Lecture 6. Chapter 8: Robust Stability and Performance Analysis for MIMO Systems. Eugenio Schuster.

Size: px
Start display at page:

Download "Lecture 6. Chapter 8: Robust Stability and Performance Analysis for MIMO Systems. Eugenio Schuster."

Transcription

1 Lecture 6 Chapter 8: Robust Stability and Performance Analysis for MIMO Systems Eugenio Schuster Mechanical Engineering and Mechanics Lehigh University Lecture 6 p. 1/73

2 6.1 General configuration with uncertainty [8.1] For our robustness analysis we use a system representation in which the uncertain perturbations are pulled out into a block-diagonal matrix, (6.1) = diag{ i } = 1... i... where each i represents a specific source of uncertainty. Lecture 6 p. 2/73

3 3 w 1 z 2 (a) Original system with multiple perturbations Lecture 6 p. 3/73

4 1 2 3 w z N (b) Pulling out the perturbations Figure 1: Rearranging an uncertain system into the N -structure Lecture 6 p. 4/73

5 y u N z w Figure 2: N -structure for robust performance analysis If we also pull out the controller K, we get the generalized plant P, as shown in Figure 3. For analysis of the uncertain system, we use the N -structure in Figure 2. Lecture 6 p. 5/73

6 u y w P z u v K Figure 3: General control configuration (for controller synthesis) Lecture 6 p. 6/73

7 Consider Figure 1 where it is shown how to pull out the perturbation blocks to form and the nominal system N. N is related to P and K by a lower LFT (6.2) N = F l (P,K) = P 11 +P 12 K(I P 22 K) 1 P 21 Similarly, the uncertain closed-loop transfer function from w to z, z = Fw, is related to N and by an upper LFT, (6.3) F = F u (N, ) = N 22 +N 21 (I N 11 ) 1 N 12 Lecture 6 p. 7/73

8 To analyze robust stability of F, we can then arrange the system into the M -structure of Figure 4, where M = N 11 is the transfer function from the output to the input of the perturbations. u y M Figure 4: M -structure for robust stability analysis Lecture 6 p. 8/73

9 6.2 Representing uncertainty [8.2] As usual, each individual perturbation is assumed to be stable and is normalized, (6.4) σ( i (jω)) 1 ω For a complex scalar perturbation we have δ i (jω) 1, ω, and for a real scalar perturbation 1 δ i 1. Since the maximum singular value of a block diagonal matrix is equal to the largest of the maximum singular values of the individual blocks, it then follows for = diag{ i } that (6.5) σ( i (jω)) 1 ω, i 1 Note that has structure, and therefore in the robustness analysis we do not want to allow all s.t. (6.5) is satisfied. Lecture 6 p. 9/73

10 6.2.1 Parametric uncertainty The representation of parametric uncertainty discussed for SISO systems carries straightforward over to MIMO systems. Lecture 6 p. 10/73

11 6.2.2 Unstructured uncertainty We define unstructured uncertainty as the use of a full complex perturbation matrix, usually with dimensions compatible with those of the plant, where at each frequency any (jω) satisfying σ( (jω)) 1 is allowed. Six common forms of unstructured uncertainty are shown in Figure 5. In Figure 5(a), (b) and (c) are shown three feedforward forms; additive uncertainty, multiplicative input uncertainty and multiplicative output uncertainty: (6.6) (6.7) (6.8) Π A : G p = G+E A ; E a = w A a Π I : G p = G(I +E I ); E I = w I I Π O : G p = (I +E O )G; E O = w O O Lecture 6 p. 11/73

12 In Figure 5(d), (e) and (f) are shown three feedback or inverse forms; inverse additive uncertainty, inverse multiplicative input uncertainty and inverse multiplicative output uncertainty: (6.9) (6.10) (6.11) Π ia : G p = G(I E ia G) 1 ; E ia = w ia ia Π ii : G p = G(I E ii ) 1 ; E ii = w ii ii Π io : G p = (I E io ) 1 G; E io = w io io The negative sign in front of the E s does not really matter here since we assume that can have any sign. denotes the normalized perturbation and E the actual perturbation. We have here used scalar weights w, so E = w = w, but sometimes one may want to use matrix weights, E = W 2 W 1 where W 1 and W 2 are given transfer function matrices. Lecture 6 p. 12/73

13 w A A w ia ia G (a) G (d) w I I w ii ii (b) + + G + + (e) G w O O w io io G (c) + + G + + Figure 5: (a) Additive uncertainty, (b) Multiplicative input uncertainty, (c) Multiplicative output uncertainty, (d) Inverse additive uncertainty, (e) Inverse multiplicative input uncertainty, (f) Inverse multiplicative output uncertainty (f) Lecture 6 p. 13/73

14 6.3 ObtainingP,N andm [8.3] WI y I u v K u + + G W P z - w + + Figure 6: System with multiplicative input uncertainty and performance measured at the output Lecture 6 p. 14/73

15 Example 1: System with input uncertainty (Figure 6). We want to derive the generalized plant P in Figure 3 which has inputs [u w u] T and outputs [y z v] T. By writing down the equations or simply by inspecting Figure 6 (remember to break the loop before and after K) we get (6.12) P = [ ] 0 0 WI W P G W P W P G G I G Next, we want to derive the matrix N corresponding to Figure 2. First, partition P to be compatible with K, i.e. Lecture 6 p. 15/73

16 (6.13) (6.14) P 11 = [ 0 0 W P G W P ], P 12 = [ WI W P G P 21 = [ G I], P 22 = G and then find N = F l (P,K) using (6.2). We get (6.15) N = [ WI KG(I +KG) 1 W I K(I +GK) 1 W P G(I +KG) 1 ] W P (I +GK) 1 ] Alternatively, we can derive N directly from Figure 6 by evaluating the closed-loop transfer function from inputs [ u w ] after K). to outputs [ y z ] (without breaking the loop before and Lecture 6 p. 16/73

17 For example, to derive N 12, which is the transfer function from w to y, we start at the output (y ) and move backwards to the input (w) using the MIMO Rule (we first meet W I, then K and we then exit the feedback loop and get the term (I +GK) 1 ). The upper left block, N 11, in (6.15) is the transfer function from u to y. This is the transfer function M needed in Figure 4 for evaluating robust stability. Thus, we have M = W I KG(I +KG) 1 = W I T I. Lecture 6 p. 17/73

18 6.4 Robust stability & performance [8.4] 1. Robust stability (RS) analysis: with a given controller K we determine whether the system remains stable for all plants in the uncertainty set. 2. Robust performance (RP) analysis: if RS is satisfied, we determine how large the transfer function from exogenous inputs w to outputs z may be for all plants in the uncertainty set. In Figure 2, w represents the exogenous inputs (normalized disturbances and references), and z the exogenous outputs (normalized errors). We have z = F( )w, where from (6.3) (6.16) F = F u (N, ) = N 22 +N 21 (I N 11 ) 1 N 12 Lecture 6 p. 18/73

19 We here use the H norm to define performance and require for RP that F( ) 1 for all allowed s. A typical choice is F = w P S p (the weighted sensitivity function), where w P is the performance weight (capital P for performance) and S p represents the set of perturbed sensitivity functions (lower-case p for perturbed). Lecture 6 p. 19/73

20 In terms of the N -structure in Figure 2 our requirements for stability and performance are (6.17) (6.18) (6.19) (6.20) NS def N is internally stable NP def N 22 < 1; and NS RS def F = F u (N, ) is stable, 1; and NS RP def F < 1,, 1; and NS Lecture 6 p. 20/73

21 Remark 1 Allowed perturbations. For simplicity below we will use the shorthand notation (6.21) and max to mean for all s in the set of allowed perturbations, and maximizing over all s in the set of allowed perturbations. By allowed perturbations we mean that the H norm of is less or equal to 1, 1, and that has a specified block-diagonal structure. Lecture 6 p. 21/73

