- Design Objective For A Plant Modelled As A 3rd Order System G S Of The Form G 8 T18 1 T98 1 Obtain A Feedback 1 (83.31 KiB) Viewed 43 times
[kg,T1, T2]-caplant (123456) Question 1.4. Record the values of kg, T, and T3. Define the plant, G(), in transfer function form by G-ti (kg. [T1+T2, T1+T2, 1, 0]) Obtain the open-loop poles of the plant by pole (G) Plot the plant poles on the s-plane by pzmap (G): Question 1.5. Record the transfer function and comment on it. Record and comment on the plant poles. Include the print-out of the s-plane plot and explain it. Briefly discuss the implications of the open-loop dynamics for the performance of the closed loop control of this plant. What other system dynamics may affect the closed loop performance? Question 1.6. State the Nyquist criterion for this system planıt. Note that this question does not ned for the Nyquist plot which will be analyzed later in the assignment. Question 1.7. Predict the characteristics of the open loop step response. Explain why. Obtain the open loop impulse response by ispulse (G): Question 1.8. Include the print-out of this response. Verify the response and explain its characteristics". *The characteristics refer to the general character of the response, not just the explicit "characteristics" returned by a right-click on the MATLAB plot 2 Proportional controller design by root-locus The root locus displays the locii of the closed loop system poles for a controller K(8) = kp as the value of ky incronses from zero for the system shown in Figure 1. R(3) G(a) Y() Figure 1: Closed loop system with proportional controller Plot the root locus of the plant by rlocus (C) Left clicking at a point on the root locus provides the gain, poles, overshoot, damping and natural frequency Question 2.1. Use the mouse on the root locus screen plot to determine the limiting value of l, for system stability. Calculate the value by hand using the Routh-Hurwitz criterion. Show your workings. The command sgrid draws in lines of constant damping ratio and natural frequency Wn. Use the command axis equal to get equal scaling on the real and imaginary axes.
Question 2.2. Include the print-out of the root locus. Verify the plot and explain its characteristics. Question 2.3. From the root locus plot, determine whether both design specifications are simultaneously possible using proportional feedback alone. Show the area where the dominant poles must lie in order to meet both the specifications on the root locus plot. Record the value of key at the minimum required value of ws- Record the poles for this value of kp. Determine ( and the predicted overshoot. Enter the required value of kp into the MATLAB workspace as the variable kp. Calculate the closed loop transfer function T(B) = kpG(8) 1+kG(a) by T-feedback (kp.G,1) Obtain the closed loop damping ratio, natural frequency and poles by damp (T) Now display the closed loop system step response by step (T): From the plot, measure, and tp. Question 2.4. Record and comment on the closed loop transfer function. Record the closed loop damping ratio, natural frequency and poles and compare these to the values predicted from the root locus. Include the print-out of the step response. Record 0, and tp. Comment on the response. Does your design meet the specifications? Compare the overshoot and peak time to the values predicted from the root locus. 3 Frequency domain analysis In this section, the system with the proportional controller K(8) = kp for the system shown in Figure 1 will be analysed using frequency domain techniques. Obtain the plant Bode plot by bode (G) : Question 3.1. Include a print-out of the Bode plot. Verify the plot and explain its main characteristics. Confirm that the gain margin is equal to the limiting value of gain for closed loop stability obtained in the previous section. Input the value of kp that meets the peak time performance specification obtained in the previous section into the MATLAB workspace as the variable kp. Now obtain the Bode plot of the open loop compensated system L()=G(E)K (8) = k_G(3) by bode (kpG): Obtain the gain margin GM, phase margin PM, and associated frequencies w_180 and we by [GM.PM, w180,wc] - - margin (kpG) You can convert GM to decibels by 20= log10 (GM) By right clicking on the plot and choosing ‘Minimum Stability Margins' from the Characteristics' menu option, the phase and gain margins will be shown on the plot. Clicking on these points will display the numerical values. Question 3.2. Include a print-out of the Bode plot of the open-loop compensated system. Show the gain and phase margins on the plot. Comment on the plot. Obtain the Nyquist plot of the uncompensated system by nyquist (G): To restrict the range of frequencies displayed to, for example, 0.5 <w < 500, type nyquist (G, 1.5,500}); Use the command
