This paper proposes an approach to algorithmically synthesize control strategies for discrete-time nonlinear uncertain systems based on reachable set computations using the ellipsoidal calculus. For given ellipsoidal initial sets and bounded ellipsoidal disturbances, the proposed algorithm iterates over conservatively approximating and LMI-constrained optimization problems to compute stabilizing controllers. The method uses first-order Taylor approximation of the nonlinear dynamics and a conservative approximation of the Lagrange remainder.

1. Control of Uncertain Nonlinear Systems
Using Ellipsoidal Reachability Calculus
Leonhard Asselborn
Dominic Groß
Olaf Stursberg
Institute of Control and System Theory
University of Kassel
l.asselborn@uni-kassel.de
dgross@uni-kassel.de
stursberg@uni-kassel.de
www.control.eecs.uni-kassel.de

2. Problem Setting
Problem:
• discrete-time nonlinear system: xk+1 = f (xk , uk ) + Gvk
• unknown but bounded additive disturbance and input contraints
• steer the uncertain nonlinear system from an initial set into a terminal
region T
Solution Approach:
• Successive linearization with conservative error representation
• Conservative reach set approximation
• Offline algorithmic synthesis of a feedforward and feedback control by
semidefinite programming
Literature:
•
Kurzhanskiy, Varaiya 2007: linear system
dynamic with disturbances, controller synthesis
•
Holzinger, Scheeres 2012: nonlinear system
dynamic, controller synthesis
•
q0
q3
x2
Scholte, Campbell 2003: nonlinear system
dynamic with disturbances, no controller
synthesis
Introduction
Problem Definition
Approximation
T
Method
Example
q1
X3 q2
x1
X2
Conclusion
X0
X1
Appendix
2

3. Reach Set Computation - Basics
1. Set representation:
Ellipsoid: E = ε(q, Q) = x ∈ Rn |(x − q)T Q−1 (x − q) ≤ 1 ,
Q = QT > 0, q ∈ Rn
Polytope: P = {x ∈ Rn | Kx ≤ b, K ∈ Rnp ×n , b ∈ Rnp }.
2. Dynamical system:
xk+1 = f (xk , uk ) + Gvk ,
(1)
x0 ∈ X0 = ε(q0 , Q0 ) ⊂ Rn ,
(2)
uk ∈ U = {u ∈ Rm | Ruk ≤ b},
vk ∈ V = ε(0, Σ) ⊆ Rn
3. One-step reachable set:
Xk+1 = {x | xk ∈ Xk , uk ∈ U, vk ∈ V : xk+1 = f (xk , uk ) + Gvk }
= F (Xk , U) ⊕ GV
(3)
⊕: Minkowski addition.
Introduction
Problem Definition
Approximation
Method
Example
Conclusion
Appendix
3

4. Problem Definition
Problem 1
Determine a control law uk = κ(xk , k), for which it holds that:
uk ∈ U ∀ k ∈ {0, 1, . . . , N − 1}, N ∈ N, xk ∈ Xk
and that it stabilizes the nonlinear system from an initial set X0 = ε(q0 , Q0 ) in
a finite number N of time steps into an ellipsoidal terminal set T = ε(0, T )
which is parametrized by T ∈ Rn×n and is centered in the origin 0:
˜
∃ N ≥ 0 : Xk+1 = F (Xk , Uk ) ⊕ GV
XN ⊆ T = ε(0, T ), k ∈ {0, 1, . . . , N − 1}
Introduction
Problem Definition
Approximation
Method
Example
Conclusion
Appendix
4

5. Preliminaries for Solution
•
The set valued operator F (Xk , U) may return a possibly nonconvex set.
•
A conservative approximation by convex reachable sets is chosen:
ˆ
Xk ⊆ Xk
•
ˆ
Structure of the control law for xk ∈ Xk = ε(qk , Qk ):
uk = Hk ek + dk , ek = xk − qk
•
The set valued mapping of the control law
¯
Uk = Hk Ek + dk ⊆ U,
with the error ellipsoid
ˆ
Ek = Xk − qk = ε(0, Qk ).
Introduction
Problem Definition
Approximation
Method
Example
Conclusion
Appendix
5

