Igor Val Danilov
email: igor_val.danilov@acci.center

Sandra Mihailova
email: sandra.mihailova@rsu.lv

New Findings in Education: Primary Data Entry in Shaping Intentionality and Cognition

Keywords: coherent intelligence, emotional contagion, interactional synchrony, social entrainment, embodied cognition I

One of the most intriguing questions in cognitive science is the Primary Data Entry (PDE) problem: where does social knowledge come from? After birth, an organism meets with reality, which is transcendental, staying beyond any experience and understanding of this pure reason. Basic approaches to the study of cognition are cognitivism, connectionism, and embodied dynamicism, The weakness of cognitivism and connectionism is the need to introduce an initial set of social phenomena of the specific community to trigger this system: the PDE problem. There is also a same gap in knowledge in embodied dynamicism approach, this interpretation of a dynamic system is not accurate. The current paper presents the Model of Coherent Intelligence (MCI) and its neural foundation. The analysis of recent empirical data proposes new insights on the origin of intentionality: (i) cognition begins from separating sensory stimuli: Long-Term Potentiation can be induced in neurons of particular Modality-Specific gateways (ignoring other stimuli)-selective induction promotes selective sensitivity to the chaos of stimuli. (ii) Neurons can learn spike-timing-dependent plasticity in social interaction: immature neurons learn the timing code to modulate certain synaptic strength, which triggers either Long-Term Potentiation or Long-Term Depression. The hypothesis of the MCI argues that social interaction shapes organisms' intentionality, promoting similar categorization of stimuli in intimately related individuals with shared social routine and interests. This approach provides a wide range of possibilities for developing a human-computer interface.

INTRODUCTION

The academic knowledge on the study of mind historically and conceptually has settled three main approaches within cognitive science: cognitivism, connectionism, and embodied dynamicism [START_REF] Thompson | Mind in Life : biology, phenomenology, and the sciences of mind[END_REF]. Many theories of mind combine all three approaches, where they co-exist in various hybrid forms. The more interesting of them are the Embodied dynamic system [START_REF] Thompson | Mind in Life : biology, phenomenology, and the sciences of mind[END_REF], the theory of innate intersubjectiivity and innate foundations of neonatal imitation [START_REF] Delafield-Butt | Theories of the development of human communication[END_REF], the theory of natural pedagogy [START_REF] Csibra | Natural Pedagogy[END_REF], and the theory of sensitivities and expectations [START_REF] Waxman | Early Word-Learning and Conceptual Development[END_REF]. All these theories are plausible; the current paper observes different views to engage a gap in knowledge.

According to Thompson [START_REF] Thompson | Mind in Life : biology, phenomenology, and the sciences of mind[END_REF], cognitivism (the metaphor is the mind as digital computer) and connectionism (the mind as neural network), in different ways, appeal to the same computational principle of cognition. This principle based upon processing a signal within neural networks. This computational principle certainly requires the primary data entry as a necessary initial condition to launch processing. None algorithm and/or a sequence of instructions may perform the computation of any process without corresponding to the specific situation inputs, that should substitute variables and parameters of the formulas. The algorithm remains just a set of mathematical variables without this input. This argument may mean the necessity to input an initial set of social phenomena of the specific community to trigger this system -the Primary Data Entry (PDE) problem [START_REF] Val | Imitation or Early Imitation: Towards the Problem of Primary Data Entry[END_REF].

According to embodied cognitivists, the mind is an autonomous system by its self-organizing and selfcontrolling dynamics, which does not have inputs and outputs in the usual sense, and determines the cognitive domain in which it operates [START_REF] Thompson | Mind in Life : biology, phenomenology, and the sciences of mind[END_REF] [6][7]. This approach is grounded on the dynamical hypothesis [START_REF] Van Gelder | The dynamical hypothesis in cognitive science[END_REF]. However, this interpretation of a dynamic system is not accurate. Why does the dynamic system need PDE:

Argument A. According this approach embodied features of cognition deeply depend upon characteristics of the physical body. If the agent's beyond-the-brain body plays a significant causal role, then the primary data yet makes sense.

Argument B. In mathematics, a dynamic systems model is a set of evolution equations. It means that entering primary data is required. The dynamic system may not begin its life cycle without introducing initial conditions corresponding to specific situation inputs and parameters.

Argument C. The dynamical system hypothesis [START_REF] Van Gelder | The dynamical hypothesis in cognitive science[END_REF] have not claimed the lack of initial conditions. Dynamicists track primary data less than dynamic changes inside. However, it does not mean that primary data do not exist and do not necessary.

Given these above arguments, the PDE problem must be considered in the onset of cognition. The embodied dynamic system approach tends to solve the above-noted gap by introducing the notion of dynamically embodied information [START_REF] Thompson | Mind in Life : biology, phenomenology, and the sciences of mind[END_REF]. Although, to introduce this concept, it is necessary to explain the categorization of reality through intentionality. According to embodied cognition approach, symbols encode the local topological properties of neuronal maps [START_REF] Thompson | Mind in Life : biology, phenomenology, and the sciences of mind[END_REF], a dynamic action pattern. The sensorimotor motor network yields pairing of the binary cue stimulus with the particular symbol saved in the structures and processes that embody meanings. 'Representational "vehicles" are temporally extended patterns of activity that can crisscross the brainbody-world boundaries, and the meanings or contents they embody are brought forth or enacted in the context of the system's structural coupling with its environment [1, p.36].' This idea requires introducing the nature of intentionality. In a multi-stimuli environment, the stimulus-consequence pair is unpredictable due to the many stimuli claiming to be associated with the embodied dynamic information randomly. The bond of stimulus-consequence pair of a social phenomenon in the sensorimotor network requires categorizing reality by the nervous system before applying the innate reflex about this social phenomenon to a specific case. Therefore, dynamically embodied information can be useful if intentionality is already in place. Meanwhile, the embodied dynamic system introduces intentionality without a biological and / or physical basis. The theory of natural pedagogy [START_REF] Csibra | Natural Pedagogy[END_REF], and the theory of sensitivities and expectations [START_REF] Waxman | Early Word-Learning and Conceptual Development[END_REF], as well as many others may not solve the problem of PDE [START_REF] Val | Knowledge Sharing through Social Interaction: Towards the Problem of Primary Data Entry[END_REF].

According to Trevarthen and Delafield-Butt [START_REF] Delafield-Butt | Development of human consciousness[END_REF], primary consciousness develops in embryogenesis and is the first operative in early fetal life. 'Consciousness as "acting with knowledge" requires a nervous system that regulates prospective perception in intentional engagement with the world [10, p. 22]'. In the first trimester, patterns of sensory regulation of movements of the fetus' body and limbs gain affective evaluation and sensitivity for sounds and rhythms of other human presence [START_REF] Delafield-Butt | Development of human consciousness[END_REF]. It means that the pure nervous system should already possess intentionality as well as initial knowledge about social reality: human sounds and rhythms also yield meanings. Even if fetuses can hear different sounds and feel rhythms from outside of the womb, this does not mean that they alone (independently) can process their meanings.

Searle et al. [START_REF] Searle | Intentionality: An Essay in the Philosophy of Mind[END_REF] argued that intentionality is the mental power of minds to represent or symbolize things, properties, and states of affairs. According to Crane [START_REF] Crane | Intentionalism." The Oxford Handbook to the Philosophy of Mind[END_REF], mental states or events or processes which have objects in this sense are traditionally called 'intentional,' and 'intentionality' is, for this reason, the general term for this defining characteristic of thought. The meaning of directed action implies the purpose of the action, which first requires the categorization of reality. It's a dichotomy of what happens first. Current knowledge does not solve it.

Tomasello [START_REF] Tomasello | Becoming human: A theory of ontogeny[END_REF], through the study on ontogenesis and phylogenesis, introduced the hypothesis of gradually increasing social bond development in children referred to time slices: (1) emotion sharing from the birth, (2) joint intentionality from the nine-month revolution, (3) collective intentionality at around three years of age, (4) reason and responsibility. Tomasello [START_REF] Tomasello | Becoming human: A theory of ontogeny[END_REF] introduced the beginning of cognition through the newborns' basic motive force of sharing intentionality. However, the mechanism of such emotion coordination is not clear because it is grounded on emotion sharing [START_REF] Tomasello | Becoming human: A theory of ontogeny[END_REF]. Whether or not protoconversations imply understanding emotional states. Many researchers, including the authors, believe that the hypothesis about the universality of emotional expressions is formed by limited experimental methods, since other research designs show the opposite outcome [START_REF] Jack | Culture and facial expressions of emotion[END_REF]- [START_REF] Gendron | Emotion Perception in Hadza Hunter-Gatherers[END_REF]. There is no evidence of a genetic mechanism that can link meaning in mind with certain social reality to apply an appropriate emotional pattern to a specific situation. Even if one assumes that the hypothesis of universal emotional expressions proves innate emotional patterns together with their meanings; even if newborns may alone recognize the basic facial expressions of caregivers and the specific situation to apply them; but in this case, newborns do not have time for such a "training course", because they demonstrate their achievements already in the first hours of life [START_REF] Val | Social Interaction in Knowledge Acquisition: Advanced Curriculum. Critical Review of Studies Relevant to Social Behavior of Infants[END_REF]. If there is no innate mechanism, then, apparently, emotional contagion can occur between individuals without their awareness [START_REF] Val | Knowledge Sharing through Social Interaction: Towards the Problem of Primary Data Entry[END_REF][19] [21]; it can happen even without awareness of the emotional stimuli existence [21]. Section II discusses the hyperscanning and near-infrared spectroscopy studies' outcome, showing brainto-brain synchronization. Section III presents, for the first time, the hypothesis of the neural foundation of shared intentionality. Section IV elaborates all findings, describing their meanings.

