A Medley of Potpourri

Friday, September 29, 2023

Model predictive control

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Model_predictive_control

Model predictive control (MPC) is an advanced method of process control that is used to control a process while satisfying a set of constraints. It has been in use in the process industries in chemical plants and oil refineries since the 1980s. In recent years it has also been used in power system balancing models and in power electronics. Model predictive controllers rely on dynamic models of the process, most often linear empirical models obtained by system identification. The main advantage of MPC is the fact that it allows the current timeslot to be optimized, while keeping future timeslots in account. This is achieved by optimizing a finite time-horizon, but only implementing the current timeslot and then optimizing again, repeatedly, thus differing from a linear–quadratic regulator (LQR). Also MPC has the ability to anticipate future events and can take control actions accordingly. PID controllers do not have this predictive ability. MPC is nearly universally implemented as a digital control, although there is research into achieving faster response times with specially designed analog circuitry.

Generalized predictive control (GPC) and dynamic matrix control (DMC) are classical examples of MPC.

Overview

The models used in MPC are generally intended to represent the behavior of complex and simple dynamical systems. The additional complexity of the MPC control algorithm is not generally needed to provide adequate control of simple systems, which are often controlled well by generic PID controllers. Common dynamic characteristics that are difficult for PID controllers include large time delays and high-order dynamics.

MPC models predict the change in the dependent variables of the modeled system that will be caused by changes in the independent variables. In a chemical process, independent variables that can be adjusted by the controller are often either the setpoints of regulatory PID controllers (pressure, flow, temperature, etc.) or the final control element (valves, dampers, etc.). Independent variables that cannot be adjusted by the controller are used as disturbances. Dependent variables in these processes are other measurements that represent either control objectives or process constraints.

MPC uses the current plant measurements, the current dynamic state of the process, the MPC models, and the process variable targets and limits to calculate future changes in the dependent variables. These changes are calculated to hold the dependent variables close to target while honoring constraints on both independent and dependent variables. The MPC typically sends out only the first change in each independent variable to be implemented, and repeats the calculation when the next change is required.

While many real processes are not linear, they can often be considered to be approximately linear over a small operating range. Linear MPC approaches are used in the majority of applications with the feedback mechanism of the MPC compensating for prediction errors due to structural mismatch between the model and the process. In model predictive controllers that consist only of linear models, the superposition principle of linear algebra enables the effect of changes in multiple independent variables to be added together to predict the response of the dependent variables. This simplifies the control problem to a series of direct matrix algebra calculations that are fast and robust.

When linear models are not sufficiently accurate to represent the real process nonlinearities, several approaches can be used. In some cases, the process variables can be transformed before and/or after the linear MPC model to reduce the nonlinearity. The process can be controlled with nonlinear MPC that uses a nonlinear model directly in the control application. The nonlinear model may be in the form of an empirical data fit (e.g. artificial neural networks) or a high-fidelity dynamic model based on fundamental mass and energy balances. The nonlinear model may be linearized to derive a Kalman filter or specify a model for linear MPC.

An algorithmic study by El-Gherwi, Budman, and El Kamel shows that utilizing a dual-mode approach can provide significant reduction in online computations while maintaining comparative performance to a non-altered implementation. The proposed algorithm solves N convex optimization problems in parallel based on exchange of information among controllers.

Theory behind MPC

MPC is based on iterative, finite-horizon optimization of a plant model. At time $t$ the current plant state is sampled and a cost minimizing control strategy is computed (via a numerical minimization algorithm) for a relatively short time horizon in the future: $[t, t + T]$ . Specifically, an online or on-the-fly calculation is used to explore state trajectories that emanate from the current state and find (via the solution of Euler–Lagrange equations) a cost-minimizing control strategy until time $t + T$ . Only the first step of the control strategy is implemented, then the plant state is sampled again and the calculations are repeated starting from the new current state, yielding a new control and new predicted state path. The prediction horizon keeps being shifted forward and for this reason MPC is also called receding horizon control. Although this approach is not optimal, in practice it has given very good results. Much academic research has been done to find fast methods of solution of Euler–Lagrange type equations, to understand the global stability properties of MPC's local optimization, and in general to improve the MPC method.

