Root-Mean-Square Gains of Switched Linear Systems:
A Variational Approach ?
Michael Margaliot a and João P. Hespanha b
a
b
School of Electrical Eng.–Systems, Tel Aviv University, Israel 69978
Dept. of Electrical and Computer Eng., University of California, Santa Barbara, CA 93106-9560, USA
Abstract
We consider the problem of computing the root-mean-square (RMS) gain of switched linear systems. We develop a new
approach which is based on an attempt to characterize the “worst-case” switching law (WCSL), that is, the switching law
that yields the maximal possible gain. Our main result provides a sufficient condition guaranteeing that the WCSL can be
characterized explicitly using the differential Riccati equations (DREs) corresponding to the linear subsystems. This condition
automatically holds for first-order SISO systems, so we obtain a complete solution to the RMS gain problem in this case.
Key words: Switched and hybrid systems, bilinear control systems, optimal control, maximum principle, algebraic Riccati
equation, differential Riccati equation, Hamilton-Jacobi-Bellman equation.
1
Let S denote the set of all piecewise constant switching
laws. Many important problems in the analysis and design of switched systems can be phrased as follows.
Introduction
Consider the switched linear system
ẋ = Aσ(t) x + Bσ(t) u,
y = Cσ(t) x,
(1)
where x ∈ Rn , u ∈ Rm , y ∈ Rk , and the switching
signal σ : R+ → {1, 2, . . . , l} is a piecewise constant
function specifying, at each time instant t, the index
of the currently active system. Roughly speaking, (1)
models a system that can switch between the l linear
sub-systems:
ẋ = Ai x + Bi u,
y = Ci x,
(2)
for i = 1, . . . , l. Note that we consider subsystems with
no direct input-to-output term. To avoid some technical difficulties, we assume throughout that each linear
subsystem is a minimal realization.
? An abridged version of this manuscript was presented
at the IEEE Conf. on Decision and Control (CDC’07).
Prof. Hespanha’s research was supported by NSF Grants
ECS-0242798 and CCR-0311084. Corresponding author:
Dr. Michael Margaliot, School of EE-Systems, Tel Aviv University, Israel 69978. Tel: +972 3 640 7768; Fax +972 3 640
5027.
Email addresses: michaelm@eng.tau.ac.il (Michael
Margaliot), hespanha@ece.ucsb.edu (João P. Hespanha).
Preprint submitted to Automatica
Problem 1 Given S 0 ⊆ S and a property P of dynamic
systems, determine whether the switched system (1) satisfies property P for every σ ∈ S 0 .
For example, when u ≡ 0, P is the property of asymptotic stability of the origin, and S 0 = S, Problem 1 specializes into the following problem.
Problem 2 [1,2] Is the switched system (1) asymptotically stable under arbitrary switching laws?
Solving Problem 1 is difficult for two reasons. First, the
set S 0 is usually huge, so exhaustively checking the system’s behavior for each σ ∈ S 0 is impossible. Second, it
is entirely possible that each of the subsystems satisfies
property P , yet the switched system admits a solution
that does not satisfy property P . Thus, merely checking
the behaviors of the subsystems is not enough.
A general approach for addressing Problem 1 is based
on studying the “worst-case” scenario. We say that σ̃ ∈
S 0 is the worst-case switching law (WCSL) in S 0 , with
respect to property P , if the following condition holds:
if the switched system satisfies property P for σ̃, then it
satisfies property P for any σ ∈ S 0 . Thus, the analysis of
24 January 2008
property P under arbitrary switching signals from S 0 is
reduced to analyzing the behavior of the switched system
for the specific switching signal σ̃.
