Backstepping Controller Synthesis for Piecewise Polynomial Systems: A Sum of Squares Approach

Outline Introduction The Main Result Example Conclusion
Backstepping Controller Synthesis for Piecewise
Polynomial Systems:
A Sum of Squares Approach
Behzad Samadi Luis Rodrigues
Department of Mechanical and Industrial Engineering
Concordia University
ACC 2009, St. Louis, Missouri

Outline of Topics
1 Introduction
2 The Main Result
3 Example
4 Conclusion

Main Contributions
To introduce a new class of hybrid systems: piecewise
polynomial (PWP) systems in strict feedback form

Main Contributions
To introduce a new class of hybrid systems: piecewise
polynomial (PWP) systems in strict feedback form
To formulate the controller synthesis problem as a convex
feasibility problem

Piecewise Polynomial Systems
The dynamics of a PWP system:
˙x(t) = fi (x(t)), if x(t) ∈ Pi (♦)
where x(t) ∈ Rn denotes the state vector and fi (x) ∈ Rn are
polynomial functions of x.

˙x(t) = fi (x(t)), if x(t) ∈ Pi (♦)
The cells, Pi , i ∈ I = {1, . . . , M}, partition a subset of the
state space X ⊂ Rn such that ∪M
i=1Pi = X, Pi ∩ Pj = ∅, i = j,
where Pi denotes the closure of Pi .

˙x(t) = fi (x(t)), if x(t) ∈ Pi (♦)
The cells, Pi , i ∈ I = {1, . . . , M}, partition a subset of the
state space X ⊂ Rn such that ∪M
i=1Pi = X, Pi ∩ Pj = ∅, i = j,
where Pi denotes the closure of Pi .
Each cell is described by
Pi = {x|Ei (x) ≻ 0}
where Ei (x) ∈ Rpi is a vector polynomial function of x and “≻”
represents an elementwise inequality.

Lyapunov Stability
Theorem
For the PWS system ˙x(t) = fi (x(t)) with fi (x) : Pi → Rn for i = 1, . . . , M continuous in x and
locally bounded, if there exists a continuous function V (x) such that
V (0) = 0
V (x) > 0 for all x = 0 in X
t1 ≤ t2 ⇒ V (x(t1)) ≥ V (x(t2))
then x = 0 is a stable equilibrium point. Moreover if there exists a continuous function W (x)
such that
W (0) = 0
W (x) > 0 for all x = 0 in X
t1 ≤ t2 ⇒ V (x(t1)) ≥ V (x(t2)) +
t2
t1
W (x(τ))dτ
and
x → ∞ ⇒ V (x) → ∞
then all trajectories in X asymptotically converge to x = 0.

Smooth Lyapunov Functions
Proposition
(Smooth Lyapunov functions) The PWS system (♦) is
asymptotically stable if there exists a positive deﬁnite C1 function
V (x) and a positive deﬁnite continuous function W (x) so that
V (0) = 0, W (0) = 0 and for all x ∈ Pi , i = 1, . . . , M
∇V (x)T
fi (x) ≤ −W (x)

Piecewise Smooth Lyapunov Functions
Proposition
(PWS Lyapunov functions) The PWS system (♦) is asymptotically stable if its
vector ﬁeld is continuous in x, i.e. for any i, j ∈ {1, . . . , M} such that Pi Pj = ∅,
fi (x) = fj (x), ∀x ∈ Pi Pj
and there exists positive deﬁnite functions V (x) and W (x) so that V (0) = 0,
W (0) = 0 and
V (x) is a continuous function where
V (x) = Vi (x), x ∈ Pi
where Vi : Pi → R is a C1 function,
W (x) is a continuous function,
for all x ∈ Pi , i = 1, . . . , M
∇V (x)T
i fi (x) ≤ −W (x)

PWP system in strict feedback form
The dynamics of this new class of systems can be written in the
form



˙x1 = f1i1 (x1) + g1i1 (x1)x2, for x1 ∈ P1i1
˙x2 = f2i2 (x1, x2) + g2i2 (x1, x2)x3, for [ x1
x2 ] ∈ P2i2
...
˙xk = fkik
(x) + gkik
(x)u, for x ∈ Pkik
where x is the state vector and is divided into k subvectors:
x =
x1
x2
...
xk
∈ Rn
, xj ∈ Rnj

Structure of the regions
It is assumed that for 1 ≤ j1 < j2, the projection of each region Pj2ij2
for
ij2 = 1, . . . , Mj2 on the (x1, . . . , xj1 ) space is a subset of only one of the regions
Pj1ij1
for ij1 = 1, . . . , Mj1 .
x1 P11
vvmmmmmmmmmmmmmmmm
((QQQQQQQQQQQQQQQQ
(x1, x2) P21

P22

P23

(x1, x2, x3) P31
}}{{{{{{{{

P32
}}{{{{{{{{
!!CCCCCCCC P33
!!CCCCCCCC
(x1, x2, x3, x4) P41 P44 P42 P45 P43 P46

Equilibrium point
It is also assumed that
fji⋆
j
(0, . . . , 0) = 0, ∀i⋆
j ∈ Ij (0, . . . , 0)
where
Ij (x1, . . . , xj ) := ij
x1
...
xj
∈ Pjij

PWP Systems with Discontinuous Vector Fields
Let us start from the following subsystem:
˙x1 = f1i1 (x1) + g1i1 (x1)x2, for x1 ∈ P1i1 ,
It is assumed that there exist a polynomial Lyapunov function V1(x1),
a polynomial controller x2 = γ1(x1) and a polynomial vector
Γ1i1 (x1) ∈ Rp1 such that for i1 = 1, . . . , M1



V1(x1) − λ(x1) is SOS
−∇V1(x1)T(f1i1 (x1) + g1i1 (x1)γ1(x1)) − Γ1i1 (x1)TE1i1 (x1)
−αV1(x1) is SOS
Γ1i1 (x1) is SOS
V1(0) = 0
γ1(0) = 0
where α 0 is ﬁxed and λ(x1) is a positive deﬁnite polynomial.

