PI/PID controller stabilizing sets of uncertain nonlinear systems: an efficient surrogate model-based approach

AbstractClosed forms of stabilizing sets are generally only available for linearized systems. An innovative numerical strategy to estimate stabilizing sets of PI or PID controllers tackling (uncertain) nonlinear systems is proposed. The stability of the closed-loop system is characterized by the sign of the largest Lyapunov exponent (LLE). In this framework, the bottleneck is the computational cost associated with the solution of the system, particularly including uncertainties. To overcome this issue, an adaptive surrogate algorithm, the Monte Carlo intersite Voronoi (MiVor) scheme, is adopted to pertinently explore the domain of the controller parameters and classify it into stable/unstable regions from a low number of nonlinear estimations. The result of the random analysis is a stochastic set providing probability information regarding the capabilities of PI or PID controllers to stabilize the nonlinear system and the risk of instabilities. The minimum of the LLE is proposed as tuning rule of the controller parameters. It is expected that using a tuning rule like this results in PID controllers producing the highest closed-loop convergence rate, thus being robust against model parametric uncertainties and capable of avoiding large fluctuating behavior. The capabilities of the innovative approach are demonstrated by estimating robust stabilizing sets for the blood glucose regulation problem in type 1 diabetes patients

Power electronic converters design and control based on non-linear dynamical models with veri_ed controllability via set theory in control

ABSTRACT: In this work a design procedure of Power Electronic Converters (PECs) is proposed, which is based on system knowledge and systems theory. The proposed design procedure is composed by 4 stages and 10 steps and allows to obtain all PEC parameters and its control structure such that established system operating requirements are satisfied. In this proposed design procedure, system controllability is tested based on set theory in control. The main achievement of this design procedure is to allow the PEC design taking into account its inherent dynamical nature and with verified controllability, but without fixing any control structure. The proposed design procedure is applied to a Boost DCDC converter as illustrative example to show its applicability. Then, proposed procedure design is applied to a case of study: a Three-Leg Split-Capacitor Shunt Active Power Filter is successfully designed

Towards a framework for the development of control-oriented multiscale models of dynamical systems: semibatch emulsion polymerization case study

ABSTRACT: This work develops a framework for the construction of a control-oriented model from a multiscale perspective, using a semibatch emulsion polymerization process as a case study. First, a so-called full multiscale model (considering the macro-, meso-, and micro-scopic scales) was developed which is composed of a set of Partial/Ordinary Differential Equations and a kinetic Monte Carlo simulation (PDE/ODE - kMC). Then, to obtain a reduced-order representation of the multiscale model, Variance Algebra concepts are used as a tool for representing, at the mesoscopic scale, a disperse-phase system from which only statistical information is available. After that, a dataset considering several process operational conditions is built to capture the main dynamics at the microscopic scale. This dataset is used to derive a closed-form model of the microscopic state variables by adopting a statistical modeling approach. The final obtained control-oriented model is composed of a set of ODEs comprising the macroscopic and the mesoscopic scales that can be solved by using standard ODEs integration schemes, whereas the microscopic scale variables are conveniently defined as some of the system outputs, represented by a set of algebraic equations. In order to consistently solve the full multiscale model, a numerical scheme based on the Finite Element Method is developed capturing the nonlinear evolution of the Particle Size Distribution (PSD). The validity of the obtained reduced-order model is verified through several simulations with respect to the system inputs. Finally, the multiscale control-oriented representation is employed to perform a batch output-controllability analysis based on a set-theoretic approach. The proposed framework might be adopted as a tool for the derivation of dynamical multiscale models keeping a good balance between their tractability and predictive capability, which can constitute an advantage when implementing real-time optimization and process control

Analysis and Control of Power Electronic Converters Based on a System Zero Locations Approach

This chapter presents a procedure to design and control power electronic converters (PECs), which includes a zero-based analysis as a dynamical system response criterion for dimensioning converter passive elements. For this purpose, a nonideal boost DC-DC converter (converter considering its parasitic losses) is dynamically modeled and analyzed in steady state as an application example. The steady-state model is obtained from the average nonlinear model. The steady-state model allows deducing expressions for equilibrium conversion ratio M D and efficiency η of the system. Conditions for the converter conduction modes are analyzed. Simulations are made to see how parasitic losses affect both M D and η . Then, inductor current and capacitor voltage ripple analyses are carried out to find lower boundaries for inductor and capacitor values. The values of the boost DC-DC converter passive elements are selected taking into account both steady-state and zero-based analyses. A nonideal boost DC-DC converter and a PI-based current mode control (CMC) structure are designed to validate the proposed procedure. Finally, the boost DC-DC converter is implemented in PSIM and system operating requirements are satisfactorily verified

