The Ethics of Nonlinearity
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This models light intensity and the perceived difference in weight. It was also proposed the logarithmic perception applies to more indirect senses, e. It seems plausible it would hold for the perception of quantities like wealth, if we measure the perception of wealth by asking people to rate their wealth on a scale from 0 to We suppose this may also hold for the abstract sense of happiness, as used in philosophy and utilitarian calculus.
Specifically, we propose the percieved happiness p H is related to the raw hapiness H as. It should be noted that while we chose happiness, the argument would be the same for related or similar quantities, such as well-being. It may seem such logarithmic rescaling is just an irrelevant change of scale.
However, when we aggregate a quantity over many people, there are significant differences between using the raw quantity and the perception.
Feedforward and Feedback Control Performance Assessment for Nonlinear Systems
We suggest the raw quantity is often the more useful when aggregating. This can be easily seen in case of physical quantities, like weight. If we want to calculate total weight carried by a group of people, or total illumination created by a group of celestial objects, we can not simply add the perceived weights or perceived intensities, but we must first recover the raw quantity of stimuli and only latter sum or integrate.
Same holds for averaging. As various some of the utilitarian normative ethical theories suggest we should attempt to maximize quantities like happiness or well-being, the difference between counting the raw happiness hedons or aggregating the perceptions leads to different results. While in the non-corrected happiness calculus, we would integrate over beings and time the percepts of happiness directly, in the exponentially corrected version the integral has the form.
We can demonstrate the difference on a famous philosophical problem, taken from population ethics. In its classical formulation, the repugnant conclusions is: Then, while the conclusion is still technically true in a sense, the paradox is resolved for all practical purposes by taking into account the resource demanded by such populations. As an illustrative comparison, we can imagine an open ended subjetive quality of life scale where 1 means life of no quality, life with happiness 1.
Most likely the resource cost of existence of such an immense population would be many times greater than of the original population, even if lives barely worth living are cheaper than high quality life. If the conjecture is true, a lot of effective altruist prioritization calculations related to present problems e. As it seems higher percieved levels of happiness are approximately logarithmically dependent on resources, it is not at first sight obvious the most effective interventions would be improving lives of the poorest of the poor, as is the case if happiness perceptions are summed as linear quantities.
Note that if is greater than or equal to , then. This implies that there is no prediction error since we exactly know the future value of the effect on the process of the measured disturbance. The minimum variance control law is found by minimizing the mean square error of the output: It follows from this formula that the minimum variance controller MVC set the manipulated variables to exactly cancel the predictions; that is, Then the process output under this control scheme can be denoted by As we have assumed that there is no cross correlation among the unmeasured and measured disturbances, the prediction errors and are independent and unrelated with controller parameters.
Abstract and Applied Analysis
The difference is that it is possible to adopt different controllers for obtaining same minimum variance. Second, a more general form of nonlinear feedforward and feedback control systems is considered: They are also nonlinear and can be represented by nonlinear ARMA model as Further, we assume that the output disturbance admits a representation of the form. Multiply both sides by and substitute for all values of , , in The unmeasured output disturbance is represented as According to the definition of conditional expectation, the -step ahead prediction is Now in the aforementioned equation, we know where is the joint distribution of.
Then the prediction error for the unmeasured output disturbance is In a same manner, the prediction error for the measured output disturbance is The process output can be written as If it is possible to find the control action at time such that then the resulting controller is the minimum variance controller.
It may not be possible to implement a minimum variance controller due to the various reasons.
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For instance, it may lead to excessive manipulated variable action and may not be robust to modeling errors. However, the output variance set by minimum variance provides a theoretical lower bound on the system output and can be used as a useful guide for controller assessment. The process output under minimum variance control is given by the sum of the individual error in predicting the effect of the disturbances: It should be pointed out that the terms and are very complicated functions, and they may not be expanded in convergent time series as that in linear systems.
Therefore, it is difficult to estimate the MVPLB from the closed-loop operation data of feedforward and feedback control system by using traditional linear regression method. But we can get a conclusion that the MVPLB does not depend on the manipulated variable and only related with the most recent past unmeasured disturbance driving force and past measured disturbance driving force.
Analysis of variance ANOVA methods are a class of statistical methods that are useful in process systems engineering. Its primary task is to decompose the variance of a response variable into contributions arising from the inputs and assess the magnitude and significance of each of their contributions. Historically, the ANOVA variance decomposition techniques were used to provide variance analysis for nonlinear systems with the multidisturbance sources [ 20 ].
Coping with the Bounds: Speculations on Nonlinearity in Military Affairs
For the output of a static system such as , the relative importance of the independent inputs can be quantified by the fractional variance, and this can be calculated using an ANOVA-like decomposition formula [ 21 ]: In the same way, if we partition the variable set into two groups: For the nonlinear feedforward and feedback control systems described by Figure 1 , we separate the disturbance entering the system after time 0, say , into two groups: The first group includes all the disturbances entering the system after time and the second group includes all the disturbances entering the system up to and including time and including time starting from the initial time.
Now, we are interested in determining the sensitivity of output variations of two vector series and. Since the future behavior of is possibly dependent on initial conditions due to the nonlinearity, the initial condition must be considered before using the ANOVA-like decomposition equation. Using the well-known variance decomposition theorem, the variance of can be decomposed into two terms: The first term in above equation is the fractional contribution to the variance of from the disturbance signal and the interaction between disturbance and the initial condition.
