Application of H-infinity Robust Controller on PAC

Transcription

1 Application of H-infinity Robust Controller on PAC S.Ozana*, M.Pies* *VSB-Technical University of Ostrava, FEI Czech Republic (Tel: ; Abstract: The paper deals with the design and implementation of chosen algorithm based on modern control theory into programmable automation controllers (PAC) used for tutorial purposes. Physical model of Ball&Beam was chosen as an example of a real system to be controlled by H-inf robust controller, using a mixed sensitivity configuration scheme. It covers all the phases of design and implementation of robust controller in Matlab&Simulink, while the implementation was carried out in REX Control system with WinPAC controllers as special embedded systems, mainly WinPAC WP-884. Keywords: Matlab, Simulink, REX Control System, PAC, embedded system, robust control. INTRODUCTION. Programmable Automation Controller A programmable automation controller (PAC) is a compact controller that combines the features and capabilities of a PCbased control system with that of a typical programmable logic controller (PLC). A PAC thus provides not only the reliability of a PLC, but also the task flexibility and computing power of a PC. PACs are most often used in industrial settings for process control, data acquisition, remote equipment monitoring, machine vision, and motion control. Additionally, because they function and communicate over popular network interface protocols like TCP/IP, OLE for process control (OPC) and SMTP, PACs are able to transfer data from the machines they control to other machines and components in a networked control system or to application software and databases. A PAC at the core of an automation system can integrate more multiple fieldbus networks like RS-485, RS-232, RS-422, CAN, Ethernet, EtherNet/IP, and others..2 WinPAC-8000 with REX Control System This series, particularly WP-884, runs on Windows CE operating system, as described at It provides OPC communication to visualize trends and store the data. It can be programmed in.net platform or in Simulink environment, but the way of the design is rather different. There's a very elegant way how to program these controllers, using REX Control system as described below. REX is the multiplatform real-time control system compatible with the globally spread MATLAB/Simulink. At present, REX is implemented for MS Windows (2000 or XP recommended), Windows CE.NET and for real-time operating system Phar Lap ETS. The compatibility between REX and Matlab-Simulink is ensured by the large function block library RexLib, which exists for Matlab-Simulink and all target platforms. The control algorithm can be designed directly in Matlab-Simulink (or even simulated) or in a special RexDraw SW (part of REX). Full version of the REX control system for WinPAC-8000 (WinPAC successor) by ICPDAS comes with Windows CE 5.NET operating system. The system is purchased with a software license bound to the particular WinPAC station. The version is suitable for: Control application of medium-rate machines and processes Good price/performance ratio applications where the HMI (human machine interface) software runs on the same station as the control algorithm. REX OPC server (included in the product) is used for the communication between REX and HMI software Both centralized and distributed applications with a wide range of input/output modules Hard real-time" applications with strict requirements on the sampling period stability. Minimum achievable sampling period is 2 msec, typical minimum sampling period is 5-0 msec The main advantages of REX are the following: Matlab/Simulink compatibility. The complete control algorithm can be simulated and tuned before final implementation, further info at OPC support - visualization screens can be done in all common SCADA/HMI systems (Genesis, Labview, Indusoft, Reliance,... ) Java support. The visualization screens or applets embedded into web pages can be written in Java. The client side can be run at all common operating systems and all common web browsers. This is the main advantage compared to Microsoft ActiveX components. The visualization screens can be done also in C#.

2 The complete diagnostic and any changes in control strategy can be done remotely via Internet. More info can be found in sections Technologies/REX Control System and Technologies/Micro RexLib at Fig.. shows physical plant that served for implementation of robust algorithm on programmable automation controller. Fig. 2. Physical model of Ball&Beam 2.2 System Equations Fig.. Physical model of Ball&Beam.3 Algorithms Based on Modern Control Theory The modern control theory is a discipline dealing with formal foundations of the analysis and design of computer control and management systems. Some of the algorithms date back to the 60 s but other arose in the late 80 s and they re still having been developed recently. To demonstrate the capabilities of WinPAC controller, this paper covers the problematic of design and implementation of H-infinity robust control (mixed sensitivity problem) 2. MATHEMATICAL MODEL 2. Problem setup A ball is placed on a beam, where it is allowed to roll with degree of freedom along the length of the beam, see Fig. 2. A lever arm is attached to the beam at one end and a servo gear at the other. As the servo gear turns by an angle theta, the lever changes the angle of the beam by alpha. When the angle is changed from the vertical position, gravity causes the ball to roll along the beam. A controller will be designed for this system so that the ball's position can be manipulated. 2. Parameters, constants and variables m mass of the ball R radius of the ball g gravitational acceleration J ball's moment of inertia L length of the beam r ball position α beam angle coordinate d lever arm offset θ servo gear angle [kg] [m.s-2] [kg.m2] [rad] [rad] The Lagrangian equation of motion for the ball is given by the following: 0 Linearization of this equation about the beam angle, alpha = 0, supposing sin(α) α for small angles, gives us the following linear approximation of the system: 2.2 Transfer Function Rearranging we find the transfer function from the beam angle to the ball position: -mg m+ J R State-space Description The linearized system equations can also be represented in state-space form. This can be done by selecting the ball's position r and velocity as the state variables and the beam angle α as the input. Introducing the following terms leads to state-space representation. ; x 2t= -m g α m+ J R ; 0 Fig. 3. shows a general feedback control scheme for Ball&Beam plant. It has to be modified for application of H- infinity robust control as described in next chapter. Fig. 3. General Feedback Control Scheme

