CL Digital Control

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1 CL Digital Control Kannan M. Moudgalya Department of Chemical Engineering Associate Faculty Member, Systems and Control IIT Bombay Autumn 26 CL 692 Digital Control, IIT Bombay 1 c Kannan M. Moudgalya, Autumn Topics to be covered Modelling Signal Processing Identification Transfer function approach to control design State space approach to control design You will think digital at the end of the course CL 692 Digital Control, IIT Bombay 2 c Kannan M. Moudgalya, Autumn 26

2 2. Objectives of a Control Scheme To stabilize unstable plants To improve the performance of plants (rise time, overshoot, settling time) To remove the effect of disturbance (load) and noise If a good model is available, use feed forward control scheme If a good model is not available, use feedback control scheme Often, we don t have good models feedback control schemes are preferred CL 692 Digital Control, IIT Bombay 3 c Kannan M. Moudgalya, Autumn Schematic of Feedback Control v r e G c u G y G: denotes plant or process G c : denotes digital controller r: reference variable, setpoint v: disturbance variable y: plant output or controlled variable u: plant input or manipulated variable or control effort CL 692 Digital Control, IIT Bombay 4 c Kannan M. Moudgalya, Autumn 26

3 4. Digital Signals Digital systems deal with digital signals Digital signals are quantized in value discrete in time As or 1 refers to a range of voltages, digital signals can be made less noisy Can implement error checking protocols So digital devices became popular - impetus for advancement of digital systems Digital devices have become rugged, compact, flexible and inexpensive Modern controllers are based on digital systems CL 692 Digital Control, IIT Bombay 5 c Kannan M. Moudgalya, Autumn Noise Margin - Impetus for Growth of Digital Devices source sink If transmitted signal is received exactly, no noise Analog circuitry always has noise Digital devices have good noise margins Example: TTL Can pick up noise, voltage can increase by.4v, still considered low signal Note: if voltage decreases because of noise, no probem High: =.8.4 =.4V Low Voltage () At output At input At output: At input: Considered low Considered low if voltage <.4V if voltage <.8V Considered high Considered high if voltage > 2.4V if voltage > 2V 1 = =.4V CL 692 Digital Control, IIT Bombay 6 c Kannan M. Moudgalya, Autumn 26

4 6. Advantages of Digital Control All modern controllers are digital - even if they appear to be analog Can easily implement complicated algorithms Flexible, can easily change algorithms Low prices - can achieve complicated algorithms without much expenditure Analysis of difference equations is easier than differential equations - so easier to design digital controllers How do we connect digital controllers with real life objects, which could be analog? CL 692 Digital Control, IIT Bombay 7 c Kannan M. Moudgalya, Autumn Analog to Digital Conversion Actual & Quantised Data data time A/D Converter quantized data Quantized Sampled Data time Analog to Digital (A/D) converter produces digital signals from analog signals Higher signal frequency requires faster sampling rate But uniform sampling rate is used in an A/D converter Quantization errors Finiteness of bits - quantization errors Increase number of bits to reduce errors Falling hardware prices help achieve this Sampling rate Slow rate loss of information Fast rate computational load Analog s output is sent to digital through A/D. Reverse? CL 692 Digital Control, IIT Bombay 8 c Kannan M. Moudgalya, Autumn 26

5 8. Digital to Analog Conversion Sampled signal Discrete ZOH Signals Discrete Signals ZOH is the most popular Real life systems are analog Cannot work with binary numbers Binary D/A analog We will consider only ZOH in this course Assumption used in this course: Need to know values at all times The easiest way to handle this to use Zero Order Hold (ZOH) Complicated hold devices possible. All inputs are ZOH signals OK when the input is produced by a digital device Also OK when the input signal varies slowly CL 692 Digital Control, IIT Bombay 9 c Kannan M. Moudgalya, Autumn Magnetically Suspended Ball R L V + i M h Current through coil induces magnetic force Magnetic force balances gravity Ball is suspended in midair - 1 cm from core g Force balance: M d2 h Ki2 = Mg 2 h Voltage balance V = L di + ir Want to move to another equilibrium CL 692 Digital Control, IIT Bombay 1 c Kannan M. Moudgalya, Autumn 26

