# MIMO analysis: loop-at-a-time

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1 MIMO robustness MIMO analysis: loop-at-a-time y 1 y 2 P (s) + + K 2 (s) r 1 r 2 K 1 (s) Plant: P (s) = 1 s 2 + α 2 s α 2 α(s + 1) α(s + 1) s α 2. (take α = 10 in the following numerical analysis) Controller: K 1 = , K 2 = 1 1+α 2 1 α α 1 Control Systems II 11: 1

2 MIMO robustness MIMO analysis: loop-at-a-time y 1 y 2 P (s) + + K 2 (s) r 1 r 2 K 1 (s) Closed-loop transfer function: y1 y 2 = P (s) (I + P (s)k 1 (s)) 1 K 2 (s) = s r1 r 2. r1 r 2 Control Systems II 11: 2

3 MIMO robustness MIMO analysis: loop-at-a-time y 1 y 2 P (s) u 1 e K 2 (s) r 1 r 2 K 1 (s) Loop transfer function: e 1 = 1 s u 1. Control Systems II 11: 3

4 MIMO robustness MIMO analysis More general input perturbation analysis: Perturbations at the plant input (actuator uncertainty) u 1 = (1 + δ 1 ) e 1. u 2 = (1 + δ 2 ) e 2 y1 P (s) + + K 2 (s) r1 y 2 K 1 (s) r 2 = δ1 0 0 δ 2, Control Systems II 11: 4

5 MIMO robustness MIMO analysis = δ1 0 0 δ 2, Take 0.05 (5% uncertainty in each actuator) Step response for y 1 channel = , y 1 (nominal) y 1 (δ 1 =0.05,δ 2 = 0.05) y 2 (δ 1 =0.05,δ 2 = 0.05) Time: seconds Control Systems II 11: 5

6 MIMO robustness MIMO analysis The closed-loop system becomes unstable with = , Closed-loop characteristic polynomial: s 2 + (2 + δ 1 + δ 2 ) s + 1+δ 1 + δ 2 +(α 2 + 1)δ 1 δ 2 =0. If δ 2 = 0 the smallest destabilizing perturbation is δ 1 = 1 Choose instead: δ 1 = 1 α and δ 2 = δ 1 Control Systems II 11: 6

7 MIMO robustness MIMO analysis 2 Stability regions: UNSTABLE STABLE 0.5 δ STABLE -1 UNSTABLE -1.5 UNSTABLE δ 1 Control Systems II 11: 7

8 MIMO robustness Singular value analysis Magnitude !1 σ(l) σ(g) σ(l) σ(g) 10! Frequency [rad/s] σ(l) 1 for all ω. Performance poor in low gain direction. κ(l) 1. Very sensitive to errors in direction. Control Systems II 11: 8

9 MIMO robustness Motivating robustness example no. 2: Distillation Process [3.7.2] Distillation Process [3.7.2] Idealized dynamic model of a distillation column, [ ] G(s) = (6.49) 75s (time is in minutes) Nominal plant: Perturbed plant: y y Time [min] Figure 62: Response with decoupling controller to filtered reference input r 1 =1/(5s +1). Theperturbed plant has 20% gain uncertainty as given by (6.52). Inverse-based controller or equivalently steady-state decoupler with a PI controller (k 1 =0.7) K inv (s) = k 1 s G 1 (s) = k [ ] 1(1 + 75s) s Control Systems II 11: 9

10 MIMO robustness Nominal performance (NP). K inv (s) = k 1 s G 1 (s) = k 1(1 + 75s) s GK inv = K inv G = 0.7 s I [ ] (6.50) first order response with time constant 1.43 (Fig. 62). Nominal performance (NP) achieved with decoupling controller. Control Systems II 11: 10

11 MIMO robustness Robust stability (RS). S = S I = s s +0.7 I; T = T I = In each channel: GM=, PM= s +1 I (6.51) Input gain uncertainty (6.48) with ɛ 1 =0.2 and ɛ 2 = 0.2: u 1 =1.2u 1, u 2 =0.8u 2 (6.52) [ L I(s) = K inv G 1+ɛ1 0 = K inv G 0 1+ɛ 2 [ ] ɛ1 0 s 0 1+ɛ 2 ] = (6.53) Perturbed closed-loop poles are s 1 = 0.7(1 + ɛ 1 ), s 2 = 0.7(1 + ɛ 2 ) (6.54) Closed-loop stability as long as the input gains 1 + ɛ 1 and 1 + ɛ 2 remain positive Robust stability (RS) achieved with respect to input gain errors for the decoupling controller. Control Systems II 11: 11