3 3. H-INFINITY ROBUST CONTROL 2.2 H-infinity Robust Control Design Basic scheme with the plant P(s) and a robust controller K(s) is shown on Fig. 4. and on Fig. 5 in detail. The main function of this circuit is to handle with low-frequency disturbance d and high-frequency noise n. The resulting controller doesn t automatically meet some common conditions required on a regulation process, such as overshoot. To follow further requirements, the scheme must be extended that leads to defining so called mixed sensitivity problem describe in next chapter, or at [Dorf,Bishop] and [Zhou,Doyle]. Fig.4. Basic H-infinity scheme Fig.5. H-infinity scheme in detail Typical inputs covered by vector w are external disturbance, sensor noise and tracking signal. The outputs are divided into two groups: a) vector y measured output for a feedback, also input to the controller b) vector z regulated output that doesn t go to the controller. Mathematical description with so called packed matrix is given as follows: 2.2 Mixed Sensitivity Problem with Weighting Functions Weights (weighting functions) make it possible to define requirements on a robust controller. Before defining it is necessary to determine their initial values. Weights are defined by through transfer functions that arise from physical conditions or guesses. Robust controller is then being found for such designed weights. If the regulation process doesn t meet the requirements, it is necessary to keep modifying the weights until the result is satisfactory. W u ~ u r e K u _ W i ~ d i ~ d Fig. 6. Mixed Sensitivity Scheme for Robust Control Design Fig. 6 shows a typical scheme with a robust controller for mixed sensitivity problem. The noise d i is brought to the input of the plant G and noise d to its output. Requirements on a regulation process and its quality are evaluated according quantity of actuating value u and control error e. These requirements can be described by weights W i, W d, W e, W u. Input weights are designed according given amplitudes of disturbances and noises, for example: W i = 0,00 W d = 0,0 This means that a disturbance input d i accepts a white noise with amplitude of 0,00 and noise input d receives a white noise with amplitude of 0,0. If the noise is different from white noise, it could be described by a first order system and thus its color would be determined then. The sense of weights become clear from the scheme above, if we assume dimensionless variable (ω) on the inputs d i and d which are transformed by the weights to the physical units of the plant (for example voltage or angle). For output weights the situation is similar. Control error e and actuating value u have generally different units and it is necessary to define priorities of their minimization. On the outputs e ~, u ~ we also assume dimensionless variable (ω). Then weight /W e represents required behavior (limitation) of control error and weight /W u required behavior (limitation) of actuating value. Fig. 7 and Fig. 8 show typical courses of /W e and /W u. Parameters M s and M u determine maximal values that are expected on control error e and actuating value u. The first one affects limitation of overshoot of a regulation process, the latter expresses limitation of actuating value (for example torque moment of a servomotor). Parameter ω b makes it possible to define frequency on which the steady state of regulation process is expected, its reciprocal value means the time constant. Parameter ω bc makes it possible to define frequency limitation of actuating value (for example limitation of a servomotor on higher frequencies due to inertia). For determination of its value it is necessary to consider the fact that it depends on M u. It leads to a conclusion that time constant of limitation of actuating value is approximately M u /ω bc. Parameters ε and ε are realization constants and they should be chosen several times smaller than M s resp. M u. d i G W d d W e ~ e