II. PROBLEM: HOW DOES SOCIAL INTERACTION ENCOURAGE COGNITION

Brain-to-brain relationships shape the mind during moment-to-moment interactions [START_REF] Valencia | What binds us? Inter-brain neural synchronization and its implications for theories of human consciousness[END_REF]. The dichotomy of newborns' succeed in beginning knowing and their communicative disability challenges our knowledge on social interaction modalities [START_REF] Val | Social Interaction in Knowledge Acquisition: Advanced Curriculum. Critical Review of Studies Relevant to Social Behavior of Infants[END_REF]. We believe that understanding the problem of the intentionality emergence in an organism at the beginning can explain the problem of PDE and the onset of consciousness. This knowledge can contribute to the study of cognition because obviously if and as soon as this implicit modality occurs it continues the whole rest of life. We believe that the caregivers' intentionality forms the intentionality in newborns. Fetuses and newborns are not able to behave intentionally on their own due to the lack of meaningful (informative) sensory interaction at the beginning [START_REF] Val | Imitation or Early Imitation: Towards the Problem of Primary Data Entry[END_REF][19] [START_REF] Val | Emotions in e-Learning: The Review Promotes Advanced Curriculum by Studying Social Interaction[END_REF]. We predict an implicit modality of social interaction that provides shared intentionality at the beginning. Cooperation in a group enhances intentionality, providing categorization.

According to Valencia and Froese [START_REF] Valencia | What binds us? Inter-brain neural synchronization and its implications for theories of human consciousness[END_REF], their review of studies based on EEG-and fNIRS hyperscanning m e t h o d o l o g i e s s h o w s e v i d e n c e o f i n t e r-b r a i n synchronization in the fastest frequency bands, supporting the possibility of extended consciousness. Among hyperscanning studies, we have chosen 4 studies conducted without explicit interaction between subjects. These studies compared differences of brain-to-brain synchronization in subjects when participants solved tasks together as confronting to the condition in which: (i) the subjects solved them individually [START_REF] Szymanski | Teams on the same wavelength perform better: Inter-brain phase synchronization constitutes a neural substrate for social facilitation[END_REF][25]; (ii) the same task when interacting with a machine [START_REF] Hu | Inter-brain synchrony and cooperation context in interactive decision making[END_REF]; (iii) the individuals from another team solved the same problem [START_REF] Astolfi | Neuroelectrical hyperscanning measures simultaneous brain activity in humans[END_REF]. These studies declared an exclusion of sensory interaction between subjects. However, it should be noted that the subjects of all these studies knew about social encounters during the experiments. Therefore, instead of mental collaboration their results may simply mean an increase of brain activity due to similar emotional arousal in participants stimulated by the social encounter.

The near-infrared spectroscopy study (nonhyperscanning) on asleep newborns shows an increase of the neural response to a familiar (English language) versus unfamiliar language (Tagalog, a Filipino language) spoken by strangers in both conditions [START_REF] May | Language and the newborn brain: does prenatal language experience shape the neonate neural response to speech?[END_REF]. The language stimuli (the identical low-pass filtered sentences) were played through two speakers approximately 1.5 m from the infants' head. According to May et al., [START_REF] May | Language and the newborn brain: does prenatal language experience shape the neonate neural response to speech?[END_REF], this findings show that the newborn's neural processing of language is influenced by early language experience due to neonate brain responds to familiar versus unfamiliar language. To our mind, this outcome may lead to evidence of another inference. This experiment was not a hyperscanning technique. However, subjects were in pairs with their caregivers. Neonates classified these sound stimuli without the ability to perceive them. Sleeping newborns' brains reacted to sound stimuli that their sensing could not provide due to their brains' sensory isolation to meaningless and unfamiliar sounds. Sleepers seem to enter a standby mode, allowing them to balance the monitoring of their surroundings with sensory isolation [START_REF] Andrillon | The vigilant sleeper: neural mechanisms of sensory (de)coupling during sleep[END_REF]. Sleepers are sensitive to the semantic content of an auditory stream [START_REF] Andrillon | The vigilant sleeper: neural mechanisms of sensory (de)coupling during sleep[END_REF]- [START_REF] Legendre | Sleepers track informative speech in a multitalker environment[END_REF] and amplify relevant, meaningful stimuli [START_REF] Andrillon | The vigilant sleeper: neural mechanisms of sensory (de)coupling during sleep[END_REF] [START_REF] Legendre | Sleepers track informative speech in a multitalker environment[END_REF]. The sleeping brain retains some residual information processing capacity, which, however, does not form enduring memories [START_REF] Cox | Sound Asleep: Processing and Retention of Slow Oscillation Phase-Targeted Stimuli[END_REF]. Neonates are not able to understand even Mother's speech although her sound is familiar. Given all these, any speech for neonates is meaningless, and asleep newborns may not be sensitive to the sounds even their native tongue (the language spoken by the mother during pregnancy) in experiments when these sounds were pronounced by outsiders. However, they were sensitive to them. Sleeping newborns' brains reacted to sound stimuli that their sensing could not provide due to their brains' sensory isolation to meaningless and unfamiliar sounds. We believe that this outcome may mean the implicit modality of newborns' interaction with caregivers since any other explanation of this outcome is excluded.

Whether or not social interaction contributes to the increase in results of a group mental activity? Whether or not dyads maintain non-perceptual interaction? What are the neurobiological grounds of intentionality?

III. NEURAL FOUNDATION OF COHERENT INTELLIGENCE

A. Experiments on Problem-Solving in Groups

Recent research of 24 online experiments presented that unprimed participants show a more significant accuracy level when they complete the thought task simultaneously with confederates who are primed with the correct answer; if they were emotionally stimulated and completed the tasks without communication [START_REF] Val | Knowledge Sharing through Social Interaction: Towards the Problem of Primary Data Entry[END_REF]. Primary groups [START_REF] Cooley | Social organization: A study of the larger mind[END_REF] show empirical evidence of a more significant accuracy in problem-solving in the coherent intelligence state. In specific, we conducted 13 experiments in dyads (116 subjects) with P-value < 0,001 (probability-value in null hypothesis significance testing), and 7 experiments in primary group adults (41 subjects) with the P-value < 0,002. Experiments with 43 secondary group subjects (unfamiliar adults, M=20) show the effect only with the task of unfamiliar language translation. Non-semantic tasks-with synthetic language and two-color round symbolsdid not stimulate the effect in 2 experiments with 207 secondary group subjects (unfamiliar). These results are consistent with research Danilov et al. [START_REF] Val | Unconscious social interaction, coherent intelligence in learning[END_REF].

B. The Model of Coherent Intelligence

According to Danilov and Mihailova [START_REF] Val | Emotions in e-Learning: The Review Promotes Advanced Curriculum by Studying Social Interaction[END_REF], a supranormal environmental case-e.g., first hours after birth-stimulates supranormal sensation in dyads. This can push the inherited mechanism of social entrainment of infants to the rhythm of the mother. Both the supranormal sensation and social entrainment may stimulate the common emotional arousal. The latter is increased by the ongoing supranormal sensation and the occurring rhythm of arbitrary movements of the infant. The continuing supranormal sensation and everincreasing arousal of the infant and the mother along with the rhythm of the infant's unintentional movements stimulate early imitation and emotional contagion. The problem is how the infant capture and reproduce the kinematic of movements.

The MCI proposes that common emotional arousal together with the identical rhythm create coherent mental processes in dyads-Coherent Intelligence (Figure 1). At Sensorimotor Stage (by Piaget, or Stage 3 of the Model of Hierarchical Complexity MHC [START_REF] Commons | The fundamental issues with behavioral development[END_REF]), organisms do not maintain bilateral communication. According to Danilov and Mihailova [START_REF] Val | Emotions in e-Learning: The Review Promotes Advanced Curriculum by Studying Social Interaction[END_REF], individuals are able to interact by distinguishing perceptual signals of identical modality by their significance. This ability can contribute to ostensive cues. After all, this meaningless interaction modifies into communication when individuals imbue perceptual impulses with mutually implied meanings, cascading their signals in response to the history of relations between them [START_REF] Val | Emotions in e-Learning: The Review Promotes Advanced Curriculum by Studying Social Interaction[END_REF].

C. Neural Foundation of this MCI

Leaving aside semantic constraints, it seems consistent to say that intentionality manifests itself in intention. Specific brain regions may be engaged in shared sensory/cognitive processes irrespective of the feedback's valence and in encoding the subjective relevance of the feedback [START_REF] Oldham | The anticipation and outcome phases of reward and loss processing: A neuroimaging meta-analysis of the monetary incentive delay task[END_REF] [START_REF] Martinsa | Mapping social reward and punishment processing in the human brain: A voxel-based meta-analysis of neuroimaging findings using the Social Incentive Delay task[END_REF].

