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MSc in Computer Engineering, Cybersecurity and Artificial Intelligence Course FDE , a.a. 2019/2020, Lecture 11 Brief introduction to controllability and observability for discrete-time linear systems

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

University of Cagliari, Italy

Wednsday, 22nd April 2020

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Outline

Introduction Controllability Observability Duality

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Introduction

Introduction

• So far we have focused on the analysis of the behavior of dynamical

systems

• We are now interested in the next two fundamental problems:

1 There always exist control inputs such that we can determine the

future behavior of dynamical systems?

2 Can estimate the current and past state of a dynamical system by

measuring its outputs?

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Introduction

Introduction

• So far we have focused on the analysis of the behavior of dynamical

systems

• We are now interested in the next two fundamental problems:

1 There always exist control inputs such that we can determine the

future behavior of dynamical systems?

2 Can estimate the current and past state of a dynamical system by

measuring its outputs?

3 / 33

Introduction

Introduction

• So far we have focused on the analysis of the behavior of dynamical

systems

• We are now interested in the next two fundamental problems:

1 There always exist control inputs such that we can determine the

future behavior of dynamical systems?

2 Can estimate the current and past state of a dynamical system by

measuring its outputs?

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Introduction

Introduction

• We will answer this questions focusing on discrete-time linear systems

①(k + 1) = ❆①(k) + ❇✉(k) ②(k) = ❈①(k) + ❉✉(k) where ①(k) is the state vector; ✉(k) is the input vector; ②(k) is the output vector.

• Nowadays the almost totality of control systems are digital and therefore

practical implementations involve discrete-time control laws and algorithms.

• Fundamental properties of discrete-time and continuous-time linear

systems are similar.

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Introduction

Introduction

• We will answer this questions focusing on discrete-time linear systems

①(k + 1) = ❆①(k) + ❇✉(k) ②(k) = ❈①(k) + ❉✉(k) where ①(k) is the state vector; ✉(k) is the input vector; ②(k) is the output vector.

• Nowadays the almost totality of control systems are digital and therefore

practical implementations involve discrete-time control laws and algorithms.

• Fundamental properties of discrete-time and continuous-time linear

systems are similar.

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Introduction

Introduction

• We will answer this questions focusing on discrete-time linear systems

①(k + 1) = ❆①(k) + ❇✉(k) ②(k) = ❈①(k) + ❉✉(k) where ①(k) is the state vector; ✉(k) is the input vector; ②(k) is the output vector.

• Nowadays the almost totality of control systems are digital and therefore

practical implementations involve discrete-time control laws and algorithms.

• Fundamental properties of discrete-time and continuous-time linear

systems are similar.

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Outline

Introduction Controllability Observability Duality

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Controllability

Complete state controllability

• A dynamical system is said to be completely state controllable if it is

possible to bring the system from an arbitrary initial state ①(0) to an arbitrary final state xf in a finite time period by providing a suitable input.

• Consider the next system

①(k + 1) = ❆①(k) + ❇u(k) (1) where ❆ is an n × n matrix; ❇ is an n × 1 matrix; u(k) is a scalar input.

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Controllability

Complete state controllability

• By recalling the formula for the state response starting at the discrete-time

k = 0 where ①(0) is the initial state and xf (k) is the final desired state: ①f (k) = ❆k①(0) +

n−1

• j=0

❆k−j−1❇u(j) = ❆k①(0) + ❆k−1❇✉(0) + ❆k−2❇✉(1) + . . . + ❇u(k − 1) thus ①f (k) − ❆k①(0) = ❆k①(0) + ❆k−1❇u(0) + ❆k−2❇u(1) + . . . + ❇u(k) =

• ❇, ❆❇, ❆2❇, . . . , ❆k−1❇
• 

      u(k − 1) u(k − 2) . . . u(0)        (2)

