Self-organizing quantum robust control methods and systems for situations with uncertainty and risk

Control systems, apparatus, and methods can apply quantum algorithms to control a control object in the presence of uncertainty and/or information risk. A self-organizing controller can include a quantum inference unit that can generate a set of robust control gains for a controller that can meet the control objectives for the particular realization of the control object. In one embodiment, the quantum inference unit can include a quantum correlator configured to generate a plurality of quantum states based on a plurality of controller parameters and a correlation type. In this embodiment, the quantum inference unit can also include a quantum optimizer configured to select the correlation type of the quantum correlator and to select a quantum state from the plurality of the quantum states. The self-organizing controller can control the control object with one or more controller gains that are based on the selected quantum state.

BACKGROUND

1. Technical Field

The present disclosure relates generally to the field of automatic feedback systems for devices, machines, and systems in the presence of uncertainty and/or risk.

2. Description of the Related Technology

In classical control systems, a linear dynamic model of a control object is obtained in the form of dynamic equations, usually ordinary differential or difference equations. Based on the linear dynamic model, a controller can be designed by assuming the linear dynamic model approximates the control object. Under this approach, the control object is assumed to be relatively linear and time invariant. However, many real control objects can be time varying, strongly nonlinear, and unstable. For example, the dynamic model may include parameters (e.g., masses, inductance, aerodynamics coefficients, etc.) that depend on a changing environment or a state of the dynamic model. If the parameter variation is small and the dynamic model is stable, then a proportional-integral-derivative (PID) controller can be satisfactory.

Modern multivariable control methodologies, such as H∞or mixed-μ controllers synthesis, can account for some degree of uncertainty in the control object's parameters and/or dynamics. Like classic control systems, controller design is based on the assumption that the control object can be sufficiently approximated by a linear dynamic model. For the purpose of controller design, time variations and nonlinearities can be treated as “uncertainty” in the problem formulation. As a result, strongly time-varying and/or nonlinear control objects can result in a linear dynamic model with an unnecessarily large characterization of its uncertainty. However, for sufficiently large uncertainty, there may not be a controller that can stabilize the control object, particularly for all possible forms of uncertainty. Further, the resulting controller, if it exists, can be conservative in the sense that it accounts for the worst case realization of the uncertainty.

There are several nonlinear control techniques, such as feedback linearization, dynamic inversion, sliding mode, nonlinear damping, adaptation (for example, online parameter estimation), or the like. However, there does not appear to be a general nonlinear control methodology that takes into account formal uncertainty and performance requirements.

In quantum computing, classical computers with a Von Neumann architecture can simulate quantum algorithm gates for performing quantum algorithms and computation. A quantum algorithm can be solved using an algorithmic-based approach, wherein matrix elements of the quantum gate are calculated on demand. For example, an implementation of a Grover's search algorithm can be executed on a classic computer.

SUMMARY

One innovative aspect of the subject matter described in this disclosure can be implemented in a method that includes selecting a correlation type based on a result of a quantum genetic search algorithm. The method can also include generating quantum states based on a plurality of controller parameters and the selected correlation type. In addition, the method can include tuning a controller configured to control a control object based on at least one of the generated quantum states.

In some implementations, selecting can include simulating a closed-loop system to evaluate a candidate correlation type, evaluating a different candidate correlation type when the simulating indicates that fitness requirements are not satisfied, and returning the candidate correlation type as the selected correlation type when the fitness requirements are satisfied. Selecting can be based on evaluating fitness functions associated with different correlation types. The correlation type can include one of a spatial correlation, a temporal correlation, or a spatial-temporal correlation. Alternatively or additionally, the correlation type comprises an external correlation.

The plurality of controller parameters can include a set of controller gains. According to some implementations, generating quantum states can include generating a set of candidate quantum states based on the set of controller gains, and performing a correlation of the selected correlation type on the set of candidate quantum states from which a selected quantum state of the candidate quantum states is identified. In some of these implementations, the method can also include identifying a quantum state having the highest probabilities of the candidate quantum states as the selected quantum state. Alternatively or additionally, the method can include indentifying the selected quantum state by performing Grover's search algorithm.

The method can also include identifying one of the generated quantum states as a selected quantum state, wherein tuning is based on the selected quantum state. The quantum genetic search algorithm can include Grover's search algorithm. The generated quantum states each can include at least one real state and at least one virtual state. Generating the quantum states can provide at least one quantum state associated with a higher probability than any linear combination of the controller parameters. Alternatively or additionally, the controller parameters are based on at least one information risk production rule.

In some instances, the method can include generating two sets of controller gains with one or more proportional-integral-derivative (PID) controllers, wherein the plurality of controller parameters include the two sets of controller gains, and wherein the controller comprises a quantum PID controller.

Another innovative aspect of this disclosure can be implemented in a control system that includes a quantum inference unit configured to provide one or more controller gains to a controller configured to control a control object. The quantum inference unit includes a quantum correlator configured to generate a plurality of quantum states based on a plurality of controller parameters and a correlation type. The quantum inference unit also includes a quantum optimizer configured to select the correlation type of the quantum correlator and to select a quantum state from the plurality of the quantum states. The one or more controller gains are based on the selected quantum state.

The control system of can also include a knowledge base storing at least one of the plurality of candidate controller parameters. The knowledge base can be configured to provide one or more of the plurality of candidate controller parameters to the quantum inference unit. The knowledge base can stores one or more information risk production rules.

The quantum inference unit can also include a quantum encoder configured to generate a plurality of candidate quantum states based on the at least one of the plurality of controller parameters and provide the quantum correlator with the candidate quantum states. The quantum correlator can be configured to generate the plurality of quantum states based on performing a correlation of the selected correlation type on the candidate quantum states.

The quantum inference unit can also include a decoder configured to output the one or more controller gains based on decoding the selected quantum state.

The quantum optimizer can include a quantum search unit configured to perform a superposition operation, an entanglement operation, and an interference operation on the at least one of the plurality of quantum states generated by the quantum correlator. Alternatively or additionally, the quantum optimizer can include a genetic algorithm unit configured to simulate a closed-loop system and select the selected quantum state based on evaluating a fitness function associated with the selected correlation type.

The correlation type can include one of a spatial correlation, a temporal correlation, or a spatial-temporal correlation.

The control system can also include the controller. For instance, the controller can include a proportional-integral-derivative (PID) controller. In some of these instances, the controller includes a fractional PID controller. The controller can include a quantum PID controller configured to tune the control object based, and the plurality of controller parameters can be based on two sets of gains generated by two PID controllers. In some implementations, the controller can include a sliding mode controller.

Another innovative aspect of this disclosure can be implemented in a method that includes generating quantum states based on a plurality of controller parameters. The method can also include executing a quantum search algorithm on the generated quantum states to select a quantum state based on information risk. In addition, the method can include controlling a control object based on an indicator of the selected quantum state.

In some instances, executing is performed on line. Alternatively or additionally, executing can include identifying the selected state when a risk termination condition is satisfied, and generating new quantum states when the risk termination condition is not satisfied.

The method can also include adjusting at least one of the controller parameters based on an information risk production rule.

The information risk can be associated with a current state of a control system that includes the control object. The controller parameters may not be designed to control the control object in the current state of the control system.

The method can also include selecting a correlation type based on a result of performing another quantum search algorithm, wherein generating is based on the selected correlation type. Selecting can be based on evaluating fitness functions associated with different correlation types, in which the fitness functions incorporate information risk.

The quantum search algorithm can be a quantum genetic search algorithm.

Yet another innovative aspect of this disclosure can be implemented in a control system that includes a quantum inference unit configured to provide one or more controller gains to a controller configured to control a control object. The quantum inference unit includes a quantum correlator configured to generate a plurality of quantum states based on a plurality of controller parameters. The quantum inference unit also includes a quantum optimizer configured to select a quantum state from the plurality of the quantum states based on information risk. The one or more controller gains are indicative of the selected quantum state.

The control system can also include a knowledge base storing the controller parameters and one or more information risk production rules. The knowledge base can be configured to provide the one or more controller parameters to the quantum inference unit. The information risk can include the one or more information production rules. The controller parameters can include controller gains, and the knowledge base can be configured to compute the controller gains provided to the quantum inference unit based on a risk estimation signal indicative of information risk.

The quantum genetic optimizer can include a genetic algorithm unit configured to simulate a closed-loop system and select the selected quantum state based on evaluating a fitness function that incorporates information risk.