PWP Systems with Discontinous Vector Fields
A polynomial controller can then be designed for the following
subsystem
˙x1 = f1i1 (x1) + g1i1 (x1)x2, for x1 ∈ P1i1
˙x2 = f2i2 (x1, x2) + g2i2 (x1, x2)x3, for [ x1
x2 ] ∈ P2i2

A polynomial controller can then be designed for the following
subsystem
˙x1 = f1i1 (x1) + g1i1 (x1)x2, for x1 ∈ P1i1
˙x2 = f2i2 (x1, x2) + g2i2 (x1, x2)x3, for [ x1
x2 ] ∈ P2i2
Lyapunov function construction: We consider the following
candidate Lyapunov function
V2(x1, x2) = V1(x1) +
1
2
(x2 − γ1(x1))T
(x2 − γ1(x1))

Controller synthesis:
Find x3 = γ2(x1, x2), Γ2i2 (x1, x2)
such that
− ∇x1 V2(x1, x2)T
(f1i(1,2,i2)(x1) + g1i(1,2,i2)(x1)x2)
− ∇x2 V2(x1, x2)T
(f2i2 (x1, x2) + g2i2 (x1, x2)x3)
− Γ2i2 (x1, x2)T
E2i2 (x1, x2) − αV2(x1, x2) is SOS,
Γ2i2 (x1, x2) is SOS
γ2(0, 0) = 0
where α 0, i2 = 1, . . . , M2 and γ2(x1, x2) is a polynomial
function of x1 and x2.

PWP Systems with Discontinuous Vector Fields
Note that if this SOS program is feasible then the procedure can
be repeated for the next steps by adding the dynamics of xi for
i = 3, . . . , k. Assume that all SOS programs in the backstepping
procedure are feasible and at the last step the following candidate
Lyapunov function is used:
Vk(x) = Vk−1(x1, . . . , xk−1)
+
1
2
(xk − γk−1(x1, . . . , xk−1))T
(xk − γk−1(x1, . . . , xk−1))
where γk−1(x1, . . . , xk−1) is a polynomial function.

Finally, we search for a PWP control
u = γkik
(x), for x ∈ Pkik
for ik = 1, . . . , Mk. This step can be formulated as the
following SOS program:
Find u = γkik
(x), Γkik
(x)
such that
− ∇x1 V T
k (f1i(1,k,ik )(x1) + g1i(1,k,ik )(x1)x2)
− ∇x2 V T
k (f2i(2,k,ik )(x1, x2) + g2i(2,k,ik )(x1, x2)x3)
− . . . − ∇xk
V T
k (fkik
(x) + gkik
(x)u)
− Γkik
(x)T
Ekik
(x) − αVk is SOS,
Γkik
(x) is SOS
for ik = 1, . . . , Mk.

Summary:
PWP system with a discountinuous vector ﬁeld
Smooth Lyapunov function
PWP controller
In Theorem 2, we prove that the designed controller stabilizes the
whole PWP system in strict feedback form.

PWP Systems with Continuous Vector Fields
Summary:
PWP system
PWP Lyapunov function
PWP controller to make the vector ﬁeld of the closed loop
PWP system continuous
In Theorem 3, we prove that the designed controller stabilizes the
whole PWP system in strict feedback form.

Single-link Flexible-joint Robot
˙x1 = x2
˙x2 = −
MgL
I
sin(x1) −
K
I
(x1 − x3)
˙x3 = x4
˙x4 = −
Tf
J
+
K
J
(x1 − x3) +
1
J
u
where x1 = θ1, x2 = ˙θ1, x3 = θ2 and x4 = ˙θ2, u is the motor
torque and Tf = f2(x4) denotes the motor friction

Partitioning Variables
x1 in sin(x) and x4 in Tf
Tf = bmx4 + sgn(x4) Fcm + (Fsm − Fcm) exp(−
x2
4
c2
m
)
x4
f2(x4)
f2(x4)
ˆf2(x4)
-8 -6 -4 -2 0 2 4 6 8
-3
-2
-1
0
1
2
3
Three polynomial pieces for sin(x1) and two polynomial pieces for Tf
create 6 regions for the whole system.

Simulation results
Numerical simulation for a PWP controller with 6 regions:
Time
x1
Time
x2
Time
x3
Time
x4
0 2 4 60 2 4 6
0 2 4 60 2 4 6
-10
-5
0
5
10
0
1
2
3
-5
-4
-3
-2
-1
0
0
1
2
3
4

Summary of the contributions:
Strict feedback form for PWP systems was introduced.
Backstepping controller synthesis for this form of PWP
systems was formulated as an SOS program.
SOS Lyapunov functions for PWP systems with discontinuous
vector ﬁelds
PWP Lyapunov functions for PWP systems with continuous
vector ﬁelds

Backstepping Controller Synthesis for Piecewise Polynomial Systems: A Sum of Squares Approach

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