Alessandro N. Vargas

Robust stability, ℋ2 analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix

The stability and the problem of ℋ2 guaranteed cost computation for discrete-time Markov jump linear systems (MJLS) are investigated, assuming that the transition probability matrix is not precisely known. It is generally difficult to estimate the exact transition matrix of the underlying Markov chain and the setting has a special interest for applications of MJLS. The exact matrix is assumed to belong to a polytopic domain made up by known probability matrices, and a sequence of linear matrix inequalities (LMIs) is proposed to verify the stability and to solve the ℋ2 guaranteed cost with increasing precision. These LMI problems are connected to homogeneous polynomially parameter-dependent Lyapunov matrix of increasing degree g. The mean square stability (MSS) can be established by the method since the conditions that are sufficient, eventually turns out to also be necessary, provided that the degree g is large enough. The ℋ2 guaranteed cost under MSS is also studied here, and an extension to cope with the problem of control design is also introduced. These conditions are only sufficient, but as the degree g increases, the conservativeness of the ℋ2 guaranteed costs is reduced. Both mode-dependent and mode-independent control laws are addressed, and numerical examples illustrate the results.

Mode-Independent

This brief presents a control strategy for Markov jump linear systems (MJLS) with no access to the Markov state (or mode). The controller is assumed to be in the linear state-feedback format and the aim of the control problem is to design a static mode-independent gain that minimizes a bound to the corresponding H2-cost. This approach has a practical appeal since it is often difficult to measure or to estimate the actual operating mode. The result of the proposed method is compared with that of a previous design, and its usefulness is illustrated by an application that considers the velocity control of a DC motor device subject to abrupt failures that is modeled as an MJLS.

{\cal H}_{2}

This paper presents a variational method to the solution of the model predictive control (MPC) of discrete-time Markov jump linear systems (MJLS) subject to noisy inputs and a quadratic performance index. Constraints appear on system state and input control variables in terms of the first two moments of the processes. The information available to the controller does not involve observations of the Markov chain state and, to solve the problem a sequence of linear feedback gains that is independent of the Markov state is adopted. The necessary conditions of optimality are provided by an equivalent deterministic form of expressing the stochastic MPC control problem subject to the constraints. A numerical solution that attains the necessary conditions for optimality and provides the feedback gain sequence is proposed. The solution is sought by an iterative method performing a variational search using a LMI formulation that takes the state and input constraints into account

-Control of a DC Motor Modeled as a Markov Jump Linear System

SUMMARY This paper addresses the optimal solution for the regulator control problem of Markov jump linear systems subject to second moment constraints. We can characterize and obtain the solution explicitly using linear matrix inequalities techniques. The constraints are imposed on the second moment of both the system state and control vector, and the optimal solution is obtained in a computable form. To illustrate the usefulness of the approach, specially that for systems subject to abrupt variations and physical limitations, we present an application for one joint of the European Robotic Arm. Copyright

Constrained model predictive control of jump linear systems with noise and non-observed Markov state

In this technical note, the stability for time-varying discrete-time stochastic linear systems, with possibly unbounded trajectories, is associated to the existence of the long-run average cost criteria. Under controllability and observability, the stochastic system is stable in the sense that its state tends asymptotically to the origin in the mean. Further conditions related to a lower bound on the long-run average cost, or with periodic time-varying systems, provides uniform second moment stability.

Second moment constraints and the control problem of Markov jump linear systems

This paper presents an analytic, systematic approach to handle quadratic functionals associated with Markov jump linear systems with general jumping state. The Markov chain is finite state, but otherwise general, possibly reducible and periodic. We study how the second moment dynamics are affected by the additive noise and the asymptotic behaviour, either oscillatory or invariant, of the Markov chain. The paper comprises a series of evaluations that lead to a tight two-sided bound for quadratic cost functionals. A tight two-sided bound for the norm of the second moment of the system is also obtained. These bounds allow us to show that the long-run average cost is well defined for system that are stable in the mean square sense, in spite of the periodic behaviour of the chain and taking into consideration that it may not be unique, as it may depend on the initial distribution. We also address the important question of approximation of the long-run average cost via adherence of finite horizon costs.

Average Cost and Stability of Time-Varying Linear Systems

The note presents an algorithm for the average cost control problem of continuous-time Markov jump linear systems. The controller assumes a linear state-feedback form and the corresponding control gain does not depend on the Markov chain. In this scenario, the control problem is that of minimizing the long-run average cost. As an attempt to solve the problem, we derive a global convergent algorithm that generates a gain satisfying necessary optimality conditions. Our algorithm has practical implications, as illustrated by the experiments that were carried out to control an electronic dc-dc buck converter. The buck converter supplied a load that suffered abrupt changes driven by a homogeneous Markov chain. Besides, the source of the buck converter also suffered abrupt Markov-driven changes. The experimental results support the usefulness of our algorithm.

Quadratic costs and second moments of jump linear systems with general Markov chain

We demonstrate here that a necessary condition of optimality studied in a previous paper is in fact a necessary and sufficient condition of optimality for the receding horizon control problem of discrete-time Markov jump linear systems subject to noisy inputs. The performance index is quadratic and the information available to the controller does not involve observations of Markov chain states. Seqyebces of linear feedback gains that are independent of the Markov state is adopted, in accordance with the information available to the controller. We make use of an equivalent deterministic form of expressing the stochastic problem, and the complete solution given in feedback form, is obtained by dynamic programming arguments and by the benefit of some quadratic convex relations.

Optimal Control of DC-DC Buck Converter via Linear Systems With Inaccessible Markovian Jumping Modes

This paper presents an application of unscented Kalman filters (UKFs) to an automotive electronic throttle device. The motivation of this study is on estimating the position of the throttle device when measurements of the position are inaccessible, e.g., due to failures in the sensor of position. In this case, an external wattmeter is connected in the circuitry to measure the power consumed by the throttle, and this information feeds UKFs to produce the estimation for the position. Experimental data support the findings of this paper. Almost all of the brand-new vehicles based on spark-ignition combustion engines have an electronic throttle valve to control the power produced by the engine. The electronic throttle has a unique sensor for measuring the position of the throttle valve, and this feature can represent a serious problem when the sensor of position fails. As an attempt to prevent the effects of a failure from such a sensor, we present an algorithm (UKF) combined with the use of an additional sensor, i.e., a wattmeter. The wattmeter is detached from the throttles structure but is arranged to measure the electric power consumed by the throttle. Measurements of the power consumption then feed the UKF. This filter then produces an estimation of the position of the throttle valve. Experimental data illustrate the practical benefits of our approach.

Optimality Condition for the Receding Horizon Control of Markov Jump Linear Systems with Non-observed Chain and Linear Feedback Controls

The paper presents a control strategy for an automotive electronic throttle body, a device largely used into vehicles to increase the efficiency of the combustion engines. The synthesis of the proposed controller is based on a linear matrix inequality (LMI) formulation, which allows us to deal with uncertainties on the measurements of the position of the throttle valve. The LMI approach generates a suboptimal solution for the robust ℋ2 static output feedback control problem, and the corresponding suboptimal control gain was evaluated in practice to control the valve position of the throttle. The usefulness of the approach has been verified not only by numerical simulations but also by real experiments taken in a laboratory prototype.