Regional MPC with active set updates

Genetic negative feedback loops are essential and recurrent biological motifs. They are traditionally described with N-dimensional competitive dynamical systems, composed of highly non-linear Hill functions. The stability property of their unique steady state usually determines the global dynamical behavior: homeostasis under stability or emergence of oscillations. When homeostasis conditions are disrupted, undesired oscillations can emerge and may lead to various diseases. This paper presents a classical affine control strategy that is able to stabilize the unstable steady state of the disrupted system and suppress undesired oscillations. For biological purpose, this control is designed as simple as possible in order to reduce the use of devices and the complexity of the biological set-up. For this reason, the control law only depends on the measurement of a unique gene and only acts on its own expression. Due to the complexity of this controlled dynamical system, a new methodology, based on the construction of successive hyperrectangles of the state space that act as Lyapunov function level-sets, is proposed in order to prove global convergence and global stability results. Despite its apparent simplicity, this affine control law is shown to globally stabilize the disrupted system and recover homeostasis.

The optimal solution to a linear model predictive control (MPC) problem is usually understood as the optimal input for the current state, i.e., a point in state space. However, the solution at a particular point contains more information and provides an optimal affine control law for a polytopic region in state-space. This control law and polytope can be used as long as the closed-loop system stays in the polytope. In the present paper we extend the polytopes to larger, nonlinearly constrained regions that result if optimality is no longer required, but only stability and feasibility of the closed-loop system are enforced. We show fewer quadratic programs need to be solved than in the optimal case, and the effort for evaluating the control laws is reduced compared to an earlier approach.

International audience

Among the appropriate methods to describe the morphology of a robotic system, we note the development of a nonlinear dynamic model defined by the equations of the robot’s motion. Thus, the control design method can be based on the linearization of the nonlinear model. Therefore, in this work, we rely on the difference between the nonlinear model of Lagrangian robotic systems and the approximated linear dynamics to obtain a dynamical model that satisfies a certain Lipschitz condition. Thus, we introduce an affine PD-based control law in the obtained dynamic model to develop by three methods the stability conditions using the Schur complement, the Young inequality and the S-procedure lemma. These conditions are described by Linear Matrix Inequalities (LMIs). Finally, we present some results by adopting, as an illustrative example, the manipulator robot with two degrees of freedom, as a Lagrangian robotic system.

Using a Taylor series expansion, nonlinear control systems are approximated about desired equilibrium points in state space as linear control systems. The techniques of the linear regulator problem are then used to calculate the linear state variable feedback gains needed to keep the system at the desired equilibrium point. Liapunov's second method is the basis for determining stable regions of operation about the equilibrium points. When driving the system from one equilibrium point to another, the piecewise linearization method is also used but with intermediate equilibrium points successively utilized. Enough equilibrium points are used so that the system is stable throughout the region of interest. Thus, a nonlinear system is controlled using linear state variable feedback. A small digital computer is required to implement this strategy.

Reducing the computational effort of MPC with closed-loop optimal sequences of affine laws

A dynamical system is assumed to be governed by a set of ordinary differential equations subject to control. The set of points in state space from which there exist permissible controls that can transfer these points to a prescribed target set in a finite time interval is called a capture set. The task of determining the capture set is studied in two contexts. first, in the case of the system subject to a single control vector; and second, in the case of the system subject to two control vectors each operated independently. In the latter case, it is assumed that one controller's aim is to cause the system to attain the target, and the other's is to prevent that from occurring. Sufficient conditions are developed that, when satisfied everywhere on the interior of some subset of the state space, ensure that this subset is truly a capture set. A candidate capture set is assumed to have already been predetermined by independent methods. The sufficient conditions developed herein require the use of an auxiliary scalar function of the state, similar to a Lyapunov function. To ensure capture, five conditions must be satisfied. Four of these constrain the auxiliary state function. Basically, these four conditions require that the boundary of the controllable set be an envelope of the auxiliary state function and that that function be positive inside the capture set, approaching zero value as the target set is approached. The final condition tests the inner product of the gradient of the auxiliary state function with the system state velocity vector. If the sign of that inner product can be made negative everywhere within the test subset, then that subset is a capture set.