The RMS gain can also be computed by solving a suitable Riccati equation [14]. This equation arises because
the Hamilton-Jacobi-Bellman equation, characterizing a
suitable dissipation function [15], admits a solution with
a quadratic form for linear systems. Fix T ∈ (0, ∞), a
symmetric matrix PT ∈ Rn×n , and consider the DRE:
The WCSL for Problem 2 (that is, the “most destabilizing” switching law) can be characterized using variational principles (see the survey paper [3]). This idea
originated in the pioneering work of E. S. Pyatnitskii on
the celebrated absolute stability problem [4,5] (see also
[6–9]). The basic idea is to embed the switched system
in a more general bilinear control system. 1 Then, the
“most destabilizing” switching law can be characterized
as the solution to a suitable optimal control problem.
For second-order systems, this problem can be explicitly
solved using the generalized first integrals of the subsystems [11,12].
Ṗ (t) = −Sγ (P (t)),
Theorem 1 [16] g(T ) = inf{γ ≥ 0 : I(0) = [0, T ]}.
The next result, which is a special case of [17,
Thm. 4.1.8], provides a sufficient condition for P (t) to
be monotonic.
Theorem 2 If Ṗ (T ) ≤ 0 then Ṗ (t) ≤ 0, for all t ∈
I(PT ).
The algebraic Riccati equation (ARE) associated with
the DRE (4) is:
Sγ (P ) = 0.
(5)
We use standard notation. Vectors are denoted by boldface letters, and the transpose of a vector v is denoted
by v 0 . Matrices are denoted by capital letters. If P, Q are
two symmetric matrices, then P > Q means that P − Q
is positive-definite.
Theorem 3 [16] Consider the linear system (3),
where A is Hurwitz. Fix γ > 0 and denote R := γ −2 BB 0 .
If γ > g(∞) then the ARE (5) admits symmetric and
positive definite solutions P − , P + ∈ Rn×n , referred to
as the stabilizing and antistabilizing solutions, respectively, such that A + RP − and −(A + RP + ) are Hurwitz.
Moreover, P + > P − .
Preliminaries
It is possible to express the solution to the DRE (4)
using P − and P + .
In this section, we describe some known results on the
RMS gain of (non-switched) linear systems and the associated Riccati equations.
Theorem 4 [16] Suppose that the conditions of Theorem 3 hold, and that PT − P + is nonsingular. Define Q := A + RP + , and
Consider the linear system
ẋ = Ax + Bu,
y = Cx,
(4)
where Sγ (P ) := P A + A0 P + C 0 C + γ −2 P BB 0 P .
Let I(PT ) ⊆ [0, T ] denote the maximal time interval for
which the solution P (t) exists.
In this paper, we use a similar approach for studying the
root-mean-square (RMS) gain problem. Our main result
shows that if a certain condition holds, then the WCSL
can be obtained by switching between the l differential
Riccati equations (DREs) corresponding to the l linear
subsystems. This condition automatically holds for firstorder SISO systems, so we obtain a complete solution to
the RMS gain problem for this case.
2
P (T ) = PT ,
0
Λ(t) :=eQ(t−T ) (PT − P + )−1 + (P + − P − )−1 eQ (t−T )
(3)
− (P + − P − )−1 ,
with (A, B) controllable and (A, C) observable. For T ∈
(0, +∞], let L2,T denote the set of functions f (·) such
1/2
R
T 0
< ∞. The RMS
f
(t)f
(t)dt
that ||f ||2,T :=
0
t ≤ T.
(1) The solution to (4) is P (t) = P + + (Λ(t))−1 , for
all t ∈ I, where I ⊂ (−∞, T ] is an interval on
which Λ is nonsingular.
(2) If PT − P + < 0, then P (t) exists for all t ≤ T and
limt↓−∞ P (t) = P − .
(3) If PT −P + 6≤ 0, the solution P (t) has a finite escape
time, 2 that is, there exist τ ∈ (−∞, T ) and z ∈ Rn
such that limt↓τ z 0 P (t)z = +∞.
gain over [0, T ] of (3) is defined by g(T ) := inf{γ ≥ 0 :
||y||2,T ≤ γ||u||2,T , ∀ u ∈ L2,T }, where y is the output
of (3) corresponding to u with x(0) = 0.
