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MSc in Computer Engineering, Cybersecurity and Artificial Intelligence Course FDE , a.a. 2019/2020, Lecture 13 State feedback control design by eigenvalue assignment

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

University of Cagliari, Italy

Monday, 27th April 2020

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Outline

Introduction A necessary and sufficient condition for state feedback design for arbitrary eigenvalue assignment Design procedure Design with reference input

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Introduction

Introduction

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

• Matlab Simulink block diagram (open-loop system dynamics):

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Introduction

Introduction

We consider a discrete time linear model which could be: An intrinsic discrete-time system. Example: Dynamics of a bank savings account with fixed interest rates; A discretized linear system: A continuous-time linear system which is measured and actuated at a given sampling rate. Example: DC electric motor; An intrinsic discrete time nonlinear system linearized around an equilibrium/operating point. A continuous time nonlinear system linearized around an equilibrium/operating point and discretized.

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Introduction

Introduction

We consider a discrete time linear model which could be: An intrinsic discrete-time system. Example: Dynamics of a bank savings account with fixed interest rates; A discretized linear system: A continuous-time linear system which is measured and actuated at a given sampling rate. Example: DC electric motor; An intrinsic discrete time nonlinear system linearized around an equilibrium/operating point. A continuous time nonlinear system linearized around an equilibrium/operating point and discretized.

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Introduction

Introduction

We consider a discrete time linear model which could be: An intrinsic discrete-time system. Example: Dynamics of a bank savings account with fixed interest rates; A discretized linear system: A continuous-time linear system which is measured and actuated at a given sampling rate. Example: DC electric motor; An intrinsic discrete time nonlinear system linearized around an equilibrium/operating point. A continuous time nonlinear system linearized around an equilibrium/operating point and discretized.

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Introduction

Introduction

We consider a discrete time linear model which could be: An intrinsic discrete-time system. Example: Dynamics of a bank savings account with fixed interest rates; A discretized linear system: A continuous-time linear system which is measured and actuated at a given sampling rate. Example: DC electric motor; An intrinsic discrete time nonlinear system linearized around an equilibrium/operating point. A continuous time nonlinear system linearized around an equilibrium/operating point and discretized.

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Introduction

Assumptions and main objective

• We now assume to have full access to the state of the system
• In particular, all state variables are measured at discrete intervals of time
• Our goal is to design a proportional, full state feedback control u(k) = −Kx(k)

where K is a row-vector to modify the stability properties of the system (or just an equilibrium point if it is a linearized model)

• The method allows to tailor the transient response of the system by assigning

desired eigenvalues to the closed-loop system, i.e., assignment of desired time-constants to the modes of the system.

• The method we are about to present can be easily generalized for

multi-input/multi-output systems where K is, in general a r × n rectangular matrix

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Introduction

Assumptions and main objective

• We now assume to have full access to the state of the system
• In particular, all state variables are measured at discrete intervals of time
• Our goal is to design a proportional, full state feedback control u(k) = −Kx(k)

where K is a row-vector to modify the stability properties of the system (or just an equilibrium point if it is a linearized model)

• The method allows to tailor the transient response of the system by assigning

desired eigenvalues to the closed-loop system, i.e., assignment of desired time-constants to the modes of the system.

• The method we are about to present can be easily generalized for

multi-input/multi-output systems where K is, in general a r × n rectangular matrix

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Introduction

Assumptions and main objective

• We now assume to have full access to the state of the system
• In particular, all state variables are measured at discrete intervals of time
• Our goal is to design a proportional, full state feedback control u(k) = −Kx(k)

where K is a row-vector to modify the stability properties of the system (or just an equilibrium point if it is a linearized model)

• The method allows to tailor the transient response of the system by assigning

desired eigenvalues to the closed-loop system, i.e., assignment of desired time-constants to the modes of the system.

• The method we are about to present can be easily generalized for

multi-input/multi-output systems where K is, in general a r × n rectangular matrix

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Introduction

Assumptions and main objective

• We now assume to have full access to the state of the system
• In particular, all state variables are measured at discrete intervals of time
• Our goal is to design a proportional, full state feedback control u(k) = −Kx(k)

where K is a row-vector to modify the stability properties of the system (or just an equilibrium point if it is a linearized model)

• The method allows to tailor the transient response of the system by assigning

desired eigenvalues to the closed-loop system, i.e., assignment of desired time-constants to the modes of the system.