The quantum inference unit can also include a quantum encoder configured to generate a plurality of candidate quantum states based on the controller parameters and provide the quantum correlator with the candidate quantum states. The quantum correlator can generate the plurality of quantum states based on performing a correlation on the candidate quantum states.

The quantum inference unit can also include a decoder configured to decode the one or more controller gains from the selected quantum state.

The control system of can also include the controller. The controller can include one of a proportional-integral-derivative (PID) controller, a fractional PID controller, and a sliding mode controller.

The quantum inference unit can be a quantum fuzzy inference unit configured for providing one or more controller gains to a fuzzy controller.

DETAILED DESCRIPTION

Generally described, aspects of the present disclosure relate to controlling a control object. More specifically, aspects of the present disclosure relate to self-organizing control of an uncertain control object. In some embodiments, a self-organizing controller can include n knowledge bases of candidate controllers and a quantum inference unit. Each of the n knowledge bases can provide a set of candidate controller gains based on online measurements of the control object and/or the environment. The quantum inference unit can perform online processing of the n sets of controller gains of the n knowledge bases to generate a set of robust control gains to tune a controller, in which n is a positive integer. The robust controller gains can be controller gains of a controller that can meet the control objectives for the particular realization of the control object.

The quantum inference unit can generate robust controller gains by processing the n sets of controller gains by using quantum computing operations, including the operations of (i) superposition, (ii) quantum correlations (entanglement), (iii) interference, or any combination thereof. For example, the quantum inference unit can encode each controller gain generated by the n knowledge bases into a quantum state. In some embodiments, the quantum state of a controller gain is a superposition of a real controller state |0and a virtual controller state |1. The real controller state |0can correspond to the controller gain generated corresponding to a knowledge base, and a virtual controller state |1can correspond to a separate control gain (a “virtual gain”) computed based on the real controller state |0. Using the real and virtual states as building blocks, the quantum inference unit can correlate two or more of the encoded quantum states to create a superposition signal of correlated states. The superposition signal can be used to determine a robust controller gain. For example, the quantum interference unit can perform an interference operation, which selects a correlated state. For instance, the correlated state can be selected based on having the highest probability. The selected correlated quantum state can be decoded into a controller gain, with which a controller included in a feedback loop with the control object can be tuned to control the control object.

Advantageously, some embodiments of the self-organizing controllers described herein are configured to compensate for uncertainty and/or risk that not accounted for by any of the n individual knowledge bases. For example, the quantum inference unit can compress redundant data and/or extract hidden data from the n knowledge bases. Data can be compressed and/or extracted by utilizing real and virtual gains, correlating and/or superposing various controller quantum states, and extracting a state from the superposition of correlated quantum states. In this way, the quantum inference unit can combine features of the each independent knowledge base and/or generate a robust controller gain for a situation not addressed by any individual knowledge base by extracting information from the collection of knowledge bases. Non-limiting examples of such situations include, for example, component aging, sensor failure, and the like.

The controllers described herein can be generated and/or tuned automatically offline and/or online. For example, a correlation type can be selected from a variety of correlation types. The correlation type used by such a self-organizing controller can effect performance, and the optimal correlation type can be problem dependent. Factors that can influence the optimal selection of the correlation can include the characteristics of the control object, the control objectives and/or constraints, the environment, the like, or any combination thereof. Moreover, some of these factors can change over time. These changes can be drastic. Thus, online tuning can be beneficial. Alternatively or additionally, the self-organizing controller can use risk estimation to alter its behavior. Utilizing risk estimation can improve robustness to uncertainty.

Self-organizing control can include any combination of features or quantum/genetic algorithms disclosed in U.S. patent application Ser. No. 11/313,077, filed Dec. 20, 2005, titled “METHOD AND DEVICE FOR PERFORMING A QUANTUM ALGORITHM TO SIMULATE A GENETIC ALGORITHM,” and published as U.S. Patent Publication 2008/0140749 on Jun. 12, 2008, which is hereby incorporated by reference in its entirety. Additionally, self-organizing control can include any combination of features of quantum and/or genetic algorithms disclosed in U.S. patent application Ser. No. 09/979,348, filed Jan. 23, 2002, titled “METHOD AND HARDWARE ARCHITECTURE FOR CONTROLLING A PROCESS OR FOR PROCESSING DATA BASED ON QUANTUM SOFT COMPUTING,” and issued as U.S. Pat. No. 7,383,235 on Jun. 3, 2008, which is hereby incorporated by reference in its entirety.

Self-organizing control based on soft computing and quantum computing techniques can be implemented on a quantum computer or simulated, for example, using classical efficient simulation methods of quantum algorithms on computers with a classical architecture, such as a von Neumann architecture.

FIG. 1shows an example closed-loop system100that includes a control object108, a self-organizing controller102having an intelligent quantum tuning unit104and a controller106, one or more sensors110, a risk estimator128, a comparator132, and a summer134. The closed-loop system100can control the control object108in the presence of uncertainty and/or risk. The closed-loop system100is one example of a control system.

The control object108can be referred to as “a plant” or “a process” in control theory literature. The control object108can be a system having relationships between certain inputs u* and certain outputs y*, in which it is desired that the outputs y* satisfy control objectives. Non-limiting examples of a control object108can include a temperature controller configured to control a temperature of a room, a vehicle such as an aircraft, a robotic system, one or more computing systems configured to control data traffic over a communication network, a medical device such as a prosthesis, a biological system such as a genetically-modified bacteria, and the like. Example inputs u* in a control system that includes an aircraft as a control object can include signals to motors that deflect movable surfaces and/or flaps on the aircraft's wings and/or tail and/or signals to affect throttle settings. Example outputs y* of an aircraft can include the aircraft's position, velocity, orientation, rate of rotation, the like, or any combination thereof.

The input-output relationships of the control object108can be dynamic relationships. For instance, the outputs y* can be generated based on past outputs and/or past inputs in real-time. The dynamics of the control object108can be unstable, for example, small perturbations of the inputs u* and/or states of the control object108can lead to large perturbations in the states of the control object108that do not dampen out with time. For example, a high-performance aircraft can be dynamically unstable and thus require constant or near constant adjustment of the aircraft's control surfaces, flaps, and engine throttle to maintain stable flight. Additionally, the dynamics of control objects, such as an aircraft, can be nonlinear. Nonlinear dynamics can lead to instabilities that grow more rapidly than instabilities in systems with linear dynamics. Moreover, some control objects that have nonlinear dynamics behave differently with operation conditions. For example, an aircraft can behave quite differently at low speeds and low altitudes than at high speeds and high altitudes.

When a control object exhibits unsatisfactory dynamics (for example, instabilities and/or nonlinearities), the controller106can generate a control signal to adjust the output y* within a suitable predefined range. For instance, in the closed-loop system100ofFIG. 1, the controller106can receive an error signal e from the control object108and provide the control object108a control signal u that is generated based on the error signal e. The error signal e can represent a tracking error that can measure a difference between a reference signal r (for example, the desired values of outputs y* of the control object108) and measurements y. In this case, the controller106can generate the control signal u to adjust the error signal e to a sufficiently small value. In some implementations, the controller106can independently receive the reference signal r and the measurements y. These implementations can allow for the integration of a feed forward controller.

The controller106can provide a control signal u to the control object108to cause desired dynamic behavior that achieves one or more of the control objectives. Control objectives can include maintaining stability and/or achieving a specified level of performance, in terms of output tracking, disturbance rejection, the like, or any combination thereof. Performance can be specified by one or more of a number of metrics. Table 1 shows non-limiting example performance measures that can be included as part of the control objectives. Alternatively or additionally, performance can be quantified according to H∞loop-shaping techniques and/or can be quantified as being a function of one or more of the following: rise time, settling time, overshoot, the error signals and their integral norms, control signals and their integral norms, pole locations, and the like. Further, control objectives can include constraints on, for example, the control signal and/or the states of the control object.