12 MIMO robustness Robust performance (RP). Performance with model error poor (Fig. 62) SISO: NP+RS RP (within a factor of 2) MIMO: NP+RS RP (arbitrarily large violation) RP is not achieved by the decoupling controller Robustness conclusions [3.7.3] Multivariable plants can display a sensitivity to uncertainty (in this case input uncertainty) which is fundamentally different from what is possible in SISO systems [ ] y 1 Nominal plant: Perturbed plant: y Time [min] Control Systems II 11: 12

13 General control problem formulation (3.8 in S&P) (weighted) exogenous inputs w P (weighted) exogenous outputs z u control signals K v sensed outputs Figure 63: General control configuration for the case with no model uncertainty The overall control objective is to minimize some norm of the transfer function from w to z, for example, the H norm. The controller design problem is then: Find a controller K which based on the information in v, generates a control signal u which counteracts the influence of w on z, therebyminimizingthe closed-loop norm from w to z. Control Systems II 11: 13

14 Calculating the generalized plant The routines in MATLAB for synthesizing H and H 2 optimal controllers assume that the problem is in the general form of Figure 63 Example: One degree-of-freedom feedback control configuration. d r + u + - K G + y y m + + Figure 64: One degree-of-freedom control configuration n (weighted) exogenous inputs w P (weighted) exogenous outputs z u control signals K v sensed outputs Control Systems II 11: 14

15 Calculating the generalized plant r d + u + y - K G + y m + + n Equivalent representation of Figure 64 where the error signal to be minimized is z = y r and the input to the controller is v = r y m w { d r n P G z u v K Figure 65: General control configuration equivalent to Figure 64 Control Systems II 11: 15

16 Calculating the generalized plant w = w 1 w 2 w 3 = d r ; z = e = y r; v = r y m = r y n n (6.55) z = y r = Gu + d r = Iw 1 Iw 2 +0w 3 + Gu v = r y m = r Gu d n = = Iw 1 + Iw 2 Iw 3 Gu and P which represents the transfer function matrix from [ w u] T to [ z v] T is [ ] I I 0 G P = (6.56) I I I G Note that P does not contain the controller. Alternatively, P can be obtained from Figure 65. Control Systems II 11: 16

17 Calculating the generalized plant Remark. In MATLAB we may obtain P via simulink, or we may use the sysic program in the Robust Control toolbox. The code in Table 2 generates the generalized plant P in (6.56) for Figure 64. Table 2: Matlab program to generate P % Uses the Robust Control toolbox systemnames = G ; % G is the SISO plant. inputvar = [d(1);r(1);n(1);u(1)] ; % Consists of vectors w and u. input to G = [u] ; outputvar = [G+d-r; r-g-d-n] ; % Consists of vectors z and v. sysoutname = P ; sysic; Control Systems II 11: 17

18 Calculating the generalized plant w { d r n P G z u v K Control Systems II 11: 18

19 Control Systems II 11: 19 Calculating the generalized plant P u v w { K G z n r d >> [A,B,C,D] = linmod( P_diagram ); >> P = ss(a,b,c,d); >> P = minreal(p);

20 Including weights To get a meaningful controller synthesis problem, for example, in terms of the H or H 2 norms, we generally have to include weights W z and W w in the generalized plant P,see Figure 66. P w W w w z z P W z K Figure 66: General control configuration for the case with no model uncertainty That is, we consider the weighted or normalized exogenous inputs w, andtheweightedornormalized controlled outputs z = W z z. Theweightingmatrices are usually frequency dependent and typically selected such that weighted signals w and z are of magnitude 1, that is, the norm from w to z should be less than 1. Control Systems II 11: 20