Principles of MPC

Model predictive control is a multivariable control algorithm that uses:

an internal dynamic model of the process
a cost function J over the receding horizon
an optimization algorithm minimizing the cost function J using the control input u

An example of a quadratic cost function for optimization is given by:

J = \sum_{i = 1}^{N} w_{x_{i}} (r_{i} - x_{i})^{2} + \sum_{i = 1}^{M} w_{u_{i}} {Δ u_{i}}^{2}

without violating constraints (low/high limits) with

x_{i}

i

^th controlled variable (e.g. measured temperature)

r_{i}

i

^th reference variable (e.g. required temperature)

u_{i}

i

^th manipulated variable (e.g. control valve)

w_{x_{i}}

: weighting coefficient reflecting the relative importance of

x_{i}

w_{u_{i}}

: weighting coefficient penalizing relative big changes in

u_{i}

etc.

Nonlinear MPC

Nonlinear model predictive control, or NMPC, is a variant of model predictive control that is characterized by the use of nonlinear system models in the prediction. As in linear MPC, NMPC requires the iterative solution of optimal control problems on a finite prediction horizon. While these problems are convex in linear MPC, in nonlinear MPC they are not necessarily convex anymore. This poses challenges for both NMPC stability theory and numerical solution.

The numerical solution of the NMPC optimal control problems is typically based on direct optimal control methods using Newton-type optimization schemes, in one of the variants: direct single shooting, direct multiple shooting methods, or direct collocation. NMPC algorithms typically exploit the fact that consecutive optimal control problems are similar to each other. This allows to initialize the Newton-type solution procedure efficiently by a suitably shifted guess from the previously computed optimal solution, saving considerable amounts of computation time. The similarity of subsequent problems is even further exploited by path following algorithms (or "real-time iterations") that never attempt to iterate any optimization problem to convergence, but instead only take a few iterations towards the solution of the most current NMPC problem, before proceeding to the next one, which is suitably initialized. Another promising candidate for the nonlinear optimization problem is to use a randomized optimization method. Optimum solutions are found by generating random samples that satisfy the constraints in the solution space and finding the optimum one based on cost function.

While NMPC applications have in the past been mostly used in the process and chemical industries with comparatively slow sampling rates, NMPC is being increasingly applied, with advancements in controller hardware and computational algorithms, e.g., preconditioning, to applications with high sampling rates, e.g., in the automotive industry, or even when the states are distributed in space (Distributed parameter systems). As an application in aerospace, recently, NMPC has been used to track optimal terrain-following/avoidance trajectories in real-time.

Explicit MPC

Explicit MPC (eMPC) allows fast evaluation of the control law for some systems, in stark contrast to the online MPC. Explicit MPC is based on the parametric programming technique, where the solution to the MPC control problem formulated as optimization problem is pre-computed offline. This offline solution, i.e., the control law, is often in the form of a piecewise affine function (PWA), hence the eMPC controller stores the coefficients of the PWA for each a subset (control region) of the state space, where the PWA is constant, as well as coefficients of some parametric representations of all the regions. Every region turns out to geometrically be a convex polytope for linear MPC, commonly parameterized by coefficients for its faces, requiring quantization accuracy analysis. Obtaining the optimal control action is then reduced to first determining the region containing the current state and second a mere evaluation of PWA using the PWA coefficients stored for all regions. If the total number of the regions is small, the implementation of the eMPC does not require significant computational resources (compared to the online MPC) and is uniquely suited to control systems with fast dynamics. A serious drawback of eMPC is exponential growth of the total number of the control regions with respect to some key parameters of the controlled system, e.g., the number of states, thus dramatically increasing controller memory requirements and making the first step of PWA evaluation, i.e. searching for the current control region, computationally expensive.

Robust MPC

Robust variants of model predictive control are able to account for set bounded disturbance while still ensuring state constraints are met. Some of the main approaches to robust MPC are given below.

Min-max MPC. In this formulation, the optimization is performed with respect to all possible evolutions of the disturbance. This is the optimal solution to linear robust control problems, however it carries a high computational cost. The basic idea behind the min/max MPC approach is to modify the on-line "min" optimization to a "min-max" problem, minimizing the worst case of the objective function, maximized over all possible plants from the uncertainty set.
Constraint Tightening MPC. Here the state constraints are enlarged by a given margin so that a trajectory can be guaranteed to be found under any evolution of disturbance.
Tube MPC. This uses an independent nominal model of the system, and uses a feedback controller to ensure the actual state converges to the nominal state. The amount of separation required from the state constraints is determined by the robust positively invariant (RPI) set, which is the set of all possible state deviations that may be introduced by disturbance with the feedback controller.
Multi-stage MPC. This uses a scenario-tree formulation by approximating the uncertainty space with a set of samples and the approach is non-conservative because it takes into account that the measurement information is available at every time stage in the prediction and the decisions at every stage can be different and can act as recourse to counteract the effects of uncertainties. The drawback of the approach however is that the size of the problem grows exponentially with the number of uncertainties and the prediction horizon.
Tube-enhanced multi-stage MPC. This approach synergizes multi-stage MPC and tube-based MPC. It provides high degrees of freedom to choose the desired trade-off between optimality and simplicity by the classification of uncertainties and the choice of control laws in the predictions.

