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### ME3514_State_Space_MATLAB_RAB

Course: ME 3514, Fall 2011
School: Virginia Tech
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Word Count: 1113

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To ME3514 State-Space model Dynamic Systems State-Space State Space Modeling: is a method to find the response of dynamic systems that can be easily implemented on the computer, e.g. MATLAB. State-Space equations: consists of two type of equations, e.g. &quot;state equations&quot; and &quot;output equations&quot;, and they take the form output equations State equations x1 (t ) a11 x1 (t ) a12...

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To ME3514 State-Space model Dynamic Systems State-Space State Space Modeling: is a method to find the response of dynamic systems that can be easily implemented on the computer, e.g. MATLAB. State-Space equations: consists of two type of equations, e.g. "state equations" and "output equations", and they take the form output equations State equations x1 (t ) a11 x1 (t ) a12 x2 (t ) a1n xn (t ) b11u1 (t ) b12u2 (t ) b1r ur (t ) x2 (t ) a21 x1 (t ) a22 x2 (t ) a2 n xn (t ) b21u1 (t ) b22u2 (t ) b2 r ur (t ) xn (t ) an1 x1 (t ) an 2 x2 (t ) ann xn (t ) bn1u1 (t ) bn 2u2 (t ) bnr ur (t ) Output equations y1 (t ) c11 x1 (t ) c12 x2 (t ) c1n xn (t ) d11u1 (t ) d12u2 (t ) d1r ur (t ) y2 (t ) c21 x1 (t ) c22 x2 (t ) c2 n xn (t ) d 21u1 (t ) d 22u2 (t ) d 2 r ur (t ) ym (t ) cm1 x1 (t ) cm 2 x2 (t ) cmn xn (t ) d m1u1 (t ) d m 2u2 (t ) d mr ur (t ) R. Burdisso 1 ME3514 State-Space Equations State-Space equations: are expressed in matrix form as x A x B u y C x D u where x(t ) state vector y (t ) output vector u (t ) input vector A state matrix B input matrix C output matrix D state direct transmission R. Burdisso 2 ME3514 Example Example: Write state-space equation of the system my (t ) by (t ) ky (t ) u (t ) IC : yo and yo Select the states as: x1 y x2 y Then, Then x1o yo then I.C. are x2 o yo R. Burdisso 3 ME3514 Matlab: Unit step input EX.1 Problem: Ex 5-2 sys=SS(A B C D) sys=SS(A,B,C,D) step(sys) step(A,B,C,D) >> A=[-1 -1;6.5 0]; >> B=[1 1; 1 0]; >> C=[1 0; 0 1]; >> D=[0 0; 0 0]; [ ; ]; >> sys=ss(A,B,C,D); >> step(sys) >> grid >> title('Unit Step Response') Comments: 4 curves: (1) y when input is j u1 ( ) y1 p just (3) y2 when input is just u1 (2) y1 when input is just u2 ( ) y p j (4) y2 when input is just u2 R. Burdisso 4 ME3514 Matlab: Unit step input EX. 2 same problem as above sys=ss(A,B,C,D) (A B C D) t=ti:dt:tf step(sys, t) >> A=[-1 -1;6.5 0]; >> B=[1 1; 1 0]; >> C=[1 0; 0 1]; >> D=[0 0; 0 0]; [ ; ]; >> sys=ss(A,B,C,D); >> t=0:0.1:15; >> step(sys,t) >> grid Comments: time vector is specified by user R. Burdisso 5 ME3514 Matlab: Unit step input EX. 3 same problem as above sys SS(A,B,C,D) sys=SS(A B C D) step(A, B, C, D, iu) step(A, B, C, D, iu, t) or [y,x,t]=step(A, B, C, D, iu) or [y,x,t]=step(A, B, C, D, iu, t) >> A=[-1 -1;6.5 0]; [ ; ]; >> B=[1 1; 1 0]; >> C=[1 0; 0 1]; >> D=[0 0; 0 0]; >> step(A,B,C,D,1) >> grid >> title('Step Response Plots (u1=unit step input, u2=0)') il ( l ( i i )) >> >> figure; >> step(A,B,C,D,2) >> grid >> title( Step Response Plots (u1=0, u2=unit step input) ) title('Step (u1=0 input)') Comments: (1) iu is index of input vector . iu=1 means that u1=unit step, u2=0; Iu=2 means that u1=0, u2=unit step (2) compare these four curves with that of Ex.1 h f ih h fE 1 R. Burdisso 6 ME3514 Matlab: Unit Impulse EX. 4 same A, B, C, D as above but response to impulse sys=ss(A B C D) sys=ss(A,B,C,D) impulse(sys) impulse(A,B,C,D) >> A [-1 -1;6 5 0]; A=[-1 -1;6.5 >> 1; B=[1 1 0]; >> C=[1 0; 0 1]; >> D=[0 0; 0 0]; >> sys=ss(A,B,C,D); >> i impulse(sys) l ( ) >> grid R. Burdisso 7 ME3514 Matlab: Arbitrary Input EX. 5 same A, B, C, D as above Problem: if considering both u1=unit step input and u2=unit step input, what is y1 or y2 ? sys=ss(A,B,C,D) u=[u1;u2] [y, t]=lsim(sys, u, t) y1=[1 0] y y1 [1 0]*y' y2=[0 1]*y' Plot (u is input column vector) (need just y1 from output vector y, y : transpose of y) y y': (need just y2 from output vector y) must use plot function >> A=[-1 -1;6.5 0]; >> B=[1 1; 1 0]; B [1 1 0] >> C=[1 0; 0 1]; >> D=[0 0; 0 0]; >> sys=ss(A,B,C,D); >> t=0:0.1:15; >> u=[ones(size(t));ones(size(t))]; >> lsim(sys,u,t); >> [y,t]=lsim(sys,u,t); >> y1=[1 0]*y'; >> y2=[0 1]*y ; 1]*y'; >> plot(t,y1,'b:',t,y2,'g-') >> grid>> R. Burdisso 8 ME3514 Matlab: Arbitrary Input Y2 when u1+u2 Y1 when u1+u2 Comments: Comparing these two curves to Ex.1, y1 here= upper left curve + upper right curve in Ex. 1 y2 here= lower left curve + lower right curve in EX.1 2 h l l ft l i ht i EX 1 R. Burdisso 9 ME3514 Matlab: Arbitrary Input EX. 6 same A, B, C, D as above Problem: only u1=unit step input, u2=0 Y2 when only u1 sys=ss(A,B,C,D) u=[u1;u2] [y. t]=lsim(sys, u, t) y1=[1 0]*y' y [ ] y y2=[0 1]*y' Plot >> A=[-1 -1;6.5 0]; >> B=[1 1; 1 0]; B [1 1 0] Y1 when only u1 >> C=[1 0; 0 1]; >> D=[0 0; 0 0]; >> sys=ss(A,B,C,D); y ( ) >> t=0:0.1:15; >> u=[ones(size(t));zeros(size(t))]; >> [y,t]=lsim(sys,u,t); >> y1=[1 0]*y'; >> y2=[0 1]*y ; 1]*y'; >> plot(t,y1,'b:',t,y2,'g-') >> grid R. Burdisso Y1 when only u1 h l 1 Comments: Are these two curves same to the left two curves in EX.1? 10 ME3514 Matlab: Initial Conditions (using initial) EX. 7 Problem: Ex 5-3 Only IC [y,x,t]=initial(A,B,C,D,x0,t) >> A=[0 1; -2 -3]; >> B=[0; 0]; B [0; >> C=[1 0; 0 1]; >> D=[0; 0]; >> t=0:0.01:6; >> x0=[0.1; 0.05]; >> [ t] i iti l(A B C D 0 t) [y,x,t]=initial(A,B,C,D,x0,t); >> y1=[1 0]*y'; >> y2=[0 1]*y'; >> plot(t,y1,'b:',t,y2,'g-') >> grid g >> title('Response to Initial Condition') >> xlabel('t (s)') >> ylabel('y1 and y2') Comments: (1) Initial conditions column vector x0 (2) If need response to input + IC, how? lsim(sys,u,t,x0) R. Burdisso 11 ME3514 Matlab: Arbitrary Input (using lsim) EX. 8 Problem: Ex. 7 Only IC Response to Initial Condition 0.12 y=lism (A, B, C,D, u, t, x0) This i for state-space model only i is f 0.1 01 0.08 0.06 >> A=[0 1; -2 -3]; >> B=[0; 0]; B [0; >> C=[1 0; 0 1]; >> D=[0; 0]; >> t=0:0.01:6; >> x0=[0.1; 0.05]; >> u=[zeros(size(t))]; [ ( i (t))] >> y=lsim(A,B,C,D,u, t,x0); >> y1=[1 0]*y'; >> y2=[0 1]*y'; >> plot(t,y1,'b:',t,y2,'g-') p ( y y g ) >> grid >> title('Response to Initial Condition') >> xlabel('t (s)') >> ylabel('y1 and y2') y1 and y2 0.04 0.02 0 -0.02 -0.04 0 04 -0.06 0 1 2 3 t (s) 4 5 6 Comments: (1) Same results? (2) If need response to input + IC, how? lsim(sys,u,t,x0) R. Burdisso 12
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