A survey of recent results in quantized and eventbased nonlinear control
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Abstract
Constructive nonlinear control design has undergone rapid and significant progress over the last three decades. In this paper, a review of recent results in this important field is presented with a focus on interdisciplinary topics at the interface of control, computing and communications. In particular, it is shown that the nonlinear smallgain theory provides a unified framework for solving problems of quantized feedback stabilization and eventtriggered control for nonlinear systems. Some open questions in quantized and networked nonlinear control systems are discussed.
Keywords
Nonlinear systems nonlinear control quantized control eventbased control smallgain theory1 Introduction
Stabilization, a fundamentally important topic in control theory, seeks to find a feedback control law that renders a dynamical system stable at an equilibrium point of interest. Early efforts in tackling this problem for nonlinear systems resulted in nonlinear feedback algorithms that are restricted in their use to certain classes of nonlinear systems. Examples of such algorithms include sliding mode control and absolute stability based methods. For instance, when sliding mode control is applied to design stabilizing controllers, nonlinearities (which may be uncertain) are assumed to fall into the input spanned space, i.e., some kind of matching conditions are required. Latest developments in sliding mode control methods have relaxed this assumption for special classes of nonlinear systems but with limited success for higherdimensional nonlinear systems with unmatched uncertainties. Global stabilization of nonlinear timeinvariant systems differs from its linear counterpart and leads to early results that often assume global Lipschitz condition on system nonlinearities using state and output feedback. With these restrictions of early nonlinear feedback algorithms in mind, linear thinking yields limited success in solving the stabilization problem of nonlinear systems. The development of fundamentally nonlinear feedback design tools has thus become a hot topic in nonlinear control. Starting from the late 1980s followed by the publication of the survey paper[1], significant progress has been made in nonlinear stabilization. Over the last three decades, many innovative ideas and methods have been proposed by numerous researchers for the local, semiglobal and global stabilization of nonlinear systems[2, 3, 4, 5, 6, 7, 8]. Other byproducts of this collective effort by many researchers include advances in other important topics in nonlinear control such as output regulation of nonlinear systems[9, 10, 11, 12, 13, 14, 15], optimal nonlinear control, and output feedback control[16, 17, 18]. Due to space limitation and the limited knowledge of the authors, a tough choice must be made here and we will selectively discuss topics tied to our recent research and cite relevant results which are closely tied to our chosen topics.
The layout of the paper is as follows. Section 2 first states the formulation of the stabilization problem and then reviews some early results in nonlinear control algorithms. Section 3 presents the basics of nonlinear smallgain theory and some recent developments in quantized feedback stabilization of nonlinear systems by means of smallgain theorems. Section 4 focuses on the emerging topic of eventbased nonlinear control that aims to update controllers only when some events occur. Both centralized and decentralized systems with event triggers will be discussed. Smallgain based solutions to eventbased nonlinear control design will be presented. Finally, Section 5 closes this review article with some brief concluding remarks and open problems.
2 Early results in nonlinear feedback stabilization
2.1 Problem statement
In the sequel, we first address the existence of a statefeedback control law for the stabilization of nonlinear system (1). Then, we will present some tools that allow us to construct explicitly stabilizing control laws.
2.2 Explicit design algorithms
 Theorem 1. Assume that V is a smooth CLF for system (1). Then, a feedback control law u = μ(x)that globally asymptotically stabilizes the system takes the following form:$$\mu (x) = \left\{ {\begin{array}{*{20}c} {  \frac{{L_f V(x) + \sqrt {\left( {L_f V\left( x \right)} \right)^2 + \left( {L_g V\left( x \right)} \right)^4 } }} {{L_g V(x)}},} \\ {if L_g V(x) \ne 0} \\ {0, otherwise.} \\ \end{array} } \right.$$(6)
