Duarte Dj Guerreiro Tomé Antunes

Self-triggered linear quadratic control

a b s t r a c t Self-triggered control is a recently proposed paradigm that abandons the more traditional periodic time- triggered execution of control tasks with the objective of reducing the utilization of communication resources, while still guaranteeing desirable closed-loop behavior. In this paper, we introduce a self- triggered strategy based on performance levels described by a quadratic discounted cost. The classical LQR problem can be recovered as an important special case of the proposed self-triggered strategy. The self-triggered strategy proposed in this paper possesses three important features. Firstly, the control laws and triggering mechanisms are synthesized so that a priori chosen performance levels are guaranteed by design. Secondly, they realize significant reductions in the usage of communication resources. Thirdly, we address the co-design problem of jointly designing the feedback law and the triggering condition. By means of a numerical example, we show the effectiveness of the presented strategy. In particular, for the self-triggered LQR strategy, we show quantitatively that the proposed scheme can outperform conventional periodic time-triggered solutions.

Rollout event-triggered control : beyond periodic control performance

Cyber-Physical Systems (CPSs) resulting from the interconnection of computational, communication, and control (cyber) devices with physical processes are wide spreading in our society. In several CPS applications it is crucial to minimize the communication burden, while still providing desirable closed-loop control properties. To this effect, a promising approach is to embrace the recently proposed event-triggered control paradigm, in which the transmission times are chosen based on well-defined events, using state information. However, few general event-triggered control methods guarantee closed-loop improvements over traditional periodic transmission strategies. Here, we provide a new class of event-triggered controllers for linear systems which guarantee better quadratic performance than traditional periodic time-triggered control using the same average transmission rate. In particular, our main results explicitly quantify the obtained performance improvements for quadratic average cost problems. The proposed controllers are inspired by rollout ideas in the context of dynamic programming.

Volterra Integral Approach to Impulsive Renewal Systems: Application to Networked Control

We analyze impulsive systems with independent and identically distributed intervals between transitions. Our approach involves the derivation of novel results for Volterra integral equations with positive kernel. We highlight several applications of these results, and show that when applied to the analysis of impulsive systems they allow us to (i) provide necessary and sufficient conditions for mean square stability, stochastic stability and mean exponential stability, which can be equivalently tested in terms of a matrix eigenvalue computation, an LMI feasibility problem, and a Nyquist criterion condition; (ii) assess performance of the impulsive system by computing a second moment Lyapunov exponent. The applicability of our results is illustrated in a benchmark problem considering networked control systems with stochastically spaced transmissions, for which we can guarantee stability for inter-sampling times roughly twice as large as in previous papers.

Stability of networked control systems with asynchronous renewal links: An impulsive systems approach

We consider networked control systems in which sensors, actuators, and controller transmit through asynchronous communication links, each introducing independent and identically distributed intervals between transmissions. We model these scenarios through impulsive systems with several reset maps triggered by independent renewal processes, i.e., the intervals between jumps associated with a given reset map are identically distributed and independent of the other jump intervals. For linear dynamic and reset maps, we establish that mean exponential stability is equivalent to the spectral radius of an integral operator being less than one. We also prove that the origin of a non-linear impulsive system is (locally) stable with probability one if its local linearization about the zero equilibrium is mean exponentially stable, which justifies the importance of studying the linear case. The applicability of the results is illustrated through an example using a linearized model of a batch-reactor.

Dynamic programming formulation of periodic event-triggered control: Performance Guarantees and co-design

While potential benefits of choosing the transmissions times in a networked control system based on state or event information have been advocated in the literature, few general methods are available that guarantee closed-loop improvements over traditional periodic transmission strategies. In this paper, we propose event-triggered controllers that guarantee better quadratic discounted cost performance than periodic control strategies using the same average transmission rate. Moreover, we show that the performance of a method in the line of previous Lyapunov based approaches is within a multiplicative factor of periodic control performance, while using less transmissions. Our approach is based on a dynamic programming formulation for the co-design problem of choosing both transmission decisions and control inputs in the context of periodic event-triggered control for linear systems. A numerical example illustrates the advantages of the proposed method over traditional periodic control.

