A Launch Vehicle Application
2013 American Control Conference (ACC) Washington, DC, USA, June 17-19, 2013
Self-Scheduled and Structured H∞ Synthesis : a Launch Vehicle Application. David Saussié1 , Quentin Barbès2 and Caroline Bérard2 Abstract— This paper presents a new application of structured H∞ synthesis to tune self-scheduled controllers. Newly available M ATLAB-based tools allow to tune fixed-structure linear controllers while satisfying H∞ constraints. Moreover multi-model synthesis capabilities can extend their application to self-scheduled controllers. This technique is successfullyw1 applied to the attitude control of a launch vehicle in atmospheric ascent phase.
I. I NTRODUCTION
w2
Initiated by Zames [1] and further developed by Doyle [2], H∞ synthesis has ever since offered an efficient frequencydomain framework to design SISO and MIMO controllers [3]. In its most basic form, H∞ control boils down to a stabilization and disturbance rejection problem. Classical frequency-domain techniques such as SISO loop-shaping and gain/phase margin analysis are thus easily extended to the MIMO case. In this framework, design requirements areu expressed through H∞ gain constraints (as weighting filters). Efficient algorithms based upon a state-space solution [2], or an LMI characterization [4], have been proposed and implemented in the M ATLAB Robust Control Toolbox [5]. In spite of great qualities, H∞ synthesis still suffers from several practical limitations. The order of H∞ controllers is always as high as the order of the augmented system; thus controller reduction has to be performed to lower the complexity. Moreover no particular structure arises from the controller and physical interpretation by engineers are not easy. This led to techniques that tried to directly tune arbitrary control architectures while satisfying H∞ constraints. The most noticeable contributions in the field are the HIFOO toolbox [6] and the hinfstruct M ATLAB function [7], which is part of the Robust Control Toolbox [5]; the latter function is based upon nonsmooth optimizers presented in [8], [9]. Depending on the system, an unique controller might not be sufficient to ensure performance on the whole domain of evolution; control engineers have therefore resorted to gain-scheduling [10] to overcome this problem. Controller parameters are then updated accordingly to scheduling variables representative of the operating point. Classical gainscheduling consists in designing a set of LTI controllers corresponding to a given set of linearized models around equilibrium points [11]–[13] which are then interpolated 1 D. Saussié is with Electrical Engineering Departement, Polytechnique Montréal, QC H3T 1J4, Canada. [email protected] 2 Q. Barbès and C. Bérard are with the Mathematics, Computer Science, Automatics Departement, ISAE 31055 Cedex 4 Toulouse, France.
{quentin.barbes, caroline.berard}@isae.fr 978-1-4799-0176-0/$31.00 ©2013 AACC
versus scheduling variables which may be exogenous and/or endogenous variables. A posteriori interpolation can however become a challenging question when dynamic compensators are considered [14], [15] and implementation problems can occur [16], [17]. To avoid such problems, a self-scheduling P ( s ) technique was proposed in [18], [19] to choose a priori the scheduling formula. This paper extends the work done in [20]. Taking advanW3multi-model ( s) tage of new capabilities of hinfstruct, we propose to use structured H∞ synthesis to design directly a self-scheduled controller. The paper is organized as follows. W2 ( s ) Section 2 discusses the formulation of self-scheduling in the structured H∞ synthesis framework. Section 3 depicts the launch vehicle problem (mathematical model and requireW1 ( s ) ments). Finally, Section 4 shows the application of our selfscheduling technique.
