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MSc in Computer Engineering, Cybersecurity and Artificial Intelligence Course FDE , a.a. 2019/2020, Lecture 15 Design procedure for state observers and observer-based feedback

• Prof. Mauro Franceschelli
• Dept. of Electrical and Electronic Engineering

University of Cagliari, Italy

Monday, 4th May 2020

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Outline

Design of state observers Design procedure Deadbeat response Separation principle Reduced/minimum order state observer

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Design of state observers

Introduction

• In the previous lecture we discussed asymptotic state observers
• Observers which estimate x(k + 1) from observation of y(k − 1) and u(k − 1)

are called prediction observers

• We now discuss how to systematically design the feedback gain Ke of an

asymptotic observer

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Design of state observers

Introduction

• In the previous lecture we discussed asymptotic state observers
• Observers which estimate x(k + 1) from observation of y(k − 1) and u(k − 1)

are called prediction observers

• We now discuss how to systematically design the feedback gain Ke of an

asymptotic observer

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Design of state observers

Introduction

• In the previous lecture we discussed asymptotic state observers
• Observers which estimate x(k + 1) from observation of y(k − 1) and u(k − 1)

are called prediction observers

• We now discuss how to systematically design the feedback gain Ke of an

asymptotic observer

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Design of state observers

State observer design

Consider the next discrete-time linear system: ①(k + 1) = ❆①(k) + ❇✉(k) ②(k + 1) = ❈①(k) where ❆ is an n × n matrix; ❇ is an n × r matrix; ❈ is an n × 1 matrix;

• We now discuss the state observer design for the single output case

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Design of state observers

State observer design

• We assume that the system is both completely state controllable and

completely observable.

• Thus, the inverse of the observability matrix

OT =

• ❈ T, ❆T❈ T, . . . (❆n−1)T❈ T

exists.

• Also, we consider a system controlled by full-state observer-based feedback

u(k) = −K ˜ ①(k) where K is a rectangular matrix (multi-input case) and is a design input.

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Design of state observers

State observer design

• We assume that the system is both completely state controllable and

completely observable.

• Thus, the inverse of the observability matrix

OT =

• ❈ T, ❆T❈ T, . . . (❆n−1)T❈ T

exists.

• Also, we consider a system controlled by full-state observer-based feedback

u(k) = −K ˜ ①(k) where K is a rectangular matrix (multi-input case) and is a design input.

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Design of state observers

State observer design

• We assume that the system is both completely state controllable and

completely observable.

• Thus, the inverse of the observability matrix

OT =

• ❈ T, ❆T❈ T, . . . (❆n−1)T❈ T

exists.

• Also, we consider a system controlled by full-state observer-based feedback

u(k) = −K ˜ ①(k) where K is a rectangular matrix (multi-input case) and is a design input.

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Design of state observers

State observer design

Matlab Simulink block diagram of a discrete-time linear system with full-order state observer and full-state observer-based feedback

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Design of state observers

State observer design

• The state observer dynamics is given by:

˜ ①(k + 1) = ❆˜ ①(k) + ❇✉(k) + ❑ e(y(k) − ˜ ②(k)) ˜ ②(k) = ❈ ˜ ①(k) (1)

• Where ˜

①(k) is the estimated state of the system, i.e., the observers’ state, and matrix ❑ e is a feedback gain which weights the error between the measure

• utputs of the system and the current outputs estimated by the observer.

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Design of state observers

Observable canonical form

• Since the system is completely observable, thus there exists a similarity transformation P

which puts the system into the observable canonical form with ①(k) = P③(k) ③(k + 1) = ˆ ❆③(k) + ˆ ❇✉(k) ②(k) = ˆ ❈③(k) where ˆ ❆ =          . . . −a0 1 . . . −a1 1 . . . −a2 . . . . . . . . . ... . . . . . . . . . −an−2 . . . 1 −an−1          ˆ ❈ = 0, 0, 0, . . . 0, 1

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Design of state observers

Observable canonical form

The coefficients a0, a1, . . . , an−1 are the coefficients of the characteristic polynomial of matrix ❆ det(λI − ❆) = λn + an−1λn−1 + . . . + a2λ2 + a1λ + a0 with αn = 1 (monic polynomial) by convention.