Outside areas involved in this processing, additional brain areas are specifically engaged according to the particular communicative modality [START_REF] Enrici | Intention Processing in Communication: A Common Brain Network for Language and Gestures[END_REF]. According to Tettamanti et al. [START_REF] Tettamanti | Effective connectivity gateways to the Theory of Mind network in processing communicative intention[END_REF], Intention Processing Network (IPN) involves the medial prefrontal cortex, precuneus, bilateral posterior superior temporal sulcus, and temporoparietal junctions. Depending on different social interaction modalities, the IPN is complemented by activation of additional brain areas, reflecting different Modality-Specific (M-S) input gateways to the IPN [START_REF] Tettamanti | Effective connectivity gateways to the Theory of Mind network in processing communicative intention[END_REF]. The M-S gateways mediate the structural and semantic decoding of stimuli and provide M-S information [START_REF] Tettamanti | Effective connectivity gateways to the Theory of Mind network in processing communicative intention[END_REF]. Sensory inputs of a specific modality can activate the precise association of certain sensorimotor networks with specific brain emotion circuits [START_REF] Nummenmaa | Emotional speech synchronizes brains across listeners and engages large-scale dynamic brain networks[END_REF].

We believe that this emotion-motion coherence can involve the particular cognitive process of a high order. When two or more organisms are in common emotional arousal and simultaneously in the interactional synchrony, then these two different experiences may meet each other in high-order cognitive processing. Emotional arousal can trigger evolutionarily old brain circuits, which interact with high-order cognitive and linguistic processing [START_REF] Nummenmaa | Emotional speech synchronizes brains across listeners and engages large-scale dynamic brain networks[END_REF]. It seems uncontroversial to say that infants' pure nervous system may experience emotions, but only primitive ones related to survival, such as those associated with hunger and pain. However, newborns cannot express emotions themselves appropriately to a specific social case on their own, even though they possess inherited neuronal patterns of primitive emotional impressions. They also cannot understand the expression of others' emotions (as is discussed above). They are only able of experiencing primitive emotions. Research on insects-organisms in stage 3 of MHC [START_REF] Commons | The fundamental issues with behavioral development[END_REF] like human newborns-assumes that they also experience emotions [START_REF] Bateson | Agitated honeybees exhibit pessimistic cognitive biases[END_REF]. Researchers argued that agitated honeybees exhibit pessimistic cognitive biases: 'Whether animals experience human-like emotions is controversial and of immense societal concern. The next reason is that animals cannot provide subjective reports of how they feel, emotional state can only be inferred using physiological, cognitive, and behavioral measures. In humans, negative feelings are reliably correlated with pessimistic cognitive biases, defined as the increased expectation of bad outcomes. Recently, mammals and birds with poor welfare have also been found to display pessimistic-like decision making, but cognitive biases have not thus far been explored in invertebrates [START_REF] Bateson | Agitated honeybees exhibit pessimistic cognitive biases[END_REF].'

In parallel, interactional synchrony stimulates a sensorimotor network engaging neural networks responsible for communicative intention processing (including highorder cognitive and linguistic processing) [START_REF] Enrici | Intention Processing in Communication: A Common Brain Network for Language and Gestures[END_REF]. Neural networks of emotional excitation and the sensorimotor networks are separately connected to many different M-S gateways. Meanwhile their coherence intersects in certain M-S gateways of each organism depending on (i) pattern of neural circuit engaged through emotional excitation and (ii) pattern of the sensorimotor network [START_REF] Enrici | Intention Processing in Communication: A Common Brain Network for Language and Gestures[END_REF].

We propose a rough hypothesis of how Long-Term Potentiation (LTP) can be induced only in particular M-S gateways, retaining information about the certain received stimulus. Different areas of the brain exhibit different forms of LTP, their types depend on a number of factors, such as age and the neuron's anatomic location. However, the common processes are the same for all. The simple nature of Hebbian learning, based only on the coincidence of pre-and post-synaptic activity, LTP is persistent, lasting from several minutes to many months, and it is this persistence that separates LTP from other forms of synaptic plasticity [START_REF] Abraham | How long will long-term potentiation last?[END_REF]. Spike-timing-dependent plasticity (STDP)-that involves the pairing of pre-and postsynaptic action potentials (APs)causes a variation of LTP or Long-Term Depression (LTD) [START_REF] Tazerart | A spike-timing-dependent plasticity rule for dendritic spines[END_REF]. The timing between pre-and postsynaptic APs modulates synaptic strength, triggering LTP or LTD [START_REF] Tazerart | A spike-timing-dependent plasticity rule for dendritic spines[END_REF]. The sign and magnitude of the change in synaptic strength depend on the relative timing between spikes of two connected neurons (the pre-and postsynaptic neuron) [START_REF] Tazerart | A spike-timing-dependent plasticity rule for dendritic spines[END_REF]. The structural organization of excitatory inputs supporting STDP remains unknown [START_REF] Tazerart | A spike-timing-dependent plasticity rule for dendritic spines[END_REF]. Even though the ensemble of emotion-motion integrated networks weakly stimulates the intersected neurons in their junction with M-S gateways. If all M-S gateways also simultaneously receive weak stimulation from the receptors (due to the chaos of stimuli received by the pure nervous system), then this multi-signal contributes to LTP in the neurons of particular M-S gateway at the junction of this emotion-motion ensemble due to the effect of the synaptic cooperativity, because of the following. LTP can be induced either by strong tetanic stimulation of a single pathway to a synapse, or cooperatively via the weaker stimulation of many. Neurons from the gateways in the connections of these networks receive cooperative stimulation. Induction of cooperativity can ensure LTP.

According to Tazerart et al. [START_REF] Tazerart | A spike-timing-dependent plasticity rule for dendritic spines[END_REF], the synaptic cooperativity of only two neighboring synaptic inputs onto spines in the basal dendrites of L5 pyramidal neurons extends the pre-post timing window that can trigger potentiation. The engaged M-S gateways retain a certain stimulus, while other M-S gateways (also of the same sensory modality) remain depressed without keeping information of other stimuli. Therefore, specific M-S gateways are sensitive, and all these organisms respond to specific sensory modalities. Figure 2 shows a schematic picture of this process. The induced emotion and sensorimotor networks (they are red in the picture) together activate certain M-S gateway even with weak stimulation of sensory input. The different colors of M-S gateways refer to different sensory modalities. At this point, the analysis encounters the ground of the PDE problem of how immature neurons learn the timing code to modulate certain synaptic strength, which triggers either LTP or LTD. Because the structural organization of excitatory inputs supporting STDP remains unknown [START_REF] Tazerart | A spike-timing-dependent plasticity rule for dendritic spines[END_REF].

The study of the PDE problem leads to the analysis of the axiomatic foundations of Psychology, Sociology, and Neuroscience-the basic notions that form these sciencesfrom the perspectives of the actual scientific paradigm. The question of "how can the blank mind begin to learn from social interaction" is reduced to "how immature neurons learn the timing code to modulate certain synaptic strength, that triggers either LTP or LTD". The sign and magnitude of the change in synaptic strength depend on the relative timing between spikes of two connected neurons (the pre-and postsynaptic neuron). How can neurons of an immature organism (even a newborn) learn the structural organization of excitatory inputs that support STDP? The further arguments show why we believe that the entanglement state of neurons can contribute to simultaneous LTP in neurons.

The daily routine develops neural patterns of primitive emotions and sensorimotor neural patterns in infants. Their everyday coherence with the social world forms various integrated neural patterns of different emotions from the existing ensemble of emotion scripts in their community. We believe that caregivers contribute to the formation of emotion scripts and, consequently, shaping of specific neural patterns in infants. Obviously, adults experienced intentionality before their coherent mental process began with newborns. Life experience taught them particular emotion scripts and defined their precise motion kinematics, that formed more elaborated sensorimotor patterns. In routine cooperation with newborns, a caregiver enters in interactional synchrony with a newborn, under the influence of supranormal stimuli, being in social entrainment. Therefore, the similar M-S gateways are excited in the dyad. Meanwhile, the adult's current intentionality has already triggered a particular network that includes current emotion patterns and sensorimotor patterns. Part of it corresponds to a primitive complex emotionsensorimotor network in the newborn with similar M-S gateways. This newborn's primitive network is less developed, although it is similar to the part of the adult's integrated complex network. It can be assumed that the neurons in the connections of different excited emotion patterns and sensorimotor patterns go into the entangled state because they receive LTP, being induced cooperatively via many stimulations. In such a manner, the adult and infant neurons behave as a single unit. Therefore, specific M-S gateways are sensitive in dyads, and these organisms equally respond to specific sensory modalities. The induction of t-LTP and t-LTD in single spines follows a bidirectional Hebbian STDP learning rule [START_REF] Tazerart | A spike-timing-dependent plasticity rule for dendritic spines[END_REF]. Hebbian theory claims that an increase in synaptic efficacy arises from the learning process. The PDE problem in a chaos of stimuli requires a teaching mechanism from the beginning. The entanglement state of neurons is a possible option of how infants' neurons learn spike-timing-dependent plasticity. In the entanglement state, the behavior of the neurons of a mature organism determines and trains the neurons of a newborn.

