matlae
3-1.什么是线性系统？其主要的特征是什么？
答：凡是用线性微分方程来描述其动态特性的系统称为线性系统。
特征：可运用叠加原理进行计算。
3-2.
（1）f(t)-k*y(t)=md2y/dt2
(2 ) f(t )-k1*(k2*y(t)-f(t)/k2-k1)-k2*(y(t)*k1-f(t)/k1-k2)=md2y(t)/dt2
3-3
(a)设i1为流过R1的电流，i为总电流，则有
U0=Ri+1/C2*∫idt
Ui-U0=i1*R1
Ui-U0=1/C1*∫(i-i1)dt
化简得
C1*R2*(U0)¨+(1+R2/R1+C1/C2)*(U0)’+U0/C2*R1
=C1*R2 (Ui)¨=(R2/R1+C1/C2)Ui’+Ui/C2*R1
(b)
设电流为I,则有
Ui=U0+R1*i+1/C1*∫idt
U0=1/C2*∫idt+ R2*i
3-4
M=J(θ)+Cm*¨θ’+Rk*(Rθ’-x)
K(Rθ-x)=mx¨+Cx’
消去中间x得
m*Jθ(4)+(m*Cm+Cj) θ(3)+R2*k*m+Cm*c+Jk) (θ) ¨+k(c*R2+Cm) θ’
=m*M¨+c*M’+k*M
3-5
（1）(s3+15s2+50s+500)Y(S)=(s2+2s)U(s) (2) (5s2+25s)Y(S)=(0.5s)U(s) G=Y(S)/U(S) G=Y(S)/U(S)
num=[1 2 0] num=[0.5 0]
num = num =
1 2 0 0.5000 0 >> den=[1 15 50 500] den=[5 25 0]
den = den =
1 15 50 500 5 25 0
>> G=tf(num,den) >> G=tf(num,den)
Transfer function: Transfer function:
s^2 + 2 s 0.5 s
------------------------- ------------
s^3 + 15 s^2 + 50 s + 50 5 s^2 + 25 s
(3) (s2+3s+6+4*1/s)Y(S)=(4s)U(s)
G=Y(S)/U(S)
num=[4 0]
num =
4 0
>> den=[1 3 6 4]
den =
1 3 6 4
>> G=tf(num,den)
Transfer function:
4 s
---------------------
s^3 + 3 s^2 + 6 s + 3
3-6
由传递函数定义得
Xi=1/s
Y=1/s-1/(s+2)+2/(s+1)
Y/Xi=(2s2+6s+2)/(s2+3s+2)
3-8
(1) num=[1 35 291 1093 1700]
num =
1 35 291 1093 1700
>> den=[1 298 0 2541 4684 5856 4629 1700]
den =
1 298 0 2541 4684 5856 4629 1700
>> G=tf(num,den)
Transfer function:
s^4 + 35 s^3 + 291 s^2 + 1093 s + 1700
--------------------------------------------------------------
s^7 + 298 s^6 + 2541 s^4 + 4684 s^3 + 5856 s^2 + 4629 s + 1700
sys_zpk=zpk(sys_tf)
Zero/pole/gain:
(s+25) (s+4) (s^2 + 6s + 17)
-------------------------------------------------------------------------------
--
(s+298) (s^2 + 1.379s + 0.5664) (s^2 + 0.4945s + 1.032) (s^2 - 1.902s + 9.758)
(2) num=15*[1 1]
num =
15 15
>> den=conv(conv([1 3],[1 5]),[1 15])
den =
1 23 135 225
>> sys=tf(num,den)
Transfer function:
15 s + 15
--------------------------
s^3 + 23 s^2 + 135 s + 225
>> z=-1;
p=[-3 -5 -15];
k=15;
sys=zpk(z,p,k)
Zero/pole/gain:
15 (s+1)
------------------
(s+3) (s+5) (s+15)
(3)
sys1=zpk([0,-1,-2,-2],[-1,1],100);
sys2=tf([1,3,2],[1,2,5,2]);
sys3=tf(1,[1,2,4]);
sys4=tf(1,[1,2,4])
sys=sys1*sys2*sys3*sys4
Zero/pole/gain:
100 s (s+1)^2 (s+2)^3
----------------------------------------------------------------
(s+1) (s+0.4668) (s-1) (s^2 + 2s + 4)^2 (s^2 + 1.533s + 4.284)
>> sys_tf=tf(sys)
Transfer function:
100 s^6 + 800 s^5 + 2500 s^4 + 3800 s^3 + 2800 s^2 + 800 s
------------------------------------------------------------------------------ s^9 + 6 s^8 + 24 s^7 + 56 s^6 + 91 s^5 + 74 s^4 - 4 s^3 - 104 s^2 - 112 s – 32 3-9
3-9 (1) 解
>> A=[5,2,1,0;0,4,6,0;0,-3,-5,0;0,-3,-6,-1];
>> B=[1;2;3;4];
>> C=[1,2,5,2];
>> D=zeros(1,1)
>> sys=ss(A,B,C,D)
a =
x1 x2 x3 x4
x1 5 2 1 0
x2 0 4 6 0
x3 0 -3 -5 0
x4 0 -3 -6 -1
b =
u1
x1 1
x2 2
x3 3
x4 4
c =
x1 x2 x3 x4
y1 1 2 5 2
d =
u1
y1 0
Continuous-time model.