6 1. Magnetically Suspended Ball R L + V i h M Linearize RHS: i 2 h = i2 s + 2 i h (is, ) i i2 h (is, ) h 2 Model equations: M d2 h Ki2 = Mg 2 h V = L di + ir In deviation variables: = M d2 2 M d2 h 2 g = Mg Ki2 s [ ] i 2 = K h i2 s = i2 s + 2 i s i i2 s h h 2 s Substitute and simplify M d2 h = K 2 d 2 h = K 2 M [ i 2 s + 2 i s i i2 s h 2 s i 2 s 2 hs h K i s i. 2 M Voltage balance in deviation: V = L d i + R i ] h i2 s CL 692 Digital Control, IIT Bombay 11 c Kannan M. Moudgalya, Autumn Magnetically Suspended Ball - Continued 3 Force balance: d 2 h 2 = K M Voltage balance: i 2 s 2 hs h K i s i 2 M V = L d i + R i Define new variables = h = ḣ = i u = V d = d = K M i 2 s h 2 s 2 K M d = R L + 1 L u In matrix form: d = 1 K i 2 s 2 K M h 2 s M R L This is of the form i s i s ẋ(t) = F x(t) + Gu(t) + u 1 L CL 692 Digital Control, IIT Bombay 12 c Kannan M. Moudgalya, Autumn 26

7 12. Magnetically Suspended Ball - Continued 4 d = 1 K i 2 s 2 K M h 2 s M R L i s + u M Mass of ball.5 Kg L Inductance.1 H R Resistance 1 Ω K Coefficient.1 g Acceleration due to gravity 9.81 m/s 2 Equilibrium Distance.1 m i s Current at equilibrium 7A 1 L d = u CL 692 Digital Control, IIT Bombay 13 c Kannan M. Moudgalya, Autumn Model of a Flow System A dh(t) = Q i (t) k h(t) Initially at steady state = A d = Q is k Linearise with Taylor s series approximation: d h(t) Initial condition: = k 2A h(t) + 1 A Q i(t) at t =, h(t) = or h(t) =. It can be written as, ẋ = F x + Gu Known as state space equation. CL 692 Digital Control, IIT Bombay 14 c Kannan M. Moudgalya, Autumn 26

8 14. IBM Lotus Domino Server Clients access the database of s maintained by the server through Remote Procedure Calls (RPCs). Number of RPCs, denoted as RIS has to be controlled. If the number of RIS becomes large, the server could be overloaded, with a consequent degradation of performance. If RIS is less, the server is not being used optimally. Not possible to regulate RIS directly. Regulation of RIS may be achieved by limiting the maximum number of users (MaxUsers) who can simultaneously use the system. Because of stochastic nature, difficult to come up with analytic model. Obtained through expt., data collection, curve fitting (identification). y(k) = RIS(k) RIS u(k) = MaxUsers(k) MaxUsers y(k + 1) =.43y(k) +.47u(k) CL 692 Digital Control, IIT Bombay 15 c Kannan M. Moudgalya, Autumn Motivation for Discrete Model Real systems are continuous Digital controller s view of the process: Receives sampled signals. Sends out sampled signals Thus views the process as a sampled system Hence, to determine the control effort, a discrete model of the plant is required Discrete model relates the system variables as a function of their values at previous time instants No value required/used in between sampling instants. Time derivatives have no meaning We will next explain how to arrive at discrete models CL 692 Digital Control, IIT Bombay 16 c Kannan M. Moudgalya, Autumn 26

If every Bounded Input produces Bounded Output, system is externally stable equivalently, system is BIBO stable

1. External (BIBO) Stability of LTI Systems If every Bounded Input produces Bounded Output, system is externally stable equivalently, system is BIBO stable g(n) < BIBO Stability Don t care about what unbounded