# Learn2Control Laboratory

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1 Learn2Control Laboratory Version 3.2 Summer Term

2 This Script is for use in the scope of the Process Control lab. It is in no way claimed to be in any scientific way complete or unique. Errors should be reported to: or 2

3 1 BASICS OF SYSTEM DYNAMICS AND CONTROL 4 Block diagrams and the concept of feedback control 4 Dynamic response of transfer elements (dynamic operators) 4 Simple control loop The proportional element (P-element) or pure gain The integrator (I-element) The first-order time delay (PT1-element) The dead time element 8 Transfer behavior of closed control loops and block diagram algebra 9 Linear controller elements 11 Behavior of the closed control loop 13 2 LINEAR STATE FEEDBAC CONTROL (CONCERNING DYN 23) State Feedback Control by Pole Assignment State Feedback Control by Ackermann s Formula Eigenvalue and Eigenvector Assignment by State Feedback Control Linear Quadratic Regulator (LQR) Choice of the Filter Matrix 20 3 PROJECTS DYN23 - Control of a chemical reactor The chemical process DYN26 - Modeling and Analysis of a Reactor System Description of the reactor system User interface 24 4 PREPARATION 25 General project setup 25 Experimental modeling 25 Theoretical modeling 25 Controller design 25 3

4 1 Basics of system dynamics and control Block diagrams and the concept of feedback control A block diagram is a sequence of causes and effects. Each block contains a description of a cause/effect relationship between its inputs (cause) and the resulting output (effect). A typical control system consists of a physical dynamic system (process) and a controller. The process has an output (controlled variable y) that is required to follow some desired value over time (set point w) and has a control input that can be set externally (manipulated variable u). The output deviation (control error e) is calculated by subtracting the measured value of the controlled variable y from the desired value of w. Based on this deviation the controller calculates the manipulated variable u, such that y is corrected. The objective of the feedback controller is to compensate the disturbance z and to follow the trajectory of the desired variable w(t) as fast and exact as possible. Figure 1 shows the block diagram of the unity feedback system described above. Controller Process Figure 1: Block diagram of the unity feedback system Dynamic response of transfer elements (dynamic operators) In general the dynamic response of a transfer element with input u and output y can be described by: y( t) ( u( )); t (1) In this lab, we consider linear time-invariant (LTI) transfer functions only. By Laplace transformation of (1) in the specific form t y( t) u( ) g( t ) d, (2) 0 we obtain the following transfer function: Y( s) G( s) (2) U ( s) Figure 2: Block diagram and transfer function of a general transfer element The so called transfer element or dynamic operator can be described by its response to a test input signal. Since it is simple to realize, the unit step 1(t) is commonly used as the input for the response experiment. 4

5 1( t t 0 0 t t ) 1 t t 0 0 Figure 3: Unit step 1(t) The corresponding output signal h(t) is called (unit) step response. For simple transfer elements the shape of the response can be outlined in the respective block of a block diagram element. Thus, we achieve a simple and descriptive symbol of the transfer element. The frequency response G(j) can be described as the system s response to a sinusoidal input signal. For a unit amplitude sinusoidal input, the output will be a sinusoid with a magnitude A and a phase shift. From the complex transfer function G(j) of the real variable (frequency of the sinusoid) we can describe the frequency response analytically. The magnitude is obtained by and the phase shift by ReG ( j) 2 ImG ( j ) 2 G( j) Im G( j) ( j) arctan. Re G( j) The Bode plot constitutes a convenient way to represent the frequency response characteristics. The magnitude A is plotted in decibel units (linear scale) versus the logarithm of the frequency, while the phase shift is plotted in degrees (linear scale) versus log(). The decibel unit (db) is defined as (3) (4) G( j) 20 log G( j) (5) db so that the decibel scale converts a logarithmic scale for the magnitude into a linear scale. Simple control loop As shown in Figure 1, a control loop consists of two elements; controller and process. Each can consist of a number of simple transfer elements. These simple transfer elements are also called the canonical form of the transfer function. In the following text we describe some relevant simple elements The proportional element (P-element) or pure gain The input-output relation of the proportional element, also called gain is described by y(t) = u(t). (6) By Laplace-transformation we obtain the following transfer function: G s Y U s s The response from a unit step input signal is (7) h(t) = 1(t) (8) 5