It is well-known that g(∞) = ||C(sI − A)−1 B||∞ ,
where ||Q(s)||∞ := supRe(s)≥0 ||Q(s)|| is the H∞ norm
of the transfer matrix Q(s) [13].
Theorem 4 implies in particular that g(T ) ≤ g(∞) for
all T . Indeed, seeking a contradiction, suppose that there
1
For a recent and comprehensive presentation of bilinear
systems, see [10].
2
2
See [17,18] for some related considerations.
exists a time T such that g(T ) > g(∞). Fix γ such
that g(∞) < γ < g(T ). Theorem 3 implies that for
this γ the ARE admits solutions P − , P + > 0. Part (2)
of Theorem 4 implies that the solution P (t) to the DRE,
with PT = 0, exists for all t ≤ T . Theorem 1 now
yields g(T ) ≤ γ, which is a contradiction.
control
3
Now suppose that c1 b1 = a1 , c2 > 0, and b2 = a2 /c2 . In
this case, g1 (∞) = g2 (∞) = 1, so g1 (T ), g2 (T ) ≤ 1. Yet,
||y||
arbitrarily large by taking a large
we can make ||u||2,T
2,T
enough c2 . Thus, gS (T ) can be made arbitrarily large,
even though both subsystems have RMS gain ≤ 1. 2
||y||22,T
= (2/T )
||u||22,T
The RMS gain of (1), over some set of switching signals S 0 ⊆ S, is defined by
gS 0 (T ) := inf {||y||2,T ≤ γ||u||2,T , ∀ u ∈ L2,T , ∀ σ ∈ S 0 },
γ≥0
Theorem 5 [23] Fix γ > max1≤i≤l {gi (∞)}. If there
exist i, j ∈ {1, . . . , l} such that Pi+ 6≥ Pj− then gS1 (∞) ≥
γ. If Pi+ > Pj− for all i, j then gS1 (∞) ≤ γ.
By definition, gS (T ) ≥ max1≤i≤l {gi (T )}, where gi (T )
is the RMS gain of the ith linear subsystem. Consider
the switched system (1) with u ≡ 0. It is well-known
that global asymptotic stability of the individual linear subsystems is necessary, but not sufficient for global
asymptotic stability of the switched system for every σ ∈
S [1]. We assume from here on that the switched system with u ≡ 0 is globally uniformly asymptotically
stable (GUAS), that is, there exist λ1 , λ2 > 0 such
that ||x(t)|| ≤ λ1 ||x(0)||e−λ2 t , for all t ≥ 0, σ ∈ S,
and x(0) ∈ Rn . In particular, this implies of course
that Ai , i = 1, . . . , l, are all Hurwitz. The next example, adapted from [23], shows that even in this case, the
RMS gain of the switched system can be very different
from that of the subsystems.
Thus it is possible to determine whether gS1 (∞) ≤ γ
or gS1 (∞) ≥ γ by analyzing the relationship between
the stabilizing and antistabilizing solutions to the AREs
associated with the subsystems.
In this paper, we use a variational approach to address
the problem of computing the RMS gain.
4
Variational approach
P
For z = (z1 , . . . , zl )0 , let A(z) := li=1 zi Ai , B(z) :=
Pl
Pl
i=1 zi Bi , and C(z) :=
i=1 zi Ci . Our starting point
is to replace (1) with the more general control system:
Example 1 Consider the system (1) with l = 2, n =
k = m = 1 and a1 , a2 < 0. Each subsystem is asymptotically stable and since n = 1 the switched system is
GUAS.