axis equal to get equal scaling on the real and imaginary axes. You can use the zoom facility to get an accurate display of the region of interest, but use the axis equal command again after doing so. Now obtain the Nyquist plot of the open-loop compensated system k G(a). Right clicking on the plot and choosing "Minimum Stability Margins from the 'Characteristics' menu option will show the phase and gain margins on the plot. Clicking on these points will display the numerical values. Question 3.3. Include a print-out of the Nyquist plot (including the gain and phase margins) of the open-loop compensated system. Verify the plot by calculating and showing the asymptotes. Is the closed loop system stable? Now obtain the Nichols chart by nichols(): Place lines of closed loop magnitude and phase on the chart by ngrid Obtain the Nichols plot of the open-loop compensated system kGs). Clicking on the line will display the frequency at a particular point. The gain and phase margins can also be displayed as for the Bode and Nyquist plots. Question 3.4. Include a print-out of the Nichols chart of the open-loop compensated system k_G(s). Show the gain and phase margins on the plot. Show on the plot the maximum closed loop gain, Mr, resonant phase lag, 0r, and maximum phase lag. Record the frequencies that these occur. Also show the closed loop bandwidth, to Now calculate the closed loop transfer function T() k G(8) 1+ k,G( by T-feedback (kp-G, 1) Now display the cloud loop system frequency response by bode (T); Question 3.5. Include a print-out of the closed loop frequency response. Record M. Or, W, W. and maximum phase lag and its frequency. Compare these val ose predicted by the Nichols chart. Now calculate the closed loop sensitivity function 1 S(8) 1+k,G(8) by S-feedback (1, kpC) Now display the closed loop system sensitivity frequency magnitude response by sigma (S): Question 3.6. Include a print-out of the closed loop system sensitivity frequency magnitude response. Com- ment on the response and discuss its significance. Record the maximum gain of the sensitivity function, M5- Compare this value to that predicted by the Nyquist plot. 4 P+D controller design PID controllers are the most commonly used type of industrial control. They are usually designed by trial and error using a few simple rules. Question 4.1. What will be the system type number if a PID controller is used on this plant? For zero steady state error (to unity step references) for this plant, is an integrator required in the controller? If there is a steady disturbance at the plant input, is integral action required in the controller for zero steady state error? Show why. *The system type number is the number of open-loop poles at the origin i.e. ats-0
For some mechanical control applications, the output rate is available for measurement using tachometers and other sensors. However, for many applications, such a signal is simply not available, or prohibitively expensive to obtain. Question 4.2. Assume that the output rate (derivative of the output), y can be measured. Use the Routh- Hurwitz criterion to determine the stabilizing range of gains, ky and ko where your controller is a P+D controller of the form K () = kp + skg. Show your workings. On a graph of ks against kp, plot the stability region(s). Relate the plot to the tuning rules for P+D terms. If the output derivative signal is not available then the derivative signal y(t) should be approximated by 08 + 1 Y (8) where 1/a is greater than the system bandwidth. Making 1/a too large means that high frequency noise in the measurement signal of y(t) is amplified. The closed loop system is shown in Figure 2. The controller K (8) for this exercise is a P+D controller ska kp +sky + okp) K®) = kp + 08 + 1 (Y8+1 K() R(S) G(8) Y (3) Figure 2: Closed loop system with P+D controller Now design the controller. Firstly, define a. A suggested value is a =0.02 alpha-0.02 Define a proportional gain term, kp, in the MATLAB workspace in a variable kp. The value used in the previous two sections will be appropriate. Similarly, define a derivative gain term, kg, in a variable kd. Start with ks quite small. Now define the transfer function, Kpd(a), of the controller by Kpd-tf([kd+alpha+kp. kp], [alpha, 1]) Obtain the system open loop transfer function L(8) = G()Kpa(®) by L-KpdG Now obtain the system closed loop transfer function G(8)Kpd() T(8) L(a) 1+G(8)Kpd() 1+ L) by T-feedback (KpdG,1) Now display the closed loop system step response by step(T); If the response does not meet the specifications, change kp and kd and repeat the procedure until a satisfactory design is obtained. Question 4.3. Include the print-out of the step response of your final design. From the plot, measure o, and tp. Does your design meet the specifications? Is it a good response? Record the final values of kp and ke- Record the transfer function Kpd(s), the system open loop transfer function L(a) and the closed loop transfer function (). Obtain the Bode plot of the controller transfer function Kpd(8) by
