EL8223 Homework 14 Solution
1. It is required to determine for each of the following systems whether it is input-to-state stable
(ISS):
(a) ẋ = Ax + Bf (u) with x ∈ Rn , A being a stable matrix, and f being some continuous
nonlinear function : ISS if f (0) = 0 : can be demonstrated easily using the Lyapunov
function V = xT P x with P being a symmetric positive definite matrix chosen such that
P A + AT P = −I. Hence, V̇ = −|x|2 + 2xT P Bf (u) ≤ − 21 |x|2 + 2|P Bf (u)|2 . Note that
if A is a stable matrix, we can find a symmetric positive definite matrix P to solve the
equation P A + AT P = −Q with Q being any given symmetric positive definite matrix.
(b) ẋ = −x + xu with x ∈ R : not ISS : consider u = 2, for instance, in which case x diverges
to ∞ for any nonzero initial condition.
(c) ẋ = −x3 + x2 u1 − xu2 + u1 u2 with x ∈ R and u = [u1 , u2 ]T : ISS : simple to demonstrate
using the Lyapunov function V = 12 x2 and the inequality pq ≤ a1 pa + 1b q b that holds for
any positive real numbers p, q, a, b satisfying a1 + 1b = 1.
2. Define z2 = x2 + x21 , z3 = x3 + x1 , and z4 = x4 + x1 . Hence,
ẋ1
ż2
ż3
ż4
=
=
=
=
(z2 − x21 )(z3 − x1 )(z4 − x1 )2
u1 + 2x1 x2 x3 x24
u2 + x2 x3 x24 + x2 x3
u3 + x2 x3 x24 + x31 .
(1)
Using the ISS Lyapunov function V1 = 21 x21 , it can be seen that the x1 subsystem is ISS with
inputs being (z2 , z3 , z4 ). To show this, expand out ẋ1 , note that the highest power of x1 that
occurs in the expression for ẋ1 is −x51 and upper-bound the remaining terms using the usual
inequality that pq ≤ a1 pa + 1b q b which holds for any positive real numbers p, q, a, b satisfying
1
1
a + b = 1.
Defining the control laws as:
u1 = −z2 − 2x1 x2 x3 x24
u2 = −z3 − x2 x3 x24 − x2 x3
u3 = −z4 − x2 x3 x24 − x31 ,
(2)
each of the z2 , z3 , and z4 subsystems are globally uniformly asymptotically stable. Hence,
the system can be viewed as a cascade interconnection of a globally uniformly asymptotically
stable system with state [z2 , z3 , z4 ]T and a ISS system with state x1 and input [z2 , z3 , z4 ]T .
Therefore, the overall system is globally uniformly asymptotically stable under the control
laws defined above.
3. The (x1 , x2 ) subsystem is ISS with input x3 and state xa = [x1 , x2 ]T . The x3 subsystem is
ISS with input [x1 , x2 ]T . We can use V1 = xTa P xa and V2 = 21 x23 . Here, P is a symmetric
positive definite matrix such that P A + AT P = −Q is a symmetric negative definite matrix
1
with A =
0
1
. We get:
−1 −1
V̇1 = −xTa Qxa + 2kxTa P Bx3
1
V̇2
1
= −xTa Qxa + 2kxTa Q 2 Q− 2 P Bx3
1
≤ − xTa Qxa + 2B T P Q−1 P Bk 2 x23
2
= −x43 + x3 x31
3
3
3
1
≤ −x43 + x43 + x41 = − x43 + x41 .
4
4
4
4
(3)
(4)
where B = [0, 1]T . Hence, from the above Lyapunov inequalities, α1 (r) = λmin (P )r2 , α1 (r) =
λmax (P )r2 , α2 (r) = α2 (r) = 12 r2 , α1 (r) = 21 λmin (Q)r2 , σ1 (r) = 2B T P Q−1 P Bk 2 r2 , α2 (r) =
3 4
3 4
4 r , and σ2 (r) = 4 r where λmin (X) and λmax (X) denote respectively the minimum and
maximum eigenvalues of a matrix X. Hence, the gains of the (x1 , x2 ) and the x3 subsystems
respectively are evaluated as:
s
λmax (P ) 4k1 B T P Q−1 P B
−1
γ1 (r) = α−1
|k|r
1 ◦ α1 ◦ α1 ◦ k1 σ1 (r) =
λmin (P )
λmin (Q)
1
−1
4
γ2 (r) = α−1
2 ◦ α2 ◦ α2 ◦ k2 σ2 (r) = k2 r
(5)
where k1 and k2 are any numbers larger than 1. By the small gain theorem, we know that if
γ1 ◦ γ2 (r) < r for all r > 0, then the overall system is globally asymptotically stable. This
yields the condition
s
1
λmax (P ) 4k1 B T P Q−1 P B
(6)
|k|k24 r < r
λmin (P )
λmin (Q)
from which a range of values of k for which the overall system is globally asymptotically stable
directly follows as (note that k1 and k2 can be picked to be any numbers larger than 1, i.e.,
can be arbitrarily close to 1):
s
λmin (P )
λmin (Q)
|k| <
(7)
λmax (P ) 4B T P Q−1 P B
2