Commercially available MPC software

Commercial MPC packages are available and typically contain tools for model identification and analysis, controller design and tuning, as well as controller performance evaluation.

A survey of commercially available packages has been provided by S.J. Qin and T.A. Badgwell in Control Engineering Practice 11 (2003) 733–764.

MPC vs. LQR

Model predictive control and linear-quadratic regulators are both expressions of optimal control, with different schemes of setting up optimisation costs.

While a model predictive controller often looks at fixed length, often graduatingly weighted sets of error functions, the linear-quadratic regulator looks at all linear system inputs and provides the transfer function that will reduce the total error across the frequency spectrum, trading off state error against input frequency.

Due to these fundamental differences, LQR has better global stability properties, but MPC often has more locally optimal[?] and complex performance.

The main differences between MPC and LQR are that LQR optimizes across the entire time window (horizon) whereas MPC optimizes in a receding time window, and that with MPC a new solution is computed often whereas LQR uses the same single (optimal) solution for the whole time horizon. Therefore, MPC typically solves the optimization problem in a smaller time window than the whole horizon and hence may obtain a suboptimal solution. However, because MPC makes no assumptions about linearity, it can handle hard constraints as well as migration of a nonlinear system away from its linearized operating point, both of which are major drawbacks to LQR.

This means that LQR can become weak when operating away from stable fixed points. MPC can chart a path between these fixed points, but convergence of a solution is not guaranteed, especially if thought as to the convexity and complexity of the problem space has been neglected.

Earthscope

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Earthscope

The EarthScope project was an National Science Foundation (NSF) funded earth science program that, from 2003-2018, used geological and geophysical techniques to explore the structure and evolution of the North American continent and to understand the processes controlling earthquakes and volcanoes. The project had three components: USArray, the Plate Boundary Observatory, and the San Andreas Fault Observatory at Depth. Organizations associated with the project included UNAVCO, the Incorporated Research Institutions for Seismology (IRIS), Stanford University, the United States Geological Survey (USGS) and National Aeronautics and Space Administration (NASA). Several international organizations also contributed to the initiative. EarthScope data are publicly accessible.

Observatories

There were three EarthScope observatories: the San Andreas Fault Observatory at Depth (SAFOD), the Plate Boundary Observatory (PBO), and the Seismic and Magnetotelluric Observatory (USArray). These observatories consist of boreholes into an active fault zone, global positioning system (GPS) receivers, tiltmeters, long-baseline laser strainmeters, borehole strainmeters, permanent and portable seismographs, and magnetotelluric stations. The various EarthScope components will provide integrated and highly accessible data on geochronology and thermochronology, petrology and geochemistry, structure and tectonics, surficial processes and geomorphology, geodynamic modeling, rock physics, and hydrogeology.

Seismic and Magnetotelluric Observatory (USArray)

USArray, managed by IRIS, is a 15-year program to place a dense network of permanent and portable seismographs across the continental United States. These seismographs record the seismic waves released by earthquakes that occur around the world. Seismic waves are indicators of energy disbursement within the earth. By analyzing the records of earthquakes obtained from this dense grid of seismometers, scientists can learn about Earth structure and dynamics and the physical processes controlling earthquakes and volcanoes. The goal of USArray is primarily to gain a better understanding of the structure and evolution of the continental crust, lithosphere, and mantle underneath North America.

The USArray is composed of four facilities: a Transportable Array, a Flexible Array, a Reference Network, and a Magnetotelluric Facility.

The Transportable Array is composed of 400 seismometers that are being deployed in a rolling grid across the United States over a period of 10 years. The stations are placed 70 km apart, and can map the upper 70 km of the Earth. After approximately two years, stations are moved east to the next site on the grid – unless adopted by an organization and made a permanent installation. Once the sweep across the United States is completed, over 2000 locations will have been occupied. The Array Network Facility is responsible for data collection from the Transportable Array stations.

The Flexible Array is composed of 291 broadband stations, 120 short period stations, and 1700 active source stations. The Flexible Array allows sites to be targeted in a more focused manner than the broad Transportable Array. Natural or artificially created seismic waves can be used to map structures in the Earth.

The Reference Network is composed of permanent seismic stations spaced about 300 km apart. The Reference Network provides a baseline for the Transportable Array and Flexible Array. EarthScope added and upgraded 39 stations to the already existing Advanced National Seismic System, which is part of the Reference Network.