bode (Kpd) Question 4.4. Include the print-out of the Bode plot. What do you notice about it? Obtain the Bode plot of the system open loop transfer function L(8) by bode (L) Obtain the gain margin GM, phase margin Pm, and associated frequencies weg and wep by [GM.PM. Wcg. Wcp] - margin (L) You can convert CM to decibels by 20log10 (GM) Question 4.5. Record the gain margin, phase margin, and associated frequencies. Compare these to the values obtained in the previous section and explain the difference. Include the print-out of the Bode plot and show the gain margin and phase margin on the Bode plot. Obtain the closed loop poles and zeros by pole (T) zero(T) Plot the closed loop system poles and zers on the 8-plane by | pzmap (1): Question 4.6. Include the print-out of the s-plane plot. Record the closed loop poles and zeros. Discuss their values and their relationship to the design criteria. . 5 Pole placement using state feedback In this section, a state feedback controller is designed that will place the closed loop eigenvalues at desired locations. The assumption is made here that the system states are directly measurable using sensing devices. This, however, is frequently not a practical or cost-effective proposition. In such cases, it is possible to construct state estimators (or observers) which estimate the values of the states. For those who are interested, state estimator theory can be found in many textbooks. A state space equivalent representation of the plant is given by G(8) = C(al - A)-1B+D and this can be obtained by G88-98 (G) Question 5.1. Record the state space representation produced by MATLAB. There are an infinite number of equivalent state space representations. If S is any square invertible matrix with dimension equal to the number of states, then equivalent representations of (A,B,C,D) are given by (SAS-, SB, CS-1,D). We can convert the MATLAB state space representation to the representation given in the course Modelling of Dynamic Systems (also see the Appendix of your course notes), by setting S= 0, where is a matrix known as the observability matrix (this can be proved by means of the Cayley-Hamilton Theorem, see Appendix for details). The observability matrix can also provide a test for the observability of the plant. The observability matrix is defined (for a 3rd order system) с O= CA CA and can be obtained by S-obav (Gas.a, Gss.c) The state space system is transformed into the representation given in the Appendix of your course notes by Gs3-38283 (Gas,s) You can check that this state space representation is equivalent to the original plant by tf (Gas)
Question 5.2. Record the transformed state space representation and the transformation/observability matrix S. From G(), calculate the state space representation by hand, and compare the results to those given by MATLAB Define a MATLAB vector P which contains the desired closed loop pole positions of the closed loop system Place the two dominant poles at positions required to meet the design objective (from Section 1), and place the third pole at least a factor of 5 away from the complex pole pair. For example, if the desired pole positions are {-3+2, -16), define P-[-3+2j: -3-2j: -16] Question 5.3. Record and explain how you calculated the desired pole positions. Now obtain the state feedback controller gain matrix K = [K1, K2, Kg such that the control input u(t) is u(t)=-(K111(t) + K322(t) + K373(t)) or in matrix form u=-Kr and the closed loop eigenvalues, i.e. the eigenvalues of A - BK are those specified in vector P. This can be done in MATLAB by the command K - place (Gss.a, Gss.b,P) Question 5.4. Record the values of K. Calculate the state feedback controller gain matrix, K, by hand using the method in your notes and verify the MATLAB result. Check the closed loop eigenvalues by eig(Gs3.a-Gas.b*K) and ensure that they are equal to the values specified in P. y(t) r(t) B S (t) (t) А Figure 3: Closed loop system with state feedback controller Obtain the closed loop state space system for state feedback by Tss-38 (Gss.a-Gas. b*K,G83. b*K(1), Css.c, Gas.d) Obtain the closed loop transfer function by T-tf (Tas) Obtain step response of the closed loop system shown in Figure 3 by step(Tas) Question 5.5. Include the print-out the closed loop step response. From the plot, measure of and tp. Does your design meet the specifications? If not, explain why. Question 5.6. Briefly give the main disadvantages of state feedback.
