A Deeper Look at LPV
Stephan Bohacek
USC
General Form of Linear Parametrically
Varying (LPV) Systems
x(k+1) = A(k)x(k) + B(k)u(k)
z(k) = C(k)x(k) + D(k) u(k)
(k+1) = f((k))
linear parts
nonlinear part
xRn
u Rm
 - compact
A, B, C, D, and f are continuous functions.
How do LPV Systems Arise?
• Nonlinear tracking
(k+1)=f((k),0) – desired trajectory
(k+1)=f((k),u(k)) – trajectory of the system under control
Objective: find u such that
| (k)-(k) | 0 as k  .
A(k)
Define x(k) = (k) - (k)
x(k+1) = A(k)x(k) + B(k)u(k)


(k+1)= f((k),0) + f((k),0) ((k)- (k)) + fu((k),0) u(k)
B(k)
How do LPV Systems Arise ?
• Gain Scheduling
A(k)


x(k+1) = g(x(k), (k), u(k))  gx(0,(k),0) x(k) + gu(0,(k),0) u(k)
B(k)
(k+1) = f(x(k), (k), u(k)) – models variation in the parameters
Objective: find u such that |x(k)| 0 as k 
Types of LPV Systems
Different amounts of knowledge about f lead to a different types of LPV systems.
• f()  - know almost nothing about f (LPV)
• |f()- |< - know a bound on rate at which  varies (LPV with rate limited
parameter variation)
• f() - know f exactly (LDV)
•  is a Markov Chain with known transition probabilities (Jump Linear)
• f() where f() is some known subset of  (LSVDV)
– f()={0, 1, 2,…, n}
nominal
type 1
failure
type n
failure
ball of radius 
centered at n
– f()={B(0,), B(1,), B(2,),…, B(n ,)}
nominal
type 1
failure
type n
failure
Stabilization of LPV Systems
Packard and Becker, ASME Winter Meeting, 1992.
Find SRnn and ERmn such that

S

A S + B

 

C S


D S
(A S + B E )T (C S)T
E
In this case,
(D S)T 
S
I
I
x(k+1) = (A+B (ES-1)) x(k)
(k+1) = f((k))


 >0


for all 
is stable.
If  is a polytope, then solving the LMI for all  is easy.
Cost
For LTI systems, you get the exact cost.
x(0) X x(0) = k[0,] |Cj[0,k](A+BF)x(0)|2
+ |DF(j[0,k](A+BF))x(0)|2
where X = ATXA - ATXB(DTD + BTXB)-1BTXA + CC
For LPV systems, you only get an upper bound on the cost.
xT X x  k[0,] |C(k)j[0,k](A(j)+B(j)F)x|2
+ |D(k)F(j[0,k](A(j)+B(j)F))x|2
}
where X=S-1
If the LMI is not solvable, then
• the inequality is too conservative,
• or the system is unstabilizable.
depends
on 
LPV with Rate Limited Parameter Variation
Wu, Yang, Packard, Berker, Int. J. Robust and NL Cntrl, 1996
Gahinet, Apkarian, Chilali, CDC 1994
Suppose that | f()-  | <  and
S = i{1,N} bi() Si
E = i{1,N} bi() Ei
where Si Rnn, Ei Rmn and {bi}
is a set of orthogonal functions
such that |bi () - bi (+)| < .
T
T
T
 S + i{1,n} iSi (A S + B E ) (C S) (D S)

 AS + B E
S


C S
I


D S
I
then
We have assumed
solutions to the LMI have
a particular structure.



