Mixed Logical Dynamical System Research Articles

Hybrid zonotopes: A new set representation for reachability analysis of mixed logical dynamical systems

This article presents a new set representation named the hybrid zonotope that is equivalent to the union of 2N constrained zonotopes – convex polytopes – through the addition of N binary zonotope factors. The major contribution of this manuscript is a closed-form solution for exact forward reachable sets of discrete-time, linear hybrid systems modeled as mixed logical dynamical systems. The proposed approach captures the worst-case exponential growth in the number of convex sets required to represent the nonconvex reachable set while exhibiting only linear growth in the complexity of the hybrid zonotope set representation. Redundancy removal techniques are provided that leverage binary trees to store the combinations of binary factors of the hybrid zonotope that map to nonempty convex subsets. Numerical examples show the hybrid zonotope’s ability to compactly represent nonconvex reachable sets with an exponential number of features. Furthermore, the hybrid zonotope is shown to be closed under linear mappings, Minkowski sums, generalized intersections, and halfspace intersections.

Robust Successor and Precursor Sets of Hybrid Systems Using Hybrid Zonotopes

This letter presents identities for calculating robust successor and precursor sets of discrete-time linear hybrid systems. The proposed technique relies on generating a set containing all possible state transitions of a dynamic system over a region of interest, named the state-update set. Forward and backward reachability can then be performed using only projection, intersection, and Minkowski difference set operations with the state-update set. It is shown how state-update sets may be defined as hybrid zonotopes for mixed logical dynamical systems and linear systems in closed loop with model predictive control.

A Kalman filter-based Hybrid Model Predictive Control Algorithm for Mixed Logical Dynamical Systems: Application to Optimized Interventions for Physical Activity.

Hybrid Model Predictive Control (HMPC) is presented as a decision-making tool for novel behavioral interventions to increase physical activity in sedentary adults, such as Just Walk. A broad-based HMPC formulation for mixed logical dynamical (MLD) systems relevant to problems in behavioral medicine is developed and illustrated on a representative participant model arising from the Just Walk study. The MLD model is developed based on the requirement of granting points for meeting daily step goals and categorical input variables. The algorithm features three degrees-of-freedom tuning for setpoint tracking, measured and unmeasured disturbance rejection that facilitates controller robustness; disturbance anticipation further improves performance for upcoming events such as weekends and weather forecasts. To avoid the corresponding mixed-integer quadratic problem (MIQP) from becoming infeasible, slack variables are introduced in the objective function. Simulation results indicate that the proposed HMPC scheme effectively manages hybrid dynamics, setpoint tracking, disturbance rejection, and the transition between the two phases of the intervention (initiation and maintenance) and is suitable for evaluation in clinical trials.

Hybrid Model Predictive Control for Hybrid Electric Vehicle Energy Management Using an Efficient Mixed-Integer Formulation

This paper proposes to use hybrid model predictive control (HMPC) for energy management in hybrid electric vehicle (HEV) using an efficient formulation. HEV has two sources of energy - electric motor and internal combustion engine (ICE) - allowing it an additional degree of freedom to optimize the ratio between the use of two energy sources. HEV energy management is crucial to exploit its potential to reduce fuel consumption and emissions. However, it is challenging to achieve the optimal solution along with the fast dynamics of HEVs. In this paper, instead of using a nonlinear, computationally expensive dynamic model of HEV, a piecewise afne (PWA) model is used to depict its behavior, implemented with multiple linear regression. The validation of the developed model is taken with standard drive cycles. Based on the PWA model, an optimal control problem of HMPC was formulated as mixed integer linear programming (MILP). Conventionally, mixed logical dynamical (MLD) system has been used in the process control field, including early studies of HEV control. This paper applies Big-M formulation which is efficient in its problem size and tight inequality relation for integer variables. The performance of the HMPC controller was examined in a simulated environment based on MATLAB/Simulink HEVP2 application. As a result, HMPC shows superior control performance than the equivalent consumption minimization strategy (ECMS).

