# پیشرفت های اخیر در MPC اتفاقی و توسعه پایدار

کد مقاله | سال انتشار | مقاله انگلیسی | ترجمه فارسی | تعداد کلمات |
---|---|---|---|---|

29100 | 2004 | 13 صفحه PDF | سفارش دهید | محاسبه نشده |

**Publisher :** Elsevier - Science Direct (الزویر - ساینس دایرکت)

**Journal :** Annual Reviews in Control, Volume 28, Issue 1, 2004, Pages 23–35

#### چکیده انگلیسی

Despite the extensive literature that exists on predictive control and robustness to uncertainty, both multiplicative (e.g. parametric) and additive (e.g. exogenous), very little attention has been paid to the case of stochastic uncertainty. Yet this arises naturally in many control applications, for example when models are identified using least squares procedures. More generally, stochastic uncertainty is a salient feature in other key areas of human endeavour, such as sustainable development. Sustainability refers to the strategy of encouraging development at current time without compromising the potential for development in the future. Inevitably, modelling the effects of sustainable development policy over a horizon of say 30 years involves a very significant random element, which has to be taken into account when assessing the optimality of any proposed policy. Model Predictive Control (MPC) is ideally suited for generating constrained optimal solutions and as such would be an ideal tool for policy assessment. However, this calls first for suitable extensions to the stochastic case. The aim of this paper is to review some of the recent advances in this area, and to provide a pilot study that demonstrates the efficacy of stochastic predictive control as a tool for assessing policy in a sustainable development problem concerning allocation of public research and development budgets between alternative power generation technologies. This problem has been considered in earlier work, but only in the context of a single-shot, open-loop optimisation. Similarly, the consideration of stochastic predictive control methodologies has previously been restricted to general hypothetical control problems. The current paper brings together this body of work, proposes suitable extensions, and concludes with a closed-loop study of predictive control applied to a sustainable development policy assessment problem.