Emotion sharing indicates implicit modality of social interaction. The entanglement state of neurons in the certain M-S gateways is a possible option of how infants' neurons learn STDP. This involvement of similar networks and the sensibility of the certain M-S gateways lasts as long as is necessary to teach the immature nervous system. The entanglement state of these neurons ensures their immediate response to the specific stimulus, regardless of the spatial division of organisms. Therefore, specific M-S gateways are sensitive, and these organisms equally respond to specific sensory modalities.

Knowledge about consciousness is being developed by studying the interaction of neurons, whose dimensions are similar to objects studied from the perspective of quantum mechanics scale. Knowledge about their behavior may complement the set of social interaction modalities. The generation of entanglement between increasingly macroscopic and disparate systems is an ongoing effort in quantum science [START_REF] Thomas | Entanglement between distant macroscopic mechanical and spin systems[END_REF]. Recent studies have shown that the behavior of objects 15 micrometers in size is consistent with the quantum world's laws, such as the phenomenon of quantum entanglement [START_REF] Yin | Satellite-based entanglement distribution over 1200 kilometers[END_REF]- [START_REF] Sillanpää | Spooky quantum entanglement goes big in new experiments[END_REF]. An entangled state was generated even between a millimeter-sized dielectric membrane and an ensemble of 109 atoms [START_REF] Thomas | Entanglement between distant macroscopic mechanical and spin systems[END_REF]. In comparison, a neuron's nucleus has a diameter of 3 to 18 micrometers, and a neuron has a size of 4 to 100 micrometers. The quantum entanglement state of particles is not something rare in Nature. Quantum systems can become entangled through various types of interactions, for the moment we know: (a) light's polarization. One of the most commonly used methods is spontaneous parametric downconversion to generate a pair of photons entangled in polarization; (b) decay cascade of the bi-exciton. Fiber coupler to confine and mix photons, photons emitted from decay cascade of the bi-exciton in a quantum dot; (c) Hong-Ou-Mandel effect; (d) in the earliest tests of Bell's theorem, the entangled particles were generated using atomic cascades. An exciting question is how to create a quantum entanglement state of living cells. According to Marletto et al. [START_REF] Marletto | Entanglement between living bacteria and quantized light witnessed by Rabi splitting[END_REF], experiments show entanglement between living sulfur bacteria with quantized light. Whether or not neurons of different organisms become entangled through similar stimulation of their emotion networks?

Entanglement is an essential property of multipartite quantum systems, characterized by the inseparability of quantum states of objects regardless of their spatial separation [START_REF] Thomas | Entanglement between distant macroscopic mechanical and spin systems[END_REF]. Recent study tested quantum entanglement over unprecedented distances, beaming entangled pairs of photons to three ground stations across China-each separated by more than 1200 kilometers. 'Yin et al. used the Micius satellite, which was launched last year and is equipped with a specialized quantum optical payload. They successfully demonstrated the satellite-based entanglement distribution to receiver stations separated by more than 1200 km [START_REF] Yin | Satellite-based entanglement distribution over 1200 kilometers[END_REF]'.

We believe that these findings may contribute to an advanced curriculum, specifically in facilitating the teaching of young children.

IV. CONCLUSIONS

The analysis of recent empirical data yields a hypothesis of beginning cognition-the Model of Coherent Intelligence and its neural foundation of how the pure nervous system distinguishes sensory stimuli. This hypothesis postulates two new ideas of the PDE basis: (1) cognition begins from a separation of sensory stimuli: LTP can only be induced in neurons of particular M-S gateways (not all)-selective induction promotes selective sensitivity to the chaos of stimuli. (2) Neurons can learn STDP in social interaction by repeating the timing code of other organisms' mature neurons to modulate certain synaptic strength, which triggers either LTP or LTD. We believe that the MCI shapes intentionality in intimately related individuals. Coherent Intelligence is the integration of M-S gateways of particular brain areas, which contributes to different organisms' sensibility to similar sensory inputs. The 24 online experiments show empirical evidence of the MCI. We believe that this approach may contribute to study the mind and, in specific, to understanding the appearance of intentionality and the Primary Data Entry problem. We believe that these findings may contribute to an advanced curriculum, specifically in teaching 2-to 3-year-old children. Further research should also examine whether the MCI can provide a contactless interaction of the computer with neuronal circuits, in which the computer would become a part of extended mind. This approach provides a wide range of possibilities for developing a human-computer interface.

AUTHORS CONTRIBUTION

Igor Val Danilov formulated the hypothesis and wrote the first draft of the manuscript. Igor Val Danilov and Sandra Mihailova improved the text over several iterations.

Figure 1 .

 1 Figure 1. The Model of Coherent Intelligence.

Figure 2 .

 2 Figure 2. The Scheme of M-S Gateway Activation.

ACKNOWLEDGMENT

The authors express their gratitude to Iegor Reznikoff, Emeritus Professor Université de Paris Ouest Nanterre, France, for commenting this paper.


Marcelo M Morato
email: marcelomnzm@gmail.com

Victor M Cunha
email: moreiracunha.victor@gmail.com

Tito L M Santos
email: tlsantos@ufba.br

Julio E Normey-Rico

Olivier Sename
email: olivier.sename@gipsa-lab.fr

Robust Nonlinear Predictive Control through qLPV embedding and Zonotope Uncertainty Propagation

Keywords: Robust Model Predictive Control, Quasi-Linear Parameter Varying Systems, Zonotopes, Set-based Control, Constraint Tightening

This paper presents a novel robust Model Predictive Control (MPC) algorithm for nonlinear systems represented through quasi-Linear Parameter Varying (qLPV) models. The nominal MPC predictions are made considering a frozen scheduling parameter guess, which is computationally cheaper than nonlinear predictions, while zonotopes bound the disturbance propagation along the prediction. These sets are computed with respect to the bounds of the variation of scheduling parameters, offering reduced conservatism of the closed-loop dynamics and ensuring input-to-state stability and recursive feasibility properties. A DC-DC converter benchmark example is used to illustrate the advantages of the proposed method.

INTRODUCTION

Model Predictive Control (MPC) is well established and widely used for the optimal regulation of constrained dynamic systems [START_REF] Camacho | Model predictive control[END_REF]). Yet with great practical value, the standard MPC design was originally attached to the idea of a nominal linear timeinvariant (LTI) prediction model. Albeit nominal LTI MPC inherently offering a certain degree of robustness, it lacks guarantees of recursive feasibility or closed-loop stability at the presence of disturbances.

Since the 00's, literature has provided robust MPC tools, such as terminal ingredients [START_REF] Mayne | Constrained model predictive control: Stability and optimality[END_REF] and integral quadratic constraints [START_REF] Cisneros | A dissipativity formulation for stability analysis of nonlinear and parameter dependent MPC[END_REF], which enable input-to-state stability and constraint satisfaction for bounded disturbances. Nonetheless, the application of robust nonlinear MPC (NMPC) is not trivial and comes with increased numerical burden, which becomes an impediment for real-time applications [START_REF] Allgöwer | Nonlinear model predictive control[END_REF].

In parallel to the theoretical establishment of robust MPC, the Linear Parameter Varying (LPV) toolkit has been brought to focus [START_REF] Mohammadpour | Control of linear parameter varying systems with applications[END_REF][START_REF] Sename | Robust control and linear parameter varying approaches: application to vehicle dynamics[END_REF]. LPV models represent nonlinear dynamics with the use of known, bounded scheduling parameters ρ. Recent advances on NMPC algorithms conceived through quasi-LPV (qLPV) embedding have been presented, e.g. [START_REF] Morato | Model predictive control design for linear parameter varying systems: A survey[END_REF] and references therein, enabling fast operation. The elegance of the qLPV embedding is that the "full-blown" nonlinear predictions can be replaced by sequential linear predictions [START_REF] Abbas | A new approach to robust MPC design for LPV systems in input-output form[END_REF][START_REF] González Cisneros | Nonlinear model predictive control for models in quasi-linear parameter varying form[END_REF].

With respect to this context, we investigate the problem of robust NMPC based on nominal predictions obtained through quasi-Linear Parameter Varying (qLPV) embedding, as done in [START_REF] Morato | Novel qLPV MPC design with least-squares scheduling prediction[END_REF][START_REF] Cisneros | Fast nonlinear MPC for reference tracking subject to nonlinear constraints via quasi-LPV representations[END_REF]. Based on the one-steap-ahead disturbance propagation, we apply tightened constraints over the nominal qLPV predictions, enforcing the satisfactions of the performance requirements by the real system trajectories [START_REF] Santos | A constraint-tightening approach to nonlinear model predictive control with chance constraints for stochastic systems[END_REF][START_REF] Köhler | A novel constraint tightening approach for nonlinear robust model predictive control[END_REF]. The main novelty of the proposed approach is that it does not require sequential QPs or estimation of the future scheduling variables, as in the prior, but binds the prediction error within zonotopes.

