlter assuming a constant standard deviation for the measurement noise, equal to 300.

Evaluate the estimation errors at each time instant in both

cases, and compare them to the expected values provided by

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Exercise D

Assume now that a new input-output data set is available, relative to the same model as in exercise A, but with a

time-varying parameter a ₂ such that a ₂ (k) = 0.45 + 0.3 sin(2πk/20).

Apply the Kalman ﬁlter and compare the results to those obtained by using the ﬁlter adopted in exercise A (in which matrix A is constant).

Evaluate the true state estimation errors of both ﬁlters and

compare them to the expected values provided by the ﬁlters.

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DATA

Data contained in lab3.mat refer to the following variables:

parameters of the model:

a ₁ , a ₂ , a ₃ , b ₁ , b ₂ , b ₃ , c ₁ , c ₂ , c ₃ , d ₁ , d ₂ , d ₃ ;

process variances σ _w ² and σ _v ² : sw2 and sv2, respectively;

input uA and output yA sequences to be used in exercises A and B: the i-th column refers to the i-th variable (e.g., uA(:,1) corresponds to u1(t) for t = 0, 1, . . . , 100);

sequences of true state variables xtrueA in exercises A, B and C, to be used uniquely to compute the true state estimation errors;

input uA and output yC sequences to be used in exercise C;

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Thanks

DIEF- laura.giarre@unimore.it Tel: 059 2056322

giarre.wordpress.com

representation, in which a state variable represents the number of individuals in the corresponding class.

Systems and Control Theory Lecture Notes

Laura Giarr´ e

Lab Session 3: Kalman

The evolution of a cattle breeding can be described by a population dynamic model in input-state-output

representation, in which a state variable represents the number of individuals in the corresponding class.

We consider a model with the following classes:

x 1 : young females; x 4 : young males x 2 : adult females; x 5 : adult males x 3 : old females; x 6 : old males

If x i (k) represents the number of heads in the i-th class

during the year k, the population evolution is described by the

equations x(k + 1) = Ax(k) + u(k) + w(k)

with

A =

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎣

a 1 a 2 a 3 0 0 0

b 1 0 0 0 0 0

0 b 2 b 3 0 0 0 c 1 c 2 c 3 0 0 0

0 0 0 d 1 0 0

0 0 0 0 d 2 d 3

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎦

the coeﬃcients a i and c i , i = 1, 2, 3, are the birthrates of female and male heads, respectively, from cows belonging to the class x i ;

the coeﬃcients b i and d i , i = 1, 2, 3, are the surviving rates of female heads of the class x i and of male heads of the class x i + 3, respectively;

u i (k) is the buying and selling balance of heads in the i-th class, during the year k;

w i (k) represents the discrepancy between the true evolution of

the i-th class and the nominal one predicted by the model.

Assume that we have data from an annual census of the cattle breeding, reporting the total number of female and male

heads over a period of 101 years

y 1 (k) = x 1 (k) + x 2 (k) + x 3 (k) + v 1 (k) y 2 (k) = x 4 (k) + x 5 (k) + x 6 (k) + v 2 (k) (k = 0, 1, ..., 100)

where the variables v j (k), j = 1, 2, represent the errors made

in the census.

Exercise A

Assume that w i (k), i = 1, . . . , 6 and v j (k), j = 1, 2, can be modeled as realizations of discrete-time stochastic white

processes, independent with each other, with zero mean and such that

the variance of processes w i (k) is equal to σ w 2 ;

the variance of processes v j (k) is equal to σ v 2

Design and implement a Kalman ﬁlter providing an estimate of the population in each class, by using the input data u(t) and the measurements y(t).

Plot the obtained estimates and compare them with the true evolution of each class.

Compute the state estimation error 

x(k)ˆx(k) 2 and

compare it with the corresponding expected value provided by the Kalman ﬁlter (the square root of the trace of the

covariance matrix P.)