#### مقدمه انگلیسی

Model Predictive Control (MPC) solves, in a receding horizon manner, a series of open-loop optimisation problems, and thus provides tractable solutions to an infinite horizon constrained optimal control problem. As such it is ideally suited for use in a range of control applications, most of which are subject to constraints (both dynamic equality constraints and physical inequality constraints). However, MPC has enormous potential for application to a much wider class of problems in human endeavour where the aim is to maximise suitably defined benefit while keeping risk and cost within constraints. Many such examples can be found in applications of economics and finance, and also in a related area that is progressively gaining in importance: quantitative assessment of policy in sustainable development. Despite the grave concern about the effects that reckless development now will have on future generations (e.g. through the accumulation of atmospheric CO2 emissions and the consequent damaging influence on climate), most attempts made so far at decision-making for sustainability are based on qualitative or “quasi-quantitative” assessments. What is needed is a purely quantitative assessment that allows a direct objective comparison of one policy against another. MPC has the potential to provide such an assessment but this development has been restricted due to its predominantly deterministic formulation. This is a major handicap because of the strongly stochastic nature of problems in sustainable development (and also in finance and many other problems in economics). Predicting how current development will, in say 30 years, affect the potential for development by our children or even generations beyond, inevitably introduces randomness which, given some statistical regularity, leads to a modelling exercise with a strong stochastic flavour. It is somewhat curious to note that early development of MPC, such as self-tuning controllers for minimum variance (Astrom & Wittenmark, 1973) or generalized minimum variance (Clarke & Gawthrop, 1975), were specifically designed to handle random processes. However, optimisation with respect to a single statistic, be it the variance or the expected value (dealt with for example in a more recent paper by Batina, Stoorvogel, & Weiland, 2001), converts the problem to what amounts to a deterministic optimisation and therefore largely removes the stochastic element of the problem. To a significant degree this is avoided in the statistical framework of Whittle (1971), which later on was shown to lead to a convex second order cone program (e.g. Lobo, Vandenberghe, Boyd, & Lebret, 1998). This formulation considered linear inequality constraints with random coefficients and required that the constraints hold true with a probability greater than a given threshold. A closely related approach was adopted by Van Hessem and Bosgra, 2002 and Van Hessem and Bosgra, 2002, who consider the problem of disturbance rejection in constrained MPC. The presence of constraints invalidates the assumptions of the separation principle, which would only allow consideration of the effects of past uncertainty during estimation and would ignore uncertainty altogether during prediction. The methodology of Van Hessem and Bosgra, 2002 and Van Hessem and Bosgra, 2002 was based on confidence ellipsoids in the space of the random variables defining the stochastic disturbance. Statistical confidence has of course been used before to examine the effects of probabilistic uncertainty; for example, Cloud and Kouvaritakis (1986) considered the use of confidence ellipsoids in representing the effects of identification noise on model parameters. However, irrespective of whether the uncertainty is in the model parameters or future disturbances, it becomes necessary to use a projection transformation from the confidence ellipsoid to predicted outputs (in Van Hessem and Bosgra, 2002 and Van Hessem and Bosgra, 2002) or to the Nyquist plane (in Cloud & Kouvaritakis, 1986), and this transformation renders the results conservative to a significant degree. To overcome these shortcomings, and also to consider the stochastic prediction and optimisation problem in its most general form, recent work (Cannon, Kouvaritakis, & Huang, 2004; Kouvaritakis, Cannon, & Tsachouridis, 2004) followed the methodology developed in the context of sustainable development in Kouvaritakis (2000). In the context of an integrated programme, this work developed a Tool for Integrated Policy Assessment (TIPA), which identified measurable indices (outputs) and common instruments (inputs). The approach therefore enabled quantitative comparisons of one policy against another; and it was possible to consider, in a non-conservative way, the probabilistic effects of uncertainty on both constraints and predicted cost. The context was the assessment of sustainable development policy in respect of allocation of public research and development budgets between alternative technologies for electrical power generation. Under the SAPIENT programme (Kouvaritakis, 2000), the starting point for this work was the derivation of a full econometric model, Prometheus, which modelled the effect of shocks (the discrete time equivalent of impulse changes to the common instruments superimposed on a baseline) applied to the budget allocation for each of 15 alternative technologies (listed in Appendix A) on eight measurable indicators (listed in Appendix B). Monte Carlo simulations, which used Prometheus to predict the effect of impacts on indicators at the end of a 30-year horizon, established the validity of a linear model in which the impact coefficients were random variables. Further investigation determined that the distribution of coefficients could be modelled as jointly normal (with a given mean and co-variance matrix); despite its convenience, this assumption was found not to have a significant effect on the optimal solution of the problem. It was then possible in Kouvaritakis (2002) to state the optimisation problem as a static constrained maximisation of the probability that the cumulative value (over the 30-year horizon) of an indicator, identified as the primary indicator, is greater than a given threshold. By analogy the constraints on the remaining indicators, referred to as secondary, were also probabilistic, and required that the probability of the cumulative value (over 30 years) of the secondary indicators being greater than given thresholds should be greater than or equal to pre-specified probabilities. Under hedging conditions (namely for probability values greater than 0.5, as opposed to smaller probabilities which correspond to gambling), it was conjectured that the above stochastic optimisation was convex and thus admitted a unique solution. This seminal work, although derived in the different context of sustainable development, lays the foundations for a meaningful formulation of stochastic MPC. It clearly needed to be extended in several ways: (i) the relationship considered between the shocks and the cumulative values of the primary and secondary indices (at the end of the 30-year horizon) was static; (ii) the optimisation was single-shot in the sense that it allowed only for a single budget allocation over the entire prediction horizon; (iii) although it was assumed that such policy would be implemented in a receding horizon fashion, this assumption was only implicit in that no concern was expressed for the effect of such implementation on closed-loop feasibility, stability and performance. Appropriate extensions with respect to (i) and (ii) were proposed in Cannon et al. (2004), where attention was still focused on the cumulative values of indicators at the end of a 30-year horizon, but where Prometheus was used to identify a model which represents the dynamic effects of shocks on the indicators. It was then possible to account for several predicted future budget allocations within the open-loop optimisation framework. The aim of the work was not to carry out an exhaustive study of the full sustainable development with 15 inputs and 8 outputs, but rather to illustrate the benefits of dynamic models and the exploitation of the extra degrees of freedom introduced through the use of a multiple-shot optimisation. Accordingly Monte Carlo simulations performed on Prometheus were