Switched Affine Systems Research Articles

On the distinguishability and observer design for single-input single-output continuous-time switched affine systems under bounded disturbances with application to chaos-based modulation

Switched Affine Systems (SAS׳s) is a class of Hybrid Systems composed of a collection of Affine Systems (AS׳s) and a switching signal that determines, at each time instant, the evolving affine subsystem. This paper is concerned with the observability and observer design for single-input single-output (SISO) SAS׳s under unknown perturbation, for the case that no information about the switching signal is available. It is firstly demonstrated that in the presence of disturbances every pair of AS׳s is always indistinguishable from the continuous output, meaning that it is not possible to infer the evolving AS by using only the information provided by the output of the SAS. Nevertheless, by taking advantage of the knowledge on the disturbance bound, new distinguishability conditions are derived, making possible to distinguish the evolving AS. By using these new distinguishability conditions, an observer scheme for SISO SAS׳s, subject to unknown switching signal and unknown perturbations, is presented. Such an observer scheme determines in finite-time the evolving AS. Furthermore, it estimates both the state of the system and the disturbance. Finally, the proposed observer scheme is effectively applied for a non-autonomous chaotic modulation application, which is an attractive method for spread-spectrum secure communication in which the message is fed as a disturbance to a chaotic SAS and the output is then transmitted through an open channel to a receiver, which is an observer algorithm that recovers the message (the disturbance) from the output signal.

An LMI Approach to Full- and Reduced-Order Switched Luenberger Observers for Switched Affine Systems

This work deals with reduced-order Luenberger-like observer design and control for switched affine systems. Formulation of state-dependent and dynamic output-dependent switching laws for those systems was already addressed, however, in a full-order treatment. The proposed procedure defines an estimator system only for the unmeasured part of the system state. Linear matrix inequalities for obtaining output gain matrices and for composing a reduced-order Luenberger observer-based switching law are proposed. Finally, the results are validated by design and simulation for DC–DC converters.

Design of Stabilizing Switching Laws for Mixed Switched Affine Systems

This technical note presents stability analysis and stabilization for a general class of switched systems characterized by nonlinear functions. The proposed approach is composed of approximating the switched nonlinear system with a switched affine system that has a mixture of controlled and autonomous switching behavior. Utilizing a joint polyhedral partitioning approach, a stabilizing switching law based on quadratic Lyapunov functions and with considering the autonomous switching between polyhedral regions is proposed. To ensure the decrease of the overall Lyapunov function, two approaches are proposed: 1) guarantee continuity of the Lyapunov function over boundaries of polyhedral regions and 2) relax the continuity requirement by using additional matrix inequalities. The second approach is less conservative but with more variables and matrix inequalities than in the first method. With fixing one scalar variable, the stabilization conditions will have the form of linear matrix inequalities (LMIs). Further, the sufficient conditions for stabilizing the original switched nonlinear system using the proposed switching schemes are presented. Finally, through two examples, the performance of the proposed stabilization methods is demonstrated.

Dynamic output feedback [formula omitted] control of continuous-time switched affine systems

This paper deals with output feedback switching function control design for continuous-time switched affine systems, assuring global asymptotic stability of a desired equilibrium point. The set of all attainable equilibrium points is provided. More specifically, the main purpose is to design a full order switched affine filter together with a switching rule assuring not only global asymptotic stability, but also a guaranteed H∞ performance level. The design conditions do not require that the open-loop subsystems be stable and outperform other techniques available in the literature to date. The results are compared with recent ones based on a quadratic Lyapunov function and on a max-type Lyapunov function. Numerical examples illustrate the theoretical results and are used for comparisons.

Model reduction of switched affine systems

This paper addresses model reduction and extends balanced truncation to the class of switched affine systems with endogenous switching. The switched affine system is rewritten as a switched linear one with state resets that account for the affine terms. Balanced truncation can then be applied to each mode dynamics, independently. As a result, different reduced state vectors are associated with the different modes, and reset maps are here appropriately redefined so as to account and compensate for this mismatch, possibly preserving the continuity of the output. The overall behavior of the reduced switched system is determined by both the selected reduction per mode and the adopted reset maps. In this paper, we consider a stochastic setting and propose a randomized method for the selection of the reduced order. The performance of the proposed approach is illustrated through a multi-room temperature control example.

Design and implementation of a robust switching strategy for DC–DC converters

In this study, a novel robust switching rule is designed for DC–DC converters utilising switched systems’ theory. The converter dynamic is described in the form of linear switched affine systems with constant external input and then, sliding-mode switching controller is developed to ensure robust asymptotic stability of the desired equilibrium point for the system. The inherent robustness of the sliding-mode switching rule leads to efficient regulation of the output voltage under load variations. Simulation results of the buck converter are presented to demonstrate the superiority of the proposed method compared with some of existing schemes in the literature. Finally, experimental results from a laboratory buck circuit are reported to confirm the applicability of the introduced scheme.