6. Local Conservative Linearization of the Dynamics (1)
•
¯
First order Taylor series expansion with ξk = [xk , uk ]T and ξk = [qk , p]T :
¯
∃z ∈ αξk + (1 − α)ξk | α ∈ [0, 1] :
∂f (ξk )
¯
f (xk , uk ) = f (ξk ) +
∂ξk
•
¯
¯
(ξk − ξk ) + L(ξk , z)
¯
ξk =ξk
¯
Representation of the nonlinear dynamics with the error term L(ξk , z):
¯
xk+1 ∈ Ak (xk − xk ) + Bk (uk − uk ) + L(ξk , z) + f (¯k , uk ) + Gvk
¯
¯
x ¯
Introduction
Problem Definition
Approximation
Method
Example
Conclusion
Appendix
6

7. Local Conservative Linearization of the Dynamics (2)
•
¯
Overapproximation of L(ξk , z) by an ellipsoid Lell (ξk ) = ε(lk , Lk ):
˜
Xk+1 := Ak (Xk − qk ) ⊕ Bk (U − p) ⊕ Lell (ξk ) ⊕ f (qk , p) ⊕ GV ⊇ Xk+1
•
The set valued local closed-loop dynamics for the conservative locally
linearized system:
¯
˜
ˆ
Xk+1 =(Ak + Bk Hk )Xk ⊕ GV ⊕ Lell (ξk ) . . .
(4)
+ f (qk , p) − (Ak + Bk Hk )qk + Bk dk − Bk p
Introduction
Problem Definition
Approximation
Method
Example
Conclusion
Appendix
7

8. Ellipsoidal Calculus
Affine transformation of an ellipsoid ε(q, Q) by A ∈ Rn×n and b ∈ Rn×1 to an
ellipsoid:
A · ε(q, Q) + b = ε(A · q + b, AQAT )
˜k+1 of a set of ellipsoids is in general not an ellipsoid, but
The Minowski sum X
ˆ
it can be tightly enclosed by an ellipsoid Xk+1 :
˜
ˆ
Xk ⊆ Xk ⊆ Xk
ˆ
i.e. Xk is an over approximation of the exact reachable set Xk .
˜
Xk
Xk
ˆ
Xk
Introduction
Problem Definition
Approximation
Method
Example
Conclusion
Appendix
8

9. Problem Reformulation (1)
Effect of the chosen control law:
1. dk should steer the center point qk to the origin (feedforward); only
consider the center point dynamic in eq. (4):
qk+1 = Ak (xk − qk )|xk =qk + Bk (uk − p)|uk =dk + f (qk , p) + lk
= Bk (dk − p) + f (qk , p) + lk
ˆ
2. Hk should lead to a contraction of the ellipsoid Xk (feedback); consider
the error dynamic:
¯
∃z ∈ αξk + (1 − α)ξk | α ∈ [0, 1] :
¯
ek+1 = (Ak + Bk Hk )ek + Gvk + L(ξk , z) − lk .
corresponding set-valued mapping:
¯
ˆ
ˆ
Xk+1 − qk+1 = (Ak + Bk Hk ) (Xk − qk ) ⊕GV ⊕ Lell (ξk ) − lk ,
Ek+1
Introduction
Ek
Problem Definition
Approximation
Method
Example
Conclusion
Appendix
9

10. Problem Reformulation (2)
Notation for the difference ellipsoid before Minkowski addition:
ˇ
Ek+1 := (Ak + Bk Hk )Ek
The original problem is solved, if Hk and dk are chosen in any k such that the
difference ellipsoid Ek converges to an ellipsoid with constant volume and
shape, and such that the center point of the reach set ellipsoid converges to
the origin.
Introduction
Problem Definition
Approximation
Method
Example
Conclusion
Appendix
10