Generally speaking, the control law μ in (6) may not be smooth everywhere. It is shown that under certain small control property, the control law μ is “almost smooth”, i.e., is continuous at x = 0 and smooth everywhere else. Such a small control property is defined as follows[19]:
For each ε > 0, there exists a constant δ > 0 such that, for any x satisfying 0 < x < δ, there is some u with u < ε such that L_{ f }V(x) + uL_{ g }V(x) < 0.
CLFs have been used widely in the literature of modern nonlinear control, e.g., in adaptive nonlinear control[21, 22, 23], robust nonlinear control[24], nonlinear optimal control[24, 25], nonlinear timedelay systems[26, 27], and multiagent systems[28], to name only a few.
It should be mentioned that CLF is a generalization of Lyapunov function from a dynamical system without controls to a nonlinear control system. As is well documented in the past literature, the construction of both a Lyapunov function and a CLF for a nonlinear dynamical system is far from being trivial. Nonetheless, for some important classes of nonlinear control systems, tools are available for generating a CLF and a stabilizing control law. Also, it is worth noting that we may construct stabilizing controllers for specific classes of nonlinear systems without assuming the existence of a CLF. We will review some of these existing tools in the remainder of this section.
 Theorem 2. A control law that globally asymptotically stabilizes the cascade system (7)–(8) takes the form:where c_{2} is an arbitrary positive constant. Moreover, \({V_2}({x_1} + {x_2}) = {V_1}({x_1}) + {1 \over 2}{({x_2}  {\mu _1}({x_1}))^2}\) is a CLF for the cascaded system (7) and (8).$$\matrix{{u =  {c_2}({x_2}  {\mu _1}({x_1}))  {f_2}({x_1},{x_2}) + {{\partial {\mu _1}} \over {\partial {x_1}}}{f_1}({x_1},{x_2})} \hfill \cr{\quad  {{\partial {V_1}({x_1})} \over {\partial {x_1}}}\int_0^1 {{f_1}} ({x_1},{\mu _1}({x_1}) + \lambda ({x_2}  {\mu _1}({x_1})){\rm{d}}\lambda} \hfill \cr}$$(9)
It should also be mentioned that there have been several research publications devoted to relaxing the smoothness of virtual control laws μ_{1} and/or the smoothness of system nonlinearities[40, 41, 42].
3 Smallgain method and applications
3.1 Basics of nonlinear smallgain theory
The smallgain method is a tool for constructive nonlinear feedback design particularly suited for nonlinear and interconnected control systems with parametric and dynamic uncertainties[23, 36, 43, 44]. Take the system (7) and (8) as an example. When x_{1}system is considered as a dynamic uncertainty driven by x_{2} with unknown state x_{1} and dynamics f_{1}, conventional Lyapunov designs as presented above are not directly applicable. Additionally, it is not clear how to apply traditional approximation techniques such as neural networks and fuzzy systems theory to approximate “nonlinear dynamic uncertainties” represented by f_{2}(x_{1}, x_{2}).
Undoubtedly, ISS has become a fundamental tool for solving many analysis and synthesis problems in nonlinear systems, as documented in the tutorial paper by Sontag[7]. Its important role in advancing the state of the art in robust nonlinear control with respect to dynamic uncertainties has led to the introduction of generalized/nonlinear smallgain theorems. The following provides a quick review.
 Theorem 3[43, 46]. Under one of the following equivalent smallgain conditions:$${\gamma _1} \circ {\gamma _2}(s) < s,\quad \forall s > 0$$(16)the interconnected system (14) and (15) is ISS when v is considered as the input.$${\gamma _2} \circ {\gamma _1}(s) < s,\quad \forall s > 0$$(17)
As well documented in the literature, the global stabilization of the system (7) and (8) with partialstate x_{2} information can be addressed from a smallgain perspective. The crucial difference with other Lyapunov designs is that system (7) and (8) is now treated as an interconnected system. The only knowledge we need is that the x_{1}subsystem is ISS with a known ISSgain, say, γ_{1} of class K_{∞.} In order to invoke the smallgain theorem, we only need to show that a feedback law of the form u = κ(X_{2}) can be designed to render the x_{2} system ISS with a gain γ_{2} that is strictly smaller than \(\gamma _1^{ 1}\) so that the smallgain condition (16) or (17) holds. Such a result is referred to as gain assignment theorem[43]. Theorem 4 states mild assumptions under which the global stabilization problem for system (7) and (8) with incomplete state and dynamics information is solvable.