TABLE 1Criteria of control qualityPerformance MeasureIntegral of squared-error:ISE=∫0T⁢e2⁡(t)⁢ⅆtIntegrated absolute error:IAE=∫0T⁢e⁡(t)⁢ⅆtIntegrated of time-absolute-error:ITAE=∫0T⁢t⁢e⁡(t)⁢ⅆtIntegrated of time-squared-error:ITSE=∫0T⁢t⁢⁢e2⁡(t)⁢ⅆt

One challenge of control system design is achieving the control objectives in the presence of uncertainty. In control system design, there can be a tradeoff between achievable performance and robustness to uncertainty. For instance, increasing the performance of the closed-loop system can typically decrease robustness to uncertainty. Uncertainty can be associated with the control object, the environment, control objectives, the like, or any combination thereof. Control object uncertainty can include uncertainty in the parameters of the control object (for example, mass of the aircraft), in the dynamics of the control object (for example, unmodeled flexible modes of the wings of the aircraft), due to variations of dynamics over time (for example, due to component wear and/or fatigue), due to failures and damage (for example, sensor and/or actuator failures), the like, or any combination thereof. Environmental uncertainty can include uncertainty in stochastic disturbances acting on the control object (for example, wind turbulence), sensor noise, sensor and/or actuator delays, the like, or any combination thereof. Control objective uncertainty can include uncertainty in a reference signal, changes in the control objectives after, for example, failures or faults (for example, after sustaining battle damage, an aircraft flight controller may relax its performance requirements in order to avoid aggressive maneuvers that may further damage the aircraft), the like, or any combination thereof.

A second challenge in control system design is compensating for risk by adjusting the controller gains, controller structure, control objectives, the like, and combinations thereof. Risk can include an occurrence of certain risk conditions and/or an information risk increment. Examples of risk conditions include, for example, actuator failure (such as an engine-out), severe environmental conditions (such as icing), hostile combat conditions, unreliable sensor measurements, and the like. The information risk increment can indicate the expected harm in relying on given information. For example, given a controller designed for a certain linear model of the control object, the information risk associated the controller can be the expected harm in using the control signal generated by the controller. Information risk increment can be based on the probability of a condition (for example, a probability that the controller is destabilizing), the expected loss if the harm occurs, observations (for example, sensor measurements), the like, or any combination thereof. When the information risk increment is large, it can be desirable to change the control structure and/or controller gains. Where the information risk increment is small, it can be desirable to utilize the controller. More detail regarding generating an estimate of the risk will be discussed below.

Because of tradeoffs between performance and robustness in situations with large uncertainty and/or information risk, conventional controllers may not be able to achieve satisfactory performance and/or stability for a number of variations of the uncertainty. A set of candidate controllers may be able to satisfy the control objectives in one or more of the number of variations of the uncertainty that conventional controls may not be able to achieve satisfactory performance and/or stability. If the set of candidate controllers can be constructed, a system can be configured to select a desired candidate controller online.

However, it may be difficult to construct the set of candidate controllers offline and/or online. First, the range of uncertainty may not be known. For example, in the context of an aircraft control object, it may be difficult to predict all variations of possible failure cases. Second, even if the range of uncertainty is known, it may be practically infeasible to design a set of candidate controllers to covers all possible realizations of the uncertainty.

To compensate for uncertainty and/or risk, an intelligent quantum tuning unit can construct and/or tune a controller by extracting knowledge from online data and from one or more knowledge bases. Knowledge bases can provide information concerning a specific subject matter. Examples of knowledge bases include controllers that provide information related to achieving the control objectives for the given measurements and for a certain realization of uncertainty. If there is a set of n candidate controllers, where n is a positive integer, a tuning unit can tune and/or construct a controller based on the combined outputs of n candidate controllers.

Referring back toFIG. 1, the control object108can receive inputs u* and generate outputs y*. The inputs u* can represent the true inputs received by the control object108. Thus, inputs u* can be based on the control signal u and exogenous disturbances m.FIG. 1depicts an example in which the control signal u and exogenous disturbances m combine additively via summer134to generate inputs u*. However, in other implementations, the control signal u and exogenous disturbance m may affect the control object108differently. For example, the control signal u and the exogenous disturbances m can affect the control object108through separate input channels. Alternatively or additionally, there may be a delay associated with a path from control signal u to the input u*.

Outputs y* can be detected and/or measured by one or more sensors110to generate measurements y, which can be provided to the self-organizing controller102. The sensor(s)110may be noisy and/or have time delays. The self-organizing controller102can receive a risk estimation signal p, a control signal u, an error signal e, or any combination thereof. In other implementations, the control signal u can be obtained internally within the self-organizing controller102because the self-organizing controller102can generate the control signal u. The self-organizing controller102can generate the control signal u to influence the control object108in a way that satisfies the control objectives, even in the presence of uncertainty and/or risk.

The risk estimation signal ρ can indicate an occurrence of certain risk conditions and/or an information risk increment Δρ. Risk conditions can include events such as severe structural damage, sensor failure, actuator failure, the like, or any combination thereof. In connection with information risk increment, information divergence between probability density functions can be represented by a Kullback-Leibler information measure:

For defined loss functions W({tilde over (W)}) and probability density functions p(x,θ)[{tilde over (p)}(x,θ)] the average information risk can be computed by the following equation:
r(W2)({tilde over (r)}({tilde over (W)}2))=∫∫W2p(x,θ)dxdθ(∫∫{tilde over (W)}2{tilde over (p)}(x,θ)dxdθ(Equation 2)

Then information the risk increment Δρ={tilde over (r)}−r can be represented by the following equation:

In some models of control objects, a structure can be represented as a set of random parameters x=(x1, . . . , xn) in the presence of a disturbed parameter θ. For an experimental (in a statistical sense) vector of random values x=(x1, . . . , xn), the disturbed parameter θ can be represented by a probability density function of model parameters {tilde over (p)}(x,θ), which can approximate unknown (in general case) p(x,θ). The function p(x,θ) can represent the probability of an unpredicted situation. The function {tilde over (p)}(x,θ) can represent the approximated estimation of the unpredicted control situation. An output of the risk estimator128can be estimated, for example, by Equation 3 using Equation 1 for information divergence, using the probability density functions p(x,θ) and {tilde over (p)}(x,θ). The probability density functions p(x,θ) and {tilde over (p)}(x,θ) can be generated from corresponding Fokker-Planck-Kolmogorov equations, for example, as described in S. V. Ulyanov, M. Feng, V. S. Ulyanov, K. Yamafuji, T. Fukuda and F. Arai, Stochastic analysis of time-invariant non-linear dynamic systems. Part 1: The Fokker-Planck-Kolmogorov equation approach in stochastic mechanics, Probabilistic Engineering Mechanics, Vol. 13, Issue 3, July 1998, pp. 183-203, which is hereby incorporated by reference in its entirety. For example, the solution of a corresponding Fokker-Planck-Kolmogorov equation for a linearized model of a control object can give p(x,θ) and a real nonlinear model can give {tilde over (p)}(x,θ). The risk estimator128can estimate the information risk increment Δρ of uncertainty from linearization methods, for example, using Equation 3.

The risk estimation signal ρ can be generated by the risk estimator128and/or by the self-organizing controller102. In some embodiments, the risk estimation signal ρ can be generated at the direction of a human operator, a vehicle health monitoring system, a weather monitoring system, a high-level mission planner, the like, or any combination thereof. For example, such systems can provide information regarding the occurrence of certain risk conditions. Further, the risk estimation signal ρ can be generated based on monitoring. For instance, the risk estimation signal ρ can be generated based on a performance measure, for example, one of the performance measures in Table 1. As such, the actual performance of the controller106can be monitored in real-time. Detection of poor performance can cause the self-organizing controller102to take precautionary and/or emergency action (for example, relax performance requirements, use minimal control effort, the like, or any combination thereof). The risk estimation signal p can be based at least partly on Equations 1-3.

As illustrated inFIG. 1, the error signal e can be generated by a comparator132configured to subtracts measurements y from the reference signal r. The reference signal r can represent the desired value of the output y*. The reference signal r can be generated by an exogenous system, which can be referred to as “a reference generator” in the control literature. The exogenous system can include a human (for example, a pilot) and/or an outer control loop (for example, a guidance system). The error signal e can represent the quality of control, in which small values of the error signal e can represent a good quality of control.

The self-organizing controller102can include an intelligent quantum tuning unit104and a controller106. The intelligent quantum tuning unit104can generate robust controller gains KQI=[kQI,1. . . kQI,nC]T(where nCcan represent the number of tunable controller parameters of controller106) based the risk estimation signal ρ, the measurements y, the control signal u, the error signal e, or any combination thereof. The robust controller gains KQIcan tune the controller106.

The controller106can be implemented in a variety of ways.FIGS. 2A and 2Bare block diagrams illustrating examples in which the controller106is a proportional-integral-derivative (PID) type controller106a,106b.FIG. 2Ashows an example PID controller106a. A PID controller can process the error signal e and generate the control signal u based on the error signal e, an integral functional of the error signal e, a derivative functional of the error signal e, or any combination thereof. For example, the PID controller106acan generate the control signal u according to following the time-domain equation:

The controller106acan generate the control signal u according to the following Laplace domain equation:

In Equations 4 and 5, controller gain kPcan represent a proportional gain; controller gain kIcan represent an integral gain; controller gain kDcan represent a derivative gain; U(s) and E(s) can represent the Laplace transform of the time-domain signals u(t) and e(t), respectively; and C(s) can represent the transfer function of the controller. Equation 5 does not include initial conditions for ease of description.

