Linear Hybrid Systems Research Articles

Experimental Validation of a Multi Model PI Controller for a Non Linear Hybrid System in LabVIEW

In this paper a real time Single Spherical Tank Liquid Level System (SSTLLS) has been chosen for investigation. This paper describes the design and development of a Multi Model PI Controller (MMPIC) using classical controller tuning techniques for a single spherical nonlinear tank system. System identification of these different regions of nonlinear process are done using black box modeling, which is identified to be nonlinear and approximated to be a First Order Plus Dead Time (FOPDT) model. A proportional and integral controller is designed using LabVIEW and Chen-Hrones-Reswick (CHR), Zhuang-Atherton (ZA), and Skogestad’s Internal Model Controller (SIMC) tuning methods are implemented in real time. The paper provides the details about the data acquisition unit, shows the implementation of the controller, and comparision of the results of PI tuning methods used for an MMPI Controller.

Linear matrix inequality-based analysis of the discrete-continuous nonlinear multivariable systems

The E.M. Fridman method for analysis of the hybrid linear systems by passing to a system with sawtoothed delay and using the nonstationary Lyapunov-Krasovskii functionals and descriptor variables was extended to the nonlinear multivariable Lur'e systems. Consideration was given to the discrete control in the form of a feedback with bounded above variable step of discretization. At that, in the system equations the control function was multiplied by a bounded scalar nonlinear function. This case corresponds to numerous oscillators such as the "pendulum on cart" system. On the basis of the classical results obtained by V.A. Yakubovich on losslessness of the S-procedure, the problem of estimating the upper boundary of the discretization step comes to analyzing the system of linear matrix inequalities for solvability.

Eventual Generic Observability in Linear Hybrid Systems with discrete dynamic modeled by Petri nets

The observability property of Linear Hybrid Systems LHS has been studied in the past from different perspectives. In this work, the observability of LHS is addressed for the case in which the switching signal is unknown, but there are some rules that determine the switching possibilities, codified as a Petri net. By taking advantage of the knowledge of such switching rules, the LHS can be observable under some general conditions. Here, a couple of algorithms are provided in order to translate the information that can be obtained from the continuous output into the Petri net framework, combining thus the continuous and discrete output information of the LHS, and allowing the detection of the discrete state (location) under some relaxed conditions.

A Study on Solving Guard and Invariant Set Intersection in Zonotope-based Reachability of Linear Hybrid Systems

Solving the problem of guard and invariant intersection when deciding for zonotopes to represent reachable sets of linear hybrid systems is still challenging. This is due to the fact that the intersection operation causes the loss of the zonotopic form. Furthermore a tradeoff must be found between computational efficiency and the tightness of the approximation. In this paper, we provide an overview, improve on some methods as well as carry out a comparative evaluation of different methods for zonotope/hyperplane intersection.These methods are then evaluated in combination with different clustering techniques in the context of guard intersection for the reachability analysis of two-tank and colliding-masses benchmarks. We thereafter propose zonotope/halfspace and zonotope/polyhedron intersection methods as solution for handling invariants inside continuous modes. Experimental evaluation however reveals that embedded in the reachability computational process, these methods do differ in performances as compared to when they are used as standalone functions.

Solutions Stability of Initial Boundary Problem, Modeling of Dynamics of Some Discrete Continuum Mechanical System

The solution stability of an initial boundary problem for a linear hybrid system of differential equations, which models the rotation of a rigid body with two elastic rods located in the same plane is studied in the paper. To an axis passing through the mass center of the rigid body perpendicularly to the rods location plane is applied the stabilizing moment proportional to the angle of the system rotation, derivative of the angle, integral of the angle. The external moment provides a feedback. A method of studying the behavior of solutions of the initial boundary problem is proposed. This method allows to exclude from the hybrid system of differential equations partial differential equations, which describe the dynamics of distributed elements of a mechanical system. It allows us to build one equation for an angle of the system rotation. Its characteristic equation defines the stability of solutions of all the system. In the space of feedback-coefficients the areas that provide the asymptotic stability of solutions of the initial boundary problem are built up.

A decidable class of planar linear hybrid systems

In this paper, we show the decidability of a new subclass of linear hybrid automata. These automata are planar, that is, consist of two state variables, monotonic along some direction in the plane and have identity resets, that is, the values of the continuous variables are not reset to a different value on a discrete transition. Our proof uses a combination of a tree construction capturing the edge-to-edge reachability and a finite bisimulation construction. Our class strictly contains the class of two dimensional piecewise constant derivative systems for which decidability of the reachability problem is known.