6 Thus, the unit step response of a proportional element is a step with the amplitude of. The resulting block diagram symbol is shown below: Figure 4: Block diagram of a proportional element The magnitude and phase shift of the frequency response of the P-element are constant over all positive frequencies (G(j) =, (j) = 0). The magnitude Bode plot is a horizontal line parallel to the frequency axis with a distance of db, the phase Bode plot is the 0 -degree line The integrator (I-element) The input/output relation of the integrator is described by Laplace transformation results into The respective unit step response is y t t ut dt y0. (9) I 0 G s I. s (10) h(t) = I t. (11) Thus, the unit step response of an integrator is a ramp. The integration coefficient I can be read from the step response as shown in Figure 5. In addition the corresponding block diagram of the I-element is displayed. Figure 5: Step response (left) and block diagram (right) of an integrator With s=j we obtain the transfer function of the integrator from Equation (10) as The magnitude and phase shift are G I j j I G j j 2 (12) (13) 6

7 The Bode diagram of the integrator is shown in Figure 6. The Bode plot of the magnitude is a straight line with a slope of -20dB/decade and a magnitude of I,dB at =1. The phase shift is constant (j) =-90 for all. Figure 6: Bode diagram of the integrator The first-order time delay (PT1-element) The first order time delay is described by the following differential equation: The resulting unit step response is: t yt ut. T y (14) t ht 1 e T (15) The system does not response instantaneously to a unit step input; in fact the output approaches to the final value asymptotically. Theoretically the process output can reach this value only in infinite time. Thus, the dynamic character of the PT 1 -element is to delay the output response u(t). The step response and block diagram of the PT 1 -element is shown in Figure 7. The time constant T can be identified from a tangent at the origin of the diagram graphically. Figure 7: Unit step response and block diagram of the PT 1 -element 7

8 The Laplace transformation of the differential equation (14) yields TsY s Ys Us. (16) The resulting transfer function of the PT 1 -element is Gs 1 st (17) and with s=j G j. 1 jt (18) We have a magnitude and phase shift of G j j arctan T 1 T 2. (19) Figure 8: Bode diagram of the first order time delay The resulting Bode diagram is shown in Figure 9. The diagram is divided into two regions by 1 the corner or break or cutoff frequency. The magnitude of the first-order system can be T approximated by locating the corner frequency, drawing the two asymptotes y= db and a straight line with a slope of -20dB/decade to meet at the corner frequency, and sketching a fair 1 curve that passes through -3dB at and that is asymptotic to theses lines. T The previously described transfer functions P- and I-blocks are elementary and can be not made of simpler blocks. Reformulating the transfer function of the PT 1 -block (17) leads to Y Ts 1 s Us Ys From this equation it can be seen, that the PT 1 element is not elementary, but composed of the elementary P- and I-elements The dead time element A dead time element delays the input signal by the time T t. It is commonly used to describe transport delays. The magnitude of the input signal is not changed by the dead time, but the phase shift decreases at higher frequencies. Thus, the dead time has a great effect on the (20) 8

9 stability of the closed control loop as well as on the ability of the control loop to be fast. The transfer function, magnitude, and phase shift are described by G s) e j T t s ( G j j T t (degrees) In the Bode plot the phase shift at = 1/T t is degrees. Transfer behavior of closed control loops and block diagram algebra As mentioned before, the controlled system (process part of a closed control loop) generally consists of a set of interconnected simple transfer elements. Thus, the total controlled system may be of high-order and show complex transfer behavior. As a basic principle we can divide controlled systems in systems with integrator and systems without integrator. The step response of integrating systems grows over time until some physical constraint is reached. Thus, these systems are called system without saturation. In contrast, systems without integrator are called systems with saturation and grow over time to a finite value only. The step response of both system types can be seen in Figure 10. (21) Figure 10: Step response of a high-order system with saturation (left) and without saturation (right) For the analysis of complex problems it is necessary to reduce a particular network of transfer elements, e.g. to obtain the transfer function of a set of interconnected transfer elements. As shown in Figure 11 we introduce block diagram algebra for linear time-invariant systems. The algebraic operations are applied to the transfer functions in the frequency domain. To obtain the time-domain representation of the transfer function we can apply the inverse Laplace transformation. 1. Parallel connection 2. Serial connection 9