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Design of state observers

Observable canonical form

• The similarity transformation P which brings the completely observable system

with scalar output into the observable canonical form is: P = (W O)−1 where OT =

• ❈ T, ❆T❈ T, . . . (❆n−1)T❈ T

is the full rank observability matrix and W =        a1 a2 . . . an−1 1 a2 a3 . . . 1 . . . . . . . . . . . . . . . an−1 1 . . . 1 . . .       

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Design of state observers

State observer design

• Now, by substituting ˜

①(k) = P ˜ ③(k) also in the state observer dynamics given by: ˜ ①(k + 1) = ❆˜ ①(k) + ❇✉(k) + ❑ e(y(k) − ˜ y(k)) ˜ ②(k) = ❈ ˜ ①(k) (2) we obtain ˜ ③(k + 1) = P−1❆P˜ ③(k) + P−1❇✉(k) + P−1❑ e(y(k) − ˜ y(k)) ˜ ②(k) = ❈P˜ ③(k) (3)

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Design of state observers

State observer design

• The error dynamics of the estimated state ❡(k) = ③(k) − ˜

③(k) becomes ❡(k + 1) = ③(k + 1) − ˜ ③(k + 1) = P−1❆P③(k) + P−1❇✉(k) − P−1❆P˜ ③(k) − P−1❇✉(k) − P−1❑ e(②(k) − ˜ ②(k)) = P−1❆P③(k) + P−1❇✉(k) − P−1❆P˜ ③(k) − P−1❇✉(k) − P−1❑ e(❈P③(k) − ❈P˜ ③(k)) = P−1❆P(③(k) − ˜ ③(k)) − P−1❑ e❈P(③(k) − ˜ ③(k)) = P−1❆❡(k) − ❑ e❈P❡(k) =

• P−1❆P − P−1❑ e❈P
• ❡(k)

= P−1 (❆ − ❑ e❈) P❡(k) Thus, we need to design matrix Ke so that matrix (❆ − ❑ e❈) has the desired eigenvalues which determine the convergence rate of the state estimation error to zero.

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Design of state observers

State observer design

• The error dynamics of the estimated state ❡(k) = ③(k) − ˜

③(k) becomes ❡(k + 1) = ③(k + 1) − ˜ ③(k + 1) = P−1❆P③(k) + P−1❇✉(k) − P−1❆P˜ ③(k) − P−1❇✉(k) − P−1❑ e(②(k) − ˜ ②(k)) = P−1❆P③(k) + P−1❇✉(k) − P−1❆P˜ ③(k) − P−1❇✉(k) − P−1❑ e(❈P③(k) − ❈P˜ ③(k)) = P−1❆P(③(k) − ˜ ③(k)) − P−1❑ e❈P(③(k) − ˜ ③(k)) = P−1❆❡(k) − ❑ e❈P❡(k) =

• P−1❆P − P−1❑ e❈P
• ❡(k)

= P−1 (❆ − ❑ e❈) P❡(k) Thus, we need to design matrix Ke so that matrix (❆ − ❑ e❈) has the desired eigenvalues which determine the convergence rate of the state estimation error to zero.

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Design of state observers

State observer design

• Now, choose

ˆ Ke = P−1Ke =      δ0 δ1 . . . δn−1     

• It holds

P−1Ke❈P =      δ0 δ1 . . . δn−1      . . . 1 =        . . . δ0 . . . δ1 . . . . . . . . . . . . . . . δn−2 . . . δn−1       

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Design of state observers

State observer design

• Thus

P−1 (A − Ke❈) P = P−1AP − P−1Ke❈P =        . . . −a0 − δ0 1 . . . −a1 − δ1 . . . . . . . . . . . . . . . −an−2 − δn−2 . . . 1 −an−1 − δn−1        Thus matrix (A − Ke❈) due to the invariance of eigenvalues to similarity transformation has characteristic polynomial det(λ■ − A + Ke❈) = λn + (δn−1 + an−1)λn−1 + (δn−2 + an−2)λn−2 + (δn−3 + an−3)λn−3 + . . . + (δ1 + a1)λ + δ0 + a0

