FRTN10 Exercise 3. Specifications and Disturbance Models

Transcription

1 FRTN0 Exercie 3. Specification and Diturbance Model 3. A feedback ytem i hown in Figure 3., in which a firt-order proce if controlled by an I controller. d v r u 2 z C() P() y n Figure 3. Sytem in Problem 3.. a. Verify that the cloed-loop ytem i table. b. Sketch the Bode amplitude diagram of the Gang of Four for the feedback ytem: PC PC = T, P PC = P S, C PC = C S, PC = S Baed on the diagram, anwer the following quetion: Up to approximately what frequency can the proce output track the reference value? Can the feedback ytem reject a contant input load diturbance? What i the maximum amplification from meaurement noie to the control ignal? c. Calculate what effect a inuoidal output diturbance v = in(0.5t) ha on the proce output. d. Extend the block diagram to explicitly model that v i a inuoidal diturbance with frequency 0.5 rad/. 3.2 A continuou-time tochatic proce y(t) ha the power pectrum Φ y (ω). The proce can be repreented by a linear filter G() that ha unit-intenity white noie v a input. Determine the linear filter when a. b. Φ y (ω) = Φ y (ω) = a 2 ω 2 a 2, a> 0 a 2 b 2 (ω 2 a 2 )(ω 2 b 2, a, b> 0 )

2 Exercie A linear ytem with two input and one output ha the tate-pace decription [ ] [ ] ẋ = x u y = [ 0 3] x Auming that u and u 2 are independent, zero-mean, unit intenity white noie procee, calculate the tationary variance of y. 3.4 Conider a miile travelling in the air. It i propelled forward by a jet force u along a horizontal path. The coordinate along the path i z. We aume that there i no gravitational force. The aerodynamic friction force i decribed by a imple model a f = k żv, where v are random variation due to wind and preure change. Combining thi with Newton econd law, m z = u f, where m i the ma of the miile, give the input-output relation z k m ż = m (u v). a. Expre the input-output relation in tate-pace form. b. The diturbance v ha been determined to have the pectral denity Φ v (ω) = k 0 ω 2 a 2, k 0, a> 0 Expand your tate-pace decription o that the diturbance input can be expreed a white noie. 3.5 (*) Thi problem build on Problem 3.4. a. Aume that the poition meaurement i ditorted by an additive error n(t), y(t) = z(t)n(t) Write down the tate-pace equation for the ytem, auming that n(t) i white noie with intenity 0., i.e. Φ n (ω) = 0.. b. Solve the ame problem, thi time with ω 2 Φ n (ω) = 0. ω 2 b 2, b> 0 c. Solve the problem with Φ n (ω) = 0. ω 2 b 2, b> 0 2

3 Exercie (*) Conider an electric motor with the tranfer function G() = from input current to output angle. () There are two different diturbance cenario: (i) (ii) Y() = G()(U() W()) Y() = G()U() W() In both cae, ẇ(t) = v(t), where v(t) i a unit diturbance, e.g., an impule. a. Draw block diagram of the two cae. b. Convert both cae into tate-pace form. c. Give a phyical interpretation of w(t) in both cae. 3

4 Solution to Exercie 3. Specification and Diturbance Model 3. a. The cloed-loop tranfer function from r to y i given by with two LHP pole in. T = () 2 b. The other three cloed-loop tranfer function are P S = () 2, C S = 2 () 2, S = (2) () 2 The four Bode amplitude diagram are plotted below. T P S Magnitude (ab) Magnitude (ab) Frequency (rad/) Frequency (rad/) C S S Magnitude (ab) Magnitude (ab) Frequency (rad/) Frequency (rad/) From the T plot, we ee that z can track r up to approx. ω = rad/. Ye, from the P S plot, we ee that the gain from d to z approache 0 when ω 0. From the C S plot, we ee that the maximum gain from n to u i 2. c. The gain from n to z at ω = 0.5 rad/ i given by S(i0.5) = =

