Robust Moment Closure Method for the Chemical Master Equation

The Chemical Master Equation (CME) is used to stochastically model biochemical reaction networks, under the Markovian assumption. The low-order statistical moments induced by the CME are often the key quantities that one is interested in. However, in most cases, the moments equation is not closed; in the sense that the first $n$ moments depend on the higher order moments, for any positive integer $n$. In this paper, we develop a moment closure technique in which the higher order moments are approximated by an affine function of the lower order moments. We refer to such functions as the affine Moment Closure Functions (MCF) and prove that they are optimal in the worst-case context, in which no a priori information on the probability distribution is available. Furthermore, we cast the problem of finding the optimal affine MCF as a linear program, which is tractable. We utilize the affine MCFs to derive a finite dimensional linear system that approximates the low-order moments. We quantify the approximation error in terms of the $$ induced norm of some linear system. Our results can be effectively used to approximate the low-order moments and characterize the noise properties of the biochemical network under study

Dual Rate Control for Security in Cyber-physical Systems

We consider malicious attacks on actuators and sensors of a feedback system which can be modeled as additive, possibly unbounded, disturbances at the digital (cyber) part of the feedback loop. We precisely characterize the role of the unstable poles and zeros of the system in the ability to detect stealthy attacks in the context of the sampled data implementation of the controller in feedback with the continuous (physical) plant. We show that, if there is a single sensor that is guaranteed to be secure and the plant is observable from that sensor, then there exist a class of multirate sampled data controllers that ensure that all attacks remain detectable. These dual rate controllers are sampling the output faster than the zero order hold rate that operates on the control input and as such, they can even provide better nominal performance than single rate, at the price of higher sampling of the continuous output

On l∞ performance optimization: linear switched systems and systems with cone constraints

The l∞ performance of Linear Time-Invariant (LTI) systems has been one of the corner stones of the robust control theory for over the past 30 years. The l∞ performance has been studied mostly for LTI systems and the scarcity of the results for other types of systems is prominent in this area. This dissertation aims to depart from LTI systems and investigate the l∞ performance for other classes of systems. In particular, the l∞ performance of Linear Switched Systems (LSS) and of linear systems with cone constraints is studied in the first and second part of this dissertation, respectively. Part I: In Part I, we first consider the worst-case l∞ induced norm computation of LSS. That is, sup_σ‖G_σ‖, where G_σ is a LSS, σ is the switching sequence, and the norm, ‖.‖, is the l∞ induced norm. This problem can be linked to robustness of systems when the switching is arbitrary. We provide lower and upper bounds of this quantity. These bounds are hard to compute and in general conservative. Hence, we narrow our attention to special classes of LSS by defining the classes of input, output, and input-output LSS and show that for these classes, exact expressions for the worst-case l∞ induced norm can be found. Moreover, we introduce the class of generalized input-output LSS and show how their l∞ gains can be computed exactly via Linear Programming (LP). The class of generalized input-output LSS proves to be a sufficiently rich class as it is dense in the set of all stable LSS. We further derive new stability and stabilizability conditions and control synthesis in terms of LP utilizing generalized input-output LSS. The other extreme from the worst-case norm is the minimal norm, i.e., inf_σ ‖G_σ‖. The interest in this type of problem is motivated by situations where there may be limited sensor and/or actuator resources for filtering and control. We show that for Finite Impulse Response (FIR)switching systems the minimizing switching sequence can be chosen to be periodic. For input-only or output-only switching systems an exact characterization of the minimal l∞ gain is provided, and it is shown that the minimizing switching sequence is constant, which, as also shown, is not true for input-output switching. Moreover, we study Markov Linear Switched Systems (MLSS). These are LSS whose switching sequence is a Markov process. We introduce the notion of the stochastic l∞ gain and provide exact expression to compute it. However, this computation is challenging, as we show, and hence we resort to a more relaxed but tractable notion of l∞ mean performance. We provide tractable computation and control synthesis method with respect to the l∞ mean performance. Part II: Part II of this dissertation deals with the l∞ gain of linear systems with positivity type of constraints. The study of such systems is well justified as there are many physical problems in which some variables are restricted be non-negative (or non-positive); examples can be found in biology, economics, and many other areas. We consider the case when the output is forced to be in the positive l∞ cone when the input is in this cone. This reflects as, so-called, an external positivity constraint on the system. As we point out, if such a constraint is imposed on the closed loop map, finding an optimal controller is LP and hence a tractable problem. If, on the other hand, the constraint known as internal positivity is sought, we show that a dynamic controller offers no advantage over a static one. These results can be used to obtain an optimal (static) state feedback controller. However, designing an optimal output feedback controller (which is static) is a harder problem and in general leads to a bilinear program. We show that this bilinear program can be reduced to LP, if the null space of the measurement matrix is invariant under multiplication by diagonal matrices. Besides the positive systems mentioned above, we consider the case where only the input is restricted to be in the positive cone of l∞, denoted by l∞+, and seek to characterize the induced norm from l∞+ to l∞. We stress here that no positivity constraint is imposed on the system itself. As an example, consider a positive nonlinear system with positive input that is linearized about a point other than origin. The linearized model is no longer a positive system as it is not linearized about the origin. Its inputs, however, remain positive and hence fit into this class of problems. We obtain an exact characterization of this norm (the induced norm from l∞+ to l∞ which can be used to synthesis a controller minimizing the induced norm from l∞+ to l∞ via LP