10 3. Negative feedback 4. Positive feedback Figure 11: Block diagram algebra From a series connection of n PT 1 -elements we can obtain a system of order n, the so called PT n -element. By connecting an additional I-element in series, we obtain a high-order system of order n+1 without saturation. The step response of a high-order system with saturation (left) and without saturation (right) is shown in Figure 10. From Figure 11 it can be seen, that the resulting transfer function of a serial connection is obtained by multiplying the single transfer functions. Thus, the frequency response of as serial connection is G j G jg j... G j 1 (22) 2 With the complex number theory we obtain the magnitude and phase shift the frequency response of the serial connection as: G j G j G j... G j 2 n 1 (23) j j j... j. 1 2 n n (24) Because the logarithm of a product is the sum of the logarithms of each term we obtain: G j G j G j... G j. db (25) 1 db 2 n db From this relation we can derivate a useful property of the serial connection algebra in conjunction to the Bode plot representation of the frequency response. The calculation of the frequency response of a serial connection is reduced to summation (and subtraction) of magnitudes (in decibels) and phases (in degrees) of the individual elements. This property is an important simplification for controller design, because the Bode plot of the serial connection of controller and process can be obtained by a graphical method. 10

11 By multiplying several transfer functions we get a general transfer function in form of a rational function of s. b Gs a b s... b s n 0 1 n m 0 a1s... ams (26) The difference of the order of the numerator polynomial n and denominator polynomial m is called the difference order d = (m-n) of the transfer function. From the difference order we can conclude to the Bode plot of the transfer function. For high frequencies (j ) the phase shift amounts to d*-90 and slope of the magnitude is d* -20 db/decade. The magnitude and phase shift of low frequencies (j 0) can be obtained from the difference order in a similar manner. Values of s such that the numerator of G(s) equals zero are called zeros. Values of s such that the denominator of G(s) equals zero, thus the transfer function G(s) has the value infinity, are called poles. If the poles are real numbers (Im(s)=0) we can describe G(s) as a serial connection of individual PT 1 -elements. The reciprocal value of the poles p are equal to the time constant T of an individual PT 1 -element. From the step response h(t) of the PT 1 -element (15) it can be seen, that the step response of a transfer function with a positive pole and therefore with a positive exponent grows to infinity. Thus, systems with positive poles are unstable. Linear controller elements The most common basic linear controller elements are the P-controller, I-controller, and D- controller. They can be combined to build more complex controllers. The P-controller calculates a manipulated variable u(t) proportional to the value of the control deviation (error) e(t): U(t) = P e(t) G(s) = P (27) The I-controller calculates the manipulated variable in proportion to the integral of the error: u t t e d; Gs I 0 The advantage of the I-controller is that the integration does not stop until the error is zero. The manipulated variable of the D-controller is proportional to the derivative of the error. Thus, the manipulated variable is proportional to the rate of change of the error. Therefore the D-controller can react immediately in the case of high changes of the error. u t D e d d t s I (28) G(s) = D s (29) In practice you will frequently find parallel connections of these controller elements. A PIcontroller is obtained by a parallel connection of a P- an I-element, and PID-controller consists of a parallel connection of a P-, I-, and D-element. The resulting transfer functions of the more complex controllers are: 11

12 G p s PI - controller I s Gs I p Ds. s PID controller It is common to replace the three gains P, I, and D by the proportional gain R, the integral time constant T N, and the derivative time constant T V. With the relations R P TN we can easily derive the transfer function of a PID controller: G s R R I TV D R (30) 1 1 T s V (31) TN s The response of a PID controller to a unit input step of e(t) is shown in Figure 12. Consider that the real response of the D-controller has a small delay. Figure 12: Step response of a (real) PID-controller The block diagram and Bode plot of a PID controller is shown in Figure 13. Figure 13: Block diagram and Bode plot of a PID controller 12