5 d. The inuoidal ignal can be generated by a ytem with pole in ±0.5i, e.g., 2, ee below w d r u 2 v z C() P() y n 3.2 Φ y (ω) i an even, calar, non-negative function. Thu we can factor it into a. Φ y (ω) = G(iω)Φ v (ω)g( iω) where G() ha it pole and zeroe in the left half-plane and Φ v (ω) = (white noie). So the linear filter i Φ y (ω) = b. In the ame way, we get Φ y (ω) = a 2 ω 2 a 2 Φ e(ω) = a 2 b 2 (ω 2 a 2 )(ω 2 b 2 ) Φ e(ω) G() = a a = G() = a iω a a iω a ab (iω a)(iω b) ab ( iω a)( iω b) ab ( a)(b) [ ] π π Let Π x = be the tationary tate covariance E xx T. Since the π 2 π 22 ytem i table (the A-matrix ha eigenvalue λ = 3, λ 2 = 2), Π x i given by the Lyapunov equation [ ][ 5 3 π π π 2 π 22 AΠ x Π x A T BRB T = 0 ] [ ][ π π π 2 π ] [ ] [ ] =

6 with the olution The output ha the variance [ ] Π x = 7 3 E y 2 = E(Cx)(Cx) T = C E(xx T ) C T = CΠ x C T = a. To make a tate-pace decription, we let, e.g., x = z, x 2 = ż = ẋ = x 2, ẋ 2 = m (u k x 2 v). In matrix form: ( ) 0 ẋ = 0 k m z = ( 0) x. x( 0 m ) ( ) 0 u v, m b. We want to find a filter H uch that Φ v (ω) = H(iω) 2 Φ e (ω) Thu H() = k0 a, which i equivalent to v av = k 0 e. Adding a new tate x 3 = v to the tate-pace decription, give ẋ 3 = ax 3 k 0 e and ẋ = 0 k m m x m u 0 0 a k0 e z = ( 0 0) x, Φ e (ω) = 3.5 a. With {A, B, C, N} according to the olution of problem 3.4, we have ẋ = Ax Bu Ne y = Cxn where n ha pectral denity Φ n = 0.. b. A noie ignal with the pecified pectral denity i given by the output of a linear ytem with white noie input that ha pectral denity Φ wn = 0.. The tranfer function of the ytem i G n () = b = b b = b b b 6

7 In tate-pace form thi can be expreed a ẋ 4 = bx 4 bw n n = x 4 w n Combining the noie model with our original ytem give the expanded tate-pace decription: ( ) ( ) ( )( A 0 B N 0 ẋ = x u 0 b 0 0 b ( ) y = C xw n, Φ ω n = 0. e w n ) Note that the diturbance can be decribed uing a tranfer function and white noie of any pectral denity. For intance, it i often convenient to aume white noie with a pectral denity of. In thi cae, the tranfer function of the ytem would be G n () = 0. b The expanded tate pace decription would then need to be adjuted to account for thi. c. Now, the tranfer function of the noie model i G n () = b. In tate-pace form, thi i ẋ 4 bx 4 = w n. The expanded ytem become ( ) ( ) ( )( ) A 0 B N 0 e ẋ = x u 0 b 0 0 y = ( C ) x, Φ ω n = 0. A in ubproblem b, the diturbance can be decribed uing a tranfer function and white noie of any pectral denity. Auming white noie with a pectral denity of, the tranfer function of the ytem would be G n () = 0. b w n 3.6 a. (i) v u w () y 7

8 (ii) v u () w y v(t) i a unit diturbance b. (i) A { }} {{ }} { ẋ = 0 x u 0 v y = ( 0 0) x. }{{} C B 0 (ii) A {}} {{ }} { ẋ = 0 0 x u 0 v y = ( 0 ) x. }{{} C c. (i) w(t) could be an offet current on the input to the motor, and/or a tep diturbance in the load. (ii) In thi cae w(t) could be a meaurement diturbance, i.e., an additive error (contant) in the angle meaurement. It could alo be interpreted a a load diturbance on the proce output. A controller could remove the effect from a load diturbance on the proce output, but not a contant meaurement diturbance, o the interpretation make a difference. B 0 8