Robust ControlRobust Control

Robust ControlRobust Control

U i M d li U i• Uncertainty: Modeling Uncertainty• Disturbances: External Disturbances, ,

Noise

• Classical Control: Relative StabilityG h• Gain Margin, Phase Margin

• Optimal Control vs. Robust Control

2 Multivariable Linear 2. Multivariable Linear Systemsy

SISO:Linear Time-Invariant Systems

( ) ( ) ( )x t Ax t Bu t+( ) ( ) ( )( ) ( ) ( )

x t Ax t Bu ty t Cx t Du t

= += +

: 1, :1 1, :1 1: , : 1, :1 , :1 1

xx n u yA n n B n C n D

× × ×× × × ×: , : 1, :1 , :1 1x x x xA n n B n C n D× × × ×

( ) (0) ( ) ( )sX s x AX s BU s− = +

( ) ( )1 1

( ) (0) ( ) ( )( ) ( ) (0) ( )sX s x AX s BU ssI A X s x BU s

− −

− = +− = +

( ) ( )( ) ( )

1 1

1 11 1

( ) (0) ( )

( ) (0) ( )

X s sI A x sI A BU s

x t sI A x sI A BU s− −− −

= − + −

⎡ ⎤ ⎡ ⎤= − + −⎣ ⎦ ⎣ ⎦L L( ) ( )( ) ( ) ( )⎣ ⎦ ⎣ ⎦

SISO:Linear Time-Invariant Systems

( ) ( ) [ ]1 11 1 1

2 2

( ) (0) ( )n n

x t sI A x sI A BU s

A t A t

− −− − −⎡ ⎤ ⎡ ⎤= − + − ⊗⎣ ⎦ ⎣ ⎦

⎡ ⎤

L L L

( ) 11

2 !At A t A tsI A e I At

n−− ⎡ ⎤− = = + + + + +⎣ ⎦L

( ) (0) ( )At At

t

x t e x e Bu t= + ⊗( )

0(0) ( )

tAt A te x e Bu dτ τ τ−= + ∫

( )

0( ) ( ) ( ) (0) ( ) ( )

tAt A ty t Cx t Du t Ce x C e Bu d Du tτ τ τ−= + = + +∫

SISO:Impulse ResponseSISO:Impulse Response

( ) ( )u t tδ= [ ]( ) ( ) 1U s tδ= =L

( ) ( ) ( ) ( )Y s G s U s G s= =

[ ]1( ) ( ) ( )y t G s g t−= =L

[ ] [ ] [ ] [ ]1 1 1 1( ) ( ) ( ) ( ) ( ) ( )y t Y s G s U s G s U s− − − −= = = ⊗L L L L[ ] [ ] [ ] [ ]

0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )t

y

g t u t g t u dτ τ τ= ⊗ = −∫

SISO:Impulse ResponseSISO:Impulse Response

( )

If we assume zero initial conditions,

( ) ( ) ( ) ( ) ( )t A ty t Cx t Du t C e Bu d Du tτ τ τ+ −= + = +∫

( )0

( )

( ) ( ) ( ) ( ) ( )

( ) ( )t A t

y t Cx t Du t C e Bu d Du t

Ce B D t u dτ

τ τ

δ τ τ τ+ −

= + = +

= + −

∫∫ ( )

0( ) ( )

0( )( )

At tCe B D tg t

δ ≥⎧ += ⎨

∫

( )00

g tt⎨ <⎩

Complete solution:tA ∫0( ) (0) ( ) ( )tAty t Ce x g t u dτ τ τ= + −∫

Linear Time Varying SystemLinear Time-Varying System

Linear Time Varying SystemLinear Time-Varying System

Linear Time Invariant SystemsLinear Time-Invariant Systems

( ) ( ) ( )( ) ( ) ( )

x t Ax t Bu ty t Cx t Du t

= ++( ) ( ) ( )

: 1, : 1, : 1x u y

y t Cx t Du tx n u n y n

= +× × ×, ,

: , : , : , :x u y

x x x u y x y u

y

A n n B n n C n n D n n× × × ×

( )( ) ( ) ( ) (0) ( ) ( )tAt A ty t Cx t Du t Ce x C e Bu d Du tτ τ τ−+ + +∫ ( )

0( ) ( ) ( ) (0) ( ) ( )y t Cx t Du t Ce x C e Bu d Du tτ τ= + = + +∫

Impulse ResponseImpulse Response( ) ( ) ( )Y G U( ) ( ) ( )Y s G s U s=

[ ]1 ( ) ( )G s g t− =L

[ ] [ ] [ ] [ ]1 1 1 1( ) ( ) ( ) ( ) ( ) ( )y t Y s G s U s G s U s− − − −= = = ⊗L L L L[ ] [ ] [ ] [ ]

0( ) ( ) ( ) ( )

tg t u t g t u dτ τ τ= ⊗ = −∫

Impulse ResponseImpulse Response

2 Input 2 Output System2-Input 2-Output System

1 11 12 1( ) ( ) ( ) ( )( ) ( ) ( )

