Module and method for generating a control input for a nonlinear system

Abstract

This disclosure describes a computing module for reducing communication burden between a controller and a physical system comprising a non-linear system by which only a m-length string is transmitted between the controller and the physical system. Further, the computing module introduces a self-adjustable performance mechanism that flexibly adjusts symmetric or asymmetric performance boundaries of the nonlinear system to accommodate various type of initial conditions, providing a solution to an underlying tracking problem.

Description

MODULE AND METHOD FOR GENERATING A CONTROL INPUT FOR ANONLINEAR SYSTEMCROSS-REFERENCE TO RELATED APPLICATION

[0001] This application claims the benefit of priority to Singapore patent application no. 10202302959Y which was filed on 19 October 2023, the contents of which are hereby incorporated by reference in its entirety for all purposes.TECHNICAL FIELD

[0002] This application relates to a computing module for reducing communication burden between an encoder and a decoder of a non-linear system by which only a m-length string is transmitted between the encoder and the decoder. Further, the computing module introduces a self-adjustable performance mechanism that flexibly adjusts symmetric or asymmetric performance boundaries of the nonlinear system to accommodate various type of initial conditions, providing a solution to an underlying tracking problem.BACKGROUND

[0003] Practical systems, such as those used in aerial robotics or manipulator grasping systems often operate under strict computational constraints. Such systems also need to meet stringent performance requirements due to hardware limitations and task-specific demands. As a result, event-based control has garnered significant attention in recent year's due to its ability to optimize resource usage. Its advantages over traditional periodic sampling techniques have motivated further research into systematic design approaches. Hence, in view of increasing system complexities, those skilled in the art are constantly looking for ways to balance communication resource efficiency with guaranteed system performance.

[0004] A critical challenge in event-based control lies in the encoding and decoding processes that occur during communication between the control box (e.g., an encoder module) and actuator (e.g., a decoder module). Transmitting a large number of digits for each communication between the control box and the actuator may be unnecessary, particularly when the control signal’ s magnitude is small. In situations where communication channels have limited bandwidth, the transmission of lengthy codewords can lead to delays and an increased risk of packet loss.

[0005] Consequently, it becomes essential to explore strategics that reduce communication resources from the perspective of the encoding-decoding scheme. By optimizing the communication process, especially for small-magnitude signals, it is possible to reduce transmission delays and mitigate the risk of data loss, improving overall system efficiency. Although numerous existing protocols have sought to reduce unnecessary signal transmission, they often neglect the complexities associated with encoding large control signals into lengthy codewords. Additionally, the transmission of sensitive system signals over public channels poses security risks. Therefore, those skilled in the art have focused their efforts on exploring resource-saving strategies from an encoding -decoding perspective, with a focus on reducing the communication burden without compromising system performance.

[0006] Another issue that concerns those skilled in the art is the maintenance of system performance, particularly in event-triggered control schemes where tracking errors may be introduced due to the nature of the triggering mechanism. Event-triggered control systems are often designed with the assumption that prescribed performance criteria arc met from the beginning of operation. However, in practice, initial conditions frequently violate these constraints, rendering existing methods ineffective since they do not account for scenarios outside the allowable set. This creates a singularity problem, necessitating the development of event-triggered control systems that can ensure prescribed performance, even when initial conditions deviate from expected parameters. Various solutions have since been proposed to address these limitations such as funnel boundaries and tuning functions. However, these methods arc not without drawbacks. Specifically, they suffer from overshoot performance trade-offs, constraints on performance distribution symmetry, and the lack of control over tracking errors during the initial phase.

[0007] In addition to the above, those skilled in the art have proposed various communication protocols, such as the 1 -bit relative threshold protocol and the 2 -bit strategy for switching between fixed and relative thresholds. However, these protocols face challenges in responding promptly to significant signal changes and in recalibrating decoders to avoid signal distortion. These limitations highlight the need for more effective and generalized event- triggered control systems that can regulate initial tracking errors and provide robust performance in a broader range of applications.

[0008] Despite efforts of those skilled in the art, the development of more effective and generalized event-triggered control systems that can regulate initial tracking errors and provide robust performance in a broader range of applications remains a complex and unresolved challenge in the field.SUMMARY

[0009] In one aspect, the present application discloses a computing module for generating a control input for a nonlinear system. The computing module comprises a processing unit and a non-transitory media readable by the processing unit whereby the media stores instructions that when executed by the processing unit causes the processing unit to compute a first tracking error eras a difference between a first system state xrof the nonlinear system and a desired reference trajectory ydand to constrain the first tracking error ei within a dynamically adjusted performance range defined by a lower boundary Ei(t~) and an upper boundary Eu(t~) . The processing unit then maps the first tracking error ei to a transformed error variable zxby applying a logarithmic transformation function, wherein the transformation function normalizes the first tracking error exwith respect to the lower boundary E;(t)and the upper boundary Eu(t) and computes a second tracking error e2as a difference between a second system state x2of the nonlinear system and a virtual control input a±of the nonlinear system, the virtual control input being directly influenced by the transformed error variable zx. The processing system then proceeds to generate the control input v(t), wherein the control input is directly influenced by the second tracking error e2.

[0010] In a further embodiment of this one aspect, the computing module then directs the processing unit to encode the control input into a m-bit string using a basc-p number system when it is determined that a control signal error Av exceeds a predetermined condition, and to then transmit the encoded control input to a decoder module.

[0011] In yet another embodiment of this one aspect, the computing module then causes the decoder module to direct a processing unit provided within the decoder module to upon receiving the encoded control signal, to decode the encoded control signal using a decoding rule to obtain a decoded control input ud(t) , whereby the decoding rule applies the predetermined condition to a previously received decoded control input based on the mostsignificant bit in the encoded control signal, and to apply the decoded control inputto the nonlinear system.

[0012] In another aspect, the present application discloses a method for generating a control input for a nonlinear system using a computing module. The disclosed method comprises the steps of computing a first tracking erroras a difference between a first system state x±of the nonlinear system and a desired reference trajectory yd, constraining the first tracking error ei within a dynamically adjusted performance range defined by a lower boundary £j(t) and an upper boundary Eu(t), and mapping the first tracking error ci to a transformed error variable by applying a logarithmic transformation function, wherein the transformation function normalizes the first tracking error e, with respect to the lower boundary £'((t)and the upper boundary Eu(t). The method then computes a second tracking error e2as a difference between a second system state x2of the nonlinear system and a virtual control input a1of the nonlinear system, the virtual control input being determined based on the transformed error variable z, ; and generates the control input based on the second tracking error e2.

[0013] In a further embodiment of this another aspect, the method further comprises the steps of encoding the control input into a m-bit string using a base-p number system when it is determined that a control signal error Av exceeds a predetermined condition and transmitting the encoded control input to a decoder module.