8 6 Robustness analysis In this section, the robustness of the design to uncertainty in the parameters (kg,T1,T) is analyzed. To do this we make use of the uncertainty modelling tools in the MATLAB Robust Control Toolbox. Type help robust for details of these. We will approximate the uncertainty set by a multiplicative uncertainty model and then use the small gain theorem to check the robustness. We start by setting up the uncertain model. Define a 40% uncertainty on each of your plant model parameters, (kg,T.,12) by first redefining (kg,T1, 12) as the nominal values (kgo, T10, T20): kgo-kg:T10-T1; T20-T2: and then defining a 40% uncertainty on each of the plant model parameters: kg - ureal ('kg', kgo, 'Percentage'.40); T1 - ureal('ti',T10, 'Percentage'. 40); T2 - ureal (T2', T20, 'Percentage'. 40); Define the uncertain plant Gu(a) in transfer function form by Gu-tf (kg, [T1 T2, T1+T2, 1, 0]) so that Gu() € Op where kg G (1) ={G(a) = 3T;8+11138 +1) 1 : 0.Giko 5 kg 5 1.kyo, O.GL10 575 14T;0, 0.6790 5 To $ 1.413} is the set of parametric uncertainty models. Next, you will determine a multiplicative uncertainty model set G so that G, CG where G= {GO®)(1+W.(8)4(): max (4(jw)| <1} so that G. EG. This can be done in MATLAB by generating a random sample of LTI models from Gp, obtaining the frequency response of those models and fitting a function W.(s) to cover those responses. In other words we want to calculate a function W.(s) so that AGW) < 1 for all Aw where Gr(B) - G (8) A(8) = (3) W.(8)Go) Create a sample array of 80 LTI models chosen randomly by Garray - usample (Gu, 80); Plot the array magnitude Bode plots by bodenag Garray, 'b-- Gu. NominalValue ,'I' .1.1,1000)) Get the frequency response of those models into an array of frequency responses by Garrayg - frd (Garray, logspace (-1,3,60)); Next, you will use the cover function. This function fits an uncertain model of a given order to a set of LTI responses You will use ucover to fit an uncertain model to the array of frequency responses which covers all behaviours in the array by the multiplicative uncertainty model G. Choose the nominal value of Gy, that is G, as center of the cover, and use a first order system to model the maximum frequency distribution of the unmodelled dynamics given, from (3), by A(8)W:(8) Gu() - Go(a) Go(a) by the code orderWt - 1; [Gays, Info] - ucover (Garrayg. Gu. NominalValue, orderüt,'Input Mult'); Wt - Info.W1 : where it is assumed that Aw) < 1 for all possible uncertainty models A() and frequencies w. Thus Gu(jw) - Golw) W. (jw) for all w (4) Gow) and all G in the array.
