Research

Research interests

I am broadly interested in the area of control theory and optimization, with an emphasis on dynamical systems with state and control constraints. My current research interests are:

Model predictive control
Hybrid systems
Scenario optimization
Data-driven analysis and control
Linear embedding of nonlinear systems
Set-theoretic methods

Model predictive control

Model Predictive Control (MPC) is a powerful control technology for general constrained systems due to its ability to handle hard constraints on the system. In the classic MPC, the control action is determined by optimizing the future evolution of the trajectories. However, many engineering systems such as power, water distribution, traffic and manufacturing systems consist of a group of interacting subsystems that may transmit information among one another. The mathematical models of these systems are huge and complex, resulting in difficult or even intractable MPC optimization problems, which have to be solved online with real-time requirements (e.g., limited computing power). To circumvent this computational issue, we have developed several MPC approaches for different situations.

Distributed MPC with coupled dynamics and constraints: Coupled dynamics often occurs in networked systems in which individual subsystems are equipped with computational devices and communicate with others over a network. To tackle this issue, we have proposed a distributed MPC approach which converts the overall MPC problem into a group of smaller problems that can be solved efficiently in a distributed fashion. Besides coupled dynamics, constraint coupling is also a common problem that can be found in many applications, such as water distribution systems with limits on overall input/output, and Heating, Ventilation, and Air Conditioning (HVAC) systems with limits on total energy input and airflow rate. In the presence of globally coupled constraints, achieving global optimality becomes challenging. One possible way is to solve the dual problem of the MPC problem. We have developed distributed MPC approaches to alleviate this challenge using a distributed Alternating Direction Multiplier Method (ADMM) and a distributed Nesterov-like gradient algorithm. Both of the two proposed approaches require a finite-time consensus algorithm, whose convergence speed depends on the minimal polynomial of the weighted graph.

Z. Wang, C.J Ong. Distributed MPC of constrained linear systems with time-varying terminal sets, Systems & Control Letters, 88: 14-23, 2016
Z. Wang, C.J. Ong, Distributed model predictive control of linear discrete-time systems with local and global constraints, Automatica, 81, 184-195, 2017.
Z. Wang, C.J. Ong, Accelerated distributed MPC of linear discrete-time systems with coupled constraints, IEEE Transactions on Automatic Control, 63(11), 3838 - 3849, 2018.
Z. Wang, C.J. Ong, Speeding up finite-time consensus via minimal polynomial of a weighted graph - a numerical approach, Automatica, 93, 415–421, 2018.

Complexity vs. Optimality: An alternative method to reduce online computation is to use input parameterization and curtail the number of degrees of freedom in the MPC problem at the expense of optimality. We have proposed a scheme to reduce complexity using singular value decomposition (SVD), where the control inputs are parameterized by a reduced set of variables that contain the maximal amount of energy.

C.J. Ong, Z. Wang. Reducing variables in Model Predictive Control of linear system with disturbances using singular value decomposition, Systems & Control Letters, 71: 62-68, 2014.

MPC of hybrid systems: Hybrid behaviors, which is the interaction of continuous and discrete dynamics, often appear in cyber-physical systems. Controlling hybrid systems is very challenging in general. An important family of hybrid systems are switched linear systems which consist of a finite set of linear dynamics (called modes) and a switching rule. We have proposed a MPC approach for such systems with a minimal dwell-time restriction.

C.J. Ong, Z. Wang, and M. Dehghan. Model predictive control for switching system with dwell-time restriction, IEEE Transactions on Automatic Control, 61(12), 4189-4195, 2016.

Computation of safe and invariant sets

For safety-critical applications, it is of significant importance to ensure that the system operates within safety constraints. To formally verify safety, one needs the characterization of safe sets where the system never escapes the given safety constraints. The concept of positively invariant sets in dynamical systems can be leveraged to characterize such sets. Indeed, when a subset of the safety constraint set is positively invariant, it can be considered as a safe set. In particular, the maximal invariant set inside the safety set coincides with the largest safe set.

Non-convex constraints: In real applications, safety constraints are often non-convex. For instance, in the case of obstacle avoidance, by removing the obstacles, the safe region is a non-convex constraint set. Although the literature on invariant set computation is large, computing the maximal safety set is still a challenging problem in the presence of non-convex constraints even for linear systems. Recently, we have proposed a method for computing the maximal invariant set of linear systems subject to a class of non-convex constraints that include semialgebraic sets by solving a set of linear matrix inequalities. We have also shown that this method can be extended to certain nonlinear systems and switched linear systems.

Z. Wang, R.M. Jungers, and C.J. Ong, Computation of the maximal invariant set of discrete-time linear systems subject to a class of non-convex constraints, Automatica, 125: 109463, 2021.