• The method we are about to present can be easily generalized for

multi-input/multi-output systems where K is, in general a r × n rectangular matrix

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Introduction

Assumptions and main objective

• We now assume to have full access to the state of the system
• In particular, all state variables are measured at discrete intervals of time
• Our goal is to design a proportional, full state feedback control u(k) = −Kx(k)

where K is a row-vector to modify the stability properties of the system (or just an equilibrium point if it is a linearized model)

• The method allows to tailor the transient response of the system by assigning

desired eigenvalues to the closed-loop system, i.e., assignment of desired time-constants to the modes of the system.

• The method we are about to present can be easily generalized for

multi-input/multi-output systems where K is, in general a r × n rectangular matrix

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Introduction

Introduction

Graphical representation of the closed-loop system with full-state feedback control

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Introduction

Why proportional state feedback

• Proportional feedback is a popular and robust control method, very

simple to implement on embedded digital systems.

• To control a dynamical system by digital means the main limiting factor

is the sampling rate.

• If the sampling rate is too high, i.e., the sampling interval is too short,

then the embedded system might not be fast enough to perform analogue to digital conversion, process the signal by computing the control u(k) and convert back with digital to analogue conversion.

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Introduction

Why proportional state feedback

• Proportional feedback is a popular and robust control method, very

simple to implement on embedded digital systems.

• To control a dynamical system by digital means the main limiting factor

is the sampling rate.

• If the sampling rate is too high, i.e., the sampling interval is too short,

then the embedded system might not be fast enough to perform analogue to digital conversion, process the signal by computing the control u(k) and convert back with digital to analogue conversion.

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Introduction

Why proportional state feedback

• Proportional feedback is a popular and robust control method, very

simple to implement on embedded digital systems.

• To control a dynamical system by digital means the main limiting factor

is the sampling rate.

• If the sampling rate is too high, i.e., the sampling interval is too short,

then the embedded system might not be fast enough to perform analogue to digital conversion, process the signal by computing the control u(k) and convert back with digital to analogue conversion.

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Outline

Introduction A necessary and sufficient condition for full state feedback design for arbitrary eigenvalue assignment Design procedure Design with reference input

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Full state feedback design

State feedback design

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

• Let matrix A have spectrum σol = {λ1, λ2, . . . , λn}

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Full state feedback design

State feedback design

• Consider now a full-state feedback u(k) = −K①(k) with K a 1 × n matrix

(single input case). The dynamics of system (1) become ①(k + 1) = ❆①(k) + ❇u(k) = ❆①(k) + ❇(−K①(k)) = (❆ − ❇❑) ①(k) (2)

• The state transition matrix is now ❆ − ❇❑ and has spectrum

σcl = {µ1, µ2, . . . , µn} ⇒ state feedback changes the stability properties of a system.

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Full state feedback design

State feedback design: A necessary and sufficient condition

• Under what conditions can we design matrix K such that the spectrum of

matrix (❆ − ❇❑) has the desired eigenvalues σcl = {µ1, µ2, . . . , µn}?

• A necessary and sufficient condition for the existence of a matrix K which

allows to assign arbitrary eigenvalues to matrix (❆ − ❇❑), assuming unbounded control input, is that the system defined by the pair (❆, ❇) is completely state controllable.

• A discrete time linear system is completely state controllable if its controllability

matrix T =

• ❇, ❆❇, . . . ❆n−1❇
• has full rank, i.e., rank(T ) = n.

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Full state feedback design

State feedback design: A necessary and sufficient condition

• Under what conditions can we design matrix K such that the spectrum of

matrix (❆ − ❇❑) has the desired eigenvalues σcl = {µ1, µ2, . . . , µn}?

• A necessary and sufficient condition for the existence of a matrix K which

allows to assign arbitrary eigenvalues to matrix (❆ − ❇❑), assuming unbounded control input, is that the system defined by the pair (❆, ❇) is completely state controllable.

