- Mch3008 Control Systems Laboratory Lab 7 Stability And Root Locus Controller Design Laboratory 3 Stability And Root Loc 1 (68.49 KiB) Viewed 73 times
- Mch3008 Control Systems Laboratory Lab 7 Stability And Root Locus Controller Design Laboratory 3 Stability And Root Loc 2 (69.32 KiB) Viewed 73 times
- Mch3008 Control Systems Laboratory Lab 7 Stability And Root Locus Controller Design Laboratory 3 Stability And Root Loc 3 (79.15 KiB) Viewed 73 times
- Mch3008 Control Systems Laboratory Lab 7 Stability And Root Locus Controller Design Laboratory 3 Stability And Root Loc 4 (55.36 KiB) Viewed 73 times
s=tf('s')
Gs=zpk(-7,[0 -5 -15 -20],1)
zeros=zero(Gs)
poles=pole(Gs)
rlocus(Gs)
isstable(Gs)
zeta=0.69
wn=4.34
sgrid(zeta,wn)
[K,polse]=rlocfind(Gs)
sys=feedback(K*Gs,1)
step(sys)
I have etered sisotools after this and entered edit
archictecture than changed G to my Gs
Can you please help me with the last question kind regards
MCH3008 CONTROL SYSTEMS LABORATORY LAB 7 STABILITY AND ROOT LOCUS CONTROLLER DESIGN LABORATORY 3: STABILITY AND ROOT LOCUS CONTROLLER DESIGN 1. OBJECTIVES This lab aims to design proportional controller for certain specifications using root locus and use to study the stability of systems. The root locus of an (open-loop) transfer function H(s) is a plot of the locations (locus) of all possible closed-loop poles with proportional gain K and unity feedback. K H(s) The closed-loop transfer function is: Y (8) KH(s) = R(8) 1+ KH(s) and thus the poles of the closed-loop poles of the closed-loop system are values of s such that 1 + KH(s)=0. If we write H(s)=b(s)/a(s), then this equation has the form: a(s) + Kb(s) = 0 a(s) + b(s) = 0 et n = order of a(s) and m = order of b(s) (the order of a polynomial is the highest power of s that appears in it). We will consider all positive values of K. In the limit as K → 0, the poles of the closed-loop system are a(s)=0 or the poles of H(s). In the limit as K→ ∞, the poles of the closed-loop system are b(s)=0 or the zeros of H(s). No matter what we pick K to be, the closed-loop system must always have n poles, where n is the number of poles of H(s). The root locus must have n branches; each branch starts at a pole of H(s) and goes to a zero of H(s). If H(s) has more poles than zeros m <n and we say that H(s) has zeros at infinity. In this case, the limit of H(s) as s→ ∞ is zero. The number of zeros at infinity is nm, the number of poles minus the number of zeros, and is the number of branches of the root locus that go to infinity (asymptotes). Since the root locus is actually the locations of all possible closed-loop poles, from the root locus we can select a gain such that our closed-loop system will perform the way we want. If any of the selected poles are on the right half plane, the 'closed-loop system will be unstable. The poles that are
MCH3008 CONTROL SYSTEMS LABORATORY LAB 7 STABILITY AND ROOT LOCUS CONTROLLER DESIGN closest to the imaginary axis have the greatest influence on the closed-loop response, so even though the system has three or four poles, it may still act like a second or even first order system depending on the location(s) of the dominant pole(s). y (¹) Allowable tolerance Figure 0.05 or 1 0.02 0.5 shows the transient response characteristic of a second order system for a unit step. It is possible to design controller satisfying certain percent overshoot (PO) and peak time (t) using root locus. PO and the t are functions of the damping ratio (3) and natural frequency (las (1.1) |S=069 and (1.2) 1.1 Required Matlab Commands W₁n₁ = 4.34 (>>help functionname gives detailed information) roots(num) - find polynomial roots locus(sys) - calculates and plot the locus of the