The Magnetotelluric Facility is composed of seven permanent and 20 portable sensors that record electromagnetic fields. It is the electromagnetic equivalent of the seismic arrays. The portable sensors are moved in a rolling grid similar to the Transportable Array grid, but are only in place about a month before they are moved to the next location. A magnetotelluric station consists of a magnetometer, four electrodes, and a data recording unit that are buried in shallow holes. The electrodes are oriented north-south and east-west and are saturated in a salt solution to improve conductivity with the ground.

Plate Boundary Observatory (PBO)

The Plate Boundary Observatory PBO consists of a series of geodetic instruments, Global Positioning System (GPS) receivers and borehole strainmeters, that have been installed to help understand the boundary between the North American Plate and Pacific Plate. The PBO network includes several major observatory components: a network of 1100 permanent, continuously operating Global Positioning System (GPS) stations many of which provide data at high-rate and in real-time, 78 borehole seismometers, 74 borehole strainmeters, 26 shallow borehole tiltmeters, and six long baseline laser strainmeters. These instruments are complemented by InSAR (interferometric synthetic aperture radar) and LiDAR (light detection and ranging) imagery and geochronology acquired as part of the GeoEarthScope initiative. PBO also includes comprehensive data products, data management and education and outreach efforts. These permanent networks are supplemented by a pool of portable GPS receivers that can be deployed for temporary networks to researchers, to measure the crustal motion at a specific target or in response to a geologic event. The Plate Boundary Observatory portion of EarthScope is operated by UNAVCO, Inc. UNAVCO is a non-profit, university-governed consortium that facilitates research and education using geodesy.

San Andreas Fault Observatory at Depth (SAFOD)

The San Andreas Fault Observatory at Depth (SAFOD) consists of a main borehole that cuts across the active San Andreas Fault at a depth of approximately 3 km and a pilot hole about 2 km southwest of San Andreas Fault. Data from the instruments installed in the holes, which consist of geophone sensors, data acquisition systems, and GPS clocks, as well as samples collected during drilling, will help to better understand the processes that control the behavior of the San Andreas Fault.

Data Products

Data collected from the various observatories are used to create different types of data products. Each data product addresses a different scientific problem.

P-Wave Tomography

Tomography is a method of producing a three-dimensional image of the internal structures of a solid object (such as the human body or the earth) by the observation and recording of differences in the effects on the passage of energy waves impinging on those structures. The waves of energy are P-waves generated by earthquakes and are recording the wave velocities. The high quality data that is being collected by the permanent seismic stations of USArray and the Advanced National Seismic System (ANSS) will allow the creation of high resolution seismic imaging of the Earth's interior below the United States. Seismic tomography helps constrain mantle velocity structure and aids in the understanding of chemical and geodynamic processes that are at work. With the use of the data collected by USArray and global travel-time data, a global tomography model of P-wave velocity heterogeneity in the mantle can be created. The range and resolution of this technique will allow investigation into the suite of problems that are of concern in the North American mantle lithosphere, including the nature of the major tectonic features. This method gives evidence for differences in thickness and the velocity anomaly of the mantle lithosphere between the stable center of the continent and the more active western North America. This data is vital for the understanding of local lithosphere evolution, and when combined with additional global data, will allow the mantle to be imaged beyond the current extent of USArray.

Receiver Reference Models

EarthScope Automated Receiver Survey (EARS), has created a prototype of a system that will be used to address several key elements of the production of EarthScope products. One of the prototype systems is the receiver reference model. It will provide crustal thickness and average crustal Vp/Vs ratios beneath USArray transportable array stations.

Ambient Seismic Noise

The main function of the Advanced National Seismic System (ANSS) and USArray, is to provide high quality data for earthquake monitoring, source studies and Earth structure research. The utility of seismic data is greatly increased when noise levels, unwanted vibrations, are reduced; however broadband seismograms will always contain a certain level of noise. The dominant sources of noise are either from the instrumentation itself or from ambient Earth vibrations. Normally, seismometer self noise will be well below the seismic noise level, and every station will have a characteristic noise pattern that can be calculated or observed. Sources of seismic noise within the Earth are caused by any of the following: the actions of human beings at or near the surface of the Earth, objects moved by wind with the movement being transferred to the ground, running water (river flow), surf, volcanic activity, or long period tilt due to thermal instabilities from poor station design.