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Controllability

Complete state controllability

• By recalling the formula for the state response starting at the discrete-time

k = 0 where ①(0) is the initial state and xf (k) is the final desired state: ①f (k) = ❆k①(0) +

n−1

• j=0

❆k−j−1❇u(j) = ❆k①(0) + ❆k−1❇✉(0) + ❆k−2❇✉(1) + . . . + ❇u(k − 1) thus ①f (k) − ❆k①(0) = ❆k①(0) + ❆k−1❇u(0) + ❆k−2❇u(1) + . . . + ❇u(k) =

• ❇, ❆❇, ❆2❇, . . . , ❆k−1❇
• 

      u(k − 1) u(k − 2) . . . u(0)        (2)

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Controllability

The controllability matrix

• Notice that each matrix B, AB, ..., Ak−1B is a column vector.

①f (k) − ❆k①(0) =

• B, AB, A2B, . . . , Ak−1B
• 

    u(k − 1) u(k − 2) . . . u(0)     

• Now, let k = n, i.e., the order of the system. If

rank

• B, AB, A2B, . . . , An−1B
• = n

then the columns of matrix

• B, AB, A2, . . . , An−1B
• span the whole n-dimensional space.
• Matrix T =
• B, AB, A2B, . . . , An−1B
• is commonly called the controllability matrix.
• In literature for discrete-time systems it is also referred to as the reachability matrix.

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Controllability

The controllability matrix

• Notice that each matrix B, AB, ..., Ak−1B is a column vector.

①f (k) − ❆k①(0) =

• B, AB, A2B, . . . , Ak−1B
• 

    u(k − 1) u(k − 2) . . . u(0)     

• Now, let k = n, i.e., the order of the system. If

rank

• B, AB, A2B, . . . , An−1B
• = n

then the columns of matrix

• B, AB, A2, . . . , An−1B
• span the whole n-dimensional space.
• Matrix T =
• B, AB, A2B, . . . , An−1B
• is commonly called the controllability matrix.
• In literature for discrete-time systems it is also referred to as the reachability matrix.

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Controllability

The controllability matrix

• Notice that each matrix B, AB, ..., Ak−1B is a column vector.

①f (k) − ❆k①(0) =

• B, AB, A2B, . . . , Ak−1B
• 

    u(k − 1) u(k − 2) . . . u(0)     

• Now, let k = n, i.e., the order of the system. If

rank

• B, AB, A2B, . . . , An−1B
• = n

then the columns of matrix

• B, AB, A2, . . . , An−1B
• span the whole n-dimensional space.
• Matrix T =
• B, AB, A2B, . . . , An−1B
• is commonly called the controllability matrix.
• In literature for discrete-time systems it is also referred to as the reachability matrix.

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Controllability

The controllability matrix

• Notice that each matrix B, AB, ..., Ak−1B is a column vector.

①f (k) − ❆k①(0) =

• B, AB, A2B, . . . , Ak−1B
• 

    u(k − 1) u(k − 2) . . . u(0)     

• Now, let k = n, i.e., the order of the system. If

rank

• B, AB, A2B, . . . , An−1B
• = n

then the columns of matrix

• B, AB, A2, . . . , An−1B
• span the whole n-dimensional space.
• Matrix T =
• B, AB, A2B, . . . , An−1B
• is commonly called the controllability matrix.
• In literature for discrete-time systems it is also referred to as the reachability matrix.

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Controllability

Complete state controllability: a criterion

If the rank of the controllability matrix T is n then from any initial state ①0 and arbitrary final state ①f there exists an unbounded control signal u(0), u(1), . . . , u(n − 1) which brings the dynamical system from the state x0 to the state xf in n steps. Thus the system is said to be completely state controllable in n steps.

• Thus, the condition rank(T ) = n is proven by construction to be at least

a sufficient condition for complete state controllability. In fact, it is also necessary.

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Controllability

Complete state controllability: a criterion

If the rank of the controllability matrix T is n then from any initial state ①0 and arbitrary final state ①f there exists an unbounded control signal u(0), u(1), . . . , u(n − 1) which brings the dynamical system from the state x0 to the state xf in n steps. Thus the system is said to be completely state controllable in n steps.

