Definitions of systems, closed-loop systems, initial conditions, causality, existence and uniqueness properties are all slightly modified in this setting. For the definition and discussion of ambient spaces see Section 3.2on page 39 in Chapter 3.
Definition 4.23. Given normed signal spaces 𝒰, 𝒴 and 𝒲 ≜ 𝒰 × 𝒴, a system 𝑄 is defined to be the set:
𝔅𝑄≜ {𝑤 ∈ 𝒲𝑎−⊕ 𝒲𝑎+ ∣ 𝑤 = (𝑢, 𝑦) is an input-output pair of 𝑄
}
(4.72)
which satisfies the assumption that any input-output pair 𝑤 ∈ 𝔅𝑄 is defined over a
maximal interval (−𝑇1, 𝑇2) with both 𝑇1 and 𝑇2 belong to (0,∞], and that if 𝑇1 (resp.,
𝑇2) is finite, then∥𝑤∥(𝜏,0]→ ∞ (resp., ∥𝑤∥[0,𝜏 )→ ∞) as 𝜏 tends to −𝑇1 (resp., 𝑇2) from
up (resp., below).
A system 𝑄 represented by the set 𝔅𝑄(see (4.72)) is said to be time-invariant if 𝑤∈ 𝔅𝑄
implies 𝜎𝜏𝑤 ∈ 𝔅𝑄 for all 𝜏 ∈ ℝ with 0 ∈ (𝑎 − 𝜏, 𝑏 − 𝜏) (where dom(𝑤) = (𝑎, 𝑏) and
𝜎𝜏 is the shift operator defined by (𝜎𝜏𝑤)(⋅) = 𝑤(⋅ + 𝜏). Otherwise, 𝑄 is said to be
time-variant.
The following is the definition of causality for a system defined in the ambient space: Definition 4.24. A system 𝑄 represented by the set 𝔅𝑄(see (4.72)) is said to be causal
if, ∀(𝑢, 𝑦𝑢), (𝑣, 𝑦𝑣)∈ 𝔅𝑄,∀𝑡 ∈ dom(𝑢, 𝑣),
𝑢∣(−∞,𝑡]∩dom(𝑢,𝑣) = 𝑣∣(−∞,𝑡]∩dom(𝑢,𝑣)⇒ 𝔅𝑢𝑄∣(−∞,𝑡]∩dom(𝑢,𝑣)= 𝔅𝑣𝑄∣(−∞,𝑡]∩dom(𝑢,𝑣) where 𝔅𝑢
𝑄={𝑤 ∈ 𝒲𝑎 ∣ ∃𝑦 ∈ 𝒴𝑎 s.t. 𝑤 = (𝑢, 𝑦)∈ 𝔅𝑄}.
Note that any operator Φ :𝒰𝑎+ → 𝒴𝑎+ can be regarded as a special system in the sense of Definition 4.23, i.e., 𝔅Φ={𝑤 = (𝑢, 𝑦) ∈ 𝒲𝑎−⊕ 𝒲𝑎+ ∣ 𝑦∣(−∞,0]= 𝑢∣(−∞,0] = 0, 𝑅+𝑦 =
Φ(𝑅+𝑢)}. We say the operator Φ is causal if and only if the corresponding system 𝔅Φ
is causal. For convenience, the special definition of a causal operator is stated below. Given normed signal spaces𝒰 and 𝒴, an operator Φ : 𝒰+
𝑎 → 𝒴𝑎+ is said to be causal if,
{ ∀ 𝑢, 𝑣 ∈ 𝒰+ 𝑎, ∀𝑡 ∈ dom(𝑢, 𝑣) ∩ dom(Φ𝑢, Φ𝑣), : [ 𝑢∣[0,𝑡]= 𝑣∣[0,𝑡] ⇒ (Φ𝑢)∣[0,𝑡]= (Φ𝑣)∣[0,𝑡] ]
Definition 4.25. Given a system 𝑄 represented by the set 𝔅𝑄 (see (4.72)), its past
trajectories is defined by
𝔅−𝑄≜ 𝑅−𝔅𝑄={𝑤−∈ 𝒲𝑎− ∣ ∃ 𝑤+∈ 𝒲𝑎+, s.t. 𝑤−∧𝑤+∈ 𝔅𝑄}. (4.73)
Here ∧ denotes concatenation at time 0 (see (3.8) on page50). The system 𝑄 is said to have the existence property if ∀𝑤−∈ 𝔅−𝑄,∀𝑢+∈ 𝒰𝑎+, ∃𝑦+∈ 𝒴𝑎+ such that
∃ ˆ𝑤+∈ 𝒲𝑎+, 𝑤−∧ ˆ𝑤+ ∈ 𝔅𝑄, (𝑢+, 𝑦+)(𝑡) = ˆ𝑤+(𝑡),∀𝑡 ∈ dom(𝑢+, 𝑦+, ˆ𝑤+)
and the uniqueness property if ∀𝑤− ∈ 𝔅−𝑄,∀𝑤+ = (𝑢+, 𝑦+) ∈ 𝒲𝑎+,∀ ˜𝑤+ = (˜𝑢+, ˜𝑦+) ∈
𝒲+ 𝑎,
𝑤−∧𝑤+∈ 𝔅𝑄, 𝑤−∧ ˜𝑤+ ∈ 𝔅𝑄, 𝑢+= ˜𝑢+ ⇒ 𝑦+ = ˜𝑦+
and is well-posed if it has both the existence and uniqueness properties.
