New trends in modeling and control of hybrid systems

Model predictive control of hybrid systems : stability and robustness

This paper illustrates how an adaptive model predictive controller for the class of hybrid systems affected by parametric uncertainties can be synthesized. Specifically, we investigate the case where the hybrid system is modeled as a Piecewise Affine (PWA) system where certain elements of the PWA state-update matrices are expressed as symbolic parameters with unknown, but bounded values. For such parametric uncertain PWA systems we show that the control law can be constructed in a way such that all system states are pushed towards a pre-defined target set in the minimal possible number of time steps for all possible realizations of the uncertainty. The robust minimum-time control problem is solved using parametric programming techniques which leads the control law in a form of a look-up table. We illustrate that if the values of the parameters (unknown at the time of the synthesis of the control law) are measured on-line, the resulting feedback policy guarantees time-optimal convergence of the systems states towards a chosen terminal set.

Hybrid systems are dynamic systems involving the presence of continuous and discrete variables in the models. Consequently, the optimization methods used by the predictive control scheme are confronted to combinatorial aspects. Among the various paradigms allowing building hybrid models, some classes of systems lend themselves particularly to the development of predictive control laws, especially in their discrete time formulation. A commonly used class of hybrid systems is piecewise affine systems (PWA) which are defined by a piecewise affine dynamic over non-overlapping regions of the state-input space. The effort of research is mainly devoted to techniques leading to a reduction of the combinatorial complexity involved in the optimization problem that is associated with the predictive control law. The purpose of these techniques consists in avoiding the enumeration of all the possibilities for the discrete variables on the prediction horizon, which generates a (very) great number of sub problems to be solved to find the searched optimum at each sampling time.

Hybrid systems have steadily grown in popularity over the last few decades because they ease the task of modeling complicated nonlinear systems. Legged locomotion, robotic manipulation, and additive manufacturing are representative examples of systems benefiting from hybrid modeling. They are also prime examples of repetitive processes; gait cycles in walking, product assembly tasks in robotic manipulation, and material deposition in additive manufacturing. Thus, they would also benefit substantially from Iterative Learning Control (ILC), a class of feedforward controllers for repetitive systems that achieve high performance in output reference tracking by learning from the errors of past process cycles. However, the literature is bereft of ILC syntheses from hybrid models. The main thrust of this dissertation is to provide a boradly applicable theory of ILC for deterministic, discrete-time hybrid systems, i.e. piecewise defined (PWD) systems. In summary, the three main gaps addressed by this dissertation are (1) the lack of compatibility between existing hybrid modeling frameworks and ILC synthesis techniques, (2) the failure of ILC based on Newton's method (NILC) for systems with unstable inverses, and (3) the lack of inversion and stable inversion theory for piecewise affine (PWA) systems (a subset of PWD systems). These issues are addressed by (1) developing a closed-form representation for PWD systems, (2) developing a new ILC framework informed by NILC but with the new ability to incorporate stabilizing model inversion techniques, and (3) deriving conventional and stable model inversion theories for PWA systems.

Simultaneous state and input estimation of hybrid systems with unknown inputs

This paper is concerned with the problem of designing event-triggered controllers for discrete-time piecewise affine (PWA) systems with norm-bounded uncertainties. Event-triggered scheme transmits signals only when the eventtriggering condition is violated. Due to its advantage in saving communication resources, it is utilized to design the state feedback controllers for piecewise affine systems. A quadratic cost function is introduced to quantifying the control performance. An optimization problem for minimizing the upper bound of the quadratic cost function is formulated in the form of linear matrix inequalities (LMIs). By solving the optimization problems a unique state feedback controller can be obtained. The resulting closed-loop system stability is guaranteed while the number of transmitted signals is reduced. A numerical example is given to show the effectiveness of our methods.

