Lukas Rusar

Multivariable gain scheduled control of two funnel liquid tanks in series

This paper presents a concept of the continuous-time gain scheduled 2DOF control for the nonlinear system of two funnel liquid tanks in series, based on linearization of nonlinear equations of the system about selected operating points. Discussed technique extends the idea of design via linearization approach. In order to achieve desired stability and performance requirements, a linear feedback controller is designed at each point. Based on this strategy, polynomial method is taken into account. Specifically, the exact pole placement method as well as weighting matrix, dividing weights among numerators of transfer function matrices of subcontrollers guarantee control quality. The parameters of the resulting family of linear controllers are scheduled as functions of reference variables, resulting in a single feedback matrix controller. The derived gain scheduled controller is implemented and applied on the illustrative nonlinear model of two funnel liquid tanks in series.

Nonlinear Gain Scheduled Controller For A Sphere Liquid Tank.

In this paper, we develop a gain-scheduled controller for a nonlinear plant of the sphere liquid tank. The proposed strategy is based on linearization of the nonlinear state equation around selected operating points. This methodology allows to apply any linear control design method. In particular, we discuss a polynomial method to achieve desired stability and performance requirements. Following this idea, both 1DOF and 2DOF control configurations are considered. The linear design methods are applied at each operating point in order to arrive at a set of linear control laws. Additionally, the parameters of the resulting family of linear controllers are scheduled as functions of reference variable, resulting in a single controller. Nonlocal performance of gain scheduled controller for the nonlinear model is checked by mathematical simulation.

Multimodel Approach In State-Space Predictive Control.

This paper presents a multimodel approach to control nonlinear systems. The system of the inverted pendulum which has one control input and two measured outputs was chosen as an exemplar process. This system is an example of the nonlinear process with a sampling period in order of milliseconds. The state-space predictive control of the system described by CARIMA model was chosen as a control method. This paper presents a description of the inverted pendulum nonlinear mathematical model, its linearization and the control signal calculation using predictor-corrector method. The results compare three methods of linearized model combination. All of the simulations were done in Matlab. INTRODUCTION Process control area offers a variety of different processes with different level of complexity. Even the sampling period can be very different. This paper focuses on nonlinear processes with fast sampling period. The complex and fast processes need a modern control method to control them effectively such as model predictive control (Bobál 2008). This method predicts the output values on the chosen time interval based on the mathematical model of the controlled process. The model of the inverted pendulum is described by the state-space CARIMA mathematical model for the single-input multi-output (SIMO) system (Bars et al. 2011; Wang 2009). The predictive control method uses a minimization of the cost function, which is usually in quadratic form, to calculate the control signal. The quadratic form of the cost function minimize the differences between the reference value and the output value and the control signal increments. The predictive control method offers a possibility to constrain the process variables then the chosen predictor-corrector minimization method can be used to minimize the cost function (Camacho and Bordons 2004; Maciejowski 2002; Rossiter 2003). However, the chosen state-space predictive control method uses a linear CARIMA mathematical model for prediction of the output values but the chosen process of the inverted pendulum has nonlinear behaviour. That means we have to linearize the nonlinear model first. the nonlinear behaviour of the process can be described with a set of the linear models. The final linear model used to output values prediction is obtained by combination of two or more linear models out of the set of the linearized models according to the current output value (Albertos Peréz and Sala 20014; Hangos et al. 2004). This paper is divided into the following sections. The model of the inverted pendulum is described in the first section. The predictive control and the calculation of the control signal are described next. The final sections shows the results of the research and the conclusion. MATHEMATICAL MODEL OF THE CONTROLLED SYSTEM The controlled system in this paper is represented by Amira PS600 inverted pendulum system which is shown at figure 1. The pendulum rod of this system is attached to the cart which is driven by a servo motor (Amira 2000; Chalupa and Bobál 2008). Figure 1 : Amira PS600 Inverted Pendulum system The servo motor produces the input force (input variable) that move with the cart. Position of the cart is the first measured output variable and the pendulum angle is the second measured output variable.. The figure 2 shows the analysis of the forces acting in the system (Amira 2000; Chalupa and Bobál 2008). Proceedings 32nd European Conference on Modelling and Simulation ©ECMS Lars Nolle, Alexandra Burger, Christoph Tholen, Jens Werner, Jens Wellhausen (Editors) ISBN: 978-0-9932440-6-3/ ISBN: 978-0-9932440-7-0 (CD) Figure 2 : Analysis of the inverted pendulum The angle of the pendulum rod is expressed as φ, the input force produced by DC motor is symbol F, symbol ls means distance between pendulum gravity centre and rotation centre of the pendulum, weight of the cart is expressed as M0, weight of the pendulum is expressed as M1 and g is gravity acceleration constant. The equations (1) and (2) describe the horizontal and the vertical forces that the pendulum causes on cart.

