For many applications, PIECE-WISE AFFINE (PWA) SYSTEMs are made from approximation of nonlinear dynamics by AFFINE subSYSTEMs. Approximation and non-exact stability analysis are the main disadvantages of derived PWA SYSTEM and as a solution, a new class of hybrid SYSTEMs are introduced in this article. The proposed class is derived directly from switched model and without averaging then it is named as Direct PWA (DPWA). The new hybrid class has AFFINE subSYSTEMs and regulable and/or constant switching boundaries. Many applications such as power electronics and process engineering can be modeled as proposed class. Some theorems are presented for stability analysis that are based on quadratic Lyapunov function. The problem of stability analysis and controller design is redounded to convex optimization in form of linear matrix inequality. The proposed model is compared with conventional hybrid models. Proposed model is used for modeling and stability analysis of a dc-dc resonant power converter.

This article aims to derive new sufficient conditions to guarantee the stability of piecewise AFFINE SYSTEMs with time-varying delay (PWA-TVD). The set of delay-dependent linear matrix inequality (LMI) describes the novel stability criteria. This approach considers the PWA-TVD SYSTEM with a time-delayed state-dependent switching signal. The newly suggested Lyapunov-Krasovskii functional (L-K-F) and improved estimation of its derivative have a crucial role in decreasing the complexity and conservatism of the proposed stability results. The suggested L-K-F belongs to the current and time-delayed states, the integral of the states over the time-varying delay, and time derivation of the states. A new inequality was used to obtain an upper bound (UB) for the time derivation of the Lyapunov functional. Then based on this UB, less conservative results are achieved. The theoretical results are applied to the numerical examples. The results confirm the effectiveness of the presented method. The conservative index is the maximum admissible UB of time delay.

In this paper, a predictive control based on the proposed hybrid model is designed to control the fluid height in a three-tank SYSTEM with nonlinear dynamics whose operating mode depends on the instantaneous amount of SYSTEM states. The use of nonlinear hybrid model in predictive control leads to a problem of mixed integer nonlinear programming (MINLP) which is very complex and time consuming to solve. One way to solve this problem is to approximate nonlinear equations with linear or piecewise AFFINE (PWA) expressions. The linear approximation often has a large error in calculating the operating modes and the SYSTEM states. The PWA approximation produces less error than the linear approximation, but its computational load is much higher. In this study, with the aim of reducing the computational load, a closed form model has been obtained for the equations of the three-tank SYSTEM in each of the modes. The resulting SYSTEM is an PWA, each mode being described by an PWA expression. Predictive control of this SYSTEM is a mixed integer linear programming problem that can be solved by conventional solvers. To evaluate the performance of the proposed method and the possibility of using it online, the optimal control input sequence is calculated using MOSEK commercial solver in MPT toolbox, and at any sampling time only the first member of the sequence is applied to the precise modeled three-tank SYSTEM in the Simulink/Stateflow environment. The simulation results indicate that the proposed controller performs the tracking efficiently and the constraints on the SYSTEM states are also satisfied.

This paper presents a new approach to estimate and to enlarge the domain of attraction for a planar continuous-time piecewise AFFINE SYSTEM. Various continuous Lyapunov functions have been proposed to estimate and to enlarge the SYSTEM’s domain of attraction. In the proposed method with a new vision and with the aids of a discontinuous piecewise quadratic Lyapunov function, the domain of attraction at the origin is enlarged by designing a state feedback controller. This paper shows that the continuity of the Lyapunov function on the boundaries, increases the conservativeness in estimating the domain of attraction, and gives more powerful search ability to the domain of attraction estimation algorithm by relaxing this continuity condition. The simulation results show the superiority of the proposed method so that using this method the larger estimation of the domain of attraction is obtained than continuous one.

In this paper, a novel optimal control design method by discontinuous quadratic Lyapunov function and continuous quadratic Lyapunov function for 2-dimensional piecewise AFFINE SYSTEMs via semi-definite programming with LMI constraints is proposed. At the first, an upper bound for a quadratic cost function for a stable closed-SYSTEM is obtained. Then after, considering a state-feedback control approach, not only sufficient conditions for the stability of the closed-loop SYSTEM but also the upper bound of the cost function are obtained. The optimization problem is formulated as a semi-definite programming with bilinear constraints (BMI). Some variables in BMIs are searched by genetic algorithm, so the bilinear constraints are converted to linear constraints and the controller coefficients are calculated. The effectiveness of the proposed method is verified by numerical examples.

In this article, stability analysis for Stochastic Piecewise AFFINE SYSTEMs which are a subclass of stochastic hybrid SYSTEMs is investigated. Here, additive noise signals are considered that does not vanish at equilibrium points. These noises will prohibit the exponential stochastic stability discussed widely in literature. Also, the jumps between the subSYSTEMs in this class of stochastic hybrid SYSTEMs are state-dependent which make stability analysis more complex. The presented theorem considering both additive noise and state-dependent jumps, gives upper bounds for the second stochastic moment or variance of Stochastic Piecewise Nonlinear SYSTEMs trajectories and guarantees that stable SYSTEMs have a steady state probability density function. Then, linear case of such SYSTEMs is studied where the stability criterion is obtained in terms of Linear Matrix Inequality (LMI) and an upper bound on state covariance is obtained for them. Next, to validate the proposed theorem, solving the Fokker Plank equations which describes the evolution of probability density function, is addressed. A solution for the problem of boundary conditions that arises from jumps in this class of SYSTEMs is given and then with finite volume method the corresponding partial differential equations are solved for a case study to check the results of the presented theorem numerically.

A new method based on piecewise AFFINE approximation is proposed to model a distillation column and a novel robust MPC is addressed for PWA SYSTEMs considering multiple prediction trajectories. Distillation columns have highly nonlinear and complex behavior. However, even in a rigorous dynamical model a number of model simplifications are included. Piecewise AFFINE maps have universal approximation properties which are useful for modeling of nonlinear SYSTEMs in a wide range of operation. Model predictive control for PWA SYSTEMs faces multiple prediction trajectories at each sample time due to different SYSTEM dynamics over prediction horizon. Thus, the computational burden increases exponentially with the prediction horizon length. In order to decrease the computational burden, a suboptimal method is used to solve MIQP problems in MPC for PWA SYSTEMs in presents of model uncertainty and disturbances. A real distillation column of a debutanizer unit in South Pars Gas refineries is modeled with PWA method and validated using the appropriate nonlinear model. A tube-based model predictive control proposed in such a way that the optimization problem can be solved considering multiple prediction trajectories at each sample time. In the proposed method, the computational burden is increased linearly by the prediction horizon length.

The main contribution of this paper is to present the SYSTEMatic and low-complexity translation techniques between a class of hybrid SYSTEMs referred to as automaton-based DHA and piecewise AFFINE (PWA) SYSTEMs. As an starting point the general modeling framework of the automaton-based DHA is represented which models the controlled and uncontrolled switching phenomena between linear continuous dynamics including discrete and continuous states, inputs and outputs. The basic theoretical definitions on the state trajectories of the proposed DHA with forward and backward evolutions which yield forward and backward piecewise AFFINE (FPWA and BPWA) SYSTEMs are given. Next, the well-posedness and equivalency properties are proposed and the sufficient conditions under which the wellposedness property is achieved with the automaton-based DHA and PWA SYSTEMs are given. It is shown that the graphical structure of the proposed automaton-based DHA makes it possible to obtain analytically the equivalent PWA SYSTEM with a polynomial complexity in contrast to the existing numerical translation techniques via decomposed structure of the DHA with an exponential complexity. Examples are presented to confirm the effectiveness of the proposed translation techniques.