13 In this lab we employ a PI controller. The integral time constant T N of the controller is set equal to the greatest time constant T 2 of the controlled process. Thus, we can cancel the slowest time constant from the transfer function to an I-element (32). This method is called PI compensation of the largest process time constant. The value of the proportional gain R determines the dynamics of the controlled system. (See next section). G ( s) G s R s 1 R(1 TN s) S R ( s). 1T1 s 1T2s TN s 1T1 s TN s Behavior of the closed control loop T T 2 N The dynamic behavior of a control loop can be described by the disturbance rejection and by the reference reaction. Reference reaction is defined as the reaction of the closed loop system to a change in the set point or reference w. The disturbance rejection is the reaction of the closed loop system to a disturbance variable z. (See Figure 1). In general, a change in the values of the reference or disturbance signal causes a transition state of the controlled variable. The run of the controlled variable may be a damped oscillation or non-periodic. If the controlled variable oscillates or rises, the closed loop is unstable and thus useless. To obtain a well operating control loop, we set up the requirements to the transfer behavior of a closed control loop: I. The closed control loop is stable, e.g. the response from a step change of the reference or disturbance signal must converge to a finite value for t. II. The closed loop exhibits high steady state accuracy, e.g. the control deviation e has to be sufficiently small in steady state. III. The response from changes in reference or disturbance should be damped sufficiently, e.g. it should show damped oscillations only. IV. The closed control loop is sufficiently fast, e.g. a step in reference or disturbance should be rejected in short time. It is obvious, that the requirements III and IV are contradictory. Similarly the requirement I and II are contrary to each other because a high accuracy can be obtained by a high gain or by an I-element. Both turn the closed loop towards instability. Thus, the controller design is an optimization problem. The stability of a closed control system can be tested from the transfer functions of the process G S (s) and the controller G R (s). If G S (s) and G R (s) are rational functions, the control loop is stable if and only if all poles of the closed loop transfer function s have a negative real component. Zeros and poles can be complex conjugated numbers. The complex number with the smallest absolute value determines the duration of the transition. The ratio of real and imaginary part determines the damping (the greater the ratio, the better the damping, no imaginary part = non-periodic transition). The poles of high-order can only be obtained by numerical or graphical methods. One method is to conclude stability and the damping characteristics of the closed loop system from the Bode plot of the open loop system G S (s)g R (s) (See Figure 14). (32) 13

14 Figure 14: Bode plot of an open loop system The magnitude plot of the open loop intersects the 0-dB line at the frequency D (crossover frequency). The amount by which the phase exceeds -180 at the crossover frequency is called phase margin R. The critical frequency is the frequency at the intersection of the phase plot and the line. The distance of the magnitude plot from the 0-dB-line is called gain margin A R. R 180 o db. A G ) G ( ) (33) D R S ( R For our lab example we can estimate the stability and damping from the phase margin of the open loop by the following propositions: If the gain of the open loop is positive and the open loop includes less than two I-elements, then the closed loop is stable, if the phase margin is positive If the slope of the magnitude plot is at least 20dB/decade around the crossover frequency, then we can apply: Overshoot of the closed loop step response [%] + phase margin [ ] = 70. For a phase margin of R =55 the overshooting of the closed loop step response is approximately 15%. By varying the gain of the open loop (of the controller R ) we can move the magnitude up or down to reach the required phase margin. 14

15 2 Linear State Feedback Control (concerning DYN 23) In this section approaches to linear state feedback control are discussed. We consider the linear time invariant system of the form: n with x, measured. p u, p n and m y, m n. It is assumed that all dynamic states can be Figure 2-1: Classical control structure In Figure 2-1 the classical control structure is depicted. The structure of the state feedback controller is shown in Figure 2-2. Figure 2-2: State feedback control structure For a time invariant, linear control law the vector of the manipulated variables reads as: Replacing u in equation 2.1 yields: The idea behind state feedback control is to assign the eigenvalues and therefore the dynamic behavior of the closed loop system. From the analysis of linear systems it becomes clear that only controllable eigenvalues can be determined by state feedback. From equation (2.3) it follows that the controller matrix R has to be chosen such that arbitrary disturbances in the initial conditions x 0 are rejected. The term Mw can be considered as an open loop control action and has to be added as it is required for a controller that it is also able to realize set point changes for the controlled variables. State feedback control can be used for disturbance decoupling. Consider the dynamic linear system: 15