[0014] In yet another embodiment of this another aspect, the decoder module upon receiving the encoded control signal, the decoder module performs the steps of decoding the encoded control signal using a decoding rule to obtain a decoded control input ud(t), whereby the decoding rule applies the predetermined condition to a previously received decoded control input based on the most significant bit in the encoded control signal, and applying the decoded control input ud(t) to the nonlinear system.BRIEF DESCRIPTION OF THE DRAWINGS

[0015] Various embodiments of the present disclosure are described below with reference to the following drawings:Figure 1 illustrates a block diagram of a system for generating a control input for a nonlinear system in accordance with embodiments of the present disclosure;Figure 2 illustrates a block diagram of a processing system for performing embodiments of the present disclosure;Figure 3 illustrates a flowchart of a process for generating a control input for a nonlinear system in accordance with embodiments of the present disclosure;Figure 4a illustrates a plot showing a classical prescribed performance;Figure 4b illustrates a plot showing a self-adjustable prescribed performance in accordance with embodiments of the present disclosure;Figure 5 illustrates a plot showing simulation results of a computing module that implemented the self-adjustable prescribed performance and control mechanism under different threshold strategies in accordance with embodiments of the present disclosure;Figure 6 illustrates a plot showing the triggering times for the self-adjustable prescribed performance and the coded event triggered control mechanism under various threshold strategies in accordance with embodiments of the present disclosure;Figure 7 illustrates a plot showing simulation results of a computing module that implemented the self-adjustable prescribed performance and control mechanism under different threshold strategies when external disturbances are introduced during the 5-10s time period in accordance with embodiments of the present disclosure; andFigure 8 illustrates a plot showing the triggering times for the self-adjustable prescribed performance and the coded event triggered control mechanism under various threshold strategies when external disturbances are introduced during the 5-10s time period in accordance with embodiments of the present disclosure;DETAILED DESCRIPTION

[0016] The following detailed description is made with reference to the accompanying drawings, showing details and embodiments of the present disclosure for the purposes of illustration. Features that are described in the context of an embodiment may correspondingly be applicable to the same or similar features in the other embodiments, even if not explicitly described in these other embodiments. Additions and / or combinations and / or alternatives as described for a feature in the context of an embodiment may correspondingly be applicable to the same or similar feature in the other embodiments.

[0017] Tn the context of various embodiments, the articles “a”, “an” and “the” as used with regard to a feature or element include a reference to one or more of the features or elements.

[0018] In the context of various embodiments, the term “about” or “approximately” as applied to a numeric value encompasses the exact value and a reasonable variance as generally understood in the relevant technical field, c.g., within 10% of the specified value.

[0019] As used herein, the term “and / or” includes any and all combinations of one or more of the associated listed items.

[0020] As used herein, “comprising” means including, but not limited to, whatever follows the word “comprising”. Thus, use of the term “comprising” indicates that the listed elements are required or mandatory, but that other elements are optional and may or may not be present.

[0021] As used herein, “consisting of’ means including, and limited to, whatever follows the phrase “consisting of’. Thus, use of the phrase “consisting of’ indicates that the listed elements are required or mandatory, and that no other elements may be present.

[0022] One skilled in the art will recognize that certain functional units in this description have been labelled as modules or computing modules throughout the specification. The person skilled in the art will recognize that a module may be implemented as circuits, logic chips or any sort of discrete component. Still further, one skilled in the art will also recognize that a module may be implemented in software which may then be executed by a variety of processor architectures. In embodiments of the disclosure, a module may also comprise computer instructions or executable code that may instinct a computer processor to cany out a sequence of events based on instructions received. The choice of the implementation of the modules is left as a design choice to a person skilled in the art and does not limit the scope of the claimed subject matter in any way.

[0023] The present disclosure describes a Coded Event-Triggered Control (CEC) mechanism and a Scalable Prescribed Performance (SPP) framework aimed at reducing communication resources between a control box (i.e. , an encoder module) and an actuator (i.e., a decoder module), while also addressing the Entry' Capture Problem face by nonlinear systems. The CEC requires only an m-length string in a base-p number system to be transmitted for each communication, significantly lowering the communication burden compared to existing event-triggered control methods. This approach is particularly impactful from an encoding-decoding perspective as it offers a more efficient approach to data transmission in control systems.

[0024] In addition, the proposed SPP framework introduces flexibility in adjusting performance boundaries based on initial conditions, distinguishing it from previous approaches. By incorporating auxiliary functions, the SPP makes the control design applicable to any initial conditions, including cases where initial constraints are violated. This flexibility allows the CEC to handle both symmetric and asymmetric performance constraints, regulating tracking errors into an allowable set within a customizable and bounded time frame. As a result, the initial-condition constraints are effectively removed, leading to improved transient performance and broader applicability across various control systems.

[0025] A class of strict-feedback nonlinear systems may be described by the following equation:y = x±... equation (1) whs an unknown but continuous function enotes an unknown time-varying controlcoefficient, uare the system’s input and output respectively.

[0026] Based on equation (1) above, a tracking error can be defined as:is defined as the desired trajectory. The CEC mechanism is designed such that for nonlinear systems (as described in equation (1)), under any initial conditions: (i) all signals in the closed-loop systems are bounded; (ii) only a coded m-length string is required for each communication between CEC and actuator when the event-trigger condition is satisfied; and (iii) For any initial condition, the tracking error e^t) can be regulated into the prescribed allowable set in a customizable manner before a given time and then be constrained within there.

[0027] It should be noted that for any initial tracking condition (including initial constraint violation), if the tracking error e1(t) is fully constrained by the prescribed performance boundaries right after a user-given settling time T > 0, ...equation (2)then, such error tracking performance is considered to be an Entry Capture Problem, where — ei(t) and eu(tj denote lower and upper prescribed boundaries, respectively.

[0028] The Entry Capture Problem represents a frequently encountered tracking situation in that the prescribed performance is involved right after a given time T, whereas, in most existing works, these performance constraints should be satisfied from the beginning of the system’s operation. One typical example is a flight system which is often released from any condition (including initial-constraint violation) but is required to track and interact with a target in prescribed performance for successful task completion. Therefore, addressing such an Entry Capture Problem is interesting yet more challenging since it needs the control design applicable for more general cases.

[0029] A number of assumptions are made in the present disclosure. a. The first assumption is that the desired trajectory yci(t') and its time derivatives up to (n+1) th-order are known, bounded, and piece-wise continuous. b. The second assumption is that the nonlinear functi , n is unknown but certain crude information is available so thatwhere > 0 is an unknown constant and 0((TJ > 0 is a known smooth function. c. The third assumption is that the control coefficient i = 1, ... , n is unknown and time-varying, but away from zero, that is, there exist positive constants gi and Ji such that Without loss of generality, it is assumed thatthe signs of <7((Tf) are known and are all positive.

[0030] In general, it should be noted that the first assumption is widely adopted in tracking control of nonlinear systems, the second assumption indicates that some core functions can be easily extracted based only on some crude system information, which is reasonable and in line with current practice and the third assumption guarantees the controllable condition of a nonlinear system as described in equation (1).

[0031] A block diagram of system 100 for generating a control input for a nonlinear system in accordance with embodiments of the present disclosure is illustrated in Figure 1. As shown, system 100 generally comprises controller 101 and physical system 110 - which includes a system that is being controlled. Controller 101 is communicatively coupled to physical system 1 10 via network 109 which may comprise a wired and / or wireless communication network. The exact details of network 109 is omitted for brevity as it is well known to one skilled in the art.

[0032] In operation, coded event-triggered control (CEC) module 102 (as contained in controller 101) is configured to generate a control signal v(t) to regulate plant module 116’ s behavior (as contained within physical system 110), whereby plant module 116 may comprise nonlinear systems such as, but is not limited to, various types of equipment, physical systems or processes that arc being controlled by controller 101. Control signal v(l) is continuously monitored by event detector module 108, which is configured to determine whether an error between the current and previous control signals exceeds a predetermined threshold. In embodiments of the disclosure, the threshold is determined by the parameters / ? and a> as stored in module 106. These parameters arc used to define the sensitivity of the event-triggering mechanism used in controller 101. If event detector module 108 determines that the error exceeds the threshold, an event is triggered, and the control signal v(t) is passed to encoder 104 for encoding.