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Design of state observers

State observer design

Now, let the desired eigenvalues for matrix (A − Ke❈) be µ1, µ2, . . . , µn, thus the desired characteristic polynomial of matrix (A − Ke❈) is det(λ■ − A + Ke❈) = (λ − µ1)(λ − µ2)(λ − µ3) . . . (λ − µn) = λn + αn−1λn−1 + αn−2λn−2 . . . + α1λ + α0 Thus, equating the coefficients corresponding to terms of the same order of the desired characteristic polynomial and the characteristic polynomial of the observer error dynamics we get α0 = a0 + δ0 α1 = a1 + δ1 α2 = a2 + δ2 . . . = . . . αn−1 = an−2 + δn−2 αn = an−1 + δn−1

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Design of state observers

State observer design

By choosing δ0 = α0 − a0 δ2 = α1 − a1 δ3 = α2 − a2 . . . = . . . δn−2 = αn−2 − an−2 δn−1 = αn−1 − an−1 and recalling that P−1Ke =      δ0 δ1 . . . δn−1      it follows Ke = P      δ0 δ1 . . . δn−1     

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Design of state observers

State observer design

Thus, since the system is completely observable, matrix O is non-singular (and thus P is non-singular) and the observer feedback gain can be chosen as Ke = P      δ0 δ1 . . . δn−1      = (W O)−1      δ0 δ1 . . . δn−1      which assigns a characteristic polynomial with desired eigenvalues det(λ■ − ❆ + ❑ e❈) = (λ − µ1)(λ − µ2)(λ − µ3) . . . (λ − µn) to the state observer’s error dynamics ❆ − ❑ e❈.

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Outline

Design of state observers Design procedure Deadbeat response Separation principle Reduced/minimum order state observer

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Design procedure

Design procedure

• We now summarize the design steps for a full-order asymptotic state
• bserver

Consider the next discrete-time linear system: ①(k + 1) = ❆①(k) + ❇✉(k) ②(k + 1) = ❈①(k) where ❆ is an n × n matrix; ❇ is an n × r matrix; ❈ is an n × 1 matrix;

• Let matrix A have spectrum {λ1, λ2, . . . , λn}
• Let µ1, µ2, . . . , µn be the desired eigenvalues of the error dynamics of the state
• bserver.

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Design procedure

Design procedure

• We now summarize the design steps for a full-order asymptotic state
• bserver

Consider the next discrete-time linear system: ①(k + 1) = ❆①(k) + ❇✉(k) ②(k + 1) = ❈①(k) where ❆ is an n × n matrix; ❇ is an n × r matrix; ❈ is an n × 1 matrix;

• Let matrix A have spectrum {λ1, λ2, . . . , λn}
• Let µ1, µ2, . . . , µn be the desired eigenvalues of the error dynamics of the state
• bserver.

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Design procedure

Design procedure

Step 1: Verify that the observability matrix O is full rank, i.e., rank(OT) = rank(

• ❈ T, ❆T❈ T, . . . (❆n−1)T❈ T

) = n Step 2: Build matrix P = (W O)−1 where W =        a1 a2 . . . an−1 1 a2 a3 . . . 1 . . . . . . . . . . . . . . . an−1 1 . . . 1 . . .       

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Design procedure

Design procedure

Step 1: Verify that the observability matrix O is full rank, i.e., rank(OT) = rank(

• ❈ T, ❆T❈ T, . . . (❆n−1)T❈ T

) = n Step 2: Build matrix P = (W O)−1 where W =        a1 a2 . . . an−1 1 a2 a3 . . . 1 . . . . . . . . . . . . . . . an−1 1 . . . 1 . . .       

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Design procedure

Design procedure

Step 3: Compute the coefficients of the characteristic polynomial corresponding to matrix ❆ det(λ■ − ❆) = (λ − λ1)(λ − λ2)(λ − λ3) . . . (λ − λn) = λn + an−1λn−1 + an−2λn−2 . . . + a1λ + a0λ Step 4: Compute the characteristic polynomial corresponding to the desired eigenvalues of matrix ❆ − ❑ e❈: det(λ■ − ❆ + ❑ e❈) = (λ − µ1)(λ − µ2)(λ − µ3) . . . (λ − µn) = λn + αn−1λn−1 + αn−2λn−2 . . . + α1λ + α0

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Design procedure

Design procedure

Step 3: Compute the coefficients of the characteristic polynomial corresponding to matrix ❆ det(λ■ − ❆) = (λ − λ1)(λ − λ2)(λ − λ3) . . . (λ − λn) = λn + an−1λn−1 + an−2λn−2 . . . + a1λ + a0λ Step 4: Compute the characteristic polynomial corresponding to the desired eigenvalues of matrix ❆ − ❑ e❈: det(λ■ − ❆ + ❑ e❈) = (λ − µ1)(λ − µ2)(λ − µ3) . . . (λ − µn) = λn + αn−1λn−1 + αn−2λn−2 . . . + α1λ + α0