• A discrete time linear system is completely state controllable if its controllability

matrix T =

• ❇, ❆❇, . . . ❆n−1❇
• has full rank, i.e., rank(T ) = n.

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Full state feedback design

Proof of sufficiency

• We now prove sufficiency of the condition, a proof of necessity can be found in
• K. Ogata ”Discrete time control systems”, Prentice-Hall int., Chapter 6.
• Let {µ1, µ2, . . . µn} be a set of desired complex eigenvalues for matrix

(❆ − ❇❑).

• Consider a similarity transformation P which puts the system into controllable

canonical form

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Full state feedback design

Proof of sufficiency

• We now prove sufficiency of the condition, a proof of necessity can be found in
• K. Ogata ”Discrete time control systems”, Prentice-Hall int., Chapter 6.
• Let {µ1, µ2, . . . µn} be a set of desired complex eigenvalues for matrix

(❆ − ❇❑).

• Consider a similarity transformation P which puts the system into controllable

canonical form

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Full state feedback design

Proof of sufficiency

• We now prove sufficiency of the condition, a proof of necessity can be found in
• K. Ogata ”Discrete time control systems”, Prentice-Hall int., Chapter 6.
• Let {µ1, µ2, . . . µn} be a set of desired complex eigenvalues for matrix

(❆ − ❇❑).

• Consider a similarity transformation P which puts the system into controllable

canonical form

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Full state feedback design

Proof of sufficiency

Thus P = T W where T =

• ❇, ❆❇, . . . ❆n−1❇
• is the full rank controllability matrix and

W =        a1 a2 . . . an−1 1 a2 a3 . . . 1 . . . . . . . . . . . . . . . an−1 1 . . . 1 . . .        where a0, . . . , an are the coefficients of the characteristic polynomial of matrix ❆ det(λI − ❆) = λn + an−1λn−1 + . . . + a2λ2 + a1λ + a0

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Full state feedback design

Proof of sufficiency

By computing the new equivalent model with state variables ①(k) = P③(k), we have that the new matrices are in controllable canonical form ˆ ❆ = P−1❆P =          1 . . . 1 . . . . . . . . . . . . . . . ... . . . . . . . . . 1 −a0 −a1 −a2 . . . −an−2 −an−1          ˆ ❇ = P−1❇ =        . . . 1       

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Full state feedback design

Proof of sufficiency

• Now, choose

ˆ K = KP = δn−1 δn−2, . . . , δ0

• The equivalent system is

③(k + 1) = ˆ ❆③(k) − ˆ ❇ ˆ K③(k) =

• ˆ

❆ − ˆ ❇ ˆ K

• ③(k)

(3) with ˆ ❇ ˆ K =        . . . 1        δn−1 δn−2, . . . , δ0

• =

       . . . . . . . . . . . ., . . . . . . 0, . . . , δn−1 δn−2, . . . , δ0       

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Full state feedback design

Proof of sufficiency

Thus, ˆ ❆ − ˆ ❇ ˆ K =        1 . . . . . . . . . . . . . . . . . . 0, . . . 1 −an−1 −an−2 . . . −a0        −        . . . . . . . . . . . . . . . . . . 0, . . . δn−1 δn−2, . . . δ0        ˆ ❆ − ˆ ❇ ˆ K =        1 . . . . . . . . . . . . . . . . . . 0, . . . 1 −an−1 − δn−1 −an−2 − δn−2 . . . , −a0 − δ0       

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Full state feedback design

Proof of sufficiency

The characteristic polynomial of ˆ A − ˆ ❇ ˆ K is det(λ■ − ˆ A+ ˆ ❇ ˆ K) = det               λ −1 . . . λ . . . . . . . . ., . . . . . . 0, . . . , −1 an−1 + δn−1 an−2 + δn−1, . . . , λ + a0 + δ0               which is equal to det(λ■ − ˆ A + ˆ ❇ ˆ K) = 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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Full state feedback design

Proof of sufficiency

Now, the desired eignevalues for matrix ˆ ❆ − ˆ ❇ ˆ K are µ1, µ2, . . . , µn, thus the desired characteristic polynomial of matrix ˆ A − ˆ ❇ ˆ K is det(λ■ − ˆ ❆ + ˆ ❇ ˆ K) = (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 closed loop state transition matrix ˆ ❆ − ˆ ❇ ˆ K) 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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Full state feedback design

