MECH 522 Foundations in Control Engineering

Clarence W. de Silva, Ph.D., P.Eng. Professor of Mechanical Engineering The University of British Columbia

e-mail: desilva@mech.ubc.ca http:// www.mech.ubc.ca/~ial

C.W. de Silva

Presentation Part 5

Part 5 Controllability And

Observability

PLAN • Definitions of Controllability and observability (for

linear, time-invariant systems)

• Criteria of checking controllability and observability: 1. Modal representation (distinct eigenvalue case) 2. General criterion 3. Pole-zero cancellation

• Examples

• Companion Form and Controllability

• Stabizability and Detectability

• Output Controllability

Definitions

Controllability and Observability Controller: State feedback (constant-gain) Questions: • Can the state (vector) be changed from an arbitrary value to any specified value in finite time? (Controllability) • If not all the states in the system are available for measurement (observation), can the full state vector be determined from the outputs (or, measurable states) over a finite duration? (Observability) Note: Specified (future) state can be made the origin of state space (no loss of generality).

Definition of Controllability and Observability

Definition 1: A linear system is controllable at time t0 if there is an input u that transfers any arbitrary state to the origin (x = 0) in finite time t1. If this is true for any t0 then the system is said to be completely controllable. Definition 2: A linear system is observable at time t0 if we can completely determine the state vector from the output measurements y over the duration [t0, t1] for finite t1. If this is true for any t0 then the system is said to be completely observable.

)( 0tx

)( 0tx

Criteria of Checking Controllability and Observability

Criteria for Checking Controllability and Observability

Time-invariant (i.e., constant-parameter) linear system: Criteria for Checking Controllability/Observability: Criterion 1: Use of Modal Representation—Distinct Eigenvalues Criterion 2: General, Uses Matrices A, B, and C Criterion 3: Pole-Zero Cancellation (Uses Transfer Function Representation)

BuAxx +=DuCxy +=

Criterion 1: Use of Modal Representation

Valid for the Case of Distinct Eigenvalues Consider: Time-invariant (i.e., constant-parameter) linear system: ; Apply the modal transformation: M = modal matrix (matrix of independent eigenvectors) * Controllable iff M-1B has all non-zero rows. Justification: Then, all modes are influenced (controlled) by u * Observable iff C M has all non-zero columns. Justification: Then, all modal responses are present in measured y Note: Iff means “If and Only If”

BuAxx += DuCxy +==x Mq

BuMJqq 1−+=

DuCMqy +=

Criterion 2: Use of Matrices A, B, and C

Generally Valid Controllable iff Rank [ B | A B|…|An-1B] = n (i.e., Rank of “Controllability Matrix” = n) Observable iff Rank (i.e., Rank of “Observability Matrix” = n) Proof for Controllability (“Only If” Part): From Definition 1, for controllability (with t0 = 0 and t = t1), (from finite series expansion of the matrix exponential) RHS should be able to form any arbitrary x(0) Tterms AjB together should span n-dimensional state space

nT1nTTT =− ]|...||[ CACAC

∫ −+=1

11

t

0

tt d0 τττ )()( )( Buexe0 AA ∫ −−=1t

0

d0 τττ )()( Buex A

12 n 1

0 1 2 10

[ ( ) ( ) ( ) ... ( ) ] ( )t

n dα τ α τ α τ α τ τ τ−−= + + + +∫ I A A A Bu

τττα d1n

0j

t

0

j1

)()(∑ ∫−

=

= uBA jj

1n

0j

j vBA∑−

−

=

Criterion 2 (Cont’d): Use of Matrices A, B, and C

Proof for Observability (“Only If” Part): Without loss of generality, take t0 = 0. For observability, x(0) must be completely solvable for finite (but arbitrary t) Its coefficient matrix should be convertible to an invertible (i.e., non-singular) nth order matrix, using arbitrary choices of finite t Note: Coefficient matrix = linear combination of We need Note: If X is a real matrix, X and XT have the same rank. Note: “If” part can be proved by starting from other end

)()()( ttt DuCxy += DuBuexeC AA ++= ∫ − ])()([ )( τττ d0t

0

tt

DuBueCyxCe AA −−= ∫ −t

0

tt dt0 τττ )()()( )( )()()(])(...)()([ ufyxAAIC −=+++ −− t0tt 1n

1n10 tααα

1, ,.., n−C CA CA

10 1 1[ ( ) ( ) ... ( ) ]n

nt t tα α α −−+ + +C CA CA

1Rank[ | | ... | ]T T T n T n− =C A C A C

Criterion 3: Pole-Zero Cancellation

BuAxx +=System: ; Use Laplace domain. (a) Controllability (x(0) arbitrary, not relevant): should not have any pole-zero cancellation Rationale: The cancelled pole (mode) cannot be controlled by U(s) (b) Observability (u arbitrary, not relevant): should not have any pole-zero cancellation Rationale: The cancelled pole (mode) cannot contribute to Y(s); hence Y(s) cannot reconstruct x, which depends on that mode too.

= +y Cx Du

( ) 1( ) ( )s s s−= −X I A BU1( )s −−I A B

( ) 1( ) [ (0) ( )]s s s−= −Y C I A x + BU1( )s −−C I A

Examples

Example 1

State space model: ⇒ Criterion 2: Controllability matrix Det P = 0 ⇒ Rank P < 2 (Rank P = 1) ⇒ uncontrollable. Observability matrix Det Q ≠ 0 ⇒ Rank Q = 2 ⇒ observable.

