Model reduction in the presence of uncertainty in model parameters

This short note describes how to extend a certain class of existing model reduction techniques to take into account uncertainty in model parameters. The key idea of this extension is that the reduced-order model should not only contain the model parameters, but that the reduction procedure itself has to be geared for dealing with parametric uncertainty. This goal is achieved by augmenting the vector of inputs to the system with the uncertain parameters and by performing model reduction on the augmented system. It is shown that error bounds for the reduced-order model can be computed if the underlying system is linear with respect to the states, parameters, and inputs. A comparison between the presented technique and a conventional approach is made via two examples.     

Passivity preserving model reduction via interpolation of spectral zeros

An algorithm is developed for passivity preserving model reduction of linear time invariant systems. Implementation schemes are described for both medium scale (dense) and large scale (sparse) applications. The algorithm is based upon interpolation at selected spectral zeros of the original transfer function to produce a reduced transfer function that has the specified roots as its spectral zeros. These interpolation conditions are satisfied through the computation of a basis for a selected invariant subspace of a certain block matrix which has the spectral zeros as its spectrum. Explicit interpolation is avoided and passivity of the reduced model is established, instead, through satisfaction of the necessary conditions of the Positive Real Lemma. It is also shown that this procedure indirectly solves the associated controllability and observability Riccati equations and how to select the interpolation points to give maximal or minimal solutions of these equations. From these, a balancing transformation may be constructed to give a reduced model that is balanced as well as passive and stable.     

Model reduction of systems with zeros interlacing the poles

The problem of reducing a system with zeros interlacing the poles (ZIP) on the real axis is considered. It is proved that many model reduction methods, such as the balanced truncation, balanced residualization, suboptimal and optimal Hankel approximations, inherit the ZIP property. Properties of the Hankel singular values of ZIP systems are also listed.     

Model reduction and model predictive control of energy-efficient buildings for electrical heating load shifting

In France, buildings account for a significant portion of the electricity consumption (around 68%), due to an important use of electrical heating systems. This results in high peak load in winter and causes tensions on the production-consumption balance. In view of reducing such fluctuations, advanced control systems (including the Model Predictive Control framework) have been developed to shift heating load while maintaining indoor comfort and taking advantage of the building thermal mass. In this paper, a framework for developing optimisation-based control strategies to shift the heating load in buildings is introduced. The balanced truncation method and a time-continuous optimisation method were used to develop a real-time control of the heating power. These two methods are well suited for control problems and yield precise results. The novelty of the approach is to use reduced models derived from advanced building simulation software. A simulation case study demonstrates the controller performance in the synthesis of a predictive model-based optimal energy management strategy for a single-zone test building of the “INCAS” platform built in Le Bourget-du-Lac, France, by the National Solar Energy Institute (INES). The controller exhibits excellent performance, reaching between 6 and 13% cost reduction, and can easily be applied in real-time.     

Principle of frequency-domain balanced structure in linear systems and model reduction

In this paper a novel model reduction technique for linear time-invariant systems is presented. The proposed technique is based on a conceptual viewpoint regarding the balancing of the controllability and observability Gramians of a multivariable system in a given range of frequency. The conditions for the stability of the reduced model are also provided. From a real-time applicability viewpoint, the frequency-domain balanced structure provides an attractive approach to the model reduction of large scaled systems. The simulation results establish the effectiveness of this proposed method compared to the effectiveness of existing techniques.     

Comparison of theoretical and computational characteristics of dimensionality reduction methods for large-scale uncertain systems

Synthesizing optimal controllers for large scale uncertain systems is a challenging computational problem. This has motivated the recent interest in developing polynomial-time algorithms for computing reduced dimension models for uncertain systems. Here we present algorithms that compute lower dimensional realizations of an uncertain system, and compare their theoretical and computational characteristics. Three polynomial-time dimensionality reduction algorithms are applied to the Shell Standard Control Problem, a continuous stirred-tank reactor (CSTR) control problem, and a large scale benchmark problem, where it is shown that the algorithms can reduce the computational effort of optimal controller synthesis by orders of magnitude. These algorithms allow robust controller synthesis and robust control structure selection to be applied to uncertain systems of increased dimensionality.     