22 6.5 Robust stability of them -structure [8.5] Consider the uncertain N -system in Figure 2 for which the transfer function from w to z is, as in (6.16), given by (6.22) F u (N, ) = N 22 +N 21 (I N 11 ) 1 N 12 Suppose that the system is nominally stable (with = 0), that is, N is stable. We also assume that is stable. Thus, when we have nominal stability (NS), the stability of the system in Figure 2 is equivalent to the stability of the M -structure in Figure 4 where M = N 11. Lecture 6 p. 22/73

23 Theorem 1 Determinant stability condition (Real or complex perturbations). Assume that the nominal system M(s) and the perturbations (s) are stable. Consider the convex set of perturbations, such that if is an allowed perturbation then so is c where c is any real scalar such that c 1. Then the M -system in Figure 4 is stable for all allowed perturbations (we have RS) if and only if (6.23) Nyquist plot of det(i M(s) (s)) does not encircle the origin, (6.24) det(i M(jω) (jω)) 0, ω, (6.25) λ i (M ) 1, i, ω, Lecture 6 p. 23/73

24 Theorem: Generalized (MIMO) Nyquist theorem. Let P ol denote the number of open-loop unstable poles in L(s). The closed-loop system with loop transfer function L(s) and negative feedback is stable if and only if the Nyquist plot of det(i +L(s)) i) makes P ol anti-clockwise encirclements of the origin, and ii) does not pass through the origin. Note By Nyquist plot of det(i +L(s)) we mean the image of det(i +L(s)) as s goes clockwise around the Nyquist D-contour. Lecture 6 p. 24/73

25 6.6 RS for unstructured uncertainty [8.6] Theorem 2 RS for unstructured ( full ) perturbations. Assume that the nominal system M(s) is stable (NS) and that the perturbations (s) are stable. Then the M -system in Figure 4 is stable for all perturbations satisfying 1 (i.e. we have RS) if and only if (6.26) σ(m(jω)) < 1 w M < 1 Lecture 6 p. 25/73

26 Small Gain Theorem. Consider a system with a stable loop transfer function L(s). Then the closed-loop system is stable if (6.27) L(jω) < 1 ω where L denotes any matrix norm satisfying AB A B, for example the singular value σ(l). Lecture 6 p. 26/73

27 6.6.1 Application of the unstructured RS-condition [8.6.1] We will now present necessary and sufficient conditions for robust stability (RS) for each of the six single unstructured perturbations in Figure 5 with (6.28) E = W 2 W 1, 1 To derive the matrix M we simply isolate the perturbation, and determine the transfer function matrix (6.29) M = W 1 M 0 W 2 from the output to the input of the perturbation, where M 0 for each of the six cases becomes (disregarding some negative signs which do not affect the subsequent robustness condition) is given by Lecture 6 p. 27/73

28 (6.30) G p = G+E A : M 0 = K(I +GK) 1 = KS G p = G(I +E I ) : M 0 = K(I +GK) 1 G = T I G p = (I +E O )G : M 0 = GK(I +GK) 1 = T G p = G(I E ia G) 1 : M 0 = (I +GK) 1 G = SG G p = G(I E ii ) 1 : M 0 = (I +KG) 1 = S I G p = (I E io ) 1 G : M 0 = (I +GK) 1 = S Theorem 2 then yields (6.31) RS W 1 M 0 W 2 (jω) < 1, w For instance, from second equation in (6.30) and (6.31) we get for multiplicative input uncertainty with a scalar weight: (6.32) RS G p = G(I +w I I ), I 1 w I T I < 1 Lecture 6 p. 28/73

29 Note that the SISO-condition (5.15) (6.33) RS = T < 1/ w I, ω follows as a special case of (6.32). Similarly, (5.20) (6.34) RS S < 1/ w oi, ω follows as a special case of the inverse multiplicative output uncertainty in (6.30): (6.35) RS G p = (I w io io ) 1 G, io 1 w io S < 1 In general, the unstructured uncertainty descriptions in terms of a single perturbation are not tight. Lecture 6 p. 29/73

30 6.7 RS with structured uncertainty [8.7] SAME UNCERTAINTY D D 1 D 1 M D NEW M: DMD 1 Figure 7: Use of block-diagonal scalings, D = D Lecture 6 p. 30/73

31 Consider now the presence of structured uncertainty, where = diag{ i } is block-diagonal. To test for robust stability we rearrange the system into the M -structure and we have from (6.26) (6.36) RS if σ(m(jω)) < 1, ω We have here written if rather than if and only if since this condition is only necessary for RS when has no structure (full-block uncertainty). To reduce concervativism introduce the block-diagonal scaling matrix (6.37) D = diag{d i I i } where d i is a scalar and I i is an identity matrix of the same dimension as the i th perturbation block, i (Figure 7). This clearly has no effect on stability. Lecture 6 p. 31/73

32 (6.38) RS if σ(dmd 1 ) < 1, ω This applies for any D in (6.37), and therefore the most improved (least conservative) RS-condition is obtained by minimizing at each frequency the scaled singular value, and we have (6.39) RS if min D(ω) D σ(d(ω)m(jω)d(ω) 1 ) < 1, ω where D is the set of block-diagonal matrices whose structure is compatible to that of, i.e, D = D. Lecture 6 p. 32/73

33 6.8 The structured singular value [8.8] The structured singular value (denoted Mu, mu, SSV or µ) is a function which provides a generalization of the singular value, σ, and the spectral radius, ρ. We will use µ to get necessary and sufficient conditions for robust stability and also for robust performance. How is µ defined? A simple statement is: Find the smallest structured (measured in terms of σ( )) which makes det(i M ) = 0; then µ(m) = 1/ σ( ). Mathematically, (6.40) µ(m) 1 = min { σ( ) det(i M ) = 0 for structured } Lecture 6 p. 33/73

34 Clearly, µ(m) depends not only on M but also on the allowed structure for. This is sometimes shown explicitly by using the notation µ (M). Remark. For the case where is unstructured (a full matrix), the smallest which yields singularity has σ( ) = 1/ σ(m), and we have µ(m) = σ(m). Lecture 6 p. 34/73

35 Example 1 Full perturbation ( is unstructured). Consider (6.41) = The perturbation (6.42) M = [ [ ][ ][ = 1 σ 1 v 1 u H 1 = = 0 0 [ ] [ = ] H ] [ ] = with σ( ) = 1/ σ(m) = 1/3.162 = makes det(i M ) = 0. Thus µ(m) = when is a full matrix. ] Lecture 6 p. 35/73

36 Note that the perturbation in (6.42) is a full matrix. If we restrict to be diagonal then we need a larger perturbation to make det(i M ) = 0. This is illustrated next. Example 1 continued. Diagonal perturbation ( is structured). For the matrix M in (6.41), the smallest diagonal which makes det(i M ) = 0 is [ 1 0 ] (6.43) = with σ( ) = Thus µ(m) = 3 when is a diagonal matrix. Lecture 6 p. 36/73

37 Definition 1 Structured Singular Value. Let M be a given complex matrix and let = diag{ i } denote a set of complex matrices with σ( ) 1 and with a given block-diagonal structure (in which some of the blocks may be repeated and some may be restricted to be real). The real non-negative function µ(m), called the structured singular value, is defined by µ(m) = 1 (6.44) min{k m det(i k m M ) = 0, σ( ) 1} If no such structured exists then µ(m) = 0. A value of µ = 1 means that there exists a perturbation with σ( ) = 1 which is just large enough to make I M singular. A larger value of µ is bad as it means that a smaller perturbation makes I M singular, whereas a smaller value of µ is good. Lecture 6 p. 37/73

38 Properties of the structured singular value: Check Sections 8.8.1, and in the book. Lecture 6 p. 38/73

39 6.9 RS - Structured uncertainty [8.9] Consider stability of the M -structure in Figure 4 for the case where is a set of norm-bounded block-diagonal perturbations. From the determinant stability condition: (6.45) det(i M(jω) (jω)) 0, ω,, σ( (jω)) 1 ω This is just a yes/no" condition. To find the factor k m by which the system is robustly stable, we scale the uncertainty by k m, and look for the smallest k m that yields borderline instability," namely (6.46) det(i k m M(jω) (jω)) = 0 By definition, this value is k m = 1/µ(M). We obtain the following necessary and sufficient condition for stability. Lecture 6 p. 39/73