[0033] Encoder 104 then encodes the control signal v(t) into a binary string Sk- comprising a compact encoded signal that consists of bits that represent the magnitude and sign of the control error. The binary string is then transmitted by controller 101 over network 109 to physical system 110 which contains a nonlinear system such as plant module 116. This encoding process reduces the need for continuous communication, as the control signals areonly sent when an event occurs, thus saving bandwidth and processing power in networked control systems.

[0034] Upon receiving the encoded signal Sk, decoder module 1 14 (at physical system 1 10) then reconstructs the control signal u(t). This is achieved by applying the similar parameters / ? and M as previously used by event detector 108 to decode the magnitude and sign of the control signal. In embodiments of the disclosure, parameters / ? and m may be preloaded into storage module 112 during an initialization phase of system 100. This decoded signal u(t) may then be used as the actual control input to adjust the behavior of plant module 116. Plant module 116 which represents the physical nonlinear system being controlled, responds to the control input u(t) by altering its state, such as adjusting its position, velocity, or other dynamic variables.

[0035] Plant module’s current state %j(t) is then provided to CEC module 102. CEC module 102 then generates the next control signal based on this updated state information. Based on the above, it can be seen that system 100 is designed such that it ensures that control signals are only updated when necessary, thus optimizing resource usage while maintaining precise control over the plant.

[0036] In embodiments of the disclosure, in order to generate a control input for plant module 116, CEC module 102 is configured to compute a first tracking erroras a difference between a first system state xrof the plant module 116 and a desired reference trajectory yd. Once this is done, CEC module 102 constrains the first tracking error ei within a dynamically adjusted performance range defined by a lower boundary E^t) and an upper boundary Eu(t). The definition of the lower boundary Ei(t) and the upper boundary Eu(t) will be described in greater detail in the later parts of this disclosure. CEC module 102 then maps the first tracking error er to a transformed error variable zxby applying a logarithmic transformation function, wherein the transformation function normalizes the first tracking error exwith respect to the lower boundary £j(t) and the upper boundary Eu(t) . A second tracking error e2is then computed by CEC module 102 as a difference between a second system state x2°f the plant module 116 and a virtual control input a1of plant module 116, the virtual control input being determined based on the transformed error variable z1. The control input v(t) to be provided to encoder 104 is then generated based on this second tracking error e2.

[0037] In accordance with embodiments of the present disclosure, a block diagram representative of components of processing system 200 to carry out the computing and processing functions in accordance with embodiments of the disclosure is shown in Figure 2. These components of processing system 200 may be integrated within any of the modules shown in Figure 1 such as in controller 101, CEC module 102, encoder 104, event detector 108, decoder 114 or in any other functional modules described in this disclosure. One skilled in the art will recognize that the exact configuration of each processing system provided within these modules may be different and the exact configuration of processing system 200 may vary and the arrangement illustrated in Figure 2 is provided by way of example only.

[0038] In embodiments of the disclosure, processing system 200 may comprise controller201 and user interface 202. User interface 202 is arranged to enable manual interactions between a user and the computing module as required and for this purpose includes the input / output components required for the user to enter instructions to provide updates to each of these modules. A person skilled in the art will recognize that components of user interface202 may vary from embodiment to embodiment but will typically include one or more of display 240, keyboard 235 and optical device 236.

[0039] Controller 201 is in data communication with user interface 202 via bus 215 and includes memory 220, processing unit or processor 205 mounted on a circuit board that processes instructions and data for performing the method of this embodiment, an operating system 206, an input / output (I / O) interface 230 for communicating with user interface 202 and a communications interface, in this embodiment in the form of a network card 250. Network card 250 may, for example, be utilized to send data from these modules via a wired or wireless network to other processing devices or to receive data via the wired or wireless network. Wireless networks that may be utilized by network card 250 include, but are not limited to, Wireless-Fidelity (Wi-Fi), Bluetooth, Near Field Communication (NFC), cellular networks, satellite networks, telecommunication networks, Wide Area Networks (WAN) and etc.

[0040] Memory 220 and operating system 206 are in data communication with processor 205 via bus 210. The memory components include both volatile and non-volatile memory and more than one of each type of memory, including Random Access Memory (RAM) 223, ReadOnly Memory (ROM) 225 and a mass storage device 245, the last comprising one or more solid-state drives (SSDs). One skilled in the art will recognize that the memory components described above comprise non-transitory computer-readable media and shall be taken to comprise all computer-readable media except for a transitory, propagating signal. Typically, the instructions are stored as program code in the memory components but can also be hardwired. Memory 220 may include a kernel and / or programming modules such as a software application that may be stored in either volatile or non-volatile memory.

[0041] Herein the term “processor” or “processing unit” is used to refer generically to any device or component that can process such instructions and may include: a microprocessor, a processing unit, a microcontroller, a programmable logic device or other computational device. That is, processor 205 may be provided by any suitable logic circuitry for receiving inputs, processing them in accordance with instructions stored in memory and generating outputs (for example to the memory components or on display 240). In this embodiment, processor 205 may be a single core or multi-core processor with memory addressable space. In one example, processor 205 may be multi-core, comprising — for example — an 8 core CPU. In another example, it could be a cluster of CPU cores operating in parallel to accelerate computations.

[0042] A flowchart which sets out the process for generating a control input for a nonlinear system using a computing module in accordance with embodiments of the present disclosure is illustrated in Figure 3. In embodiments of the disclosure, process 300 as illustrated in Figure 3 may be performed by controller 101, CEC module 102 or any of the modules or combination of the modules provided within controller 101 and / or system 110.

[0043] Process 300 begins at step 302 with process 300 computing a first tracking error eLas a difference between a first system state x±of the nonlinear system and a desired reference trajectory yd. At step 304, process 300 then constrains the first tracking error ei within a dynamically adjusted performance range defined by a lower boundary E((t) and an upper boundary Eu(t) . Process 300 subsequently goes on to map the first tracking error erto a transformed error variable zxand does so by applying a logarithmic transformation function to the first tracking error ex, wherein the transformation function normalizes the first tracking error exwith respect to the lower boundary ^(tj and the upper boundary EU(E) . This is performed at step 306. A second tracking error e2is then computed by process 300 as adifference between a second system state x2of the nonlinear system and a virtual control input arof the nonlinear system whereby the virtual control input is directly influenced by the transformed error variable zx. This takes place at step 308. At step 310, process 300 then generates the control input, based on the second tracking error e2wherein the control input is directly influenced by the second tracking error e2.

[0044] In embodiments of the disclosure, at step 312, process 300 may encode the control input into a m-bit string using a base-p number system when it is determined that a control signal error Av exceeds a predetermined condition and then proceed to transmit the encoded control input to a decoder module.

[0045] In embodiments of the disclosure, at a specific time t, the lower boundary ^(t) may be obtained based on a difference between an outcome of a lower auxiliary function ^(t) and a lower Tunnel Prescribed Performance constraint function, wherein the lower auxiliary function (t) is defined as a function that has an initial value that is less than a summation of an initial value of the first tracking error e1(0) and an initial value of the lower Tunnel Prescribed Performance constraint function e;(0), has no value after a settling time T, and has a time derivative function that has no value after the settling time T.