Optimal complexity reduction of polyhedral piecewise affine systems

This paper focuses on the NP-hard problem of reducing the complexity of piecewise polyhedral systems (e.g. polyhedral piecewise affine (PWA) systems). The results are fourfold. Firstly, the paper presents two computationally attractive algorithms for optimal complexity reduction that, under the assumption that the system is defined over the cells of a hyperplane arrangement, derive an equivalent polyhedral piecewise system that is minimal in the number of polyhedra. The algorithms are based on the cells and the markings of the hyperplane arrangement. In particular, the first algorithm yields a set of disjoint (non overlapping) merged polyhedra by executing a branch and bound search on the markings of the cells. The second approach leads to non-disjoint (overlapping) polyhedra by formulating and solving an equivalent (and well-studied) logic minimization problem. Secondly, the results are extended to systems defined on general polyhedral partitions (and not on cells of hyperplane arrangements). Thirdly, the paper proposes a technique to further reduce the complexity of piecewise polyhedral systems if the introduction of an adjustable degree of error is acceptable. Fourthly, the paper shows that based on the notion of the hyperplane arrangement PWA state feedback control laws can be implemented efficiently. Three examples, including a challenging industrial problem, illustrate the algorithms and show their computational effectiveness in reducing the complexity by up to one order of magnitude.     

Optimization of a thin film deposition process using a dynamic model extracted from molecular simulations

This study presents and demonstrates an algorithm for computing a dynamic model for a thin film deposition process. The proposed algorithm is used on high dimensional Kinetic Monte Carlo (KMC) simulations and consists of applying principal component analysis (PCA) for reducing the state dimension, a self organizing map (SOM) for grouping similar surface configurations and simple cell mapping (SCM) for identifying the transitions between different surface configuration groups. The error associated with this model reduction approach is characterized by running more than 1000 test simulations with highly dynamic and random input profiles. The global error, which is the normalized Euclidean distance between the simulated and predicted states, is found to be less than 1% on average relative to the test simulation results. This indicates that our reduced order dynamic model, which was developed using a rather small simulation set, was able to accurately predict the evolution of the film microstructure for much larger simulation sets and a wide range of process conditions. Minimization of the deposition time to reach a desired film structure has also been achieved using this model. Hence, our study showed that the proposed algorithm is useful for extracting dynamic models from high dimensional and noisy molecular simulation data.     

On the balanced truncation and coprime factors reduction of Markovian jump linear systems

The paper deals with the balanced truncation and coprime factors reduction of Markovian jump linear (MJL) systems, which can have mode-varying state, input, and output dimensions. We develop machinery for balancing mean square stable MJL system realizations using generalized Gramians and strict Lyapunov inequalities, and provide an improved a priori upper bound on the error induced in the balanced truncation process. We also generalize the coprime factors reduction method and, in doing so, extend the applicability of the balanced truncation technique to the class of mean square stabilizable and detectable MJL systems. We provide tools to establish mean square stabilizability and detectability of the considered MJL systems. In addition, a notion of right-coprime factorization of MJL systems and methods to construct such factorizations are given. The error measure in the coprime factors reduction approach, while still norm-based, does not directly capture the mismatch between the nominal system and the reduced-order model, as is the case in the balanced truncation approach where mean square stable models are considered. Instead, the error measure is given in terms of the distance between the coprime factors realizations, and thus has an interpretation in terms of robust feedback stability. The paper concludes with an illustrative example which demonstrates how to apply the coprime factors model reduction approach.     

IDD-based model validation of biochemical networks

This paper presents efficient techniques for the qualitative and quantitative analysis of biochemical networks, which are modeled by means of qualitative and stochastic Petri nets, respectively. The analysis includes standard Petri net properties as well as model checking of the Computation Tree Logic and the Continuous Stochastic Logic. Efficiency is achieved by using Interval decision diagrams to alleviate the well-known problem of state space explosion, and by applying operations exploiting the Petri structure and the principle of locality. All presented techniques are implemented in our tool IDD-MC which is available on our website.     