ẋ = A(v)x + B(v)u,
y = C(v)x,
v ∈ V,
(6)
where V is the set of measurable
functions ofPtime takn
l
ing values in the set W := z ∈ Rl : zk ≥ 0, k=1 zk =
o
1 . For v(t) ≡ ei , where ei ∈ Rl is the ith unit vec-
Fix T > 0, x(0) = 0, the switching signal σ(t) = 1
for t ∈ [0, T /2), σ(t) = 2 for t ∈ [T /2, T ], and the control
u(t) = 1 for t ∈ [0, T /2), u(t) = 0 for t ∈ [T /2, T ]. A
calculation yields
y(t) =
y 2 (t)dt
0
Several authors considered the RMS gain of switched
systems over sets of switching signals with a sufficiently
large dwell time between consecutive discontinuities.
Upper bounds for the RMS gain were derived in [24–
26]. Hespanha [23] provided a complete solution to the
problem of computing gS1 (∞), where S1 is the set of
switching signals with no more than a single switch.
where y is the solution to (1) corresponding to u, σ,
with x(0) = 0. This can be interpreted as the “worstcase” energy amplification gain for the switched system,
over all possible input and switching signals in S 0 . Computing gS 0 (T ) is an important open problem in the design and analysis of switched systems [19]. In particular,
calculating induced gains is the first step toward the application of robust control techniques to switched systems [20–22].
c1 b1 (exp(a1 t) − 1)/a1 ,
c2 b1 exp(a2 (t − T /2))z/a1 ,
T
= c21 b21 (3 + T a1 + exp(T a1 ) − 4 exp(a1 T /2))/(T a31)
+ c22 b21 (exp(a1 T /2) − 1)2 (exp(a2 T ) − 1))/(T a2 a21 ).
RMS gain of switched systems
Z
tor in Rl , (6) becomes (2). Thus, every solution of the
switched system (1) is also a solution of (6).
The RMS gain of (6) over [0, T ] is defined by gb (T ) :=
inf{γ ≥ 0 : ||y||2,T ≤ γ||u||2,T , ∀u ∈ L2,T , ∀v ∈ V},
where y is the solution to (6) corresponding to u, v,
with x(0) = 0.
t ∈ [0, T /2),
t ∈ [T /2, T ],
where z := exp(a1 T /2) − 1. Note that y(t) does not depend on b2 . Thus, for this particular switching signal and
For z ∈ W, let Sγ (P, z) := P A(z) + A0 (z)P +
3
0, for t ∈ (τ1 , T ]. By (8), Ṗv ∗ (τ1− ) ≤ Ṗv ∗ (τ1+ ) ≤ 0, and
we conclude that Ṗv ∗ ≤ 0, t ∈ (τ2 , τ1 ). Proceeding in
this way yields Ṗv ∗ (t) ≤ 0, for almost all t ∈ Iv ∗ . 2
γ −2 P B(z)B 0 (z)P + C 0 (z)C(z). We can now state our
main result.
Theorem 6 Fix arbitrary T ∈ [0, ∞) and γ >
max1≤i≤l {gi (T )}. Consider the problem of maximizing
RT
J(u, v) := 0 (y 0 (τ )y(τ ) − γ 2 u0 (τ )u(τ ))dτ along the
trajectories of (6), with x(0) = 0, u ∈ L2,T and v ∈ V.
Let Pv (·) denote the solution of the equation
Ṗ (t) = −Sγ (P (t), v(t)),
P (T ) = 0,
Let x∗ denote the solution of (6) corresponding to v =
v ∗ and an arbitrary u ∈ L2,T . Proposition 1 implies
that x∗ is also a solution of the switched system (1), and
that Pv ∗ (t) is obtained by switching appropriately between the DREs Ṗ = −Si (P ), i = 1, . . . , l, corresponding to the l linear subsystems (see [27,28] for some related considerations).
