Robust Control: A Note on the Timing of Model Uncertainty

In this note a one-state, one-control variable quadratic linear problem with robust control and discount factor is developed to examine the optimal response of the first-period control to changes in future model uncertainty. A change in future model uncertainty has an effect on the optimal first-period control response going in the same direction as the one caused by an equal size change in current model uncertainty. However, both analytical and numerical results show that such effect is much lower than the one derived from a change in current model uncertainty. Moreover, such effect is even much lower as the change in model uncertainty moves farther away into the future. Finally, the infinite horizon result confirms the reinforcing nature of the effects on the optimal first-period control response of current and future changes in model uncertainty.The author thanks P. Ruben Mercado, David A. Kendrick and an anonymous referee for useful comments on earlier versions of this note. This research work was completed while the author was working at the Centro de Investigación y Docencia Económicas (CIDE) in Mexico City. The content of this note is only responsibility of the author. Any opinions expressed here in no way reflect comments nor suggestions made by the Board of Governors or any other member of the Bank of Mexico.     

Optimal Policy Response with Control Parameter and Intercept Covariance

Parameter uncertainty and the interaction between the uncertain parameters are important aspects of economic policy. In this work, I develop an analytical one-state variable, one-control variable model with two uncertain parameters (the control parameter and the intercept) and a nonzero covariance. I characterize the effect of changes in each of the covariance components on the optimal expected control variable. I found that the nature of the optimal policy maker’s response depends on the specific changing component of the covariance, the sign of the correlation coefficient and the sign of the optimal expected control variable when the covariance is zero. I obtain the conditions under which the effect of the covariance is considerable. This work complements previous studies by providing a complete set of cases and conditions for an aggressive or cautionary optimal policy maker’s response to changes in each covariance component. Finally, the importance of the analytical results is shown for the regulation of a stock pollutant leading to global warming.      

Robust high-order repetitive control: Optimal performance trade-offs

High-order repetitive control has previously been introduced to either improve the robustness for period-time uncertainty or reduce the sensitivity for non-periodic inputs of standard repetitive control schemes. This paper presents a systematic, semidefinite programming based approach to compute high-order repetitive controllers that yield an optimal trade-off between these two performance criteria. The methodology is numerically illustrated through trade-off curves for various controller orders and levels of period-time uncertainty. Moreover, existing high-order repetitive control approaches are shown to correspond to specific points on these curves.     

Monetary Policy Under Time-Varying Uncertainty Aversion

In this paper we develop a framework to analyze the optimal policy of an inflation-targeting monetary authority that is not fully confident about its model and the degree of mistrust changes over time as the structure of the economy changes. These changes can include structural breaks as well as price, output or real exchange shocks. We use robust control to denote the degree of uncertainty aversion of the policy maker and a Markov chain to capture the time-varying nature of the uncertainty aversion. We find that in general a more aggressive interest rate policy is the optimal response to: (i) more uncertainty aversion and (ii) higher likelihood that the uncertainty aversion may appear in the future. Moreover, we find that the policy maker’s welfare decreases when there is an increase in uncertainty aversion. However, the transition probabilities in the Markov-chain have ambiguous effects on the policy maker expected losses.     

Origins of robust control: Early history and future speculations

A revolution in control theory thought happened in the early 1970s when the dominant focus of research shifted from optimality to robustness in response to unexpected failures of optimal control theory to produce feedback control designs capable of tolerating normal differences between design models and reality. The robustness concept has since become such an integral part of present day control theory that it is difficult to imagine that time long ago when the concept lacked a clear mathematical representation and the tools of multivariable robustness analysis were yet to be identified. We shall revisit that time to examine the events that facilitated, and necessitated, this remarkable paradigm shift. Next, looking to the future, we will consider how failures of robust control designs to cope with incorrect uncertainty estimates are beginning to spur control theorists to consider data-driven problem formulations for estimation and control that tacitly question the roles of basic concepts like true model and uncertainty bounds, stochastic noise models and even Bayesian probability. We will discuss how and why Karl Popper’s scientific logic of unfalsification seems to be emerging as a central concept in these data-driven problem formulations, and how the unfalsification concept might again shift the focus of mathematical research in the areas of estimation and control.     

Scheduling and control of failure prone systems with demand uncertainty and job overlaps

Given that the overlapping of jobs is permitted, the paper studies the scheduling and control of failure prone production systems,i, e.so-called settings with demand uncertainty and job overlaps. Because a variable demand resource is revolved in the production and corrective maintenance control problems of the system, which switched randomly between zero and a maximum level, it is difficult to obtain the analytical solutions of the optimal single hedging point policy. An asymptotic optimal scheduling policy is presented and a double hedging point policy is offered to control simultaneously the production rate and the corrective maintenance rate of the system. The corresponding analytical solutions and approximate solutions are obtained. Considering the relationship of production, corrective maintenance and demand variable, an approximate optimal single hedging point control policy is proposed. Numerical results are presented.     

