Bayu Jayawardhana

On the Mathematical Structure of Balanced Chemical Reaction Networks Governed by Mass Action Kinetics

Motivated by recent progress on the interplay between graph theory, dynamics, and systems theory, we revisit the analysis of chemical reaction networks described by mass action kinetics. For reaction networks possessing a thermodynamic equilibrium we derive a compact formulation exhibiting at the same time the structure of the complex graph and the stoichiometry of the network, and which admits a direct thermodynamical interpretation. This formulation allows us to easily characterize the set of positive equilibria and their stability properties. Furthermore, we develop a framework for interconnection of chemical reaction networks, and we discuss how the formulation leads to a new approach for model reduction.

Passivity of nonlinear incremental systems : Application to PI stabilization of nonlinear RLC circuits

It is well known that if the linear time invariant system xdot = Ax + Bu, y = Cx is passive the associated incremental system xtilde = Axtilde + Butilde, ytilde = Cxtilde, with (middot) = (middot) - (middot)*, u*, y*, the constant input and output associated to an equilibrium state x* , is also passive. In this paper, we identify a class of nonlinear passive systems of the form x = f(x) +gu, y = h(x) whose incremental model is also passive. Using this result we then prove that general nonlinear RLC circuits with convex and proper electric and magnetic energy functions and passive resistors with monotonic characteristic functions are globally stabilizable with linear PI control

Coordination of Passive Systems under Quantized Measurements

In this paper we investigate a passivity approach to collective coordination and synchronization problems in the presence of quantized measurements and show that coordination tasks can be achieved in a practical sense for a large class of passive systems.

The Circle Criterion and Input-to-State Stability

In this paper, the focus is on absolute stability and input-to-state stability of the feedback interconnection of an infinite-dimensional linear system Σ and a nonlinearity Φ : dom(Φ) ⊂ Lloc(R+, Y ) → L 2 loc(R+, U), where dom(Φ) denotes the domain of Φ and U and Y (Hilbert spaces) denote the input and output spaces of Σ, respectively (see Figure 1, wherein v is an essentially bounded input signal). The system Σ is assumed to belong to the rather general class of well-posed systems (see, for example, [11, 13] and the references therein) and the nonlinearity is assumed to satisfy a (generalized) sector condition. In the literature on the circle criterion for infinite-dimensional systems (see, for example, [3, 4, 5, 7, 9, 12], and the references therein), the emphasis is usually on Lor L-stability and global asymptotic or global exponential stability (or some variants thereof) of feedback systems of the type shown in Figure 1, with a static sector-bounded nonlinearity Φ in the feedback path. The new contribution of this paper as compared to the previous literature is twofold. (i) In addition to static nonlinearities, we include a class of dynamic nonlinearities which may exhibit bias, but still satisfy a generalized pointwise sector condition.

On the Internal Model Principle in the Coordination of Nonlinear Systems

The role of the internal model principle is investigated in this paper for the coordination of relative-degree-one and relative-degree-two nonlinear systems. For relative-degree-one systems that are incrementally (output-feedback) passive, we propose internal-model-based distributed control laws which guarantee output synchronization to an invariant manifold driven by autonomous synchronized internal models. For relative-degree-two systems, we consider a different internal-model-based distributed control framework for solving a formation control problem where the agents have to track a reference signal available only to the leader agent. In both cases, the local controller is also able to reject the disturbance signals generated by a local exosystem.

Brief paper: Tracking and disturbance rejection for fully actuated mechanical systems

In this paper, we solve the tracking and disturbance rejection problem for fully actuated passive mechanical systems. We assume that the reference signal r and its first two derivatives r@?,r@? are available to the controller and the disturbance signal d can be decomposed into a finite superposition of sine waves of arbitrary but known frequencies and an arbitrary L^2 signal. We combine the internal model principle with the ideas behind the Slotine-Li adaptive controller. The internal model-based adaptive controller that we propose causes the closed-loop state trajectories to be bounded, and the tracking error and its derivative to converge to zero, without any prior knowledge of the plant parameters. An important part of our results is that we prove the existence and uniqueness of the state trajectories of the closed-loop system.