[0046] In embodiments of the disclosure, at the specific time t, the upper boundary Fu(t) may be obtained based on a summation between an outcome of an upper auxiliary function and an upper Tunnel Prescribed Performance constraint function, wherein the upper auxiliary functionis defined as a function that has an initial value that is more than a difference between the initial value of the first tracking error Cj fO) and an initial value of the upper Tunnel Prescribed Performance constraint function eu(0), and has a value more or equal to a value of the lower auxiliary function, has no value after a settling time T, and has a time derivative function that has no value after the settling time T.

[0047] In embodiments of the disclosure, the upper auxiliary function ) may be definedl by the following expressi hen the specific timet is between 0 and the settling time T, wherein e1(0) is defined as the initial value of the first tracking error, eu(0) is defined as the initial value of the upper Tunnel Prescribed Performanceconstraint function, e([0) is defined as the initial value of the lower Tunnel Prescribed Performance constraint function, and where A and I are defined as positive constants > 1.

[0048] In embodiments of the disclosure, the lower auxiliary functionmay be defined l by the following expression when the specific timet is between 0 and the settling time T, wherein ex(0J is defined as the initial value of the first tracking error, eu(0) is defined as the initial value of the upper Tunnel Prescribed Performance constraint function, e((0) is defined as the initial value of the lower Tunnel Prescribed Performance constraint function, and where A and I are defined as positive constants > 1.

[0049] In embodiments of the disclosure the logarithmic transformation function may be defined by the following expression: z wherein z±is defined as the transformederror variable, etis defined as the first tracking error, Etis defined as the lower performance boundary, and Euis defined as the upper performance boundary.

[0050] In embodiments of the disclosure, the virtual control input armay be defined as a function that is based on a scaling factorwhose value is dependent on the upper and lower boundaries of the dynamically adjusted performance range, a proportional gain applied to the transformed error variable z1;and an adaptive error-dependent variable.

[0051] In embodiments of the disclosure, the control input v(t) may be defined as a function that is based on a proportional control variable of the second tracking error e2, and an adaptive term.

[0052] In embodiments of the disclosure, the process of encoding the control input into the m-bit string using the base-p number system further comprises the process of encoding a most significant bit of the m-bit string based on a sign of the control signal error Av, and encoding remaining m-1 bits of the m-bit string based on a magnitude of the control signal error Av using the base-p system. In this embodiment, the value p in the base-p system may be an even number.

[0053] In embodiments of the disclosure, the decoder module may be configured to upon receiving the encoded control signal to performs the process of decoding the encoded control signal using a decoding rule to obtain a decoded control input ud(t), whereby the decoding rule applies the predetermined condition to a previously received decoded control input based on the most significant bit in the encoded control signal and then apply the decoded control input ud(t) to the nonlinear' system.

[0054] Auxiliary Functions

[0055] A set of auxiliary functions are defined to regulate the tracking error WO into the prescribed allowable set in a customizable manner for a specific given time and such that the tracking error ex(t) be constrained within these functions for any initial conditions (including initial constraint violations).

[0056] To achieve this, the following lower auxiliary functionand upper auxiliary function are defined as any scalar' functions that satisfy the following properties: tinuous and bounded;defined as the settling time as defined in equation (2); and(iv) WO > W0> Vt > 0 and rju(t) = WO = 0, Vt > 0

[0057] Exemplary equations that may satisfy conditions (i)-(iv) above are as follows:where I and A > 1 are positive constants, e„(0) and 6 / (0) denote the initial values of upper and lower Tunnel Prescribed Performance constraint functions eu(t) and WO respectively which will be described in greater detail in the subsequent sections. In embodiments of thedisclosure, the exemplary equations defined above may be used as the upper auxiliary function and the lower auxiliary function r / ft) respectively.

[0058] It is useful to note that for most nonlinear systems, the initial condition of such system may initially violate the performance constraints of such systems, i.e., defined by the upper and lower Tunnel Prescribed Performance constraint functions, rendering most existing Prescribed Performance control methods inapplicable as they are not defined and suffer from singularity problems. Through the use of these auxiliary functions, the performance boundaries may then be dynamically adjusted in accordance with the initial condition. By leveraging the second condition (i.e., condition (ii) above), any given initial condition would remain within an updated and allowable set thereby ensuring the applicability of the control method.

[0059] Self-adjustable Prescribed Performance

[0060] The following Self-adjustable Prescribed Performance (SPP) is designed by introducing the above upper and lower auxiliary functions into the performance boundaries defined by the upper and lower Tunnel Prescribed Performance constraint functions respectively. Hence, the first tracking error e1(t) is therefore constrained within a dynamically adjusted performance range defined by a lower boundary £)(t) and an upper boundary Eu(tf and this is defined as: ...equation (3) where, . . .equation (4) ...equation (5)where eepresent the known upper and lower Tunnel Prescribed Performance (TPP) as described in the publication titled ‘ Saturation-tolerant prescribed control for nonlinear time-delay systems’ by the authors Ji, R., Li, D. and Ge, S. S and as published in IEEE Transactions on Fuzzy Systems in 2022. These equations are reproduced here as:

[0061] A comparison between classical prescribed performance conditions and performance boundaries defined by the SPP mechanism are plotted in Figures 4a and 4b. The following parameters were utilized for this comparison:, p0, prr, = 4, and with the initial tracking error being ex(0) = 2.

[0062] Figure 4a illustrates a plot of the initial tracking error C| (0) under classical prescribed performance conditions / boundaries and it can be seen that the initial tracking error ex(0) violates the original allowable set established byHence, existing methods are inapplicable due to the singularity problem shown in Figure 4a.

[0063] With the aid of the lower and upper auxiliary functions, the SPP mechanism may readjust the upper and lower performance boundaries according to different initial conditions such the initial tracking error is always within the updated allowable set (i.e., — £j(0) <as shown in Figure 4b.

[0064] During the initial time interval , the SPP mechanism not only ensuresthe control method applicable for any initial conditions but also provides temporary performance constraints on the tracking error. When t > T, the upper and lower performance boundaries Eu(0) and — £j(0) respectively become equivalent to the original performance boundaries, that is, Eu(t) = eu(t) and — £j(t) = — et(t) since= ^(t) = 0 as depicted in the plot of Figure 4b. Therefore, it can be seen from Figure 4b that the SPP mechanism provides an effective solution for the underlying Entry Capture Problem.

[0065] In summary, during the initial time period when he tracking errorex(t) is bounded by the upper auxiliary function pu(t) and the lower auxiliary function tp(t) in —j( )x( )u( ) hen t > T, the boundaries are similarly defined by —E^t') < e , (t) < Eu(t) which can then be rewritten as Through thisapproach, the disclosure is able to address the Entry Capture Problem.

[0066] Additionally, the SPP mechanism described above has an improved transient performance when compared to the know funnel boundary and global prescribed performancewhich often encounters an overshoot problem, as their initial performance constraints tend to infinity leading to a loose allowable set. Similarly, by tuning functions, there are no constraints on tracking errors during the initial interval, which is undesirable in practice due to safety concerns and task demands. The SPP mechanism addresses these issues by adjusting performance boundaries through the use of upper and lower auxiliary functions such that the proposed control scheme can be applied to any initial condition even if the initial constraints are violated. It provides bounded yet customizable virtual performance boundaries during the initial interval as depicted in Figure 4b, effectively avoiding the aforementioned severe overshoot performance and unconstrained behaviours. The tracking error is regulated into the allowable set within finite time and the Entry Capture Problem is therefore resolved.