Data-driven set invariance verification: For black-box systems without dynamical models, model-based approaches are no longer applicable. To deal with such systems, we have proposed data-driven methods relying on the observation of trajectories to determine almost-invariant sets, which are invariant everywhere except possibly in a small subset, with probabilistic set invariance guarantees. These techniques can be useful for safety verification of complex systems with learning-based control.

Z. Wang, R.M. Jungers, Scenario-based set invariance verification for black-box nonlinear systems, IEEE Control Systems Letters, 2020.
Z. Wang, R.M. Jungers, A data-driven method for computing polyhedral invariant sets of black-box switched linear systems. IEEE Control Systems Letters, 2020.

Linear embedding and learning dynamics from data

Linearization of nonlinear systems is one of the most fascinating research topics in systems and control. An emerging linearization method is the state immersion method (including operator-theoretic approaches like the Carleman linearization approach and the Koopman approach), which lifts or embeds a nonlinear system into a linear system with a higher dimension.

Data-driven immersion and the Koopman operator: We have proposed a data-driven immersion approach called polyflow approximation to obtain high-dimensional linear equivalents or approximations of discrete-time nonlinear systems. Our approach only takes a finite set of trajectories and hence an analytic model is not required. The mismatch between the approximate linear model and the original system is rigorously discussed with formal bounds. We also provide a Koopman-operator interpretation of this technique, which shows a link between system immersibility and the Koopman operator theory.

Z. Wang and R.M. Jungers, A data-driven immersion technique for linearization of discrete-time nonlinear systems. IFAC World Congress, 2020.

Immersion-based invariant set computation: Linear embedding of nonlinear systems facilitates system analysis and control design in the sense that it enables the use of powerful linear control techniques. For instance, we have investigated the use of the immersion approach above for computing invariant sets of nonlinear systems. With the system immersion, the invariant set of the original nonlinear system can be characterized by a lifted linear model. For general nonlinear systems, we use linear approximations because exact immersion cannot be achieved. To handle mismatch errors, we tighten the constraint set of the lifted linear model that leads to an invariant inner approximation of the invariant set of the original system.

Z. Wang, R.M. Jungers, and C.J. Ong, Computing invariant sets of discrete-time nonlinear systems via state immersion. IFAC World Congress, 2020.
Z. Wang, R.M. Jungers, and C.J. Ong, Computation of invariant sets via immersion for discrete-time nonlinear systems, 2021, under review.

Immersion-based MPC: In MPC, the control input is typically computed by solving optimization problems repeatedly online. For general nonlinear systems, the online optimization problems are non-convex and computationally expensive or even intractable. In this paper, we propose to circumvent this issue by computing a high-dimensional linear embedding of discrete-time nonlinear systems. The computation relies on an algebraic condition related to the immersibility property of nonlinear systems and can be implemented offline. With the high-dimensional linear model, we then define and solve a convex online MPC problem. We also provide an interpretation of our approach under the Koopman operator framework.

Z. Wang and R.M. Jungers, Immersion-based model predictive control of constrained nonlinear systems: Polyflow approximation. In Proceedings of the European Control Conference, 2021.

Industrial projects

Application of Big Data digital technologies in the Med-Tech context: One-click calibration

The goal is to implement an automated calibration procedure for the proton beamlines in the proton therapy system. This is a typical cyber-physical system, where the beam monitor and magnet electronic units are connected to steer the beam distribution and the action of a magnetic field onto protons is governed by the laws of electromagnetism. The main challenge of calibration is that there are strict specifications, including the beam size, its shape or its optical distribution. We have proposed a sequential linearized algorithm to achieve specifications of the beam distribution. Our algorithm has been tested on a beamline simulator developed by IBA (a world leader in proton therapy) and several real sites. With this automated calibration procedure, we have reduced the calibration time by about 50%.

Z. Wang, R.M. Jungers, Q. Flandroy, B. Herregods, C. Hernalsteens, Finite-horizon covariance control of state-affine nonlinear systems with application to proton beamline calibration. In Proceedings of the European Control Conference, Naple, Italy, 2019.

Control and optimization of multi-zone building systems

The goal is to minimize the total energy cost while maintaining thermal comfort. Commercial buildings often consist of multiple interacting thermal zones, which are dynamically coupled due to the heat transfer between adjacent zones. Besides the coupled dynamics, globally coupled constraints will also arise from the limited total supply energy and air ow rate. To tackle the these couplings and the stability issue, we have proposed distributed MPC approaches for multi-zone building systems.

Z. Wang, G. Hu. Economic MPC of nonlinear systems with non-monotonic Lyapunov functions and its application to HVAC control, International Journal of Robust and Nonlinear Control, 28(6): 2513-2527, 2018.
Z. Wang, G. Hu, and C. J. Spanos. Distributed model predictive control of bilinear HVAC systems using a convexification method. In Proceedings of the Asian Control Conference, Gold Coast, Australia, 2017.

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