deployed to derive a 2×2 dynamic model linking the application of budget shocks on Wind Turbine and Combined Cycle Gas Turbine technologies to two indicators, one measuring cumulative CO2 emissions and the other Energy Costs in a particular world region. Simulations, repeated several thousands of times, were performed to identify the mean value and covariance matrix for the vector of model parameters. The random nature of the parameters implies that predicted indicator values themselves are random, and in an Auto Regressive Moving Average type of model, this would lead to difficulties deriving from the multiplication of parameters with predicted outputs, both of which are random. To avoid this, non-parsimonious Moving Average models were preferred, but model order was kept to a minimum through the use of double-discounting. Discounting is normally used on the indicators only, with the justification that although sustainable development pays attention to future benefit (measured by the primary indicator) and cost (measured by the secondary indicators), it should accord them progressively less emphasis over a prediction horizon. However, the spending of public research and development budgets is itself a cost and thus should be subject to discounting, hence leading to double-discounting, namely discounting applied to both inputs and outputs. Both the major strength and weakness of MPC lie in the difference between open-loop optimal performance and closed-loop results. Strength because it is through receding horizon optimisation that MPC converts an intractable infinite-dimensional problem to one amenable to practical implementation; weakness because at each time instant MPC obtains optimal solutions to a problem that could differ significantly from the desired target, namely closed-loop performance. Lack of concern for the effects of a receding horizon application of open-loop optima could have catastrophic consequences in the case of sustainable development. For example, the open-loop maximisation of the probability that the cumulative value of the primary indicator is greater than a given threshold could result in the indicator achieving low predicted values in the near future, with high predicted values achieved towards the end of the horizon. If this predicted behaviour were repeated at future open-loop optimisations, then the predicted benefit would prove elusive, receding further and further into the future. The converse scenario with respect to cost could have equally adverse effects, for the climate for example, if the relevant cost-measuring indicator relates to CO2 emissions. Here, the concern is that the indicator may meet the probabilistic constraints on cumulative values through good performance in the far future, but provide poor performance at the beginning of the prediction horizon. Persistence of such a trend in a receding horizon application could result in CO2 emissions accumulating in the closed-loop at levels well above the values prescribed by the open-loop constraints. The problem of reducing the discrepancy between open- and closed-loop results in terms of feasibility, stability, and bounds on performance has been addressed in MPC, and a range of techniques have been developed. However these concern a deterministic regulation or tracking problem and as such are quite inappropriate for use in sustainable development where: (i) overshoots in terms of benefit or undershoot in terms of cost are highly desirable; (ii) randomness precludes the deployment of the usual Lyapunov stability arguments. For this reason, in Cannon et al. (2004), open-loop optimisation was divided into two phases, one computing the largest possible (in a probabilistic sense) steady-state target for the primary indicator, and the other imposing variable bounds (again in a probabilistic sense), at each prediction time instant, on the modulus of the deviation of the predicted values from the steady-state target, and subsequently minimising a suitably defined norm on the vector of bounds. The constraints on the secondary indicators were also treated in a point-wise manner, i.e. through a requirement that the indicator, at each prediction time, should be greater than a given threshold with a probability that is at least as large as a pre-specified value, however the thresholds for each indicator were taken to be constant over the entire horizon. This framework, coupled with an equality stability constraint placed on the expected values of the indicators, enabled a recursive assertion of feasibility and the statement of a Lyapunov closed-loop stability result. The development was not predicated on the sustainable development problem, but was rather illustrated in terms of a hypothetical example couched in terms of a control design problem. The purpose of the present paper is to give a brief review of the recent developments described above, and combine the open-loop sustainable development results of Cannon et al. (2004), together with the closed-loop stochastic MPC results of Kouvaritakis et al., 2003 and Kouvaritakis et al., 2004, in order to present a complete closed-loop study in stochastic MPC addressing a sustainable development problem. It is noted that, as in most recent MPC algorithms, the online optimisation deployed is open-loop. This is a paradox in so far as uncertainty reduction is primarily a closed-loop problem. It is not a paradox when one considers the exponential rise in computational complexity with prediction horizon reported in Scokaert and Mayne (1998). Quasi closed-loop approaches have been considered by Kouvaritakis (1992), and Bemporad (1999), where in essence closed-loop predictions are deployed. This can be further enriched with the use of a Youla parameter (Kouvaritakis et al., 1992), which has also been used as a means of avoiding nonlinear dependences in closed-loop formulations (Van Hessem and Bosgra, 2002 and Van Hessem and Bosgra, 2002), with the aim of reducing the effects of stochastic disturbances. However, when uncertainty is also present as randomness in the model parameters, then the derivation of suitable stochastic descriptions for the closed-loop operators is no longer straightforward and invalidates the justification of the normal distribution assumptions. Section 2 gives a brief description of the earlier sustainable development single-shot open-loop optimisation problem, while Section 3 considers its extension through the introduction of dynamics in a multiple-shot open-loop optimisation. The stochastic MPC formulation is discussed in Section 4, and suitably extended in Section 5 so as to render the approach applicable to the sustainable development problem under consideration. The simulation results and the application to Prometheus are presented in Section 6, while concluding remarks are drawn in Section 7.

#### نتیجه گیری انگلیسی

Earlier work on MPC has taken into account stochastic uncertainty through the minimisation of appropriate variances, or of the expected value of the usual MPC cost. However, such methodologies fail to take complete account of available information on random uncertainty, which has attracted very little research attention to date. Recent work considered extensions of constrained MPC in the presence of stochastic disturbances and deployed a probabilistic form of constraints. In parallel, work on policy assessment in sustainable development considered an open-loop optimisation problem which was subject to uncertainty in the coefficients of linear regressors. This problem formulation was later introduced into an MPC framework and the relevant results are reviewed in this paper. After suitable extension, the paper illustrates the efficacy of stochastic MPC in issues of sustainable development. The approach is widely applicable, and it is believed to have the potential for making useful contributions in many other areas of human endeavour (e.g. economics, finance, and of course control). Before this is possible, several issues need further development. Amongst these must be included the further exploration of assumptions about normality, the use of nonlinear models, the use of closed-loop predictions in order to reduce the effect of uncertainty on predictions, and the systematic evaluation of the potential of MPC as a multi-objective tool. All of these topics form a basis for future research.