[0067] Further, the SPP mechanism described above may be extended to other control methods by integrating the upper and lower auxiliary functions into any known performance boundaries. However, the funnel boundary and global prescribed performance rely on specific performance functions, which limits their adaptability to other control methods. Moreover, the methods known to those skilled in the art are limited to only symmetric performance distributions, but the disclosed SPP mechanism may be applied to both symmetric and asymmetric cases, thereby enhancing its applicability to more general cases.

[0068] Coded Event-triggered Control for Second-order Nonlinear System

[0069] In the following embodiment, the coded event-triggered control (CEC) module is used to generate a control signal for a second-order nonlinear system that may be defined as follows:...equation (8)where x±and x2are system states of the physical system, fi(xt) andgj(Xj), i = 1.2 , are unknown yet smooth nonlinear functions that satisfy the 2ndand 3rdassumptions made in the earlier sections, and u denotes the control input to the nonlinear system.

[0070] The tracking errors may be defined as follows: er = Xi - yd...equation (9)...equation (10) where atrepresents the virtual control input and ydrepresents the desired trajectory of the physical system. The derivation of this virtual control input will be described in greater detail in the subsequent sections. To ensure that the tracking error e, satisfies equation (3) for all t > 0, the tracking error ei is mapped to a transformed error variable zrby applying a logarithmic transformation function to the tracking error e1. The transformation function normalizes the first tracking error e, with respect to the lower boundary L](t)and the upper boundary £u(t) and the logarithmic error transformation function may be defined as: . . . equation (11)

[0071] Based on equations (4), (5) and (10), the time derivative of the transformed error variable zxmay be defined as:...equation (12) where,1(ei + Et)(Eu- ex)

[0072] is defined as a parameter that describes system performance characteristics. Based on equations (3)-(7) and property (iv) of the Auxiliary Functions, it can be derived that £■„(£) + Ei(t) > 0 and (£„(£) — e1(t))(Ei(t) + e;(t)) > 0 are bounded functions for e1(t) in the compact set= {e1(t) G E8.: —£)(£) < ex(t) < £u(t)}. Therefore, Hi0 and pxGLeo. The time derivative of -zf along equation (12) may then be defined as:ZiZi = z^ / fi + g1e2- yd) + z1g1g1a1+ zlfu2. . .equation (13)—zi9i9iai + “i

[0073] Based on the 1st to third assumptions, and Young’s inequality, the following equations are derived:where from the earlier described second assumption, <pris a known smooth function that is related to an upper bound of fr. Details of the smooth function is omitted for brevity as it is known to one skilled in the art.

[0074] Therefore, the term Etin equation (13) may be bounded by:...equation (18)where,...equation (19). . . equation (20) where from the earlier described second assumption, b;> 0 is an unknown constant that is related to the upper bound of fr.

[0075] As a result, the virtual control input armay be defined as:...equation (21). . . equation (22)are positive constants, where c1z1represents a proportional gain that is applied to the transformed error, z1019>1represents an adaptive error-dependent term that allows the system to adaptively compensate for system uncertainties or disturbances, 0Xis the estimation of 0Xwhich represents an adaptive parameter and may be obtained using an adaptive law or a Real-Time Parameter Estimation technique - where both techniques are known to oneskilled in the art, g±— l(Eu+ £)) as defined above, and z, is defined in equation (11). The values of cltrltandare used to determine the convergence rate and tracking accuracy of the system, ft should be noted that their extreme values may lead to large control inputs, making the system sensitive to external disturbances. Therefore, these parameters should be carefully selected in view of the effect they have on tracking performance and system ability.

[0076] Based on the Lyapunov function,where 61— 61— 01. Based on the earlier described third assumption, gris defined as a control coefficient that may be estimated or adaptively learning using adaptive control or estimation theory. Hence, based on equations (13), (18), (21) and (22) the time derivative of the equation above can be obtained as:

[0077] Based on the definition of 61, the following equation is obtained:0101

[0078] The time derivative may then be rewritten as:...equation (23) where a, =^012is a bounded signal. Based on the earlierdescribed third assumption, g1describes the lower boundary of g2, and g2comprises an unknown positive constant defined as a control coefficient that may be estimated or adaptively learning using adaptive control or estimation theory.

[0079] The time derivative of e2may then be derived from equations (8) and (10) as:e2= x2- a2= f2+ g2u - ar. . . equation (24) with, 5)01, which is computable, and k is the update time. Based on the definition of the control signal error Av(t), where Av(t) = v(t) — u(t), the following equation can be obtained: u(t) = v(t) - Av(t), | Av(t) | < p ...equation (26)_ pm where p — max {<uop°, <u1p1, ... , u)Sc ipSc_1} represents a positive constant, sc— — . The Lyapunov function V2may then be defined as: ■ ■ ■ equation (27)where 02=#2—®2- where 02is the estimation of 02(which will be computed in equation (31) below) and r2is a positive constant. The time derivative of V2based on equations (24) - (26) may then be derived as: ...equation (28)

[0080] Using Young’s inequality, S2can be expanded as inequalities in equations (14) - (17). Hence, it can be seen that the control input signal will be updated whenever the encoder is triggered, which indicates that a predetermined condition such as a>pP can be used to trigger the encoder under the condition |Av| pP where a>, ft and p are all bounded numbers. Hence, it can be derived from equation (26) that p is a bounded constant, leading to:...equation (29)This equation then yields:...equation (30)where,02— max {1, b2, b2}, ...equation (31)...equation (32)where from the earlier described second assumption, <p2is a known smooth function that is related to an upper bound of f2.

[0081] In view of the equations above, the actual control input v(t) may then be defined as: ...equation (33) ...equation (34)Where c2e2is defined as a proportional control variable of the second tracking error, 02e2<t>2is defined as an adaptive term, c nd 02is an adaptive law to estimate the value of02. The values of and o2will determine the convergence rate and tracking accuracy of the system. It should be noted that their extreme values may lead to large control inputs, making the system sensitive to external disturbances. Therefore, these parameters should be carefully selected in view of the effect they have on tracking performance and system ability. Through the use of equations (30) - (34), the inequality in equation (28) may be rewritten as: . equation (35)

[0082] Based on the definition of 62, the following equation is obtained:...equation (36)

[0083] When equation (36) is substituted into equation (35), this results in:...equation (37)

[0084] Based on the equations above, with regard to the second-order nonlinear system as described by equation (8) and based on the first, second and third assumptions, the virtual control cq and the actual control input v(t) have been defined in equations (21) and (33) respectively. Through the use of the proposed control method and the SPP mechanism, the tracking error e, was able to satisfy the Entry Capture Problem property since it is strictly constrained by the prescribed performance: — e((t) < er(t)' < eu(t) right after an user-given finite time T. Additionally, all the signals in the closed-loop system are bounded regardless of the initial conditions and as a result, Zeno behaviour is excluded.

[0085] By revisiting equation (37),< — 77 IA + £2> where n — > 0 . Subsequently, the equation may bedescribed as:

[0086] From the above, it can be concluded that V2is bounded which guarantees the boundedness of zr, e2, and d2. Based on the error transformation function in equation (11), the condition that — E)(t) < e, (t) < Eu(t) is satisfied due to z, E Lm. Further, as the tracking error satisfies this condition — e{(t) < ex(t) < eu(t) when t > T, this shows that cq meets the Entry Capture Problem.