40 Theorem 3 RS for block-diagonal perturbations (real or complex). Assume that the nominal system M and the perturbations are stable. Then the M -system in Figure 4 is stable for all allowed perturbations with σ( ) 1, ω, if and only if (6.47) µ(m(jω)) < 1, ω Condition (6.47) for robust stability may be rewritten as (6.48) RS µ(m(jω)) σ( (jω)) < 1, ω which may be interpreted as a generalized small gain theorem that also takes into account the structure of. Lecture 6 p. 40/73

41 Example: RS with diagonal input uncertainty Consider robust stability of the feedback system in Figure 6 for the case when the multiplicative input uncertainty is diagonal. A nominal 2 2 plant and the controller (which represents PI-control of a distillation process using the DV-configuration) is given by [ ] G(s) = τs ; [ ] K(s) = 1+τs (6.49) s (time in minutes). The controller results in a nominally stable system with acceptable performance. Assume there is complex multiplicative uncertainty in each manipulated input of magnitude Lecture 6 p. 41/73

42 (6.50) w I (s) = s s+1 On rearranging the block diagram to match the M -structure in Figure 4 we get M = w I KG(I +KG) 1 = w I T I (recall (6.15)), and the RS-condition µ(m) < 1 in Theorem 3 yields (6.51) RS µ I (T I ) < 1 w I (jω) ω, I = [ δ1 δ 2 ] This condition is shown graphically in Figure 8 so the system is robustly stable. Also in Figure 8, σ(t I ) can be seen to be larger than 1/ w I (jω) over a wide frequency range. This shows that the system would be unstable for full-block input uncertainty ( I full). Lecture 6 p. 42/73

43 Magnitude σ(t 1 I ) w I (jω) µ I (T I ) Frequency Figure 8: Robust stability for diagonal input uncertainty is guaranteed sinceµ I (T I ) < 1/ w I, ω. The use of unstructured uncertainty and σ(t I ) is conservative Lecture 6 p. 43/73

44 Example: RS of Spinning Satellite [3.7.1] Angular velocity control of a satellite spinning about one of its principal axes: (6.52) G(s) = 1 s 2 +a 2 [ s a 2 a(s+1) a(s+1) s a 2 ] ; a = 10 A minimal, state-space realization, G = C(sI A) 1 B +D, is Lecture 6 p. 44/73

45 (6.53) [ A B C D ] = 0 a 1 0 a a 0 0 a Poles at s = ±ja For stabilization: (6.54) K = I T(s) = GK(I +GK) 1 = 1 s+1 [ ] 1 a a 1 Lecture 6 p. 45/73

46 Nominal stability (NS). Two closed loop poles at s = 1 and [ ] [ ] [ ] 0 a 1 a 1 0 A cl = A BKC = = a 0 a Nominal performance (NP). Figure 9(a) Magnitude σ(g) σ(g) Frequency [rad/s] Figure 9: Singular values - Spinning satellite (6.52) Lecture 6 p. 46/73

47 σ(l) 1 ω poor performance in low gain direction g 12,g 21 large strong interaction Robust stability (RS). Check stability: one loop at a time z 1 w 1 G Figure 10: Checking stability margins one-loop-ata-time K Lecture 6 p. 47/73

48 (6.55) z 1 w 1 = L1 (s) = 1 s Good Robustness? NO GM =,PM = 90 Consider perturbation in each feedback channel (6.56) u 1 = (1+ǫ 1 )u 1, u 2 = (1+ǫ 2 )u 2 B = [ ] 1+ǫ ǫ 2 Closed-loop state matrix: [ ] A 0 a cl = A B KC = a 0 [ ][ ] 1+ǫ1 0 1 a 0 1+ǫ 2 a 1 Lecture 6 p. 48/73

49 Characteristic polynomial: det(si A cl ) = s2 +(2+ǫ 1 +ǫ 2 ) s+ }{{} a 1 +1+ǫ 1 +ǫ 2 +(a 2 +1)ǫ 1 ǫ }{{} 2 a 0 Stability for ( 1 < ǫ 1 <,ǫ 2 = 0) and (ǫ 1 = 0, 1 < ǫ 2 < ) (GM= ) But only small simultaneous changes in the two channels: for example, let ǫ 1 = ǫ 2, then the system is unstable (a 0 < 0) for 1 ǫ 1 > a Summary. Checking single-loop margins is inadequate for MIMO problems. Lecture 6 p. 49/73

50 µ(t) = σ(t) irrespective of the structure of the COMPLEX multiplicative uncertainty perturbation (full block, diagonal, repeated scalar). We can tolerate more than 100% uncertainty above 10 rad/sec. At low frequencies we have µ(t) 10, so to guarantee RS we need less than 10% uncertainty. This confirms previous results showing that real perturbations δ 1 = 0.1 and δ 2 = 0.1 yield instability. Then the use of complex rather than real perturbations is not conservative in this case, at least for diagonal uncertainty. For repeated scalar uncertainty, there is a difference between real and complex uncertainties. Lecture 6 p. 50/73

51 6.10 Robust performance [8.10] With an H performance objective, the RP-condition is identical to a RS-condition with an additional perturbation block. In Figure 11 step B is the key step. P (where capital P denotes Performance) is always a full matrix. It is a fictitious uncertainty block representing the H performance specification. Lecture 6 p. 51/73

52 Testing RP usingµ[8.10.1] Theorem 4 Robust performance. Rearrange the uncertain system into the N -structure of Figure 11. Assume nominal stability such that N is (internally) stable. Then (6.57) RP def F = F u (N, ) < 1, 1 = (N(jω)) < 1, w µ where µ is computed with respect to the structure [ ] 0 (6.58) = 0 P and P is a full complex perturbation with the same dimensions as F T. Lecture 6 p. 52/73

53 RP STEP A F < 1, 1 STEP B P F STEP C P N STEP D P N µ (N) < 1, w P 1 is RS, 1 (RS theorem) P 1 is RS, 1 is RS, 1 Lecture 6 p. 53/73

54 Summary ofµ-conditions for NP, RS, RP [8.10.2] Rearrange the uncertain system into the N - structure, where the block- diagonal perturbations satisfy 1. Introduce F = F u (N, ) = N 22 +N 21 (I N 11 ) 1 N 12 and let the performance requirement (RP) be F 1 for all allowable perturbations. Then we have: (6.59) (6.60) (6.61) (6.62) NS = N (internally) stable NP = σ(n 22 ) = µ P (N 22 ) < 1, ω, and NS RS = µ (N 11 ) < 1, ω, and NS [ ] 0 RP = µ (N) < 1, ω, =, 0 P and NS Lecture 6 p. 54/73

55 Here is a block-diagonal matrix (its detailed structure depends on the uncertainty we are representing), whereas P always is a full complex matrix. Note that nominal stability (NS) must be tested separately in all cases. What does µ = 1.1 for RP mean? Our RP-requirement would be satisfied exactly if we reduced both the performance requirement and the uncertainty by a factor of 1.1. To find the worst-case weighted performance for a given uncertainty, one needs to keep the magnitude of the perturbations fixed ( σ( ) 1). (6.63) σ( ) 1 σ(f max u(n, )(jω)) = µ s (N(jω)) Lecture 6 p. 55/73

56 To find µ s numerically, we scale the performance part of N by a factor k m = 1/µ s and iterate on k m until µ = 1. That is, at each frequency skewed-µ is the value µ s (N) which solves [ ] I 0 (6.64) µ(k m N) = 1, K m = 0 1/µ s Note that µ underestimates how bad or good the actual worst-case performance is. This follows because µ s (N) is always further from 1 than µ(n). Lecture 6 p. 56/73

57 6.11 Application: RP - input uncertainty [8.11] - W I K + G + W P I + w + z I w N z Figure 12: Robust performance of system with input uncertainty Lecture 6 p. 57/73

58 Interconnection matrix [8.11.1] On rearranging the system into the N -structure, as shown in Figure 12, we get [ ] wi T N = I w I KS (6.65) w P SG w P S where T I = KG(I +KG) 1, S = (I +GK) 1 and for simplicity we have omitted the negative signs in the 1,1 and 1,2 blocks of N, since µ(n) = µ(u N) with unitary U = [ I 0 0 I ]. For a given controller K we can now test for NS, NP, RS and RP using (6.59)-(6.63). Here = I may be a full or diagonal matrix (depending on the physical situation). Lecture 6 p. 58/73