The controller gains kP, kI, and kDcan each vary, for example, as a function of time, operation conditions, exogenous signals, or any combination thereof. For example, the controller gains kP, kI, and kDof the self-organizing controller102can vary according to the robust controller gains KQI. For example, the robust controller gains KQIcan be a vector signal comprising the signals kP, kI, and kD. The robust controller gains KQIcan be represented by the following equation:
KQI(t)=[kP(t)kI(t)kD(t)]T(Equation 6)

Due to practical considerations, in some embodiments the PID controller106amay not be implemented with pure integration or derivative action. For example, the integrator may be modified to include anti-windup, saturation, leakage, or any combination thereof. In some embodiments the derivative operator can be implemented with a “dirty derivative”, such as a realizable filter configured to approximate a derivative operator for certain operation conditions (for example, over a particular frequency range).

Some implementations of the controller106of the self-organizing controller102include a filter structure based on integer powers of the Laplace independent variable s (or z-transform independent variable z for discrete-time filters), although other implementations do not include such a filter structure. For instance,FIG. 2Bshows an example fractional-PID controller106bthat utilizes fractional powers of s. The fractional-PID control106bcan generate the control signal u based on the error signal e according to the following equation:

In Equation 7, α and β can have non-integer values. In one example, α=0.5 and β=0.5. Robust controller gains KQIcan be a vector signal comprising the signals kP, kI, and kD, for example, as described earlier in Equation 6.

The controller102of the self-organizing controller can include a nonlinear controller. For example,FIG. 2Ddepicts the controller106ofFIG. 1as sliding mode controller106c. The sliding mode controller106ccan include a sliding surface evaluator202and a fuzzy sliding mode controller204.FIG. 2Cshows an example sliding mode surface S=0 and an example set of fuzzy membership functions of the fuzzy sliding mode controller204. In this example the surface signal can be represented by the following equation:
S(e,ė)=ae+ė(Equation 8)

In Equation 8, a>0 can be a design constant. On the sliding surface S=0, the error signal e(t) can evolve according to the dynamics ė=−ae. Therefore, the error signal e can converge exponentially to zero, as desired, where the constant a controls the rate of convergence.

The design problem then becomes driving the error signal e and its derivative ė onto the sliding surface S=0. This can be accomplished by including a nonlinear component in the control law that has a discontinuity at S=0 so that the vector fields of the half spaces S>0 and S<0 both point towards S=0 in a way that causes the sliding surface to be reached in finite time and a sliding mode to occur along S=0. For example, the control law can include a nonlinear component of the form −kSMsgn(S), where sgn(·) represents the signum function.

As illustrated inFIG. 2C, the nonlinear control component can be calculated using fuzzy membership functions NB, NM, NS, ZE, PS, PM, and PB. One advantage of this approach can be that the chattering phenomenon associated with a discontinuous control signal can be avoided. Chattering can lead to instabilities by exciting high-frequency unmodeled dynamics. And as a practical concern, chattering can lead to excessive actuator wear.

Another advantage of using fuzzy membership functions can be that increasing gains −kSMsgn(S) can be used as error signal e increases beyond specified limits (for example, the NB and PB regions shown inFIG. 2C). One way that this is beneficial is that if the error signal e escapes past predefined limits it can be momentarily pushed with a greater magnitude towards a region within the limits, thereby preventing the error signal e from growing even larger. This can be advantageous for an application in which the control object's states should be kept within a prescribed neighborhood of a desired region. For example, when controlling a vehicle it may be desirable to ensure that error signal e does not exceed some distance; otherwise the vehicle (i.e., the control object) may crash into an obstacle.

Utilizing this sliding mode behavior, a sliding mode controller106cis shown inFIG. 2Das an example implementation of the controller106inFIG. 1. In certain implementations, a sliding surface evaluator202can generate a surface signal S as a function of the error signal e and provide the surface signal S to an input of the fuzzy sliding mode controller204. In some implementations, the sliding surface evaluator202can generate a surface signal S as a function of the derivate of the error signal e. Based on the value of the surface signal S, the fuzzy sliding mode control204can generate the control signal u to drive the error signal e and its derivatives onto the sliding surface S=0 in a finite time, and then maintain the error signal e and its derivatives on the sliding surface S=0. The fuzzy sliding mode controller204can include controller gains that are generated by fuzzy membership functions.

There are a number of types of controllers. For example, the controller106can be a classical linear controller, such as a proportional-integral-derivative (PID) controller, a lead-lag compensator, or the like. Moreover, the controller106can be a modern linear controller, such as an optimal linear quadratic Gaussian controller, a multivariable robust controller (for example, H∞or mixed-μ controllers), or the like. Further, the controller106can be a nonlinear controller, such a controller based on feedback linearization, dynamic inversion, sliding mode, nonlinear damping, adaptation (for example, online parameter estimation), or the like. In addition to being one of above-mentioned types of controllers, the controller106can include a combination of two or more types of controller types. For instance, a nonlinear sliding mode controller can include a linear PID-controller component.

Each of these types of controllers can include a number of control parameters that determine the controller's input-output relationship. For example, the control parameters of a PID controller can include a proportional gain kP, integral gain kI, and derivative gain kD. Thus, PID controller design can be viewed as the selection of those three gains such that the control objectives are satisfied.

FIG. 3Adepicts an illustrative block diagram of the intelligent quantum tuning unit104afor tuning the parameters of a controller. For instance, the intelligent quantum tuning104ais an example of the intelligent quantum tuning unit104ofFIG. 1configured to tune the parameters of the controller106. The intelligent quantum tuning unit104acan include a robust knowledge base302and a quantum inference unit304. The robust knowledge base302can receive the risk estimation signal ρ and/or the error signal e and generate n sets of controller gains K1-Kn, in which n is a positive integer. Each of the n sets of controller gains K1-Kncan correspond to controller gains that can achieve the control objectives for a certain realization of the uncertainty. The n sets of controller gains can be provided to the quantum inference unit304. The quantum inference unit can be configured to generate the robust controller gain KQIto tune the controller106.

The robust knowledge base302can include n knowledge bases306a-306nto generate the n sets of controller gains K1-Kn. The input to each of the n knowledge bases306a-306ncan include the risk estimation signal ρ and/or the error signal e. The output of the ithknowledge base includes the ithset of controller gains Ki.

Knowledge bases306a-306ncan generate suitable gains for a certain realization of the uncertainty given an online error signal (based at least partly on measurements y). In some embodiments, one or more of the knowledge bases306a-306ncan be developed using fuzzy, neural, fuzzy-neural, or the like techniques. However, knowledge bases306a-306ncan be developed using any suitable control design methodology to produce reliable sets of controller gains. For example, n sets of gains can be developed using traditional gain-scheduling techniques based on n models of the control object108.

With continued reference toFIG. 3A, the knowledge bases306a-306ncan include information risk production rules310a-310n, respectively. The information risk production rules310a-310ncan represent how the risk estimation signal p affects the computations of controller gains K1, K2, . . . , Kn. For example, increased levels of information risk can adjust the controller gains Kiin order to avoid instabilities. Similarly, in some embodiments decreased levels of information risk can adjust the controller gains Kiin order to avoid instabilities.

The quantum inference unit304can receive the n sets of controller gains K1-Knfor the knowledge base302and generate the robust controller gain KQI. The intelligent quantum tuning unit104can tune the controller106based on the robust controller gain KQI. More details of the quantum inference unit304are provided below, for example, in connection withFIGS. 4-10.

FIG. 3Bdepicts an illustrative block diagram of the intelligent quantum tuning unit ofFIG. 3Ain combination with an optimizer312. Otherwise like elements inFIG. 3Bcan be substantially the same as like elements inFIG. 3A. The optimizer312can receive the risk estimation signal ρ, the measurements y, and the control signal u. The optimizer312can output one or more signals that can create and/or tune elements of the robust knowledge base302and/or the quantum inference unit304. For example, in some embodiments, the optimizer312can determine the number of knowledge bases and/or the number and/or shapes of the membership functions implemented by each of the n knowledge bases306a-306n. In some embodiments the optimizer may also determine certain parameters of the quantum inference unit304, such as a correlation type, for example, as discussed later in more detail in connection withFIG. 9.