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Design procedure

Design procedure

Step 5: Compute the coefficients of matrix P−1Ke =      δ0 δ1 . . . δn−1      as δ0 = α0 − a0 δ2 = α1 − a1 δ3 = α2 − a2 . . . = . . . δn−2 = αn−2 − an−2 δn−1 = αn−1 − an−1

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Design procedure

Design procedure

Step 5: Compute the coefficients of matrix P−1Ke =      δ0 δ1 . . . δn−1      as δ0 = α0 − a0 δ2 = α1 − a1 δ3 = α2 − a2 . . . = . . . δn−2 = αn−2 − an−2 δn−1 = αn−1 − an−1

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Design procedure

Design procedure

Step 6: Compute matrix Ke = P      δ0 δ1 . . . δn−1      Step 7A:(Case with no observer-based full-state feedback). The state observer dynamics is ˜ ①(k + 1) = ❆˜ ①(k) + ❇✉(k) + ❑ e(y(k) − ˜ y(k)) ˜ ②(k) = ❈ ˜ ①(k) (4) The state observer state update equation to be implemented is ˜ ①(k + 1) = (❆ − KeC) ˜ ①(k) + ❇✉(k) + ❑ ey(k) (5)

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Design procedure

Design procedure

Step 6: Compute matrix Ke = P      δ0 δ1 . . . δn−1      Step 7A:(Case with no observer-based full-state feedback). The state observer dynamics is ˜ ①(k + 1) = ❆˜ ①(k) + ❇✉(k) + ❑ e(y(k) − ˜ y(k)) ˜ ②(k) = ❈ ˜ ①(k) (4) The state observer state update equation to be implemented is ˜ ①(k + 1) = (❆ − KeC) ˜ ①(k) + ❇✉(k) + ❑ ey(k) (5)

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Design procedure

Design procedure

Step 7B: (Case with observer-based full-state feedback, completely state controllable system). The state observer dynamics is ˜ ①(k + 1) = ❆˜ ①(k) − ❇❑ ˜ ①(k) + ❑ e(y(k) − ˜ y(k)) ˜ ②(k) = ❈ ˜ ①(k) (6) The state observer state update equation to be implemented is ˜ ①(k + 1) = (❆ − ❑ e❈ − ❇❑) ˜ ①(k) + ❑ ey(k) (7) The design is now complete.

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Design procedure

Design procedure

Step 7B: (Case with observer-based full-state feedback, completely state controllable system). The state observer dynamics is ˜ ①(k + 1) = ❆˜ ①(k) − ❇❑ ˜ ①(k) + ❑ e(y(k) − ˜ y(k)) ˜ ②(k) = ❈ ˜ ①(k) (6) The state observer state update equation to be implemented is ˜ ①(k + 1) = (❆ − ❑ e❈ − ❇❑) ˜ ①(k) + ❑ ey(k) (7) The design is now complete.

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Design procedure

Comments on the design procedure

• The state observer performance depends ultimately on the chosen feedback gain

Ke which weights the error between the measured output and the estimation

• utput by the observer.
• Matrix Ke depend upon the choice of desired eigenvalues µ1, µ2, . . . , µn.
• A choice of eigenvalues small in magnitude correspond to a large gain Ke which

amplifies the error signal.

• If there is a large measurement noise or the model is not accurate, then there is

a trade-off: A choice of of eigenvalues with larger magnitude produces a design which is more robust with respect to noise, disturbances or model uncertainties.

• In practice, one should design several feedback gain matrices and test them

numerically against noise and model uncertainties for better design robustness.

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Design procedure

Comments on the design procedure

• The state observer performance depends ultimately on the chosen feedback gain

Ke which weights the error between the measured output and the estimation

• utput by the observer.
• Matrix Ke depend upon the choice of desired eigenvalues µ1, µ2, . . . , µn.
• A choice of eigenvalues small in magnitude correspond to a large gain Ke which

amplifies the error signal.

• If there is a large measurement noise or the model is not accurate, then there is

a trade-off: A choice of of eigenvalues with larger magnitude produces a design which is more robust with respect to noise, disturbances or model uncertainties.