(7)
and let Iv ⊆ [0, T ] denote the maximal time interval for
which the solution Pv (t) exists. 3 If there exists v ∗ ∈ V
such that
We can now prove Theorem 6. Fix an arbitrary s ∈ Iv ∗
(so Pv ∗ (t) exists for all t ∈ [s, T ]). Fix also u ∈ L2,T ,
v ∈ V, x0 ∈ Rn , and let x denote the corresponding
trajectory of (6), with x(s) = x0 . For t ∈ [s, T ], denote V (x, t) := x0 Pv ∗ (t)x, and
Sγ (Pv ∗ (t), v ∗ (t)) ≥ Sγ (Pv ∗ (t), z), ∀ t ∈ Iv ∗ , ∀ z ∈ W,
(8)
then the following properties hold.
m(t) := V (x(t), t) +
(1) If Iv ∗ = [0, T ], then the problem of maximizing J(u, v) is well-defined; (u∗ (t) ≡ 0, v ∗ ) is a
maximizing pair; and gS (T ) ≤ gb (T ) ≤ γ.
(2) If the solution Pv ∗ (t) does not exist for all t ∈ [0, T ],
then J is unbounded, gb (T ) ∈ [γ, +∞], and gS (T ) ∈
[γ, +∞].
t
s
(y 0 (τ )y(τ ) − γ 2 u0 (τ )u(τ ))dτ.
(10)
Note that x and y are evaluated along the trajectory
corresponding to (u, v). However, Pv ∗ (t) is the matrix
defined by (7) with v = v ∗ . Then
ṁ = x0 Ṗv ∗ x + 2x0 Pv ∗ (A(v)x + B(v)u) + y 0 y − γ 2 u0 u
= −γ 2 (u − γ −2 B 0 (v)Pv ∗ x)0 (u − γ −2 B 0 (v)Pv ∗ x)
2
Remark 1 A calculation yields ∂∂z 2 Sγ (Pv ∗ , z) = 2H,
Pl
P
0
−2
where H :=
Pv ∗ ( li=1 Bi Bi0 )Pv ∗ .
i=1 Ci Ci + γ
Since H ≥ 0 this implies that
+ x0 (Ṗv ∗ + Sγ (Pv ∗ , v))x
≤ x0 (−Sγ (Pv ∗ , v ∗ ) + Sγ (Pv ∗ , v))x.
max S (P ∗ , z) = max Sγ (Pv ∗ , ei ),
z ∈W γ v
i∈{1,...,l}
(11)
Using (8) yields ṁ(t) ≤ 0, so m(T ) ≤ m(s) for any
admissible pair (u, v), and m(T ) = m(s) for v = v ∗
and u = u∗ := γ −2 B 0 (v ∗ )Pv ∗ x∗ .
so if condition (8) holds then there exists a piecewiseconstant v ∗ (t) taking values in {e1 , . . . , el } only. Specifically, v ∗ (t) = ei if
Sγ (Pv ∗ (t), ei ) ≥ Sγ (Pv ∗ (t), ej ),
Z
Combining (10) with the fact that Pv ∗ (T ) = 0 yields
(9)
V (z, t) =
for all j ∈ {1, . . . , l}.
Z
T
(y 0 (τ )y(τ )−γ 2 u0 (τ )u(τ ))dτ +m(t)−m(T ),
t
(12)
where y : [t, T ] → Rk is the solution to (6) for the
initial condition x(t) = z. Since the linear subsystems
are controllable, there exists a control taking x(0) = 0
to x(t) = z, for any z ∈ Rn , so (12) holds for any z ∈ Rn .
In order to prove Theorem 6, we require the following
result.
Proposition 1 If the conditions of Theorem 6 hold then
there exists a v ∗ satisfying v ∗ (t) ∈ {e1 , . . . , el } for all t ∈
Iv ∗ . The corresponding solution Pv ∗ (t) is monotonically
non-increasing on Iv ∗ .
The analysis of m(t) above and (12) yield
V (z, t) = sup{u ∈ L2,T , v ∈ V, x(t) = z :
Z T
(y 0 (τ )y(τ ) − γ 2 u0 (τ )u(τ ))dτ },
Proof. The first statement follows from Remark 1. To
(13)
prove the second statement, let τi ∈ Iv ∗ denote the
t
∗
switching times of v with · · · < τ3 < τ2 < τ1 < T .