Nonlinearity handling in MPC for Power Congestion management in sub-transmission areas

This paper proposes congestion management solutions based on Model Predictive Control (MPC) principles for transmission network zones. The contribution resides in the use of logical variables for describing the nonlinearities related to the modelling of the physical aspects in power curtailment, or some desired control goals. This allows for the representation of the nonlinear model of a sub-transmission zone with a storage device in a linear Mixed Logical Dynamical (MLD) formulation, then paving the way to the utilisation of linear mixed integer programming for the optimization problem. Moreover, part of the contribution considers supplementary temporal specifications for the energy storage device utilisation. This is modelled using additional temporal and logical variables as part of the system state and control signal. Consequently, the extended model of the power network zone is formulated as a linear MLD system and integrated to the MPC design. The proposed controllers are validated through simulations on an industrial case-study.

Switched Pinning Control for Merging and Splitting Maneuvers of Vehicle Platoons

This paper considers a velocity control problem for merging and splitting maneuvers of vehicle platoons. In this paper, an external device sends velocity commands to some vehicles in the platoon, and the others adjust their velocities autonomously. The former is pinning control, and the latter is consensus control in multi-agent control. We propose a switched pinning control algorithm. Our algorithm consists of three sub-methods. The first is an optimal switching method of pinning agents based on an MLD (Mixed Logical Dynamical) system model and MPC (Model Predictive Control). The second is a representation method for dynamical platoon formation with merging and splitting maneuver. The platoon formation follows the positional relation between vehicles or the formation demand from the external device. The third is a switching reduction method by setting a cost function that penalizes the switching of the pinning agents in the steady-state. Our proposed algorithm enables us to improve the consensus speed. Moreover, our algorithm can regroup the platoons to the arbitrary platoons and control the velocities of the multiple vehicle platoons to each target value.

Cruise Control for a Two-Wheeled Mobile Vehicle Using Its Mixed Logical Dynamical System Model

This paper proposes a cruise control for a two-wheeled mobile vehicle on a two-dimensional plane. The cruise control is executed using model predictive control (MPC), such that input and state constraints can be addressed explicitly. For real-time execution of MPC, we model the dynamics of the vehicle as a mixed logical dynamical system rather than an underlying nonlinear model. The cruise control is then reduced to a mixed integer quadratic programming problem, for which efficient solvers using the convexities are available. Simulations and experiments demonstrate the effectiveness of the proposed cruise control through a computation time comparison with the nonlinear MPC method.

Dynamic Surveillance by Multiple Agents with Fuel Constraints

The surveillance problem is to find optimal trajectories of agents that patrol a given area as evenly as possible. In this paper, we consider multiple agents with fuel constraints. The surveillance area is given by a weighted directed graph, where the weight assigned to each arc corresponds to the fuel consumption/supply. For each node, the penalty to evaluate the unattended time is introduced. Penalties, agents, and fuels are modeled by a mixed logical dynamical system model. Then, the surveillance problem is reduced to a mixed integer linear programming (MILP) problem. Based on the policy of model predictive control, the MILP problem is solved at each discrete time. In this paper, the feasibility condition for the MILP problem is derived. Finally, the proposed method is demonstrated by a numerical example.

Mixed-Integer Model Predictive Control of Hybrid Impulsive Linear Systems

We devise a model predictive control algorithm for impulsive linear systems with autonomous flow dynamics and controlled jumps. Thereby the moments of jumps are not fixed, but rather considered as decision variables. To this end, the complete system dynamics is formulated as a mixed-logical dynamical system after an appropriate discretization step. The resulting optimization problem contains both discrete and continuous decision variables, giving rise to a mixed-integer programming problem. The objective of the optimization is to steer the states into a target set. The stability is addressed through an appropriate cost function together with invariance conditions, as well as by introducing terminal constraints which are only enforced within a certain distance to the target set, thus, providing a trade-off between guaranteed convergence to the target set and computational complexity.

A Modelling and Control Approach for a Type of Mixed Logical Dynamical System Using in Chilled Water System of Refrigeration System

The chilled water system of central air conditioning is a typical hybrid system. The dynamic adjustment of cooling capacity based on hybrid system can achieve accurate temperature control and real-time energy saving. Mixed logical dynamical (MLD) systems have advantage for solving constrained optimization problems of this type of case by numerical methods. This paper proposes a novel modified type of MLD system, which enhances the model applicability and improves the switching flexibility and control effect for the framework. In order to meet the needs of cooling capacity and energy saving, the optimal control problem is transformed as MIQP problem by defined performance index. As a numeral example of application, the model and control method is used in pumps group control for variable water volume in chilled water system of central air conditioning. At last, the dynamic and energy-saving effects of the system are simulated, which shows the ideal control results.