In some embodiments, the optimizer312is used offline. For example, the optimizer312can be a design tool used to develop the self-organizing controller102. The optimizer312can include soft and/or quantum computation based optimizers. Further, in some embodiments, the optimizer312can be used online to adjust the intelligent quantum tuning unit104. This mode of operation can be useful to optimize controller performance, for example, in the presence of drastic failures. For instance, it may be desirable to adjust the robust knowledge base302and/or the quantum inference unit304in response to a sensor failure. Accordingly, in some embodiments, the risk estimation signal p and/or other exogenous signals can cause the optimizer312to perform one or more optimization routines. This mode of operation can be advantageous in situations when it would be infeasible to run the optimizer312continuously or at set periods.

In some embodiments, it is difficult and/or impractical to use the optimizer312with the actual control object108. Thus, in some embodiments the control object108used during the design and/or tuning phase is an experimental and/or simulation model of the actual control object. For example, the self-organizing controller102can be designed with software by using MATLAB/Simulink® models of the dynamics of the actual control object, and then the results of the optimizer312are used to program a processor capable of controlling the actual control object.

FIG. 4is an example block diagram of the quantum inference unit304for processing the n sets of controller gains K1-Knand generating the robust controller gain KQI. The quantum inference unit304can include a normalizer402, a quantum encoder404, a quantum correlator406, a quantum optimizer410(which can select a correlation type), and a decoder412, or any combination thereof.

The normalizer402can provide the normalized control gains418to the quantum encoder404. The quantum encoder can encode the n sets of controller gains into corresponding quantum states420. The quantum encoder404can include a probability database416configured to encode normalized controller gains418to quantum states420. The quantum states420can be provided to the quantum correlator406. The quantum correlator406can generate a superposition signal422based on correlations of selected states. Correlation and selection can be based on a correlation type signal426. The superposition signal422can be provided to the quantum optimizer410. The quantum optimizer410can be a quantum genetic optimizer that is configured to execute one or more quantum genetic search algorithms. The quantum optimizer410can generate one or more robust quantum states424that relate to at least one of the combinations in the superposition signal422. As shown inFIG. 4, the quantum optimizer410can also generate the correlation selection signal426, which can be used by the quantum correlator406to select the correlation type. The robust quantum states424can be provided to the decoder412. The decoder412can decode the quantum states424to generate the robust controller gain KQthat can be used to tune the controller106to control the control object108.

FIG. 5is an illustrative flow diagram of a process500for generating the robust controller gains KQI. The robust controller gains KQIcan be the gains generated by the intelligent quantum tuning unit104for tuning the controller106. At block502, n sets of controller gains are obtained. Each set of the n controller gains Ki(i=1, . . . , n) can satisfy the control objectives for a range of realizations of the uncertainty. The n sets of controller gains can be obtained, for example, from a robust knowledge base302, a data store, a separate device, or any combination thereof.

At block504, the n sets of controller gains can be normalized such that each gain is within a certain bound. For example, the normalizer402(FIG. 4) can receive n sets of controller gains K1, K2, . . . , Knand can output normalized controller gains418that are bounded by a specified range. In particular, some embodiments can normalize the n sets of controller gains K1, K2, . . . , Knsuch that each controller gain Kitakes on a value on the closed and bounded interval [0, 1]. For instance, the output of the normalizer410can be represented by the following equation:

In Equation 9,Kcan represent normalized controller gains, Kmaxi(i=1, . . . , n) can represent a maximum value of the Ki. The divisor Kmaxican be obtained by simulation results, experimental results, by other prior knowledge such as physical limitations, or any combination thereof. In some embodiments, for example, as shown inFIG. 4, the values for Kmaxi(i=1, . . . , n) can be stored in a max-K database414. In some embodiments, the maximum value of Kican be calculated by querying the robust knowledge base302and/or be obtained from another device. Normalization can improve quantum encoding. However, in some embodiments, normalization can be omitted.

At block506, the normalized gains418can be encoded into a set of quantum states, in which each quantum state corresponds to a normalized controller gain. For example, the quantum encoder404(FIG. 4) can receive the normalized gains418and generate a set of quantum states420. Encoding conventional controller gains to quantum states can allow for extraction and/or compression of information contained in the aggregate of the n sets of controller gains. One example encoding process506will be described in more detail in connection withFIG. 6.

At block508, a superposition of quantum states can be generated by selectively correlating individual quantum states. For example, the quantum correlator406(FIG. 4) can obtain the quantum states420. The quantum correlator406can generate the superposition signal422, for example, as described in connection withFIG. 6. The superposition signal422can be indicative of one or more correlations of selected quantum states. The selected quantum states can be an encoded quantum state corresponding to a controller gain correlated with one or more encoded quantum states corresponding to other related controller gains (for example, a proportional gain can be correlated with encoded quantum states of an integral gain and/or other gains), past/future values of the same controller gains, or any combination thereof. The type of correlation performed on the selected quantum states can be selected offline and/or online.

At block510, certain quantum states of the superposition signal can be selected to form the robust quantum state based on selection criteria. The robust quantum state can correspond to the quantum state of the robust controller gains KQIused to tune the controller106. For example, the quantum optimizer410(FIG. 4) can be configured to receive the superposition signal422and generate the robust quantum states424. The quantum optimizer410can select the robust quantum states by executing quantum algorithms, quantum genetic algorithms, the like, or any combination thereof. The robust quantum states can be optimal quantum states. More detail regarding selecting the robust quantum states and executing these algorithms will be described in connection withFIGS. 8-10.

At block512, the robust intelligent quantum state is decoded for generating the robust controller gain KQIused in tuning the controller106. For example, the decoder412(FIG. 4) can receive the robust quantum states424and generate the robust controller gains KQIby decoding the robust quantum state.

At block514, a controller can be tuned based on the decoded robust controller gain KQI. For example, as illustrated inFIG. 1, the intelligent quantum tuning unit104can provide the controller106with the robust controller gains KQI.FIG. 1shows that the signal line of KQI“pierces” the block of the controller106. The piercing signal symbol can represent that the controller106is tuned based on the robust controller gains KQI. A piercing signal, such as the robust controller gains KQIsignal line (FIG. 1), can represent the piercing signal tunes or adjusts certain parameters of the system that it pierces, such as controller106(FIG. 1).

FIG. 6is an example flow diagram for a process506of encoding controller gains into corresponding quantum states. The process506can be performed at block506of the process500in some implementations. The quantum states can include quantum superpositions of real states |0and virtual states |1i,j, where i=1, . . . , n; j=1, . . . , nc; and nccan be the number of controller parameters corresponding to n controller. The generated quantum states can be considered building blocks for a self-assembling algorithm. The real state |0can correspond to a controller gain generated by a knowledge base, and the virtual state |1can correspond to a virtual controller gain that can subsequently be determined based on another knowledge base.

At block602, controller gains corresponding to the real controller gains are obtained. For example, the quantum encoder404(FIG. 4) can receive normalized control gains represented by Equation 9, in whichKi=[k1,j. . .knc,j]T(i=1, . . . , n) can represent a nc-vector;ki,j(j=1, . . . , nc) can represent a scalar representing the jth normalized controller parameter of knowledge base i; and nccan represent the number of controller parameters. For instance, for the PID controller106a(FIG. 2A), the normalized controller gains can take on the formKi=[kP,ikI,ikD,i]Twith nc=3.

At block604, the real controller gains can be associated with their real quantum states |0i,j. For example, the quantum encoder404(FIG. 4) can associate a real quantum state |0i,jwith each normalized controller gainki,j. At block606, a probability pi,j0can be associated with each real state |0i,jbased on the value of the real controller gain. For example, the probability database416(FIG. 4) can provide the probability pi,j0=Pi,j{|0i,j}, where Pi,j{|0i,j} can represent the probability of real state |0i,j. The probability database416can be queried for such probabilities. In some embodiments, the probability database416can provide the probability pi,j0by evaluating a function that approximates the function Pi,j{|0i,j}. Probability databases can be generated based on offline simulation results. For example, simulations can provide data to generate histograms on the number occurrences of certain values of the controller gains. This data can in turn be used to generate estimates of the probability of certain gain values.

