Stabilization of linear systems over networks with bounded packet loss and its use in model predictive control

This paper considers stabilization of discrete-time linear systems, where network exists for transmitting the sensor and controller information, and arbitrary bounded packet loss occurs in the sensor–controller link and the controller–actuator link. The stabilization of this system is transformed into the robust stabilization of a set of systems. The stability result for this system is specially applied on model predictive control (MPC) that explicitly considers the satisfaction of input and state constraints. Two synthesis approaches of MPC are presented, one parameterizing the infinite horizon control moves into a single state feedback law, the other into a free control move followed by the single state feedback law. Two simulation examples are given to illustrate the effectiveness of the proposed techniques.     

Using hash tables to manage the time-storage complexity in a point location problem: Application to explicit model predictive control

The online computational burden of linear model predictive control (MPC) can be moved offline by using multi-parametric programming, so-called explicit MPC. The solution to the explicit MPC problem is a piecewise affine (PWA) state feedback function defined over a polyhedral subdivision of the set of feasible states. The online evaluation of such a control law needs to determine the polyhedral region in which the current state lies. This procedure is called point location; its computational complexity is challenging, and determines the minimum possible sampling time of the system. A new flexible algorithm is proposed which enables the designer to trade off between time and storage complexities. Utilizing the concept of hash tables and the associated hash functions, the proposed method solves an aggregated point location problem that overcomes prohibitive complexity growth with the number of polyhedral regions, while the storage–processing trade-off can be optimized via scaling parameters. The flexibility and power of this approach is supported by several numerical examples.     

Certifying spatially uniform behavior in reaction–diffusion PDE and compartmental ODE systems

We present a condition that guarantees spatial uniformity for the asymptotic behavior of the solutions of a reaction–diffusion PDE with Neumann boundary conditions. This condition makes use of the Jacobian matrix of the reaction terms and the second Neumann eigenvalue of the Laplacian operator on the given spatial domain, and eliminates the global Lipschitz assumptions commonly used in mathematical biology literature. We then derive numerical procedures that employ linear matrix inequalities to certify this condition, and illustrate these procedures on models of several biochemical reaction networks. Finally, we present an analog of this PDE result for the synchronization of a network of identical ODE models coupled by diffusion terms. From a systems biology perspective, the main contribution of the paper is to blend analytical and numerical tools from nonlinear systems and control theory to derive a relaxed and verifiable condition for spatial uniformity of biological processes.     

An adaptive learning regulator for uncertain minimum phase systems with undermodeled unknown exosystems

The design of an adaptive learning regulator is addressed for uncertain minimum phase linear systems (with known bounds, known upper bound on system order, known relative degree, known high frequency gain sign) and for unknown exosystems (with unknown order, uncertain frequencies). On the basis of a known bound on system uncertainties and a known bound on the modeled exosystem frequencies, a new adaptive output error feedback control algorithm is proposed which guarantees exponential convergence of both the output and the control input errors into residual bounds which decrease as the exosystem modeling error decreases. Exponential convergence of both errors to zero is obtained when the regulator exactly models all exosystem excited frequencies, while asymptotic convergence of both errors to zero is achieved when the actual exosystem is overmodeled by the regulator. The new algorithm generalizes existing learning controllers since, in the case of periodic references and/or disturbances, the knowledge of the period is not required.     

Improving the state estimation for optimal control of stochastic processes subject to multiplicative noise

Computational models for the neural control of movement must take into account the properties of sensorimotor systems, including the signal-dependent intensity of the noise and the transmission delay affecting the signal conduction. For this purpose, this paper presents an algorithm for model-based control and estimation of a class of linear stochastic systems subject to multiplicative noise affecting the control and feedback signals. The state estimator based on Kalman filtering is allowed to take into account the current feedback to compute the current state estimate. The optimal feedback control process is adapted accordingly. The resulting estimation error is smaller than the estimation error obtained when the current state must be predicted based on the last feedback signal, which reduces variability of the simulated trajectories. In particular, the performance of the present algorithm is good in a range of feedback delay that is compatible with the delay induced by the neural transmission of the sensory inflow.     

Sensorless supervision of linear dynamical systems: The Feed-Forward Command Governor approach

This paper proposes a novel class of Command Governor (CG) strategies for input and state-related constrained discrete-time LTI systems subject to bounded disturbances in the absence of explicit state or output measurements. While in traditional CG schemes the set-point manipulation is undertaken on the basis of either the actual measure of the state or its suitable estimation, it is shown here that the CG design problem can be solved, with limited performance degradation and with similar properties, also in the case that such an explicit measure is not available. This approach, which will be referred to as the Feed-Forward CG (FF-CG) approach, may be a convenient alternative CG solution in all situations whereby the cost of measuring the state may be a severe limitation, e.g. in distributed or decentralized applications. In order to evaluate the method proposed here, numerical simulations on a physical example have been undertaken and comparisons with the standard state-based CG solution reported.     

Partial-realization theory for linear switched systems —A formal power series approach

The paper presents partial-realization theory and a realization algorithm for linear switched systems. The results are similar to partial-realization theory of linear and bilinear systems. Our main tool is the theory of rational formal power series.     

Feedback control of nonlinear stochastic systems for targeting a specified stationary probability density

In the present paper, an innovative procedure for designing the feedback control of multi-degree-of-freedom (MDOF) nonlinear stochastic systems to target a specified stationary probability density function (SPDF) is proposed based on the technique for obtaining the exact stationary solutions of the dissipated Hamiltonian systems. First, the control problem is formulated as a controlled, dissipated Hamiltonian system together with a target SPDF. Then the controlled forces are split into a conservative part and a dissipative part. The conservative control forces are designed to make the controlled system and the target SPDF have the same Hamiltonian structure (mainly the integrability and resonance). The dissipative control forces are determined so that the target SPDF is the exact stationary solution of the controlled system. Five cases, i.e., non-integrable Hamiltonian systems, integrable and non-resonant Hamiltonian systems, integrable and resonant Hamiltonian systems, partially integrable and non-resonant Hamiltonian systems, and partially integrable and resonant Hamiltonian systems, are treated respectively. A method for proving that the transient solution of the controlled system approaches the target SPDF as t→∞ is introduced. Finally, an example is given to illustrate the efficacy of the proposed design procedure.     

Stochastic system transformation using generalized orthonormal basis functions with applications to continuous-time system identification

This paper studies the system transformation using generalized orthonormal basis functions that include the Laguerre basis as a special case. The transformation of the deterministic systems is studied in the literature, which is called the Hambo transform. The aim of the paper is to develop a transformation theory for stochastic systems. The paper establishes the equivalence of continuous and transformed-discrete-time stochastic systems in terms of solutions. The method is applied to the continuous-time system identification problem. It is shown that using the transformed signals the PO-MOESP subspace identification algorithm yields consistent estimates for system matrices. An example is included to illustrate the efficacy of the proposed identification method, and to make a comparison with the method using the Laguerre filter.     

Set-valued Lyapunov functions for difference inclusions

The paper relates set-valued Lyapunov functions to pointwise asymptotic stability in systems described by a difference inclusion. Pointwise asymptotic stability of a set is a property which requires that each point of the set be Lyapunov stable and that every solution to the inclusion, from a neighborhood of the set, be convergent and have the limit in the set. Weak set-valued Lyapunov functions are shown, via an argument resembling an invariance principle, to imply this property. Strict set-valued Lyapunov functions are shown, in the spirit of converse Lyapunov results, to always exist for closed sets that are pointwise asymptotically stable.