Control Of Piecewise Linear Systems Research Articles

Pseudo-Hopf Bifurcation for a Class of 3D Filippov Linear Systems

We consider a nongeneric family of 3D Filippov linear systems with a discontinuity plane that have two parallel tangency lines, such that the region between them is the sliding region. We are interested in finding under what conditions the family has a crossing limit cycle, when the sliding region changes its stability. We call this phenomenon the pseudo-Hopf bifurcation. This class of systems is motivated by piecewise-linear control systems which have not yet been treated in the context of crossing limit cycles.

Two-Parameter Boundary Equilibrium Bifurcations in 3D-Filippov Systems

This work concerns the analysis of boundary equilibrium bifurcations (BEBs) in 3D piecewise-smooth systems of Filippov type. In particular, we consider a family whose vector fields are linear on both sides of the switching boundary and which exhibit two parallel tangency lines each containing a cusp point. This configuration is observed in piecewise-linear control systems in which the control action is discontinuous such as the sliding mode control. We consider a general system of this class and then derive a canonical form to reduce the number of system parameters. The general objective in this work is, from the canonical form, to perform an analysis of the equilibria, stability, sliding dynamics and BEBs. The main result is the classification of the BEBs and its unfoldings in the sliding vector field. This and others results obtained on the existence and stability of equilibria are applied in a practical example involving the control of DC–DC buck power converter.

On the small-time controllability of discontinuous piece-wise linear systems

This note presents a necessary and sufficient condition for small time controllability of a specific class of piece-wise linear systems. These systems are linear in four areas determined in the state space by two hyperplanes with a nonempty intersection. On the hyperplanes the systems appear to be discontinuous. Our main result extends the classical Kalman criterion for controllability to the case of piece-wise linear control systems with controls taking values in closed convex cones.

Adaptive Control of Piecewise Linear Systems with State Feedback for Output Tracking

AbstractA piecewise linear system consists of a set of linear time‐invariant (LTI) subsystems, with a switching sequence specifying an active subsystem at each time instant. This paper studies the adaptive control problem of single‐input, single‐output (SISO) piecewise linear systems. By employing the knowledge of the time instant indicator functions of system parameter switches, a new controller structure parametrization is proposed for the development of a stable adaptive control scheme with reduced modeling error in the estimation error signal used for parameter adaptive laws. This key feature is achieved by the new control scheme's ability to avoid a major parameter swapping term in the error model, with the help of indicator functions whose knowledge is available in many applications. A direct state feedback model reference adaptive control (MRAC) scheme is presented for such systems to achieve closed‐loop signal boundedness and small output tracking error in the mean square sense, under the usual slow system parameter switching condition. Simulation results on linearized NASA GTM models are presented to demonstrate the effectiveness of the proposed scheme.

Adaptive Control of Piecewise Linear Systems: the State Tracking Case

Nonlinear controlled systems at multiple operating points are modeled as piecewise linear systems, where changes in operating points are modeled as switches between constituent linearized systems. This note studies the adaptive state feedback for state tracking control problem for such systems. Piecewise linear reference model systems are used for generating desired state trajectories and their stability properties are studied. Adaptive state feedback control schemes are developed, and their stability and tracking performance are analyzed and evaluated by simulation examples. It is shown that exponential tracking performance can be achieved if the reference input is sufficiently rich and the switches are sufficiently slow.

H ∞ Control of Piecewise-Linear Systems Under Unreliable Communication Links

This paper considers the problem of H ∞ control of piecewise-linear control systems under unreliable communication links. Due to the limited bandwidth of the channels, signal transmission delays and data packet losses can occur between the plant and the controller. In the presence of random signal transmission delays and data packet losses, a piecewise controller is designed to stabilize the piecewise-linear system in the sense of mean square and also achieve a prescribed H ∞ disturbance attenuation performance based on piecewise-quadratic Lyapunov–Krasovskii functionals. It is shown that the H ∞ controllers can be designed by solving a set of linear matrix inequalities (LMIs) that are numerically feasible. Moreover, the controller design method is further extended to uncertain case where the system matrices’ uncertainties are represented in polytopic frameworks. Finally, an example is provided to illustrate the effectiveness of the developed theoretical results.

Robust control of piecewise linear systems: A case study in sheet flow control

This paper presents a control design approach for robust sheet control in a printer paper path. By splitting up the control problem into low level motor control loops and a high level sheet control loop, a hierarchical control structure is obtained. Based on the piecewise linear model of the high level sheet dynamics, the control design is formulated in the H ∞ framework. The resulting dynamic output feedback controllers guarantee stability and performance of the closed-loop system. To show the effectiveness of the control design approach in practice, the robust sheet controllers are successfully implemented on an experimental paper path setup.