Figure 1: Time-lapse screen capture showing ambient occlusion values calculated in real-time. We present a single-pass hardware accelerated method to compute ambient occlusion values in real-time on dynamic character skins. Ambient occlusion at a point is the cosine weighted integral of the visibility function. Assuming the scene is enclosed in a spherical area source light, ambient occlusion corresponds to the amount of light that will reach each point. Unfortunately, computing ambient occlusion is time-consuming. It requires sampling the hemisphere around each point to determine visibility. Our method takes a data driven approach to approximating ambient occlusion in real-time. It builds a set of linear functions over several subsets of poses, then produces the ambient occlusion using a blend of these functions. Because we are using linear functions our output is smooth, fast to evaluate, and easy to implement in a vertex or fragment shader. Several researchers have examined precomputation of illumination. [Sloan et al., 2005] used spherical harmonics to compress low-frequency transfer functions at points on a static mesh. This method handles a wide range of low-frequency phenomena. However, it is limited to local changes in geometry, and does not handle the type of deformation seen in character animation. [James and Twigg, 2005] precompute illumination data at several points in state space, then blend this data as they deform the mesh. Our method is similar, but we provide a means to minimize error variance within local regions of state space. Recently, [Bunnell, 2005] proposed a multi-pass method for calculating ambient occlusion in real-time. This method makes approximations to the shape of elements, and error is dependent on the number of passes. Ambient occlusion is a function of vertex positions, which in turn are functions of the character’s joint angles. Therefore, ambient occlusion for character skins is also a function of joint angles. It is desirable to represent this function linearly, because linear functions can be computed easily in hardware. Unfortunately, ambient occlusion can change nonlinearly over a wide range of poses. The key observation of our method is that ambient occlusion changes smoothly for small changes in joint angles. Rather than building one large linear function, our method builds several linear functions, each local to a dierent region of pose space. These

Identification of determinism in noisy neuronal systems

Entropy notions for ε-incremental practical stability and incremental stability of deterministic nonlinear systems under disturbances are introduced. The entropy notions are constructed via a set of points in state space which induces the desired stability properties, called an approximating set. We provide conditions on the system which ensures that the approximating set is finite. Lower and upper bounds for the two estimation entropies are computed. The construction of the finite approximating sets induces a robust state estimation algorithm for systems under disturbances using quantized and time-sampled measurements.

We consider evolving networks in which each node can have various associated properties (a state) in addition to those that arise from network structure. For example, each node can have a spatial location and a velocity, or it can have some more abstract internal property that describes something like a social trait. Edges between nodes are created and destroyed, and new nodes enter the system. We introduce a "local state degree distribution" (LSDD) as the degree distribution at a particular point in state space. We then make a mean-field assumption and thereby derive an integro-partial differential equation that is satisfied by the LSDD. We perform numerical experiments and find good agreement between solutions of the integro-differential equation and the LSDD from stochastic simulations of the full model. To illustrate our theory, we apply it to a simple model for osteocyte network formation within bones, with a view to understanding changes that may take place during cancer. Our results suggest that increased rates of differentiation lead to higher densities of osteocytes, but with a smaller number of dendrites. To help provide biological context, we also include an introduction to osteocytes, the formation of osteocyte networks, and the role of osteocytes in bone metastasis.

In this paper, we simplify the dynamics of a bounding quadrupedal robot by using a planar and conservative model which has a rigid body and two massless virtual spring legs to separately simulate the effects of the front and hind legs. The quantitative formulation of the model is derived by using the Lagrangian method. We proceed to search for stable operation points in state space, and the local system behaviors of which are then approximated by a trained neural network model. Next, a model predictive controller is utilized to stabilize the model. In simulation results, the cont vroller succeeded in balancing the model in a dynamic bounding gait without any form of energy input. This research may serve as a guideline for real quadrupedal robots with actuators to create energy conservative gaits.

Holonomic quantum computation exploits a quantum state's non-trivial,\nmatrix-valued geometric phase (holonomy) to perform fault-tolerant computation.\nHolonomies arising from systems where the Hamiltonian traces a continuous path\nthrough parameter space have been well-researched. Discrete holonomies, on the\nother hand, where the state jumps from point to point in state space, have had\nlittle prior investigation. Using a sequence of incomplete projective\nmeasurements of the spin operator, we build an explicit approach to universal\nquantum computation. We show that quantum error correction codes integrate\nnaturally in our scheme, providing a model for measurement-based quantum\ncomputation that combines the passive error resilience of holonomic quantum\ncomputation and active error correction techniques. In the limit of dense\nmeasurements we recover known continuous-path holonomies.\n

Clustering based multiple state–space projections

We have developed a new nonlinear filter that is superior to particle filters in five ways: (1) it exploits smoothness; (2) it uses an exact solution of the Fokker-Planck equation in continuous time; (3) it uses an FFT to compute the effect of process noise at discrete times; (4) it uses the adjoint method to compute the optimal density of points in state space to represent the smooth conditional probability density, and (5) it uses Bayes' rule exactly by exploiting the exponential family of probability densities. In contrast to particle filters, which do not exploit smoothness, the new filter does not use importance sampling or Monte Carlo methods. The new non-particle filter should be superior to particle filters for a broad class of practical problems. In particular, the new filter should dramatically reduce the curse of dimensionality for many (but not all) important real world nonlinear filter problems.