Controlled Lorenz model of fluid convection

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Event-triggered control

In convention setups, the feedback laws are implemented in a time-triggered fashion such that two transmission instants are separated by (at most) the MATI. Although this strategy is appeal-ing from the implementation point of view, it is not obvious that time-triggering is appropriate for NCS. First, the transmission interval is usually designed such that the closed-loop stability is guaranteed in all possible situations. To do so, the design has to be carried out based on the worst case scenario which may result in a small MATI bound. The second reason is that the time-triggered approach has a blind nature since the transmission instants are generated regardless the system state. This may lead to an inefficient usage of the comp utation and the communication resources. Intuitively, if the system has reached a desired equilibrium point and no disturbance is acting on the plant, there is no need to close the loop and to calculate a new control input, but the time-triggered paradigm keeps doing so. In the last decades, many researchers suggested to develop alternative implementation policies such that the amount of transmissions is adapted to the current plant state. This may allow to significantly redu ce the usage of the communication and computation resources. ETC has been proposed in this context.

Hybrid model

Consider the case where the controller communicates with the plant via a digital channel. In this section, we derive a model of event-triggered control systems in the case where the feedback law only has access to an output of the plant for the sake of generality (like in [22], [30], [87]). Consider the nonlinear plant model x˙p = fp(xp, u), y = gp(xp), (1.1).
where xp ∈ Rnp is the plant state, u ∈ Rnu is the control input, y ∈ Rny is the measured output of the plant. Assume that the plant is stabilized by general dynamic controller of the form x˙c = fc(xc, y), u = gc(xc, y), (1.2).
where xc ∈ Rnc is the controller state. Note that, by setting u = gc(y), we obtain a static controller. Since the feedback loop is closed via a digital channel, the plant output and the control input are sent only at some transmission instants ti, i ∈ Z≥0, see Figure 1.2. At each transmission instant, the plant output is sent to the controller which computes a new control input that is instantaneously transmitted to the plant. We assume that this process is performed in a synchronous manner and we ignore the computation times and the possible transmission delays. In that way, we obtain x˙p = fp(xp, uˆ) x˙c = fc(xc, yˆ) u = gc(xc, yˆ) ˙ = 0 yˆ ˙ = 0 uˆ yˆ(t+) = y(t ) t ∈ [ti, ti+1] t ∈ [ti, ti+1]t ∈ [ti, ti+1] t ∈ [ti, ti+1] (1.3).
where yˆ an d uˆ respectively denote the last transmitted values of the plant output and of the control input. We assume that zero-order-hold devices are used to generate the sampled values yˆ and uˆ between two successive transmission instants which leads to yˆ = 0 and uˆ = 0 for almost all t ∈ [ti, ti+1], i ∈ Z≥0. Other types of holding function can be considered ([64]) but we do not investigate those in this thesis. After each transmission instant, yˆ and uˆ are reset to the actual values of y and u, respectively. We introduce the network-induced error e := (ey , eu) ∈ Rne , where ey := yˆ − y (1.4).

Event-triggering mechanisms

The main objective of the ETC problem is to design the flow and t he jump sets of system (1.5), i.e. the triggering condition, to guarantee the closed-loop stability, to reduce the number of transmissions, and to ensure the existence of a uniform strictly positive lower bound on the inter-transmission times. In what follows, we present some common techniques in the literature to design the flow and the jump sets.

State-dependent threshold

In many control system applications, asymptotic stability properties are required for the closed-loop system. To achieve this goal, the triggering condition threshold should be a function of the system state and not a fixed constant as in the previous tec hnique. Most of the existing results on this triggering mechanism assume that the full state measurement can be accessed by the controller, e.g. [43, 44, 57, 61, 70, 89, 119] and the references therein. Consequently, the feedback law is designed based on the full state information and we have that, in view of (1.3), yˆ = xˆp, u = gc(ˆxp) and the sampling induced error becomes ex = xˆp − xp. (1.8).
In this context, many strategies have been proposed in the literature to construct the flow and jump sets for the hybrid system (1.5). We present here the result in [103] which is one of the common techniques in the literature. Furthermore, we will start from this result to establish our triggering mechanism later. The idea in [103] is to first assume that the state feedback law u = k(ˆxp) renders the closed-loop system x˙p = fp(xp, k(xp + ex)) (1.9).