roots of 1+kG(s) = 0 [K.Boles] = locfind(sys) - find the gain K and its corresponding poles for a given roots locus. sgrid(zeta, Wn) - draw lines (boundaries) of zeta and Wn. feedback (G,H) - computes a closed-loop model G for the feedback loop 1. LABORATORY OBJECTIVES 2.1 Finding Roots and Root Locus For the giv following transfer functions, calculate poles and zeros and show your codes, results (poles and zeros) and comment on the stability of system in the blanks. For each G(s) plot the root locus. (Hint: use roots(), tf) and locus functions) ld PO=100 exp [(-5)/√(1-5²] 11-01-5 t
MCH3008 CONTROL SYSTEMS LABORATORY LAB 7 STABILITY AND ROOT LOCUS CONTROLLER DESIGN Code: Poles and Zeros: Stability: -1.9053 +1.2837i s=tf('s') Gs=1/(s^3+4*s^2+6*5+1) Stable G(s)= -1.9053 - 1.2837i s³ +4s² +6s+1 zeros-zera (Gs.) poles-pole (GS) rlocus (Gs) isstable(Gs) -0.1895 + 0.0000i s=tf('s') Zeros= -2 s+2 stable GS=(5+2)/(s^2+2 5+1) G(s) = Polse= -1, -1 s²+2s+1 zeros-zern(GS) Beles-Rele(GS) clocus.(GS) isstable(Gs) s=tf('s') zeros = unstable s²+s+1 Gs-(s^2+s+1)/(s* (s^2+ -0.5000+ 0.8660i G(s) = 4*s+1)) zeros-zero(Gs) Rales-Rale(Gs) clecus(GS) isstable(Gs.) -0.5000-0.8660i s(s² + 4s+1) poles 0 0.2679 2.2. A feedback control system with certain design criteria Consider an open-loop system which has a transfer function of Y(s) s+7 H(s) = U(s) s(s+5)(8 +15)(8 +20) Design a feedback control system with proportional (gain K) for PO less than 5% and to less than 1 sec. Follow the steps given below: Open a new script in Matlab and use s-tf('s) to write the transfer function H(s) as a variable named sys. Plot root locus. Lastly, limit your axes using axis([-22 3 -15 151). Write your code in the following blank. After you run your codes, you will have the plot given below. Root Locus s-tf('s') GS-zak-7, [0 -5 -15 -20],1) zeces-zace(as) Roles-rale (GS) clocus (GS) isstable (Gs) zsta=0.69 0=4.34 serid(zstawn) [Kapolse]-clocfind (Gs) sys-feedback(K*Gs,1) RealAxis (seconds) -3.7321,-
To satisfy the design criteria as: a. Find using equation 1.1 in order to have a percent overshoot (PO) less than 5% 5=0.69 b. Find using equation 1.2 in order to have a peak time (tp) less than 1 sec. 4= 4.34 According to your solutions write appropriate zeta (and wn (n) in following Matlab code and add it to the script. You will see the following plot. Root Locus zeta = wn = sgrid(Zeta, Wn) -20 -15 On the plot you get, the two dotted lines indicate pole locations with your : in between these lines, the poles will have higher than 5. The semicircle indicates pole locations with a natural frequency Ra Going back to our problem, to make the overshoot less than 5%, the poles have to be in between the two white dotted lines. And to make the peak time shorter than 1 second, the poles have to be outside of the white dotted semicircle. So now we know only the part of the locus outside of the semicircle and in between the two lines are acceptable. All the poles in this location are in the left-half plane, so the closed-loop system will be stable. We need a proportional controller to move the poles to the desired region. Use the flocfind command in MATLAB to choose the desired poles on the locus. Click on the plot the point where you want the closed-loop pole to be. Select the points indicated in the plot to satisfy the design criteria. FoxLoc [K.Boles] = xlocfind(sys) 5 Amb
In order to find the step response, you need to know the closed-loop transfer function. Write K into code below and apply to your system. K = sys_cl = feedback(K*sys, 1) step(sys_c1) Prove your design with Simulink using transfer function block to construct H(s) and apply a step input. Show the block diagram and output Y(s).