11. Solution based on Semidefinite Programming (1)
Center point convergence
To account for the time-varying linearized dynamic (Ak , Bk ) we use flexible
ˆ
Lyapunov functions [1] to obtain convergence of the center points of Xk to 0:
V
T
T
qk+1 M qk+1 − ρqk M qk ≤ αk
αk ≤
max
i∈{1,...,k}
ω i αk−i
k
Ellipsoidal shape convergence
ˇ
Shape matrix of the ellipsoid Ek+1 :
(Ak + Bk Hk )Qk (Ak + Bk Hk )T
constrained by a symmetric matrix to be optimized in SDP:
Sk ≥ (Ak + Bk Hk )Qk (Ak + Bk Hk )T
Input constraints
¯
Ensure, that Uk is inside of U in every k. Cast original constraints
R(Hk ek + dk ) ≤ b, ∀ek ∈ Ek into an LMI by the use of [8, 10]:
(b − ri dk )In
1
2
T T
Qk Hk ri
Introduction
1
2
ri Hk Qk
≥0
∀i = {1, . . . , nc }
b − r i dk
Problem Definition
Approximation
Method
Example
Conclusion
Appendix
11

12. Solution based on Semidefinite Programming (2)
Semidefinite programm to be solved in any k:
min
Sk ,Hk ,dk ,αk
trace(Sk )
(5)
s. t. :
 T
T
qk+1 M qk+1 − ρqk M qk ≤ αk

center point convergence:
q
= Bk (dk − p) + f (qk , p) + lk
 k+1

αk ≤ maxi∈{1,...,k} ω i αk−i

trace(Sk ) ≤ trace(Qk )

ellipsoidal shape convergence:
Sk
(Ak + Bk Hk )T

≥0

(Ak + Bk Hk )
Q−1
k


1
 (b − r d )I
2
ri Hk Qk 
i k n

input constraints:
≥ 0, ∀i = {1, . . . , nc }.
1
 Q 2 H T rT
b − r i dk
k
Introduction
Problem Definition
k
i
Approximation
Method
Example
Conclusion
Appendix
12

13. Algorithm for Controller Synthesis (1)
Given: f (x0 , u0 ), F (X0 , U), X0 , U, V as well as T, ρ, ω, α0 , kmax and M .
Define: k := 0
ˆ
while Xk T and k < kmax do
•
¯
Apply linearization procedure to get Ak , Bk , Lell (ξk )
•
Solve the optimization problem (5)
if no feasible Hk , dk is found do
stop algorithm (synthesis failed)
end if
•
Evaluate system dynamic (4) and compute the over-approximating
ˆ
ellipsoid Xk+1
•
k := k + 1
end while
return Hk , dk
Introduction
Problem Definition
Approximation
Method
Example
Conclusion
Appendix
13

14. Algorithm for Controller Synthesis (2)
Lemma 1
ˆ
If the control algorithm terminates with Xk ⊆ T, k ≤ kmax , the original
problem is successfully solved and a control law (5) exists which steers the
initial state x0 ∈ X0 into the target set T in N steps for any possible
disturbance vk ∈ V. Furthermore, the input constraint uk ∈ U holds for all
ˆ
0 < k < N , and the center point qk of the reach set Xk asymptotically
converges to the origin.
Proof (sketch):
ˆ
• Successful termination of the control algorithm implies that XN ∈ T and
consequently xN ∈ XN ⊆ T holds for all initial states x0 ∈ X0 and all
disturbances vk ∈ V.
ˆ
• By construction xk ∈ Xk implies that ek ∈ Ek . It follows, that the input
constraint uk ∈ U holds at each time step, if uk = Hk ek + dk ∈ U holds
for all ek ∈ Ek .
• The use of flexible Lyapunov functions imply that V (xk ) = q T M qk can be
k
ˆ
used to ensure asymptotic convergence of the center point qk of Xk (cf.
Lemma III.4 in [1]) over the considered horizon N .
Introduction
Problem Definition
Approximation
Method
Example
Conclusion
Appendix
14

15. Control of a Three-Tank-System: Dynamic Model
•
state: height of fluid inside each tank
•
input: valve for in- and outflow with nonlinear characteristic curve
•
disturbance: additive measurement uncertainty
•
task: fill empty tank system to reference height
System:
Introduction
Problem Definition
Approximation
Method
Example
Conclusion
Appendix
15

