Masako Kishida

Optimal control of cellular uptake in tissue engineering

The optimal control of a distributed parameter system with reaction, diffusion, and convection is investigated. The problem is motivated by tissue engineering where the control of the uptake of growth factors (signaling molecules) is required to spatially and temporally regulate cellular processes for the growth or regeneration of a tissue. Four approaches for solving the optimal control problem are compared: (i) basis function expansion, (ii) method of moments, (iii) internal model control, and (iv) model predictive control. This comparison suggests that these approaches should be combined to solve the optimal control problem for multiple spatial dimensions.

Efficient Polynomial-Time Outer Bounds on State Trajectories for Uncertain Polynomial Systems Using Skewed Structured Singular Values

Outer bounds for the evolution of state trajectories of uncertain systems are useful for many purposes such as robust control, state and parameter estimation, model invalidation, safety evaluation, fault diagnosis, and experimental design. Obtaining tight outer bounds is, however, a challenging task. This technical note proposes a new approach to obtaining such bounds for discrete-time polynomial systems with uncertain initial state, uncertain parameters, and bounded disturbances. To obtain outer bounds, the nonlinear map describing the uncertain dynamical system is represented via linear fractional transformation. The bounds on the trajectories are obtained by computing polynomial-time upper and lower bounds for the skewed structured singular value of the linear fractional transformation. Algorithms with different tradeoffs between computational complexity and conservatism are outlined. The tradeoffs as well as efficiency of the approach are illustrated in a numerical example, which shows small conservatism of the obtained bounds.

State-constrained optimal spatial field control for controlled release in tissue engineering

Distributed parameter control problems involving manipulation within the spatial domain arise in a variety of applications including vibration control, active noise reduction, epidemiology, tissue engineering, and cancer treatment. A state-constrained spatial field control problem motivated by a biomedical application is solved in which the manipulation occurs over a spatial field and the state field is constrained both in spatial frequency and by a partial differential equation (PDE) that effects the manipulation. An optimization algorithm combines three-dimensional Fourier series, which are truncated to satisfy the spatial frequency constraints, with exploitation of structural characteristics of the PDEs. The computational efficiency and performance of the optimization algorithm are demonstrated in a numerical example, for which the spatial tracking error is almost entirely due to the limitation on the spatial frequency of the manipulated field. The numerical results suggest that optimal control approaches have promise for controlling the release of macromolecules in tissue engineering applications.

A model-based approach for the construction of design spaces in quality-by-design

An algorithm is proposed for the characterization of the set of allowable real parametric uncertainties that achieve output specifications for nonlinear systems. The approach first expands the system output by a multivariate polynomial or rational function of perturbations of the real parameters, which is then written as a linear fractional transformation. Bounds on the uncertainty set are computed for the expansion using the skewed structured singular value. The proposed algorithm is demonstrated to be effective for the tracking problem of determining the allowable speed and angle for throwing a ball to land within a target and for a pharmaceutical crystallization in which an alternative solution method is available.

Ellipsoid bounds on state trajectories for discrete-time systems with time-invariant and time-varying linear fractional uncertainties

Polynomial-time algorithms are proposed for computing tight ellipsoidal bounds on the state trajectories of discrete-time linear systems with time-varying or time-invariant linear fractional parameter uncertainties and ellipsoidal uncertainty in the initial state. The approach employs linear matrix inequalities to determine an initial estimate of the ellipsoid, which is improved by the subsequent application of the skewed structured singular value. Tradeoffs between computational complexity and conservatism are discussed for the three algorithms. Small conservatism for the tightest bounds is observed in numerical examples used to compare the algorithms.