16 where d is an unknown disturbance vector. The matrix E can be chosen such that the disturbances cannot be observed at the output y of the system. Summarized, the state feedback controller has the following important properties: 1. The state feedback controller has a similar feedback structure as known from classical controllers. 2. In contrast to the classical control structure, the state feedback controller uses no comparison between the set point and the current values of the controlled variables in order to keep this difference small. 3. The state feedback controller uses the complete state vector, which is in general not available. 4. The state feedback controller can be used for disturbance decoupling. The design problem of the state feedback controller reduces to calculate the controller matrix R. Different approaches have been developed in the past. We will discuss 2 approaches to the design by pole assignment and the design by minimizing a quadratic cost function. Additionally the filter matrix M has to be chosen properly. As this design is independent of the controller design it is discussed afterwards. 2.1 State Feedback Control by Pole Assignment To simplify the discussion on the design methods for the controller matrix R only the initial behavior of the closed loop system is considered, i.e. the filter matrix M = 0. Thus, the state space representation of the closed loop system reads as: Hence, the closed loop system is described by an autonomous differential equation system. From stability analysis of linear systems it is already known that the autonomous system is asymptotically stable if all eigenvalues have a real part < 0. The controller matrix has to be determined such that the eigenvalues of (A - BR) are shifted to the desired values. R can be calculated by comparing the coefficients of the characteristic polynomials of (A - BR) and the desired eigenvalues R, : This equation results in n equations for the np elements of R. For single input systems there is a unique solution while for multiple input systems an infinite number of solutions exists as in the resulting nonlinear equations more unknowns than equations exist. The additional degrees of freedom can be used to define the eigenvalues and the eigenvectors of the closed-loop system. For a single input system the formula of Ackermann is a suitable approach to shift certain eigenvalues. 16

17 2.2 State Feedback Control by Ackermann s Formula If the system is controllable and it is demanded that the closed loop system (A - br) fulfills the characteristic polynomial the controller matrix r can be determined as by: The vector t 1 represents the last row of the inverse alman controllability matrix This method is, due to the necessary calculation of the controllability matrix, numerically disadvantageous for higher order systems. 2.3 Eigenvalue and Eigenvector Assignment by State Feedback Control It has already been mentioned that for multiple input systems degrees of freedom are left by the pole assignment comparing the coefficients of the characteristic polynomials. Additional to the eigenvalues or poles, respectively, the assignment of the eigenvectors of the closed loop system offers the possibility to use the left degrees of freedom for controller design. The first question to be answered is: which eigenvalues can be shifted? Let s assume that i is an eigenvalue of A - BR. Then it holds: v i defines the right eigenvector corresponding to the eigenvalue left eigenvector of the closed-loop system. The matrix B is given. Then the following two matrices exist: i and w i is the respective B L is the left pseudo inverse (Moore Penrose Inverse) and N B defines the annihilator of B. B As the matrix N L has n linear independent rows (i.e. is of full rank) the following B transformation of the right eigenvector is allowed: 17

18 Replacing (2.13) in (2.11) yields: which describes the decomposition into a part completely independent on the control matrix R and a part determined by the control matrix R. If an eigenvalue cl i exists such that a right eigenvector ~ W which fulfills equation (2.14) always exists. T i If such w ~ i w ~ n p exist, then i is always an eigenvalue of the closed loop system and cannot be changed by state feedback as it holds: cl cl Hence, i is also an eigenvalue of the open loop system, i. e. i i is an uncontrollable eigenvalue. Therefore, only controllable eigenvalues can be shifted by state feedback. Another question to be discussed is: which eigenvectors can be chosen? To answer this question let s consider the equation: 18