Exercise B

By using the Matlab function dlqe, design and implement the asymptotic Kalman ﬁlter relative to the same problem of

exercise A.

Compare the estimates provided by the asymptotic Kalman ﬁlter to those of the time-varying Kalman ﬁlter used in

exercise A.

Discuss the role of the Kalman ﬁlter initialization.

Exercise C

Consider the same evolution of the state variables as in

exercise A (with the same input signal u(t)), but assume that a diﬀerent set of measurements y(t) is available. The

measurement errors aﬀecting these new data have a standard deviation equal to σ v

(k) = 3000.95 k ; j = 1, 2.

Apply the Kalman ﬁlter again, using these new measurements.

Compare the estimates with those provided by a Kalman ﬁlter assuming a constant standard deviation for the measurement noise, equal to 300.

Evaluate the estimation errors at each time instant in both

cases, and compare them to the expected values provided by

Exercise D

Assume now that a new input-output data set is available, relative to the same model as in exercise A, but with a

time-varying parameter a 2 such that a 2 (k) = 0.45 + 0.3 sin(2πk/20).

Apply the Kalman ﬁlter and compare the results to those obtained by using the ﬁlter adopted in exercise A (in which matrix A is constant).

Evaluate the true state estimation errors of both ﬁlters and

compare them to the expected values provided by the ﬁlters.

DATA

Data contained in lab3.mat refer to the following variables:

parameters of the model:

a 1 , a 2 , a 3 , b 1 , b 2 , b 3 , c 1 , c 2 , c 3 , d 1 , d 2 , d 3 ;

process variances σ w 2 and σ v 2 : sw2 and sv2, respectively;

input uA and output yA sequences to be used in exercises A and B: the i-th column refers to the i-th variable (e.g., uA(:,1) corresponds to u1(t) for t = 0, 1, . . . , 100);

sequences of true state variables xtrueA in exercises A, B and C, to be used uniquely to compute the true state estimation errors;

input uA and output yC sequences to be used in exercise C;

Thanks

DIEF- laura.giarre@unimore.it Tel: 059 2056322

If x _i (k) represents the number of heads in the i-th class

a ₁ a ₂ a ₃ 0 0 0

b ₁ 0 0 0 0 0

0 b ₂ b ₃ 0 0 0 c ₁ c ₂ c ₃ 0 0 0

0 0 0 d ₁ 0 0

0 0 0 0 d ₂ d ₃

the coeﬃcients a _i and c _i , i = 1, 2, 3, are the birthrates of female and male heads, respectively, from cows belonging to the class x _i ;

the coeﬃcients b _i and d _i , i = 1, 2, 3, are the surviving rates of female heads of the class x _i and of male heads of the class x _i + 3, respectively;

u _i (k) is the buying and selling balance of heads in the i-th class, during the year k;

w _i (k) represents the discrepancy between the true evolution of

y ₁ (k) = x ₁ (k) + x ₂ (k) + x ₃ (k) + v ₁ (k) y ₂ (k) = x 4 (k) + x 5 (k) + x 6 (k) + v 2 (k) (k = 0, 1, ..., 100)

where the variables v _j (k), j = 1, 2, represent the errors made

Assume that w _i (k), i = 1, . . . , 6 and v j (k), j = 1, 2, can be modeled as realizations of discrete-time stochastic white

the variance of processes w _i (k) is equal to σ _w ² ;

the variance of processes v _j (k) is equal to σ _v ²

Compute the state estimation error

x(k)ˆx(k) ² and

measurement errors aﬀecting these new data have a standard deviation equal to σ _v

(k) = 3000.95 ^k ; j = 1, 2.

time-varying parameter a ₂ such that a ₂ (k) = 0.45 + 0.3 sin(2πk/20).

a ₁ , a ₂ , a ₃ , b ₁ , b ₂ , b ₃ , c ₁ , c ₂ , c ₃ , d ₁ , d ₂ , d ₃ ;

process variances σ _w ² and σ _v ² : sw2 and sv2, respectively;