[0087] Other tracking signals in the closed-loop system are also proved to be bounded as the adaptive laws described in equation (22) and (34), and 02remain bounded. Thevirtual control input aAdefined in equation (21) belongsMoreover, by equation (33), the boundedness of actual control signal v(t) is guaranteed. From equations (9) and (10), it can be derived that xrand x2remain bounded. All the closed loop signals are therefore bounded for the second-order nonlinear system under any initial condition.

[0088] To illustrate the exclusion of Zeno behaviour, the time derivative of Av(t) during the inter-execution interval is given as:Where k is defined as the trigger sequence, that is k = 1,2 ... Based on equation (33), this results in■ Tn accordance with the first, second and thirdAssumptions, prescribed performance functions eu(t) and e;(t) as defined in equations (6)- (7), and the fact that e2is governed by equation (24) and other closed loop signals remain bounded, the boundedness of v is ensured. For convenience, the upper bound of |v| can be specified by a positive constant K. Therefore, the following equation can be obtained as:this indicates that a strictlypositive constant t* =exists such that:with t * denoting minimal inter-transmission interval. Hence, this shows that the Zeno behavior is excluded.

[0089] Coded Event-triggered Control for high-order system

[0090] In this section, the proposed CEC will be extended to nth-order systems. Instead of repeating the step-by-step procedure, its control design is omitted here for conciseness since its stability analysis is identical to the second-order system as described in equation (8).

[0091] The tracking errors may be defined as follows: et= %i — yd... equation (38)...equation (39) where ai-1represents virtual control inputs. In order to ensure that exfulfils the SPP mechanism, the similar error transformation function is utilized. zx= In (£;+ei) . . . equation (40)Ell~el

[0092] The virtual control inputs a±may be defined as: ...equation (41)1 ...equation (42) and the adaptive laws are updated by:...equation (43). . . equation (44) for i — 2, ... , n — 1, where c;, r;, and <yi, i — 1, ..., n — 1 are positive constants, and 0(- is the estimation of 0j, and <J> i is a computable function where 0j and d’j arc:e;(t)) > 0 are bounded functions for e1(t) in the compact set flei= {e-^t) G IR: — Ei(t) < ex(t) < £u(t)}. Therefore, Pi0 and pxG LmSimilar to the previous section, the actual control input v(t) and the adaptive law are defined as follows: . . . equation (49)... equation (50) where cn,rn, an> 0, 6nis the estimation of 6nandis a computable function. Through the use of equations (45) - (48), the following equations are defined:6n— max{l, bf, b2, ■ ■■ , b(Q ...equation (51)+ (Affn-J2+ e2_! + < / >2...equation (52)[0093 J The Lyapunov function L2may then be defined as:where dt=Similar to the stability analysis performed in the previous section, it can be derived that:Based on the definition of the control signal error Av (I) , |Av(t) | < p , where p =... , MScpSc] represents a positive bounded constant. As such, the following equation can be defined as:

[0094] From the above, it can be concluded that Fn(t) is bounded which guarantees the boundedness of z1;e2, en, 61, ... , 6nE Lm. Based on the error transformation function in equation (40), the condition that —£)(£) < ex(t) < Eu(t) is satisfied due to z,Further, as the tracking error satisfies this condition — et(C) < er(t) < eu(t) when t > T, this shows that ermeets the Entry Capture Problem.

[0095] Other tracking signals in the closed-loop system are also proved to be bounded. From equations (43)-(45) and (50)-(51), and the definition of 0j which leads to 0j and 0, E Leo for i — 1, By equations (41 ) and (42) and (49), the boundedness of control input signals ar, ... , an-r, v is guaranteed. Considering from equations (37) and (39), x1, ... , xnbelong to Lm. Hence, all the signals in the closed-loop system are bounded.

[0096] To illustrate the exclusion of Zeno behaviour, the time derivative of Av(t) before the next triggering is given as:Av(t) = v(t) - v(tk) = v(f), tk< t < tk+1, where k = 1,2 ... Based on equation (49), this results in v(t) = +yn dv (fc+1)^lk-° drn{k)As all the signals in the closed-loop system are bounded, from the first, second and third Assumptions, prescribed performance functions eu(t) and e([t) as defined in equations (6)-(7), i>(t) can be bounded by the following equation:|Ai)(t) | < k,tk< t < tk+1,where k — 1,2, ... Given that lim | A v | = 0 and lim |Av| = a>pP, this indicates that a strictly C-tfc t-’tfc+l positive constant t* —exists such that:with t* denoting minimal inter-transmission interval. Hence, this shows that the Zeno behavior is excluded.

[0097] Coded Event-triggered Scheme

[0098] The coded event-triggered scheme is utilized to ensure that only a coded m-length string is used for communication between a CEC module and / or a controller module and a physical system (which includes a nonlinear system) when an event-trigger condition is satisfied. The coded event-triggered scheme (CES) may be defined by the following equations:where v(t) is the actual control input that is developed by the SPP mechanism described in the previous sections, i.e., at equations (33) and (49), the control signal error, Av(t), may be defined asepresents a control input provided to the nonlinear system, tkrepresents the update time, and p is an even number for a base-p system (i.e., Binary, Octal number system). The selection of the parametersand / ? (i.e. the parameters for the predetermined condition defined in equation (54)) are based on the following rules:pm where q G {0,1, — , sc— 1}, sc— — with m being the length of the encoded string to be transmitted. It should be noted that the values of p and m are detennined by the user of the system. Based on equation (55), the region that |u(t) | belongs to is then classified, and the value of b = q. For example, if IT1< |u(t) | < n2, then q = 1. Hence if b = q is determined based on the above law, s;- has to be selected such that it is able to satisfy equation (56). For example, if b = q = 1 , and m = 3 , then Sjx= 0, Sj2— 1, Sj3= 0. Further q is defined as time-varying value which is related to the current value of |u(t) |. Hence, as there will be variations of sc, the following requirement is introduced 0 = 7T0< nq < ••• < 7TS T< TTSC= -l-oo, when the level of the control input is measured, where mare positive constants chosen by designers, and SjkE {0,1, ..., p — 1} denotes each digit value of the base- pi number system at a £-th updated time.

[0099] Based on the triggering condition in equation (54), the sign of Av(t) has to be determined in order to ensure accurate encoding-decoding procedures. As shown in the encoding procedure defined in equation (55) and in the decoding procedure (which will be described in a later section), if Av(t) > 0, then sm fc< otherwise sm k> p Hence, under the assumption that t E (tk, tk+1), k — 0,1, ..., the encoded string Skused for the communication between the controller / encoder and the physical system / nonlinear system is constructed as follows:where Skis a string with m-bits, i.e., a m-length string, that is to be transmitted whenever the predetermined condition in equation (54) is met. Based on equations (54) and (56), string Skcan be described as a sign-and-magnitude representation of the threshold for the control signal error Av(t) . Specifically, sm kdenotes a sign digit, whose value determines the sign of Av(t)(e.g., Further, the magnitude of thethreshold for Av(t) is derived by the remaining digital numbers, i.e., sm-l k, ..., sl k.