• Thus, the condition rank(T ) = n is proven by construction to be at least

a sufficient condition for complete state controllability. In fact, it is also necessary.

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Controllability

Complete state controllability: necessity

• To prove necessity, assume that rank(T ) < n.
• By the Cayley-Hamilton theorem, for an arbitrary k the term AkB can be

expressed as a linear combination of B, AB, An−1B. Thus it would follow

• B, AB, A2B, . . . , Ak−1B
• < n
• Therefore, the vectors =
• B, AB, A2B, . . . , An−1B
• do not span the

n-dimensional state space and thus there exists some xf such that it is not possible to have x(k) = xf for all k. Thus the condition rank(T ) = n is also necessary.

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Controllability

Complete state controllability: necessity

• To prove necessity, assume that rank(T ) < n.
• By the Cayley-Hamilton theorem, for an arbitrary k the term AkB can be

expressed as a linear combination of B, AB, An−1B. Thus it would follow

• B, AB, A2B, . . . , Ak−1B
• < n
• Therefore, the vectors =
• B, AB, A2B, . . . , An−1B
• do not span the

n-dimensional state space and thus there exists some xf such that it is not possible to have x(k) = xf for all k. Thus the condition rank(T ) = n is also necessary.

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Controllability

Case with vector input

• If ✉(k) is a vector, thus the terms AkB of the controllability matricx are

matrices, it can be proven similarly that if the n × nr controllability matrix has rank(

• B, AB, A2B, . . . , An−1B
• ) = n

it is still a necessary and sufficient condition.

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Controllability

Determination of control sequence

• If the system defined by the pair (A, B) is completely state controllable,

then the control sequence to bring the system from state ①(0) to state ①f (n) in n steps is simply:        u(n − 1) u(n − 2) . . . u(0)        = T −1 (①f (n) − ❆n①(0)) (3)

• Note: If more then n steps are taken, or the input is not scalar, then the

sequence is not unique.

• Thus, it is possible to optimize the input with respect parameters of

interests such control energy, amplitude, etc..

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Controllability

Determination of control sequence

• If the system defined by the pair (A, B) is completely state controllable,

then the control sequence to bring the system from state ①(0) to state ①f (n) in n steps is simply:        u(n − 1) u(n − 2) . . . u(0)        = T −1 (①f (n) − ❆n①(0)) (3)

• Note: If more then n steps are taken, or the input is not scalar, then the

sequence is not unique.

• Thus, it is possible to optimize the input with respect parameters of

interests such control energy, amplitude, etc..

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Controllability

Determination of control sequence

• If the system defined by the pair (A, B) is completely state controllable,

then the control sequence to bring the system from state ①(0) to state ①f (n) in n steps is simply:        u(n − 1) u(n − 2) . . . u(0)        = T −1 (①f (n) − ❆n①(0)) (3)

• Note: If more then n steps are taken, or the input is not scalar, then the

sequence is not unique.

• Thus, it is possible to optimize the input with respect parameters of

interests such control energy, amplitude, etc..

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Controllability

Complete output controllability

• In practical applications it might be of interest to control the output,

rather than the state.

• Compelte state controllability is neither necessary nor sufficient for

complete output controllability

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Controllability

Complete output controllability

• A dynamical system is qualitatively said to be completely output

controllable if it is possible to bring the system from an arbitrary initial

• utput ②(0) to an arbitrary final output yf (k) in a finite time period by

providing a suitable input.

• Consider the next system

①(k + 1) =❆①(k) + ❇u(k) ②(k) =❈①(k) where ❆ is an n × n matrix; ❇ is an n × 1 matrix; ❈ is an p × n matrix.