In the nonlinear setting, the majority of uncertainty propagation methods, such as Lipschitz constants and Kfunctions, result in exceeding conservatism, limiting performances and reducing the closed-loop domain of attraction [START_REF] Marruedo | Input-to-state stable MPC for constrained discrete-time nonlinear systems with bounded additive uncertainties[END_REF]. Nonetheless, the use of zonotopes comes as an alternative, offering representation simplicity and flexibility, as well as low implementation cost. These sets are symmetrical, convex, and compact polyhedrons, originally used for state estimation and fault detection [START_REF] Pourasghar | Interval observer versus set-membership approaches for fault detection in uncertain systems using zonotopes[END_REF].

In this paper, we provide a new method for the uncertainty propagation considering nonlinear systems with nominal qLPV predictions, applying the mean value zonotope extension framework from [START_REF] Alamo | Guaranteed state estimation by zonotopes[END_REF]; [START_REF] Cunha | Nonlinear robust predictive control with distrubance propagation via zonotopes[END_REF]. Additionally, we "robustify" the parameterdependent terminal ingredients from González [START_REF] González Cisneros | Nonlinear model predictive control for models in quasi-linear parameter varying form[END_REF], which now ensure input-to-state stability and recursive feasibility properties even in the case of (bounded) additive disturbances and model prediction uncertainties. The main restricting hypothesis of the procedure is the necessity of bounds upon the variation of the scheduling variables between sampling instants.

The rest of this paper is organised as follows. Sec. 2 provides preliminary discussions, the problem setup, and the qLPV terminal ingredients. Sec. 3 details the use of zonotopes to bound the qLPV nominal prediction errors. Sec. 4 illustrates a DC-DC converter case study application of the method, together with comparisons to other NMPC algorithms from the literature. General conclusions are drawn in Section 5.

Notation. The index set N [a,b] represents {i ∈ N | a ≤ i ≤ b}, with 0 ≤ a ≤ b.
The identity matrix of size j is denoted as I j ; I {j} denotes the j-th row of I j ; col{•} denotes the vectorization of the entries and diag{v} denotes the diagonal matrix generated with the line vector v. 1 n×m stands for the n × m vector of unit entries; The value of a given variable v(k) at time instant k + i, computed based on the information available at instant k, is denoted as v(k+i|k). K refers to the class of positive and strictly increasing scalar functions that pass through the origin. A C 1 function f is such that it is differentiable with continuous derivatives. In this case, ∇ T f : R m → R n×m denotes its Jacobian matrix. Consider sets A, B ⊂ R n , C ⊂ R m and a matrix R ∈ R n×m . The Minkowski set addition is defined by

A ⊕ B := {a + b | a ∈ A , b ∈ B}, while the Pontryagin set difference is defined by A B := {a | a ⊕ B ⊆ A}. A linear mapping is RA = {y ∈ R n : y = Ra, a ∈ A ⊆ R m , R ∈ R n×m }, while the cartesian product holds as A × C = {z ∈ R n+m : z = (a T c T ) T , a ∈ A, c ∈ C}.The unitary m-dimensional box is denoted B m ∞ = {ξ ∈ R m : ||ξ|| ∞ ≤ 1}. The set of real compact intervals is given by I = {[a, b], a, b ∈ R, a ≤ b}.
An interval matrix J ∈ I n×m has mid(J) and rad(J) denoting its middle point and radius, respectively. ν denotes the 2-norm.

ROBUST NMPC THROUGH QLPV EMBEDDING

The Nonlinear System

We consider the following discrete-time nonlinear system:

x(k + 1) = f (x(k), u(k)) + Ew(k) ,
(1) being k ∈ N the sampling instant, x ∈ R nx the vector of states, u ∈ R nu the vector of control inputs, and w ∈ R nw an unknown additive disturbance, assumed bounded to a compact set W ⊂ R nw is such way that w(k) ≤ w, ∀k. The states x are measurable at all sampling instants.

The system in Eq. ( 1) is subject to state and input constraints, which define admissible operation, as follows:

x ∈ X := {x j ∈ R :

x j ≤ x j , ∀j ∈ N [1,nx] } and u ∈ U := {u i ∈ R : u i ≤ u i , ∀i ∈ N [1,nu] }.
Without loss of generality, the origin is considered an equilibrium point of Eq. ( 1), in such way that f (0, 0) = 0. Moreover, the nonlinearity f (x, u) is of class C 1 over Z := X × U. The constraint set are also expressed as:

X := {x ∈ R nx : H x x ≤ g x } , U := {u ∈ R nu : H u u ≤ g u } .

qLPV Embedding

The nonlinear system satisfies the Linear Differential Inclusion property [START_REF] Shamma | An overview of LPV systems[END_REF], which means that it can be embedded under a qLPV formalism. Suppose that

∃H(x, u) ⊆ R nx×nx , ∀(x, u) ∈ Z s.t. f (x, u) := H(x, u) [ x u ]
T . Then, Eq. ( 1), for all (x, u) ∈ Z, equivalent to:

x(k + 1) = A(ρ(k))x(k) + B(ρ(k))u(k) + Ew(k) , (2) ρ(k) = f ρ (x(k)) ∈ P .
(3)

The model in Eq. ( 2) is qLPV, with an endogenous nonlinear function f ρ (•) that results in the scheduling parameters ρ(k), which are bounded and known at each instant k. We note, nonetheless, that they are generally unknown for any future instant k + j , ∀j ∈ N [1,∞] . We consider the following set:

P := ρ j ∈ R | ρ j ≤ ρ j ≤ ρ j , ∀j ∈ Z [1,nρ] .
Assumption 1. The scheduling variables ρ(k) evolve along the horizon with bounded rates of variation. This is: δρ(k+ 1) = (ρ(k + 1) -ρ(k)) ∈ δP, with:

δP := δρ j ∈ R : δρ j ≤ δρ j ≤ δρ j , ∀j ∈ N [1,np] . (4)
Remark 1. Assuming bounds upon the variation of scheduling parameters between sampling instants is quite reasonable in many real LPV applications [START_REF] Jungers | MPC for LPV systems with bounded parameter variations[END_REF]. Nonetheless, if the qLPV-embedded nonlinear dynamics are s.t. δρ is unbounded, the proposed method can be deployed considering uncertainty propagation as based on the whole scheduling set P, which leads to further conservatism, but with no loss of generality.

Closed-loop Paradigm

We consider that the qLPV system is regulated according to the following LPV state-feedback control law:

u(k) = π (x(k), v(k)) = K(ρ(k))x(k) + v(k) , ( 5
)
where the virtual input v(k) is used to ensure constraints satisfaction, and the parameter-dependent feedback gain K(ρ(k)) attenuates the propagation of disturbances. The closed-loop dynamics are given by:

x(k + 1) = (A(ρ) + B(ρ)K(ρ)) x(k) + B(ρ)v(k) + Ew(k) , = A π (ρ)x(k) + B(ρ)v(k) + Ew(k) . ( 6
)
We note that the process constraints x T (k) u T (k) ∈ Z can be expressed in terms of the closed-loop dynamics and virtual entry as

x(k) v(k) ∈ Z π , with: Z π := z ∈ R nx+nu : H x 0 H u K(ρ) H u z ≤ g x g u .(7)
The control law in Eq. ( 5) will be defined by an MPC algorithm based on a frozen prediction model, which gives the sequence of future virtual inputs v [k,k+Np-1] = {v(k|k) . . . v(k + N p -1|k)} as output, considering a prediction horizon of N p steps. The MPC recursive feasibility property ensured through terminal ingredients (Theorem 1) is not violated by the time-varying nature of Z π , since the energy-dissipation arguments hold for the wider set Z.

We proceed by detailing disturbance propagation, the predictive control design, and the used terminal ingredients.

Disturbance Propagation

Given an initial condition x(k) ∈ R n , the future trajectory of the closed-loop system in Eq. ( 6) evolves, for j ≥ 0, as:

x(k + j) = φ π (j, x(k), v [k,k+j-1] , w [k,k+j-1] , ρ [k,k+j-1] ) := A j (ρ [k,k+j-1] )x(k) + B j (ρ [k,k+j-1] )v [k,k+j-1] + E j w [k,k+j-1] , (8)
where

w [k,k+j] = col{w(k + j|k)}, ∀j = N [0,j] gives the future disturbances. It is worth noting that ρ [k,k+j] = col{ρ(k + j|k)}, ∀j = N [0,j]
gives the future values of the scheduling parameters, which are unknown (function of the future states). Matrices A j , B j and E j are analytically given in (González [START_REF] González Cisneros | Nonlinear model predictive control for models in quasi-linear parameter varying form[END_REF].

Since both ρ [k,k+j-1] and w [k,k+j-1] are unknown from the viewpoint of sampling instant k, we pursue the development of an MPC algorithm based on nominal prediction with a "frozen" qLPV scheduling guess, as if ρ(k) would remain constant along the future horizon, and as if the disturbances were null. These nominal predictions are:

x(k + j|k) = φ π (j, x(k), v [k,k+j-1] , 0, ρ(k)1 np×j ) (9) = A j (I ρ (k))x(k) + B j (I ρ (k))v [k,k+j-1] ,
where I ρ (k) := ρ(k)I np×j denotes the frozen scheduling guess that replaces the real future scheduling values.