>> sys_tf=tf(sys)
Transfer function:
28 s^3 - 181 s^2 + 317 s + 46
-------------------------------
s^4 - 3 s^3 - 11 s^2 + 3 s + 10
>> sys_zpk=zpk(sys)
Zero/pole/gain:
28 (s+0.1346) (s^2 - 6.599s + 12.21) -------------------------------------
(s-5) (s+2) (s+1) (s-1)
（2）解
>> A=[2,2,1;1,3,1;1,2,2];
>> B=[3;3;4];
>> C=[1,0,0];
>> D=zeros(1,1);
>> sys=ss(A,B,C,D)
a =
x1 x2 x3
x1 2 2 1
x2 1 3 1
x3 1 2 2
b =
u1
x1 3
x2 3
x3 4
c =
x1 x2 x3
y1 1 0 0
d =
u1
y1 0
Continuous-time model. >> sys_zpk=zpk(sys)
Zero/pole/gain:
3 (s-1) (s-0.6667)
------------------
(s-1)^2 (s-5)
>> sys_tf=tf(sys)
Transfer function:
3 s^2 - 5 s + 2
----------------------
s^3 - 7 s^2 + 11 s - 5
3-10
3-12
(1)sys1=tf([1 1],[1 3 4]);
sys2=tf([1 3 5],[1 4 3 2 1]);
sys=sys1*sys2
Transfer function:
s^3 + 4 s^2 + 8 s + 5
-------------------------------------------------
s^6 + 7 s^5 + 19 s^4 + 27 s^3 + 19 s^2 + 11 s + 3
(2)
sys1=tf([1 1],[1 3 4]);
sys2=tf([1 3 5],[1 4 3 2 1]);
sys=sys1*sys2;
sys_ss=ss(sys_tf)
a =
x1 x2 x3 x4 x5 x6 x7 x8 x9
x1 -6 -3 -1.75 -0.7109 -0.2891 0.01563 0.2031 0.2187 0.25
x2 8 0 0 0 0 0 0 0 0
x3 0 4 0 0 0 0 0 0 0
x4 0 0 4 0 0 0 0 0 0
x5 0 0 0 2 0 0 0 0
x6 0 0 0 0 1 0 0 0 0
x7 0 0 0 0 0 2 0 0 0
x8 0 0 0 0 0 0 1 0 0
x9 0 0 0 0 0 0 0 0.25 0
b =
u1
x1 8
x2 0
x3 0
x4 0
x5 0
x6 0
x7 0
x8 0
x9 0
Continuous-time model.