Predictive control of the magnetic levitation model

This paper presents a possible way to control the a very fast nonlinear systems. The system of the magnetic levitation was chosen as an exemplar process. This is an example of the process with a sampling period in order of milliseconds. We chose a predictive control method to control this system. The state-space CARIMA mathematical model is used for prediction of the output values. This paper describes the magnetic levitation model, its linearization, prediction of the output values and a calculation of the control signal by using a predictor-corrector method which turned out to be the best solution out of the selected ones. The results compare several optimization methods to achieve the fastest calculation of the control signal. All of the simulation was done in Matlab.

State-Space Predictive Control Of Inverted Pendulum Model.

This paper presents a possible way to control the a very fast nonlinear systems. The system of the inverted pendulum was chosen as an exemplar process. This is an example of the nonlinear single-input multi-output process with a sampling period in order of milliseconds. The state-space predictive control was chosen as a control method and the system is described by CARIMA model. The whole process of the controller design is described in this paper. That includes a description of the inverted pendulum nonlinear mathematical model and its linearization, the inference of the output values prediction and the control signal calculation. The control signal is calculated by predictor-corrector method. The results compare several optimization methods to achieve the fastest calculation of the control signal. All of the simulation was done in Matlab. INTRODUCTION In real life we can come across with many types of processes. Many of them are nonlinear and their mathematical models are very complex. Even the sampling period can be very different. This paper focuses on the very fast processes with a sampling period in the order of milliseconds. The basic control methods may not handle with this situation with required precision so we need a more advanced method. The predictive control is a great example of the modern control method that can be used to solve the complex control problems (Bobál 2008). This method belongs to the model based control methods and the mathematical model is used for the output values prediction. This prediction is determine on the chosen time horizon that should be long enough to cover the step response of the controlled system. The model of the inverted pendulum is described by the state-space CARIMA mathematical model for the single-input multi-output (SIMO) system (Bars et al. 2011; Wang 2009). The control signal calculated by the predictive control ensures the desired output values in the near future time horizon. This is achieved by minimization of the cost function that usually has a quadratic form and it minimize the differences between the reference value and the output value and the control signal increments. If the process require some kind of the process variable constraints, several method such as quadratic programming method, fast-gradient method, predictorcorrector method etc. can be used to minimize the cost function (Camacho and Bordons 2004; Maciejowski 2002; Rossiter 2003). However, the chosen CARIMA mathematical model used to the prediction of the output values works only for the linear models so the nonlinear mathematical model of the inverted pendulum needs to be linearized. This paper is divided into the following sections. The model of the inverted pendulum is described in the first section. The predictive control and the calculation of the control signal are described next. The final sections shows the results of the research and the conclusion (Albertos Peréz and Sala 20014; Hangos et al. 2004). MATHEMATICAL MODEL OF THE CONTROLLED SYSTEM The Amira PS600 inverted pendulum system was used as the exemplar model. The photo of this system is shown at figure 1. The main parts of the system are cart driven by servo amplifier and the pendulum rod attached to the cart (Amira 2000; Chalupa and Bobál 2008). Figure 1 : Amira PS600 Inverted Pendulum system The inverted pendulum system is an example of the single-input two-output system. The force produced by the DC motor that moves with the cart is the input variable and the cart position and the angle of the pendulum rod are the output variables. The figure 2 shows the analysis of the forces acting in the system (Amira 2000; Chalupa and Bobál 2008). Proceedings 31st European Conference on Modelling and Simulation ©ECMS Zita Zoltay Paprika, Péter Horák, Kata Váradi, Péter Tamás Zwierczyk, Ágnes Vidovics-Dancs, János Péter Rádics (Editors) ISBN: 978-0-9932440-4-9/ ISBN: 978-0-9932440-5-6 (CD) Figure 2 : Analysis of the inverted pendulum The variables in the figure 2 are following. The angle of pendulum rod is φ, M0 and M1 stands for the weight of the cart and pendulum respectively, lS is a distance between centre of gravity of the pendulum and the centre of rotation of the pendulum and g is the gravity acceleration constant. Symbol F represents the force produced by the DC motor. The affect of the pendulum on the cart can be expressed as a horizontal and a vertical forces described by the equations (1) and (2)