Explicit hybrid MPC for the lateral stabilization of electric vehicle system

PWA Modelling and Co-ordinated Continuous and Logical Control of a Laboratory Scale Plant with Hybrid Dynamics

An algorithm for nonlinear optimal generalized predictive functional control is defined for controlling discrete-time piecewise affine systems. The main piecewise affine form is transferred into a corresponding state-dependent system form. The principal piecewise affine system's hybrid properties are retained, and both the continuous and switching dynamics are combined in the same system description. This method enables the use of nonlinear generalized predictive functional control for this hybrid system class. In the generalized predictive functional control method, different classical controller structures can be employed in the feedback loop, such as PI, PID or other classical transfer-function structures. The loop controller is defined here to have a PI structure, and its selected linear transfer-functions set is multiplied by gains that are found to minimize a GPC type of cost-index. The simulation results are shown using a model of a continuous stirred tank reactor.

Cooperative control of multiple autonomous underwater vehicles (AUVs) plays an important role on marine scientific investigation and marine development. The formation of multi-AUV can significantly enhance applications on the marine sampling, imaging, surveillance and communications. Compared to the formation control of multi-robot, the formation control of multi-AUV is particularly difficult, especially on controlling attitude and direction of AUV; what is more, the communication method among AUVs is acoustic. When communication distance increases, the communication qualities deteriorate quickly; this mainly makes time-delay, signal attenuation and distortion. Although formation control of multiple AUVs obtains a wide range of attention in recent years, the fruits on formation control problem are less than ones on land multi-robot problems. For example, Fiorelli conducted a collaborative and adaptive sampling research of multi-AUV at the Monterey Bay [; Yu and Ura carried out the cable-based modular fast-moving and obstacle-avoidance experiments, and presented an interconnected multi-AUV system with three-dimension sensors. On the aspect of formation control framework [2-, [ proposed a four-layer cooperative control strategy based on hierarchical structure; [ proposed a hierarchical control framework based on hybrid model. In addition, Yang converted a nonholonomic system to a chain one and designed a controller to implement a leader-follower formation for multiple AUVs in [. The formation control for multiple autonomous underwater vehicles is rather different than the control methods for other vehicles, because the formation control for AUVs is of its characteristics, such as the large-scale distribution in space. The finite-time consensus controller designing based on finite-time control and consensus problem has important theoretical and practical significance. The decentralized controller methods for the autonomous underwater vehicle are applied more and more, but they ignore the coupling relationship between them. Another method is that an AUV is modeling as an agent, but this method ignores attitude characteristics of AUVs (pitch, roll and yaw). In this paper, we consider the cooperative control problem in three dimensional spaces. Finite-time formation for Autonomous Underwater Vehicles (AUVs) with constraints on communication range is investigated. We proposed a two-layer finite-time consensus control law, to avoid leading to collapse on formation because of failure leader, all AUVs are arrayed in the same level and each AUV can obtain global formation information. Finally, the simulation results show the effectiveness of the control strategy.

Piecewise affine (PWA) system has wide applications in circuit and control systems. It is considered as a powerful tool to study nonlinear systems, as well as switched and hybrid systems. This thesis presents our research findings in control and estimation of PWA systems. We begin with the examination of the stability of PWA systems. We introduce a piecewise homogeneous polynomial Lyapunov function approach to achieve less conservative stability conditions by using two types of power transformations. Both continuous-time and discrete-time PWA systems are considered. The approach leads to less conservative stability analysis than existing results. In addition to these stability results based on halfspace representation, we also find ways to address the stability of PWA systems based on vertex representation. Specifically, we incorporate vertex information of local partition and utilize parameterized Lyapunov functions to deduce less conservative conditions for stability. Next, we address the controllability and reachability of a class of PWA systems. Based on a general classification method, explicit necessary and sufficient conditions in terms of system parameters for controllability and reachability are presented. We show that controllability and reachability can be asserted in finite steps in some situations. We also briefly discuss the general cases. We then turn to the controller design problems for discrete-time PWA systems. We first consider the state feedback control for PWA systems. By incorporating partition information of the system and applying S-procedure, linear matrix inequality ii ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library

