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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
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