59 RP with input uncertainty for SISO system [8.11.2] For a SISO system, conditions (6.59)-(6.63) with N as in (6.65) become (6.66) (6.67) (6.68) (6.69) NS = S, SG, KS and T I are stable NP = w P S < 1, ω RS = w I T I < 1, ω RP = w P S + w I T I < 1, ω Lecture 6 p. 59/73

60 RP for2 2distillation process [8.11.3] Consider again the distillation process example from Chapter 3 (Motivating Example No. 2) and the corresponding inverse-based controller: [ G(s) = ; K(s) = s+1 s G(s) 1 (6.70) The controller provides a nominally decoupled system with (6.71) where L = l I, S = ǫi and T = ti l = 0.7 s,ǫ = 1 1+l = s s+0.7, t = 1 ǫ = 0.7 s+0.7 = s+1 ] Lecture 6 p. 60/73

61 We have used ǫ for the nominal sensitivity in each loop to distinguish it from the Laplace variable s. Weights for uncertainty and performance: (6.72) w I (s) = s s+1 ; w P(s) = s/ s The weight w I (s) may approximately represent a 20% gain error and a neglected time delay of 0.9 min. w I (jω) levels off at 2 (200% uncertainty) at high frequencies. The performance weight w P (s) specifies integral action, a closed-loop bandwidth of about 0.05 [rad/min] (which is relatively slow in the presence of an allowed time delay of 0.9 min) and a maximum peak for σ(s) of M s = 2. Lecture 6 p. 61/73

62 6 4 RP µ 2 RS NP Frequency Figure 13: µ-plots for distillation process with decoupling controller Lecture 6 p. 62/73

63 NS Yes. NP With the decoupling controller we have σ(n 22 ) = σ(w P S) = s/ s+0.7 (dashed-dot line in Figure 13 NP is OK.) RS Since in this case w I T I = w I T is a scalar times the identity matrix, we have, independent of the structure of I, that µ I (w I T I ) = w I t = 0.2 5s+1 (0.5s+1)(1.43s+1) and we see from the dashed line in Figure 13 that RS is OK. RP Poor. Lecture 6 p. 63/73

64 6.12 µ-synthesis anddk-iteration [8.12] The structured singular value µ is a very powerful tool for the analysis of robust performance with a given controller. However, one may also seek to find the controller that minimizes a given µ-condition: this is the µ-synthesis problem: min K µ(n) DK-iteration [8.12.1] At present there is no direct method to synthesize a µ-optimal controller. However, for complex perturbations a method known as DK-iteration is available. It combines H -synthesis and µ-analysis, and often yields good results. The starting point is the upper bound on µ in terms of the scaled singular value µ(n) min D D σ(dnd 1 ) Lecture 6 p. 64/73

65 The structured singular value µ for complex perturbations is bounded by (6.73) (6.74) ρ(m) µ(m) σ(m) = δi (δ is a complex scalar): µ(m) = ρ(m) (6.75) is a full complex matrix: µ(m) = σ(m) Consider any matrix D which commutes with : that is D = D. Then (6.76) µ(dm) = µ(md) and µ(dmd 1 ) = µ(m) Lecture 6 p. 65/73

66 Improved upper bound. Define D to be the set of matrices D which commute with (i.e., satisfy D = D). Then (6.77) µ(d) min D D σ(dmd 1 ) This optimization is convex in D, i.e., has only one global minimum. The inequality is indeed an equality if there are three or fewer blocks in. Numerical evidence suggests that the bound is tight. Lecture 6 p. 66/73

67 The idea is to find the controller that minimizes the peak value over frequency of this upper bound, namely (6.78) min (min K D D DN(K)D 1 ) by alternating between minimizing DN(K)D 1 with respect to either K or D (while holding the other fixed). 1. K-step. Synthesize an H controller for the scaled problem, min K DN(K)D 1 with fixed D(s). 2. D-step. Find D(jω) to minimize at each frequency σ(dnd 1 (jω)) with fixed N. 3. Fit the magnitude of each element of D(jω) to a stable and minimum phase transfer function D(s) and go to Step 1. Lecture 6 p. 67/73

68 Example: µ-synthesis withdk-iteration [8.12.4] Simplified distillation process (6.79) G(s) = 1 75s+1 [ ] The uncertainty weight w I I and performance weight w P I are given in (6.72), and are shown graphically in Figure 14. The objective is to minimize the peak value of µ (N), where N is given in (6.65) and = diag{ I, P }. We will consider diagonal input uncertainty (which is always present in any real problem), so I is a 2 2 diagonal matrix. P is a full 2 2 matrix representing the performance specification. Lecture 6 p. 68/73

69 10 2 Magnitude w P w I Frequency Figure 14: Uncertainty and performance weights. Notice that there is a frequency range ( window ) where both weights are less than one in magnitude. Lecture 6 p. 69/73

70 Iteration No. 1. Step 1: With the initial scalings, D 0 = I, the H software produced a 6 state controller (2 states from the plant model and 2 from each of the weights) with an H norm of γ = Step 2: The upper µ-bound gave the µ-curve shown as curve Iter. 1 in Figure 15, corresponding to a peak value of µ= Step 3: The frequency-dependent d 1 (ω) and d 2 (ω) from Step 2 were each fitted using a 4th order transfer function. d 1 (w) and the fitted 4th-order transfer function (dotted line) are shown in Figure 16 and labelled Iter. 1. Iteration No. 2. Step 1: With the 8 state scaling D 1 (s) the H software gave a 22 state controller and D 1 N(D 1 ) 1 = Iteration No. 3. Step 1: With the scalings D 2 (s) the H norm was only slightly reduced from to Lecture 6 p. 70/73

71 1 Iter. 1 Iter. 2 Iter. 3 Optimal µ Frequency [rad/min] Figure 15: Change in µ during DK-iteration Lecture 6 p. 71/73

72 Magnitude 10 0 Initial th order fit Iter. 1 Optimal Iter Frequency [rad/min] Figure 16: Change in D-scale d 1 during DKiteration Lecture 6 p. 72/73

73 1 0.8 y y Time [min] Figure 17: Setpoint response for µ- optimal controller K 3. Solid line: nominal plant. Dashed line: uncertain plant G 3 Lecture 6 p. 73/73

ROBUST STABILITY AND PERFORMANCE ANALYSIS* [8 # ]

ROBUST STABILITY AND PERFORMANCE ANALYSIS* [8 # ] ROBUST STABILITY AND PERFORMANCE ANALYSIS* [8 # ] General control configuration with uncertainty [8.1] For our robustness analysis we use a system representation in which the uncertain perturbations are

More information

FEL3210 Multivariable Feedback Control

FEL3210 Multivariable Feedback Control FEL3210 Multivariable Feedback Control Lecture 6: Robust stability and performance in MIMO systems [Ch.8] Elling W. Jacobsen, Automatic Control Lab, KTH Lecture 6: Robust Stability and Performance () FEL3210

More information

Lecture 7 (Weeks 13-14)

Lecture 7 (Weeks 13-14) Lecture 7 (Weeks 13-14) Introduction to Multivariable Control (SP - Chapters 3 & 4) Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 7 (Weeks 13-14) p.

More information

FEL3210 Multivariable Feedback Control

FEL3210 Multivariable Feedback Control FEL3210 Multivariable Feedback Control Lecture 5: Uncertainty and Robustness in SISO Systems [Ch.7-(8)] Elling W. Jacobsen, Automatic Control Lab, KTH Lecture 5:Uncertainty and Robustness () FEL3210 MIMO

More information

Model Uncertainty and Robust Stability for Multivariable Systems

Model Uncertainty and Robust Stability for Multivariable Systems Model Uncertainty and Robust Stability for Multivariable Systems ELEC 571L Robust Multivariable Control prepared by: Greg Stewart Devron Profile Control Solutions Outline Representing model uncertainty.