• In practice, one should design several feedback gain matrices and test them

numerically against noise and model uncertainties for better design robustness.

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Design procedure

Comments on the design procedure

• The state observer performance depends ultimately on the chosen feedback gain

Ke which weights the error between the measured output and the estimation

• utput by the observer.
• Matrix Ke depend upon the choice of desired eigenvalues µ1, µ2, . . . , µn.
• A choice of eigenvalues small in magnitude correspond to a large gain Ke which

amplifies the error signal.

• If there is a large measurement noise or the model is not accurate, then there is

a trade-off: A choice of of eigenvalues with larger magnitude produces a design which is more robust with respect to noise, disturbances or model uncertainties.

• In practice, one should design several feedback gain matrices and test them

numerically against noise and model uncertainties for better design robustness.

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Design procedure

Comments on the design procedure

• The state observer performance depends ultimately on the chosen feedback gain

Ke which weights the error between the measured output and the estimation

• utput by the observer.
• Matrix Ke depend upon the choice of desired eigenvalues µ1, µ2, . . . , µn.
• A choice of eigenvalues small in magnitude correspond to a large gain Ke which

amplifies the error signal.

• If there is a large measurement noise or the model is not accurate, then there is

a trade-off: A choice of of eigenvalues with larger magnitude produces a design which is more robust with respect to noise, disturbances or model uncertainties.

• In practice, one should design several feedback gain matrices and test them

numerically against noise and model uncertainties for better design robustness.

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Design procedure

Comments on the design procedure

• The state observer performance depends ultimately on the chosen feedback gain

Ke which weights the error between the measured output and the estimation

• utput by the observer.
• Matrix Ke depend upon the choice of desired eigenvalues µ1, µ2, . . . , µn.
• A choice of eigenvalues small in magnitude correspond to a large gain Ke which

amplifies the error signal.

• If there is a large measurement noise or the model is not accurate, then there is

a trade-off: A choice of of eigenvalues with larger magnitude produces a design which is more robust with respect to noise, disturbances or model uncertainties.

• In practice, one should design several feedback gain matrices and test them

numerically against noise and model uncertainties for better design robustness.

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Outline

Design of state observers Design procedure Deadbeat response Separation principle Reduced/minimum order state observer

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Deadbeat response

Deadbeat response

• Also in state observers we can design a deadbeat response, i.e., an error

dynamics which converges to zero as fast as possible which means in at most n steps (completely observable system)

• Simply choose the desired eigenvalue assignment for the state observer error

dynamics as {0, 0, . . . , 0}, i.e., all zero eigenvalues.

• Apply the full-order state observer design procedure
• It will follow that the natural response of the state observer vanishes in at most

n steps (❆ − ❑ e❈)n =        . . . . . . . . . . . . . . . . . . 0, . . . . . . ,       

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Deadbeat response

Deadbeat response

• Also in state observers we can design a deadbeat response, i.e., an error

dynamics which converges to zero as fast as possible which means in at most n steps (completely observable system)

• Simply choose the desired eigenvalue assignment for the state observer error

dynamics as {0, 0, . . . , 0}, i.e., all zero eigenvalues.

• Apply the full-order state observer design procedure
• It will follow that the natural response of the state observer vanishes in at most

n steps (❆ − ❑ e❈)n =        . . . . . . . . . . . . . . . . . . 0, . . . . . . ,       

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Deadbeat response

Deadbeat response

• Also in state observers we can design a deadbeat response, i.e., an error

dynamics which converges to zero as fast as possible which means in at most n steps (completely observable system)

• Simply choose the desired eigenvalue assignment for the state observer error

dynamics as {0, 0, . . . , 0}, i.e., all zero eigenvalues.

• Apply the full-order state observer design procedure
• It will follow that the natural response of the state observer vanishes in at most

n steps (❆ − ❑ e❈)n =        . . . . . . . . . . . . . . . . . . 0, . . . . . . ,       

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Deadbeat response

Deadbeat response

• Also in state observers we can design a deadbeat response, i.e., an error

dynamics which converges to zero as fast as possible which means in at most n steps (completely observable system)

• Simply choose the desired eigenvalue assignment for the state observer error

dynamics as {0, 0, . . . , 0}, i.e., all zero eigenvalues.