Theorem 4. Assume that the x_{1}system is ISS with a gain γ_{1} of class K_{∞}. It is further assumed that f_{2} (x_{1}, x_{2}) is dominated by σ_{1}(x_{1})+ σ_{2}(x_{2}) with σ_{ i } being locally Lipschitz and positive semidefinite. Then, the global stabilization of system (7) and (8) is solvable by continuous partialstate feedback law u = κ(X_{2}).
The above result was initially developed in [43] and has been applied to various control problems[36, 44, 46, 47].More recently, it has been extended to the context of nonlinear feedback stabilization with quantized signals[48, 49, 50, 51, 52].
3.2 Application to quantized feedback stabilization
The convergence of control and communications has led to many new control problems of practical interest. Quantized stabilization with quantized signals is just one of them. A quantizer is a nonlinear operator that converts a signal from a continuous region to a discrete set of numbers, and thus is a discontinuous function. This special feature poses severe technical challenges to quantized controller design for both linear and nonlinear systems, e.g., [53, 54, 55, 56, 57] for quantized stabilization of linear systems and [58, 59, 60, 61] for extensions to nonlinear systems.
Despite its theoretic importance and practical relevance, quantized feedback stabilization of nonlinear systems has received little attention as of today. There are several technical obstacles one needs to overcome. First and foremost, when quantization is introduced at the levels of output measurement and/or control input, the feedback control law to be implemented will be discontinuous with respect to the state variable. The interplay of discontinuity with the nonlinearity and dimensionality of the system leads to an immediate bottleneck of the use of recursive feedback design tools such as backstepping. Second, dynamic quantization is often preferable compared with logarithmic quantization in handling the problem caused by the finite word length of the quantizers in networked control systems. The key idea of dynamic quantization is to adjust dynamically the range of the quantizer through “zoomingin” and “zoomingout” phases. To avoid finiteescape phenomenon during the “zoomingout” phase, a common feature of the existing work[50, 61, 62] is that forward completeness and smalltime normobservability are assumed for the (openloop) unforced system. Clearly, these assumptions severely limit the class of nonlinear systems we can address for quantized stabilization. Third, the closedloop system with quantized control is discontinuous and often hybrid. Stability analysis of such systems is still a hot topic of research. Last but not least, when large uncertainty occurs, the quantized feedback control of nonlinear systems is still a little explored research arena.
It should be mentioned that the generalized outputfeedback form (18)–(21) was first introduced in [17] in the absence of quantization and disturbance input d, and is an extension of the conventional outputfeedback form with only outputnonlinearities[21, 35].
The control objective is to find, if possible, a quantized outputfeedback control law that drives the output signal to within an arbitrarily small neighborhood of the origin, while keeping the boundedness of all the closedloop system signals.
 Assumption 1. The zsystem is ISS and has a positivedefinite and radially unbounded ISSLyapunov function V_{ z } that satisfies the implicationwhere \(\gamma _z^y,\gamma _z^d\) and α_{ z } are classK_{∞} functions.$${V_z}(z) \geq \max \{\gamma _z^y(\left\vert y \right\vert),\;\gamma _z^d(\left\vert d \right\vert)\} \Rightarrow \nabla {V_z}(z)\dot z \leq  {\alpha _z}(\left\vert z \right\vert)$$
 Assumption 2. For each i = 1, 2, ⋯, n, the uncertain function Δ_{ i } is overbounded by a classK_{∞} function \({K_\infty}\), i.e.,$$\left\vert {{\Delta _i}(y,z,d)} \right\vert \leq {\psi _{{\Delta _i}}}(\left\vert {(y,z,d)} \right\vert).$$