Definition 4.26. Given a system 𝑄 represented by the set 𝔅𝑄 (see (4.72)), the graph
𝒢𝑤−
𝑄 for any given past trajectory 𝑤− ∈ 𝔅−𝑄 is a subset of 𝒲𝑎+, which contains all of
𝑤+∈ 𝒲𝑎+ defined over a maximal interval [0, 𝑇 ) with 0 < 𝑇 ≤ ∞ such that 𝑤−∧𝑤+ ∈
𝔅𝑄, and if 𝑇 =∞ then 𝑤+∈ 𝒲+, and if 𝑇 is finite then ∥𝑤+∥[0,𝜏 ) → ∞ as 𝜏 tends to
𝑇 from below.
Definition 4.27. Given a system 𝑄 represented by the set 𝔅𝑄 (see (4.72)), we define
𝔖𝑄 the initial state space of 𝑄 at initial time 0 as the quotient set 𝔅−𝑄/ ∼ (i.e., 𝔖𝑄 ≜
𝔅−𝑄/∼). While the equivalence relation ∼ on 𝔅−𝑄 (see (4.73)) is defined by
𝑤− ∼ ˜𝑤− if and only if 𝑄𝑤−(𝑢 +) = 𝑄𝑤˜−(𝑢+), ∀𝑢+∈ 𝒰𝑎+ where 𝑤−, ˜𝑤− ∈ 𝔅−𝑄 and 𝑄𝑤−(𝑢+) ≜ { 𝑦+∈ 𝒴𝑎+
𝑤−∧(𝑢+, 𝑦+)∈ 𝔅𝑄} and the set
𝑄𝑤˜−(𝑢
+) is similarly defined.
The equivalence class of 𝑤−∈ 𝔅−𝑄 is [𝑤−] ≜{𝑤˜− ∈ 𝔅−𝑄 ∣ ˜𝑤−∼ 𝑤−}∈ 𝔖𝑄. The size of
[𝑤−]∈ 𝔖𝑄 is defined by 𝜒([𝑤−]) ≜ inf ˜
𝑤−∈[𝑤−]{∥ ˜
𝑤−∥}. (thus defined 𝜒(⋅) is a real-valued function on 𝔖𝑄.)
From the equivalence relation ∼, for any initial state 𝑥0 ∈ 𝔖𝑄, we can define the set
𝑄𝑥0(𝑢
+) by:
𝑄𝑥0(𝑢
+) ≜ 𝑄𝑤−(𝑢+), ∀𝑢+ ∈ 𝒰𝑎+. (4.74)
where 𝑤−∈ 𝔅−𝑄 is any element in 𝑥0.
If the system 𝑄 is well-posed, then, for every 𝑤−∈ 𝔅−𝑄, 𝑄𝑤−(⋅) is an operator from 𝒰+
𝑎
to 𝒴+
𝑎. This in turn implies that, for every 𝑥0 ∈ 𝔖𝑄, 𝑄𝑥0(⋅) is an operator from 𝒰𝑎+ to
𝒴+ 𝑎.
