Piecewise Linear Control Law Research Articles

Energy Storage Controller Synthesis for Power Systems With Temporal Logic Specifications

We present an energy storage controller synthesis method for power systems with respect to metric temporal logic (MTL) specifications. The power systems with both constant impedance loads and constant power loads are modeled as a set of differential–algebraic equations. After a fault is cleared, with uncertainties in the fault clearing time, the generator machine angles and rotor speed deviations will enter a set of postfault initial states. We use the robust neighborhood approach to cover this set using the initial robust neighborhoods of finitely many simulated postfault trajectories. These simulated postfault trajectories meet the frequency regulation requirements specified in MTL as they are driven by the optimal control input signals obtained through a functional gradient descent approach. In this way, all the possible postfault trajectories with the given uncertainties in the fault clearing time are guaranteed to satisfy the MTL specification. Furthermore, we learn a piecewise linear control law from the data of the simulated trajectories to generate a feedback controller.

Shaping the PDF of the state variable based on piecewise linear control for non-linear stochastic systems

In this paper, we propose a shape-control scheme for non-linear stochastic dynamical systems to enable us to shape the probability density function (PDF) of the state variable. First, we derive the PDF analytical expression using the Fokker-Planck-Kolmogorov (FPK) equation obtained from stochastic systems. Then, we control the PDF shape by devising a piecewise linear control law whose parameters are calculated using the conjugated gradient method. Finally, we perform contrast simulation experiments to validate the effectiveness and superiority of the proposed algorithm.

Stabilisation of constrained uncertain systems by multi-parametric optimisation

This paper aims to construct a controlled invariant set while determining a stabilising piecewise linear control law for constrained uncertain systems. The proposed approach is based on the equivalence between the considered system and its Euler auxiliary system (EAS). The control law is obtained by the resolution of a constrained finite time optimal control (CFTOC) problem. Due to the exhaustive computing demands, a multi-parametric formulation and an offline resolution of the problem are considered to extend the set parameters variation.

On improving the convergence speed of linear continuous-time systems under constrained control: Sylvester equation approach

A controller design methodology is proposed for linear continuous-time systems with asymmetrical constrained control. Because of the trade-off between the size of the domain of attraction and convergence speed, one can improve the convergence speed during the system evolution. The closed-loop system starts with bad performances and a large domain of attraction and ends with good performances near the origin. The objective of this work is to overcome some discontinuities resulting from a piecewise linear control law previously proposed to improve the convergence speed of a linear system during its evolution. The proposed method is based on the use of the Sylvester equation together with a quadratic Lyapunov function. It guaranties the improvement of the speed of convergence without any discontinuities on the control vector components and respecting always the imposed asymmetric constraints without saturations.

On Improving the Convergence Rate of Linear Continuous-Time Systems Under Constrained Control With State Observers

A controller design methodology for linear continuous-time systems using a state observer is proposed. In particular, the objective is to smooth the discontinuity resulting from a piecewise linear control law previously proposed, which improves the convergence rate of systems under asymmetrically constrained control. The proposed method is based on the use of a non quadratic asymmetrical Lyapunov function. Both full-order observers and minimal order observers are considered to reconstruct the inaccessible state. The speed of convergence of the controlled system and the observer is improved, with smooth control vector components, that respect always the imposed asymmetric constraints.

On improving the convergence rate of linear continuous‐time systems subject to asymmetrically constrained control

This paper solves the problem of controlling linear continuous‐time systems subject to control signals constrained in magnitude (maybe asymmetrically). A controller design methodology is proposed, based on using an asymmetric Lyapunov function, that avoids the discontinuities in the control vector components resulting from the application of a piecewise linear control law previously proposed. The proposed method gives improved speed of convergence without discontinuities of the control vector components, respecting always the imposed asymmetric constraints. An example illustrates the approach.

On improving the performance with bounded continuous feedback laws

We present controller design methods to smooth the discontinuity resulting from a piecewise linear control law which was proposed to improve the convergence performance for systems with input constraints. The continuous control laws designed in this paper are explicit functions of the state and are easily implementable. We also show that the convergence performance can be further improved by using a saturated high-gain feedback law. The efficiency of the proposed methods is illustrated with the PUMA 560 robot model.

PIECEWISE LINEAR CONSTRAINED CONTROL FOR CONTINUOUS-TIME SYSTEMS: THE MAXIMAL ADMISSIBLE DOMAIN

This paper presents an application of a piecewise linear control law in order to obtain the maximal domain of the admissible initial states. This is done in an homothetic way to the initial domain generated by the imposed dynamics on the closed-loop of a linear continuous-time system.

Piecewise-Linear Robust Control of Systems with Input Constraints

Robust stabilisation of an uncertain system subject to input constraints is addressed without making open-loop stability assumptions. A local approach is taken to find a robust control law and a set of initial conditions that can be stabilised. A piecewiselinear control law is described. It is generated by a parametrised algebraic Riccati equation or a parametrised linear matrix inequality determinant maximisation problem. Once system trajectories are sufficiently close to the origin, a certain performance level is ensured through a guaranteed cost control. Simple and tractable design algorithms are proposed and illustrated by a numerical example.

Robust time-optimal control of constrained linear Systems

A version of dynamic programming, which computes level sets of the value function rather than the value function set itself, is used to design robust non-linear controllers for linear, discrete-time, dynamical systems subject to hard constraints on controls and states. The controller stabilizes the system and steers all trajectories emanating in a prescribed set to a control invariant set in minimum time. For the robust regulator problem, the control invariant terminal set is a neighborhood, preferably small, of the origin; for the robust tracking problem, the control invariant terminal set is a neighborhood of the invariant set in which the tracking error is zero. Two non-linear controllers which utilize the level sets of the value function, are described. The first requires the controller to solve, on-line, a modest linear program whose dimension is approximately the same as that of the control variable. The second decomposes each level set into a set of simplices; a piecewise linear control law, affine in each simplex, is then constructed.

Vibration control by limiting the maximum axial forces in space trusses

The authors propose a method of vibration control based on limiting the maximum axial forces in the active members of an adaptive truss. The actuators simulate linear elastic-perfectly plastic-linear elastic behavior and consume the vibrational energy as work. The method is applicable to both statically determinate as well as indeterminate truss structures. However, for energy efficient control of statically indeterminate trusses, extra actuators may be needed on the redundant bars. An energy formulation relating the various control parameters is derived to get an estimate of the control time. Since the simulation of linear elastic-perfectly plastic-linear elastic behavior requires a piecewise linear control law, a general analytical solution is not possible. Numerical simulation by step-by-step integration is performed to simulate the control of an example truss structure. The problems of application to statically indeterminate trusses and optimal actuator placement are identified for future work.