Set membership identification of switched linear systems with known number of subsystems

This paper addresses the problem of robust identification of a class of discrete-time linear hybrid systems, switched linear models, in a set membership framework. Given a finite collection of input/output data from a noisy process the objective is twofold: (i) establish whether this data was generated by a system that switches amongst an a priori known number of subsystems, and (ii) in that case identify a suitable set of linear models along with a switching sequence that can explain the available experimental information. Our main result shows that these problems are equivalent to minimizing the rank of a matrix whose entries are affine in the optimization variables, subject to a convex constraint imposing that these variables are the moments of an (unknown) Borel measure with finite support. The use of well known (tight) convex relaxations of rank allows for further reducing the problem to a semidefinite optimization that can be efficiently solved. In the second part of the paper we extend these results to handle sensor failures that result in corrupted input/output measurements. Assuming that these failures are infrequent, we show that the problem can be recast into an optimization form where the objective is to simultaneously minimize the rank of a matrix and the number of nonzero rows of a second one. In both cases, appealing to well known convex relaxations of rank and sparsity leads to overall semidefinite optimization problems that can be efficiently solved. These results are illustrated with multiple examples showing substantially improved identification performance in the presence of noise and sensor faults.

Progressive Learning Machine: A New Approach for General Hybrid System Approximation.

As the most important property of neural networks (NNs), the universal approximation capability of NNs is widely used in many applications. However, this property is generally proven for continuous systems. Most industrial systems are hybrid systems (e.g., piecewise continuous), which is a significant limitation for real applications. Recently, many identification methods have been proposed for hybrid system approximation; however, these methods only operate in linear hybrid systems. In this paper, the progressive learning machine-a new learning algorithm based on multi-NNs-is proposed for general hybrid nonlinear/linear system approximation. This algorithm classifies hybrid systems into several continuous systems and can approximate any hybrid system with zero output error. The performance of the proposed learning method is demonstrated via numerical examples and with experimental data from real applications.

Decoupling Control for Bearingless Synchronous Reluctance Motor Based on Fuzzy Inverse Model

For the bearingless synchronous reluctance motor (BSRM) is a multivariable, strong coupling, multi-input and multi-output system, based on the adaptive inverse control theory, a decoupling control method based on the T-S fuzzy inverse model identification is put forward in this paper. According to the input and output information of the system, a fuzzy inverse model of the motor control system is established, then making the inverse model and the original control system in series forms pseudo linear hybrid system to realize the approximate linearization and dynamic decoupling of the motor control system. Building the composite system and proceeding research in the Matlab/Simulink environment, the simulation results show that the control strategy can realize dynamic decoupling among the electromagnetic torque subsystem and the radial suspension force subsystem and among thex- andy-direction of the suspension force, and with excellent static and dynamic performance and adaptive ability.

Analysis and synthesis of linear hybrid systems with state-triggered jumps

This paper investigates first the stability and L2-gain problems for a class of linear hybrid control systems with state-triggered jumps by using an extended Lyapunov-like function (ELF) technique. The obtained results are then applied to facilitate the joint design of a hybrid controller (linear controller plus reset law) for a continuous-time linear time-invariant (LTI) plant. The hybrid H∞ synthesis conditions are expressed in linear matrix inequalities (LMIs) with two line search parameters. An inverted pendulum example is used to illustrate the effectiveness of the proposed approach.

Model-based synthesis of control software from system-level formal specifications

Many embedded systems are indeed software-based control systems , that is, control systems whose controller consists of control software running on a microcontroller device. This motivates investigation on formal model-based design approaches for automatic synthesis of embedded systems control software. We present an algorithm, along with a tool QKS implementing it, that from a formal model (as a discrete-time linear hybrid system ) of the controlled system ( plant ), implementation specifications (that is, number of bits in the Analog-to-Digital , AD, conversion) and system-level formal specifications (that is, safety and liveness requirements for the closed loop system ) returns correct-by-construction control software that has a Worst-Case Execution Time (WCET) linear in the number of AD bits and meets the given specifications. We show feasibility of our approach by presenting experimental results on using it to synthesize control software for a buck DC-DC converter, a widely used mixed-mode analog circuit, and for the inverted pendulum.

Design of Robust Internal Models for a Class of Linear Hybrid Systems

This paper focuses on the design of internal models for a class of hybrid linear SISO systems and exosystems. The internal model design takes into consideration observability properties of the exosystem that have significance in the hybrid setting. In particular, visibility and invisibility of the so-called hybrid steady-state generator are discussed, as well as uniform observability properties.