Assumption 3. The unforced system (18)–(21) with u = 0 is forward complete and smalltime normobservable with y as the output[50,62].
 Assumption 4. The quantizer q_{ μ } satisfies the following propertywhere M, δ are positive constants, Mμ is the range of the quantizer, and δμ is the maximum quantization error for all u in the range of the quantizer. As usual, μ is called “zooming” variable.$$\left\vert {{q_\mu}(u)  u} \right\vert \leq {\delta _\mu},\quad {\rm{if}}\left\vert u \right\vert \leq M\mu$$
Obviously, system (25)–(29) is a higherorder variant of the system (7)–(8) with x_{1} = (ζ, z) appended with more than one nonlinear integrators.

Theorem 5. Under Assumptions 1–4, the quantized outputfeedback control problem is solvable for nonlinear systems transformable to the generalized outputfeedback form (18)–(21).

Remark 1. Under some mild conditions, the controller can be fine tuned to achieve asymptotic convergence of the state signals to the origin. See [63] for some initial results.
4 Eventbased nonlinear control
The study of eventtriggered control has recently attracted considerable attention within the control systems community. A usually considered eventtriggered control system can be viewed as a sampleddata system in which data sampling is triggered by external events depending on the realtime system state, and may not be periodic. Compared with the traditional periodic sampling, eventtriggered sampling takes into account the system behavior between the sampling time instants and has been proved to be quite useful in reducing the waste of computation and communication resources in networked control systems. Eventtriggered sampling also provides solutions to sampleddata control of nonlinear systems, for which periodic sampling may not work very well.
Significant contributions have been made to the literature of eventtriggered control, e.g., [64, 65, 66, 67, 68, 69, 70, 71] and the references therein. Specifically, in [65, 69], impulsive control methods are developed to keep the states of firstorder stochastic systems inside certain thresholds. In [72, 73], prediction of the realtime system state between the sampling time instants was employed to generate the control signal, and the prediction is corrected by datasampling when the difference between the true state and the predicted state is too large. For eventbased control of nonlinear systems, Tabuada[67] considered the systems which admit controllers to guarantee the robustness with respect to the sampling errors. Then, the event trigger is designed such that the sampling error is bounded by a specific threshold (depending on the realtime system state) for convergence of the system state. Marchand et al. [74] proposed a universal formula for eventbased stabilization of general nonlinear systems affine in the control by extending Sontag’s result for continuoustime stabilization[19]. Tallapragada and Chopra[75] proposed a Lyapunov condition for tracking control of nonlinear systems. The designs have been extended to distributed networked control[76, 77], outputfeedback control and decentralized control[71] and systems with quantized measurements[78], to name a few. The reader may consult the nice tutorial[79] for the recent developments of eventtriggered control and selftriggered control. For practical implementation of eventtriggered control, infinitely fast sampling should be avoided, i.e., the intervals between the sampling time instants should be lower bounded by some positive constant[80]. In the context of eventbased control, due to the hybrid nature, the forward completeness of the closedloop system is a complex issue.
4.1 Smallgain based eventtriggering controllers

Case 1. S = Z_{+} and lim_{k→x} t_{ k } < ∞, which means Zeno behavior[81].

Case 2. S = Z_{+} and lim_{k→∞} t_{ k } = ∞. In this case, x(t) is defined on [0, ∞).

Case 3. S is a finite set {0, ⋯, k*} with k* ∈ Z_{+}, i.e., there is a finite number of sampling time instants. In this case, \({t_{k\ast}} < {T_{\max}}\) and we set \({t_{k\ast + 1}} = {T_{\max}}\) for convenience of discussions.
It should be noted that, in any case, x(t) is defined for all \(t \in [0,{T_{max}})\). With an appropriate event trigger design, we will show that inf_{k∈S}{t_{ k }_{+1} − t_{ k }} > 0, which means that Case 1 is impossible. Also, by means of smallgain arguments, we will prove that T_{max} = ∞ for Case 3.
If ω(t) is not adjustable, then the eventtriggered control problem is reduced to the measurement feedback control problem. The basic idea of eventtriggered control is to adjust ω(t) online with an appropriate datasampling strategy, to realize asymptotic convergence of x(t), if possible. From this point of view, the structure of the closedloop system can be represented with the block diagram shown in Fig. 2.
 Assumption 5. System (36) is ISS with ω as the input, i.e., there exist \(\beta \in {\cal K}{\cal L}\) and \(\gamma \in {\cal K}\) such that for any initial state x(0) and any measurable and locally essentially bounded ω, it holds thatfor all t ≥ 0.$$\left\vert {x(t)} \right\vert \leq \max \{\beta (\left\vert {x(0)} \right\vert ,t),\gamma ({\left\Vert w \right\Vert _\infty})\}$$(37)
The data sampling event is not triggered if for some specific \({k^{\ast}} \in {{\bf{Z}}_ +},\;x({t_{k\ast}}) = 0\) or \(\{t > {t_k}:\rho (\vert x(t)\vert)  \vert x(t)  x({t_k})\vert = 0\} = \phi\). Note that, under the assumption of \(f(0,v(0)) = 0,\;if\;x({t_{{k^{\ast}}}}) = 0\), then \(u(t) = v(x({t_{{k^{\ast}}}})) = 0\) keeps the system state at the origin for all \(t \in [{t_{{k^{\ast}}}},\infty)\).