More information

MIMO analysis: loop-at-a-time

MIMO analysis: loop-at-a-time 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:

More information

Structured Uncertainty and Robust Performance

Structured Uncertainty and Robust Performance Structured Uncertainty and Robust Performance ELEC 571L Robust Multivariable Control prepared by: Greg Stewart Devron Profile Control Solutions Outline Structured uncertainty: motivating example. Structured

More information

Chapter Robust Performance and Introduction to the Structured Singular Value Function Introduction As discussed in Lecture 0, a process is better desc

Chapter Robust Performance and Introduction to the Structured Singular Value Function Introduction As discussed in Lecture 0, a process is better desc Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology c Chapter Robust

More information

Structured singular value and µ-synthesis

Structured singular value and µ-synthesis Structured singular value and µ-synthesis Robust Control Course Department of Automatic Control, LTH Autumn 2011 LFT and General Framework z P w z M w K z = F u (F l (P,K), )w = F u (M, )w. - Last week

More information

Lecture 3. Chapter 4: Elements of Linear System Theory. Eugenio Schuster. Mechanical Engineering and Mechanics Lehigh University.

Lecture 3. Chapter 4: Elements of Linear System Theory. Eugenio Schuster. Mechanical Engineering and Mechanics Lehigh University. Lecture 3 Chapter 4: Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 3 p. 1/77 3.1 System Descriptions [4.1] Let f(u) be a liner operator, u 1 and u

More information

Frequency methods for the analysis of feedback systems. Lecture 6. Loop analysis of feedback systems. Nyquist approach to study stability

Frequency methods for the analysis of feedback systems. Lecture 6. Loop analysis of feedback systems. Nyquist approach to study stability Lecture 6. Loop analysis of feedback systems 1. Motivation 2. Graphical representation of frequency response: Bode and Nyquist curves 3. Nyquist stability theorem 4. Stability margins Frequency methods

More information

6.241 Dynamic Systems and Control

6.241 Dynamic Systems and Control 6.241 Dynamic Systems and Control Lecture 17: Robust Stability Readings: DDV, Chapters 19, 20 Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology April 6, 2011 E. Frazzoli

More information

Uncertainty and Robustness for SISO Systems

Uncertainty and Robustness for SISO Systems Uncertainty and Robustness for SISO Systems ELEC 571L Robust Multivariable Control prepared by: Greg Stewart Outline Nature of uncertainty (models and signals). Physical sources of model uncertainty. Mathematical

More information

Control System Design

Control System Design ELEC ENG 4CL4: Control System Design Notes for Lecture #11 Wednesday, January 28, 2004 Dr. Ian C. Bruce Room: CRL-229 Phone ext.: 26984 Email: ibruce@mail.ece.mcmaster.ca Relative Stability: Stability

More information

Robust Performance Example #1

Robust Performance Example #1 Robust Performance Example # The transfer function for a nominal system (plant) is given, along with the transfer function for one extreme system. These two transfer functions define a family of plants

More information

A brief introduction to robust H control

A brief introduction to robust H control A brief introduction to robust H control Jean-Marc Biannic System Control and Flight Dynamics Department ONERA, Toulouse. http://www.onera.fr/staff/jean-marc-biannic/ http://jm.biannic.free.fr/ European

More information

Classify a transfer function to see which order or ramp it can follow and with which expected error.

Classify a transfer function to see which order or ramp it can follow and with which expected error. Dr. J. Tani, Prof. Dr. E. Frazzoli 5-059-00 Control Systems I (Autumn 208) Exercise Set 0 Topic: Specifications for Feedback Systems Discussion: 30.. 208 Learning objectives: The student can grizzi@ethz.ch,

More information

Chapter Stability Robustness Introduction Last chapter showed how the Nyquist stability criterion provides conditions for the stability robustness of

Chapter Stability Robustness Introduction Last chapter showed how the Nyquist stability criterion provides conditions for the stability robustness of Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology c Chapter Stability

More information

Robust Control 3 The Closed Loop

Robust Control 3 The Closed Loop Robust Control 3 The Closed Loop Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University /2/2002 Outline Closed Loop Transfer Functions Traditional Performance Measures Time

More information

Introduction. Performance and Robustness (Chapter 1) Advanced Control Systems Spring / 31

Introduction. Performance and Robustness (Chapter 1) Advanced Control Systems Spring / 31 Introduction Classical Control Robust Control u(t) y(t) G u(t) G + y(t) G : nominal model G = G + : plant uncertainty Uncertainty sources : Structured : parametric uncertainty, multimodel uncertainty Unstructured

More information

Robust fixed-order H Controller Design for Spectral Models by Convex Optimization

Robust fixed-order H Controller Design for Spectral Models by Convex Optimization Robust fixed-order H Controller Design for Spectral Models by Convex Optimization Alireza Karimi, Gorka Galdos and Roland Longchamp Abstract A new approach for robust fixed-order H controller design by

More information

Analysis of SISO Control Loops

Analysis of SISO Control Loops Chapter 5 Analysis of SISO Control Loops Topics to be covered For a given controller and plant connected in feedback we ask and answer the following questions: Is the loop stable? What are the sensitivities

More information

Robust Control. 2nd class. Spring, 2018 Instructor: Prof. Masayuki Fujita (S5-303B) Tue., 17th April, 2018, 10:45~12:15, S423 Lecture Room

Robust Control. 2nd class. Spring, 2018 Instructor: Prof. Masayuki Fujita (S5-303B) Tue., 17th April, 2018, 10:45~12:15, S423 Lecture Room Robust Control Spring, 2018 Instructor: Prof. Masayuki Fujita (S5-303B) 2nd class Tue., 17th April, 2018, 10:45~12:15, S423 Lecture Room 2. Nominal Performance 2.1 Weighted Sensitivity [SP05, Sec. 2.8,

More information

CHAPTER 5 ROBUSTNESS ANALYSIS OF THE CONTROLLER

CHAPTER 5 ROBUSTNESS ANALYSIS OF THE CONTROLLER 114 CHAPTER 5 ROBUSTNESS ANALYSIS OF THE CONTROLLER 5.1 INTRODUCTION Robust control is a branch of control theory that explicitly deals with uncertainty in its approach to controller design. It also refers

More information

Input-output Controllability Analysis

Input-output Controllability Analysis Input-output Controllability Analysis Idea: Find out how well the process can be controlled - without having to design a specific controller Note: Some processes are impossible to control Reference: S.

More information

ROBUST CONTROL THROUGH ROBUSTNESS ENHANCEMENT

ROBUST CONTROL THROUGH ROBUSTNESS ENHANCEMENT ROBUST CONTROL THROUGH ROBUSTNESS ENHANCEMENT Control Configurations And Two-Step Design Approaches Carles Pedret i Ferré Submitted in partial fulfillment of the requirements for the degree of Doctor Engineer

More information

Chapter 9 Robust Stability in SISO Systems 9. Introduction There are many reasons to use feedback control. As we have seen earlier, with the help of a

Chapter 9 Robust Stability in SISO Systems 9. Introduction There are many reasons to use feedback control. As we have seen earlier, with the help of a Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology c Chapter 9 Robust

More information

Richiami di Controlli Automatici

Richiami di Controlli Automatici Richiami di Controlli Automatici Gianmaria De Tommasi 1 1 Università degli Studi di Napoli Federico II detommas@unina.it Ottobre 2012 Corsi AnsaldoBreda G. De Tommasi (UNINA) Richiami di Controlli Automatici

More information

Return Difference Function and Closed-Loop Roots Single-Input/Single-Output Control Systems

Return Difference Function and Closed-Loop Roots Single-Input/Single-Output Control Systems Spectral Properties of Linear- Quadratic Regulators Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 2018! Stability margins of single-input/singleoutput (SISO) systems! Characterizations

More information

Chapter 7 Interconnected Systems and Feedback: Well-Posedness, Stability, and Performance 7. Introduction Feedback control is a powerful approach to o

Chapter 7 Interconnected Systems and Feedback: Well-Posedness, Stability, and Performance 7. Introduction Feedback control is a powerful approach to o Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology c Chapter 7 Interconnected

More information

Topic # Feedback Control

Topic # Feedback Control Topic #4 16.31 Feedback Control Stability in the Frequency Domain Nyquist Stability Theorem Examples Appendix (details) This is the basis of future robustness tests. Fall 2007 16.31 4 2 Frequency Stability

More information

MAE 143B - Homework 9

MAE 143B - Homework 9 MAE 43B - Homework 9 7.2 2 2 3.8.6.4.2.2 9 8 2 2 3 a) G(s) = (s+)(s+).4.6.8.2.2.4.6.8. Polar plot; red for negative ; no encirclements of, a.s. under unit feedback... 2 2 3. 4 9 2 2 3 h) G(s) = s+ s(s+)..2.4.6.8.2.4

More information

Robust Control of Flexible Motion Systems: A Literature Study

Robust Control of Flexible Motion Systems: A Literature Study Robust Control of Flexible Motion Systems: A Literature Study S.L.H. Verhoeven DCT Report 2009.006 APT536-09-6288 Supervisors: Dr. ir. J.J.M. van Helvoort Dr. ir. M.M.J. van de Wal Ir. T.A.E. Oomen Prof.ir.