There is a clear mismatch between the nominal predictions (9) and real qLPV trajectories (8). Therefore, in order to guarantee recursive feasibility of the MPC, one-step ahead disturbance propagation sets S(j) ⊆ R n , j = 0 . . . N p are developed [START_REF] Santos | A constraint-tightening approach to nonlinear model predictive control with chance constraints for stochastic systems[END_REF].

Condition 1. The one-step-ahead disturbance propagation sets S(j), j ∈ N [0,Np] , must be compact and satisfy the following conditions:

(1) S(0) contains EW.

(2) For all

x a , x b ∈ R nx , v ∈ R nu , ρ a , ρ b ∈ P and j ∈ N [1,Np] , with (x a , v) ∈ Z π (S(j -1) × {0}), (x b -x a ) ∈ S(j -1), and (ρ b -ρ a ) ∈ δP, the set S(j) must be such that (A π (ρ b )x b + B(ρ b )v) - (A π (ρ a )x a + B(ρ a )v) ∈ S(j). With x a = x(k + 1|k) = A(ρ a )x + B(ρ a )v, x b = x(k + 1) = A(ρ b )x+B(ρ b )v, ρ a = ρ(k-1) and ρ b = ρ(k), we have x b -x a ∈ EW and (ρ b -ρ a ) ∈ δP. It follows by induction from Condition 1 that x(k + j|k + 1) ∈ x(k + j|k) ⊕ S(j - 1), j = 1 . . . N p + 1 for any admissible sequence v [k,k+j-1] .
The sets S(j) thus bound the one-step-ahead prediction error between the nominal MPC predictions and the real trajectories.

We note that, in the linear setting, the smallest sets that satisfy Condition 1 can be directly computed through linear expansions of the closed-loop dynamics [START_REF] Ferramosca | Robust MPC for tracking zone regions based on nominal predictions[END_REF]. Anyhow, for generic nonlinear systems, there are no efficient algorithms for the exact computation of these sets [START_REF] Köhler | A novel constraint tightening approach for nonlinear robust model predictive control[END_REF], which means that more conservative strategies, taking into account upper bounds on the uncertainty propagation through the nonlinear system, must be sought.

One simple alternative for the computation of these sets in the nonlinear case is to bound the disturbance propagation with Lipschitz constants, e.g. Theorem 1 of (Morato et al., 2020a). In this case, for nonlinear closed-loop dynamics

x(k + 1) = f π (x(k), v(k)) + w(k), Lipschitz constant Γ ∈ R satisfying inequality f π (x a , v) -f π (x b , v) ≤ Γ x a -x b for any (x T a , v T ) T , (x T b , v T ) T ∈ Z π and S(0) := {x ∈ R nx : x ≤ w} ⊇ EW.
Then, the Lipschitz sets S(j), ∀j ∈ N [0,Np] that satisfy the one-step-ahead disturbance propagation condition are:

S(j) := x ∈ R nx : x ≤ Γ j w , ∀j ∈ N [0,Np] . (10)
Lipschitz disturbance propagation sets, in the form of Eq. ( 10), are often quite conservative, since they inherently consider the worst-case disturbance propagation in all directions. Less conservative sets can be obtained by zonotope-based disturbance propagation methods, as described in [START_REF] Cunha | Nonlinear robust predictive control with distrubance propagation via zonotopes[END_REF]. Furthermore, predictions that rely directly on nonlinear models result in nonlinear and non-convex optimisation problems, which are computationally costly and do not guarantee global optimality. Herein, we provide an extension of the zonotopic disturbance propagation method considering the qLPV setting, developing zonotopes S(j) that satisfy Condition 1. Remark 2. The qLPV embedding in Eq. ( 2) stands for a realisation of the nonlinear dynamics with only statedependent scheduling parameter. This framework holds for many applications. Nevertheless, in some cases, the scheduling parameters are state-and input-dependent, i.e. ρ = f ρ (x, u). Thereby, alternative formulations to Condition 1 should be sought, since the uncertainty will propagate along the horizon depending on the future control inputs, which may belong to a wide set, thus leading to poor performances. One option is to represent the system as nonlinear parameter varying (NLPV), with an explicit input-dependent nonlinearity that can be handled with Lipschitz propagation, as done in (Morato et al., 2020a).

Controller Design

This process must be regulated in such way that the state trajectories are steered to the origin, in an admissible manner, despite disturbances and the discrepancy between the nominal prediction model and the real qLPV model. As discussed, we apply a robust MPC scheme, for which contracted constraints are used. Considering an initial constraint set Z π (0) = Z π , the following sets for j ∈ N [1,Np] are iteratively taken as:

Z π (j + 1) = Z π (j) (S(j) × {0}) . ( 11
)
Therefore, at each sampling instant k, we measure the state x(k), compute the scheduling parameter ρ(k), and solve the following optimization problem, which embeds the performance objectives of the system, as well as the operational constraints:

min v [k,k+Np -1] V (x(k + N p |k)) (12) + Np-1 j=0 (x(k + j|k), v(k + j|k)) s.t. x(k + j + 1|k) = A π (ρ(k))x(k + j|k) (13) +B(ρ(k))v(k + j|k), j ∈ N [0,Np-1] (x T (k + j|k) v T (k + j|k)) T ∈ Z π (j), j ∈ N [0,Np-1] x(k + N p |k) ∈ X f , (14)
where

(x, v) = x 2 Q + v 2 R is a quadratic stage cost, V (•) is a terminal cost and X f is a terminal set. We denote v [k,k+Np-1]
as the optimal solution of this problem, of which the first entry v (k|k) is applied to the process according to Eq. ( 5).

Terminal Ingredients

We proceed by detailing the terminal ingredients and the feedback gain such that input-to-state stability and recursive feasibility properties are ensured.

In order to ensure these properties, first of all, the cost functions and V must be K-class bounded, as details [START_REF] Mayne | Constrained model predictive control: Stability and optimality[END_REF]:

(x, v) ≥ β ( x ), ∀x ∈ X , β (•) ∈ K-class , (15) 0 ≤ V (x) ≤ β V ( x ), ∀x ∈ X , β V (•) ∈ K-class . (16)
In this paper, we choose parameter-dependent quadratic terminal ingredients, being X f := x | x T P (ρ)x ≤ 1 an ellipsoidal terminal set and a sub-level Lyapunov terminal cost: V (x) = x T P (ρ)x. Likewise, we consider a terminal qLPV feedback K(ρ), which is given in terms of the positive-definite matrix P (ρ).

The terminal set X f must be robust positively invariant regarding the closed-loop dynamics of Eq. ( 6), as provide the following Theorems. Theorem 1. [START_REF] Mayne | Constrained model predictive control: Stability and optimality[END_REF] Suppose ∃u = K(ρ)x. Consider that the MPC is given by Eq. ( 12), with a terminal state set given by X f (ρ) and a terminal cost V (x, ρ). Then, input-to-state stability is ensured if the following conditions hold ∀ρ ∈ P: (C1) The origin lies in the interior of X f (ρ); (C2) Any consecutive state to x, in closed-loop given by (A(ρ) + B(ρ)K(ρ)) x lies within X f (ρ); (C3) The discrete Lyapunov equation is verified within this invariant set, this is, ∀ x ∈ X f (ρ) and ∀ ρ ∈ P and

∀ δρ ∈ δP: V ((A(ρ + B(ρ)K(ρ)) x, ρ + δρ) -V (x, ρ) ≤ -x T Qx -x T (K(ρ) T RK(ρ)x.
(C4) The image of the nominal feedback lies within the admissible control domain: K(ρ)x ∈ U , ∀ρ ∈ P. (C5) The terminal set X f (ρ) is a subset of X .

Assuming that the initial solution of the MPC problem v [0,Np-1] is feasible, then, the MPC is recursively feasible, stabilizing the state origin. Proof 1. Refer to [START_REF] Mayne | Constrained model predictive control: Stability and optimality[END_REF]. Theorem 2. Robust Positively Invariant Terminal set Assume that there exists an ellipsoidal terminal set X f . X f is a robust positively invariant set iff, for any x ∈ X f and ρ ∈ P, i.e. x T P (ρ)x ≤ 1, it follows that:

x T A T π (ρ)P (ρ + δρ)A π (ρ)x ≤ 1 . (17) Proof 2.
It is trivial to verify Eq. ( 17) by using x(k), x(k + 1) ∈ X f , ∀k ∈ N. Theorem 3. Terminal Ingredients The conditions (C1)-(C5) of Theorem 1 and the inequality of Theorem 2 are satisfied if there exist a symmetric parameter-dependent positive definite matrix P (ρ) : R np → R nx×nx , a parameter-dependent rectangular matrix W (ρ) : R np → R nu×nx , and a scalar 0 21) and the BMI (18) hold for w † given as the vertices of S(N p ), and for all ρ ∈ P and δρ ∈ δP, under the minimization of log det{Y (ρ)}.