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Controllability

Output controllability

• By recalling the formula for the output response starting at the discrete-time

k = 0 where ②(0) is the initial output and yf (k) is final desired output at time k: ② f (k) − ❈❆k①(0) =

n−1

• j=0

❈❆k−j−1❇u(j) = ❈❆k−1❇✉(0) + ❈❆k−2❇✉(1) + . . . + ❈❇u(k − 1) =

• ❈❇, ❈❆❇, ❈❆2❇, . . . , ❈❆k−1❇
• 

    u(k − 1) u(k − 2) . . . u(0)      (4)

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Controllability

Output controllability

• Thus, for k = n the condition

rank(

• ❈❇, ❈❆❇, ❈❆2❇, . . . , ❈❆n−1❇
• ) = p

(5) where p is the number of outputs, can be shown to be necessary and sufficient for complete output controllability in the case there is influence of the input directly in the output, i.e., no matrix D.

• Complete state controllability implies complete output controllability only if the

p rows of matrix ❈ are linearly independent.

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Controllability

Output controllability

• Thus, for k = n the condition

rank(

• ❈❇, ❈❆❇, ❈❆2❇, . . . , ❈❆n−1❇
• ) = p

(5) where p is the number of outputs, can be shown to be necessary and sufficient for complete output controllability in the case there is influence of the input directly in the output, i.e., no matrix D.

• Complete state controllability implies complete output controllability only if the

p rows of matrix ❈ are linearly independent.

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Controllability

Complete output controllability

• Now consider the next system

①(k + 1) = ❆①(k) + ❇u(k) ②(k) = ❈①(k) + ❉u(k) (6) where ❆ is an n × n matrix; ❇ is an n × r matrix; ❈ is an p × n matrix; ❉ is an p × r matrix.

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Controllability

Output controllability

• By recalling the formula for the output response starting at the discrete-time

k = 0 where ②(0) is the initial state and yf (k) is the final desired state: ② f (k) − ❈❆k①(0) =

n−1

• j=0

❈❆k−j−1❇u(j) + ❉✉(k) = ❈❆k−1❇✉(0) + ❈❆k−2❇✉(1) + . . . + ❈❇u(k − 1) + ❉✉(k) =

• ❉, ❈❇, ❈❆❇, ❈❆2❇, . . . , ❈❆k−1❇
• 

      u(k) u(k − 1) u(k − 2) . . . u(0)        (7)

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Controllability

Output controllability

• Thus, for k = n the condition

rank(

• ❉, ❈❇, ❈❆❇, ❈❆2❇, . . . , ❈❆n−1❇
• ) = p

(8) can be shown to be necessary and sufficient for complete output controllability in the case there is influence of the input directly in the output, i.e., no matrix D.

• Thus, the presence of matrix ❉ always helps in regard to complete output

controllability.

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Controllability

Output controllability

• Thus, for k = n the condition

rank(

• ❉, ❈❇, ❈❆❇, ❈❆2❇, . . . , ❈❆n−1❇
• ) = p

(8) can be shown to be necessary and sufficient for complete output controllability in the case there is influence of the input directly in the output, i.e., no matrix D.

• Thus, the presence of matrix ❉ always helps in regard to complete output

controllability.

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Controllability

Controllability for continuous-time systems

• The structure of the controllability matrix T is the same for both discrete-time

and continuous time linear systems

• More precise categories of the notion of controllability for discrete time systems

are which we do not discuss now in detail are: Complete controllability ⇒ Reachability ⇒ Controllability to the origin ⇒ Stabilizability.

• In continuous-time linear systems we talk simply of controllability instead of

complete controllability because in any given finite time T we have a continuous u(t) instead of a finite number of samples, thus there no need to distinguish between th existing different notions of controllability in the literature.

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Outline

Introduction Controllability Observability Duality

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Observability

Observability

• A system is said to be completely observable if every initial state x(0)

can be determined from the observation of a finite number of output samples ②(k). Thus the system is completely observable if any state transition affects the outputs of the system.

• The concept of observability is useful to solve the issue of

determining/reconstructing unmeasurable state variables. Thus,as we will see later in this course, it is also strictly related to general estimation problems in dynamical systems including the identification of faults.

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Observability

Observability

• A system is said to be completely observable if every initial state x(0)

can be determined from the observation of a finite number of output samples ②(k). Thus the system is completely observable if any state transition affects the outputs of the system.