Mixed logical dynamical model-based MPC for yaw stability control of distributed drive electric vehicles

This paper proposes a hybrid model predictive control (MPC) controller for yaw stability control of distributed drive electric vehicles based on the nonlinear dynamic model. To reduce the computational burden during the optimization process of MPC problem, the vehicle model is formulated by approximating the magic tire model with a set of piecewise linear functions. The dynamic model is equivalently translated into a mixed logical dynamical (MLD) system which is used to design the hybrid MPC controller. To improve the energy efficiency of the system while tracking the reference signals, the input signals are restricted in the high efficiency range of in-wheel motors. Finally, simulation results of different inputs and the comparison of closed-loop and open-loop system indicate the suggested hybrid MPC control method is able to fleetly track the reference and stabilize the vehicle effectively for the test maneuvers.

Optimal control of intelligent vehicle longitudinal dynamics via hybrid model predictive control

This paper presents an innovative application of the hybrid model predictive control (HMPC) scheme to optimally regulate the intelligent vehicle longitudinal velocity. For autonomous velocity regulation, the intelligent vehicle needs to be operated in two distinct modes (drive and brake) and because of the mode-dependent constraints on accelerations and decelerations by considering the comfort of passengers, the intelligent vehicle longitudinal dynamics control process can be regarded as a constrained hybrid dynamical system. Thus, in this study, the intelligent vehicle longitudinal dynamics is approximated as a two-mode discrete-time mixed logical dynamical (MLD) system. Using this approximation, a hybrid model predictive controller, which allows us to optimize the switching sequences of the operation modes (binary control inputs) and the torques acted on the wheels (continuous control inputs), is tuned based on online mixed-integer quadratic programming. Numerical simulation analysis is conducted for a sport utility vehicle to demonstrate the effectiveness of the proposed control method. Finally, the explicit representation of the HMPC is computed to control the intelligent vehicle in real-time, and the experimental results are presented to show the applicability of the proposed controller.

Ellipsoidal Lyapunov-based hybrid model predictive control for mixed logical dynamical systems with a recursive feasibility guarantee

This paper proposes an ellipsoidal hybrid model predictive control approach to solve the robust stability problem of uncertain hybrid dynamical systems modelled by the mixed logical dynamical framework. In this approach, the traditional terminal equality constraint is replaced by an ellipsoid that results in a maximal positive invariant set for the closed-loop system. Then, a Lyapunov decreasing condition along with the robustness criterion is introduced to the optimization problem to achieve the robust stability of the closed-loop system. As the main advantages, the ellipsoidal terminal set proposed in this paper attains a larger domain of attraction along with the recursive feasibility guarantee. Moreover, the stability and robustness constraints are achieved by a lower prediction horizon, which leads to a smaller dimension optimization problem. In addition, to reduce the computational complexity of the corresponding optimization problem, a suboptimal version of the proposed algorithm is introduced. Finally, numerical and car suspension system examples show the capabilities of the proposed method.

On the complexity and dynamical properties of mixed logical dynamical systems via an automaton‐based realization of discrete‐time hybrid automaton

SummaryModeling of hybrid systems using mixed logical dynamical (MLD) systems is an art. The MLD framework often introduces numerous constraints and auxiliary binary and continuous variables, which, in turn, increase the computational complexity of the optimization problems. This paper presents an automaton‐based realization for discrete‐time hybrid automaton (DHA) with both controlled and uncontrolled switching phenomena by which it is attempted to develop efficient translation techniques to MLD systems and reduce the total number of decision variables in the MLD model. Based on this DHA model, a modified version of MLD systems, which is called extended MLD (EMLD) is formally defined and represented. EMLD is derived based on the concept of forward evolution in which the one‐step delay between the change of discrete states and continuous dynamics of the existing evolution in the conventional MLD systems is eliminated. The result is that the size of EMLD model in terms of the number of variables, which determines the complexity of the synthesis problems on MLD systems, is reduced. The dynamical properties such as the nonblocking, determinism, and well‐posedness of the proposed DHA and its equivalent MLD models are investigated, and the necessary and sufficient conditions are derived. The effectiveness of the proposed ideas is shown by the numerical examples.