[0100] When the CES condition in equation (54) is met at time tk, Skis then broadcasted to the physical system. When the physical system receives Sk, the system will adopt thefollowing decoder process to update the physical system’s control input ud(t) with the aid of the last control input and parameters stored:...equation (59)where ud(0) = v(t0), and the selections of f> and co follow the same rules in equations (55)- (57), which are also stored in the physical system. Based on the triggering condition in equation (54) and the decoder rule in equation (59), it can be derived that u(t) = ud(t).

[0101] Exemplary Embodiment

[0102] In this exemplary embodiment, it was assumed that p = 2 and m = 3. This results in a Binary number system and 3 -bit binary string is used for each communication between the controller and the physical system, where each digit value of the Binary number system has a value of either 1 or 0. The encoding and decoding processes in equations (58) and (58) are executed as shown in Table 1 below whenever the condition in equation (54) is triggered.Table 1

[0103] The CES described above was designed from an encoding-decoding viewpoint by which only a concise m-lcngth string is transmitted for each communication instead of the raw control input and the value of m can be chosen by those skilled in the art. This approach notonly reduces the bandwidth required but also provides a balanced strategy between system performance and network constraints. It should be noted that the proposed CES adopts a piecewise increasing threshold in relation to |u(t) | and approaches a constant threshold when |u(t) | is excessively large. This dynamic threshold facilitates more accurate tracking performance due to the lower threshold when |u(t) | is small and ensures a rapid response to the signal’s change when |u(t) | is large with the aid of the constant threshold. This is shown in equations (55) - (57).

[0104] Therefore, the signal distortion problem commonly observed in fixed and relative thresholds is effectively alleviated since the CES has a good balance between system performance and resource constraints. In summary, the CES approach disclosed not only alleviates the signal distortion problem but also addresses the challenge of codesigning an event-trigger scheme with encoding -decoding rules. As a result, the CES provides a leaner communication protocol, which is friendly for practical systems with limited communication bit resources. Additionally, by encrypting the control signal into an encoded string Sk, this inherently prevents potential cyber threats as the message may not be decoded without the knowledge of encoding-decoding rules.

[0105] Simulation Results

[0106] To illustrate the effectiveness of the CEC described above, some comparative simulations have been performed based on the following second-order nonlinear system....equation (60)where, / iCq) = x( + 0.1 cos(0.5x^), f2Cx2) - 4x, x2+ x, e 'x2+ 0.05 sin(x1x2), ^l(xt) = 5 + 0.5 sin(x1), and g2(x2~) = 3 + 0.2 cos(x1x2).

[0107] Based on the 2ndassumption, it can be derived that,+ 0.1 with bx= 0.3 with b2= 3 and (p2— x2+ x2+ 1.

[0108] The initial condition is set by x1(0) = 1.5 and x2(0) = 0. The desired tracking trajectory is yd= sin(0.5t) . The control parameters are selected as: m = 3 (i. e., / ? = 0, 1, 2, 3 and a> — 0.3, 0.4, 0.5,15,0.002, I = 0.6, 2. - 2, 8 — 0.5, p0— 1, pm— 0.4, and <; = 1. If these values are much larger than the above- selected values, although this causes the convergence speed of the system to improve, the system will become sensitive to external disturbances. If these values arc much smaller than the above-selected ones, this results in a system convergence speed that is too slow.

[0109] In this simulation, the performance of the disclosed CEC with the SPP mechanism is compared with the performance of a control design based on relative threshold and switching threshold.[001 10] In an embodiment of the disclosure (labelled as “Case 1”), the simulation results of the disclosed CEC under different threshold strategies are shown in Figures 5 and 6. From Figure 5, it can be seen that the initial tracking error e, (0) violates the initial performance constraints —e((t) and eu(t) , leading to the classical prescribed control methods being inapplicable. Through the use of the disclosed CEC, e1(0) can be regulated into the allowable set within the given time T. With the aid of the SPP mechanism, bounded yet customizable virtual performance boundaries Eu(t) and £;(£) are established during the initial interval.

[0111] Therefore, the severe overshoot or unconstrained behaviours over [0, T) are effectively avoided. On the other hand, the triggering times of three threshold strategics (c.g., CEC scheme - 602, relative threshold scheme - 604, switching threshold scheme - 606) are shown in Figure 6, and the number of triggering events and bit assumptions are summarized in the second and third columns of Table 2 under the column titled “Case 1”, in which the initial signal transmission was excluded. As the value of m = 3 in the CES, it indicates that only a 3- bit string is transmitted for each communication when the event condition is satisfied. Compared with the relative threshold and the switching threshold where control inputs should be encoded by 8-bit strings before each transmission, the disclosed CES consumes fewer bits while maintaining the tracking performance, which is significant for the practical system with limited communication bandwidth.Table 2

[0112] In a further embodiment of the disclosure (labelled as “Case 2”), to further illustrate the effectiveness of the disclosed CES, the introduction of external disturbance d(t) — 2 cos 0.5t during 5-10s is considered in the second row of equation (60). The comparative results under different threshold strategies are shown in Figures 7 and 8. Due to the signal distortion problem associated with the relative threshold strategy, the tracking performance as illustrated in Figure 7 is severely degraded with e1(t) violating performance constraints at 5.24s. This makes the control design suffer from the singularity problem. Figure 7 also shows that the proposed CES and switching threshold strategy was able to achieve a good balance between tracking performance and network constraints. Therefore, the disclosed approach ensures that the tracking errors remain within the allowable set even when external disturbances are introduced. However, from the fourth and fifth columns of Table 2 (under the column “Case 2”) and the triggering times illustrated in Figure 8 (e.g., CEC scheme - 802, relative threshold scheme - 804, switching threshold scheme - 806), it can be seen that the CES scheme consumes fewer bit resources compared to the switching threshold scheme. On the other hand, for each communication case, only an encrypted 3 -bit string was transmitted through the public network rather than the sensitive real control input in the switching threshold strategy. Therefore, the proposed control design simultaneously reduces the communication burden and addresses any security concerns.

[0113] Numerous other changes, substitutions, variations, and modifications may be ascertained by the skilled in the art and it is intended that the present application encompass all such changes, substitutions, variations, and modifications as falling within the scope of the appended claims.

Claims

CLAIMS:

1. A computing module for generating a control input for a nonlinear system comprising: a processing unit; and a non-transitory media readable by the processing unit, the media storing instructions that when executed by the processing unit causes the processing unit to: compute a first tracking error evas a difference between a first system state x1of the nonlinear system and a desired reference trajectory yd: constrain the first tracking error ei within a dynamically adjusted performance range defined by a lower boundaryand an upper boundary Fu(t); map the first tracking error ei to a transformed error variable zrby applying a logarithmic transformation function, wherein the transformation function normalizes the first tracking error e1with respect to the lower boundary £j(t)and the upper boundary MO; compute a second tracking error e2as a difference between a second system state x2of the nonlinear system and a virtual control input arof the nonlinear system, the virtual control input being directly influenced by the transformed error variable zx; and generate the control input v(t), wherein the control input is directly influenced by the second tracking error e2.

2. The module according to claim 1, wherein at a specific time t, the lower boundary Ei(t) is obtained based on a difference between an outcome of a lower auxiliary function MO and a lower Tunnel Prescribed Performance constraint function, wherein the lower auxiliary functionis defined as a function that: has an initial value that is less than a summation of an initial value of the first tracking error e^O) and an initial value of the lower Tunnel Prescribed Performance constraint function e£(0), has no value after a settling time T, and has a time derivative function that has no value after the settling time T.

