{'fraction_non_alphanumeric': 0.04607873054623131, 'fraction_numerical': 0.018614586512053707, 'mean_word_length': 4.787637969094923, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Robust Global Adaptive Exponential Stabilization of Discrete-Time Systems with Application to Freeway Traffic Control

This paper is devoted to the development of adaptive control schemes for uncertain discrete-time systems, which guarantee robust, global, exponential convergence to the desired equilibrium point of the system. The proposed control scheme consists of a nominal feedback law, which achieves robust, global, exponential stability properties when the vector of the parameters is known, in conjunction with a nonlinear, dead-beat observer. The obtained results are applicable to highly nonlinear, uncertain discrete-time systems with unknown constant parameters. The applicability of the obtained results to real control problems is demonstrated by the rigorous application of the proposed adaptive control scheme to uncertain freeway models.


Introduction
Adaptive control for discrete-time systems has been studied in many works (see for instance [8,29,30,31]) and in many cases it is a direct extension of adaptive control schemes for continuoustime systems (see [16]). Although discrete-time systems allow a direct study of the limitations of adaptive control schemes (see for example [28]), the major shortcoming of many adaptive control methodologies is that the closed-loop system does not exhibit an exponential convergence rate to the desired equilibrium point of the system, even if the nominal feedback law achieves global exponential stability properties when the parameters are precisely known.
This work is devoted to the development of adaptive control schemes for uncertain discrete-time systems, which guarantee robust, global, exponential convergence to the desired equilibrium point of the system. The idea is simple: use a nominal feedback law, which achieves robust, global, exponential stability properties when the vector of the parameters is known, in conjunction with a nonlinear, dead-beat observer. The dead-beat observer (designed using an extension of the methodology described in [12]) achieves the precise knowledge of the vector of unknown parameters after a transient period; then the states of the closed-loop system are robustly led to the desired equilibrium point with an exponential rate by the nominal feedback law. The proposed adaptive scheme does not require the knowledge of a Lyapunov function for the closed-loop system under the action of the nominal feedback stabilizer.
The obtained results are applicable to highly nonlinear, uncertain discrete-time systems with unknown constant parameters. The applicability of the obtained results to real control problems is demonstrated by the rigorous application of the proposed adaptive control scheme to uncertain freeway models.
Traffic congestion in freeways leads to serious degradation of the infrastructure causing excessive delays, and impacting traffic safety and the environment. Extensive research has been conducted to investigate and develop traffic control measures which can tackle this phenomenon. However, measures such as ramp metering, variable speed limits or dynamic route guidance have to be driven by appropriate control strategies in order to achieve their target. Traffic control strategies such as nonlinear optimal control [2,5] and Model Predictive Control [1,9] have been extensively studied but they are highly demanding from the computational point of view. However, the efficiency of traffic operations can also be enhanced by explicit feedback control approaches such as the pioneering I-type regulator ALINEA [23] and its extensions [26,27], as well as other proposed feedback control algorithms in [10,11,24,25]. These explicit feedback control strategies should guarantee local stability properties for the desired uncongested equilibrium point of the freeway model.
A Lyapunov approach was adopted in [14,15], which led to the robust, global exponential stabilization of the uncongested equilibrium point of a nonlinear freeway model. The nonlinear freeway model considered in [14,15] is a generalization of various freeway models (see [4,5,19]), which are special cases of the model used in [14,15]. However, the nonlinear feedback stabilizer demands the knowledge of several model parameters, which are usually unknown. The present work proposes an adaptive control scheme, which is based on a dead-beat nonlinear observer and guarantees the robust, global exponential convergence rate to the desired uncongested point of the freeway model.
The structure of the present work is as follows. Section 2 is devoted to the development of the robust, global, exponential adaptive control scheme for nonlinear uncertain discrete-time systems. The obtained results are applied rigorously in Section 3 to uncertain freeway models for the robust, global, exponential attractivity of the (unknown) desired uncongested equilibrium point of the freeway model. The concluding remarks of the paper are given in Section 4.
Notation. Throughout this paper, we adopt the following notation:

Exponential Stabilization Of Systems With Unknown Parameters
is a non-empty set, We next consider discrete-time systems with uncertain constant parameters and outputs. Consider the discrete-time system: is a locally bounded mapping. We assume that S x   is an equilibrium point for system (2.2) and D d  is a vanishing perturbation, i.e., there exist vectors and for all Assumption (H1) is a standard assumption, which guarantees the existence of a robust global exponential stabilizer when the vector of the parameters    is known. Assumptions (H2)-(H3) are equivalent to complete, robust observability assumption of  from the output given by (2.3) (see also [12]).
Then the dynamic feedback stabilizer achieves the following:

1)
There exist constants . Since It follows from (2.6) and the triangle inequality that the following inequality holds: Combining (2.7) and the fact that . By virtue of assumption (i), we get: . Combining (2.9), (2.10) and the fact that Noticing that assumption (iii) guarantees that is RGES for the uncertain system (2.1). The proof is complete.  We are now ready to provide the proof of Theorem 2.1.
be an arbitrary sequence and let , and, consequently, assumption (H2) guarantees that . Using induction and the previous argument, it follows that Therefore, assumption (i) of Lemma 2.1 is a consequence of assumption (H1).
Assumption (ii) of Lemma 2.1 follows from the fact that , it follows that assumption (iii) of Lemma 2.2 holds.