• Apply the full-order state observer design procedure
• It will follow that the natural response of the state observer vanishes in at most

n steps (❆ − ❑ e❈)n =        . . . . . . . . . . . . . . . . . . 0, . . . . . . ,       

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Deadbeat response

Deadbeat response

• The deadbeat response is the fastest achievable by a state observer
• There is a trade-off: To bring the state estimation error to zero as fast as

possible the output measurement errors need to be sufficiently small, otherwise the estimation error could be amplified.

• Choosing eigenvalues small in magnitude but greater than zeros ameliorates the

effect of noise on estimation errors.

• If the available model is not accurate, the deadbeat response will also contain

estimation errors.

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Outline

Design of state observers Design procedure Deadbeat response Separation principle Reduced/minimum order state observer

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Separation principle

The separation principle

• So far we have established that the error dynamics of the state observer is not

affected by a full-state observer-based feedback.

• Is the designed full-state feedback affected by the choices we make in the
• bserver design process? short answer: no, but only for linear systems

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Separation principle

The separation principle

• So far we have established that the error dynamics of the state observer is not

affected by a full-state observer-based feedback.

• Is the designed full-state feedback affected by the choices we make in the
• bserver design process? short answer: no, but only for linear systems

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Separation principle

The separation principle

Consider the next discrete-time linear system: ①(k + 1) = ❆①(k) + ❇✉(k) ②(k + 1) = ❈①(k) where ❆ is an n × n matrix; ❇ is an n × r matrix; ❈ is an n × p matrix;

• Assume that the system is completely observable and completely state

controllable

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Separation principle

The separation principle

• Let the system be controlled by a full-state observer-based feedback

✉(k) = −❑ ˜ ①(k)

• The state update equation becomes

①(k + 1) = ❆①(k) − ❇❑ ˜ ①(k)

• If we add and subtract −❇❑①(k), it holds

①(k + 1) = ❆①(k) − ❇❑①(k) + ❇❑①(k) − ❇❑ ˜ ①(k) = ❆①(k) − ❇❑①(k) + ❇❑ (①(k) − ˜ ①(k))

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Separation principle

The separation principle

• Let the system be controlled by a full-state observer-based feedback

✉(k) = −❑ ˜ ①(k)

• The state update equation becomes

①(k + 1) = ❆①(k) − ❇❑ ˜ ①(k)

• If we add and subtract −❇❑①(k), it holds

①(k + 1) = ❆①(k) − ❇❑①(k) + ❇❑①(k) − ❇❑ ˜ ①(k) = ❆①(k) − ❇❑①(k) + ❇❑ (①(k) − ˜ ①(k))

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Separation principle

The separation principle

• Let the system be controlled by a full-state observer-based feedback

✉(k) = −❑ ˜ ①(k)

• The state update equation becomes

①(k + 1) = ❆①(k) − ❇❑ ˜ ①(k)

• If we add and subtract −❇❑①(k), it holds

①(k + 1) = ❆①(k) − ❇❑①(k) + ❇❑①(k) − ❇❑ ˜ ①(k) = ❆①(k) − ❇❑①(k) + ❇❑ (①(k) − ˜ ①(k))

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Separation principle

The separation principle

• Thus, by substituting ❡(k) = ①(k) − ˜

①(k) ①(k + 1) = ❆①(k) − ❇❑①(k) + ❇❑①(k) − ❇❑ ˜ ①(k) = ❆①(k) − ❇❑①(k) + ❇❑❡(k)

• Recall that the state observer error dynamics is

❡(k + 1) = (❆ − ❑ e❈) ❡(k)

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Separation principle

The separation principle

• It follows that

①(k + 1) ❡(k + 1)

• =

❆ − ❇❑ ❇❑ 0n×n ❆ − ❑ e❈ ①(k) ❡(k)

• Since the state update matrix is upper block triangular, the eigenvalues of the

matrix are equal to the eigenvalues of the matrix blocks in the diagonal, i.e., (❆ − ❇❑) and (❆ − ❑ e❈).