Proof of sufficiency

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 ˆ K = KP = δn−1 δn−2, . . . , δ0

• it follows

K = ˆ KP−1 =

• δn−1

δn−2, . . . , δ0

• P−1

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Full state feedback design

Proof of sufficiency

Thus, if matrix T is non-singular (and thus P is non-singular), i.e., the system is completely state controllable, then we just constructed a feedback matrix gain K = δn−1 δn−2, . . . , δ0

• P−1

which assigns a characteristic polynomial with desired eigenvalues det(λ■ − ˆ ❆ + ˆ ❇ ˆ K) = (λ − µ1)(λ − µ2)(λ − µ3) . . . (λ − µn) to the closed loop state transition matrix ˆ ❆ − ˆ ❇ ˆ K, thus proving that compelte state controllability is a sufficient condition for the existence of matrix K for an arbitrary eigenvalue assignment.

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Outline

Introduction A necessary and sufficient condition for full state feedback design for arbitrary eigenvalue assignment Full state feedback Design procedure Design with reference input

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Full state feedback Design procedure

Design procedure

• There are many ways to design a feedback matrix gain K which assigns

the desired eigevalues to a closed loop system.

• Here we propose one the simple and systematic method to assign

eigenvalues to single input systems.

• A generalization of this method for multi-input and multi-output systems

can be found in K. Ogata ”Discrete time control systems”, Prentice-Hall int., Appendix C.

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Full state feedback Design procedure

Design procedure

• There are many ways to design a feedback matrix gain K which assigns

the desired eigevalues to a closed loop system.

• Here we propose one the simple and systematic method to assign

eigenvalues to single input systems.

• A generalization of this method for multi-input and multi-output systems

can be found in K. Ogata ”Discrete time control systems”, Prentice-Hall int., Appendix C.

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Full state feedback Design procedure

Design procedure

• There are many ways to design a feedback matrix gain K which assigns

the desired eigevalues to a closed loop system.

• Here we propose one the simple and systematic method to assign

eigenvalues to single input systems.

• A generalization of this method for multi-input and multi-output systems

can be found in K. Ogata ”Discrete time control systems”, Prentice-Hall int., Appendix C.

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Full state feedback Design procedure

Design procedure

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

• Let λ1, λ2, . . . , λn be the eigenvalues of matrix A.
• Let µ1, µ2, . . . , µn be the desired eigenvalues of the closed loop system

under full state feedback control u(k) = −K①(k).

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Full state feedback Design procedure

Design procedure

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

• Let λ1, λ2, . . . , λn be the eigenvalues of matrix A.
• Let µ1, µ2, . . . , µn be the desired eigenvalues of the closed loop system

under full state feedback control u(k) = −K①(k).

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Full state feedback Design procedure

Design procedure

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

• Let λ1, λ2, . . . , λn be the eigenvalues of matrix A.
• Let µ1, µ2, . . . , µn be the desired eigenvalues of the closed loop system

under full state feedback control u(k) = −K①(k).

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Full state feedback Design procedure

Design procedure

Step 1: Verify that the controllability matrix T is full rank, i.e., rank(T ) = rank(

• ❇, ❆❇, . . . ❆n−1❇
• ) = n

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

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Full state feedback Design procedure

Design procedure

Step 1: Verify that the controllability matrix T is full rank, i.e., rank(T ) = rank(

• ❇, ❆❇, . . . ❆n−1❇
• ) = n

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

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Full state feedback 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 ❆ − ❇❑: det(λ■ − ❆ + ❇K) = (λ − µ1)(λ − µ2)(λ − µ3) . . . (λ − µn) = λn + αn−1λn−1 + αn−2λn−2 . . . + α1λ + α0

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Full state feedback 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 ❆ − ❇❑: det(λ■ − ❆ + ❇K) = (λ − µ1)(λ − µ2)(λ − µ3) . . . (λ − µn) = λn + αn−1λn−1 + αn−2λn−2 . . . + α1λ + α0

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Full state feedback Design procedure

Design procedure

Step 5: Compute the coefficients of matrix ˆ K =

• δn−1

δn−2, . . . , δ0

• 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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Full state feedback Design procedure

Design procedure

Step 5: Compute the coefficients of matrix ˆ K =

• δn−1

δn−2, . . . , δ0

• 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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Full state feedback Design procedure

Design procedure

Step 6: Compute matrix K = ˆ KP−1 Step 7: Let u(k) = −K①(k) The design is now complete.