2

212

11 2

xyuxxx

uxx

=+−−=

+−=

2 01 1

− = − −

A11

=

B [ ]0 1=C

,

1 2[ , ]

1 2−

= = − P B AB

0 1[ , ]

1 1T T T −

= = − Q C A C

Example 1 (Cont’d)

Criterion 1: Eigenvalues: ⇒ Eigenvectors: For For Modal matrix

2 0( 2)( 1) 0

1 1λ

λ λ λ λλ

+ − = ⇒ − = + + = +

I A I A

2,1 21 −=−= λλ

1 1

1 0 0 01:

1 0 0 1ab

λ

= − = ⇒ =

ξ

2 2

0 0 0 12 : 0

1 1 0 1a

a bb

λ

= − = ⇒ − = ⇒ = − ξ

0 11 1

=

M

Example 1 (Cont’d) Controllability Check: ⇒ uncontrollable Note: Specifically, mode with eigenvalue -1 is uncontrollable (but the system is “stabilizable” because the uncontrolled mode is stable. See later). Observability Check: ⇒ observable

,

1 11 1 1 1 1 1 1 011 0 1 0 1 0 1 1( 1)

− −− − − = = ⇒ = = −−

M M B

[ ] [ ]0 10 1 1 1

1 1

= =

CM

Example 1 (Cont’d) Criterion 3: Controllability Check: Straightforward linear algebra gives ⇒ ⇒ ⇒ pole-zero cancellation ⇒ system is uncontrollable. Note: Specifically, mode with eigenvalue λ1= -1 is uncontrollable (but, stabilizable; confirms the previous result). Observability Check: Straightforward linear algebra: ⇒ no cancellation ⇒ observable.

[ ]1 1( ) 1 ( 2)( 1)( 2)

s ss s

−− = − ++ +

C I A

2 01 1

ss

s+

− = + I A

1 1 01( )1 2( 1)( 2)

ss

ss s− +

− = − ++ + I A

1 1 11 ( 1)( )1 1( 1)( 2) ( 1)( 2)

s ssss s s s

− + +− = = ++ + + +

I A B

Example 2

What do you think of system: Third state equation: This is uncoupled and does not contain an input term. Note: In fact, this state is a mode (with eigenvalue +2) which is unstable. ⇒State x3 is uncontrollable. ⇒The system is uncontrollable and also unstabilizable (see

later)

,

1 2 1 2 31 1 3 2 2

0 0 2 0 03 2 7 6 1

u

− − − − = +

x x

1 2 3 44 3 2 1

= − − − −

y x

3 32x x=

Example 3 What do you think of system: ; From output equation: Only the states and are present in the output variables. From state equations: These two states are uncoupled from the rest, and they in fact correspond to two modes (of eigenvalues +1 and -3). They will not carry complete information about the other two states ( and ), which depend on the other two modes too. ⇒States and are unobservable System is unobservable Note: To check whether the system is “detectable” we find the other two eigenvalues using MATLAB -2, +5 undetectable (because an unobservable mode—eigenvalue +5—is unstable).

,

2x 4x

3x1x1x 3x

1 2 3 4 10 1 0 0 24 3 2 1 30 0 0 3 4

u

= + −

x x [ ]0 1 0 3= −y x

Companion Form and Controllability

Companion Form and Controllability

Input-output differential equation State variables: ⇒ Companion form:

,

1

1 01 ...n n

nn n

d y d ya a y udt dt

−

− −+ + + =

1 11 2, ,..., / .n n

nx y x y x d y dt− −= = =

0 1 2 1

0 1 0 . . . 00 0 1 . . . 0 00 0 0 . . . 0 0. . . . . . . , . . . . . . . 00 0 0 . . . 1 1

. . . na a a a −

= = − − − −

A B

Companion Form and Controllability Controllability matrix: Note: Det is unchanged if we add a multiple of a column to another column Det P = Det I =1 ⇒ Rank P = n ⇒ system is controllable.

Conclusion: If a system can be expressed in the companion form, then it is controllable. Note: Other names companion form: Control canonical form, controllable canonical form, and phase-variable canonical form

,

12

1 2 1

0 0 0 . 10 0 .

10 11

n

n n n

aa a a

−

− − −

= − − − + • •

P

Other Considerations: Stabilizability; Detectability;

Output Controllability

Stabilizability and Detectability Stabilizability: If all the unstable modes are controllable, the system is stabilizable. Note 1: If a mode is controllable but unstable, that mode can be stabilized through feedback control Note 2: A unstable system can be uncontrollable yet stabilizable (if all the unstable modes are controllable) Detectability: If all the unstable modes are observable, the system is detectable. Note 1: If a mode is observable but unstable, that mode is present in the output and hence it can be estimated. Note 2: An unstable system can be unobservable yet detectable.

,

Output Controllability Note: What we presented before was the condition for “Complete State Controllability” Often we are primarily interested in controlling the output. Then “Complete Output Controllability” is applicable. Essentially, this concerns the controllability of “Cx+Du” Then, without having to give a complete proof, the condition for output controllability may be expressed as: Output Controllable iff Rank [ CB | CA B|…|CAn-1B | D ] = m Where, m = order of y

,