9/17/2015. Contents. ELEC-E8101 Digital and Optimal Control (5 cr), autumn 2015

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1 ELEC-E8101 Digital and Optimal Control (5 cr), autumn 2015 Lectures Fridays at , room AS2 Lecturer: Kai Zenger, TuAS-house, room 3567, kai.zenger(at)aalto.fi Exercise hours Wednesdays at room Tu 1 AND Thursdays at room AS3 (problem solving help, questions, doing exercises with friends., informal, Laskutupa ) Contents Introduction, discrete-time control problem, the z-transform Discrete-time systems, properties and analysis Controller design: pole placement by state space methods PID-controller and its discretized versions, the sampling theorem, choosing the sampling interval Discrete approximations of continuous-time controllers Stochastic disturbance models Optimal prediction and minimum variance control Dynamic programming Linear quadratic controllers (LQ, LQG); discrete-time and continuous-time cases Assistants: Juho Lindholm, juholind(at)mappi.helsinki.fi Jyrki Parkkinen, jyrki.parkkinen(at)aalto.fi Passing the course: Final exam or two intermediate exams. Final exam: maximum 30 points (5 problems, max 6p. each). Alternatively two intermediate (Mid term) exams. Each has three problems, max. 5p./problem. Maximum of the two intermediate exams 30 p. Bonus points of (not mandatory) homeworks. 6 homework problems, altogether max 6 p. The bonus points are added to determine the final grade. The bonus points are true bonus, because the grade limits are not scaled up because of the possibility to get bonus points. To pass the course 15 points is always enough. Otherwise the grade limits can vary a bit. Intermediate exams are suggested to pass the course. They are organised during the last week of each lecture period (2 hours). During the second exam time you can choose, after seeing the problems, whether you want to do intermediate exam or full exam. The full exams later are three hours in lenght; otherwise they are similar to the full exam during the lecture period. In the full exam you can (re)do intermediate exam 1 or 2. Bonus points and intermediate exam results are valid one year, until the course lectures start again (autumn 2016). Exams: Fri at 12:00-14:00 AS2 (intermediate exam 1) Wed at 14:00-16:00 AS2 (intermediate 2, full exam) Mon at 16:30-19:30 AS2 (full exam) Mon at 16:30-19:30 (full exam) Department of Automation and Systems Technology 1

2 Please register to the course (oodi), if you have not done that already. You do not have to register to intermediate exams. You have to register to full exams, starting from the exam Important: Course substitutions: - if you have done the old courses AS Digital control AND AS Model-Based Control Systems, you do not have to do ELEC-E8101 Digital and Optimal Control (it will be substituted by those two courses) - if you have done only AS Digital control or only AS Model- Based Control Systems, you have to do ELEC-E8101 Digital and Optimal Control in a normal manner. The first part is however very similar to Digital Control course. If you know Digital control well, you can relax in the first period; it is suggested however that you do the homework problems to get bonus points. ELEC-E8101 Introduction Study material Lecture slides (in English, ordinary and print versions) in MyCourses (mycourses.aalto.fi) Exercises with solutions (English) in MyCourses Web learning environment (in Finnish): /index.html Note: The above old web course material is a good introduction, but does not cover the whole ELEC-E8101. It also contains some material (e.g. polynomial controllers), which are not discussed in ELEC-E8101. The lectures, lecture slides and exercises with solutions cover the whole course. It is not absolutely necessary to purchase the book. In each lecture, topics to read for the following session are given. Note: not all slides are presented during the lectures; some are left for home study. Homework assignments (six altogether) are given regularly. They appear in MyCourses Department of Automation and Systems Technology ELEC-E8101 Introduction Books for support: Åström K. J., Wittenmark B.: Computer Controlled Systems - Theory and Design (3rd ed.), Prentice-Hall, (textbook) Analog (continuous time) control system Process controlled by an analog controller Franklin, Powell, Workman: Digital Control of Dynamic Systems. Third edition, Addison Wesley, Ogata: Discrete-Time Control Systems, 2nd ed., Prentice-Hall, 1994 Kuo, B. C. : Digital Control Systems (2nd ed.), Oxford University Press, Santina, M. S., Stubberud, A. R., Hostetter, G. H.: Digital Control System Design, Saunders College Publishing,