16. Control of a Three-Tank-System: Parameters
Dynamics with xk = xk + xref :
˜

x
˜
x1,k + τ · k1 · u2 − k2 · tanh(˜1,k − x2,k ) + v1,k
1,k




x
˜
x
˜
= x2,k + τ · k3 · tanh(˜1,k − x2,k ) − k4 · tanh(˜2,k − x3,k ) · (1 + u2 ) + v2,k 
2,k


x
x3,k + τ · k5 · tanh(˜2,k − x3,k ) + k6 · u2 − k7 · tanh(˜3,k ) + v3,k
x
˜
3,k

xk+1

Initial set: X0 = ε −xref , 1e
−3

2
0
0



0
0.4113
0 , xref = 0.3091 ,
2
0.2067
0
2
0
Lyapunov function: M = I3


0.2
Disturbance set : V = ε [0, 0, 0]T , 1e−6  0
0
0
0.2
0

0
0  ,
0.2
Input constraints: U = {ui ∈ [0, 1], i ∈ {1, 2, 3}}
Introduction
Problem Definition
Approximation
Method
Example
Conclusion
Appendix
16

17. Control of a Three-Tank-System: Results (1)
0.4
0.3
0.2
0.1
•
Implementation with
Matlab 7.12.0, YALMIP
3.0, and SeDuMi 1.3.
•
x3
Termination after 280 time
steps in 916s.
•
0
Reachability computations:
ellipsoidal toolbox ET [11].
−0.1
−0.2
−0.3
−0.4
0.5
0
x1
0.4
−0.5
0
0.2
−0.2
−0.4
x2
ˆ
Reachable sets Xk
Trajectory of the nonlinear dynamic
Target set T
Introduction
Problem Definition
Approximation
Method
Example
Conclusion
Appendix
17

18. Control of a Three-Tank-System: Results (2)
0.6
0.5
0.5
0
0.4
x1
1
−0.5
0
5
10
15
20
25
30
35
40
520
525
530
535
540
545
550
555
560
520
525
530
535
540
545
550
555
560
520
525
530
535
540
545
550
555
560
t
1
0.5
x2
1
0.5
0
0
−0.5
−0.5
40
0
5
10
15
20
25
30
35
t
0.6
0.3
x3
0.4
0.2
0.2
0
−0.2
0.1
0
5
10
15
20
25
30
35
40
t
Introduction
Problem Definition
Approximation
Method
Example
Conclusion
Appendix
18

19. Conclusion
Summary:
•
Algorithm for the stabilization of an uncertain nonlinear system
•
Reach set approximation by using conservative linearization
•
Stablization problem reformulated as an iterative problem
•
Numerical example to show a successful application
Future work:
•
Tighter approximation of the Lagrange remainder
•
Consider probabilistic modeling of the disturbances in order to get a
probabilistic success rate of the control law
Introduction
Problem Definition
Approximation
Method
Example
Conclusion
Appendix
19

20. References
[1]
M. Lazar:
Flexible Control Lyapunov Functions;
American Control Conference, pp. 102-107, 2009.
[2]
A. Kurzhanski, I. Valyi:
Ellipsoidal Calculus for Estimation and Control;
Birkh¨user, 2009.
a
[3]
T. O. Apostol:
Calculus, Vol. I;
Xerox College Publishing, 1967.
[4]
M. Althoff:
Reachability Analysis and its Application to the Safety Assessment of Autonomous Cars;
Disertation, TU Muenchen, 2010.
[5]
S. Boyd, L. E. Ghaoui, E. Feron, V. Balakrishnan:
Linear Matrix Inequalities in System and Control Theory;
SIAM Studies in Applied Mathematics, 1994.
[6]
A. A. Kurzhanskiy, P. Varaiya:
Reach set computation and control synthesis for discrete-time dynamical systems with disturbances;
Automatica, Vol. 47, pp. 1414-1426, 2011.
[7]
M. Althoff, O. Stursberg, M. Buss:
Reachability Analysis of Nonlinear Systems with Uncertain Parameters using Consverative Linearization;
IEEE Conf. on Decision and Control, pp. 4042-4048, 2008.
[8]
P. J. Goulart, E. C. Kerrigan, J. M. Maciejowski:
Optimization Over State Feedback Policies for Robust Control with Constraints;
Automatica, Vol. 42, Issue 4, pp. 523-533, 2006.
[9]
S. V. Rakovic, E. C. Kerrigan, D. Q. Mayne, J. Lygeros:
Reachability Analysis of Discrete-Time Systems with Disturbances;
IEEE Trans. on Automatic Control, Vol. 51, pp. 546-561,2006.
Introduction
Problem Definition
Approximation
Method
Example
Conclusion
Appendix
20