Predictive Control Of A Series Of Multiple Liquid Tanks Substituted By A Single Dynamics With Time-Delay.

The article focuses on control of a system consisting of a series of liquid tanks. Accumulation of individual dynamics causes, that the overall system exhibits high order behaviour. Another effect is a summation of slow responses of individual systems on an input signal leading to a significant time gap in reaction time of the whole system. In order to make control operations more straightforward and increase calculation speed, the mathematical description of gathered dynamics was approximated into a simplified form containing timedelay. The resulting form of the system is regulated by a predictive controller with time-delay compensation. The whole process is simulated in the Matlab environment.

1DOF Gain Scheduled PH Control Of CSTR.

Motivated by the rich dynamics of chemical processes, we present a gain scheduled control strategy for a pH neutralization occurring inside continuously stirred tank reactor built on a linearization of a nonlinear state equation about selected operating points. Firstly, we address the problem of a selection of scheduling variable. Based on this, an extra scheduling mechanism is presented to simulate the behavior of a nonlinear process using a linear model. Specifically, the proposed step aims at extending the region of validity of linearization by introducing a parametrized linear model, which enables to construct linear controller at each point. Finally, the parameters of resulting family of linear controllers are scheduled as functions of the reference variable, resulting in a single scheduling controller.

State-Space Predictive Control of Two Liquid Tanks System

This paper presents a process control method called the predictive control used to control a nonlinear process about a selected operating point. The system of the two funnel liquid tanks in series is chosen as an exemplar process. The predictive control is used in its state-space modification for CARIMA mathematical model. This paper describes the linearization process of the nonlinear system at the operating point and a process of a control signal calculation. The designed controller is verified on the process without and with a time-delay.

Multivariable Gain Scheduled Control of Four Tanks System: Single Scheduling Variable Approach

Motivated by the special class of nonlinear systems, we introduce a scheduling technique that aims at extending the region of validity of linearization by designing an extra scheduling mechanism. Specifically, we show, how a simplification of a control problem may results in a considerably difficult scheduling procedure. In particular, the scheduling problem in the context of a nonlinear model of four tanks is addressed. The main innovation consists in the use of auxiliary scheduling variables dealing with the problem of limited number of output variables. This allows to construct a linear feedback controller at each operating point. Additionally, an integral control which ensures desired stability and performance requirements is presented. The resulting method has been integrated into the gain scheduled control design of the four tanks system and has shown a great performance through the operating range.

Nonlinearity and Time-Delay Compensations in State-Space Model Based Predictive Control

In this paper a promising series of modifications of predictive control has been combined in order to extend the functionality of principles of predictive control via linearization. Based on this approach a linear model predictive controller is designed at each point to achieve desired local stability and performance requirements leading to guaranteed functionality through the whole operating range of a nonlinear system. In addition, a compensating technique has been applied in order to deal with the system dynamic burdened with a delayed control input. The improved predictive controller has been implemented and applied on illustrative examples of tank system.