3. The module according to claim 2, wherein at the specific time t, the upper boundary £u(t) is obtained based on a summation between an outcome of an upper auxiliary function r)u(t) and an upper Tunnel Prescribed Performance constraint function, wherein the upper auxiliary function i]u(t) is defined as a function that:has an initial value that is more than a difference between the initial value of the first tracking error e1(0) and an initial value of the upper Tunnel Prescribed Performance constraint function eu( 0 ), and has a value more or equal to a value of the lower auxiliary function, has no value after a settling time T, and has a time derivative function that has no value after the settling time T.

4. The module according to claim 3, whereby the upper auxiliary function pu(t) is defined by the following expression: , when the specifictime t is between 0 and the settling time T, wherein e1(0) is defined as the initial value of the first tracking error, eu(0) is defined as the initial value of the upper Tunnel Prescribed Performance constraint function, e;(0) is defined as the initial value of the lower Tunnel Prescribed Performance constraint function, and where A and I are defined as positive constants > 15. The module according to claim 3, whereby the lower auxiliary functionis defined by the following expression:=when the specific time t is between 0 and the settling time T, wherein e1(0) is defined as the initial value of the first tracking error, eu(0) is defined as the initial value of the upper Tunnel Prescribed Performance constraint function, e((0) is defined as the initial value of the lower Tunnel Prescribed Performance constraint function, and where A and I are defined as positive constants > 1.

6. The module according to claim 1, wherein the logarithmic transformation function is defined by the following expression: , wherein z±is defined as thetransformed error variable, exis defined as the first tracking error, Etis defined as the lower performance boundary, and Euis defined as the upper performance boundary.

7. The module according to claim 1, wherein the virtual control input a±is defined as a function that is based on a scaling factorwhose value is dependent on the upper and lower boundaries of the dynamically adjusted performance range, a proportional gain applied to the transformed error variable z1, and an adaptive error-dependent variable.

8. The module according to claim 1, wherein the control input v(t) is defined as a function that is based on a proportional control variable of the second tracking error e2, and an adaptive term.

9. The module according to claim 1, further comprising instructions for directing the processing unit to: encode the control input into a m-bit string using a base-p number system when it is determined that a control signal error Av exceeds a predetermined condition; and transmit the encoded control input to a decoder module.

10. The module according to claim 9, wherein the instruction for directing the processing unit to encode the control input into the m-bit string using the base-p number system further comprises instructions for directing the processing unit to: encode a most significant bit of the m-bit string based on a sign of the control signal error Av; and encode remaining m-1 bits of the m-bit string based on a magnitude of the control signal error Av using the base-p system.1 1. The module according to claim 10, wherein a value p in the base-p system comprises an even number.

12. The module according to claim 9, whereby the decoder module comprises instructions for directing a processing unit provided within the decoder module to upon receiving the encoded control signal, to: decode the encoded control signal using a decoding rule to obtain a decoded control inputwhereby the decoding rule applies the predetermined condition to a previously received decoded control input based on the most significant bit in the encoded control signal; and apply the decoded control input u^t) to the nonlinear system.

13. A method for generating a control input for a nonlinear system using a computing module, the method comprising:computing a first tracking error e±as a difference between a first system state x±of the nonlinear system and a desired reference trajectory yd; constraining the first tracking error ei within a dynamically adjusted performance range defined by a lower boundary Ei(t) and an upper boundary Eu(t); mapping the first tracking error ei to a transformed error variable by applying a logarithmic transformation function, wherein the transformation function normalizes the first tracking error exwith respect to the lower boundary E;(t)and the upper boundary MO; computing a second tracking error e2as a difference between a second system state x2of the nonlinear system and a virtual control input arof the nonlinear system, the virtual control input being directly influenced by the transformed error variable zx; and generating the control input v(t), wherein the control input is directly influenced by the second tracking error e214. The method according to claim 13, wherein at a specific time t, the lower boundary E;(t) is obtained based on a difference between an outcome of a lower auxiliary function Th(t) and a lower Tunnel Prescribed Performance constraint function, wherein the lower auxiliary functionis defined as a function that: has an initial value that is less than a summation of an initial value of the first tracking error e^O) and an initial value of the lower Tunnel Prescribed Performance constraint function e;(0), has no value after a settling time T, and has a time derivative function that has no value after the settling time T.

15. The method according to claim 14, wherein at the specific time t, the upper boundary Eu(t) is obtained based on a summation between an outcome of an upper auxiliary functionand an upper Tunnel Prescribed Performance constraint function, wherein the upper auxiliary function ?]u(t) is defined as a function that: has an initial value that is more than a difference between the initial value of the first tracking error e^O) and an initial value of the upper Tunnel Prescribed Performance constraint function eu(0), and has a value more or equal to a value of the lower auxiliary function, has no value after a settling time T, andhas a time derivative function that has no value after the settling time T.

16. The method according to claim 15, whereby the upper auxiliary functionis defined by the following expression: , when the specifictime t is between 0 and the settling time T, wherein e, (Oj is defined as the initial value of the first tracking error, eu(0) is defined as the initial value of the upper Tunnel Prescribed Performance constraint function, e;(0) is defined as the initial value of the lower Tunnel Prescribed Performance constraint function, and where A and I are defined as positive constants > 1.

17. The method according to claim 15, whereby the lower auxiliary function rjL(t) is defined by the tollowing expression: T]i{t)+ when the specifictime t is between 0 and the settling time T, wherein e^O) is defined as the initial value of the first tracking error, eu(0) is defined as the initial value of the upper Tunnel Prescribed Performance constraint function, e£(0) is defined as the initial value of the lower Tunnel Prescribed Performance constraint function, and where A and I are defined as positive constants > 1.

18. The method according to claim 13, wherein the logarithmic transformation function is defined by the following expression: wherein zxis defined as thetransformed error variable, eris defined as the first tracking error, Etis defined as the lower performance boundary, and Euis defined as the upper performance boundary.

19. The method according to claim 13, wherein the virtual control input aris defined as a function that is based on a scaling factorwhose value is dependent on the upper and lower boundaries of the dynamically adjusted performance range, a proportional gain applied to the transformed error variable z1;and an adaptive error-dependent variable.

20. The method according to claim 13, wherein the control input v(t) is defined as a function that is based on a proportional control variable of the second tracking error e2, and an adaptive term.

21. The method according to claim 13, further comprising the steps of:encoding the control input into a m-bit string using a base-p number system when it is determined that a control signal error Av exceeds a predetermined condition; and transmitting the encoded control input to a decoder module.

22. The method according to claim 21, wherein the step of encoding the control input into the m-bit string using the base-p number system further comprises the steps of: encoding a most significant bit of the m-bit string based on a sign of the control signal error Av; and encoding remaining m-\ bits of the m-bit string based on a magnitude of the control signal error Av using the base-p system.

23. The method according to claim 22, wherein a value p in the base-p system comprises an even number.

24. The method according to claim 21, whereby the decoder module upon receiving the encoded control signal, the decoder module performs the steps of: decoding the encoded control signal using a decoding rule to obtain a decoded control input ud(t), whereby the decoding rule applies the predetermined condition to a previously received decoded control input based on the most significant bit in the encoded control signal; and applying the decoded control input ud(t) to the nonlinear system.

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