21. References
[10] A. Nemirovskii:
Advances in Convex Optimization: conic programming;
Int. Congress of Mathematicians, pp. 413-444, 2006.
[11] A. A. Kurzhanskiy, P. Varaiya:
Ellipsoidal Toolbox;
http://code.google.com/p/ellipsoids, 2006.
[12] A. Girard, C. L. Guernic, O. Maler:
Efficient Computation of reachable sets of linear time-invariant systems with inputs;
Hybrid Systems: Computation and Control, Springer-LNCS, Vol. 3927, pp. 257-271, 2006.
[13] A. Girard, C. L. Guernic:
Efficient reachability analysis for linear systems using support functions;
Proc. 17th IFAC World Congress, pp. 8966-8971, 2008.
[14] A. A. Kurzhanskiy, P. Varaiya:
Ellipsoidal Techniques for Reachability Analysis of Discrete-Time Linear Systems;
IEEE Trans. on Automatic Control, Vol. 52(1), pp. 26-38, 2007.
[15] M. J. Holzinger, D. J. Scheeres:
Reachability Results for Nonlinear Systems with Ellipsoidal Initial Sets;
IEEE Trans. on Aerospace and Electronic Systems, Vol. 48(2), pp. 1583-1600, 2012.
[16] E. Scholte, M. E. Campbell:
A Nonlinear Set-Membership Filter for On-line Applications;
Int. J. of Robust and Nonlinear Control, Vol 13(15), pp.1337-1358, 2003.
[17] F. Yang, Y. Li:
Set-Membership Fuzzy Filtering for Nonlinear Discrete-Time Systems;
IEEE Trans. on Systems, Man, and Cybernetics, Vol. 40(1), pp. 116-124, 2010.
[18] T. Dang:
Approximate Reachability Computation for Polynomial Systems;
Hybrid Systems: Control and Computation, Springer-LNCS, Vol. 3927, pp 138-152, 2006.
[19] O. Stursberg, B.H. Krogh:
Efficient Representation and Computation of Reachable Sets for Hybrid Systems;
Hybrid Systems: Control and Computation, Springer-LNCS, Vol. 2623, pp 482-497, 2003.
Introduction
Problem Definition
Approximation
Method
Example
Conclusion
Appendix
21

22. Input Constraint
By substituting (5) into the input constraint in (1) it can be seen that the input constraint holds for all ek ∈ Ek if:
R(Hk ek + dk ) ≤ b, ∀ek ∈ Ek
(6)
By row-wise maximization of the linear inequality in (6), the universal quantifier can be eliminated and the following condition is obtained:
max ri w ≤ b − ri dk , ∀i = {1, . . . nc }
w∈W
m
W = {w ∈ R
| w = Hk ek , ek ∈ Ek }
(7)
(8)
−1
The ellipsoid Ek can be mapped into a unit ball ||zk ||2 ≤ 1 by the change of variables according to zk = Q 2 ek . This yields:
k
W =
w ∈ R
1
w = Hk Q 2 zk ,
k
m
zk 2 ≤ 1
(9)
The maximization problem (7) subject to (8) can be recast as follows [8]:
max ri w =
w∈W
1
ri Hk Q 2
k
2 ≤ b − ri d k
(10)
Finally, the Euclidean norm in (10) can be expressed as LMI [10], resulting in:

(b − ri dk )In

1

T T
Q 2 Hk ri
k

1
ri Hk Q 2 
k  ≥ 0

b − ri d k
∀i = {1, . . . , nc }
(11)
Thus, (6) and (11) are identical and the proposition follows. Note that this results in nc LMI constraints.
Introduction
Problem Definition
Approximation
Method
Example
Conclusion
Appendix
22