3 Digital (discrete, discrete-time) control system To think about... r(tk) + _ e(tk) Controller y(tk) u(tk) D/A A/D u(t) Process y(t) y(t) The system contains both analog and discrete signals (a hybrid system) Basic problem already in analog control: how do you treat these kinds of systems analytically? Are the traditional time domain and frequency domain methods available now? Can they be modified? How do you design digital controllers? What should be taken into account in implementation? In A/D-block the analog signal is sampled ; in D/A-block the discretetime signal is changed into an analog one (hold). To think about... Is it so that a digital controller only imitates the corresponding analog controller and the result is somewhat worse then (due to loosing information in discretization)? Do discrete-time systems have properties that the corresponding analog systems do not have? Yes, but this is not so simple: sometimes a discrete controller can perform better than the analog one; on the other hand dicrete-time systems have anomalies that do not have a correspondence in the analog world. An example of controller design (only an example:no need to learn very carefully) Controller design of a switched-mode power supply (Buck-type; output voltage smaller than input) The switch operates at high frequency, e.g. 100 khz. By changing the on/off time ratio of the switch (duty cycle) power is transferred into load and the output voltage level is controlled. Department of Automation and Systems Technology 3

By connecting AC/DC power supplies in parallel, and providing a battery for back-up a suitable power system for a Telecom load (for example) is obtained.

4 By connecting AC/DC power supplies in parallel, and providing a battery for back-up a suitable power system for a Telecom load (for example) is obtained. Modern industrial power systems are complex and difficult; System blocks are connected in series and in parallel. They are provided with input filters (EMI), loads are changing and complex. A difficult multivariable system, MIMO = multiple input, multiple output. C.f. SISO = single input, single output Systematic methods are needed = theory is needed! Today s magic buzzword is digitalization! Negative feedback = feedback control, RC-circuit is the load. The output voltage is compared to reference, and a discrete-time PID-controller is used to control the switch. Systems can be described by block diagrams PWM (pulse width modulation) transforms the controller output signal to the duty cycle of the switch; basically that belongs to the operation of the actuator Input voltage. V g and load current i load are disturbances from control viewpoint. Robustness means the ability of the controller to tolerate disturbances and modeling errors. 4

5 By writing the dynamic equations of the system blocks the performance can be studied. A lead/lag compensator has been designed for the process by using the Bode diagram as a design tool. The insufficient phase margin (2 degrees) of the open loop has been improved to (50 degrees). Stability must always be guaranteed. G ( s) G c s l 1 1 z s s 1 Gcm 33.8, z ( rad/ s) 1257( rad/ s), ( rad/ s) l cm p p But the compensator can be approximated by a PID-controller: The control algorithm can be discretized directly for computer based control (details are presented later in the course) The operating frequency of the measurement and control are now new concepts that must be considered. Here the sampling frequency is 20 khz meaning that the sampling time is 1/20000 s. 5

The performance of the controller can be simulated. Below the output voltage and inductor current are shown. The zoomed figures show clearly the operation of the switch.