For a well-posed system 𝑄, if 𝑄 is causal, then we have 𝑄𝑥0 is a causal operator from
𝒰+
𝑎 to 𝒴𝑎+.
The notion of locally input to output stability is defined as follows.
Definition 4.28. The system 𝑄 is said to be locally input to output stable if, and only if, it is well-posed and causal, and there exist 𝑑 > 0 and functions 𝛽∈ 𝒦ℒ and 𝛾 ∈ 𝒦∞ such that, ∀𝑥0 ∈ 𝔖𝑄, ∀𝑢0+∈ 𝒰+,∀𝑡 ≥ 0
max{𝜒(𝑥0),∥𝑢0+∥} ≤ 𝑑 ⇒ ∣(𝑄𝑥0𝑢0+)(𝑡)∣ ≤ 𝛽 (𝜒(𝑥0), 𝑡) + 𝛾(∥𝑢0+∥[0,𝑡))
where the real-valued function 𝜒(⋅) is defined in Definition 4.27.
Note that a potentially weaker definition might merely require that the above condition hold only for all 𝑡∈ [0, 𝑇𝑥0,𝑢0+), (where [0, 𝑇𝑥0,𝑢0+) is the maximal interval over which
𝑄𝑥0𝑢
0+ is defined). However, this definition turns out to be equivalent to the one given
above. Indeed, by standard facts from differential equations (see e.g., [Sontag, 1998a], [Sontag, 1998b, Proposition C.3.6, p. 481]), since the right-hand side is bounded on a maximal interval, we have that the left-hand side is also bounded on the maximal interval and therefore that the maximal interval should be [0,∞).
The following is the definition of a closed-loop system:
Definition 4.29. Given normed signal spaces 𝒰, 𝒴 and 𝒲 ≜ 𝒰 × 𝒴 (such as 𝑊 = 𝐿∞(ℝ, ℝ𝑚+𝑝)). Let the sets 𝔅𝑃 and 𝔅𝐶 represent the subsystems 𝑃 (plant) and 𝐶
(controller), respectively. Consider the standard feedback configuration shown in Fig- ure 3.1 on page 38 that satisfies equations (3.1). Then the closed-loop system [𝑃, 𝐶] represented by the set 𝔅𝑃//𝐶 is defined by
𝔅𝑃//𝐶 ≜ {(𝑤0, 𝑤1)∈ 𝒲𝑎2 ∣ 𝑤0 is input, 𝑤1 ∈ 𝔅𝑃 is output, 𝑤0− 𝑤1 ∈ 𝔅𝐶} (4.75)
which satisfies the assumption that any input-output pair (𝑤0, 𝑤1) ∈ 𝔅𝑃//𝐶 is defined
over a maximal interval (−𝑇1, 𝑇2) with both 𝑇1 and 𝑇2 belong to (0,∞], and that if 𝑇1
(resp., 𝑇2) is finite, then∥(𝑤0, 𝑤1)∥(𝜏,0]→ ∞ (resp., ∥(𝑤0, 𝑤1)∥[0,𝜏 )→ ∞) as 𝜏 tends to
−𝑇1 (resp., 𝑇2) from up (resp., below).
For the closed-loop system [𝑃, 𝐶] represented by the set 𝔅𝑃//𝐶, we can similarly define the initial state space 𝔖𝑃//𝐶 at initial time 0 in terms of Definition4.27. And the closed- loop system [𝑃, 𝐶] has the existence property, the uniqueness property, and the well- posedness property if and only if the set 𝔅𝑃//𝐶has the existence property, the uniqueness
property, and the well-posedness property, respectively, according to Definition 4.25. Note that for any 𝑠0 ∈ 𝔖𝑃//𝐶 and 𝑤0+ ∈ 𝒲𝑎+, we have defined a set Π𝑠𝑃//𝐶0 (𝑤0+)
i.e.,
Π𝑠0
𝑃//𝐶(𝑤0+) =
{
𝑤1+∈ 𝒲𝑎+ ∣ (𝑤0−, 𝑤1−)∧(𝑤0+, 𝑤1+)∈ 𝔅𝑃//𝐶,∀(𝑤0−, 𝑤1−)∈ 𝑠0}
If the closed-loop system [𝑃, 𝐶] is well-posed, then Π𝑠0
𝑃//𝐶(⋅) defines an operator from
𝒲+
𝑎 to 𝒲𝑎+.