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Full state feedback Design procedure

Design procedure

Step 6: Compute matrix K = ˆ KP−1 Step 7: Let u(k) = −K①(k) The design is now complete.

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Full state feedback Design procedure

Design procedure

Step 6: Compute matrix K = ˆ KP−1 Step 7: Let u(k) = −K①(k) The design is now complete.

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Full state feedback Design procedure

Important remark

• In the design of control system for a discretized model, the design is

dependent on the sampling time T chosen to sample the continuous-time system.

• A change of sampling time changes the eigenvalues of the discrete-time

model.

• Thus, a change in sampling time requires a redesign of the feedback

matrix gain K and, eventually K0.

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Full state feedback Design procedure

Important remark

• In the design of control system for a discretized model, the design is

dependent on the sampling time T chosen to sample the continuous-time system.

• A change of sampling time changes the eigenvalues of the discrete-time

model.

• Thus, a change in sampling time requires a redesign of the feedback

matrix gain K and, eventually K0.

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Full state feedback Design procedure

Important remark

• In the design of control system for a discretized model, the design is

dependent on the sampling time T chosen to sample the continuous-time system.

• A change of sampling time changes the eigenvalues of the discrete-time

model.

• Thus, a change in sampling time requires a redesign of the feedback

matrix gain K and, eventually K0.

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Outline

Introduction A necessary and sufficient condition for full state feedback design for arbitrary eigenvalue assignment Full state feedback Design procedure Design with reference input

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Design with reference input

Design with reference input

• In practical applications we often wish to set a given operating point for a

plant/process/machine by providing a reference signal (for instance, a reference speed or temperature). Consider the next Simulink block diagram

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Design with reference input

Design with reference input

• The input to the dynamical system under full state feedback is now

u(k) = −K①(k) + K0r(k), where r(k) is the reference signal and K0 is the desired gain with respect to the reference.

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Design with reference input

Design with reference input

• Example 1: when controlling a crane with a joystick we provide a reference with

the angular position of the stick which translates in a large movement speed of the crane. (We wish y(k) = βr(k))

• Example 2: when controlling the crane by setting numerically its speed, we wish

to have a unitary gain, i.e., the system tracks the reference signal with no gain (We wish y(k) = r(k))

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Design with reference input

Design with reference input

• Important: Full state feedback changes the stability properties of a system and

its transient behavior by changing its eigenvalues but it also changes the gain of a system.

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Design with reference input

Design with reference input

Consider the next discrete-time linear system: ①(k + 1) = ❆①(k) + ❇✉(k) y(k) = ❈①(k) where ❆ is an n × n matrix; ❇ is an n × 1 matrix; ❈ is a 1 × n matrix. Now, let u(k) = −K①(k) + K0r(k) where K0 is a scalar constant and r(k) is a scalar reference signal.

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Design with reference input

Design with reference input

It follows ①(k + 1) = ❆①(k) + ❇(−K①(k) + K0r(k)) = (❆−❇K) ①(k) + ❇K0r(k) Thus, if the system is completely state controllable the eigenvalues of the state transition matrix can be arbitrarily assigned by choosing a suitable K, also they can be set independently from K0.

• Now, by a proper full state feedback control the closed loop system is

asymptotically stable and admits a steady state y(k) = y for a constant reference signal r(k) = r and sufficiently large k (for instance, more than 5 times the largest time constant).

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Design with reference input

Design with reference input

It follows ①(k + 1) = ❆①(k) + ❇(−K①(k) + K0r(k)) = (❆−❇K) ①(k) + ❇K0r(k) Thus, if the system is completely state controllable the eigenvalues of the state transition matrix can be arbitrarily assigned by choosing a suitable K, also they can be set independently from K0.