• It follows that we can design separately the full-state feedback and state
• bserver and assign arbitrarily their eigenvalues if the system is completely state

controllable and completely observable. ⇒ This fact is known as the separation principle

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Separation principle

The separation principle

• By the separation principle, we can design a full-state feedback as if the full

state is accessible and assign the desired eigenvalues based solely on specifications

• n transient and steady-state behavior of the linear dynamical system
• Then, we can design a state observer and assign its eigenvalues which enable

the fastest convergence to zero of the error dynamics while considering the effect

• f noise and disturbances on estimation errors
• Note that the separation principle holds only if the dynamical system is linear
• The separation principle can be applied on a linearized system but the system

trajectory of both the observer and the system needs to be close to the considered equilibrium point, i.e., the approach is sensitive to initialization errors.

• If the system is nonlinear, one needs to first design the feedback control, then

design an the observer and finally identify a Lyapunov function which guarantees the stability of the closed-loop autonomous system.

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Separation principle

The separation principle

• By the separation principle, we can design a full-state feedback as if the full

state is accessible and assign the desired eigenvalues based solely on specifications

• n transient and steady-state behavior of the linear dynamical system
• Then, we can design a state observer and assign its eigenvalues which enable

the fastest convergence to zero of the error dynamics while considering the effect

• f noise and disturbances on estimation errors
• Note that the separation principle holds only if the dynamical system is linear
• The separation principle can be applied on a linearized system but the system

trajectory of both the observer and the system needs to be close to the considered equilibrium point, i.e., the approach is sensitive to initialization errors.

• If the system is nonlinear, one needs to first design the feedback control, then

design an the observer and finally identify a Lyapunov function which guarantees the stability of the closed-loop autonomous system.

35 / 45

Separation principle

The separation principle

• By the separation principle, we can design a full-state feedback as if the full

state is accessible and assign the desired eigenvalues based solely on specifications

• n transient and steady-state behavior of the linear dynamical system
• Then, we can design a state observer and assign its eigenvalues which enable

the fastest convergence to zero of the error dynamics while considering the effect

• f noise and disturbances on estimation errors
• Note that the separation principle holds only if the dynamical system is linear
• The separation principle can be applied on a linearized system but the system

trajectory of both the observer and the system needs to be close to the considered equilibrium point, i.e., the approach is sensitive to initialization errors.

• If the system is nonlinear, one needs to first design the feedback control, then

design an the observer and finally identify a Lyapunov function which guarantees the stability of the closed-loop autonomous system.

35 / 45

Separation principle

The separation principle

• By the separation principle, we can design a full-state feedback as if the full

state is accessible and assign the desired eigenvalues based solely on specifications

• n transient and steady-state behavior of the linear dynamical system
• Then, we can design a state observer and assign its eigenvalues which enable

the fastest convergence to zero of the error dynamics while considering the effect

• f noise and disturbances on estimation errors
• Note that the separation principle holds only if the dynamical system is linear
• The separation principle can be applied on a linearized system but the system

trajectory of both the observer and the system needs to be close to the considered equilibrium point, i.e., the approach is sensitive to initialization errors.

• If the system is nonlinear, one needs to first design the feedback control, then

design an the observer and finally identify a Lyapunov function which guarantees the stability of the closed-loop autonomous system.

35 / 45

Separation principle

The separation principle

• By the separation principle, we can design a full-state feedback as if the full

state is accessible and assign the desired eigenvalues based solely on specifications

• n transient and steady-state behavior of the linear dynamical system
• Then, we can design a state observer and assign its eigenvalues which enable

the fastest convergence to zero of the error dynamics while considering the effect

• f noise and disturbances on estimation errors
• Note that the separation principle holds only if the dynamical system is linear
• The separation principle can be applied on a linearized system but the system

trajectory of both the observer and the system needs to be close to the considered equilibrium point, i.e., the approach is sensitive to initialization errors.

• If the system is nonlinear, one needs to first design the feedback control, then

design an the observer and finally identify a Lyapunov function which guarantees the stability of the closed-loop autonomous system.

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Outline

Design of state observers Design procedure Deadbeat response Separation principle Reduced/minimum order state observer

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Reduced/minimum order state observer

Reduced/minimum order state observer

• One drawback of the full-order observer is that the full-state of the system is

estimated from its output even if some of its state variables can be measured directly.

• If the measurement process is accurate, noise is low, the addition of the

necessary sensors is acceptable in the design, then measuring directly the state variables which can be measured and estimate the others can provide greater

• performance. If noise is high, it is preferable to use a full-order state observer.
• In such scenario, we can design a reduced order state observer, i.e., a state
• bserver with a number of state variables less then those of the dynamical system.
• Clearly, if the we can measure m out of n state variables, then a state observer
• f order n − m is said to be a minimum order state observer.