< λ ∈ R such that Y (ρ) = (P (ρ)) -1 > 0, W (ρ) = K(ρ)Y (ρ) such that LMIs (19)-(
  λY (ρ) 0 (1 -λ) A(ρ)Y (ρ) + B(ρ)W (ρ) w † Y (ρ + δρ)   > 0 , (18)
   Y (ρ) (A(ρ)Y (ρ) + B(ρ)W (ρ)) Y (ρ + δρ) Y (ρ) 0 Q -1 W (ρ) 0 0 R -1    ≥ 0 , ( 19
)
u 2 i I {i} W (ρ) Y (ρ) ≥ 0, i ∈ N [1,nu] , (20) x 2 j I {j} Y (ρ) I T {j} Y T (ρ) Y (ρ) ≥ 0, j ∈ N [1,nx] . ( 21
)
The proof of Theorem 3 is provided in the Appendix. This Theorem ensures a positive definite parameter dependent matrix P (ρ) which is used to compute the MPC terminal ingredients V (•) and X f such that input-to-state stability of the closed-loop is guaranteed. Furthermore, when the MPC is designed with these terminal ingredients, for any initial condition x(0) ∈ X f , the MPC is recursively feasible for all k > 0.

Remark 3. The BMI in Theorem 3 can be solved through simple bisection search over the optimization plane since 0 < λ ≤ 1, by construction, as argues [START_REF] Yang | An optimal approach to output-feedback robust model predictive control of LPV systems with disturbances[END_REF].

Remark 4. LMIs ( 20)-( 21) in Theorem 3 require box-type constraints on each u i and x j . Nonetheless, these can be converted into more generic polyhedral constraints, if necessary. We chose box-type constraints for the simplicity of the proofs. Remark 5. Theorem 3 provides infinite-dimensional inequalities, which must hold ∀ ρ ∈ P and ∀ δρ ∈ δP.

In practice, the solution can be found by enforcing the inequalities over a sufficiently dense grid of points (ρ, δρ) along the P × δP plane. Then, the solution can be verified over a denser grid. The parameter-dependency of P may be dropped if the system is quadratically stabilizable, but this may result in quite conservative performances. Remark 6. The solution of Theorem 3 is a parameterdependent Y (ρ) = np j=1 ρ j Y j . The online procedure, nonetheless, depends on a inversion of Y (ρ) to compute V , K(ρ) and X f , at each sampling instant.

UNCERTAINTY PROPAGATION VIA ZONOTOPES

Zonotopes are a particular class of convex, compact, and symmetrical polyhedrons [START_REF] Pourasghar | Interval observer versus set-membership approaches for fault detection in uncertain systems using zonotopes[END_REF], which can be described by a Minkowsky sum of line segments or by the affine image of a unitary box B ng ∞ :

Z = {G, c} = c ⊕ GB ng ∞ , ( 22
)
where c ∈ R n is the center of the zonotope and its shape is given by the rows of G ∈ R n×ng (generators). The number of generators n g ≥ n defines the complexity of the zonotope; n = n g defines a parallelotope.

The use of zonotopes for set-based state estimation and disturbance propagation is tied to their simplicity and to the numerical efficiency of linear transformations and Minkowsky sums of zonotopes [START_REF] Alamo | Guaranteed state estimation by zonotopes[END_REF], as well as the Pontryagin difference of a polyhedron by zonotope [START_REF] Alvarado | Model Predictive control for tracking constrained linear systems[END_REF].

Given Z 1 = {G 1 , c 1 }, Z 2 = {G 2 , c 2 } ⊆ R n , R ∈ R m×n , it follows that: RZ 1 = {RG 1 , Rc 1 } , Z 1 ⊕ Z 2 = {(G 1 G 2 ), c 1 + c 2 } . ( 23
)
We aim to compute zonotopic reachable sets S(j), ∀j ∈ N [0,Np] that satisfy Condition 1. For such, we develop Theorem 4, based on Lemma 1. Lemma 1. Consider a centered zonotope X = M B ng ∞ ⊆ R m , an interval matrix J ∈ I n×m , and a zonotope family Z = JX. A zonotopic inclusion is defined by:

(Z) := mid(J)X ⊕ P B ng ∞ , ( 24
)
where P is a diagonal matrix such that:

P ii = ng j=1 m k=1 rad(J) ik |M kj |, ∀i ∈ N [1,n] . ( 25
)
It holds that Z ⊆ (Z). Proof 3. Refer to [START_REF] Rego | Guaranteed methods based on constrained zonotopes for set-valued state estimation of nonlinear discrete-time systems[END_REF]. Theorem 4. Consider the qLPV system described in Eq. ( 6), with constraints (x(k), v(k)) ∈ Z π , disturbances w(k) ∈ W and scheduling parameters ρ(k

) ∈ P, δρ(k) ∈ δP. Let Z ⊆ R nx+nu , S 0 ⊆ R nx be zonotopes and A, ∆ A ∈ I nx×nx , ∆ B ∈ I nx×nu interval matrices satisfying EW ⊆ S 0 , Z π ⊆ Z, A π (ρ a ) ∈ A, A π (ρ b ) -A π (ρ a ) ∈ ∆ A and B(ρ b ) -B(ρ a ) ∈ ∆ B for all ρ a , ρ b ∈ P, ρ b -ρ a ∈ δP.
The sets S(j), j ∈ N [0,Np] defined iteratively by S(0) = S 0 and S(j

) = V ⊕ (AS(j -1)), j ∈ N [1,Np] , (26)
where V = ((∆ A ∆ B ) Z), satisfy Condition 1.

Proof 4. The first condition is satisfied by design. Considering x a , x b ∈ R nx , v ∈ R nu , ρ a , ρ b ∈ R nρ , as given in the second condition for some j ∈ N [1,N p] , and

∆ j = (A π (ρ b )x b + B(ρ b )v) -(A π (ρ a )x a + B(ρ a )v), we have: ∆ j = (A π (ρ b ) -A π (ρ a ))x a + A π (ρ b )(x b -x a ) + (B π (ρ b ) -B π (ρ a ))v ∈ (∆ A ∆ B ) Z π ⊕ AS(j -1) ⊆ V ⊕ (AS(j -1)) = S(j).
Therefore, the sets S(j) satisfy Condition 1. Remark 7. In the case of qLPV systems with A π (ρ) and B(ρ) affine on ρ, it follows that A π (ρ(k + 1)) -A π (ρ(k)) = Āπ (δρ(k + 1)) and B(ρ(k + 1)) -B(ρ(k)) = B(δρ(k + 1)), with Āπ (•) and B(•) being linear mappings. Then, the interval matrices ∆ A and ∆ B can be computed directly from δP. In the case of non-affine models, a simple alternative to convert them into affine by augmenting the number of scheduling parameters, which may result in slightly more conservatism. Remark 8. Bounds on scheduling parameters' variations δP can be obtained based on their dependence on the states ρ(k) = f ρ (x(k)). Specifically, this can be done by placing bounds on δρ = f ρ (f (x, u) + Ew) -f ρ (x) for (x, u), (f (x, u)+Ew, u) ∈ Z and w ∈ W, either by interval arithmetics or optimisation. Remark 9. Due to the zonotope inclusion and Minkowsky sum, the number of generators of the zonotopes S(j) increase by 2n x for each iteration. Methods for complexity reduction can be used to restrict the number of generators of each S(j) to a predefined value [START_REF] Scott | Constrained zonotopes: A new tool setbased estimation and fault detection[END_REF]. Remark 10. We stress that the qLPV model in Eq. ( 2) requires a linear nominal prediction model, as in Eq. ( 9), which changes at each sampling instant k, since it is based on a frozen prediction for the future scheduling parameters. Therefore, the propagation of disturbances along the horizon is crucial, but may yield conservative sets S(j). This is nonetheless expected, since we are using a linear time-invariant model to make predictions on a nonlinear system. Nonlinear predictions can result in less conservative sets S(j), but also require more online computational cost, as discussed in Section 2.4.

APPLICATION EXAMPLE

For illustration purposes of the proposed method, we consider a nonlinear model from [START_REF] Lazar | Input-to-state stabilizing sub-optimal nmpc with an application to dc-dc converters[END_REF], which represents a DC-DC Buck-Boost converter system. The nonlinear dynamics are:

f (x, u) = x 1 + α 1 x 2 + (β 1 -γ 2 x 2 ) u -α 2 x 1 + α 3 x 2 + (β 2 + γ 1 x 1 ) u , (27)
which can be embedded into the qLPV form of Eq. ( 2) with:

A(ρ) = 1 α 1 -α 2 α 3 , B(ρ) = β 1 -ρ 2 β 2 + ρ 1 , (28)
f ρ (x) = γ 1 0 0 γ 2 x , E = I nx . ( 29
)
The states x 1 and x 2 represent, respectively, the inductor current and the output tension, while u is a duty-cycle input signal. The model parameters are:

α 1 = 0.0541Ω -1 , α 2 = 0.1033Ω, α 3 = 0.9909, β 1 = 2.619A, β 2 = 0.2400V , γ 1 = 0.2273Ω and γ 2 = 0.1190Ω -1 .
The state and input constraints are:

x(k) ∈ X = {x ∈ R 2 : x ∞ ≤ x = 1} and u(k) ∈ U = {u ∈ R : |u| ≤ u = 0.
1}. The scheduling parameter constraints are:

ρ ∈ P = ρ ∈ R 2 : |ρ 1 | ≤ γ 1 x max , |ρ 2 | ≤ γ 2 x max . (30)
In order to compute the bounds on δρ, one can use either constrained optimisation or interval arithmetic methods as described in Remark 8. The tightest bounds were obtained through constrained optimisation, which gives:

δP := δρ ∈ R 2 : -0.0859 -0.0249 ≤ δρ ≤ 0.0805 0.0195 .( 31
)
The system operates subject to additive load disturbance bounded to the box w(k) ∈ W := w ∈ R 2 : w ∞ ≤ 0.02 .