At block608, the probability pi,j1of the virtual state |1i,jcan be calculated based on the probability pi,j0of the real state. The virtual state |1i,jcan be a state corresponding to a virtual gainki,j1that can subsequently be extracted from the knowledge base. For example, the quantum encoder404(FIG. 4) can calculate the probability pi,j1based on the relationship pi,j0+pi,j1=1 (for example, by computing pi,j1=1−pi,j0). At block610, a virtual controller gainki,j1corresponding to virtual state |1i,jcan be determined. For example, the quantum encoder404can determine the virtual gainki,j1based on the following equation:
ki,j1=Pi,j−1{pi,j0}  (Equation 10)

In Equation 10, Pi,j−1can represent the inverse map of the probability Pi,j. In some embodiments, the probability Pi,jand/or the inverse map Pi,j−1can be calculated offline and stored in the probability database416(FIG. 4). In some embodiments, the virtual gainski,j1can be determined by evaluating a function that approximates the inverse map Pi,j−1.

At block612, a quantum state |ki,jcan be generated based the real state |0i,jand the virtual state |1i,jand their corresponding probabilities. For example, the quantum encoder404(FIG. 4) can generate the quantum state based on the following equation:
|ki,j=ai,j0|0+ai,j1|1(Equation 11)

In Equation 11, ai,j0can represent the square root of the probability associated with a real state pi,j0(i.e. ai,j0=√{square root over (pi,j0)}) and ai,j1can represent the square root of the probability associated with a virtual state pi,j1(i.e., ai,j1=√{square root over (pi,j1)}).

After quantum encoding, the quantum state signal can be processed to generate the superposition signal for extracting knowledge from the dynamic behavior of controller and/or the control object. Applying quantum operations can extract additional quantum information hidden in correlation classical states of control laws. As a result, the n knowledge bases can be unified as a robust knowledge base302.

Hidden information can be extracted by applying quantum correlations to controller responses. These quantum correlations can also reduce extra information.FIGS. 7A-7Dillustrate various ways of performing quantum correlations. The type and form of quantum correlation can coordinate the controller of gains of the n knowledge bases.

FIG. 7Aillustrates an example of spatial correlation, in which quantum states of different controller parameters are correlated.FIG. 7Ashows that at time tiquantum states |kPand |kDcan be correlated. The corresponding spatially correlated quantum state can be given by the following equation:
|kPkD=|kP{circle around (x)}|kD(Equation 12)

In Equation 12, the {circle around (x)} operator can represent the tensor product operator.

In contrast,FIG. 7Billustrates an example of temporal correlation, in which the quantum states of a controller parameter at different points in time are correlated.FIG. 7Billustrates that quantum states of |kPcan be correlated at different times tiand tj. For example, the temporal correlation can be represented by the following equation:
|kP(ti)kP(tj)=|kP(ti){circle around (x)}|kP(tj)(Equation 13)

Correlations can include a combination of the spatial and temporal correlations. For example,FIG. 7Cillustrates an example of spatial-temporal correlation.FIG. 7Cshows that a quantum state |kPat time tican be correlated with another quantum state |kDat a different time tj. The resulting correlation can be represented by the following equation:
|kP(ti)kD(tj)=|kP(ti){circle around (x)}|kD(tj)(Equation 14)

Quantum correlations can include internal correlations and/or external correlations. Internal and/or external correlations can be performed in connection with temporal correlations, spatial correlations spatial-temporal correlations, or any combination thereof.FIG. 7Dillustrates that quantum correlations can include internal correlation and external correlation. In an internal correlation, quantum states from the same knowledge base can be correlated. In contrast, an external correlation can involve correlating quantum states from different knowledge bases. For example, as shown inFIG. 7D, a correlated quantum state |kPcan be represented by the following equation:
|kP=|kP1kD1kP2kD2(Equation 15)

In Equation 15, the internal correlated components of |kPare |kP1kD1and |kP2kD2, and the external correlated components of |kPare |kP1kD1and |kP2kD2.FIG. 7Dalso shows an example internal and external correlation for correlated quantum states |kDand k1. In general, each robust controller gain kQI,iof the robust controller gains KQIcan have an associated correlated quantum state |ai,1. . . ai,ns, where each associated quantum state |ai,jcan correspond to a quantum state generated by the quantum encoder404(for example, via the process506ofFIG. 6) and nSis the number of selected quantum states for quantum correlation.

The information obtained by measurements y (FIG. 1) can be a one-sided information exchange. In particular, the amount of available extracted information can increase. In this case, states with such a property are not necessarily entangled, and the corresponding channel of information transmission can be realized using a Hadamard transform. Therefore, by using a Hadamard transform (and quantum correlation as a physical carrier of message transmission between a finite number of designed knowledge bases), it is possible to include larger amount of information in an initial quantum state by accounting for existing hidden states in a classical correlation. These effects can be taken into account, for example, during quantum correlation of control gains.

This approach makes it possible to understand more completely the solution to the problem of determining the role and influence of quantum effects on increasing the robustness level of the designed control systems. The design of knowledge base quantum self organization can be performed in two parts using a soft computing optimizer and a quantum computing optimizer. In the first part, a finite set of knowledge bases for given control situations can be developed offline based on optimizing knowledge bases with soft computing. In the second part, a quantum inference model can be applied online to realize self-organization of the responses of the n knowledge bases, which can result in a robust knowledge base302. In design using the quantum inference unit304, it may not be necessary to form new production rules to handle new types of uncertainties. For instance, it can be sufficient to obtain real-time reactions of production rules of the n knowledge bases. Advantageously, the quantum inference unit304may not require knowledge of which particular production rules are activated.

The choice of the correlation type and its corresponding form can affect the performance of the self-organizing controller. The selection of the correlation type and form is discussed below in connection with the quantum optimizer410.

FIG. 8is a block diagram of an example quantum optimizer410configured to select the quantum correlation type and select the robust controller gain KQI. The quantum optimizer410can include a quantum search unit802that is in communication with a quantum counting unit810and a genetic algorithm unit814. The quantum search unit802can receive the superposition signal422and select the robust quantum states424associated with the robust controller gains KQI. In some embodiments, the quantum search unit802can generate the correlation selection signal426for choosing the correlation type of the correlation performed by the quantum correlator406. In some embodiments, quantum optimizer410can perform any combination of the operations described herein in connection with the optimizer310.

One example of how the robust quantum states424are selected from the superposition signal will be described for illustrative purposes. Two knowledge bases can generate the controller gains K1=[k1,P, k1,I, k1,D]Tand K2=[k2,Pk2,Ik2,D]T. The correlated quantum state |kPof the superposition signal can correspond to the robust controller gains kQI,Pas represented by the following equation:

In Equation 16, the following relationships can hold: α1=a1,P0a2,P0, α1=a1,P0a2,P1, α1=a1,P0a2,P0, and α1=a1,P1a2,P0. As described earlier, ai,P0=√{square root over (pi,P0)} and can represent the corresponding square roots of the probabilities of the real controller state |0i,Pand the virtual controller state |0i,P. The quantum optimizer410can select the correlated state with the maximum probability amplitude αi*. Then the robust quantum states can have the form αi*|b1b2. The selection can be made using quantum optimizing and/or quantum search algorithms, such as quantum genetic optimization and/or quantum genetic search algorithms. The correlated state with the maximum probability amplitude αi* can be referred to as the intelligent state.

The mathematical structure of the quantum optimizer410can be described as a set of genetic and quantum operations:
QGO={C,Ev,P0,L,Ω,χ,μ,Sup,Ent,Int,Λ}(Equation 17)

In Equation 17, QGO can represent a quantum genetic optimizer; C can represent a genetic coding scheme of individuals for a given problem; Eν can represent a genetic evaluation function to compute the fitness values of the individuals; P0can represent an initial population; L can represent a size of the population; Ω can represent a genetic selection operator; χ can represent a genetic crossover operator; μ can represent a genetic mutation operator; Sup can represent a quantum linear superposition operator; Ent can represent a quantum entanglement operator (for example, quantum super-correlation); Int can represent a quantum interference operator. The operator Λ can represents termination conditions, which can include the stopping criteria as a minimum of Shannon/von Neumann entropy, the optimum of the fitness functions, the minimum risk, or any combination thereof. Accordingly, a quantum genetic optimizer can be configured to execute the three genetic algorithm operations of selection-reproduction, crossover, and mutation, and the three quantum search algorithm operations of superposition, entanglement and interference. In some implementations, a quantum genetic optimizer can include any subset of a set that includes the three genetic algorithm operations and the three quantum search algorithm operations.