Output-based feedback control of discrete-time hybrid systems is an important problem, as in practice it is rarely the case that the full state variable is available for feedback. A typical approach for output-based feedback design for linear and smooth nonlinear systems is to use certainty equivalence control, in which an observer and a state feedback controller (using the observer state) are combined. Although for linear systems and some classes of nonlinear systems, separation principles exist to justify this approach, for hybrid systems this is not the case. In this paper, we isolate a class of hybrid systems for which a systematic design procedure for certainty equivalence controllers including a separation principle will be presented. This class consists of discrete-time piecewise-affine (PWA) systems with continuous dynamics. In the design procedure, we will exploit the continuity of the PWA dynamics twice. Firstly, it will be used to establish input-to-state stability (ISS) w.r.t. measurement errors from ISS w.r.t. additive disturbances. This is a crucial step as the latter problem is much easier to tackle than the former. Secondly, continuity will be used in the observer design procedure to obtain a significantly simplified set of LMIs with respect to existing observer design approaches for PWA systems. All the design conditions will be formulated in term of LMIs, which can be solved efficiently, as is also illustrated by a numerical example.

A generalized H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> observers design is proposed for linear systems with unknown inputs. It generalizes the existing results concerning the proportional observer (PO) design and the proportional integral observer (PIO) design. The approach is based on the solutions of the algebraic constraints obtained from the unbiasedness conditions of the estimation error. The observer design is obtained from the solutions of linear matrix inequalities (LMIs). A numerical example is given to illustrate our approach.

Many physical systems today are modeled by interacting continuous and discrete states that influence the dynamic behavior. There has been an increasing interest in these types of systems during the last decade, mostly due to the growing use of computers in the control of physical processes. Hybrid system models, suitable for describing the essential dynamics of a fairly large class of physical systems in control engineering applications, contain both continuous dynamics and discrete dynamics. The authors discuss the design of efficient hybrid controllers for the platoon maneuvers on an automated highway system (AHS). For the modeling of a hybrid system including the merge and split operations, they introduce the safety distance policy for the merge and split operations. Then, the platoon system is modeled by a hybrid system. In addition, the hybrid controller for the proposed merge and split operation models are presented. Finally, they demonstrate their scenarios for platoon maneuvers using the hybrid control scheme.

Hybrid dynamical systems are those with interaction between continuous and discrete dynamics. For the analysis and control of such systems concepts and theories from either the continuous or the discrete domain are typically readapted. In this thesis the ideas from perturbation theory are readapted for approximating a hybrid system using a continuous one. To this purpose, hybrid systems that possess a two-time scale property, i.e. discrete states evolving in a fast time-scale and continuous states in a slow time-scale, are considered. Then, as in singular perturbation or averaging methods, the system is approximated by a slow continuous time system. Since the hybrid nature of the process is removed by averaging, such a procedure is referred to as dehybridization in this thesis. It is seen that fast transitions required for dehybridization correspond to fast switching in all but one of the discrete states (modes). Here, the notion of dominant mode is defined and the maximum time interval spent in the non-dominant modes is considered as the 'small' parameter which determines the quality of approximation. It is shown that in a finite time interval, the solutions of the hybrid model and the continuous averaged one stay 'close' such that the error between them goes to zero as the 'small' parameter goes to zero. To utilize the ideas of dehybridization for control purposes, a cascade control design scheme is proposed, where the inner-loop artificially creates the two-time scale behavior, while the outer-loop exponentially stabilizes the approximate continuous system. It is shown that if the origin is a common equilibrium point for all modes, then for sufficiently small values of the 'small' parameter, exponential stability of the hybrid model can be guaranteed. However, it is shown that if the origin is not an equilibrium point for some modes, then the trajectories of the hybrid model are ultimately bounded, the bound being a function of the 'small' parameter. The analysis approach used here defines the hybrid system as a perturbation of the averaged one and works along the lines of robust stability. The key technical diffierence is that though the norm of the perturbation is not small, the norm of its time integral is small. This thesis was motivated by the stick-slip drive, a friction-based micropositioning setup, which operates in two distinct modes 'stick' and 'slip'. It consists of two masses which stick together when the interfacial force is less than the Coulomb frictional force, and slips otherwise. The proposed methodology is illustrated through simulation and experimental results on the stick-slip drive.