Subject Ph. D.

A major question in engineering applications is to design a control action such that the output of a physical system is able to track a desired reference, while rejecting unwanted disturbances. When the desired reference is constant, this problem is referred to as set-point tracking problem, and it arises naturally in numerous engineering applications. This is the case, e.g., if we want to operate a power converter (physical system) at constant power (desired reference) to mention an example.

It is well-known that an integral control action is needed to robustly solve the set-point tracking problem. For instance, sufficient conditions for linear time-invariant systems can be found in [2,9]. For nonlinear systems, the theory is not as developed. However, when the plant possesses an equilibrium point and its steady-state input-output map is well defined, low- gain integral control methods can be used [3,11], relying on singular perturbation theory [5]. Yet, when applying these techniques in practice, input constraints must be taken into account.

For instance, in the presence of actuator saturation, the actual plant input may differ from the controller output. When this happens, the plant is not driven by the controller and, as a result, the states of the controller are wrongly updated causing long transients, oscillations, and even instability. This phenomenon is called controller windup [4]. Various anti-windup techniques have been proposed, resulting in a vast literature [4,12]. However, despite the effectiveness of these strategies, they primarily focus on linear plants with nonlinear actuators.

A different approach for constrained integral control of nonlinear systems has been recently proposed in [7,8]. The idea is to use tools from projected dynamical systems [1,10] to constrain the state of the integrator in a closed and convex set, where safety constraints are guaranteed. By doing this, the set- point tracking problem can be solved and safety constraints (including anti-windup guarantees) are enforced altogether. We call this new class of controllers projected integral controllers. As shown in [7,8], projected integral controllers find a wide range of applications in power systems, e.g., for the constrained output power regulation in grid-connected synchronverters [6].

The methods developed in [7,8] are extremely promising in applications, yet the theory is still in its early stages. The objective of this thesis is to extend the results from [7,8], further investigating projected integral controllers. These results will be applied to the control of power converters.

References:

[1] B. Brogliato and A. Tanwani. Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability. SIAM Review, 62(1):3-129, 2020. [2] E. Davison. The robust control of a servomechanism problem for linear time-invariant multivariable systems. IEEE Transactions on Automatic Control, 21(1):25-34, 1976. [3] C. Desoer and C.-A. Lin. Tracking and disturbance rejection of MIMO nonlinear systems with PI controller. IEEE Transactions on Automatic Control, 30(9):861-867, 1985 [4] M.V. Kothare, P.J. Campo, M. Morari, and C.N. Nett. A unified framework for the study of anti- windup designs. Actomatica, 30(12):1869-1883, 1994. [5] P. Kokotović, H.K. Khalil, and J. O'reilly. Singular Perturbation Methods in Control: Analysis and Design. SIAM, 1999. [6] P. Lorenzetti, Z. Kustanovich, S. Shivratri, and G. Weiss. The equilibrium points and stability of grid- connected synchronverters. IEEE Transactions on Power Systems, 37(2):1184-1197, 2022. [7] P. Lorenzetti and G. Weiss. PI control of stable nonlinear plants using projected dynamical systems. Automatica, 146:110606, 2022. [8] P. Lorenzetti and G. Weiss. Saturating PI control of stable nonlinear systems using singular perturbations. IEEE Transactions on Automatic Control, 68(2):867-882, 2022. [9] M. Morari. Robust stability of systems with integral control. IEEE Transactions on Automatic Control, 30(6):574-577, 1985. [10] A. Nagurney and D. Zhang. Projected Dynamical Systems and Variational Inequalities with Applications, volume 2. Springer Science & Business Media, 1995. [11] J.W. Simpson-Porco. Analysis and synthesis of low-gain integral controllers for nonlinear systems. IEEE Transactions on Automatic Control, 66(9):4148-4159, 2021. [12] S. Tarbouriech and M. Turner. Anti-windup design: An overview of some recent advances and open problems. IET Control Theory & Applications, 3(1):1-19, 2009.