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State feedback controllers

The technique proposed in Section 2.2 is also relevant in the context of state feedback control, i.e. when y = x, as the constant T in (2.10) can be used to directly tune the lower bound on the inter-transmission time (up to T in (2.11)). Although the existence of a lower bound on the inter-transmission times is guaranteed in [103], the obtained value may be subject to some conservatism and it is not explicitly predetermined as in our triggering mechanism. Furthermore, the generated amount of transmissions by our triggering mechanism are ensured to be less than or, at least, equal to those generated by conventional periodic setups in the sense of [79]. These properties of our proposed triggering mechanism extend its interest to the context of state feedback control.

Enlarging the guaranteed minimum inter-transmission time

A key challenge in the design of output feedback event-triggered controllers is to ensure the existence of a uniform strictly positive lower bound on the inter-transmission times. Although the existence of that lower bound is guaranteed by different techniques in the literature, the available expressions are often subject to some conservatism. It is therefore unclear whether the event-triggered controller has a dwell-time which is compatible with the hardware limitations. We investigate in this section how to employ the LMI conditions (3.8), (3.9) to maximize the guaranteed minimum inter-transmission time. We first state the following lemma to motivate our approach.

Table of contents :

1 Introduction 
1.1 Networked control systems
1.1.1 Control design approaches
1.1.1.1 Emulation
1.1.1.2 Co-design
1.1.1.3 Direct discrete-time
1.2 Event-triggered control
1.2.1 The idea
1.2.2 Hybrid model
1.2.3 Event-triggering mechanisms
1.2.3.1 Static threshold
1.2.3.2 State-dependent threshold
1.2.3.3 Using additional variables
1.2.4 Other state-dependent sampling paradigms
1.2.5 Output feedback control
1.2.5.1 Motivating example
1.2.5.2 Existing results
1.3 Objectives and contributions
1.4 Conclusion
2 Emulation design for nonlinear systems 
2.1 Hybrid model
2.2 Main results
2.2.1 Assumptions
2.2.2 Event-triggering condition
2.2.3 Stability results
2.2.4 Illustrative examples
2.2.4.1 Controlled Lorenz model of fluid convection
2.2.4.2 Single-link robot arm model
2.3 Case studies
2.3.1 LTI systems
2.3.1.1 Analytical results
2.3.1.2 Illustrative example
2.3.2 State feedback controllers
2.3.2.1 Analytical results
2.3.2.2 Illustrative examples
2.4 Conclusion
3 Co-design for LTI systems 
3.1 Hybrid model
3.2 Global asymptotic stabilization
3.3 Optimization problems
3.3.1 Enlarging the guaranteed minimum inter-transmission time
3.3.2 Reducing the amount of transmissions
3.4 Illustrative example
3.5 Conclusion
4 Singularly perturbed systems 
4.1 Introduction
4.2 Approximate models
4.3 Hybrid model
4.4 Assumptions
4.5 Main results
4.5.1 A first observation
4.5.2 Semiglobal practical stabilization
4.5.3 Global asymptotic stabilization
4.6 Case studies
4.6.1 A class of globally Lipschitz systems
4.6.2 Application to LTI systems
4.7 Autopilot control of an F-8 aircraft
4.8 Conclusion
5 Conclusions 
5.1 Conclusions
5.2 Contributions
5.3 Recommendations for future research
A Proofs of Chapter 4 
A.1 Proof of Theorem 4.1
A.2 Proof of Theorem 4.2
B Mathematical review 
B.1 Fundamental properties
B.2 Input-to-state stability
B.3 Hybrid dynamical systems
B.4 Miscellaneous
Bibliography 

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