PI/PID controller stabilizing sets of uncertain nonlinear systems: an efficient surrogate model-based approach

Closed forms of stabilizing sets are generally only available for linearized systems. An innovative numerical strategy to estimate stabilizing sets of PI or PID controllers tackling (uncertain) nonlinear systems is proposed. The stability of the closed-loop system is characterized by the sign of the largest Lyapunov exponent (LLE). In this framework, the bottleneck is the computational cost associated with the solution of the system, particularly including uncertainties. To overcome this issue, an adaptive surrogate algorithm, the Monte Carlo intersite Voronoi (MiVor) scheme, is adopted to pertinently explore the domain of the controller parameters and classify it into stable/unstable regions from a low number of nonlinear estimations. The result of the random analysis is a stochastic set providing probability information regarding the capabilities of PI or PID controllers to stabilize the nonlinear system and the risk of instabilities. The minimum of the LLE is proposed as tuning rule of the controller parameters. It is expected that using a tuning rule like this results in PID controllers producing the highest closed-loop convergence rate, thus being robust against model parametric uncertainties and capable of avoiding large fluctuating behavior. The capabilities of the innovative approach are demonstrated by estimating robust stabilizing sets for the blood glucose regulation problem in type 1 diabetes patients

Optimum PI/PID Controllers Tuning via an Evolutionary Algorithm

In this chapter, it is demonstrated that when using advanced evolutionary algorithms, whatever the adopted system model (SOSPD, nonminimum phase, oscillatory or nonlinear), it is possible to find optimal parameters for PID controllers satisfying simultaneously the behavior of the system and a performance index such as absolute integral error (IAE). The Multidynamics Algorithm for Global Optimization (MAGO) is used to solve the control problem with PID controllers. MAGO is an evolutionary algorithm without parameters, with statistical operators, and for the optimization, it does not need the derivatives, what makes it very effective for complex engineering problems. A selection of some representative benchmark systems is carried out, and the respectively two-degree-of-freedom (2DoF) PID controllers are tuned. A power electronic converter is adopted as a case study and based on its nonlinear dynamical model, a PI controller is tuned. In all cases, the control problem is formulated as a constrained optimization problem and solved using MAGO. The results found are outstanding

Voltage control of a DC Bus for shunt active power compensators

RESUMEN: Este artículo aborda el problema del control de tensión en el bus DC para compensadores activos de potencia conectados en paralelo (SAPC, Shunt Active Power Compensator) de tres ramas y cuatro hilos, basando la solución en su modelo matemático. A partir del modelo del SAPC se establece un modelo con información suficiente para deducir la función de transferencia que describe el proceso de carga y descarga del bus DC. Se emplea la sintonía de un controlador clásico PI con la velocidad y robustez suficientes para mantener el nivel de tensión adecuado en el bus DC, de tal modo que, la red solo suministra la potencia activa que la carga demanda. El modelo propuesto garantiza la compensación de potencia reactiva, la regulación de la tensión en el bus DC, y el equilibrio entre los dos capacitores del enlace del bus DC. Los resultados se verifican a partir del modelo matemático en MATLAB y se validan por medio de un modelo circuital implementado en PSIM.ABSTRACT: This paper deals with the DC-bus control problem for three-leg four-wires Shunt Active Power Compensators (SAPCs), based on the solution of a mathematical model. Starting with the SAPC concept a mathematical model with enough information to deduce the transfer function that describes the charging/discharging DC-bus process is proposed. A PI controller for tuning the SAPC with enough speed and robustness to keep the DC voltage in an adequate level is employed, so that it is guaranteed that the active power is only supplied on demand. The proposed model guarantees reactive power compensation, regulation of the DC bus, and balance between the two capacitors in the DC bus. The mathematical model is verified by Matlab simulations and it is validated through a circuital model implemented in PSIM

Transport map Bayesian parameter estimation for dynamical systems

AbstractAccurate online state and parameter estimation of uncertain non‐linear dynamical systems is a demanding task that has been traditionally handled by adopting non‐linear Kalman Filters or particle filters. However, in case of Kalman filters the system needs to be linearised and for particle filters the computational demand can be high. Recent advances in optimal transport theory and the application to Bayesian model updating pave the way for other approaches to system and parameter identification. They also provide a way of formulating the problem in such a way that efficient online estimation for complex systems is possible. In this work, we investigate the properties of the transport map approach when compared to standard Markov Chain Monte Carlo in an off‐line setting as a first step towards on‐line parameter estimation. We apply both approaches to an analytical exponential model and a dynamical system with seven unknown parameters subjected to ground displacement. Details on the theory of transport maps and on the used MCMC algorithm are also given.</jats:p