To confirm that W. (s) covers the multiplicative uncertainty (i.e. satisfies (1)), plot the weighting function frequency response along with the array of the difference between Go and the model array, that is (GB) - Go(s)/Go(8) that is covered by W:(8). In other words, the sample array of the multiplicative uncertainty in the system model. Do this by DeltaArray-(Garray-Gu. NominalValue)/Gu. NominalValue; bodesag (DeltaArray, '--'.Wt,'I'.1.1,1000}) Obtain the low frequency gain of W.Gw) as w0 by dcgain(W) Obtain the weighting function in a zero/pole format to get the high frequency gain ssw → by zpk (Wt) Question 6.1. Include the print-out of the magnitude frequency responses of the array (Gu() – Go())/Go(a) and W. (3). From the parameter uncertainty set (1), work out the formula for (G.() - Go())/G (8) and 30 calculate the maximum multiplicative uncertainty as w+ and w +00, and compare these values to those of W.(s). Discuss the accuracy of W.() in describing the uncertainty, and suggest how the accuracy could be improved Question 6.2. Use the small gain theorem to determine the robust stability conditions for a multiplicative uncertainty model of the form of (2). Show your workings. Now test the robust stability of your proportional controller kp designed in Section 2. Obtain the magnitude frequency response of W.(-T () by T-feedback (kp•G,1); bodemag (Wt+1,1.1,100}) :grid on Check the robust stability results for the controller by obtaining the uncertain closed loop system: Tu-feedback (KpdGu, 1); Create a sample array of 80 closed loop system models Tarray - usample (Tu,80): Find the maximum real part of the eigenvalues for the array by max(max (real (eig (Tarray)))) Now repeat the robust stability tests for the P+D controller using the closed loop system Tod-feedback (Kpd*G, 1); and uncertain cloud loop system: Tpdu-feedback (Kpd#Gu,1); Question 6.3. Discuss the robust stability results. Include the frequency plots. Finally check the closed loop robust performance of the two controllers against the original step response design objectives by step(Tarray) Question 6.4. Include the print-out of the set step responses. Briefly discuss the result. 7 Controller design for plant input disturbance rejection For the final section, the design objectives are extended so that the controller should ensure that there is zero steady state error for plant input disturbances. Use the same plant as for Section 1. Design objectives for a plant modelled as a 3rd order system, G(s), of the form ky G(8) 8(T18+ 1)(798 +1) obtain a feedback controller which achieves the performance specifications on the step response of 0, < 10% and tp < 0.5 sec, with zero steady state error to plant input and plant output disturbances. Question 7.1. Design a controller that meets the objectives. Justify your choice of controller and design method. Explain your design steps and analyze your final design. Support your design by mathematical analysis and with appropriate MATLAB tools. Discuss the advantages and limitations of your design. Ensure that you have completed Tutorial Question 1.6(vii).
Appendix A SISO Tool: an interactive MATLAB design tool (OP- TIONAL EXERCISE) MATLAB provides a graphical user interface for the interactive design of single-input/single-output (SISO) compensators by interacting with the root locus, Bode, and Nichols plots of the open-loop system. To run the tool, type sisotool (6) To import the plant or controller data into the SISO Tool, select the "Import" item from the "File" menu. Try and improve your designs using this tool. Can the step response be improved by putting derivative action into the feedback path compensator,H? Try and improve the step response by means of a pre-filter, F. A= 0 0 -G3 0 1 - Appendix B - Transformation of state space system to representation given in course notes In the Appendix of your course notes, you are given a method to convert a transfer function into a representation given by (for a 3rd order SISO system) 1 0 -02 bi B = 12 bs Č= 1 0 0 Dad where the characteristic equation 82 +2182 +228 +13=0=det(s1 - A). To transform any state space system represented by (A, B,C,D) into the representation given by (A,B,C,D), we can use the transformation A= SAS-1 B=SB C=CS- D=D where S is the observability matrix given by с S=CA CA Proof. It is clear that CS=C hence C=CS-1 Now A = SAS-, 80 ĀS = SA. Expanding AS gives А ASC -a31 - 0,4-4,42 and expanding SA gives -... SA=CA2 43 Now from the Cayley-Hamilton theorem, 4% +2,42 +224 +231 = 0, hence AS =SA.