• The concept of observability is useful to solve the issue of

determining/reconstructing unmeasurable state variables. Thus,as we will see later in this course, it is also strictly related to general estimation problems in dynamical systems including the identification of faults.

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Observability

Observability

• Consider the next autonomous system

①(k + 1) = ❆①(k) ②(k) = ❈①(k) where ❆ is an n × n matrix; ❈ is an p × n matrix;

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Observability

Observability

• Its solution is :

②(0) = ❈①(0) ②(1) = ❈❆①(0) ②(2) = ❈❆2①(0) . . . ②(k) = ❈❆k①(0) (9)

• Grouping the terms:

       ②(0) ②(1) ②(2) . . . ②(k)        =        ❈ ❈❆ ❈❆2 . . . ❈❆k        ①(0)

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Observability

Observability

• Its solution is :

②(0) = ❈①(0) ②(1) = ❈❆①(0) ②(2) = ❈❆2①(0) . . . ②(k) = ❈❆k①(0) (9)

• Grouping the terms:

       ②(0) ②(1) ②(2) . . . ②(k)        =        ❈ ❈❆ ❈❆2 . . . ❈❆k        ①(0)

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Observability

Observability matrix

• To determine a unique solution x(0), we need n linearly independent equations

       ②(0) ②(1) ②(2) . . . ②(k)        =        ❈ ❈❆ ❈❆2 . . . ❈❆k        ①(0)

• Thus, a unique solution x(0) can be found if only if the so called observability

matrix: O =        ❈ ❈❆ ❈❆2 . . . ❈❆n−1        is full rank, i.e., rank n.

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Observability

Observability matrix

• To determine a unique solution x(0), we need n linearly independent equations

       ②(0) ②(1) ②(2) . . . ②(k)        =        ❈ ❈❆ ❈❆2 . . . ❈❆k        ①(0)

• Thus, a unique solution x(0) can be found if only if the so called observability

matrix: O =        ❈ ❈❆ ❈❆2 . . . ❈❆n−1        is full rank, i.e., rank n.

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Observability

Complete observability

• Noting that the rank of a matrix and its traspose is the same, we can also state

that a necessary and sufficient condition for complete output observability is rank(O) = rank(

• ❈, ❆T❈ T, (A2)T❈ T, . . . , (An−1)T❈ T

) = n

• The proof follows similar reasoning as for complete controllability.

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Observability

Observability

• The reason we established observability for an autonomous system is because if

we measure the input, we can simply subtract its effect on the output.

• Consider the next autonomous system

①(k + 1) = ❆①(k) + ❇✉(k) ②(k) = ❈①(k) + ❉u(k) (10)

• it holds

②(k) = ❈❆k①(0) +

k−1

• j=0

❈❆k−j−1❇u(j) + ❉✉(k) and ②(k) −

k−1

• j=0

❈❆k−j−1❇u(j) + ❉✉(k) = ❈❆k①(0))

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Observability

Observability

• Thus

       ②(0) − ❉✉(0) ②(1) − ❈❇u(0) − ❉✉(1) ②(2) − ❈❆❇u(1) − ❈❇u(0) − ❉✉(2) . . . ②(n − 1) − n−2

j=0 ❈❆n−j−2❇u(j) + ❉

       =        ❈ ❈❆ ❈❆2 . . . ❈❆n−1        ①(0)

• Therefore leading to the same fundamental result which is independent from

the applied input.

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Observability

Observability for continuous-time systems

• The structure of the observability matrix O is the same for both discrete-time

and continuous time linear systems

• In continuous-time linear systems we talk simply of observability instead of

complete observability because in any given finite time T we observe a continuous y(t) instead of a finite number of samples.

• More precise categories of the notion of observability for discrete time systems

are which we do not discuss now in detail are: Complete observability ⇒ observability ⇒ reconstructability ⇒ detectability.

• Note that in the case of observability it does not make sense to distinguish in

state and output observability as for controllability, if we know the full state of the system the problem is solved trivially.