c =
x1 x2 x3 x4 x5 x6 x7 x8 x9
y1 0 0 0.3906 0.7813 1.221 1.855 0.6836 0.1953 0
d =
u1
y1 0
(3)sys=step(sys)
sys =
0.0035
0.0234
0.0679
0.1402
0.2408
0.3682
0.6881
0.8701
1.0582 1.2457 1.4257 1.5918 1.7383 1.8601
1.9535
2.0157 2.0453 2.0422 2.0075 1.9435 1.8535 1.7421 1.6140 1.4749 1.3306 1.1870 1.0496 0.9237 0.8141 0.7245 0.6579 0.6164 0.6010 0.6114 0.6467 0.7047 0.7826 0.8767
0.9829
1.0967 1.2135 1.3285 1.4373 1.5357 1.6202 1.6879 1.7365 1.7646 1.7718
1.7250 1.6740 1.6077 1.5292 1.4418 1.3493 1.2553 1.1638 1.0781 1.0015 0.9367 0.8861 0.8511 0.8328 0.8314 0.8465 0.8770 0.9214
0.9775
1.0428 1.1145 1.1895 1.2649 1.3376 1.4049 1.4643 1.5137 1.5513 1.5760 1.5873 1.5850 1.5696 1.5420 1.5037 1.4564 1.4024 1.3438 1.2831 1.2227 1.1651 1.1124 1.0665 1.0291
0.9843 0.9780 0.9826
0.9977
1.0222 1.0550 1.0946 1.1392 1.1870 1.2359 1.2841 1.3296 1.3707 1.4060 1.4342 1.4544 1.4659 1.4687 1.4628 1.4487 1.4271 1.3992 1.3662 1.3295 1.2907 1.2513 1.2129 1.1771 1.1451 1.1181 1.0970 1.0825 1.0750 1.0746 1.0811 1.0940 1.1128 1.1364 1.1638 1.1939 1.2254 1.2570 1.2875
1.3404 1.3610 1.3766 1.3869 1.3915 1.3903 1.3837 1.3721 1.3559 1.3360 1.3133 1.2887 1.2632 1.2380 1.2138 1.1918 1.1727 1.1571 1.1456 1.1385 1.1360 1.1380 1.1444 1.1548 1.1686 1.1853 1.2040 1.2241 1.2446 1.2648 1.2838 1.3010 1.3157 1.3275 1.3358 1.3406 1.3417 1.3391 1.3331 1.3240 1.3122 1.2983 1.2829
1.2501 1.2341 1.2191 1.2057 1.1944 1.1857 1.1797 1.1766 1.1765 1.1793 1.1848 1.1927 1.2026 1.2142 1.2268 1.2400 1.2533 1.2660 1.2778 1.2881 1.2967 1.3032 1.3075 1.3093 1.3088 1.3060 1.3010 1.2942 1.2858 1.2763 1.2660 1.2553 1.2447 1.2346 1.2254 1.2174 1.2109 1.2061 1.2032 1.2022 1.2031 1.2058 1.2102
1.2230 1.2309 1.2393 1.2479 1.2564 1.2644 1.2716 1.2777 1.2826 1.2861 1.2880 1.2884 1.2873 1.2848 1.2809 1.2760 1.2701 1.2636 1.2568 1.2499 1.2432 1.2369 1.2313 1.2266 1.2230 1.2205 1.2192 1.2192 1.2204 1.2227 1.2261 1.2303 1.2351 1.2404 1.2460 1.2515 1.2568 1.2618 1.2661 1.2697 1.2724 1.2741 1.2749
1.2734 1.2713 1.2685 1.2649 1.2609 1.2566 1.2521 1.2477 1.2434 1.2396 1.2363 1.2335 1.2316 1.2304 1.2300 1.2304 1.2315 1.2334 1.2358 1.2388 1.2421 1.2456 1.2492 1.2528 1.2561 1.2591 1.2617 1.2637 1.2651 1.2660 1.2661 1.2656 1.2645 1.2629 1.2608 1.2584 1.2557 1.2528 1.2499 1.2471 1.2444 1.2421 1.2401
1.2376 1.2371 1.2371 1.2376 1.2386 1.2400 1.2418 1.2438 1.2460 1.2484 1.2507 1.2529 1.2550 1.2568 1.2583 1.2594 1.2601 1.2604 1.2603 1.2598 1.2589 1.2577 1.2562 1.2545 1.2527 1.2508 1.2490 1.2472 1.2456 1.2442 1.2431 1.2423 1.2418 1.2416 1.2418 1.2423 1.2431 1.2441 1.2453 1.2467 1.2482 1.2497 1.2512
1.2538 1.2549 1.2558 1.2564 1.2567 1.2568 1.2565 1.2561 1.2554 1.2545 1.2535 1.2523 1.2511 1.2499 1.2487 1.2476 1.2467 1.2459 1.2452 1.2448 1.2446 1.2446 1.2448 1.2452 1.2458 1.2466 1.2474 1.2484 1.2493 1.2503 1.2512 1.2521 1.2529 1.2535 1.2540 1.2543 1.2544 1.2543 1.2541 1.2537 1.2532 1.2526 1.2519
1.2503 1.2496 1.2488