More information

DESIGN USING TRANSFORMATION TECHNIQUE CLASSICAL METHOD

DESIGN USING TRANSFORMATION TECHNIQUE CLASSICAL METHOD 206 Spring Semester ELEC733 Digital Control System LECTURE 7: DESIGN USING TRANSFORMATION TECHNIQUE CLASSICAL METHOD For a unit ramp input Tz Ez ( ) 2 ( z ) D( z) G( z) Tz e( ) lim( z) z 2 ( z ) D( z)

More information

Today (10/23/01) Today. Reading Assignment: 6.3. Gain/phase margin lead/lag compensator Ref. 6.4, 6.7, 6.10

Today (10/23/01) Today. Reading Assignment: 6.3. Gain/phase margin lead/lag compensator Ref. 6.4, 6.7, 6.10 Today Today (10/23/01) Gain/phase margin lead/lag compensator Ref. 6.4, 6.7, 6.10 Reading Assignment: 6.3 Last Time In the last lecture, we discussed control design through shaping of the loop gain GK:

More information

APPLICATION OF D-K ITERATION TECHNIQUE BASED ON H ROBUST CONTROL THEORY FOR POWER SYSTEM STABILIZER DESIGN

APPLICATION OF D-K ITERATION TECHNIQUE BASED ON H ROBUST CONTROL THEORY FOR POWER SYSTEM STABILIZER DESIGN APPLICATION OF D-K ITERATION TECHNIQUE BASED ON H ROBUST CONTROL THEORY FOR POWER SYSTEM STABILIZER DESIGN Amitava Sil 1 and S Paul 2 1 Department of Electrical & Electronics Engineering, Neotia Institute

More information

An Overview on Robust Control

An Overview on Robust Control Advanced Control An Overview on Robust Control P C Scope Keywords Prerequisites allow the student to assess the potential of different methods in robust control without entering deep into theory. Sensitize

More information

EL2520 Control Theory and Practice

EL2520 Control Theory and Practice So far EL2520 Control Theory and Practice r Fr wu u G w z n Lecture 5: Multivariable systems -Fy Mikael Johansson School of Electrical Engineering KTH, Stockholm, Sweden SISO control revisited: Signal

More information

CDS 101/110a: Lecture 10-1 Robust Performance

CDS 101/110a: Lecture 10-1 Robust Performance CDS 11/11a: Lecture 1-1 Robust Performance Richard M. Murray 1 December 28 Goals: Describe how to represent uncertainty in process dynamics Describe how to analyze a system in the presence of uncertainty

More information

Lecture 9: Input Disturbance A Design Example Dr.-Ing. Sudchai Boonto

Lecture 9: Input Disturbance A Design Example Dr.-Ing. Sudchai Boonto Dr-Ing Sudchai Boonto Department of Control System and Instrumentation Engineering King Mongkuts Unniversity of Technology Thonburi Thailand d u g r e u K G y The sensitivity S is the transfer function

More information

Control Systems I. Lecture 9: The Nyquist condition

Control Systems I. Lecture 9: The Nyquist condition Control Systems I Lecture 9: The Nyquist condition adings: Guzzella, Chapter 9.4 6 Åstrom and Murray, Chapter 9.1 4 www.cds.caltech.edu/~murray/amwiki/index.php/first_edition Emilio Frazzoli Institute

More information

Stability Margin Based Design of Multivariable Controllers

Stability Margin Based Design of Multivariable Controllers Stability Margin Based Design of Multivariable Controllers Iván D. Díaz-Rodríguez Sangjin Han Shankar P. Bhattacharyya Dept. of Electrical and Computer Engineering Texas A&M University College Station,

More information

Design Methods for Control Systems

Design Methods for Control Systems Design Methods for Control Systems Maarten Steinbuch TU/e Gjerrit Meinsma UT Dutch Institute of Systems and Control Winter term 2002-2003 Schedule November 25 MSt December 2 MSt Homework # 1 December 9

More information

An Internal Stability Example

An Internal Stability Example An Internal Stability Example Roy Smith 26 April 2015 To illustrate the concept of internal stability we will look at an example where there are several pole-zero cancellations between the controller and

More information

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 23: Drawing The Nyquist Plot Overview In this Lecture, you will learn: Review of Nyquist Drawing the Nyquist Plot Using

More information

Control Systems II. ETH, MAVT, IDSC, Lecture 4 17/03/2017. G. Ducard

Control Systems II. ETH, MAVT, IDSC, Lecture 4 17/03/2017. G. Ducard Control Systems II ETH, MAVT, IDSC, Lecture 4 17/03/2017 Lecture plan: Control Systems II, IDSC, 2017 SISO Control Design 24.02 Lecture 1 Recalls, Introductory case study 03.03 Lecture 2 Cascaded Control

More information

Linear Control Systems Lecture #3 - Frequency Domain Analysis. Guillaume Drion Academic year

Linear Control Systems Lecture #3 - Frequency Domain Analysis. Guillaume Drion Academic year Linear Control Systems Lecture #3 - Frequency Domain Analysis Guillaume Drion Academic year 2018-2019 1 Goal and Outline Goal: To be able to analyze the stability and robustness of a closed-loop system

More information

Singular Value Decomposition Analysis

Singular Value Decomposition Analysis Singular Value Decomposition Analysis Singular Value Decomposition Analysis Introduction Introduce a linear algebra tool: singular values of a matrix Motivation Why do we need singular values in MIMO control

More information

Robust control for a multi-stage evaporation plant in the presence of uncertainties

Robust control for a multi-stage evaporation plant in the presence of uncertainties Preprint 11th IFAC Symposium on Dynamics and Control of Process Systems including Biosystems June 6-8 16. NTNU Trondheim Norway Robust control for a multi-stage evaporation plant in the presence of uncertainties

More information

Chapter 20 Analysis of MIMO Control Loops

Chapter 20 Analysis of MIMO Control Loops Chapter 20 Analysis of MIMO Control Loops Motivational Examples All real-world systems comprise multiple interacting variables. For example, one tries to increase the flow of water in a shower by turning

More information

Additional Closed-Loop Frequency Response Material (Second edition, Chapter 14)

Additional Closed-Loop Frequency Response Material (Second edition, Chapter 14) Appendix J Additional Closed-Loop Frequency Response Material (Second edition, Chapter 4) APPENDIX CONTENTS J. Closed-Loop Behavior J.2 Bode Stability Criterion J.3 Nyquist Stability Criterion J.4 Gain

More information

Exam. 135 minutes, 15 minutes reading time

Exam. 135 minutes, 15 minutes reading time Exam August 6, 208 Control Systems II (5-0590-00) Dr. Jacopo Tani Exam Exam Duration: 35 minutes, 5 minutes reading time Number of Problems: 35 Number of Points: 47 Permitted aids: 0 pages (5 sheets) A4.