Robustness Analysis, Prediction and Estimation for Uncertain Biochemical Networks

Abstract Mathematical models of biochemical reaction networks are important tools in systems biology and systems medicine to understand the reasons for diseases like cancer, and to make predictions for the development of effective treatments. In synthetic biology, for instance, models are used for the design of circuits to reliably perform specialized tasks. For analysis and predictions, plausible and reliable models are required, i.e., models must reflect the properties of interest of the considered biochemical networks. One remarkable property of biochemical networks is robust functioning over a wide range of perturbations and environmental conditions. Plausible mathematical models of such robust networks should also be robust. However, capturing, describing, and analyzing robustness in biochemical reaction networks is challenging. First, including uncertainty in the structures, parameters, and perturbations into the model is not straightforward due to different types of uncertainties encountered. Second, robustness as well as system and thus model properties are often itself inherently uncertain, such as qualitative (i.e., nonquantitative) descriptions. Finally, analyzing nonlinear models subject to different uncertainties and with respect to quantitative and qualitative properties is still in its infancy. In the first part of this perspective article, network functions and behaviors of interest are formally defined. Furthermore, different classes of uncertainties and perturbations in the data and model are consistently described. In the second part, we review frequently used approaches and present our own recent developments for robustness analysis, estimation, and model-based prediction. We illustrate their capabilities to deal with the different types of uncertainties and robustness requirements.

Efficient polynomial-time outer bounds on state trajectories for uncertain polynomial systems using skewed structured singular values

Outer bounds for the evolution of state trajectories of uncertain systems are useful for many purposes such as robust control design, state and parameter estimation, model invalidation, safety evaluation, fault detection and diagnosis, and experimental design. Obtaining tight outer bounds for trajectory bundles of uncertain nonlinear systems subject to disturbances is a challenging task. This paper proposes a new approach to obtaining such bounds with reasonable computational requirements for discrete-time polynomial systems with uncertain initial state and fixed and/or time-varying uncertain parameters and disturbances. The map describing the dynamics of the uncertain system is reformulated as a linear fractional transformation, which is used to bound the state trajectories by employing polynomial-time algorithms for computing upper and lower bounds for the skewed structured singular value. Three different algorithms to efficiently calculate outer bounds on the evolution of the system trajectories are presented, which have different tradeoffs between computational complexity and conservatism. The tradeoffs are illustrated in numerical examples, which demonstrate the small conservatism of the tightest bounds.

Structured spatial control of the reaction-diffusion equation with parametric uncertainties

Feedback control problems for distributed parameter systems arise in a variety of physical, chemical, biological, and mechanical systems. This paper exploits the algebraic structure of the system of ordinary differential equations that arise from spatial discretization of the partial differential equation (PDE) to analyze and design feedback controllers that are robust to bounded perturbations in the parameters of the original PDE. As a prototypical problem, this paper investigates the spatial field control of a reaction-diffusion system whose spatial discretization has a state matrix that is circulant symmetric. Structured robust controllers are designed based on internal model control and mixed sensitivity optimization. The controllers are shown to be robust to inaccuracies in the spatial manipulation, even for arbitrarily fine spatial discretizations.

Optimal spatial field control of distributed parameter systems

Optimal control problems are formulated and solved in which the manipulation is distributed over a three-dimensional (3D) spatial field with constraints on the spatial variation. These spatial field control problems that arise in applications in acoustics, structures, epidemiology, cancer treatment, and tissue engineering have much higher controllability than boundary control problems, but have vastly higher degrees of freedom. Efficient algorithms are developed for computing optimal manipulated fields by combination of modal analysis and least-squares optimization over a basis function space. Small minimum control error is observed in applications to distributed parameter systems with reaction, diffusion, and convection.

On the Analysis of the Eigenvalues of Uncertain Matrices by

Based on the structured singular value μ and the skewed structured singular value v, this technical note presents several useful relationships between an uncertain matrix expressed in a linear fractional form and its eigenvalues. The results are used to derive 1) sufficient conditions to avoid bifurcations for systems with parametric uncertainties and 2) bounds on the convergence rate for uncertain stable matrices and uncertain Markov matrices. Illustrative examples are also provided.