State feedback ℋ ∞ control design of continuous‐time switched affine systems

This study deals with state feedback ℋ ∞ control design of continuous-time switched affine systems. The main purpose is to design a set of state feedback gains together with a switching function assuring global asymptotic stability of a desired equilibrium point. The equilibrium point belongs to a set of attainable ones to be determined. Moreover, the control design must take into account a pre-specified upper bound to the ℒ 2 gain from the external input to the controlled output. Two different switching functions are proposed and discussed. The first one depends only on the state and the other depends also on the external input. The results are compared with recent ones available in the literature to date, as for instance, those based on a max-type Lyapunov function and those commonly used to assure practical stability. Numerical examples illustrate the theoretical results and are used for comparisons.

Hybrid Control of DC–DC Series Resonant Converters: The Direct Piecewise Affine Approach

The control and stabilization of resonant converters are essential problems in power electronics. The conventional controller design and stability analysis for these converters are based on the linearized averaged model. Nevertheless, the state variables in resonant converters have large ac variations and the validity of the linearized average model is violated. Hence, using large signal and nonaveraged models are necessary for controller design and stability analysis. In this paper, a new hybrid controller is presented that is applicable to dc-dc series resonant converters and use neither averaging nor small signal approximation. The dc-dc resonant converters are inherently switched affine systems with constrained switching law. The proposed controller is based on the switched behavior of the converter and the concept of piecewise affine methodology. Moreover, it has switched inner and proportional-integral (PI) outer control loops and does not require a modulator. The large signal and closed loop stability analysis of the resonant converters is presented by a theorem. The minimum phase attribute of the control system is investigated by zero dynamic stability analysis. The proposed controller has less complexity in comparison to other suggested controllers and can be implemented using simple analog circuits. The simulation and experimental results show the effectiveness of the proposed method.

Model Invalidation for Switched Affine Systems with Applications to Fault and Anomaly Detection

In this paper, the model (in)validation problem is addressed for the class of switched state space models. We pose the model invalidation problem as a mixed-integer linear program and solve it using the state-of-the-art MILP solvers. Model invalidation is mainly utilized to build trust in the models obtained from system identification. However, we turn our attention to solve another important class of problems using model invalidation approach proposed in this paper. It is shown that the model invalidation approach can be utilized to detect any general fault in cyber-physical systems. Moreover, it is illustrated that knowing the fault model can reduce the complexity of fault detection approach proposed here, if the fault and system model satisfy certain conditions.

Convex Certificates for Model (In)validation of Switched Affine Systems With Unknown Switches

Checking validity of a model is a crucial step in the process of system identification. This is especially true when dealing with switched affine systems since, in this case, the problem of system identification from noisy data is known to be generically NP-Hard and can only be solved in practice by using heuristics and relaxations. Therefore, before the identified models can be used for instance for controller design, they should be systematically validated against additional experimental data. In this paper we address the problem of model (in)validation for multi-input multi-output switched affine systems in output error form with unknown switches. As a first step, we prove that necessary and sufficient invalidation certificates can be obtained by solving a sequence of convex optimization problems. In principle, these problems involve increasingly large matrices. However, as we show in the paper by exploiting recent results from semialgebraic geometry, the proposed algorithm is guaranteed to stop after a finite number of steps that can be be explicitly computed from the a priori information. In addition, this algorithm exploits the sparse structure of the underlying optimization problem to substantially reduce the computational burden. The effectiveness of the proposed method is illustrated using both academic examples and a non-trivial problem arising in computer vision: activity monitoring.

A computational approach to synthesizing guards for hybrid systems

We propose a technique for synthesizing switching guards for hybrid systems to satisfy a given state-based safety constraint. Using techniques from sum of squares (SOS) optimization, we design guards defined by semialgebraic sets that trigger mode switches, and we guarantee that the synthesized switching policy does not allow Zeno executions. We demonstrate our approach on an example of switched affine systems and on an application to traffic ramp metering.

Local Stabilization of Switched Affine Systems

This technical note considers the local stabilization problem for the class of switched affine systems. The main idea is to use an alternative representation of the switched affine system as a nonlinear system with input constraints. Switching laws can be derived by emulating locally classical controllers. It is shown that by restricting to local stabilization, the classical constraint on the existence of constant stable convex combinations may be easily avoided. The approach may be interpreted as a generalization where convex combinations defined as functions of the system state are being used. Constructive methods for deriving switching laws are proposed.

Robust Sampled – Data Control of Switched Affine Systems

This technical note considers the stabilization problem for switched affine systems with a sampled-data switching law. The switching law is assumed to be a function of the system state at sampling instants and the sampling interval may be subject to variations or uncertainty. We provide a robust switching law design that takes into account the sampled-data implementation and uncertainties. The problem is addressed from the continuous-time point of view. The method is illustrated by numerical examples.