19 From this equation it follows that all right eigenvectors must fulfill the condition: As this condition is independent of R, all possible eigenvectors are determined. These are n p equations for n unknowns. Thus, p linearly independent solutions in general exists, i.e. there are p free parameters in the solution, which are used to determine the controller matrix This calculation of the controller matrix R follows from the part of the decomposition equation which is dependent on the controller matrix: Summarizing all the eigenvectors into a matrix V: yields From the upper equation it becomes clear that R depends on the desired eigenvalues and the eigenvectors summarized in V. In the following it is discussed how to choose the design parameters for a system in order to decouple y from the unknown disturbances d by constant state feedback. Decoupling of a certain disturbance from a given output means that the controller matrix is chosen such that the disturbance becomes unobservable. Therefore, the eigenvectors that span the unobservable subspace have to be determined, i.e. C[v 1,..., v k ] = 0, V Null(C). If now the relation E spanv 1,, vk, Im(E) V is valid, the disturbances become unobservable if the eigenvectors of the closed loop system are chosen properly. This relation means that the disturbances can be described by a linear combination of eigenvectors which span an unobservable subspace, i.e.: 19

20 Disturbances can also be rejected by compensation. As for this method the disturbances have to be known, it is discussed after introducing a method to determine such unknown disturbances. 2.4 Linear Quadratic Regulator (LQR) Assuming that the system to be controlled is not at its steady state, the goal of control is to move the system state as quickly as possible to the desired stationary state. The controller matrix has to be chosen such that the movement is not too slow and does not oscillate too much and that the control energy is minimal. These two properties can be described formally by the following cost function: The weighting matrices Q x are positive semi-definite and Q u are positive definite. They provide the tuning parameters of the controller that is being derived in the following. The state vector x and the vector of manipulated variables u depend on the controller matrix R. Hence, R is the matrix of free variables for this optimization problem. In case of linear unconstrained systems the analytic solution of the stated problem can be derived. The solution consists of the so called Matrix-Riccati-Equation: and the manipulated variables of the optimal controller: The Matrix-Riccati-Equation has two solutions. The positive definite solution results in a stable closed loop system, while the negative definite solution yield in right half plane poles of the closed loop system. For the solution of the Matrix-Riccati-Equation numerous routines are available in common control software. 2.5 Choice of the Filter Matrix In order the choose the filter matrix M it is assumed that the control matrix R has already been determined, such that all eigenvalues of the closed loop system (A - BR) are in the left half complex plane. Applying the Laplace transformation to equation (2.3) it follows with Mw = u s : and with the measurement equation y = Cx: Here it is assumed that x 0 = 0. As the transfer behavior from Us to Y is of interest, this assumption is valid. The inverse in (2.29) exists, as det[si-(a - BR)] = 0 holds only if s is an eigenvalue of (A - BR). R has been chosen to shift these eigenvalues into the left half plane. Therefore, there is no eigenvalue in the right half plane including the j-axis. It follows that the inverse exists in this region. To abbreviate the derivation the following definition is applied: Therefore, (2.29) reads as: 20

21 Comparing this equation with Mw(s) = Us as previously defined, it follows: If now w(t) is given to the system, such that the final value for t t tends to w, i. e. that this stationary value is reached, then it follows from the final value theorem for the Laplace transformation With sws lim wt w lim it follows: s0 s As y w, it follows that the filter matrix can be calculated from: presumed that the controller matrix R has already been determined. In case of single input systems M is a scalar and can also be calculated by considering the stationary gain of the system. If the number of controlled variables differs from the number of manipulated variables the matrix M does not necessarily exist. As the calculation of M has been derived by considering the stationary behavior of the dynamic system it is obvious that this design method can only be applied if the set point does not vary quickly in contrast to the dynamics of the system. In this case the decoupling approach of Falb-Wolovich has to be applied. 21

22 3 Projects 3.1 DYN23 - Control of a chemical reactor In this lab course you will learn how to model a chemical process and use linear algebra and controller design fundamentals to develop simple P and PI-controllers. The experiment will start with a pre-test. There you will have to answer ten questions regarding the background of the experiment before you can proceed with the experiment. In the first task you have to finish a dynamic model of a chemical process based on eight differential equations. Your task is to develop energy, mass and component balances for a reactor and a buffer tank. In order to proceed, your model has to meet certain requirements which are checked against a reference model. Once your model is good enough you may proceed to the next task. In the last three tasks your need to develop three control structures to meet certain requirements. In task two, you start with a simple level control. Use the tools provided in task 1 to help you with your design decisions. In task three, the level control has to be expanded to include a temperature control for the reactor. In the last task, you will once again expand your control structure to incorporate a quality control loop for the concentration of the product B in the buffer tank The chemical process In this lab course you will model and analyze a chemical process for the production of a chemical compound. The reaction takes places in a continuously stirred tank reactor with a heating jacket. The product is buffered in a tank to be able to compensate flow variations. Several sensors allow the monitoring of the whole process. The feed is provided by another chemical process at the same site. As this process is not very reliable, your process has to cope with the disturbances introduced by changing feed specifications. 22

