Lecture 6 Chapter 8: Robust Stability and Performance Analysis for MIMO Systems Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 6 p. 1/73
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 w 1 z 2 (a) Original system with multiple perturbations Lecture 6 p. 3/73
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
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
u y w P z u v K Figure 3: General control configuration (for controller synthesis) Lecture 6 p. 6/73
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
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
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
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
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
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
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
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
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
(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
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
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
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
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
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
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
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
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
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
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
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
(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
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
6.7 RS with structured uncertainty [8.7] SAME UNCERTAINTY D 1 2... D 1 D 1 M D NEW M: DMD 1 Figure 7: Use of block-diagonal scalings, D = D Lecture 6 p. 30/73
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
(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
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
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
Example 1 Full perturbation ( is unstructured). Consider (6.41) = The perturbation (6.42) M = [ 2 2 1 1 [ ][ ][ 0.894 0.447 3.162 0 0.707 0.707 0.447 0.894 = 1 σ 1 v 1 u H 1 = 1 3.162 = 0 0 [ 0.707 ] 0.707 [ 0.200 0.200 = 0.100 0.100 0.707 0.707 ] H ] [0.894 0.447] = with σ( ) = 1/ σ(m) = 1/3.162 = 0.316 makes det(i M ) = 0. Thus µ(m) = 3.162 when is a full matrix. ] Lecture 6 p. 35/73
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) = 1 3 0 1 with σ( ) = 0.333. Thus µ(m) = 3 when is a diagonal matrix. Lecture 6 p. 36/73
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
Properties of the structured singular value: Check Sections 8.8.1, 8.8.2 and 8.8.3 in the book. Lecture 6 p. 38/73
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
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
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+1 1 87.8 1.4 ; 108.2 1.4 [ ] K(s) = 1+τs 0.0015 0 (6.49) s 0 0.075 (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
(6.50) w I (s) = s+0.2 0.5s+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
Magnitude 10 2 1 10 σ(t 1 I ) w I (jω) 10 0 10-1 10-2 µ I (T I ) 10-3 10-3 10-2 10-1 10 0 10 1 10 2 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
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
(6.53) [ A B C D ] = 0 a 1 0 a 0 0 1 1 a 0 0 a 1 0 0 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
Nominal stability (NS). Two closed loop poles at s = 1 and [ ] [ ] [ ] 0 a 1 a 1 0 A cl = A BKC = = a 0 a 1 0 1 Nominal performance (NP). Figure 9(a) Magnitude 10 2 10 1 10 0 10 1 σ(g) σ(g) 10 2 10 0 10 2 Frequency [rad/s] Figure 9: Singular values - Spinning satellite (6.52) Lecture 6 p. 46/73
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
(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+ǫ1 0 0 1+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
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 2 +1 0.1 Summary. Checking single-loop margins is inadequate for MIMO problems. Lecture 6 p. 49/73
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
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
6.10.1 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
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
6.10.2 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
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
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
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
6.11.1 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
6.11.2 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
6.11.3 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: [ 87.8 86.4 108.2 109.6 1 G(s) = ; K(s) = 0.7 75s+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 = 1 1.43s+1 ] Lecture 6 p. 60/73
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+0.2 0.5s+1 ; w P(s) = s/2+0.05 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
6 4 RP µ 2 RS NP 0 10-3 10-2 10-1 10 0 10 1 10 2 Frequency Figure 13: µ-plots for distillation process with decoupling controller Lecture 6 p. 62/73
NS Yes. NP With the decoupling controller we have σ(n 22 ) = σ(w P S) = s/2+0.05 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
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) 6.12.1 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
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
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
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
6.12.2 Example: µ-synthesis withdk-iteration [8.12.4] Simplified distillation process (6.79) G(s) = 1 75s+1 [ 87.8 86.4 108.2 109.6 ] 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
10 2 Magnitude 10 1 10 0 w P w I 10 1 10 3 10 2 10 1 10 0 10 1 10 2 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
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 = 1.1823. 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=1.1818. 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 = 1.0238. Iteration No. 3. Step 1: With the scalings D 2 (s) the H norm was only slightly reduced from 1.024 to 1.019. Lecture 6 p. 70/73
1 Iter. 1 Iter. 2 Iter. 3 Optimal µ 0.8 0.6 10-3 10-2 10-1 10 0 10 1 10 2 10 3 Frequency [rad/min] Figure 15: Change in µ during DK-iteration Lecture 6 p. 71/73
Magnitude 10 0 Initial 10-1 4th order fit Iter. 1 Optimal Iter. 2 10-2 10-3 10-2 10-1 10 0 10 1 10 2 10 3 Frequency [rad/min] Figure 16: Change in D-scale d 1 during DKiteration Lecture 6 p. 72/73
1 0.8 y 1 0.6 0.4 0.2 0 y 2 0 10 20 30 40 50 60 70 80 90 100 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