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Reduced/minimum order state observer

Reduced/minimum order state observer

Schematic of a reduced/minimum order observer

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Reduced/minimum order state observer

Reduced/minimum order state observer

• Let the state of the system, ①(k) be partitioned such that

①(k)T =

• xa(k)T

xb(k)TT where xa(k) is the part of the state space with m elements which can be measured directly and xb(k) is the part of the state space with n − m elements which needs to be estimated.

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Reduced/minimum order state observer

Reduced/minimum order state observer

• Thus, the system dynamics is

①a(k + 1) ①b(k + 1)

• =

❆aa ❆ab ❆ba ❆bb ①a(k) ①b(k)

• +

❇a ❇b

• ✉(k)

where ❆aa is a m × m matrix; ❆ab is a m × n − m matrix; ❆ba is a n − m × m matrix; ❆bb is a n − m × n − m matrix; ❇a is a m × r matrix; ❇a is a n − m × r matrix;

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Reduced/minimum order state observer

Reduced/minimum order state observer

• The equation related to the measurable variables is

①a(k + 1) = ❆aa①a(k) + ❆abxb(k) + ❇a✉(k) thus ❆abxb(k) = ①a(k + 1) − ❆aa①a(k) − ❇a✉(k) acts as an ”output equation”. (the right-hand side contains measured quantities)

• The part of the state which needs to be estimated evolves according to

①b(k + 1) = ❆bbxb(k) + ❆ba①a(k) + ❇b✉(k)

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Reduced/minimum order state observer

Reduced/minimum order state observer

• The equation related to the measurable variables is

①a(k + 1) = ❆aa①a(k) + ❆abxb(k) + ❇a✉(k) thus ❆abxb(k) = ①a(k + 1) − ❆aa①a(k) − ❇a✉(k) acts as an ”output equation”. (the right-hand side contains measured quantities)

• The part of the state which needs to be estimated evolves according to

①b(k + 1) = ❆bbxb(k) + ❆ba①a(k) + ❇b✉(k)

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Reduced/minimum order state observer

Reduced/minimum order state observer

• The design of the minimum order observer can be carried out using the design

procedure of the full-order observer by making the next substitutions to the

• bserver equation:

˜ ①(k) → ˜ ①b(k) ❆ → ❆bb ❇✉(k) → ❆ba①a(k) + ❇b✉(k) ②(k) → ①a(k + 1) − ❆aa①a(k) − ❇a✉(k) ❈ → ❆ab ❑ e n × m matrix → ❑ e n − m × m matrix

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Reduced/minimum order state observer

Reduced/minumum order state observer

• Full-order state observer equation

˜ ①(k + 1) = (❆ − KeC) ˜ ①(k) + ❇✉(k) + ❑ ey(k)

• Minimum-order state observer equation

˜ ①b(k + 1) = (❆bb − Ke❆ab) ˜ ①b(k) + ❆ba①a(k) + ❇b✉(k) + ❑ e(①a(k + 1) − ❆aa①a(k) − ❇a✉(k))

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Reduced/minimum order state observer

Exercise

Consider the discrete-time linear dynamical system in state space form shown next (see Assignment 3 2020). x(k + 1) =Ax(k) + Bu(k) y(k) =Cx(k) (8) where x ∈ R4, y ∈ R, u ∈ R and A =     0.8 0.1 0.1 0.1 0.8 0.1 0.1 0.8 0.1 0.1 0.1 0.8     , B =     1 0.3     , C = 1 .

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Reduced/minimum order state observer

Exercise

(a) Determine the matrix gain of a full state feedback control such that the

desired closed loop eigenvalues are: 0.8, 0.6, 0.5, 0.1.

(b) Suppose that u(k) = −Kx(k) + K0r(k) where r(k) ∈ R is a scalar reference

signal and K0 is scalar gain. Choose the gain K0 so that, for a constant reference signal, at the steady state it holds y = r

(c) Design a full-order asimptotic (Lueberger) state observer with eigenvalues

0.1, 0.1, 0.1, 0.1

(d) Is the closed-loop system with full-state observer based feedback stable?

Which are its eigenvalues?

(e) Simulate with Matlab simulink the the system under full-state observer-based

• feedback. Set the initial state of the observer to a zero vector.

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