The quantum search unit802can determine the intelligent quantum states from which the robust controller gain KQIcan be decoded. The design process of quantum search unit802can include designing a matrix of the three quantum operators: superposition (Sup), entanglement (Ent) and interference (Int). To facilitate the performance of these quantum operators, the quantum search unit802can include a superposition unit804, an entanglement unit806, and an interference unit808. Based on these operations, the structure of a quantum search unit802can implement operations execute the following equation:
|ψfin={[(Dn{circle around (x)}I)·Uf]h·(n+1H)}·{(Uf·P·Uf)·Pr·(Uf·P·Uf)}H|ψ0(Equation 18)

In Equation 18, |ψ0can represent an initial quantum state; |ψfincan represent a final quantum state; H can represent a Hadamard transform; I can represent an identity operator; Ufcan represent an entanglement (quantum correlation) matrix; P can represent a permutation operator; Pr can represent a projection operator; and Dncan represent a diffusion operator. Although, a quantum search unit802that is configured to implement the operations of Equation 18 can be based on the Grover quantum search algorithm, the quantum search unit802can be implemented using various other quantum algorithms (QA), such as the Deutsch QA, the Deutsch-Jozsa QA, the Simon QA, the Shoes QA, or any suitable control quantum search algorithm. As discussed above, one portion of the design process is the choice of the entanglement (quantum correlation) operator Uf.

Some of the operations performed by the quantum search unit802can alternatively or additionally be performed by other components of the quantum inference unit304. For example, the quantum correlator406can generate a superposition and correlation (i.e., entanglement) of selected quantum states. Accordingly, in some embodiments, at least a portion of some of the quantum operation units804-808can be distributed, shared, or duplicated across components of the intelligent quantum tuning unit104, including the optimizer unit312inFIG. 3B.

In some embodiments, a number of marked solutions is unknown. In this case, the quantum optimizer410can use a quantum counting unit810to determine the number of marked solutions in order to determine the robust controller gains KQI. The quantum counting unit810can be coupled with the quantum search unit802by line812. One surprising result of quantum mechanics is that we can find what we do not know. For example, Grover's algorithm can be used to search a phone book with N entries to find the name corresponding to a given phone number in O(√{square root over (N)}) operations, in contrast to the best known classical algorithms that require O(N) operations. A classical search algorithm would need N/2 database queries on average and in the worst case it would N−1 queries, which are both more than Grover's algorithm.

In the case of a number of marked elements M in an unsorted database of size N, the number of operations of a quantum search process increase as O(√{square root over (N/M)}). For instance, Boyer et al. have extended Grover's quantum search algorithm to the case when M>1. M. Boyer, G. Brassard, P. Hoyer, and A. Tapp, Tight bounds on quantum searching, Fortsch. Phys. Vol. 46, 1998, pp. 493-506. The simplest case is if M is known. We can run the algorithm with ┌π/4√{square root over (N/M)}┐ iterations instead of ┌π/4√{square root over (N)}┐.

One item in search space can be considered marked if there is a corresponding oracle which has the ability to identify a solution to the search problem when the oracle sees a solution. Generally, there can be two registers. The first register can store an index x to an element in a search space, and the second register can represent a single state z. With s representing the marked item, then the oracle can have the mapping |x|z|x|z⊕δzx. Thus, the oracle can recognize solutions to the search problem, in the sense that it can flip the second register when the oracle finds the solution to the problem in the first register. Accordingly, the oracle can recognize the solution when the oracle processes the solution.

When the first register is prepared in the state |xand the second register in the superposition |0−|1, then the effect of the oracle can be represented by the following equation:
|x(|0−|1)(−1)δzx|x(|0−|1)  (Equation 19)

The state of the second register is not changed by the operation of Equation 19. Accordingly, the state of the second register can be ignored, and the action of the oracle can be simplified to |x(−1)δzx|x.

A more difficult case is when a number of marked elements M is not known in advance. One problem is that if too many iterations of the quantum search algorithm are executed, then an erroneous answer may result. This problem can be handled by two different approaches. The first approach runs the quantum search unit802several times with different numbers of steps. The second approach executes the quantum counting algorithm on the quantum counting unit810to estimate the number of marked elements M, and then chooses the number of operations for the search algorithm on the quantum search unit based on the estimated number of marked elements M.

The genetic algorithm unit814can be in communication with the quantum search unit802via line816. The genetic algorithm unit814can determine parameters of quantum search unit802(for example, the correlation type of the entanglement unit808), in order to determine selected (for example, optimal) robust controller gains KQI. The genetic algorithm unit814can include a mutation unit818, a crossover unit820, a selection unit822, or any combination thereof to support the genetic operations of a genetic algorithm. The genetic algorithm unit814can also include a fitness function unit824to evaluate the fitness of the resulting populations (solutions). Example fitness functions can include criteria of control quality from Table 1. The genetic algorithm unit814can include a coding unit826for encoding/decoding tunable parameter, for example, the correlation types, superposition, and interference operators and other parameters the quantum search unit802; parameters and gains of the controller106; parameter and gains of the robust knowledge base302, the like, or any combination thereof. The coding unit826can encode tunable parameters from a binary representation to a quantum representation. Alternatively or additionally, the coding unit826can decode tunable parameters from a quantum representation to a binary representation. The genetic algorithm unit814can also include a simulator828configured to estimate the fitness of certain solutions without probing the actual control object108.

FIG. 9is an illustrative flow diagram of a process900for selecting the correlation type of the quantum correlator. At block902, the correlation types can be encoded. The genetic algorithm unit814can, for example, access the coding unit826to acquire a set of candidate correlation types and their corresponding codes. An initial correlation type and code can be chosen from the set of candidate correlation types and their corresponding codes, at block904.

At blocks906and908, the intelligent states of a closed-loop simulation can be initialized and the closed-loop system can be simulated. The intelligent states can include the robust control gains KQIfor a simulation model of the closed-loop system100. Simulations, rather than real experiments, can be convenient for running a large number of experiments. The simulator828of the genetic algorithm unit814can perform a simulation. The genetic algorithm unit814can utilize measurements y from the control object108(for example, real-time experiments and/or closed-loop system identification) as an alternative or in addition to simulation.

At block912, the results of the simulation can be evaluated to determine whether the fitness requirement is satisfied. In some embodiments, the genetic algorithm unit814can provide the fitness function unit824the results from closed loop simulation generated at block908to evaluate the correlation-type selection. The fitness requirements of an algorithm executed by the genetic algorithm unit814(for example, evaluated on the fitness function unit824) can be described by the following equation:

In Equation 20, {right arrow over (X)} can represent a vector of control object output parameters, and X0can represent restrictions on control object output parameters such that a control force and generalized entropy are minimum. As illustrative examples, Table 1 provides performance measures that can be used as part of the fitness requirements.

At block912, when the fitness requirements are satisfied, the correlation type corresponding to the closed loop simulation executed at block908can be selected as the selected correlation type. Otherwise, a different correlation type can be chosen at block914. Different intelligent states can then be searched for corresponding to the different correlation type at block916. The genetic algorithm unit814can choose a new correlation type based on the genetic operators implemented by the mutation unit818, the crossover unit820, the selection unit822, or any combination thereof. The different intelligent states can be searched for, for example, based on the quantum operators implemented by the quantum search unit802, for example, superposition, entanglement, interference, or any combination thereof. The process900can iterate through blocks908-916until the fitness requirements are satisfied at block912. When the fitness requirements are satisfied, the selected correlation type can be returned at block918.

FIG. 10is an illustrative flow diagram of a process916corresponding to a quantum search algorithm that can be used to search for intelligent state at block916of the process900. The process916can include information risk termination conditions. The process916can include initializing intelligent states at block1002. In some implementations, the initial intelligent states can be provided externally. An intelligent quantum state |xis one possible approach for selecting a quantum state based on optimal criteria from superposition of possible coding states for controller coefficient gains. In this context, an “intelligent quantum state” can represent the quantum state with minimal uncertainty, for example, the minimum uncertainty in a Heisenberg inequality. The intelligent quantum state can be correlated with solutions of quantum wave equations (for example, Schrödinger-like equations) when a wave function of a quantum system corresponds to a coherent state and the uncertainty has a global minimum. The intelligent quantum state can be generated, for example, based on the difference between von Neumann entropy and Shannon information entropy in this quantum state. Accordingly, an intelligent quantum state can be the minimum of the difference between Shannon information entropy and physical entropy von Neumann on this quantum state:
I(|Quantum state)=min(HSh−SνN)  (Equation 21)

In Equation 21, HShcan represent Shannon entropy and SνNcan represent von Neumann entropy. Thus the intelligent quantum state described in Equation 21 corresponds to a minimum of Shannon information entropy of quantum state. This minimum can correspond to a maximum probability state (according to definition of Shannon information

entropy⁢⁢HSh=-∑i⁢⁢pi⁢ln⁢⁢pi,
global minimum observed for maximum probability pi). The maximum amplitude of probability corresponding to a correlated state can be used to select the intelligent correlated (coherent) quantum state in the superposition of possible candidate quantum states. For instance, the maximum probability amplitude can correspond to the absolute value of the probability to the second power (p=|ψ|2).