More information

EECE 460. Decentralized Control of MIMO Systems. Guy A. Dumont. Department of Electrical and Computer Engineering University of British Columbia

EECE 460. Decentralized Control of MIMO Systems. Guy A. Dumont. Department of Electrical and Computer Engineering University of British Columbia EECE 460 Decentralized Control of MIMO Systems Guy A. Dumont Department of Electrical and Computer Engineering University of British Columbia January 2011 Guy A. Dumont (UBC EECE) EECE 460 - Decentralized

More information

ECSE 4962 Control Systems Design. A Brief Tutorial on Control Design

ECSE 4962 Control Systems Design. A Brief Tutorial on Control Design ECSE 4962 Control Systems Design A Brief Tutorial on Control Design Instructor: Professor John T. Wen TA: Ben Potsaid http://www.cat.rpi.edu/~wen/ecse4962s04/ Don t Wait Until The Last Minute! You got

More information

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 23: Drawing The Nyquist Plot Overview In this Lecture, you will learn: Review of Nyquist Drawing the Nyquist Plot Using the

More information

Automatic Control 2. Loop shaping. Prof. Alberto Bemporad. University of Trento. Academic year

Automatic Control 2. Loop shaping. Prof. Alberto Bemporad. University of Trento. Academic year Automatic Control 2 Loop shaping Prof. Alberto Bemporad University of Trento Academic year 21-211 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 21-211 1 / 39 Feedback

More information

Multi-Objective Robust Control of Rotor/Active Magnetic Bearing Systems

Multi-Objective Robust Control of Rotor/Active Magnetic Bearing Systems Multi-Objective Robust Control of Rotor/Active Magnetic Bearing Systems İbrahim Sina Kuseyri Ph.D. Dissertation June 13, 211 İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13,

More information

Robust Loop Shaping Controller Design for Spectral Models by Quadratic Programming

Robust Loop Shaping Controller Design for Spectral Models by Quadratic Programming Robust Loop Shaping Controller Design for Spectral Models by Quadratic Programming Gorka Galdos, Alireza Karimi and Roland Longchamp Abstract A quadratic programming approach is proposed to tune fixed-order

More information

Theory of Robust Control

Theory of Robust Control Theory of Robust Control Carsten Scherer Mathematical Systems Theory Department of Mathematics University of Stuttgart Germany Contents 1 Introduction to Basic Concepts 6 1.1 Systems and Signals..............................

More information

Control Systems I. Lecture 6: Poles and Zeros. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control D-MAVT ETH Zürich

Control Systems I. Lecture 6: Poles and Zeros. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control D-MAVT ETH Zürich Control Systems I Lecture 6: Poles and Zeros Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 27, 2017 E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/2017

More information

(Continued on next page)

(Continued on next page) (Continued on next page) 18.2 Roots of Stability Nyquist Criterion 87 e(s) 1 S(s) = =, r(s) 1 + P (s)c(s) where P (s) represents the plant transfer function, and C(s) the compensator. The closedloop characteristic

More information

Robust Control of Heterogeneous Networks (e.g. congestion control for the Internet)

Robust Control of Heterogeneous Networks (e.g. congestion control for the Internet) Robust Control of Heterogeneous Networks (e.g. congestion control for the Internet) Glenn Vinnicombe gv@eng.cam.ac.uk. University of Cambridge & Caltech 1/29 Introduction Is it possible to build (locally

More information

FREQUENCY-RESPONSE DESIGN

FREQUENCY-RESPONSE DESIGN ECE45/55: Feedback Control Systems. 9 FREQUENCY-RESPONSE DESIGN 9.: PD and lead compensation networks The frequency-response methods we have seen so far largely tell us about stability and stability margins

More information

Stability and Robustness 1

Stability and Robustness 1 Lecture 2 Stability and Robustness This lecture discusses the role of stability in feedback design. The emphasis is notonyes/notestsforstability,butratheronhowtomeasurethedistanceto instability. The small

More information

1.1 Notations We dene X (s) =X T (;s), X T denotes the transpose of X X>()0 a symmetric, positive denite (semidenite) matrix diag [X 1 X ] a block-dia

1.1 Notations We dene X (s) =X T (;s), X T denotes the transpose of X X>()0 a symmetric, positive denite (semidenite) matrix diag [X 1 X ] a block-dia Applications of mixed -synthesis using the passivity approach A. Helmersson Department of Electrical Engineering Linkoping University S-581 83 Linkoping, Sweden tel: +46 13 816 fax: +46 13 86 email: andersh@isy.liu.se

More information

Chapter 2. Classical Control System Design. Dutch Institute of Systems and Control

Chapter 2. Classical Control System Design. Dutch Institute of Systems and Control Chapter 2 Classical Control System Design Overview Ch. 2. 2. Classical control system design Introduction Introduction Steady-state Steady-state errors errors Type Type k k systems systems Integral Integral

More information

Robust stability and Performance

Robust stability and Performance 122 c Perry Y.Li Chapter 5 Robust stability and Performance Topics: ([ author ] is supplementary source) Sensitivities and internal stability (Goodwin 5.1-5.4) Modeling Error and Model Uncertainty (Goodwin

More information

The Generalized Nyquist Criterion and Robustness Margins with Applications

The Generalized Nyquist Criterion and Robustness Margins with Applications 51st IEEE Conference on Decision and Control December 10-13, 2012. Maui, Hawaii, USA The Generalized Nyquist Criterion and Robustness Margins with Applications Abbas Emami-Naeini and Robert L. Kosut Abstract

More information

We are IntechOpen, the world s leading publisher of Open Access books Built by scientists, for scientists. International authors and editors

We are IntechOpen, the world s leading publisher of Open Access books Built by scientists, for scientists. International authors and editors We are IntechOpen, the world s leading publisher of Open Access books Built by scientists, for scientists 3,350 108,000 1.7 M Open access books available International authors and editors Downloads Our

More information

3 Stabilization of MIMO Feedback Systems

3 Stabilization of MIMO Feedback Systems 3 Stabilization of MIMO Feedback Systems 3.1 Notation The sets R and S are as before. We will use the notation M (R) to denote the set of matrices with elements in R. The dimensions are not explicitly

More information

Robust Fractional Control An overview of the research activity carried on at the University of Brescia

Robust Fractional Control An overview of the research activity carried on at the University of Brescia Università degli Studi di Brescia Dipartimento di Ingegneria dell Informazione Robust Fractional Control An overview of the research activity carried on at the University of Brescia Fabrizio Padula and

More information

Control Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Jacopo Tani. Institute for Dynamic Systems and Control D-MAVT ETH Zürich

Control Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Jacopo Tani. Institute for Dynamic Systems and Control D-MAVT ETH Zürich Control Systems I Lecture 7: Feedback and the Root Locus method Readings: Jacopo Tani Institute for Dynamic Systems and Control D-MAVT ETH Zürich November 2, 2018 J. Tani, E. Frazzoli (ETH) Lecture 7:

More information

Control Systems I. Lecture 9: The Nyquist condition

Control Systems I. Lecture 9: The Nyquist condition Control Systems I Lecture 9: The Nyquist condition Readings: Åstrom and Murray, Chapter 9.1 4 www.cds.caltech.edu/~murray/amwiki/index.php/first_edition Jacopo Tani Institute for Dynamic Systems and Control

More information

ROBUST STABILITY AND PERFORMANCE ANALYSIS OF UNSTABLE PROCESS WITH DEAD TIME USING Mu SYNTHESIS

ROBUST STABILITY AND PERFORMANCE ANALYSIS OF UNSTABLE PROCESS WITH DEAD TIME USING Mu SYNTHESIS ROBUST STABILITY AND PERFORMANCE ANALYSIS OF UNSTABLE PROCESS WITH DEAD TIME USING Mu SYNTHESIS I. Thirunavukkarasu 1, V. I. George 1, G. Saravana Kumar 1 and A. Ramakalyan 2 1 Department o Instrumentation

More information

Control System Design

Control System Design ELEC ENG 4CL4: Control System Design Notes for Lecture #24 Wednesday, March 10, 2004 Dr. Ian C. Bruce Room: CRL-229 Phone ext.: 26984 Email: ibruce@mail.ece.mcmaster.ca Remedies We next turn to the question