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Observability

Observability for continuous-time systems

• The structure of the observability matrix O is the same for both discrete-time

and continuous time linear systems

• In continuous-time linear systems we talk simply of observability instead of

complete observability because in any given finite time T we observe a continuous y(t) instead of a finite number of samples.

• More precise categories of the notion of observability for discrete time systems

are which we do not discuss now in detail are: Complete observability ⇒ observability ⇒ reconstructability ⇒ detectability.

• Note that in the case of observability it does not make sense to distinguish in

state and output observability as for controllability, if we know the full state of the system the problem is solved trivially.

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Observability

Observability for continuous-time systems

• The structure of the observability matrix O is the same for both discrete-time

and continuous time linear systems

• In continuous-time linear systems we talk simply of observability instead of

complete observability because in any given finite time T we observe a continuous y(t) instead of a finite number of samples.

• More precise categories of the notion of observability for discrete time systems

are which we do not discuss now in detail are: Complete observability ⇒ observability ⇒ reconstructability ⇒ detectability.

• Note that in the case of observability it does not make sense to distinguish in

state and output observability as for controllability, if we know the full state of the system the problem is solved trivially.

29 / 33

Observability

Observability for continuous-time systems

• The structure of the observability matrix O is the same for both discrete-time

and continuous time linear systems

• In continuous-time linear systems we talk simply of observability instead of

complete observability because in any given finite time T we observe a continuous y(t) instead of a finite number of samples.

• More precise categories of the notion of observability for discrete time systems

are which we do not discuss now in detail are: Complete observability ⇒ observability ⇒ reconstructability ⇒ detectability.

• Note that in the case of observability it does not make sense to distinguish in

state and output observability as for controllability, if we know the full state of the system the problem is solved trivially.

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Outline

Introduction Controllability Observability Duality

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Duality

The principle of duality

• The concepts of controllability and observability are strictly related. Consider

the next system, which we call system (1) ①(k + 1) = ❆①(k) + ❇✉(k) ②(k) = ❈①(k) and its dual counter part, which we call system (2) ˆ ①(k + 1) = ❆T ˆ ①(k) + ❈ T ˆ ✉(k) ˆ ②(k) = ❇T ˆ ①(k) where ❆ is an n × n matrix; ❇ is an n × r matrix; ❈ is an p × n matrix;

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Duality

The principle of duality

• For system (1):

1 A necessary and sufficient condition for complete state controllability is that

rank(

• ❇, ❆❇, ❆2❇, . . . , ❆n−1❇
• ) = n

2 A necessary and sufficient condition for complete state observability is that

rank(

• ❈ T, ❆T❈ T, (❆T)2❈ T, . . . , (AT)n−1❈
• ) = n
• For system (2):

1 A necessary and sufficient condition for complete state controllability is that

rank(

• ❈ T, ❆T❈ T, (❆T)2❈ T, . . . , (AT)n−1❈
• ) = n

2 A necessary and sufficient condition for complete state observability is that

rank(

• ❇, ❆❇, ❆2❇, . . . , ❆n−1❇
• ) = n

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Duality

The principle of duality

• For system (1):

1 A necessary and sufficient condition for complete state controllability is that

rank(

• ❇, ❆❇, ❆2❇, . . . , ❆n−1❇
• ) = n

2 A necessary and sufficient condition for complete state observability is that

rank(

• ❈ T, ❆T❈ T, (❆T)2❈ T, . . . , (AT)n−1❈
• ) = n
• For system (2):

1 A necessary and sufficient condition for complete state controllability is that

rank(

• ❈ T, ❆T❈ T, (❆T)2❈ T, . . . , (AT)n−1❈
• ) = n

2 A necessary and sufficient condition for complete state observability is that

rank(

• ❇, ❆❇, ❆2❇, . . . , ❆n−1❇
• ) = n

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Duality

The principle of duality

• It follows that the observability property of a given system can be

checked by verifying the controllability property for its dual and vice versa.

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