Optimal Operation of High-Speed Trains Using Hybrid Model Predictive Control

The high-speed train operation process is highly nonlinear and has multiple constraints and objectives, which lead to a requirement for the automatic train operation (ATO) system. In this paper, a hybrid model predictive control (MPC) framework is proposed for the controller design of the ATO system. Firstly, a piecewise linear system with state and input constraints is constructed through piecewise linearization of the high-speed train’s nonlinear dynamics. Secondly, the piecewise linear system is transformed into a mixed logical dynamical (MLD) system by introducing the auxiliary binary variables. For the transformed MLD system, a hybrid MPC controller is designed to realize the precise control under hard constraints. To reduce the online computation complexity, the explicit control law is computed offline by employing the mixed-integer linear programming (MILP) technique. Simulation results validate the effectiveness of the proposed method.

Approximate MLD System Model of Switched Linear Systems for Model Predictive Control

In this paper, to reduce the computation time for solving the finite-time optimal control problem (i.e., the model predictive control (MPC) problem), an approximate model of switched linear systems is proposed. A switched linear system is one of the typical subclasses of hybrid systems, and has several applications such as networked control systems. First, an approximation of the state variable in switched linear systems is derived. Next, using auxiliary binary and continuous variables, an approximate model given by a mixed logical dynamical system model is derived. Using this approximate model, the dimension of the auxiliary continuous variable can be reduced. As a result, the online computation time in MPC can decrease. Finally, the effectiveness of the proposed method is presented by a numerical example.

Model predictive control of an air suspension system with damping multi-mode switching damper based on hybrid model

This paper presents the hybrid modeling and the model predictive control of an air suspension system with damping multi-mode switching damper. Unlike traditional damper with continuously adjustable damping, in this study, a new damper with four discrete damping modes is applied to vehicle semi-active air suspension. The new damper can achieve different damping modes by just controlling the on-off statuses of two solenoid valves, which makes its damping adjustment more efficient and more reliable. However, since the damping mode switching induces different modes of operation, the air suspension system with the new damper poses challenging hybrid control problem. To model both the continuous/discrete dynamics and the switching between different damping modes, the framework of mixed logical dynamical (MLD) systems is used to establish the system hybrid model. Based on the resulting hybrid dynamical model, the system control problem is recast as a model predictive control (MPC) problem, which allows us to optimize the switching sequences of the damping modes by taking into account the suspension performance requirements. Numerical simulations results demonstrate the efficacy of the proposed control method finally.

Semi-formal verification of closed-loop specifications in the concept design phase

Abstract The article addresses the semi-formal verification of behavioral specifications for subsystems consisting of physical parts and controllers, complemented by simulation-based integration testing. Since design errors in early phases tend to be particularly expensive, the method is tailored towards applicability in these phases. We verify behavioral specifications with proof-like credibility, or falsify them while providing a violation scenario that is reusable as a test case. The system is represented as a mixed logical dynamical (MLD) system, and specifications are expressed by a temporal logic with affine signal abstractions. The verification problem is converted into an equivalent mixed-integer linear feasibility problem solved using off-the-shelf solvers. An example illustrates the effectiveness of the method.

Building Climate Energy Management in Smart Thermal Grids via Aquifer Thermal Energy Storage Systems

This paper proposes a building energy management framework, described by mixed logical dynamical systems due to operating constraints and logic rules, together with an aquifer thermal energy storage (ATES) model. We develop a deterministic model predictive control strategy to meet building thermal energy demand. At each sampling a mixed integer quadratic optimization problem is formulated. We then provide a simulation study using an agent-based model and a geohydrological simulation environment (MODFLOW) to illustrate the performance of the framework.

A perspective-based convex relaxation for switched-affine optimal control

We consider the switched-affine optimal control problem, i.e., the problem of selecting a sequence of affine dynamics from a finite set in order to minimize a sum of convex functions of the system state. We develop a new reduction of this problem to a mixed-integer convex program (MICP), based on perspective functions. Relaxing the integer constraints of this MICP results in a convex optimization problem, whose optimal value is a lower bound on the original problem value. We show that this bound is at least as tight as similar bounds obtained from two other well-known MICP reductions (via conversion to a mixed logical dynamical system, and by generalized disjunctive programming), and our numerical study indicates it is often substantially tighter. Using simple integer-rounding techniques, we can also use our formulation to obtain an upper bound (and corresponding sequence of control inputs). In our numerical study, this bound was typically within a few percent of the optimal value, making it attractive as a stand-alone heuristic, or as a subroutine in a global algorithm such as branch and bound. We conclude with some extensions of our formulation to problems with switching costs and piecewise affine dynamics.