23 3.2 DYN26 - Modeling and Analysis of a Reactor System This project handles the fundamental methods of systems analysis. The goal of this project is to provide an opportunity to apply theoretical knowledge and to gain experience in basic principles of methods for the analysis of dynamic systems. An illustrative example of a simple continuous stirred reactor system is considered in which an equilibrium reaction of two species takes place. The practical course process is reasoned linear and divided into three large parts. The first task is to describe the dynamic system by means of three differential equations for the state variables for the concentrations for both substances and for the volume of the content of the reactor. The next step is to calculate the steady state of the system as a prerequisite of a subsequent linearization at this point. By consideration of the eigenvalues of the resulting Jacobian matrix, a statement about the stability of the system can be made. Thereafter in the third part a transfer function for the system at hand must be computed from the previous linearization. For the resulting transfer function the root locus diagram has to be computed. The graph of the root locus and step responses can also be plotted. In the given example a zero-pole cancellation can be regarded Description of the reactor system As concrete example system an continuous stirred reactor (see Fig. 3-10) is regarded, in r1 which a balance reaction of the kind A Btakes place. r2 V, ca0 M c, c, V A B V, c, c 1 A1 B1 Figure 3-10: Scheme of the Continuous stirred reactor with reversible reaction The component B is here the desired main product. The reaction inlet contains only reactive component A. For the regarded problem the relevant state variables are the concentration of the basic material A and the concentration of the product B in the reactor, at the outlet of the reactor, as well as the volume of the reactor V, respectively. The concentration of the product B is to be controlled by the processing input variables, to which the flow rate V and the concentration of the input component c A0 are available. For this simple reactor example it can be assumed, that the stirred reactor is ideal mixed and that the change of density can be neglected. The outgoing volume flow can be described with the accretion v V. The Reaction rates r 1 and r 2 of the respective fragment reaction will be described trough k1 c A and k2 c B. 23

24 3.2.2 User interface In this project tasks are modeled as a linear stepwise procedure. Each web page is associated to a subtask of the project. A web page includes the graphic representation and navigation of the project progress at the top in terms of a graph, an overview of the data structure on the left, the main section in the middle, and a help section on the right. must be chosen. After a subtask has already been finished, it is indicated by red check-sign in the navigation at the top. The fact that a further task is available can be also seen in the top. On the left hand side, an overview of the actual data is shown (this is true for all nodes of this project). Therein, the user has an overview of the plant, the desired behavior and the parameters in one figure. On the right hand side, a help window explains the meaning of the parameters for new users. If the help window is disengaged, the extra space is divided between the data overview section and the main section. 24

25 4 Preparation The posed questions should give you a hint about the knowledge tested in the beginning of the lab-experiments. General project setup 1. Explain the system under consideration! 2. How are control specifications usually given? Experimental modeling 1. What is experimental modeling? 2. How do we shape identification signals? 3. What kind of linear identification structures do you know? 4. What is special about the ARX structure? Theoretical modeling 1. What is the balance space of the system model? 2. What are the differential variables of the system? 3. What balances do we need? 4. How can the model be rendered adequate for linear control? Controller design 1. Describe the standard feedback control loop block diagram. What is the meaning of the elements and signals 2. What are the controlled resp. manipulated variables in this project? What disturbances may occur? 3. How do step responses and bode plots of typical transfer functions and controllers look like? 4. Explain terms like crossover frequency, phase margin, and roll-off rate for the frequency domain design! 5. How does an LQG controller work? 6. What is the purpose of the weighting matrices in the LQG design? 25

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