A model reduction method for biochemical reaction networks

BackgroundIn this paper we propose a model reduction method for biochemical reaction networks governed by a variety of reversible and irreversible enzyme kinetic rate laws, including reversible Michaelis-Menten and Hill kinetics. The method proceeds by a stepwise reduction in the number of complexes, defined as the left and right-hand sides of the reactions in the network. It is based on the Kron reduction of the weighted Laplacian matrix, which describes the graph structure of the complexes and reactions in the network. It does not rely on prior knowledge of the dynamic behaviour of the network and hence can be automated, as we demonstrate. The reduced network has fewer complexes, reactions, variables and parameters as compared to the original network, and yet the behaviour of a preselected set of significant metabolites in the reduced network resembles that of the original network. Moreover the reduced network largely retains the structure and kinetics of the original model.ResultsWe apply our method to a yeast glycolysis model and a rat liver fatty acid beta-oxidation model. When the number of state variables in the yeast model is reduced from 12 to 7, the difference between metabolite concentrations in the reduced and the full model, averaged over time and species, is only 8%. Likewise, when the number of state variables in the rat-liver beta-oxidation model is reduced from 42 to 29, the difference between the reduced model and the full model is 7.5%.ConclusionsThe method has improved our understanding of the dynamics of the two networks. We found that, contrary to the general disposition, the first few metabolites which were deleted from the network during our stepwise reduction approach, are not those with the shortest convergence times. It shows that our reduction approach performs differently from other approaches that are based on time-scale separation. The method can be used to facilitate fitting of the parameters or to embed a detailed model of interest in a more coarse-grained yet realistic environment.

Coordination of multi-agent Euler-Lagrange systems via energy-shaping: Networking improves robustness

In this paper, the robust coordination of multi-agent systems via energy-shaping is studied. The agents are nonidentical, Euler-Lagrange systems with uncertain parameters which are regulated (with and without exchange of information between the agents) by the classical energy-based controller where the potential energy function is shaped such that, if the parameters are known, all agents converge globally to the same desired constant equilibrium. Under parameter uncertainty, the globally asymptotically stable (GAS) equilibrium point is shifted away from its desired value and this paper shows that adding information exchange between the agents to the decentralized control policy improves the steady-state performance. More precisely, it proves that if the undirected communication graph is connected, the equilibrium of the networked controller is always closer (in a suitable metric) to the desired one than that of the decentralized controller. The result holds for all interconnection gains if the potential energy functions are quadratic, else, it is true for sufficiently large gains. An additional advantage of networking is that the asymptotic stabilization objective can be achieved by using lower gains into the loop. Some experimental results (using two nonlinear manipulators) given support to the main results of the paper

Stability of systems with the Duhem hysteresis operator

In this paper, we discuss the dissipativity property of the counterclockwise Duhem operator. Sufficient conditions on the functions which define the Duhem operator are given such that the Duhem operator has counterclockwise input-output dynamics. In particular, an explicit construction of the storage functions satisfying the counterclockwise dissipation inequality is given. The constructed storage function is also related to the underlying anhysteresis function which is commonly used to describe hysteresis in magnetic materials. The results can thus facilitate analysis of systems with the counterclockwise Duhem operator via the dissipativity approach.

A graph-theoretical approach for the analysis and model reduction of complex-balanced chemical reaction networks

In this paper we derive a compact mathematical formulation describing the dynamics of chemical reaction networks that are complex-balanced and are governed by mass action kinetics. The formulation is based on the graph of (substrate and product) complexes and the stoichiometric information of these complexes, and crucially uses a balanced weighted Laplacian matrix. It is shown that this formulation leads to elegant methods for characterizing the space of all equilibria for complex-balanced networks and for deriving stability properties of such networks. We propose a method for model reduction of complex-balanced networks, which is similar to the Kron reduction method for electrical networks and involves the computation of Schur complements of the balanced weighted Laplacian matrix.