Theorem 6. Consider the eventtriggered control system (36) with locally Lipschitz \({\bar f}\) satisfying \(\bar f(0,0)\) and ω defined in (33). If Assumption 5 is satisfied with a locally Lipschitz γ, then one can find a \(\rho \in {{\cal K}_\infty}\) such that ρ satisfies (38) and ρ^{−1} is locally Lipschitz.
The original proof of Theorem 6 can be found in [82].
4.2 Decentralized eventbased control

Objective 1. Infinitely fast sampling is avoided, i.e., for any specific x(0) and any specific μ_{ i }(0) > 0,i = 1, ⋯, N, the intervals \(t_{k + 1}^i  t_k^i\) between the eventtriggering time instants for each x_{ i }subsystem (i =1, ⋯, N) are lower bounded by a positive constant.

Objective 2. The closedloop eventtriggered system is forward complete, i.e., x(t) is defined for all t ≥ 0 and all initial condition x(0). In addition, x(t) globally asymptotically converges to the origin.
In this paper, we focus on the event trigger design, and assume, without loss of generality[21], that local feedback control laws have been designed such that each x_{ i }subsystem is inputtostate stable with the inputs e_{ i } and x_{ j } for j ≠ i.
 Assumption 6. For i =1, ⋯,N, each x_{ i }subsystem is ISS with an ISSLyapunov function \({V_i}:{{\bf{R}}^{{n_i}}} \rightarrow {{\bf{R}}_ +}\), which is locally Lipschitz on \({{\bf{R}}^{{n_i}}}\backslash \{0\}\) and satisfies$${_i}(\left\vert {{x_i}} \right\vert) \leq {V_i}({x_i}) \leq {\bar \alpha _i}(\left\vert {{x_i}} \right\vert)$$(48)where \({\underline \alpha _i},{{\bar \alpha}_i} \in {K_\infty},\chi _i^j,{\gamma _i} \in K \cup \{0\}\), and α_{ i } is continuous and positive definite.$$\matrix{{{V_i}({x_i}) \geq \mathop {\max}\limits_{j \neq i} \{\chi _i^j({V_j}({x_j})),{\gamma _i}(\left\vert {{e_i}} \right\vert)\} \Rightarrow} \hfill \cr{\nabla {V_i}({x_i}){f_i}(x,{e_i}) \leq  {\alpha _i}({V_i}({x_i}))\quad {\rm{a}}{\rm{.e}}.} \hfill \cr}$$(49)
With Assumption 6 satisfied, the largescale system (43) is ISS with e_{ i } for i = 1, ⋯, N as the inputs, if the interconnection gains \(\chi _i^j\) satisfy the cyclicsmallgain condition. If, additionally, the event triggers are designed such that Objective 1 is achieved and each e_{ i }(t) asymptotically converges to the origin, then x(t) globally asymptotically converges to the origin. In this section, we propose a new class of decentralized event triggers for the largescale nonlinear system by using ISS smallgain arguments.
The design of the event triggers depends on the dynamic behavior of the closedloop eventtriggered system. Based on the estimation of the convergence rate of the closedloop eventtriggered system, the functions φ_{ i } and ϕ_{ i } can be found for the decentralized event triggers to achieve Objectives 1 and 2.
 Lemma 1. Under Assumption 6, suppose that the largescale system composed of (43) satisfies the cyclicsmallgain condition.
 1)For each i = 1, ⋯, N, there exists \({\sigma _i} \in {{\cal K}_\infty}\) being locally Lipschitz on (0, ∞) such thatis an ISSLyapunov function of the x_{ i }subsystem, that satisfies$${\bar V_i}({x_i}) = {\sigma _i}({V_i}({x_i}))$$(52)where \(\bar \chi _i^j \in {\cal K} \cup \{0\}\) satisfies \(\bar \chi _i^j < {\rm{Id,}}{{\bar \gamma}_i} = {\sigma _i} \circ {\gamma _i}\), and α_{i′} is continuous and positive definite.$$\matrix{{{{\bar V}_i}({x_i}) \geq \mathop {\max}\limits_{j \neq i} \left\{{\bar \chi _i^j({{\bar V}_j}({x_j})),{{\bar \gamma}_i}(\left\vert {{e_i}} \right\vert)} \right\} \Rightarrow} \hfill \cr{\nabla {{\bar V}_i}({x_i}){f_i}(x,{e_i}) \leq  \alpha _i^{\prime}({{\bar V}_i}({x_i}))\quad {\rm{a}}{\rm{.e}}.} \hfill \cr}$$(53)
 2)Consider the largescale system composed of (43), (50) and (51). Suppose that (47) holds for t ∈ [0, T_{max}) for i = 1, ⋯,N. By choosing ϕ_{ i } such that \({{\bar \gamma}_i} \circ {\varphi _i} < {\rm{Id}}\) for i =1, ⋯, N, the functionsatisfies$$V(x,\eta) = \mathop {\max}\limits_{i = 1, \cdots ,N} \{{\bar V_i}({x_i}),{\eta _i}\}$$(54)for all t ∈ [0, T_{max}), where$${D^ +}V(x(t),\eta (t)) \leq  \alpha (V(x(t),\eta (t)))$$(55)for s ∈ R_{+}.$$\alpha (s) = \mathop {\min}\limits_{i = 1, \cdots ,N} \{\alpha _i^{\prime}(s),{\phi _i}(s)\}$$(56)
Based on the estimation of the convergence rate of the closedloop eventtriggered system, we summarize our main result of decentralized event trigger design in Theorem 7.
 1)