The following results are obtained with the aid of Matlab, yalmip, SDPT3, fmincon and Gurobi.

We apply our proposed zonotopic MPC with stage cost weights Q = I nx and R = I nu . The chosen prediction horizon is N p = 4. The parameter-dependent matrices W (ρ) and Y (ρ), used to compute the terminal cost V (x) =

x T Y -1 (ρ)x and feedback gain K(ρ) = W (ρ)Y -1 (ρ), found by the bisection LMI solution of Theorem 3 with λ = 0.3, are:

Y (ρ) = ρ 1 -0.3014 -0.0304 -0.2216 + ρ 2 -1.0018 -0.0559 -0.8395 , W (ρ) = ρ 1 [ 0.1130 0.0221 ] + ρ 2 [ 0.3814 0.0661 ] .
For illustration purposes, we compare the performances obtained with our method against those obtained with "full-blown" NMPC method with Lipschitz disturbance propagation [START_REF] Santos | A constraint-tightening approach to nonlinear model predictive control with chance constraints for stochastic systems[END_REF]. We note that this NMPC approach requires the solution of a nonlinear program at each sampling period, which is numerically much more costly than the online program of the proposed method, which benefits from the linear predictions of the qLPV realisation, as gave Eq. ( 9).

First and foremost, we show the disturbance propagation reachable sets S(j), ∀j ∈ N [1,Np] with Zonotope disturbance propagation. These sets are computed according to Theorem 4. In Fig. 1, we show the collection of sets S(j) over the x 1 × x 2 plane (Condition 1). Then, considering the same uniformly randomly generated disturbance vector w [0,30] with unitary seed and entries inside W, we evaluate the closed-loop performances with these two robust MPC algorithms, considering the initial condition x(0) = (0.5 -0.5) T . In Fig. 2, we show the state feasibility set X , the obtained closed-loop trajectories for 20 discrete-time samples, and the terminal set X f (parameter-dependent ellipsoid for the proposed method and polyhedron for the Lipschitz robust MPC, as detailed in [START_REF] Santos | A constraint-tightening approach to nonlinear model predictive control with chance constraints for stochastic systems[END_REF]). The results were obtained in a 2.4 GHz, 8 GB RAM Macintosh computer. We note both approaches are able to stabilise the system at the origin, as well as to ensure input feasibility u ∈ U.

In order to stress the advantages of the proposed method, we assess the obtained closed-loop trajectories in terms of performances indexes. In Tab. 1, we provide the RMS values for each state and for the stage cost (x, u), along the whole simulation. With these indexes, we can conclude that the proposed robust Zonotopic qLPV MPC method can obtain good performances, with a reasonable increase of between 13% and 25% on the RMS values compared to the "full-blown" NMPC. This performance deterioration derives from the model-process mismatches by using a "frozen qLPV scheduling guess". Anyhow, we must point out that the average online computational stress (t c index) with the proposed solution is much smaller than the NMPC one (less than 2.8 times smaller), since the nominal predictions are linear at each sampling instant, resulting in simpler, convex, optimisation problems. This indicates the flexibility of the proposed method for embedded applications, as well as its robustness against bounded additive uncertainties and the inherent model-process mismatches derived using the frozen qLPV scheduling predictions.

As a final comment, we indicate that the use of scheduling prediction guesses, as done in [START_REF] Morato | Model predictive control design for linear parameter varying systems: A survey[END_REF] would certainly refine the closed-loop performances, since the mismatches between the nominal linear prediction and the real qLPV trajectories would decrease. Moreover, the uncertainty propagation w.r.t. these mismatches would also lead to even smaller zonotopes. These predictions guesses could be formulated by iterating the MPC QP (González [START_REF] González Cisneros | Nonlinear model predictive control for models in quasi-linear parameter varying form[END_REF] or through recursive mechanisms [START_REF] Morato | Novel qLPV MPC design with least-squares scheduling prediction[END_REF]. We note that using such scheduling prediction estimates would not compromise the real-time capabilities of the proposed method, since the model predictions φ π (•) would still be linear and resulting optimization would still be convex. We also stress that the case of stochastic disturbances can be directly treated by the zonotopic inclusions of the proposed method, similarly to what is done in [START_REF] Santos | A constraint-tightening approach to nonlinear model predictive control with chance constraints for stochastic systems[END_REF]. positively invariant ellipsoid. We compare our method against a NMPC with Lipschitz disturbance propagation.

As a result, we find similar performances with smaller computational stress with the proposed scheme, which benefits from the linear predictions of the qLPV realisation. While the method is ready for embedded applications (the online stress is similar to that of a constrained QP), we still plan on investigating conservatism reduction of the solution as well as its adaptation for tracking purposes.

APPENDIX PROOF OF THEOREM 3

We begin by showing the positive invariance of the ellipsoid. Applying the S-procedure, with λ > 0 to (17) and x T P (ρ)x ≤ 1 , we get:

x T A T π (ρ)P (ρ + δρ)A π (ρ)x + λ 1 -x T P (ρ)x < 0 , which can be rewritten as:

Applying a Schur complement over P (ρ + δ) for each entry of Π leads to (18), as detail [START_REF] Limón | MPC for tracking piecewise constant references for constrained linear systems[END_REF]. This ensures Theorem 2.

Complementary, we proceed by demonstrating that the resulting P (ρ) satisfies all five conditions of Theorem 1. (C1) trivially holds due to the ellipsoidal form of X f . (C2) is verified due to the fact that X f is a sub-level set of the terminal cost V (•). Therefore, if condition (C3) is verified, (C2) is consequently ensured.

The discrete Ricatti condition (C3) is verified through the solution of LMI ( 19). Since Q -1 > 0, R -1 > 0 and Y (ρ + δρ) > 0, we can take W (ρ) = K(ρ)Y (ρ) and apply two consecutive Schur, complements. This procedures leads to: This condition can be pre and post-multiplied by x T P (ρ) and P (ρ)x, respectively, which leads to:

x T (A(ρ) + B(ρ)K(ρ))

T P (ρ + δρ) (A(ρ) + B(ρ)K(ρ)) x -x T P (ρ)x ≤ -x T Qx -x T (K(ρ)) T RK(ρ)x .

This inequality is a sufficient condition for (C3) with V (•) as a sub-level of X f .

The fourth and fifth conditions (C4-C5) are verified by the direct application of the Schur complement to Eq. ( 20) and Eq. ( 21), respectively, using W (ρ) = K(ρ)Y (ρ). They lead, respectively, to:

I {i} K(ρ) (Y (ρ)) I {i} K(ρ) T ≤ u 2 i . I T {j} (Y (ρ)) I {j} ≤ x 2 i .
Since the maximum normed F x of an x that belongs to some ellipsoid x T P x ≤ 1 is given by F T (P -1 ) F , it holds that the first inequality implies that the projection I {i} K(ρ)x (i.e. i-th control signal) is upper-bounded, in norm, by u i , which satisfies (C4). Analogously, the second inequality ensures that the projection I {j} x (i.e. j-th state) is norm-bounded by x j , which satisfies condition (C5). This concludes the proof.

Fig. 1 .

 1 Fig. 1. Zonotopic sets S(j).

Fig. 2 .

 2 Fig. 2. Closed-Loop trajectories and terminal set X f : Zonotope (left) and Lipschitz (right).

  Y (ρ)A T π (ρ)P (ρ + δρ)A π (ρ)Y (ρ) -λY (ρ) , (33) Π 12 = Y A T π (ρ)P (ρ + δρ)w † ,(34)Π 22 = (w † ) T P (ρ + δρ)w † + (λ -1) .

  ρ) + B(ρ)K(ρ)) Y (ρ) -Y (ρ) ≤ -Y (ρ)QY (ρ) -Y (ρ)(K(ρ)) T RK(ρ)Y (ρ) .

Table 1 .

 1 Performance Results

	Method	RMS{x 1 }	RMS{x 2 }	RMS{ (•)}	tc
	Lipschitz	0.3358	0.2626	0.3386	50 ms
	Zonotope	0.3813	0.3253	0.4233	17.5 ms
		5. CONCLUSIONS
	In this paper, we provided a novel MPC algorithm for
	nonlinear systems represented through qLPV models. The
	method uses a frozen scheduling parameter guess as a
	prediction model, and bounds the propagation of model-
	process mismatches and additive load disturbances with
	zonotopes. We offer LMI-solvable remedies to compute a
	stabilising LPV feedback gain and parameter-dependent
	terminal conditions, used to ensure recursive feasibility of
	the optimisation procedure. The terminal set is a robust

This work has been supported by CNPq (304032/2019 -0), CAPES (001) and ITEA3 European project (15016) EMPHYSIS.