Facility location dynamics: An overview of classifications and applications

In order to modify the current facility or develop a new facility, the dynamics of facility location problems (FLPs) ought to be taken into account so as to efficiently deal with changing parameters such as market demand, internal and external factors, and populations. Since FLPs have a strategic or long-term essence, the inherited uncertainty of future parameters must be incorporated in relevant models, so these models can be considered applicable and ready to implement. Furthermore, due to largely capital outlaid, location or relocation of facilities is basically considered as a long-term planning. Hence, regarding the way in which relevant criteria will change over time, decision makers not only are concerned about the operability and profitability of facilities for an extended period, but also seek to robust locations fitting well with variable demands. Concerning this fact, a trade-off should be set between benefits brought by facility location changes and costs incurred by possible modifications. This review reports on literature pointing out some aspects and characteristics of the dynamics of FLPs. In fact, this paper aims not only to review most variants of these problems, but also to provide a broad overview of their mathematical formulations as well as case studies that have been studied by the literature. Finally, based on classified research works and available gaps in the literature, some possible research trends will be pointed out.     

Dual adaptive control and uncertainty effects in macroeconomic systems optimization

A method is presented for the quantitative assessment of the effects of uncertainty in macroeconomic policy optimization problems. Both parameter and measurement uncertainties are considered. The method derives from an application of adaptive dual control theory which decomposes the optimal loss function into three components: deterministic, cautionary, and probing. This characterizes the welfare loss due to uncertainty and permits the study of the relative importance of caution and probing in selecting the optimal policy instruments. An interesting illustration of the method is given for two macroeconometric models derived from the same data, but each with a different level of uncertainty.     

Optimization of variable speed wind power systems based on a LQG approach

This paper focuses on a control application of optimization in wind power systems. An optimal control structure for variable speed fixed pitch wind turbines is presented. The optimality of the whole system is defined by the trade-off between the energy conversion maximization and the control input minimization that determines the mechanical stress of the drive train. The frequency separation of the short-term and the long-term variations, adopted in the wind modelling, has resulted in a two-loop control structure. The optimal problem is treated within a complete linear quadratic stochastic approach, whose effectiveness was tested on an electromechanical wind turbine simulator.     

Optimal feedback control of oil reservoir waterflooding processes

Waterflooding is a process where water is injected into an oil reservoir to supplement its natural pressure for increment in productivity. The reservoir properties are highly heterogeneous, its states change as production progresses which require varying injection and production settings for economic recovery. As water is injected into the reservoir, more oil is expected to be produced. There is also likelihood that water is produced in association with the oil. The worst case is when the injected water meanders through the reservoir, it bypasses pools of oil and gets produced. Therefore, any effort geared toward finding the optimal settings to maximize the value of this venture can never be over emphasized. Waterflooding can be formulated as an optimal control problem. However, traditional optimal control is an open-loop solution, hence cannot cope with various uncertainties inevitably existing in any practical systems. Reservoir models are highly uncertain. Its properties are known with some degrees of certainty near the well-bore region only. In this work, a novel data-driven approach for control variable (CV) selection was proposed and applied to reservoir waterflooding process for a feedback strategy resulting in optimal or near optimal operation. The results indicated that the feedback control method was close to optimal in the absence of uncertainty. The loss recorded in the value of performance index, net present value (NPV) was only 0.26%. Furthermore, the new strategy performs better than the open-loop optimal control solution when system/model mismatch was considered. The performance depends on the scale of the uncertainty introduced. A gain in NPV as high as 30.04% was obtained.     

Parameter Uncertainty and Policy Intensity: Some Extensions and Suggestions for Further Work

The effects of parameter uncertainty on optimal policy have been a matter of interest for academics, and even for some policymakers, for a long time. Two lines of literature have developed analytical results on this matter. The first line uses static models and the second dynamic models. In this dynamic line most of the results are confined to models with a single state and a single control variable. In this paper we want to encourage the analysis of more general dynamic cases. To do so, the results in the dynamic line are extended from one-state and one-control finite horizon models to models with a pair of control variables. We then discuss some of the hurdles which must be surmounted for the results to be made more general and suggests some lines for further research. JEL classification: C61; E61     

