5.2.1 System Description
We consider NCSs introduced in Chapter 3 and depicted in Fig. 3.1. In particular, we assume the plant P is stabilized by a linear time-invariant dynamic controller K that relies on a holding device J , which is a linear time-invariant system whose state is reset to the plant measurement whenever a new measurement is received. This holding device generates a continuous-time signal that feeds a linear time-invariant dynamic controller;
see Chapter 3 for further details. In this chapter, we aim the design of the controller and holding device. To this end, in the following, we consider NCSs, as in Fig. 5.1, where the network is characterized only by variable transmission intervals. This differs from previous Chapters 3 and 4, where variable transmission intervals and network delays are considered together. However, simplifying the network makes the control problem more tractable and allows us to devise a design procedure. More in details, we assume output measurements of plant P are measurable only at some time instances tk, k ∈ N>0, not known in advance. In particular, we assume that the sequence {tk}∞k=1 is strictly increasing and unbounded, and that there exist two positive real scalars T1 ≤ T2 such that
0 ≤ t1≤ T2, T1 ≤ tk+1− tk≤ T2 ∀k ∈ N>0 (5.1)
The lower bound T1 in condition (5.1) introduces a strictly positive minimum time in be-tween consecutive measurements. As such, this avoids the existence of Zeno behaviors, which are unwanted in practice [37]. Moreover, T2 defines the Maximum Allowable Trans-mission Interval (MATI).
Given plant P and the measurement setup above, the problem we solve in this chapter is to design an output feedback dynamic controller such that the closed-loop NCS is input-output exponentially stable with some required performance satisfied with the largest achievable value of T2; a formal problem statement will be provided next in the chapter. To
K P
t
ku y
oy(t
k)
J y ˆ
x
cω
y
Figure 5.1: Schematic representation of the NCS considered for the design of holding device and controller. Solid lines represent the continuous-time signals, whereas the dashed line depicts the sporadic measurements.
this end, we consider the plant P as in (3.1), and we design the controller K and holding device J . The proposed controller is an output-feedback dynamic controller as in (3.2).
The proposed holding device J is selected as follows:
J
˙ˆy(t) = cH ˆy(t) + “Exc(t) + CpBpu(t) ∀t 6= tk
ˆ
y(t+) = y(t) ∀t = tk
(5.2)
which reads as (3.3) with
H = cH + CpBpDc E = “E + CpBpCc
(5.3)
This choice of J simplifies the dynamics of the closed-loop NCS enabling the design of holding device and controller as shown next in this chapter.
5.2.2 Hybrid Modeling
The closed-loop system in Fig. 5.1 can be modeled as a linear system with jumps in ˆy. In particular, for all k ∈ N0 one obtains
To devise a design algorithm for K and J , we model the impulsive system in (5.4) into the hybrid system framework in [37]; see for a brief introduction Chapter 2. To this end, we augment the state of the closed-loop system with the auxiliary variable τ ∈ R≥0, which is a timer that keeps track of the duration of intervals in between transmissions. As in [30], to enforce (5.1), we make τ decrease as ordinary time t increases and, whenever τ = 0, reset it to any point in [T1, T2]. The whole closed-loop system composed by the states xp, xc, ˆy, and τ can be represented by the following hybrid system:
is the flow map,
is the jump map, and the the flow set C and the jump set D are defined as follows
C := Rnp+nc+ny× [0, T2], D := Rnp+nc+ny × {0} (5.8)
The set-valued jump map allows to capture all possible transmission intervals of length within T1 and T2. Specifically, the hybrid model in (5.5) is able to characterize any sequence satisfying (5.1).
At this stage, to simplify the analysis, we introduce the change of coordinates
η := y − ˆy (5.9)
which leads, by straightforward calculations, to the closed-loop hybrid system in the new coordinates:
where
Observe that, as shown in Fig. 5.2, Hcl can be interpreted as the feedback in-terconnection of two different dynamical systems Σxcl and Ση. In particular, Σxcl is a continuous-time system described by:
whereas Ση is a hybrid dynamical system given as follows:
Ση
It is worth mentioning that considering Hcl as the interconnection of Σxcl and Ση
allows us to address stability analysis of the closed-loop system by employing an approach that is reminiscent of an “input-to-state stability small gain” philosophy [48]. A conceptu-ally similar approach can be found in [16].
Σ x
clFigure 5.2: Representation of Hcl as the interconnection of the dynamical systems Σxcl and Ση.
5.2.3 Problem Statement
To formalize our control problem, we rely on the notions of exponential input-to-state stability of closed sets for a generic hybrid system H with input-to-state in Rn; see Defini-tion 2.3.1 introduced in Chapter 2 for further details.
The proposed approach aims at designing the holding device J and the controller K such that without disturbance, i.e., ω ≡ 0, the following set1
A := {0} × {0} × [0, T2] (5.16)
is exponentially stable, and, when the disturbances are nonzero, the hybrid system Hcl is input-to-state stable with respect to A. In particular, the problem we solve is as follows:
Problem 5.2.1. Given the plant P in (3.1), design
∆K:=
such that the closed-loop system satisfies the following properties with the largest achievable value of T2:
1Notice that, by definition of the system Hcland of the set A, for all x ∈ C, one has |x|A= |(xcl, η)|.
(P1) the set A is global exponentially stable when the input ω is identically zero;
(P2) when the disturbance ω is nonzero, the hybrid system Hcl in (5.10) is input-to-state stable with respect to A;
(P3) L2 stability from the disturbance ω to the performance output yo is ensured with a desired L2-gain γ.