 > 0 for all  and | |< 
i




x(k+1) = (A(k) + B(k)E(k) X(k))x(k) is stable.
where X = (S)-1
Cost
You still only get an upper bound on the cost
x(0) X(0) x(0)  k{0,} |C(k)j{0,k}(A(j)+B(j)F(k))x(0)|2
+
|D(k)F(k)(j{0,k}(A(j)+B(j)F(k)))x(0)|2
where X = (i[1,N] bi() Si)-1 and F(k) = E(k) X(k)
If the LMI is not solvable, then
• the assumptions made on S are too strong,
• the inequality is too conservative,
• or the system is unstabilizable.
Might the solution to the LMI be discontinuous?
Linear Dynamically Varying (LDV) Systems
Bohacek and Jonckheere, IEEE Trans. AC
Assume that f is known.
x(k+1) = A(k)x(k) + B(k)u(k)
z(k) = C(k)x(k) + D(k) u(k)
(k+1) = f((k))
A, B, C, D and f are continuous functions.
Def: The LDV system defined by (f,A,B) is stabilizable if there exists
F :   Z  Rmn
such that, if
then
x(k+1) = (A(k) + B(k)F((0),k)) x(k)
(k+1) = f((k))
j
|x(k+j)|  (0)(0)
|x(k)|
for some (0) <  and (0) < 1.
Continuity of LDV Controllers
Theorem: LDV system (f,A,B) is stabilizable if and only if there
exists a bounded solution X :   Rnn to the functional algebraic
Riccati equation
X = ATXA + CTC - ATXB(DTD + BTXB)-1BTXA
In this case, the optimal control is
u(k) = - (DT(k) D (k) + BT(k) X (k) B (k))-1BT (k) X (k) A(k) x(k)
and X is continuous.
Since X is continuous, X can be estimated by determining X on a grid of .
Continuity of LDV Controllers
Continuity of X implies that if |1- 2| is small, then
x oT  X  1  X  2 x o  x oT
  k
   Acl
k 0  j  0
k
T
f j  1
C
T
f k  1
k
C k
Acl j



f 1
f  1
j 0
k

 x  ...
  Acl
C
C k
Acl j

j  2  f k  2  f  2 
f  2   o
f
j 0
j 0

T
T
is small.
Which is true if
lim Acl j
 Acl j
0
f  1
f  2 
j 
k where  and  are independent of , which
A
cl



f j  
is more than stabilizability provides.
j 0
k
or
which only happened when f is stable,
LDV Controller for the Henon Map
1 k  1  1  (1.4  u k )1 k 2   2 k 
 2 k  1  0.31 k 

u k   F1 k 

 x1 k  
F2 k  

 x 2 k 
F1
F2

H Control for LDV Systems
Bohacek and Jonckheere SIAM J. Cntrl & Opt.
xk  1  A k  xk   B1 k  u k   B2 k  wk 
z k   C k  xk   D k u k 
 k  1  f  k 
Objective:
Find u  l2 such that xk   0 for w  0
zl
2   , w  l
and
2
wl
2
Continuity of the H Controller
Theorem: There exists a controller such that
zl
2   , w  l
2
wl
2
if and only if there exists a bounded solution to
T
T
T
X = CC + AXf()A - L(R)-1L
In this case, X is continuous.
X May Become Discontinuous as  is Reduced
LPV with Rate Limited Parameter Variation
Suppose that | f()-  | <  and
S = i{1,N} bi() Si
E = i{1,N} bi() Ei
where Si Rnn, Ei Rmn and {bi}
is a set of orthogonal functions
such that |bi () - bi (+)| < .
T
T
T
 S + i{1,n} iSi (A S + B E ) (C S) (D S)

 AS + B E
S


C S
I


D S
I



 > 0 for all  and | |< 
i




If the LMI is not solvable, then
• the set {bi} is too small (or  is too small),
• the inequality is too conservative,
• or the system is unstabilizable.
Linear Set Valued Dynamically Varying
(LSVDV) Systems
Bohacek and Jonckheere, ACC 2000
xk  1  A k  xk   B k u k 
 C k  xk 
z k   



D
u
k



k


 k  1  f  k 
set valued
dynamical system
A, B, C, D and f are continuous functions.
 is compact.
LSVDV systems
2
type 1 failure
f
 0
nominal
f
f
type 2 failure
1
1 - Step Cost


J  , x  : max x T AT X   A  CT C x
 f  
For example, let f()={1, 2}
 



J  , x   max xT AT X 1 A  CT C x, xT AT X 2 A  CT C x
 
alternative 1
alternative 2
Cost if Alternative 1 Occurs
xT Qxx : x  1
where Q = ATX1A + CTC
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Cost if Alternative 2 Occurs
xT Qxx : x  1
where Q = AX2A + CC
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Worst Case Cost
 