Since Pv ∗ (T ) = 0, Ṗv ∗ (T ) = −Sγ (0, v ∗ (T )) =
for all t ∈ [s, T ] and all z ∈ Rn . In other words, V is the
0 ∗
∗
−C (v (T ))C(v (T )), and Theorem 2 implies that Ṗv ∗ (t) ≤ finite-horizon “cost-to-go” function [29].
If Iv ∗ = [0, T ], then taking s = 0 and t = T in (10),
and using the fact that x(0) = 0 and Pv ∗ (T ) = 0 yields
3
By solution, we mean an absolutely continuous function P (t) that satisfies (7) for almost all t ∈ Iv .
4
J(u, v) ≤ J(u∗ , v ∗ ) = 0 for any u ∈ U and v ∈ V. By
the definition of J, this implies that gb (T ) ≤ γ. Since
every trajectory of (1) is also a trajectory of (6), gS (T ) ≤
gb (T ), so gS (T ) ≤ γ.
Corollary 1 Consider the system (1) with n = 1.
∗
For any T > 0 there exists a WCSL v ∗ (·)
: Iv →
{e1 , . . . , el } that contains no more than 2 2l switches.
Proof. The existence of a WCSL taking values in {e1 , . . . , el }
follows from Remark 1. To prove the bound on the number of switches, consider first the case l = 2. By (9),
every switching in v ∗ corresponds to a zero of the function d(r) := sγ (r, e1 )−sγ (r, e2 ), which is a second-order
polynomial in r. Since pv ∗ (t) is monotonous, d(pv ∗ (t))
can change sign no more than twice, so the number of
switches is bounded by two. For l > 2, any switching
point corresponds to a zero of one of 2l functions which
are all second-order polynomials. 2
Consider now the case where Pv ∗ (t) is not well-defined
for all t ∈ [0, T ]. It follows from the absolute continuity
and monotonicity of Pv ∗ (t) that there exists s ∈ [0, T ]
and z ∈ Rn such that limt↓s z 0 Pv ∗ (t)z = +∞. Eq. (13)
implies that for x(s) = z there exist (u∗ , v ∗ ) defined
RT
on [s, T ] that make s (y 0 (τ )y(τ )−γ 2 u0 (τ )u(τ ))dτ arbitrarily large. Since each linear subsystem is controllable,
there exists a pair (u, v) that takes the state from x(0) =
0 to x(s) = z, with v(t) = e1 , t ∈ [0, s] . Summarizing, we can find inputs (u, v) defined on [0, T ] that
make J(u, v) arbitrarily large, so γ ≤ gb (T ). It follows
from Proposition 1 that we can actually find such a pair
with v(t) ∈ {e1 , . . . , el } for all t ∈ [0, T ], so γ ≤ gS (T ).
This completes the proof of Theorem 6. 2
Example 3 Consider the switched
√ system (1) with n =
1, l = 2, a1 =√−0.8642, b1 = 40, c1 = 1.4402, a2 =
−0.0622, b2 = 0.9, c2 = 0.6831, and final time T = 15.
Corollary 1 implies that there exists a WCSL with no
more than 2 switches. Using a bisection algorithm yields
10.7573 < gb < 10.7574. The corresponding WCSL can
be written as v ∗ (t) = (1 − w∗ (t), w∗ (t))0 . Fig. 1 depicts w∗ (t) for γ = 10.7574. Note that w ∗ is piecewise
constant with two switching points.
Remark 2 It is possible to give an intuitive explanation
of the WCSL v ∗ as follows. Fix an arbitrary t ∈ int(Iv ∗ )
and > 0 sufficiently small, and denote τ := t − . Then
Pv ∗ (τ ) ≈ Pv ∗ (t) − Ṗ (t) = Pv ∗ (t) + Sγ (Pv ∗ (t), v ∗ (t)).