A decision model for choosing the optimal level of storage intemporal databases

A database allows its users to reduce uncertainty about the world. However, not all properties of all objects can always be stored in a database. As a result, the user may have to use probabilistic inference rules to estimate the data required for his decisions. A decision based on such estimated data may not be perfect. The authors call the costs associated with such suboptimal decisions the cost of incomplete information. This cost can be reduced by expanding the database to contain more information; such expansion will increase the data-related costs because of more data collection, manipulation, storage, and retrieval. A database designer must then consider the trade-off between the cost of incomplete information and the data-related costs, and choose a design that minimizes the overall cost to the organization. In temporal databases, the sheer volume of the data involved makes such a trade-off at design time all the more important. They develop probabilistic inference rules that allow one to infer missing values in spatial, as well as temporal, dimension. They then use the framework for developing guidelines for designing and reorganizing temporal databases, which explicitly includes a trade-off between the incomplete information and the data-related costs     

Robust controller design for a non-colocated spring-mass system via mixed 𝓁/∞ optimization

A successful controller design paradigm must take into account both model uncertainty and performance specifications. Model uncertainty can be addressed using the ??^∞ robust control framework. However, this framework cannot accommodate the realistic case where in addition to robustness considerations, the system is subject to time-domain specifications. We recently proposed design procedures to explicitly incorporate time-domain specifications into the ??^∞ framework. In this paper we apply these design procedures to the simple mass-spring system used as a benchmark in the 1990–1992 ACC, with the goal of minimizing the peak control effort while satisfying disturbance rejection, settling time, tracking and robustness specifications. The results show that there exist a severe trade-off between peak control action and robustness to unstructured model uncertainty.     

Hysteresis-based supervisory control with application to non-pharmaceutical containment of COVID-19

The recent COVID-19 outbreak has motivated an extensive development of non-pharmaceutical intervention policies for epidemics containment. While a total lockdown is a viable solution, interesting policies are those allowing some degree of normal functioning of the society, as this allows a continued, albeit reduced, economic activity and lessens the many societal problems associated with a prolonged lockdown. Recent studies have provided evidence that fast periodic alternation of lockdown and normal-functioning days may effectively lead to a good trade-off between outbreak abatement and economic activity. Nevertheless, the correct number of normal days to allocate within each period in such a way to guarantee the desired trade-off is a highly uncertain quantity that cannot be fixed a priori and that must rather be adapted online from measured data. This adaptation task, in turn, is still a largely open problem, and it is the subject of this work. In particular, we study a class of solutions based on hysteresis logic. First, in a rather general setting, we provide general convergence and performance guarantees on the evolution of the decision variable. Then, in a more specific context relevant for epidemic control, we derive a set of results characterizing robustness with respect to uncertainty and giving insight about how a priori knowledge about the controlled process may be used for fine-tuning the control parameters. Finally, we validate the results through numerical simulations tailored on the COVID-19 outbreak.     

Optimal back-off point determination and controller weight selection for multivariate systems under finite-horizon control

Plant economic performance is most often related to the operating point, specifically the mean values of the process variables; meanwhile, most existing performance assessment techniques involve examining the variances or covariances of the controlled variables. A combined approach is to determine the appropriate trade-off between variances of different process variables in order to operate the plant at the point that provides maximum economic benefit while satisfying the operating constraints. This problem is referred to as the minimum backed-off operating point selection, and previous works have formulated it as a non-convex constrained optimization problem. In the current work, a new technique is introduced that can provide the optimal plant operating point. Additionally, this method provides the weights for a finite horizon controller that results in the optimal trade-off in process variable variances that will allow satisfaction of the operating constraints at the optimal operating point. In this method, the plant and disturbance models for the given process are used to generate data representing possible trade-offs between process variable standard deviations. Employing a piecewise linear regression to describe the sample points of this standard deviations data allows for the operating point selection problem to be solved as a small number of linear programs. The advantages of this approach are demonstrated through the use of mathematical and simulation case studies.     

Reinforcement learning in sustainable energy and electric systems: a survey

The dynamic nature of sustainable energy and electric systems can vary significantly along with the environment and load change, and they represent the features of multivariate, high complexity and uncertainty of the nonlinear system. Moreover, the integration of intermittent renewable energy sources and energy consumption behaviours of households introduce more uncertainty into sustainable energy and electric systems. The operation, control and decision-making in such an environment definitely require increasing intelligence and flexibility in the control and optimization to ensure the quality of service of sustainable energy and electric systems. Reinforcement learning is a wide class of optimal control strategies that uses estimating value functions from experience, simulation, or search to learn in highly dynamic, stochastic environment. The interactive context enables reinforcement learning to develop strong learning ability and high adaptability. Reinforcement learning does not require the use of the model of system dynamics, which makes it suitable for sustainable energy and electric systems with complex nonlinearity and uncertainty. The use of reinforcement learning in sustainable energy and electric systems will certainly change the traditional energy utilization mode and bring more intelligence into the system. In this survey, an overview of reinforcement learning, the demand for reinforcement learning in sustainable energy and electric systems, reinforcement learning applications in sustainable energy and electric systems, and future challenges and opportunities will be explicitly addressed.     