J  , x   max xT AT X 1 A  CT C x, xT AT X 2 A  CT C x
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
The LMI Approach is Conservative
2
conservative
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Worst Case Cost
2
piece 2
1.5
1
0.5
piece 1
0
-0.5
-1
-1.5
-2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
•non-quadratic cost
•piece-wise quadratic
Piecewise Quadratic Approximation of the Cost
N
Define cones Ci : i  0, , N  such that  Ci  R n
i 0
4
C1
3
2
CN
C0
1
0
-1
-2
-3
-4
-4
-3
-2
-1
0
1
2
3
4
Define X(x,) := maxiN xTXi()x

quadratic
Piecewise Quadratic Approximation of the Cost
Given Qi : i  P, find X j : j  N such that


x T AT Qi A  I x  x T X j x
n


for x  C j   x   l cl :   0, x  1


l 1
C j - positive cone
not an LMI
Piecewise Quadratic Approximation of the Cost


max x T AT Qi A  I x  x T X 0 x
i
x  C0
3
max x T Qi x
2
i
x T Q2 x


 cone - C 0


1
0
-1
xT X 0 x
-2
-3
-3
x T Q1 x
-2
-1
0
1
2
3
Piecewise Quadratic Approximation of the Cost


max x T AT Qi A  I x  x T X 1 x x  C1
i
3


 cone - C1


2
1
0
-1
-2
-3
-3
-2
-1
0
1
2
Note : X 1 is not positive semi - definite
3
Piecewise Quadratic Approximation of the Cost
Allowing non-positive definite Xi permits good approximation.
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Piecewise Quadratic Approximation of the Cost


max x T AT Qi A  I x  X  x, 
i
max X i  X
2.5
i
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
The Cost is Continuous
Theorem: If
1. the system is uniformly exponentially stable,
2. X : Rn    R solves
X  x,    max X  A x,    x T C T C x,
 f  
3. X(x, )  0,
then X is uniformly continuous.
Hence, X can be approximated:
• partition Rn into N cones, and
• grid  with M points.
Piecewise Quadratic Approximation of the Cost
Define X(x,,T,N,M) := maxiN xTXi(,T,N,M)x
such that
X(x,,T,N,M)  maxf() X(Ax,,T-1,N,M) + xTCTCx
X(x,,0,N,M) = xTx.
Would like
X(x,,0,N,M)  X(x,) as N,M,T  
number
of cones
number of
grid points
in 
time
horizon
X can be Found via Convex Optimization
The cone centered around first coordinate axis
C1 := {x :  > 0, x = e1 + y, y1=0, |y|=1}
depends N, the number is cones
convex optimization:
X 1  , T , N , M , K  
arg min
Q  Q T R n  n
K Q1,1   log1  Q j , j 
j 1
subject to : X A x,  , T  1, N , M , K   CT C  Q
for x  C1N ,   f  
X can be Found via Convex Optimization
The cone centered around first coordinate axis
C1 := {x :  > 0, x = e1 + y, y1=0, |y|=1}
depends N, the number is cones
convex optimization:
X 1  , T , N , M , K  
arg min
Q  Q T R n  n
K Q1,1   log1  Q j , j 
j 1
subject to : X A x,  , T  1, N , M , K   CT C  Q
for x  C1N ,   f  
Theorem:
X(x,,0,N,M,K)  X(x,) as N,M,T,K  
1

In fact, 0  X  x, , T , N , M , K   X  x, , T          2 
K

related to the continuity of X
Optimal Control of LSVDV Systems
X x,     2 X  x,  
only the direction is important

u * x,     u *  x,  
the optimal control is homogeneous
u *  x  y,    u *  x,    u *  y,  
but not additive
Summary
LPV
LPV with rate limited
parameter variation
increasing
knowledge
about f
increasing
computational
complexity
optimal in
the limit
LSVDV
LDV
increasing
conservativeness
might not be
that bad
optimal