Stochastic hybrid systems with renewal transitions : moment analysis with application to networked control systems with delays

We consider stochastic hybrid systems (SHSs) for which the lengths of times that the system stays in each mode are independent random variables with given distributions. We propose an approach based on a set of Volterra equations to compute any statistical moment of the state of the SHS. Moreover, we provide a method to compute the Lyapunov exponents of a given degree, i.e., the exponential rate of decrease or increase at which statistical moments converge to zero or to infinity, respectively. We also discuss how, by computing the statistical moments, one can provide information about the probability distribution of the state of the SHS. The applicability of the results is illustrated in the analysis of a networked control problem with independently distributed intervals between data transmissions and delays.

Intercellular Variability in Protein Levels from Stochastic Expression and Noisy Cell Cycle Processes

Inside individual cells, expression of genes is inherently stochastic and manifests as cell-to-cell variability or noise in protein copy numbers. Since proteins half-lives can be comparable to the cell-cycle length, randomness in cell-division times generates additional intercellular variability in protein levels. Moreover, as many mRNA/protein species are expressed at low-copy numbers, errors incurred in partitioning of molecules between two daughter cells are significant. We derive analytical formulas for the total noise in protein levels when the cell-cycle duration follows a general class of probability distributions. Using a novel hybrid approach the total noise is decomposed into components arising from i) stochastic expression; ii) partitioning errors at the time of cell division and iii) random cell-division events. These formulas reveal that random cell-division times not only generate additional extrinsic noise, but also critically affect the mean protein copy numbers and intrinsic noise components. Counter intuitively, in some parameter regimes, noise in protein levels can decrease as cell-division times become more stochastic. Computations are extended to consider genome duplication, where transcription rate is increased at a random point in the cell cycle. We systematically investigate how the timing of genome duplication influences different protein noise components. Intriguingly, results show that noise contribution from stochastic expression is minimized at an optimal genome-duplication time. Our theoretical results motivate new experimental methods for decomposing protein noise levels from synchronized and asynchronized single-cell expression data. Characterizing the contributions of individual noise mechanisms will lead to precise estimates of gene expression parameters and techniques for altering stochasticity to change phenotype of individual cells.

Control of impulsive renewal systems: Application to direct design in networked control

We consider the control of impulsive systems with independent and identically distributed intervals between jumps. The control action and output measurement are assumed to take place only at jump times. We give necessary and sufficient conditions, in the form of LMIs, for mean square stabilizability and detectability and solve an infinite horizon quadratic optimal control problem, under appropriate stabilizability and detectability properties of the system. The class of systems considered is especially suited to model networked control systems utilizing CSMA-type protocols, with stochastic intervals between transmissions and packet drops. In this setting, the analysis and synthesis tools mentioned above are used to (i) prove that for an emulation-based design, stability of the closed-loop is preserved if the distribution of the intervals between transmissions assigns high probability to fast sampling (ii) illustrate through a benchmark example the potential advantages of controller direct-design over an emulation-based design.

Quantifying gene expression variability arising from randomness in cell division times.

The level of a given mRNA or protein exhibits significant variations from cell-to-cell across a homogeneous population of living cells. Much work has focused on understanding the different sources of noise in the gene-expression process that drive this stochastic variability in gene-expression. Recent experiments tracking growth and division of individual cells reveal that cell division times have considerable inter-cellular heterogeneity. Here we investigate how randomness in the cell division times can create variability in population counts. We consider a model by which mRNA/protein levels in a given cell evolve according to a linear differential equation and cell divisions occur at times spaced by independent and identically distributed random intervals. Whenever the cell divides the levels of mRNA and protein are halved. For this model, we provide a method for computing any statistical moment (mean, variance, skewness, etcetera) of the mRNA and protein levels. The key to our approach is to establish that the time evolution of the mRNA and protein statistical moments is described by an upper triangular system of Volterra equations. Computation of the statistical moments for physiologically relevant parameter values shows that randomness in the cell division process can be a major factor in driving difference in protein levels across a population of cells.

Stability of impulsive systems driven by renewal processes

Necessary and sufficient conditions are provided for stochastic stability and mean exponential stability of impulsive systems with jumps triggered by a renewal process, that is, the intervals between jumps are independent and identically distributed. The conditions for stochastic stability can be efficiently tested in terms of the feasibility of a set of LMIs or in terms of an algebraic test. The relation between the different stability notions for this class of systems is also discussed. The results are illustrated through their application to the stability analysis of networked control systems. We present two benchmark examples for which one can guarantee stability for inter-sampling times roughly twice as large as in a previous paper.