Therefore, a quantum oracle model can be realized by calculating amplitude probabilities corresponding to a superposition state with mixed types of quantum correlations and a selection state with maximum amplitude probability. Accordingly, the quantum oracle can have necessary information to identify a solution. For example, in Equation 16 the intelligent state between the classical states (|00, |01, |10, |11) can correspond to maximal probability amplitude αi, i=1, 2, 3, 4 from the probability amplitude set (α1, α2, α3, α4).

Otherwise, the intelligent states |xcan be initialized according to the following equation:

In Equation 22, each xican represent a randomly (or pseudo randomly) generated real number; each cican represent a randomly (or pseudo randomly) generated complex number, such that

∑i=1t⁢⁢ci2=1;
and Int1can represent a randomly (or pseudo randomly) generated unitary operator of order t.

At block1004, whether the information risk termination condition is satisfied can be determined. For example, the quantum search unit802can generate a matrix described by the following equation:

In Equation 23, each |εi,j|≦ε can be randomly (or pseudo randomly) generated such that xi,j+εi,j≠xk,j+εk,jif i1≠i1. Further, the quantum search unit802can generate a matrix described by the following equation:

In Equation 24, each Intjcan represent a unitary squared matrix of order t. The interference operator Int1can be chosen as a random unitary squared matrix of order t, whereas the interference operators for the other paths can be generated from Int1. Non-limiting examples of such matrices include the Hadamard transformation matrix Htand the diffusion matrix Dt, described above. It will be understood that many other suitable matrices can be generated. The application of entanglement and interference operators as in Equation 18 can produce a new superposition of maximal length t. According to this sequence of operations, k different superpositions can be generated from the initial superposition using different entanglement and interference operators. Each time the average entropy value can be evaluated. Selection can include holding only the superposition with a minimum average entropy value. When the superposition corresponding to the minimum average entropy is obtained, this superposition can become the new input superposition and the process can start again. The interference operator that has generated the minimum entropy superposition can be hold and Int1can be set to this operator for the new operation. The computation can stop when the minimum average entropy value falls under a predetermined threshold and/or a critical limit. At this point measurement can be simulated, from which a basis value can be extracted from the final superposition according to a square modulus of its probability amplitude. In turn, from the matrixBthe quantum search unit can generate the information risk termination conditions based on the following equation:

The information risk increment Δρ can be calculated, for example, as described above. When the termination condition is satisfied, intelligent states can be selected at block1008. For example, if Ē*<E1and the information risk increment Δρ<ρintelligent states can be selected, for example, by extracting xi*+εi*,j* from {xi*+εi*,j*,∥c′i,j*∥2}. When the termination condition is not satisfied, new intelligent states can be generated based on quantum and/or genetic algorithms at block1006. Then the process916can return to block1004. The quantum search unit802can, for example, set |xand Int1based on a Grover quantum search algorithm (which can be executed at block1004to search for j*) and/or based on any suitable genetic algorithm. For example, the quantum search unit802can set the intelligent state as |x=|output*, Int1=Intj*.

Returning toFIG. 4, the decoder412can obtain the robust quantum states424and determine the robust controller gains KQIby decoding the robust quantum states424. The ithrobust quantum state can have the form αi*|bi,1. . . bi,nSin which nScan represent the number of selected states for correlation. The ithrobust quantum state can have the maximum probability of the quantum states. The ithrobust controller gain kQI,iof KQIcan be decoded according to the following equation:

In Equation 26, CPcan represent a scaling factor that can be used to denormalize and/or optimally scale the controller gains. The optimal scaling can be determined using soft computing design techniques (for example, genetic algorithms). The scaling factor CPcan be stored and retrieved from the scaling factor database418. In some embodiments, the scaling factor CPcan be the inverse of normalizing denominator in Equation 9. In some embodiments, the robust controller gains KQIare not based on the scaling factor CP.

An example Quantum PID controller design based on quantum inference with two sets of classical PID gains K1and K2, with the classical PID gains being constant, will be described. The general structure of the example Quantum PID controller can be implemented with and/or include any combination of features described with reference to the systems, apparatus and/or methods described herein, for example, the systems, apparatus and/or methods ofFIGS. 1,2A,3A,4-10.

For two teaching conditions, the two gains K1=[kP1kD1kI1]Tand K2=[kP2kD2kI2]Tcan be designed by using a PID tuning method based on a genetic algorithm. By using an artificial stochastic noise disturbance, the gains can be obtained according to the following equation:

In Equation 27, GP, GD, GIcan represent increasing and/or decreasing coefficients that can be chosen manually. The stochastic noise can aid in generating the simulation data to generate the probability database416used to determine the quantum states |0and |1by the quantum decoder404. The quantum inference unit304can perform one or more of the following operations.

The normalizer402and/or the quantum encoder403can prepare normalized states |0and |1for current values of disturbed control signals K1and K2including (a) calculation of probability amplitudes α0, α1of states |0and |1from histograms, and (b) using the α1calculation of normalized value of state |1.

Operation 2: Choosing Quantum Correlation Type for Preparation of an Entangled State.

The quantum correlator406can perform any of the correlations described herein. For example, the quantum correlator406can perform a spatial correlation according to the following equations:
e1e2kP1,2kD1,2→kPnew·gainP;
ė1ė2kD1,2kI1,2→kDnew·gainD;
Ie1Ie2kI1,2kP1,2→kInew·gainI(Equation 28)

In Equation 28, e, ė, Ie can represent control error, derivative and integral of control error and gainP(D,I)can represent scaling factors that can be obtained by a genetic algorithm. A quantum state |a1a2a3a4a5a6=|e1e2kP1(t)kD1(t)kP2(t)kP2(t)kD2(t)can be considered as an entangled state.

Operation 3: Superposition and Entanglement.

According to the selected correlation type, the quantum optimizer410can construct superposition of entangled states.

Operation 4: Interference and Measurement:

The quantum optimizer410can select a quantum state a1a2a3a4a5a6=|e1(t)e1(t)kP1(t)kD1(t)kP2(t)kD2(t)with maximum amplitude of probability A=√{square root over (Pe1)}√{square root over (Pe2)}√{square root over (PkP1)}√{square root over (PkD1)}√{square root over (PkP2)}√{square root over (PkD2)}. A subvector |kP1(t)kD1(t)kP2(t)kD2(t)can also be selected.

The decoder412can decode the selected quantum state. The decoder412can also calculate a normalized output as a norm of the subvector of the chosen quantum state according to the following equation:

The decoder412can calculate a final (denormalized) output according to the following equation:
kPoutput=kPnew(t)·gainP,kDoutput=kDnew(t)·gainD,kIoutput=kInew(t)·gainI.  (Equation 30)

In Equation 30, scaling gains {gainP, gainI, gainD} can be identified by executing genetic algorithm. The final output can be provided to the controller.

An example Quantum PID controller design based on quantum inference with two sets of classical PID gains K1and K2is shown inFIG. 11. Referring toFIG. 11, a block diagram depicting another illustrative control system1100including a self-organizing controller for controlling a control object will be described. In some applications, in which one or more PID controllers are used, the robustness of a control system, such as the control system1100, can be increased using an intelligent quantum tuning unit104that includes a quantum inference unit304. The intelligent quantum tuning unit104can include any combination of features of the intelligent quantum tuning units described herein. In some instances, a quantum PID (QPID)1100can be tuned online based on only two sets of PID controller gains K1and K2. The controller gains1130can be generated by two non-robust PIDs1120and provided to the quantum inference unit304of the intelligent quantum tuning unit104. The quantum inference unit304can tune the QPID1110based on the controller gains1130, for example, using any combination of features of the quantum inference units described herein. Simulation results indicate that the quantum inference unit304can tune the QPID1110such that the QPID1110shows good robustness properties. For instance, the simulation results indicate QPID pole dynamic robust behavior. Accordingly, from two non-robust PIDs1120, a robust QPID1110can be created. The QPID1110can have a synergistic effect on knowledge base quantum self-organization.

CONCLUSION