3.I. The freeway model
We consider a freeway which consists of 3  n components or cells; typical cell lengths may be 200-500 m. Each cell may have an external inflow (e.g. from corresponding on-ramps), located near the cell's upstream boundary; and an external outflow (e.g. via corresponding off-ramps), located near the cell's downstream boundary (Figure 1). The number of vehicles at time The total outflow and the total inflow of vehicles of the component  ). Our first assumption states that the external (off-ramp) flows from each cell are constant percentages of the total exit flow, i.e., there exist constants 1  , such that: The constants i P are known as exit rates. Since the n -th cell is the last downstream cell of the considered freeway, we may assume that 1  n P . We also assume that , and that all exits to regions out of the network can accommodate the respective exit flows.
Our second assumption is dealing with the attempted outflows , i.e. the flows that will exit the cell if there is sufficient space in the downstream cell. We assume that there exist functions (3.4) The variable , indicates the percentage of the attempted outflow from cell 1  i that becomes actual outflow from the same cell. The function is called, in the specialized literature of Traffic Engineering (see, e.g., [4,5,6,19,20,21]), the demandpart of the fundamental diagram of the i -th cell, i.e. the flow that will exit the cell i if there is sufficient space in the downstream cell i+1. Notice that equation (3.4) for ) ( , t F out n follows from our assumption that all exits to regions out of the network can accommodate the exit flows.
, correspond to external on-ramp flows which may be determined by a ramp metering control strategy. For the very first cell 1, we assume, for convenience, that there is just one external inflow, (3.5) Our next assumption requires that the inflow of vehicles at the cell Following [4], we assume that, when the total demand flow of a cell is lower than the supply of the downstream cell, i.e. when , then the demand flow can be fully accommodated by the downstream cell, and hence we have . In contrast, when the total demand flow of a cell is higher than the supply of the downstream cell, i.e. when ), then the demand flow cannot be fully accommodated by the downstream cell, and the actual flow is determined by the supply function, i.e. we have ). Therefore, we get: are time-varying parameters. Note that, if the supply is higher than the total demand, then (3.8) yields (and by allowing them to be time-varying), we have taken into account all possible cases for the relative priorities of the inflows (and we also allow the priority rules to be time-varying); see [3,4] for freeway models with specific priority rules, which are special cases of our general approach.  .7) and (3.9) we obtain the following discrete-time dynamical system:   Assumption (H) is a technical assumption that allows a very general class of demand functions (which are also allowed to be discontinuous). A more general assumption than assumption (H) was used in [14,15], but in [14,15] it was assumed that all parameters of the model were known. More specifically, in [14,15], it was not necessary the demand functions

3.II. Global Exponential Stabilization of Freeway Models
Notice that, using definition (3.14), the control system (3.11), (3.12), (3.13) can be written in the following vector form: (3.15) Consider the freeway model (3.15) under assumption (H). Let  for the freeway model: The uncongested equilibrium point is not globally exponentially stable for arbitrary ,..., 2  ); indeed, for relatively large values of inflows ), other equilibria for model (3.15) (congested equilibria) may appear, for which the cell densities are large and can attract the solution of (3.15).
The following result (see [14,15]) is the main result in feedback design that provides the nominal feedback for the adaptive control scheme that we intend to use. The result shows that a continuous, robust, global exponential stabilizer exists for every freeway model of the form (3.15) under assumption (H).
achieves robust global exponential stabilization of the uncongested equilibrium point  x of system (3.15), i.e.,  x is RGES for the closed-loop system (3.15) with The result of Theorem 3.1 (see [14,15]) is based on the construction of a Control Lyapunov function for system (3.15) under a more general assumption than assumption (H). The feedback law provides values for the controllable inflows i v ,  ( Another feature of the present problem is that the selection of the uncongested equilibrium point may be made in an implicit way. For example, we may want the uncongested equilibrium point that guarantees the maximum outflow from the freeway. In such cases, the equilibrium position of the controllable inflows is determined as a function of the nominal values of the uncontrollable inflows and the parameters of the freeway, i.e., there exists a smooth function   . All these parameters will be denoted by

3.III. Measurements and Unknown Parameters
and are assumed to take values in a compact set . Therefore, the control system (3.11), (3.12), (3.13) can be written in the following vector form: ( .17) and (3.23)). It follows that assumption (H1) holds for system (3.24). An explicit definition of the feedback law is given by the following equations for all Therefore, the measured output is given by: , then the following equations hold:   (3.24) with output given by (3.29), (3.30), (3.31). In order to prove that assumption (H3) holds for system (3.24) with output given by (3.29), (3.30), (3.31), we need the following fact, which is a consequence of property (C5) shown in [14] and (3.20).
where  x is given by (3.24).

2)
There exists an integer Next, we notice that by virtue of (3.22) and the facts that (3.62) Using (3.25), (3.26)

Concluding Remarks
Novel results for adaptive control schemes for uncertain discrete-time systems, which guarantee robust, global, exponential convergence to the desired equilibrium point of the system, were provided in the present work. The proposed control scheme consists of a nominal feedback law, which achieves robust, global, exponential stability properties when the vector of the parameters is known, in conjunction with a nonlinear, dead-beat observer. The proposed adaptive scheme did not require the knowledge of a Lyapunov function for the closed-loop system under the action of the nominal feedback stabilizer and is directly applicable to highly nonlinear, uncertain discrete-time systems with unknown constant parameters.