6 The performance of the controller can be simulated. Below the output voltage and inductor current are shown. The zoomed figures show clearly the operation of the switch. The performance can be made faster by changing the tuning of the PID controller. That leads to increasing oscillations however. Controller design procedure Construction of the process model from physical equations or by identification. The model analysis and linearisation when needed; construction of the transfer functions Formulation of the control problem Controller design in time or frequency domain Discretization and implementation of the controller e.g. by a digital signal processor (DSP) Alternatively: discretization of the process model and controller design directly in discrete time directly Simulation To be used: Matlab/Simulink with different toolboxes... Control System Toolbox Identification Toolbox Optimization Toolbox Model Predictive Toolbox Fuzzy Toolbox Neural Networks Toolbox Symbolic Toolbox Statistical Toolbox Comsol Multiphysics 6

7 Process knowledge Measurement technology Computer technology Control Theory Challenges History of discrete-time systems 2. world war, radars, computers, numerical computation Pioneering period ~ (1961) (1962) applications (supervisory-mode, where setpoint values etc. are calculated to help the operator ) DDC ~ 1962 (Direct Digital Control = computer controls the process directly) in 1975 the first digital automation system (Honeywell TDC 2000) Programmable logic controllers (PLC) from the 70 s. Efficient signal procesors (DSP) from 90 s, embedded systems, mechatronics Wireless automation, networks, energy technology, digitalization = current trends New era: today and tomorrow belong to the digital age! Process controlled by an analog controller Process controlled by a digital controller r(t k) + _ e(tk) Controller u(t k) D/A u(t) Process y(t) A/D y(t k) y(t) 7

8 Two main design approaches: a. discretize the analog controller, b. discretize the process and do the design totally in discrete time a. Example (a): How to design a digital controller based on an analog controller Servoproblem, control of robot arm Process (double integrator): Simple servo controller: b. Department of Automation and Systems Technology Example: How to design a digital controller based on an analog controller A digital controller is derived from the analog one by replacing the differential with difference Example: How to design a digital controller based on an analog controller By using the definition of the derivative Multiplying by the Laplace-variable s corresponds to derivation in time domain (derivation operator p) Now t = kh and t = h. Let us further use the shift operator q (multiplying by q corresponds to moving forward in time; division by q corresponds to moving backwards in time) 8

Example: How to design a digital controller based on an analog controller The following approximative relationship between Laplace and Z domains is obtained Example: How to design a digital

9 Example: How to design a digital controller based on an analog controller The following approximative relationship between Laplace and Z domains is obtained Example: How to design a digital controller based on an analog controller The imitating controller becomes From the transfer function in Laplace domain an approximative pulse transfer function in Z-domain is obtained by replacing the Laplace-variable s with Example: How to design a digital controller based on an analog controller Let us include also the deadbeat-controller, which is based on digital control theory and simulate the operation of the robot arm by each controller. (Dead-beat-controller will be discussed during the course). Example: How to design a digital controller based on an analog controller Analog controller compared to digital imitating controller in different sampling frequencies 9

Example: How to design a digital controller based on an analog controller Analog controller compared to the deadbeatcontroller in different sampling frequencies Example (b): Synchronization

10 Example: How to design a digital controller based on an analog controller Analog controller compared to the deadbeatcontroller in different sampling frequencies Example (b): Synchronization Continuous process (1. order low pass filter) The corresponding discrete-time filter by using the sampling time h = 1 Details on how to obtain H(z) are shown later Department of Automation and Systems Technology Example: Synchronization Let us simulate both filters with the same input signal Example: Synchronization 10

11 Example: Synchronization With impulse and pulse inputs synchronization is even more important z-transformation A sequence for short The z-transformation is defined or Continuous systems: (differential equations/laplacetransformation) Discrete systems: (difference equations/z-transformation) Example. Discrete time impulse The sequence Example. Discrete (unit) step function Sequence In the terms is an impulse (pulse) with the strength f0 where Z-transformation: k is a time index (0,1,2,...). The absolute time at time instant k is kt, where T is the sampling interval Z-transformation: (sum of a geometric series) 11

12 Example. Pulse sequence Movement in time ( right =delay, left =prediction) Pulse sequence f(k): The sum converges in the complex plane region or It is not necessary to consider the convergence regions in what follows. Delay with one sampling interval: Prediction by one sample interval: One sample delay thus means multiplication by ; correspondingly by when the delay is n steps Compare to the derivative of Laplace-transformation in the continuous time case. 12