Theorem 7. Consider the interconnected system composed of (43), (50) and (51) subject to (45), with Assumption 6 satisfied. The two objectives of decentralized eventtriggered control are achievable if there exists a \(\bar \gamma \in {{\cal K}_\infty}\) such that \(\bar \gamma \geq {\max _{i = 1, \cdots ,N}}\{{{\bar \gamma}_i}\}\) and \(\underline \alpha _i^{ 1} \circ \sigma _i^{ 1} \circ \bar \gamma\) is locally Lipschitz for i =1, ⋯, N.
Please see [83] for the original proofs of Lemma 1 and Theorem 7.
5 Conclusions and future work
This paper has presented a review of recent results in the field of constructive nonlinear control design, with a focus on the nonlinear smallgain tools in solving the problems of quantized feedback stabilization and eventtriggered control for nonlinear systems.
 1)
Eventtriggered control of nonlinear systems with quantized and/or delayed measurements. In networked control systems, datasampling and quantization usually coexist. In the quantized control results, we use ISS gains to represent the influence of quantization error, while for eventbased control, we employ an ISS gain to represent the influence of datasampling. This creates an opportunity to develop a unified framework for eventtriggered and quantized control of nonlinear systems. Timedelays also arise from networked control systems. Note that [78] has studied eventtriggered control for linear systems with quantization and delays, also see our recent preliminary work[84, 85]. Based on the recent theoretical achievements for nonlinear systems with timedelays[86], it is of interest to study the eventtriggered control problem for nonlinear systems by taking into account the effects of timedelays.
 2)
Distributed eventtriggered control. The idea of smallgain design also bridges eventtriggered control and our recent distributed control results. In [87], it is shown that a distributed control problem for nonlinear uncertain systems can ultimately be transformed into a robust stability problem of a network of ISS subsystems. By integrating the idea in this paper, distributed control could be realized through eventtriggered information exchange. Note that such ideas have been implemented for linear systems[76, 88, 89, 90].
 3)
Datadriven nonlinear control. An emerging topic under current investigation is to develop a new class of datadriven controllers for robust optimal control of nonlinear uncertain systems, leveraging techniques from reinforcement learning and adaptive dynamic programming. Some prior results are presented in [91, 92, 93, 94] and references therein.
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