Observer-Based Control Design for Switched Affine Systems and Applications to DC–DC Converters

This paper deals with the problem of establishing a state estimator for switched affine systems. For that matter, a modification on the Luenberger observer is proposed, the switched Luenberger observer, whose idea is to design one output gain matrix for each mode of the original system. The efficiency of the proposed method relies on a simplification on estimation error which is proved always valid, guaranteeing the estimation error to asymptotically converge to zero, for any initial state and switching law. Next, a dynamic output-dependent switching law is formulated. Then, design methodologies using linear matrix inequalities are proposed, which, to the authors’s knowledge, have not yet been applied to this problem. Finally, observers for DC–DC converters are designed and simulated as application examples.

Practical stability of switched systems without a common equilibria and governed by a time-dependent switching signal

In this paper, the problem of practical stability of some classes of continuous-time switched systems is studied. The main results of this paper include some sufficient conditions concerning practical stability of continuous-time switched nonlinear systems without a common equilibria for all subsystems. In this class of switched systems, the equilibrium point varies discontinuously according to a time-dependent switching signal. So, stability with respect to a set, rather than a particular point, is discussed. Using this preliminary result, we present sufficient conditions in the form of linear matrix inequalities (LMIs) for practical stability of a particular class of switched systems without common equilibria: the switched affine systems. An illustrative example in the power system stability area is presented to show the validity of the results.

A Sparsification Approach to Set Membership Identification of Switched Affine Systems

This paper addresses the problem of robust identification of a class of discrete-time affine hybrid systems, switched affine models, in a set membership framework. Given a finite collection of noisy input/output data and some minimal a priori information about the set of admissible plants, the objective is to identify a suitable set of affine models along with a switching sequence that can explain the available experimental information, while minimizing either the number of switches or subsystems. For the case where it is desired to minimize the number of switches, the key idea of the paper is to reduce this problem to a sparsification form, where the goal is to maximize sparsity of a suitably constructed vector sequence. Our main result shows that in the case of ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> bounded noise, this sparsification problem can be exactly solved via convex optimization. In the general case where the noise is only known to belong to a convex set N, the problem is generically NP-hard. However, as we show in the paper, efficient convex relaxations can be obtained by exploiting recent results on sparse signal recovery. Similarly, we present both a sparsification formulation and a convex relaxation for the (known to be NP hard) case where it is desired to minimize the number of subsystems. These results are illustrated using two non-trivial problems arising in computer vision applications: video-shot and dynamic texture segmentation.

A numerical framework for optimal control of switched affine systems with state constraint

In this paper, we address the problem of numerical implementation of optimal control for switched affine systems with state constraints. In order to properly solve the problem, a relaxed system is introduced and the connection between the solution of this system and the solution of the initial one is established. One of the main difficulties is then related to the fact that the optimal solution is generally singular. We show that, using slack variables, a set of complementarity constraints can be used to take into account the singular nature of the solution. The optimal control problem is then reformulated as a constraint optimization problem over the Hamiltonian systems and solved via a direct method. This formulation does not require a priori knowledge on the structure (regular/singular) of the solution. In addition, state path constraints are included. Numerical simulations for power converters, both in continuous and discontinuous conduction mode, illustrate the effectiveness of the proposed methodology.

Sampled – data control of switched affine systems: a continuous-time approach

This paper considers the stabilization problem for switched affine systems with a sampled-data switching law. The switching law is assumed to be a function of the system state at sampling instants and the sampling interval may be subject to variations or uncertainty. The main idea is to provide a robust switching law design criterion that takes into account the sampled-data implementation. The problem is addressed from the continuous-time point of view using Linear Matrix Inequalities (LMIs). The method is illustrated by a numerical example.

Switched Affine Systems Using Sampled-Data Controllers: Robust and Guaranteed Stabilization

The problem of robust and guaranteed stabilization is addressed for switched affine systems using sampled state feedback controllers. Based on the existence of a control Lyapunov function for a relaxed system, we propose three sampled-data controls. Global attracting sets, computed by solving a sequence of optimization problems, guarantee practical and global asymptotic stabilization for the whole system trajectories. In addition, robust margins with respect to parameters uncertainties and non uniform sampling are provided using input-to-state stability. Finally, a buck-boost converter is considered to illustrate the effectiveness of the proposed approaches.

Sampled control design for switched affine systems: robust stabilization

AbstractThis paper deals with sampled feedback strategies applied on the class of switched affine systems without common equilibria. A distinctive feature of these systems is a cyclical behavior in steady state. From the existence of a Control Lyapunov Function for a relaxed control problem, a novel procedure enabling to compute global attracting sets for the whole system trajectories is exhibited. This procedure is extended to the case of parameter and/or switching time uncertainties leading to Input-to-State Stability results. An application on a flying capacitor converter is used as an illustrative example.