Y s G s G s U sY G U

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥

1 11 12 1

2 21 22 2

( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

Y s G s U sY s G s G s U s

Y G U G U

= = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

+1 11 1 12 2

2 21 1 22 2

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

Y s G s U s G s U sY s G s U s G s U s

= += +

1 111 12 11 12

1

( ) ( ) [ ( )] [ ( )]( ) ( ) [ ( )]

g s g s G s G sg s g s G s

− −

−

⎡ ⎤=⎢ ⎥

⎣ ⎦

L LL L 1[ ( )]G s−

⎡ ⎤⎢ ⎥⎣ ⎦21 22 21( ) ( ) [ ( )]g s g s G s⎣ ⎦ L L 22[ ( )]G s⎣ ⎦

2 Input 2 Output System2-Input 2-Output System

Example 2 1Example 2.1

Example 2 1Example 2.1

Example 2 1Example 2.1

Discrete-Time State Model and Simulation

Discrete Time State EquationsDiscrete-Time State Equations( ) ( ) ( )v t A v t B u t= +( ) ( ) ( )( ) ( ) ( )

c c

c c

v t A v t B u ty t C v t D u t

= += +

0( ) ( )0( ) ( ) ( )

tA t t A tcv t e v t e B u dτ τ τ− −= + ∫

00

0

( ) ( ) ( )

,

ct

kT T

t kT T t kT= + =

∫

( )( ) ( ) ( )kT TAT A kT T

ckTv kT T e v kT u kT e B dτ τ

+ + −+ = + ∫

Discrete Time State EquationsDiscrete-Time State Equations

( 1) ( ) ( )( ) ( ) ( )

x k Ax k Bu ky k Cx k Du k

+ = += +( ) ( ) ( )y k Cx k Du k= +

( ) ( )AT

x k v kTA

=

( )( )

AT

kT T TA kT T A

A e

B B d d Bτ τ+ + −

=

∫ ∫( )( )

0

A kT T Ac ckT

B e B d e d Bτ ττ τ+= =∫ ∫

Transfer FunctionsTransfer Functions

Transfer FunctionsTransfer Functions

Transfer FunctionsTransfer Functions

Example 2 2Example 2.2

Frequency ResponseFrequency Response

Frequency Response for SISO Systems

Frequency Response for MIMO Systems

Frequency Response for MIMO Systems

Singular Value DecompositionSingular Value Decomposition

: th columns of iU i U: th columns of iV i V

Unitary MatrixUnitary Matrix

Kronecker delta function

{ } : left singular vectoriU

{ } : right singular vectoriV

SVDSVD

{ } i l l f M{ } : singular value of i Mσ

SVDSVD

• SVD: provides a detailed picture of how the matrix operates on a vector

11 0 Vσ +⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥

11

221 2

0yn

VUSV U U U

σ ++

⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎡ ⎤= ⎣ ⎦ ⎢ ⎥⎢ ⎥y

unV +

⎣ ⎦ ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

p

i i iU Vσ +=∑1i=

SVDSVD

( )p p

i i i i i iMx U V x V x Uσ σ+ += =∑ ∑ ( )1 1

(scalar) : the length of the input in the direction

i i i i i ii i

iV x= =

+

∑ ∑( ) g p

defined by the given right singular vector i

iV

: the length of the output vector in the direction defined by t

i iV xσ +

he left singular vector iUdefined by the left singular vector : the gain for an input in the direction

i

i

Uσof a right singular vector iV

SVDSVD

⎛ ⎞

1

p

j i i i j j ji

MV U V V Uσ σ+

=

⎛ ⎞= =⎜ ⎟⎝ ⎠∑p p

i i i i i iM U V VUσ σ+

+ + +

⎝ ⎠

⎛ ⎞= =⎜ ⎟⎝ ⎠∑ ∑

1 1i i

p

M MV VU Uσ σ

= =

+ +

⎜ ⎟⎝ ⎠

⎛ ⎞= ⎜ ⎟

∑ ∑

∑1

2

j i i i j ji

M MV VU U

V U U V

σ σ

σ σ σ=

+

= ⎜ ⎟⎝ ⎠

= =

∑

2

j j j j j j j

i i i

V U U V

MM U U

σ σ σ

σ+

= =

=

: non-negative square roots of eigenvalue of or i MM M Mσ + +

SVDSVD

1i i iMV U

Vσ=

1iV

Uσ σ

=

=i i iUσ σ

SVDSVD

( ) ( ) 2MxMx Mx x xσ σ+ += ⇒ =( ) ( )

( )2 0

x

x M Mx xσ+ +⇒ − =( ) 0x M Mx x

Mx

σ⇒ =

≤ 1

if

xσ σ≤ =

≥⎧ if0 ifu

y upn

y u

n nMxn nx

σσ σ

≥⎧≥ = = ⎨ <⎩ y u⎩

SVDSVD

( ) ( ) ( )MAV MAV AV

MA M Aσ σ σ= = ≤( ) ( ) ( )

cf

V AV V

xy x y≤cf. xy x y≤

The Principle GainsThe Principle Gains

0( ) ( ) j ty t G j u e ωω= 0( ) ( )

( ) ( ) ( ) ( )p

y j

G j U Vω σ ω ω ω+=∑1

( ) ( ) ( ) ( )i i ii

G j U Vω σ ω ω ω=

=∑

Example 2 3Example 2.3

Example 2 3Example 2.3

Poles for SISO SystemsPoles for SISO Systems

Poles for MIMO SystemsPoles for MIMO Systems

Example 2 4Example 2.4

StabilityStability

StabilityStability

Internal StabilityInternal Stability

Internal StabilityInternal Stability

• Internally Stable: If all internal signals and all possible outputs g p premain bounded given that all possible inputs are boundedpossible inputs are bounded

Change of Basis: Similarity Transformations

ControllabilityControllability

ObservabilityObservability