LMI-based H ∞-optimal control with transients

In this article, the worst-case norm of the regulated output over all exogenous signals and initial states as a performance measure of the system is characterised in terms of linear matrix inequalities (LMIs). Optimal time-invariant state- and output-feedback controllers are synthesised as minimising this performance measure. The essential role in this synthesis plays a weighting matrix reflecting the relative importance of the uncertainty in the initial state contrary to the uncertainty in the exogenous signal. H _∞-optimal control with transients is shown to be actually a trade-off between H _∞-control, being optimal under unknown exogenous disturbances and zero initial state, and γ-control, being optimal under zero exogenous signal and unknown initial conditions, if and only if the weighting matrix satisfies a fundamental inequality. If this inequality is met, the performance measure is achieved and the explicit formulae for the worst-case disturbance and initial state are provided. If this inequality fails, the performance measure coincides with the H _∞-norm and the trade-off gets broken.     

Interactive evolutionary solution synthesis in fuzzy set-based preliminary engineering design

This paper describes an interactive evolutionary approach to synthesize component-based preliminary engineering design problems. This approach is intended to address preliminary engineering design as an evolutionary synthesis process, with the needs for human-computer interaction in a changing environment caused by uncertainty and imprecision inherent in the early design stages. It combines an agent-based hierarchical design representation, set-based design generation, fuzzy design trade-off strategy and interactive design adaptation into evolutionary synthesis to gradually refine and reduce the search space while maintaining solution diversity to accommodate future changes. The fitness function of solutions employed is not fixed but adapted according to elicited human value judgment and constraint change. It incorporates multi-criteria evaluation as well as constraint satisfaction. This new approach takes advantage of the different roles of computers and humans play in design and optimization. The methodology will be applicable to general multi-domain applications, with emphasis on physical modeling of dynamic systems. An automotive speedometer design case study is included to demonstrate the methodology.     

MultiETSC: automated machine learning for early time series classification

Early time series classification (EarlyTSC) involves the prediction of a class label based on partial observation of a given time series. Most EarlyTSC algorithms consider the trade-off between accuracy and earliness as two competing objectives, using a single dedicated hyperparameter. To obtain insights into this trade-off requires finding a set of non-dominated (Pareto efficient) classifiers. So far, this has been approached through manual hyperparameter tuning. Since the trade-off hyperparameters only provide indirect control over the earliness-accuracy trade-off, manual tuning is tedious and tends to result in many sub-optimal hyperparameter settings. This complicates the search for optimal hyperparameter settings and forms a hurdle for the application of EarlyTSC to real-world problems. To address these issues, we propose an automated approach to hyperparameter tuning and algorithm selection for EarlyTSC, building on developments in the fast-moving research area known as automated machine learning (AutoML). To deal with the challenging task of optimising two conflicting objectives in early time series classification, we propose MultiETSC, a system for multi-objective algorithm selection and hyperparameter optimisation (MO-CASH) for EarlyTSC. MultiETSC can potentially leverage any existing or future EarlyTSC algorithm and produces a set of Pareto optimal algorithm configurations from which a user can choose a posteriori. As an additional benefit, our proposed framework can incorporate and leverage time-series classification algorithms not originally designed for EarlyTSC for improving performance on EarlyTSC; we demonstrate this property using a newly defined, “naïve” fixed-time algorithm. In an extensive empirical evaluation of our new approach on a benchmark of 115 data sets, we show that MultiETSC performs substantially better than baseline methods, ranking highest (avg. rank 1.98) compared to conceptually simpler single-algorithm (2.98) and single-objective alternatives (4.36).

     

Dissipative control of linear discrete-time systems with dissipative uncertainty

This paper focuses on the dissipative control of uncertain linear discrete-time systems. The uncertainty under consideration is characterized by a dissipative system, which contains commonly used uncertainty structures, such as normbounded and positive real uncertainties, as special cases. We consider the design of a feedback controller which can achieve asymptotic stability and strict quadratic dissipativeness for all admissible uncertainties. Both the linear static state feedback and the dynamic output feedback controllers are considered. It is shown that the robust dissipative control problem can be solved in terms of a scaled quadratic dissipative control problem without uncertainty. Linear matrix inequality (LMI) based methods for designing robust controllers are derived. The result of this paper unifies existing results on discrete-time H and positive real control and it provides a more flexible and less conservative control design as it al ows for a bet er trade-off between phase and gain performances.