Model reduction and control of reactor–heat exchanger networks

This paper focuses on the dynamics and control of process networks consisting of a reactor connected with an external heat exchanger through a large material recycle stream that acts as an energy carrier. Using singular perturbation arguments, we show that such networks exhibit a dynamic behavior featuring two time scales: a fast one, in which the energy balance variables evolve, and a slow time scale that captures the evolution of the terms in the material balance equations. We present a procedure for deriving reduced-order, non-stiff models for the fast and slow dynamics, and a framework for rational control system design that accounts for the time scale separation exhibited by the system dynamics. The theoretical developments are illustrated with an example and numerical simulation results.     

On H2 model reduction of bilinear systems

The H₂ model reduction problem for continuous-time bilinear systems is studied in this paper. By defining the H₂ norm of bilinear systems in terms of the state-space matrices, the H₂ model reduction error is computed via the reachability or observability gramian. Necessary conditions for the reduced order bilinear models to be H₂ optimal are given. The gradient flow approach is used to obtain the solution of the H₂ model reduction problem. The formulation allows certain properties of the original models to be preserved in the reduced order models. The model reduction procedure developed can also be applied to finite-dimensional linear time-invariant systems. A numerical example is employed to illustrate the effectiveness of the proposed method.     

Model reduction of uncertain systems retaining the uncertainty structure

Model reduction of high order linear-in-parameters discrete-time systems is considered. The main novelty of the paper is that the coefficients of the original system model are assumed to be known only within given intervals, and the coefficients of the derived reduced order model are also obtained in intervals, such that the complex value sets of the uncertain original and reduced models will be optimally close to each other on the unit circle. The issue of inclusion of one value set in another is also addressed in the paper. The meaning of model reduction is defined for linear-in-parameters systems. The algorithm for obtaining the value sets of such systems is derived in the paper. Then, applying a novel approach, the infinity norm of “distance” between two polygons representing the original and the reduced uncertain systems is minimized. A noteworthy point is that by a special definition of this distance the problem is formulated as a linear semi-infinite programming problem with linear constraints, thus reducing significantly the computational complexity. Numerical example is provided.     

Structure-preserving model reduction of nonlinear building thermal models

This paper proposes an aggregation-based model reduction method for nonlinear models of multi-zone building thermal dynamics. The full-order model, which is already a lumped-parameter approximation, quickly grows in state space dimension as the number of zones increases. An advantage of the proposed method, apart from being applicable to the nonlinear thermal models, is that the reduced model obtained has the same structure and physical intuition as the original model. The key to the methodology is an analogy between a continuous-time Markov chain and the linear part of the thermal dynamics. A recently developed aggregation-based method of Markov chains is employed to aggregate the large state space of the full-order model into a smaller one. Simulations are provided to illustrate tradeoffs between modeling error and computation time.     

Identification of minimal metabolic pathway models consistent with phenotypic data

In metabolic systems, the cellular network of metabolic reactions together with constraints of (ir)reversibility of enzymes determines the space of all possible steady-state phenotypes. Analysis of large metabolic models, however, is not feasible in real-time and identification of a smaller model without loss of accuracy is desirable for model-based bioprocess optimization and control. To this end, we propose two search algorithms for systematic identification of a subset of pathways that match the observed cellular phenotype relevant for a particular process condition. Central carbon metabolism of Escherichia coli was used as a case-study together with three phenotypic datasets obtained from the literature. The first search method is based on ranking pathways and the second is a controlled random search (CRS) algorithm. Since we wish to obtain a biologically realistic subset of pathways, the objective function to be minimized is a trade-off between the error and investment costs. We found that the CRS outperforms the ranking algorithm, as it is less likely to fall into local minima. In addition, we compared two pathway analysis methods (elementary modes versus generating vectors) in terms of modelling accuracy and computational intensity. We conclude that generating vectors have preference over elementary modes to describe a particular phenotype. Overall, the original model containing 433 generating vectors or 2706 elementary modes could be reduced to a system of one to three pathways giving a good correlation with the measured datasets. We consider this work as a first step towards the use of detailed metabolic models to improve real-time optimization, monitoring, and control of biological processes.