CONTROLLABILITY & OBSERVABILITY

AC&ST

C. Melchiorri (DEI) Automatic Control & System Theory 1

AUTOMATIC CONTROL AND SYSTEM THEORY

CONTROLLABILITY & OBSERVABILITY

Claudio Melchiorri

Dipartimento di Ingegneria dell’Energia Elettrica e dell’Informazione (DEI) Università di Bologna

Email: [email protected]

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Controllability and Reachability

Controllability and Reachability Set of states reachable at time t1 starting from state x0 at time instant t0

Set of states controllable to state x1 at time t1 from time instant t0

t0 t1

x0

x1

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Controllability and Reachability The linear, time invariant MIMO system (continuous- or discrete-time):

e1

e3

e2

x1

x2 x3

x4

is completely reachable if, starting from the origin, every state can be reached in a finite amount of time with a proper input control action

is completely controllable if, starting from any state, it is possible to reach the origin in a finite amount of time by applying a proper input control action

e1

e3

e2

x1

x2 x3

x4

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Consider a discrete-time system

State reachable in k steps from “0”

n  By changing the sequence u(0), u(1), … u(k-1) we obtain the set of states reachable in k steps

Reachable states: a subspace of Rn

then:

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•  Example:

•  If

•  Note that in this case rank{V2+} = 2

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Note that the subspaces V+k{0} satisfy the sequence

This sequence, because of the Cayley-Hamilton Theorem, is stationary for k >= n steps, where n is the dimension of the state space.

Then: If a state is reachable, it can be reached in n steps at the most.

V+1 � V+

2 � . . . � V+k

Note that the inclusion sequence for V+k{0} can stop for k < n.

We define as the reachability index the smallest integer r such that

rank {[B, A B, A2 B, … Ar-1 B] } = n We define matrix P1

P1 = [B, A B, A2 B, … An-1 B] as the reachability matrix.

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Notice that the reachability index is less than n (i.e. r < n) if the system has more than an input, and at least two inputs are independent, i.e. that matrix B has at least two independent columns. In other words, if rank{B} = m, then r <= n-m+1. Moreover: 1.  V+ is the smallest A-invariant subspace containing im{B} A P1 ⊆V+ im{B} ⊆ V+ 2.  The subspace V+ is invariant with respect to changes of basis in the state

space: x = T x V+ = T V+

P1 = [B, AB, A2B, . . . An�1B]

P 01 = [B0, A0B0, A02B0, . . . A0n�1B0]

= [T�1B, T�1AT T�1B, T�1A2T T�1B, . . . T�1An�1T T�1B]

= [T�1B, T�1AB, T�1A2B, . . . T�1An�1B]

= T�1[B, AB, A2B, . . . An�1B]

= T�1P1

P1 = TP 01

A0 = T�1AT

B0 = T�1B

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Let us consider the discrete-time system:

A state x(0) is controllable to “0” in k steps if:

Set of states controllable in k steps: Controllable states: a subspace of Rn

The state x(0) is controllable in k steps if the state –Ak x(0) is reachable in k steps from the state 0

equivalent to:

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Let us consider the discrete-time system

Theorem: If P1 = [B, A B, A2 B, … An-1 B], the set of reachable states V+ in any finite interval of time is The set of controllable states V- in any finite time interval is The system is completely reachable iff im{P1} = Rn, that is if P1 is full rank.

•  Reachability implies controllability (if im{P1} = Rn, then it contains im{An}). •  Controllability does not imply reachability (if dim[im{An}] < n ). •  These conditions are equivalent if and only of A is full rank.

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Controllability and Reachability Given the continuous-time system

Theorem: If P1 = [B, A B, A2 B, … An-1 B], the set of the reachable states V+ in any finite interval of time is The set of controllable states V- in any finite time interval is Therefore, the system is completely controllable and reachable if and only if i.e. if P1 is full rank. •  R is the minimum A-invariant subspace containing im{B}

•  Matrix P1 is the so-called reachability matrix

V+ = R = im{P1}

V� = R = im{P1}

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It is possible to prove that V- , the subspace of the states controllable to the origin in any limited interval of time:

•  for continuous-time systems coincides with R (controllability = reachability)

•  for discrete-time systems in general contains Rd and coincides with Rd if Ad is non-singular.

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For discrete-time systems, the reachability and controllability properties are not equivalent. Reachability implies controllability: in fact From which Controllability does not imply reachability: for example assume that A = 0 and that rank{B} < n, then If rank{A} = n, then the two properties are equivalent.