A NEW METHODOLOGY FOR LIMIT CYCLE BIFURCATION FROM INFINITY IN N-DIMENSIONAL SYMMETRIC PIECEWISE LINEAR CONTROL SYSTEMS

The so-called “closing equations” method constitutes a suitable technique to detect limit cycles bifurcation in piecewise linear control systems. Here, this technique is adapted to detect limit cycles of great amplitude, that bifurcate from the point at infinity. The specific cases of dimension 2 and 3 are completely analyzed. In terms of the bifurcation parameter, expressions in form of power series for the period, amplitude and the characteristic multipliers of the bifurcating limit cycle are obtained.

Relaxed Optimal Control of Piecewise Linear Systems

This paper applies a dynamic programming relaxation method previously proposed by the authors to optimal control of piecewise linear systems. The method gives an algorithm to find a sub-optimal value function and control law within a user-specified distance from the optimal solution. For computational reasons, the value function used in this paper is piecewise quadratic and continuous. It cannot, in general, represent the optimal value function, but works as an approximator. The controllers found by the method are piecewise linear.

Optimization of Piecewise Linear Control Systems

A problem of optimization of piecewise linear control systems is considered. On the base of methods of solving linear optimal control problems and operations of parametric optimization a dual method of constructing optimal open-loop controls is elaborated. An algorithm of operating an optimal controller is proposed. A controller calculates in real time values of the optimal feedback realizing in the course of control process under disturbances as a result of correction of the optimal open-loop control corresponding to the previous position by operations of the described method.

Model predictive control of nonlinear systems using piecewise linear models

We consider the problem of controlling nonlinear systems which are modeled as a set of piecewise linear (PL) or affine systems using a model predictive control (MPC). The paper reviews recent results on the analysis and control of PL systems, which can model a wide range of practically relevant nonlinear systems. Using techniques from the theory of linear matrix inequalities (LMIs), we develop a multiple model MPC technique involving a sequence of local state feedback matrices, which minimize an upper bound on the ‘worst-case’ objective function. The resulting problem, which utilizes a single quadratic Lyapunov function and multiple local state-feedback matrices, can be cast as a convex optimization problem involving LMIs. Several extensions of this technique involving approximating the local regions by ellipsoids or polytopes, and their respective advantages and disadvantages, are discussed.

Robustness and Bifurcations for a Class of Piecewise Linear Control Systems

The possibilities of defining indices that measure how far a stable nonlinear control system is from losing stability are explored. Among other basic tools from bifurcation theory, the first harmonic bifurcation diagram, as an approximate method to find the actual bifurcation diagram, is used to define robustness criteria for a common class of nonlinear systems. Some global stability indices are defined and certain correspondence is found with local indices of the point at infinity, Two examples show the deep changes in the global behaviour that can be found in dealing with very common nonlinearities, as are the saturations. These changes are predicted with the proposed methodology

Global first harmonic bifurcation diagram for odd piecewise linear control systems

The describing function method (used normally as a first approximation to study the existence and the stability of periodic orbits) is used here to analyze the dependence of periodic orbits on the parameters of autonomous systems. The method can be applied to a large class of nonlinear systems but, for simplicity, attention here is paid to a class of single-input, single-output control systems with a piecewise linear character- istic function. A first approach to the bifurcation diagram associated with the periodic orbits of such systems, called a first harmonic bifurcation diagram' is obtained for two and three dimensions. This diagram in two dimensions describes all the qualitative behaviour of such systems. Although this is not the case in three dimensions, the information contained in the corresponding first harmonic bifurcation diagram is of great value. It shows much of the complexity of the periodic structure that can be found in such three-dimensional systems; in fact, part of the first harmonic bifurcation diagram coincides with the actual bifurcation diagram

Structure and control of piecewise-linear systems

Elementary algebraic topology is used in the study of the structure and optimal control of piecewise-linear and non-linear systems.

Controllability of piecewise linear systems

The paper deals with a discontinuous control system that consists of two finite-dimensional linear systems; the first describes the motion in a given half-space, the second in the complementary half-space. Necessary and sufficient conditions for local controllability are obtained.

An approach to identification of piecewise linear control systems

An approach to identification of piecewise linear control systems

Closure to “Discussions of ‘A Graphical Method for the Analysis of Piecewise Linear Control Systems, With Particular Application to Relay Controls’” (1957, Trans. ASME, 79, pp. 850–851)

Closure to “Discussions of ‘A Graphical Method for the Analysis of Piecewise Linear Control Systems, With Particular Application to Relay Controls’” (1957, Trans. ASME, 79, pp. 850–851)

A Graphical Method for the Analysis of Piecewise Linear Control Systems, With Particular Application to Relay Controls

Abstract A graphical method is described for the analysis of piecewise linear control systems. By means of it both transient and steady oscillations can be studied. After the “characteristic curves” of a system have been computed and plotted, it is shown that the effects of dead zone, solid friction, backlash, and various lead devices are easily taken into account, without further computation. The method is illustrated by a more detailed examination of the behavior of relay controls, and a few of the charts obtained by the method are given.