A new approach to seismic noise studies will be introduced with the EarthScope project, in that there are no attempts to screen the continuous waveforms to eliminate body and surface waves from the naturally occurring earthquakes. Earthquake signals are not generally included in the processing of noise data, because they are generally low probability occurrences, even at low power levels. The two objectives behind the collection of the seismic noise data are to provide and document a standard method to calculate ambient seismic background noise, and to characterize the variation of ambient background seismic noise levels across the United States as a function of geography, season, and time of day. The new statistical approach will provide the ability to compute probability density functions (PDFs) to evaluate the full range of noise at a given seismic station, allowing the estimation of noise levels over a broad range of frequencies from 0.01–16 Hz (100-0.0625s period). With the use of this new method it will be much easier to compare seismic noise characteristics between different networks in different regions.

Earthquake Ground Motion Animations

Seismometers of USArray transportable array record the passage of numerous seismic waves through a given point near the Earth's surface, and classically these seismograms are analyzed to deduce properties of the Earth's structure and the seismic source. Given a spatially dense set of seismic recordings, these signals can also be used to visualize the actual continuous seismic waves, providing new insights and interpretation techniques into complex wave propagation effects. Using signals recorded by the array of seismometers, the EarthScope project will be able to animate seismic waves as they sweep across the USArray transportable array for selected larger earthquakes. This will be able to illustrate the regional and teleseismic wave propagation phenomena. The seismic data collected from both permanent and transportable seismic stations will be used to provide these computer generated animations.

Regional Moment Tensors

The seismic moment tensor is one of the fundamental parameters of earthquakes that can be determined from seismic observations. It is directly related to earthquake fault orientation and rupture direction. The moment magnitude, Mw derived from the moment tensor magnitude, is the most reliable quantity for comparing and measuring the size of an earthquake with other earthquake magnitudes. Moment tensors are used in a wide range of seismological research fields, such as earthquake statistics, earthquake scaling relationships, and stress inversion. The creation of regional moment tensor solutions, with the appropriate software, for moderate-to-large earthquakes in the U.S. will be from USArray transportable array and Advance National Seismic System broadband seismic stations. Results are obtained in the time and the frequency domain. Waveform fit and amplitude-phase match figures are provided to allow users to evaluate moment tensor quality.

Geodetic Monitoring of the Western US and Hawaii

Global Positioning System (GPS) equipment and techniques provide a unique opportunity for earth scientists to study regional and local tectonic plate motions and conduct natural hazards monitoring. Cleaned network solutions from several GPS arrays have merged into regional clusters in conjunction with the EarthScope project. The arrays include the Pacific Northwest Geodetic Array, EarthScope's Plate Boundary Observatory, the Western Canadian Deformation Array, and networks run by the US Geological Survey. The daily GPS measurements from ~1500 stations along the Pacific/North American plate boundary provide millimeter-scale accuracy and can be used monitor the displacements of the earths crust. With the use of data modeling software and the recorded GPS data, the opportunity to quantify crustal deformation caused by plate tectonics, earthquakes, landslides and volcanic eruptions will be possible.

Time-dependent Strain

The goal is to provide models of time-dependent strain associated with a number of recent earthquakes and other geologic events as constrained by GPS data. With the use of InSAR (Interferometric Synthetic Aperture Radar), a remote-sensing technique, and PBO (Plate Boundary Observatory), a fixed array of GPS receivers and strainmeters, the EarthScope project will provide spatially continuous strain measurements over wide geographic areas with decimeter to centimeter resolution.

Global Strain Rate Map

The Global Strain Rate Map (GSRM) is a project of the International Lithosphere Program whose mission is to determine a globally self-consistent strain rate and velocity field model, consistent with geodetic and geologic field observations collected by GPS, seismometers, and strainometers. GSRM is a digital model of the global velocity gradient tensor field associated with the accommodation of present-day crustal motions. The overall mission also includes: (1) contributions of global, regional, and local models by individual researchers; (2) archive existing data sets of geologic, geodetic, and seismic information that can contribute toward a greater understanding of strain phenomena; and (3) archive existing methods for modeling strain rates and strain transients. A completed global strain rate map will provide a large amount of information which will contribute to the understanding of continental dynamics and for the quantification of seismic hazards.

Science

There are seven topics that EarthScope will address with the use of the observatories.

Convergent Margin Processes

Convergent margins, also known as convergent boundaries, are active regions of deformation between two or more tectonic plates colliding with one another. Convergent margins create areas of tectonic uplift, such as mountain ranges or volcanoes. EarthScope is focusing on the boundary between the Pacific Plate and the North American Plate in the western United States. EarthScope will provide GPS geodetic data, seismic images, detailed seismicity, magnetotelluric data, InSAR, stress field maps, digital elevation models, baseline geology, and paleoseismology for a better understanding of convergent margin processes.