21 Example: Stacked S/T/KS problem. Consider an H problem where we want to bound σ(s) (for performance), σ(t )(forrobustnessandtoavoid sensitivity to noise) and σ(ks)(to penalize large inputs). These requirements may be combined into a stacked H problem min K N(K), N = W u KS W T T (6.57) W P S where K is a stabilizing controller. In other words, we have z = Nw and the objective is to minimize the H norm from w to z. Control Systems II 11: 21

22 w W u W T z v K u G W P N Figure 67: Block diagram corresponding to generalized plant in (6.57) z 1 = W u u z 2 = W T Gu z 3 = W P w + W P Gu v = w Gu so the generalized plant P from [ w u] T to [ z v] T is 0 W u I 0 W T G P = (6.58) W P I W P G I G Control Systems II 11: 22

23 6.4.3 Partitioning the generalized plant P [3.8.3] We often partition P as P = [ P11 P 12 P 21 P 22 ] (6.59) so that z = P 11 w + P 12 u (6.60) v = P 21 w + P 22 u (6.61) In Example Stacked S/T/KS problem we get from (6.58) P 11 = [ 0 ] 0 W P I, P 12 = [ ] Wu I W T G W P G (6.62) P 21 = I, P 22 = G (6.63) Note that P 22 has dimensions compatible with the controller K in Figure 66 Control Systems II 11: 23

24 6.4.3 Partitioning the generalized plant P [3.8.3] We often partition P as P = [ P11 P 12 P 21 P 22 ] (6.59) so that z = P 11 w + P 12 u (6.60) v = P 21 w + P 22 u (6.61) In Example Stacked S/T/KS problem we get from (6.58) P 11 = [ 0 ] 0 W P I, P 12 = [ ] Wu I W T G W P G (6.62) P 21 = I, P 22 = G (6.63) Note that P 22 has dimensions compatible with the controller K in Figure 66 Control Systems II 11: 24

25 Calculating the closed-loop: w N z Figure 68: General block diagram for analysis with no uncertainty For analysis of closed-loop performance we may absorb K into the interconnection structure and obtain the system N as shown in Figure 68 where z = Nw (6.64) where N is a function of K. TofindN, firstpartition the generalized plant P as given in (6.59)-(6.61), combine this with the controller equation u = Kv (6.65) and eliminate u and v from equations (6.60), (6.61) and (6.65) to yield z = Nw where N is given by N = P 11 + P 12 K(I P 22 K) 1 P 21 = Fl (P, K) (6.66) Control Systems II 11: 25

26 Calculating the closed-loop: N = P 11 + P 12 K(I P 22 K) 1 P 21 = Fl (P, K) Here F l (P, K) denotesalowerlinear fractional transformation (LFT) of P with K as the parameter. In words, N is obtained from Figure 63 by using K to close a lower feedback loop around P.Since positive feedback is used in the general configuration in Figure 63 the term (I P 22 K) 1 has a negative sign. P w W w w z z P W z w N z K Control Systems II 11: 26

27 Robustness: a more general control structure The general control configuration in Figure 63 may be extended to include model uncertainty. Here the matrix is a block-diagonal matrix that includes all possible perturbations (representing uncertainty) to the system. It is normalized such that 1. SISO example: input multiplicative u y w I I G p w P z + G + u v K Figure 33: Plant with multiplicative uncertainty Control Systems II 11: 27

28 Robustness: a more general control structure u y w N z Figure 71: General block diagram for analysis with uncertainty included Inputs w!! 1 2 System with Actuators, Sensors and Controller Outputs z >! 1! 2! 3! 4!! 3 4 w N z (a) (b) Control Systems II 11: 28

29 Robustness: a more general control structure The block diagram in Figure 70 in terms of P (for synthesis) may be transformed into the block diagram in Figure 71 in terms of N (for analysis) by using K to close a lower loop around P.Thesame lower LFT as found in (6.66) applies, and N = F l (P, K) =P 11 + P 12 K(I P 22 K) 1 P 21 (6.71) To evaluate the perturbed (uncertain) transfer function from external inputs w to external outputs z, weuse to close the upper loop around N (see Figure 71), resulting in an upper LFT: z = F u (N, )w; (6.72) F u (N, ) = N 22 + N 21 (I N 11 ) 1 N 12 (6.73) Control Systems II 11: 29

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