In the following we give the notion of stability for closed-loop system which is derived from the notion of stability for system in Definition 4.28.
Definition 4.30. The closed-loop system [𝑃, 𝐶] represented by the set 𝔅𝑃//𝐶 with initial state space 𝔖𝑃//𝐶 is said to be locally input to output stable if, and only if, it is well-posed and causal, and there exist 𝑑 > 0 and functions 𝛽 ∈ 𝒦ℒ and 𝛾 ∈ 𝒦∞ such
that, ∀𝑠0∈ 𝔖𝑃//𝐶, ∀𝑤0+ ∈ 𝒲+, ∀𝑡 ≥ 0,
max{𝜒(𝑠0),∥𝑤0+∥} ≤ 𝑑 ⇒ ∣(Π𝑠𝑃//𝐶0 𝑤0+)(𝑡)∣ ≤ 𝛽 (𝜒(𝑠0), 𝑡) + 𝛾(∥𝑤0+∥[0,𝑡))
where the real-valued function 𝜒(⋅) is defined in Definition 4.27.
Define another set which is related to the product state in 𝔖𝑃 × 𝔖𝐶, denoted by
Π𝑥0
𝑃//𝐶(𝑤0+), for any 𝑥0 = (𝑥10, 𝑥20)∈ 𝔖𝑃 × 𝔖𝐶 and any 𝑤0+ ∈ 𝒲𝑎+, as follows:
Π𝑥0 𝑃//𝐶(𝑤0+) ≜ { 𝑤1+∈ 𝒲𝑎+ (𝑤0−, 𝑤1−)∧(𝑤0+, 𝑤1+)∈ 𝔅𝑃//𝐶, ∀ (𝑤1−, 𝑤0−− 𝑤1−)∈ 𝑥0 } (4.76)
If the closed-loop system [𝑃, 𝐶] is well-posed, then Π𝑥0
𝑃//𝐶(⋅) defines an operator from
𝒲𝑎+ to 𝒲𝑎+.
We next present several equivalent characterisation of this notion of stability as follows. Theorem 4.31. Suppose that the closed-loop system 𝔅𝑃//𝐶 is well-posed and causal.
The following four statements are equivalent:
I. The closed-loop system 𝔅𝑃//𝐶 is locally input to output stable.
II. There exist 𝑑1 > 0 and functions 𝛽1 ∈ 𝒦ℒ and 𝛾1 ∈ 𝒦∞ such that, ∀𝑠0 ∈
𝔖𝑃//𝐶, ∀𝑡 > 0, ∀𝑤0+ ∈ 𝒲+,
max{𝜒(𝑠0),∥𝑤0+∥} ≤ 𝑑1 ⇒ ∣(Π𝑠𝑃//𝐶0 𝑤0+)(𝑡)∣ ≤ 𝛽1(𝜒(𝑠0), 𝑡) + 𝛾1(∥𝑤0+∥[0,𝑡))
III. There exist 𝑑2 > 0 and functions 𝛽2 ∈ 𝒦ℒ and 𝛾2 ∈ 𝒦∞ such that, ∀𝑥0 ∈ 𝔖𝑃 ×
𝔖𝐶, ∀𝑡 > 0, ∀𝑤0+ ∈ 𝒲+,
IV. There exist 𝑑3 > 0 and functions 𝛽3 ∈ 𝒦ℒ and 𝛾3 ∈ 𝒦∞ such that, ∀𝑥0 =
(𝑥10, 𝑥20)∈ 𝔖𝑃 × 𝔖𝐶, ∀𝑡 > 0, ∀𝑤0+∈ 𝒲+, ∀𝑤1−∈ 𝑥10, ∀𝑤2− ∈ 𝑥20,
max{𝜒(𝑥0),∥𝑤0+∥} ≤ 𝑑3 ⇒ ∣(Π𝑥𝑃//𝐶0 𝑤0+)(𝑡)∣ ≤ 𝛽3(∥(𝑤1−, 𝑤2−)∥ , 𝑡) + 𝛾3(∥𝑤0+∥[0,𝑡))
Moreover, we have 𝛾1 = 𝛾2= 𝛾3, 𝑑2= 𝑑3 and 𝛽2 = 𝛽3.
Proof. Similar to the proof of Theorem3.36 on page 75.