• Now, by a proper full state feedback control the closed loop system is

asymptotically stable and admits a steady state y(k) = y for a constant reference signal r(k) = r and sufficiently large k (for instance, more than 5 times the largest time constant).

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Design with reference input

Design with reference input

• Thus, at the steady state x(k + 1) = x(k) = xe:

①e = (❆−❇K) ①e + ❇K0r = (I − ❆+❇K)−1❇K0r therefore, at the steady state the output equals ② = ❈xe = ❈(I − ❆+❇K)−1❇K0r and the gain of output with respect to the reference, i.e., the gain of closed loop system equals y r = ❈(I − ❆+❇K)−1❇K0

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Design with reference input

Design with reference input

• Thus, if we are asked to design a full state feedback control with given

specifications on stability, transient behavior (eigenvalues) and a given steady state gain β, we first design the state feedback gain K to satisfy the requirements

• f stability and transient behavior, then we design and the gain K0 as

K0 =

• ❈(I − ❆+❇K)−1❇

−1 β.

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Deadbeat control

• If a system is completely state controllable we can require that the state of the

system converges to zero as fast as possible, i.e., in at most n steps.

• Choose the desired eigenvalue assignment as {0, 0, . . . , 0}, i.e., all zero

eigenvalues.

• Apply the full state feedback control design procedure
• It will follow that

❆ − ❇K =        1 . . . . . . . . . . . . . . . . . . 0, . . . 1 . . . ,        and (❆ − ❇K)n =        . . . . . . . . . . . . . . . . . . 0, . . . . . . ,       

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Deadbeat control

• If a system is completely state controllable we can require that the state of the

system converges to zero as fast as possible, i.e., in at most n steps.

• Choose the desired eigenvalue assignment as {0, 0, . . . , 0}, i.e., all zero

eigenvalues.

• Apply the full state feedback control design procedure
• It will follow that

❆ − ❇K =        1 . . . . . . . . . . . . . . . . . . 0, . . . 1 . . . ,        and (❆ − ❇K)n =        . . . . . . . . . . . . . . . . . . 0, . . . . . . ,       

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Deadbeat control

• If a system is completely state controllable we can require that the state of the

system converges to zero as fast as possible, i.e., in at most n steps.

• Choose the desired eigenvalue assignment as {0, 0, . . . , 0}, i.e., all zero

eigenvalues.

• Apply the full state feedback control design procedure
• It will follow that

❆ − ❇K =        1 . . . . . . . . . . . . . . . . . . 0, . . . 1 . . . ,        and (❆ − ❇K)n =        . . . . . . . . . . . . . . . . . . 0, . . . . . . ,       

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Deadbeat control

• If a system is completely state controllable we can require that the state of the

system converges to zero as fast as possible, i.e., in at most n steps.

• Choose the desired eigenvalue assignment as {0, 0, . . . , 0}, i.e., all zero

eigenvalues.

• Apply the full state feedback control design procedure
• It will follow that

❆ − ❇K =        1 . . . . . . . . . . . . . . . . . . 0, . . . 1 . . . ,        and (❆ − ❇K)n =        . . . . . . . . . . . . . . . . . . 0, . . . . . . ,       

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Deadbeat control

• The deadbeat control is the fastest achievable by full state proportional

feedback.

• There is a trade-off: To bring the state of the system to zero as fast as possible

the amplitude of the required amplitude of the input may be huge (large energy require to actuate the system)

• Deadbeat control is meaningful only in discrete-time: the state is controlled to

zero only at the sampling instants, within the sampling intervals the dynamics is that of a linear system with constant input, i.e, if the system is unstable ①(t) = 0 for t ∈ (kT, (k + 1)T) where T is the sampling time.

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

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Exercise

Consider the next discrete-time linear system: ①(k + 1) = ❆①(k) + ❇✉(k) y(k) = ❈①(k) where A =     0.8147 0.6324 0.9575 0.9572 0.9058 0.0975 0.9649 0.4854 0.1270 0.2785 0.1576 0.8003 0.9134 0.5469 0.9706 0.1419]     , B =     0.5 0.1     C = [1 0]

• Design a full state feedback control such that the closed loop system has

eigenvalues: {0.8, 0.6, 0.4, 0.4} and unitary steady state gain.

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