13 The final value theorem Consider a pulse sequence f(k) and its z-transform F(z). If f(k) approaches a limit value as k approaches infinity, it holds This is the final value theorem. Compare to the continuous time case Discrete-time system Note that the difference equation Example. A first order difference equation means in absolute time Initial conditions e.g. y(0)=0, u(0)=1, u(k)=0, k >0 Calculation directly from the equation (which is directly an algorithm) Result is the pulse sequence Take the Z-transformation Initial conditions are considered to be zero when deriving the pulse transfer function, leading to 13

14 The pulse transfer function z-transformation is used to make the analysis and calculations of discrete-time systems more tractable. Compare again to the continuous time case. The results are meaningful only at sampling instants With a unit impulse as input the output (pulse response) is the inverse transformation of the pulse transfer function. Compare to the continuous time case. Note the following transformation pairs (the latter deviates somewhat from the continuous time analogue) 1. ; 2. ; In the inverse transformation similar ideas as in the continuous time case are applied (dividing terms into sums of smaller entities and then using tables) Example. No inverse transformation is found from the tables; so let us modify The inverse transformation of the bracketed term is Taking the delay into account gives finally or 14

15 Stability We can operate with pulse transfer functions as with transfer functions. E.g. the closed loop pulse transfer function can be calculated, the denominator of which is the characteristic polynomial. Its zeros are the system poles. The zeros of the numerator are the system zeros. The pulse transfer function can be divided into a sum of terms like ;the inverse transformation stays bounded when If even one term explodes, the system is unstable. Stability region: A discrete-time system is stable, when the poles (b, c, d) are located inside the unit disc. Compare to the left half plane for continuous time systems. Shift-operators A forward shift operator q (in time) Example. A difference equation (input-output representation of a system ) A backward shift operator Correspondingly (in time) can be written in which (Compare to the continuous time differential operator p) are operator polynomials. A similar equation can also be written as a function of 15

16 Note the formal relationship continuous time discrete time From state-space-representation to pulse transfer function A discrete state-space-representation is defined in an analog manner to the continuous time case; derivatives are replaced with a shift in time. However, when using p or q, we operate in time domain. When using s or z a transformation to Laplace or Z-domain has been done to deal with transfer functions. This can sound theoretical. However, in control engineering there are many design methods, in which either time domain or frequency domain (transfer functions) are used. Let us try to find the pulse transfer function. Take the Z-transformation and eliminate x. The derivation is quite analogous to the continuous time case. From input-output representation to state-space equations A similar method as in the continuous time case is available Example. The pulse transfer function is obtained as compare to the continuous time case: -state-space? -pulse transfer function? Write the equation as a function of the q-operator 16

17 Move q terms to the left side The following state representation is obtained Develop the left side as factors according to q and choose the state variables as follows and in matrix form Pulse transfer function Is there a need for discrete-time control theory? Zeros: Poles: The poles are inside the unit circle, and the system is therefore stable. Analog systems can be imitated There exists discrete-time systems without an analog correspondence Sampling can cause problems to discrete-time systems Digital controller can even beat the analog controller (fastness) Discrete-time control theory is needed! 17

18 Grounds and basic building blocks Sampling theorem (Shannon -49) Difference equations (-48) Integral transformations Z-transformation (Hurewicz -47) State-space-representation (Kalman -55) Grounds and basic building blocks Optimal control (Bellman -57) Stochastic control, LQG (Kalman -65) Algebral system theory (Rosenbrock -70) Identification, Adaptive control Robust control (-80) Soft computing, -80, fuzzy control, neural networks, cloud computing, computational sciences (-2000)... 18

CL Digital Control Kannan M. Moudgalya

CL Digital Control Kannan M. Moudgalya CL 692 - Digital Control Kannan M. Moudgalya Department of Chemical Engineering Associate Faculty Member, Systems and Control IIT Bombay kannan@iitb.ac.in Autumn 2007 Digital Control 1 Kannan M. Moudgalya,