Thus, when condition (8) holds, the choice v ∗ maximizes Pv ∗ (τ ). 2
We now explain the dynamics of this example in detail (see Fig. 2). We use the notation from Theorem 5
and the proof of Corollary 1. A calculation yields: p−
1 =
+
−
2.0000, p+
1 = 3.0003, p2 = 6.0067, and p2 = 9.9886,
and that the roots of the polynomial d(p) := sγ (p, e1 ) −
sγ (p, e2 ) are q1 := 1.4375 and q2 := 3.3098. Since d(0) =
c21 − c22 > 0, there exists a minimal time τ > 0 such
that d(pv ∗ (t)) > 0 for t ∈ (τ, T ], so v ∗ (t) = (1, 0)0
for t ∈ (τ, T ]. We know that pv ∗ (t) increases monotonically as t ↓ τ and that pv ∗ (t) → p−
1 > q1 as t ↓ −∞.
At time τ , we have pv ∗ (τ ) = q1 , so d(pv ∗ (t)) changes
sign and the WCSL is v ∗ (t) = (0, 1)0 for some interval t ∈ (ζ, τ ). Since pv∗ (τ ) < p+
2 , the solution tends to−
ward p2 > q2 as t ↓ −∞. At time ζ, we have pv ∗ (ζ) = q2 ,
and d(pv ∗ (t)) changes sign again. Finally, on the interval t ∈ (0, ζ), pv∗ (t) continues to increase, so there are
no more sign changes of d for t ∈ (0, ζ). 2
The next example demonstrates a simple application of
Theorem 6.
Example 2 Consider the switched system (1) with l =
2, A2 = A1 − αI, α > 0, B2 = B1 , and C2 = C1 . Recall
that we assume that A1 is Hurwitz, so QA1 + A01 Q = −I
admits a solution Q > 0. It is straightforward to verify
that the switched system is GUAS using the common Lyapunov function V (x) := x0 Qx. In this case, Sγ (Pv , z) =
Pv A1 + A01 Pv + γ −2 Pv B1 B10 Pv + C10 C1 − 2z2 αPv ,
so (8) holds for v ∗ (t) ≡ (1, 0)0 . Thus, Theorem 6 implies
that σ(t) ≡ 1 is a WCSL, and that gb (T ) = g1 (T ). This
is, of course, what we may expect for this particular
example. 2
5
The case n = 1
In this section, we show that Theorem 6 provides a complete and computable solution to the RMS gain problem
for first-order SISO switched linear systems
6
Conclusions
We considered the problem of computing the RMS gain
of a switched linear system. Our main result shows that
if a certain condition holds, then the switching-law that
yields the maximal gain can be described explicitly in
terms of the DREs of the linear subsystems. This condition automatically holds when n = 1, so our main result
provides a complete solution to the problem in this case.
For n = 1, the matrix sign condition (8) becomes an inequality between scalars, and it is immediate to determine v ∗ (t) for which (8) holds. We can integrate pv ∗ (t)
backwards in time from t = T to t = 0, and then gb ≤ γ
(gb ≥ γ) if pv ∗ (t) exists (does not exist) for all t ∈ [0, T ].
This yields a simple bisection algorithm for determining
the RMS gain of the switched system. It is also possible to derive a bound on the number of switches in the
WCSL.
Topics for further research include the following. More
work is needed in order to characterize the conditions
5
References
1
0.9
[1] D. Liberzon, Switching in Systems and Control.
Birkhäuser, 2003.
0.8
0.7
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“Stability criteria for switched and hybrid systems,” SIAM
Review, vol. 49, pp. 545–592, 2007.
0.6
0.5
0.4
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15
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exists a WCSL for the RMS gain problem with a uniform bound on the number of switchings. This would
imply that, under these conditions, computing the RMS
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7