Im{[B,AB,A2B, · · · , An�1B]} = V+n = IRn

Im{An} � Im{[B,AB,A2B, · · · , An�1B]} = V+ = IRn

rank{[B,AB,A2B, · · · , An�1B]} = rank{[B, 0, 0, · · · , 0]} < n

Im{An} � Im{P1}

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The main difference between reachability and controllability for discrete-time systems is that a state x(0) can be controlled to zero also without applying any input. This is due to the fact that if A is not full rank, it has at least a null eigenvalues, and therefore the corresponding eigenspace tends “naturally” to zero in a finite number of steps. Example:

x(k + 1) =

2

40 1 00 0 10 0 0

3

5x(k) �1 = �2 = �3 = 0

x(0) =

2

4111

3

5, x(1) =

2

4110

3

5, x(2) =

2

4100

3

5, x(3) =

2

4000

3

5

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Example: Let consider a diagonalizable system with a single input. In this case: Therefore the system is completely reachable iff: 1)  All the elements bi are not null 2)  All eigenvalues are not null

x(k + 1) =

2

6664

�1 00 �2

...�n

3

7775x(k) +

2

6664

b1b2...bn

3

7775u(k)

P1 =

2

6664

b1 �1b1 �21b1 . . . �n�1

1 b1b2 �2b2 �2

2b2 . . . �n�12 b2

...bn �nbn �2

nbn . . . �n�1n bn

3

7775=

2

6664

b1 00 b2

...bn

3

7775

2

6664

1 �1 �21 . . . �n�1

1

1 �2 �22 . . . �n�1

2...1 �n �2

n . . . �n�1n

3

7775

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Example: Let consider the system: In this case: The system is controllable in two steps, being However, the system is not controllable in one step, since:

V+2 = V+

3 = span

8<

:

2

4011

3

5 ,

2

4001

3

5

9=

;V+1 = span

8<

:

2

4011

3

5

9=

;

x(k + 1) =

2

40 0 00 0 00 0 1

3

5x(k) +

2

4011

3

5u(k)

Im{A2} = Im

8<

:

2

40 0 00 0 00 0 1

3

5

9=

; �

2

40 01 01 1

3

5

Im{A} = Im

8<

:

2

40 0 00 0 00 0 1

3

5

9=

; ⇥� span

8<

:

2

4011

3

5

9=

; = V+1

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Example: Let consider the system: The subspaces V-

1 and V-2 are:

Note that V+

1 and V-1 do not coincide!

x(k + 1) =

2

40 0 00 0 00 0 1

3

5x(k) +

2

4011

3

5u(k)

V�1 = span

8<

:

2

4100

3

5 ,

2

4010

3

5

9=

; V�2 = span

8<

:

2

4100

3

5 ,

2

4010

3

5 ,

2

4001

3

5

9=

;

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Example

System non fully controllable; The controllable subspace is defined by x = [1, 0, 0]T

Fully controllable system

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Controllability for discrete-time systems

•  Problem: Given a discrete-time system, compute the input sequence u(0), u(1), …, u(k-1) in order to reach the state x(k) from x(0).

•  Necessary and sufficient condition for the problem to have a solution is that

that is that the state x(k)- Ak x (0) is reachable from zero in k steps. If this is the case, then the solution is given by the n equations:

in the (k x m) unknowns:

In general, the solution is not unique (n equations in k x m unknowns):

Minimum norm solution

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Controllability for discrete-time systems

•  Since there are more solutions, it is possible to choose the “best one” according to some criteria.

For example, the minimum norm solution minimizes the control energy necessary to achieve the desires state transition

This solution, based on the (pseudo)inverse of matrix P1, is not used frequently in practice since it requires a perfect knowledge of the model parameters (matrices A and B) and no disturbances applied to the system.

kuk =

vuutk�1X

i=0

uT (i)u(i)minimization of

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Controllability for discrete-time systems - Example •  Example: It is desired to heat up to 1000c some steel bars through an oven with

five stages and increasing temperature. The cost for heating each stage is proportional to the square of its temperature. Compute the five stage temperatures in order to minimize the overall cost C = (Σι ui

2)1/2.

n  Let θI and θi+1 i = 1, …, 5 be the input/output temperatures of the bars in each stage, and let ui be the temperature of the i-th stage.

n  The heating dynamics in each stage can be described by:

n  The 5-step reachability matrix is:

θi θi+1

ui

Τ = 0 c Τ = 1000 c

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Controllability for discrete-time systems - Example

•  Assuming that the initial temperature of the steel bars is 0c , then:

•  In this case, matrix R5 is not invertible and it is necessary to use the pseudo-inverse (right pseudo-inverse, since R5 is “lower rectangular”)

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Controllability for discrete-time systems - Example

•  Since R5 is a 1 x 5 lower rectangular matrix, a 4-dimensional null space exists defined by:

Therefore, if for example y = [1000, 10, 1, 1]T, we obtain

and the cost increases from C = 1732.9 to C1 = 2000.8

Same final value

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Reachability for continuous-time systems

• Consider the continuous-time system described by;

• The state x(t) is reachable at time instant t if an input function u(·) exists such that the forced motion results

• Consider the linear operator

• Then x(t) is reachable at time t iff x(t) belongs to the range space of Rt, that represents the reachability subspace V+(t) at time t

x(t) = A x(t) +B u(t) x(0) = 0

x(t) =

Z t

0eA(t��)B u(�)d�

Rt : u(·) � x(t) =

Z t

0eA(t��)B u(�)d�

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Reachability for continuous-time systems

• By defining the reachability matrix P1 = [B, A B, A2 B, … An-1 B], where n is the dimension of the state space (order of matrix A), we have the following:

• Property: The subspace reachable at time instant t > 0, V+(t), is the

image of the reachability matrix P1

•  Note: for continuous-time systems, the reachability subspace does not depend on the length of the time interval [0, t] in which the input function u(·) is active (for discrete-time systems, V+

k depends on the time interval).

•  Note: This is true if the amplitude of the input signal may have arbitrarily large values. In practice, this cannot be true, and therefore the set of reachable states depend on the time interval and the input amplitude. The “theoretical” result is not affected, since in this case the input set is not a vector space and therefore one of the main (theoretical) assumptions is not verified.

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Controllability for continuous-time systems

• Consider the continuous-time system described by;

• Property: The controllabilty subspace V-(t), does not depend on the time interval [0, t] in which the control input is active. Moreover, V-(t) = V+(t).

• Problem: Given a continuous-time system, compute the input function u(•) in order to reach the state x(t) from x(0).

•  The problem has solution if

•  Then, the states that can be reached at time t from x(0) belong to the set

•  The input function can be computed by solving the equation

x(t) = A x(t) +B u(t) x(0) = 0

e

Atx(0) + V+

x(t)� e

Atx(0) ⇥ V+

x(t)� eAtx(0) =

Z t

0eA(t��)Bu(�)d� = Rt u(·)

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Controllability and Reachability - Example

• Consider the electric circuit:

•  In matrix form:

vu(t) J(t) R R

L CiL

vC

The system is not completely controllable if det(P1) = 0, that is if R C = L/R (the same time constant)

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Observability

•  A system is said completely observable if, knowing the input and output functions u|[0, t1], y|[0, t1] in any finite interval of time it is possible to determine the state x(0)

•  Theorem: By defining

the set of unobservable states in any finite interval of time is

The system is completely observable iff Ker{P2} = 0 that is if P2 has full rank. Q is a subspace of Rn

Matrix P2 is the observability matrix.

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Observability

•  Example:

System non fully observable: states defined by Ker{P2} = [-1, 0, 1]T

Fully observable system: (Ker{P2} = {0})

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Observability

• Define Q = ker{P2} is the maximum A-invariant subspace contained in ker{C}: for every initial state in Q, the related free motion does not affect the output y.

• For any matrix P:

That is: •  the null space of any matrix P is equal to the orthogonal complement of the

image of PT •  the image of any matrix P is equal to the orthogonal complement of the null

space of PT

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Equivalent systems

•  Two systems Σ (A, B, C, D) and Σ’ (A’, B’, C’, D’) are equivalent if a non singular matrix T exists such that x = T x’ and

•  For the reachability property, we have that:

•  If the two systems are reachable, then

and, in case of a single input

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Equivalent systems

•  For the reachability property, we have that:

•  As a matter of fact, the reachable subspaces in k steps for the two systems are

•  From this equation it follows that P’1 = T-1 P1

V 0+k = Im{[B0, A0B0, A02B0, · · · , A0k�1B0]}

= Im{[T�1B, T�1AT T�1B, · · · , T�1Ak�1T T�1B]}= Im{T�1[B, AB, · · · , Ak�1B]}= T�1V +

k

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Equivalent systems •  Example:

Check:

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Dual systems

• Consider a linear time-invariant system Σ

The dual system ΣD is defined as

Number of input of Σ = Number of output of ΣD

Number of output of Σ = Number of input of ΣD

(1)