4 For chosen parameters it is possible to evaluate transfer functions of the weights W e a W u : W W e u /W e s = M s + ω s + ω ε b b s + ωbc M = ε s + ω bc u M s ε Fig. 7. Weighting function /W e /W u M u ε Fig. 8. Weighting function /W u For designed weights it is possible to calculate acceptable controller that minimizes criteria given by the weights. Resulting frequency characteristics are shown on Fig. 9. Transfer function S(ω) represents transfer function of control error in closed loop. Transfer T(ω) represents transfer function of desired value r in closed loop. Functions S(ω) and T(ω) are complementary, thus S(ω)+T(ω)=. Their transfer functions affect the speed of regulation process. Characteristics K(ω),S(ω) determine frequency behavior of a controller and actuating value. An appropriate design should meet the condition S(ω) <=/ W e (ω) and K(ω),S(ω) <=/ W u (ω). A found robust controller might not satisfy this condition at all frequencies which is also obvious from Fig. 9. This is so because the design is a trade-off solution. If the control process has to be accelerated or an overshoot has to be reduced, the bigger actuating value should be allowed. Step-by-step experiments with the parameters of weighting functions should lead to optimal and satisfactory result. In Matlab&Simulink environment, the mixed sensitivity problem can be easily programmed, using sysic command: ω b ω bc ω ω Fig. 9. Frequency characteristics of control circuit and design of weigts W e and W u clear;close all s = tf('s'); G = ss(7.4/(s^2+e-3*s+e-6)); Ms =.5; wb = 2.5e-; eps = e-3; We = (s/ms+wb)/(s+wb*eps); Mu = 0.0; wbc = 0.; eps = Mu/00; Wu = (s+wbc/mu)/(eps*s+wbc); Wd = ss(0.0); Wi = ss(0.00); systemnames = 'G We Wu Wi Wd'; inputvar = '[di; d; u]'; outputvar = '[We; Wu; -G-Wd]'; input_to_we = '[G+Wd]'; input_to_wu = '[u]'; input_to_wi = '[di]'; input_to_wd = '[d]'; input_to_g = '[u+wi]'; cleanupsysic = 'yes'; P = sysic; nmeas=; nctrl=; gmin=0.0; gmax=000; tol=0.0; [K,g,gfin] = hinfsyn(ltisys(p.a,p.b,p.c,p.d),nmeas,nct rl,gmin,gmax,tol); [KA, KB, KC, KD] = ltiss(k); G = ss(7.4/(s^2)); K = tf(ss(ka,kb,kc,kd)); The crucial part of mixed sensitivity problem design is to create a structure according Fig. 6. The design documented in the lines of code above leads to controller K described by its transfer function as follows: >> K Transfer function: s^ s^2+0.43s s^4+42.3s^ s^ s+0.243

5 Before implementation into the controller, function of robust controller can be easily simulated in Simulink, using standard blocks with respect to nonlinearities of the model physical limits for beam angle and for ball position, see Fig.0. Fig. 0. Simulation of designed robust controller in Simulink Fig.. then shows simulation of step change of desired ball position from initial value 0,5 to 0,. The transfer function of the controller is then implemented in the REX Control System environment, see Fig. 2. Details of the content of block representing the controller can be seen on Fig.3. The representation of real plant in REX is shown on Fig. 4., it covers two I/O blocks that express connection analogue signals from the position sensor and for controlling the servo. Before uploading the scheme into the real WinPAC controller, it can be simulated in REX, to be sure of appropriate functionality, see Fig.5., compare to Fig.. Fig. 3. Screenshot of REXView simulation of robust controller in REX Fig. 4. Screenshot of REXView control of real plant Fig.. Simulated result Fig. 2. Screenshot of REXView simulation of robust controller in REX Finally, the scheme is uploaded into WinPAC controller, its fiction is documented on Fig.6. in REXView utility. Fig. 5. Screenshot of REXView result of simulation of robust controller in REX Among other capabilities, WinPAC controller provides OPC communication, thus it is possible to visualize the task in HMI/SCADA system, in this case PROMOTIC system was chosen for demonstration, see Fig. 7. and as it is described at the producer webpage

6 models that can be handled and visualized in many ways. As a typical model to be described in detail, a model of Ball&Beam has been chosen. Fig. 6. Screenshot of REXView result of controlling a real plant Programmable automation controllers make up particular group of embedded systems that provide a huge computational performance and reliability with possibility of use modern communication standards, see Fig. 8. Equation of forces equlibrium in mathematical model doesn t contain static and rolling friction. It leads to the simple design of robust controller, but it also causes different behavior of real plant compared to simulation, as it can be seen from comparison between Fig. 5. And Fig. 6. Simulation shows an overshoot unlike controlling real plant, moreover the ball might not reach the desired position due to static friction up to expectation: control error is non-zero, actuating value rises and then the ball breaks-off, causing other unwanted overshoot. Other future plan is to include uncertainty to the model. ACKNOWLEDGMENT The work and the contribution were supported by the project Grant Agency of Czech Republic GAČR 02/08/429 Safety and security of networked embedded system applications. Also supported by the Ministry of Education of the Czech Republic under Project M0567. Fig. 7. Visualization of the control process in PROMOTIC system REFERENCES Richard C. Dorf,Robert H. Bishop.(2007). Modern Control Systems. Kemin Zhou, John C. Doyle.(997). Essentials of Robust Control. Fig. 8. Programmable automation controller WP CONCLUSIONS This paper introduces new approaches and methods for implementation of real-time embedded systems at the Department of Measurement and Control at VSB-TU Ostrava with use of Matlab&Simulink and REX environment. They are applied for a number of mechatronical educational