The Enhanced Finite State Projection algorithm, using conditional moment closure and time-scale separation

© 2020 IEEE. The Chemical Master Equation (CME) is commonly used to describe the stochastic behavior of biomolecular systems. However, in general, the CME's dimension is very large or infinite, so analytical or even numerical solutions may be difficult to achieve. The truncation methods such as the Finite State Projection (FSP) algorithm alleviate this issue to some extent but not completely. To further resolve the computational issue, we propose the Enhanced Finite State Projection (EFSP) algorithm, in which the ubiquitous time-scale separation is utilized to reduce the dimension of the CME. Our approach combines the original FSP algorithm and the model reduction technique that we developed, to approximate an infinite dimensional CME with a finite dimensional CME that contains the slow species only. Unlike other time-scale separation methods, which rely on the fast-species counts' stationary conditional probability distributions, our model reduction technique relies on only the first few conditional moments of the fast-species counts. In addition, each iteration of the EFSP algorithm relies on the solution of the approximated CME that contains the slow species only, unlike the original FSP algorithm relies on the solution of the full CME. These two properties provide a significant computation advantage. The benefit of our algorithm is illustrated through a protein binding reaction example

Approximation of the Chemical Master Equation using conditional moment closure and time-scale separation

© 2019 American Automatic Control Council. To describe the stochastic behavior of biomolecular systems, the Chemical Master Equation (CME) is widely used. The CME gives a complete description of the evolution of a system's probability distribution. However, in general, the CME's dimension is very large or even infinite, so analytical solutions may be difficult to write and analyze. To handle this problem, based on the fact that biomolecular systems are time-scale separable, we approximate the CME with another CME that describes the dynamics of the slow species only. In particular, we assume that the number of each molecular species is bounded, although it may be very large. We thus write Ordinary Differential Equations (ODEs) of the slow-species counts' marginal probability distribution and of the fast-species counts' firstN conditional moments. Here, N is an arbitrary (possibly small) number, which can be chosen to compromise between approximation accuracy and the computational burden associated with simulating or analyzing a high dimensional system. Then we apply conditional moment closure and timescale separation to approximate the first N conditional moments of the fast-species counts as functions of the slow-species counts. By subsituting these functions on the right-hand side of the ODEs that describes the marginal probability distribution of the slow-species counts, we can approximate the original CME with a lower dimensional CME. We illustrate the application of this method on an enzymatic and a protein binding reaction