(2)

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Dual systems

Properties of dual systems:

Σ stable ΣD stable

Σ completely controllable ΣD completely reconstructable

Σ completely reconstructable ΣD completely controllable

Σ completely observable ΣD completely reachable

Σ completely reachable ΣD completely observable

The proof of these properties can be easily obtained from the following equations:

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Standard reachability form

Consider the MIMO linear time-invariant system (continuous- or discrete-time):

with rank(P1) = ρ < n (dim R = ρ < n): the system is NOT completely reachable. Consider the transformation matrix where the n x ρ matrix T1, is a basis matrix of V+ and the n x (n-ρ) matrix T2 makes T non singular. The following equivalent (and simpler) form is obtained for the system:

This is the standard reachability form, since the reachable part of the system is clearly identified by matrices (A’11, B’1, C’1).

A0 = T�1AT =

A0

11 A012

0 A022

�B0 = T�1B =

B0

1

0

�

C 0 = CT = [C 01, C

02] D0 = D

dim{A011} = �� , dim{B0

1} = ��m

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•  Note: By acting through the input, it is possible to modify only the first ρ components of the new state z = T x. It is not possible to apply any control action on the remaining n - ρ components: it they are not stable, an unstable behavior will always take place.

1/s

A’11

A’12

1/s

A’22

C’1

C’2

D’

B’1

+

+ +

+ + +

+

u

y

z1

z2

z1(t) = A011z1(t)+ A0

12z2(t)+ B01u(t)

z2(t) = A022z2(t)

y(t) = C 01z1(t)+ C 0

2z2(t)+ D0u(t)

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Consider the homogeneous system A trajectory originated on an invariant will remain on it. Therefore, since the reachability subspace is A-invariant, if matrix A11 is stable, a trajectory originated in this subspace will tend to the origin. Moreover, a trajectory originated out of this invariant tends to it if matrix A22 is stable. •  If A11 is stable, the invariant is called internally stable •  If A22 is stable, the invariant is called externally stable

-1 0 1 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5

-1 -0.5 0

0.5 1

1.5

V

x0

If A22 is stable: •  x2 tends to 0 •  x tends to V

x(t) = A x(t), x(0) = x0, A =

A11 A12

0 A22

�

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From the properties of dual systems we have where T is the matrix that transforms the system in the standard reachability form.

Standard observability form Consider the MIMO linear time-invariant system (continuous- or discrete-time):

with rank(P2) = ρ < n (dim Q = n - ρ > 0. The system is NOT completely observable. A transformation matrix M exists such that

This is the standard observability form, since the observable part of the system is clearly identified by matrices (A’11, B’1, C’1).

A0 = M�1AM =

A0

11 0A0

21 A022

�B0 = M�1B =

B0

1

B02

�

C 0 = CM = [C 01, 0] D0 = D

dim{A011} = �� , dim{C 0

1} = p� �

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� = (A, B, C, D) �0 = (A0, B0, C 0, D0) = (T�1AT, T�1B, CT, D)

�D = (AT , CT , BT , DT ) �0D = (TTATT�T , TTCT , T�TBT , DT )

Standard observability form

As a matter of fact, if for one of the systems we have

The following relationship holds for the transformation matrices :

Then, for its dual system:

Dual Dual

T

T-T

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Kalman decomposition

•  Consider the system ∑ = (A, B, C, D). Let R be the reachable subspace and Q the unobservable subspace. Consider the state transformation defined by the matrix

•  Where:

•  The system ∑’ = (A’, B’, C’, D’) is obtained, where

Basis of the reachable and non observable subspace

Basis of the reachable subspace

Basis of the non observable subspace

im{T1} = R � Q

im{[T1, T2]} = R

im{[T1, T3]} = Q

T4 : T non singular

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2

1

3

4

D

+

+

+ u y

Non observable sub-system

Reachable sub-system

Kalman decomposition

•  ∑2 = (A’22, B’2, C’2) reachable and observable •  ∑1 = (A’11, B’1, 0) reachable and non observable •  ∑4 = (A’44, 0, C’4) non reachable and observable •  ∑3 = (A’33, 0, 0) non reachable and non observable

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From state space to transfer functions

•  Consider the SISO linear time-invariant system

n  From the latter equations, we can identify:

- the input-output transfer function

- the free response term

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From state space to transfer functions

•  Similar results hold for MIMO systems

In fact, one obtains

where matrix is the system transfer matrix. It is simple to verify that the system transfer matrix depends only on the reachable and observable part of the system.

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