The second term is the fractional contribution to the output solely due to the uncertainties in the initial condition. Given the initial condition , conditional variance can be decomposed as where , , and. Consequently, a suitable performance index can be constructed by referring to Harris index: If the nonlinear model is stationary, then the distribution of can reach an equilibrium. For linear time series, this limiting distribution is independent of initial condition. But for a stationary nonlinear model, the limiting distribution may depend on the initial condition.
Therefore, the performance index will depend on the initial condition. If the distribution of does not depend on the initial conditions, the process is termed ergodic. In actual industry, the cases that processes are strongly nonergodic are more pathological than common cases. For an ergodic nonlinear system, in 34 will be zero for , and the variance decomposition can be expressed when as where , , and.
The performance index will turn into. Generally, we will approximate the infinite limit in above equation by some suitably large value. In Section 3 , we conclude that the MVPLB of nonlinear feedforward and feedback control systems is existent and only related with the most recent past unmeasured disturbance driving force and past measured disturbance driving force. Moreover, we have , so just is the minimum variance performance index of the nonlinear feedforward and feedback control systems. For the computation of the performance index, the principal task is to estimate the closed-loop model of nonlinear feedforward and feedback control system.
Firstly, the measured feedforward variable transfer function, given in 2 , must be estimated. Using the linear regression techniques and past values of. The model of measured disturbance can be estimated by is an estimate of the independent driving force for measured disturbance. If the process is controlled by a linear or nonlinear feedforward and feedback controller such as , then the output of closed-loop system can be written as According to the existing knowledge, any continuous can be arbitrarily well approximated by polynomial models.
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Therefore, expanding in above equation as a polynomial of degree gives the representation where and , , , and. Moreover, the output of closed-loop system can be written as a linear regression model: Then above equation can be written in the matrix form where In reality, as parameters , , and are unknown, we must consider the combined problem of structure selection and parameter estimation. To avoid losing significant terms which must be included in the final model, we are forced to consider the full model set at the beginning of the identification and then to select a subset from full model set and find the corresponding parameter.
The orthogonal least squares OLS method [ 22 ] can be used to determine the order and estimate the parameters of the model. Denote After a series of Householder transformations , have been successively applied to ; it is transformed to where , , and , and is the upper triangular matrix. Further denote Assume that the maximum of , , is achieved at.
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Then interchange the column of with the column. The procedure is terminated at stage when where is a desired tolerance. Using backward substitution, the subset model parameter estimate is computed from In addition, since the terms of unmeasured disturbance driving force are generally unmeasured, the identification will require an iterative approach.
The identification procedures can be clarified as follows. Set the initial sequence by fitting a linear model or setting the to zero, and set iteration number. Identify the nonlinear model and get the prediction errors or residuals ,.
If certain identification criteria are achieved, then the program jumps to Step 6. Otherwise, Step 4 is run. Replace the initial sequence by the prediction errors or residuals.
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Set iteration number and return to Step 2. Once the parameters of the closed-loop model are estimated, Monte Carlo MC method may be used to compute the variance decomposition. Firstly, two random vectors, and , are generated, which are two sets of simulation of multidimensional inputs that have the requisite distribution.
Then, the mean and variance of given the initial condition can be calculated by.
While in presence it does not seem feasible to test whether the raw happiness is more fundamental than the perception, it at least seems possible to observe if people's preferences are broadly consistent with the view. The minimum variance control law is found by minimizing the mean square error of the output: It is necessary to identify the model of closed-loop system to estimate the minimum variance performance index of the nonlinear system. Sign in Create an account. Then the prediction error for the unmeasured output disturbance is In a same manner, the prediction error for the measured output disturbance is The process output can be written as If it is possible to find the control action at time such that then the resulting controller is the minimum variance controller. Request removal from index.
The partial variances can be estimated as To calculate the with the different initial conditions, the average of these values can be used as the estimates of , and the performance index of nonlinear feedforward and feedback control system can be obtained. This section presents a simulation experiment to show the effectiveness of the proposed strategy. The model of nonlinear feedforward and feedback control system that we have chosen is expressed as where is process model represented by a nonlinear polynomial: The measured and unmeasured disturbances are, respectively, given by where and are sequences of independent and identically distributed normal variables with mean zero, and the variances are, respectively, 0.
Assume that the process is presently being controlled about a fixed set point by a simple proportional feedforward controller in addition to an integral feedback controller. The manipulated variable is given by Two closed-loop signal curves of different time-delay conditions , and , are shown in Figure 2. Then, the traditional linear regression method is applied to estimate the MVPLB for nonlinear forward feedback control system. It can be seen that the estimated value of model parameters and MVPLB by traditional linear regression method has larger deviation, which is always larger than the real value.
This implies the excessive estimation. It is necessary to identify the model of closed-loop system to estimate the minimum variance performance index of the nonlinear system. First, we collect the disturbance signals which can be measured and then apply the linear regression method to fit the curve to obtain the parameter of the white noise. Furthermore, we use iterative orthogonal least square method to identify the closed-loop model.
The comparison for the output signal of identified model and actual model under two different time delays is shown in Figure 3. We can see the identified model can well approximate to the real nonlinear model. It is noted that the output variance of nonlinear system is also related to the initial value. Thus, to see whether the resulting controller performance based on variance decomposition method includes the influence of the initial value or not, the output variation of closed-loop system during the period is shown in Figure 4. It can be seen that when , the distribution of the system output tends to be stable; thus we get the conclusion that the output has nothing to do with the initial value.