Control via decoupling of a class of second order linear hybrid systems

We study a terminal state control (reachability) problem for certain elastic systems of ``hybrid' type consisting of a single space dimension distributed parameter part coupled, at one endpoint of the relevant spatial, $x$, interval, to a lumped mass component. Two such systems are studied in detail.The first is a vibrating string system fixed at $x = 0$ and attached to a point mass at the right hand endpoint $x = L$. The second example concerns an Euler - Bernoulli beam system ``clamped' at $x = 0$ and attached, at $x = L$, to a mass with both translational and rotational inertia. In both cases the controls act on the mass, which is modeled by a finite dimensional system of differential equations. Analysis of the reachability problem is facilitated by a preliminary ``feedback type' transformation of the control variable which decouples the point mass from the distributed system. In both examples a concluding analysis is required to show that the component of the control generated by feedback lies in the same space as the originally applied control.

Optimal Sampled–Data State Feedback Control of Linear Systems

This paper addresses and solves two optimal state-feedback control design problems for sampled-data systems. First, we reformulate the closed-loop system as a special linear hybrid system. Then, two theorems are developed to evaluate the ℋ2 and the ℋ∞ performances of hybrid systems using specific two-point boundary value problems. Both theorems are adapted to provide optimal control conditions, based on linear matrix inequalities (LMIs), for the state-feedback problems under consideration. These results are generalised to cope with non-uniform data-rates in the communication channel between the controller and the plant.

Output Regulation of Hybrid Linear Systems with Unpredictable Jumps

In this paper the output regulation problem for a class of hybrid linear systems in the presence of uncertain time domain is considered. Uncertainty is modeled by assuming that the time domain is not known to the controller and by allowing for arbitrarily close, even simultaneous, jumps. Considering the full information setting, the geometric characterization of the relevant regulation manifolds is given. Finally, the theory is illustrated and validated by means of numerical examples.

Robust semiclassical internal model based regulation for a class of hybrid linear systems

A new robust error-feedback regulator is proposed for a class of uncertain hybrid systems with periodic jumps, using a hybrid extension of the classical internal model principle. The plant need not be minimum phase, square nor SISO. The proposed regulator contains an internal model composed by two main units, a flow internal model providing the correct input to achieve regulation during flows, and a jump internal model resetting the state of the regulator at each period. Such a structure reminds internal model design for ripple-free regulation of sampled-data systems; an important difference is that here the jump internal model must contain more copies of the relevant dynamics of the exosystem with respect to the flow internal model.

Finite data-rate feedback stabilization of switched and hybrid linear systems

We study the problem of asymptotically stabilizing a switched linear control system using sampled and quantized measurements of its state. The switching is assumed to be slow enough in the sense of combined dwell time and average dwell time, each individual mode is assumed to be stabilizable, and the available data rate is assumed to be large enough but finite. Our encoding and control strategy is rooted in the one proposed in our earlier work on non-switched systems, and in particular the data-rate bound used here is the data-rate bound from that earlier work maximized over the individual modes. The main technical step that enables the extension to switched systems concerns propagating over-approximations of reachable sets through sampling intervals, during which the switching signal is not known; a novel algorithm is developed for this purpose. Our primary focus is on systems with time-dependent switching (switched systems) but the setting of state-dependent switching (hybrid systems) is also discussed.

Internal Model Principle for Linear Systems With Periodic State Jumps

This paper deals with the problem of output regulation for a class of hybrid linear SISO systems and exosystems whose dynamics have jumps according to the value of a clock variable. The problem of designing controllers embedding an hybrid internal model of the switching exosystem in order to achieve the regulation objective is addressed. Key concepts and tools well known in the field of output regulation for continuous-time systems, such as the concept of steady-state response, of regulator equations and the internal model property, are thus generalized to the hybrid setting. Emphasis is placed on internal models and stabilizers that are robust to uncertainties entering both in the flow and jump dynamics of the hybrid controlled system. Time-varying and time-independent design principles are presented. Simulation results show the effectiveness of the proposed solutions in meaningful cases.

Stability of two-time scale linear stochastic hybrid systems

The problem of exponential mean-square stability of two-time scale, linear stochastic hybrid systems has been studied in this paper. To obtain the sufficient conditions of stability, two basic approaches of stability analysis: for one-time hybrid systems with a Markovian switching rule as well as switching rule and singularly perturbed nonhybrid systems, were combined. The Lyapunov techniques were used in both approaches.

Optimal sensor scheduling for hybrid estimation

A sensor scheduling problem was considered for a class of hybrid systems named as the stochastic linear hybrid system (SLHS). An algorithm was proposed to select one (or a group of) sensor at each time from a set of sensors. Then, a hybrid estimation algorithm was designed to compute the estimates of the continuous and discrete states of the SLHS based on the observations from the selected sensors. As the sensor scheduling algorithm is designed such that the Bayesian decision risk is minimized, the true discrete state can be better identified. Moreover, the continuous state estimation performance of the proposed algorithm is better than that of hybrid estimation algorithms using only predetermined sensors. Finally, the algorithms are validated through an illustrative target tracking example.