More information

Topic # Feedback Control Systems

Topic # Feedback Control Systems Topic #19 16.31 Feedback Control Systems Stengel Chapter 6 Question: how well do the large gain and phase margins discussed for LQR map over to DOFB using LQR and LQE (called LQG)? Fall 2010 16.30/31 19

More information

AA/EE/ME 548: Problem Session Notes #5

AA/EE/ME 548: Problem Session Notes #5 AA/EE/ME 548: Problem Session Notes #5 Review of Nyquist and Bode Plots. Nyquist Stability Criterion. LQG/LTR Method Tuesday, March 2, 203 Outline:. A review of Bode plots. 2. A review of Nyquist plots

More information

STABILITY ANALYSIS TECHNIQUES

STABILITY ANALYSIS TECHNIQUES ECE4540/5540: Digital Control Systems 4 1 STABILITY ANALYSIS TECHNIQUES 41: Bilinear transformation Three main aspects to control-system design: 1 Stability, 2 Steady-state response, 3 Transient response

More information

Lecture 2. FRTN10 Multivariable Control. Automatic Control LTH, 2018

Lecture 2. FRTN10 Multivariable Control. Automatic Control LTH, 2018 Lecture 2 FRTN10 Multivariable Control Course Outline L1 L5 Specifications, models and loop-shaping by hand 1 Introduction 2 Stability and robustness 3 Specifications and disturbance models 4 Control synthesis

More information

Analysis of Discrete-Time Systems

Analysis of Discrete-Time Systems TU Berlin Discrete-Time Control Systems 1 Analysis of Discrete-Time Systems Overview Stability Sensitivity and Robustness Controllability, Reachability, Observability, and Detectabiliy TU Berlin Discrete-Time

More information

Control of Electromechanical Systems

Control of Electromechanical Systems Control of Electromechanical Systems November 3, 27 Exercise Consider the feedback control scheme of the motor speed ω in Fig., where the torque actuation includes a time constant τ A =. s and a disturbance

More information

Didier HENRION henrion

Didier HENRION   henrion POLYNOMIAL METHODS FOR ROBUST CONTROL PART I.1 ROBUST STABILITY ANALYSIS: SINGLE PARAMETER UNCERTAINTY Didier HENRION www.laas.fr/ henrion henrion@laas.fr Pont Neuf over river Garonne in Toulouse June

More information

The loop shaping paradigm. Lecture 7. Loop analysis of feedback systems (2) Essential specifications (2)

The loop shaping paradigm. Lecture 7. Loop analysis of feedback systems (2) Essential specifications (2) Lecture 7. Loop analysis of feedback systems (2). Loop shaping 2. Performance limitations The loop shaping paradigm. Estimate performance and robustness of the feedback system from the loop transfer L(jω)

More information

Problem Set 4 Solution 1

Problem Set 4 Solution 1 Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Problem Set 4 Solution Problem 4. For the SISO feedback

More information

Subject: Optimal Control Assignment-1 (Related to Lecture notes 1-10)

Subject: Optimal Control Assignment-1 (Related to Lecture notes 1-10) Subject: Optimal Control Assignment- (Related to Lecture notes -). Design a oil mug, shown in fig., to hold as much oil possible. The height and radius of the mug should not be more than 6cm. The mug must

More information

Lecture 5: Linear Systems. Transfer functions. Frequency Domain Analysis. Basic Control Design.

Lecture 5: Linear Systems. Transfer functions. Frequency Domain Analysis. Basic Control Design. ISS0031 Modeling and Identification Lecture 5: Linear Systems. Transfer functions. Frequency Domain Analysis. Basic Control Design. Aleksei Tepljakov, Ph.D. September 30, 2015 Linear Dynamic Systems Definition

More information

Under assumptions (A1), (A2), (A3), the closed-loop system in figure() is stable if:

Under assumptions (A1), (A2), (A3), the closed-loop system in figure() is stable if: Output multiplicative uncertainty Theorem Under assumptions (A), (A2), (A3), the closed-loop system in figure() is stable if: ω \, s( jω) G( jω) K( jω) Ip G( jω) K( jω) +

More information

Zeros and zero dynamics

Zeros and zero dynamics CHAPTER 4 Zeros and zero dynamics 41 Zero dynamics for SISO systems Consider a linear system defined by a strictly proper scalar transfer function that does not have any common zero and pole: g(s) =α p(s)

More information

Control Systems. Root Locus & Pole Assignment. L. Lanari

Control Systems. Root Locus & Pole Assignment. L. Lanari Control Systems Root Locus & Pole Assignment L. Lanari Outline root-locus definition main rules for hand plotting root locus as a design tool other use of the root locus pole assignment Lanari: CS - Root

More information

Stability of CL System

Stability of CL System Stability of CL System Consider an open loop stable system that becomes unstable with large gain: At the point of instability, K( j) G( j) = 1 0dB K( j) G( j) K( j) G( j) K( j) G( j) =± 180 o 180 o Closed

More information

Robust Control, H, ν and Glover-McFarlane

Robust Control, H, ν and Glover-McFarlane Robust Control, H, ν and Glover-McFarlane Department of Automatic Control LTH, Lund University Robust Control 1 MIMO performance 2 Robustness and the H -norm 3 H -control 4 ν-gap metric 5 Glover-MacFarlane

More information

Lecture 1: Feedback Control Loop

Lecture 1: Feedback Control Loop Lecture : Feedback Control Loop Loop Transfer function The standard feedback control system structure is depicted in Figure. This represend(t) n(t) r(t) e(t) u(t) v(t) η(t) y(t) F (s) C(s) P (s) Figure

More information

Fragility via Robustness of Controllers Dedicated. to the Congestion Control in Internet Protocol

Fragility via Robustness of Controllers Dedicated. to the Congestion Control in Internet Protocol Applied Mathematical Sciences, Vol. 7, 2013, no. 88, 4353-4362 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2013.3528 Fragility via Robustness of Controllers Dedicated to the Congestion

More information

Controls Problems for Qualifying Exam - Spring 2014

Controls Problems for Qualifying Exam - Spring 2014 Controls Problems for Qualifying Exam - Spring 2014 Problem 1 Consider the system block diagram given in Figure 1. Find the overall transfer function T(s) = C(s)/R(s). Note that this transfer function

More information

Dr Ian R. Manchester

Dr Ian R. Manchester Week Content Notes 1 Introduction 2 Frequency Domain Modelling 3 Transient Performance and the s-plane 4 Block Diagrams 5 Feedback System Characteristics Assign 1 Due 6 Root Locus 7 Root Locus 2 Assign

More information

CDS 101/110a: Lecture 10-2 Control Systems Implementation

CDS 101/110a: Lecture 10-2 Control Systems Implementation CDS 101/110a: Lecture 10-2 Control Systems Implementation Richard M. Murray 5 December 2012 Goals Provide an overview of the key principles, concepts and tools from control theory - Classical control -

More information

Module 3F2: Systems and Control EXAMPLES PAPER 2 ROOT-LOCUS. Solutions

Module 3F2: Systems and Control EXAMPLES PAPER 2 ROOT-LOCUS. Solutions Cambridge University Engineering Dept. Third Year Module 3F: Systems and Control EXAMPLES PAPER ROOT-LOCUS Solutions. (a) For the system L(s) = (s + a)(s + b) (a, b both real) show that the root-locus

More information

Analysis of Discrete-Time Systems

Analysis of Discrete-Time Systems TU Berlin Discrete-Time Control Systems TU Berlin Discrete-Time Control Systems 2 Stability Definitions We define stability first with respect to changes in the initial conditions Analysis of Discrete-Time

More information

Part II. Advanced PID Design Methods

Part II. Advanced PID Design Methods Part II Advanced PID Design Methods 54 Controller transfer function C(s) = k p (1 + 1 T i s + T d s) (71) Many extensions known to the basic design methods introduced in RT I. Four advanced approaches

More information

Chapter 15 - Solved Problems

Chapter 15 - Solved Problems Chapter 5 - Solved Problems Solved Problem 5.. Contributed by - Alvaro Liendo, Universidad Tecnica Federico Santa Maria, Consider a plant having a nominal model given by G o (s) = s + 2 The aim of the

More information