( s ) S TRUCTURED H II. S ELF -S CHEDULING INGTHE ∞ F RAMEWORK We explain here how to design self-scheduled structured controller in the structured H∞ framework
K ( s)
A. Standard Form
As suggested in [21], we use the Standard Form as a standardized representation of linear systems (Fig. 1). W case where the plant and Moreover we consider the general the controller depend on parameters (θ , θ 0 ). The Standard Form consists then of two main components: c model P(s, θ ) with parameter vector θ whose • A linear Gact ( s ) G(s) components are time-varying, uncertain or a mixture of both. ref vector • A structured controller K(s, θ 0 ) scheduled w.r.t. F ( s) K ( s) θ 0 known in real-time and possibly a subvector of θ .
w u
z
P ( s, )
y
K ( s, ') Fig. 1: Standard Form for Structured H∞ Synthesis Exogenous inputs are gathered in w and regulated outputs in z, u is the command signal and y the available measure-
1593
z2 z1
y
ments. P(s, θ ) is then partitioned as: z P11 (s, θ ) P12 (s, θ ) w = y P21 (s, θ ) P22 (s, θ ) u
Moreover each H j (s) can be rewritten as H j (s) := Fl (Pi (s), K(s)) and the standard form for H(s) looks like (1)
Finally, the linear-fractional transformation (LFT) yields the closed-loop transfer function from w to z: Tzw (s) = Fl (P, K) = P11 + P12 K(I − P22 K)−1 P21
(2)
The scheduled controller K(s, θ 0 ) evolves w.r.t. to θ 0 . As traditionally done in gain-scheduling [10], different LTI controllers are designed for several models linearized around equilibrium points and are scheduled a posteriori w.r.t. some scheduling variables. As presented in [19], we plan to choose a priori the scheduling formula so it will be taken into account during the synthesis process. This choice can be oriented according to physical considerations, open-loop studies or previous work. As an example, consider a scheduling w.r.t. to parameter θ1 and an interpolation formula: K(s, θ1 ) = K0 (s) + θ1 K1 (s) + θ12 K2 (s)
(3)
Inspired by Proposition 2 from [22], the synthesis of this controller would be equivalent to the synthesis of a nonscheduled structured controller: Keq (s) = K0 (s) K1 (s) K2 (s) (4) w.r.t. the new augmented system: P11 (s, θ ) P12 (s, θ ) w P21 (s, θ ) P22 (s, θ ) z = θ1 P21 (s, θ ) θ1 P22 (s, θ ) u yeq θ12 P21 (s, θ ) θ12 P22 (s, θ ) (5) B. Structured H∞ -synthesis The H∞ -synthesis consists in finding a stabilizing controller K(s) that minimizes the H∞ -norm of the transfer function Tzw where kTzw (P, K)k∞ := max ω6=0
kzk2 = max σ¯ (Tzw ) kwk2 ω6=0
where P(s) can be inferred by input/output rearrangements of diag(P1 (s), . . . , PM (s)). As stated by [7], [21], standard H∞ algorithms cannot handle the resulting block-diagonal controller structure (Eq. 10), but, thanks to new features, hinfstruct can. This is definitely an advantage over standard H∞ synthesis; only the useful transfers are optimized over a common structured controller. C. Suggested strategies We present here different strategies depending on the nature of the problem. The hinfstruct function and the associated commands are definitely a well-suited set of tools to handle these problems. 1) Single Model Synthesis: As in [21], consider the design of a feedback controller K for the plant G under the H∞ constraints kW1 Sk∞ < 1, kW2 T k∞ < 1 (11) where S = 1/(1 + L), T = L/(1 + L) and L = GK. The diagonal structure to optimize is therefore: H(s) = diag(W1 S,W2 T )
(12)
2) Robust/Multi-Model Synthesis: Let the plant G(s, θ ), θ ∈ Θ to be uncertain. A robust controller K should then satisfy ∀θ ∈ Θ,
kW1 S(s, θ )k∞ < 1, kW2 T (s, θ )k∞ < 1
(13)
As this problem is not yet tractable, it could be approximated on a finite set of values, i.e. kW1 Sθ j k∞ < 1, kW2 Tθ j k∞ < 1 ( j = 1 . . . M)
(14)
The aggregate transfer function H(s) to optimize is therefore: H(s) = diag(. . . ,W1 Sθ j ,W2 Tθ j . . . )
(7)
where T j (s) is a closed-loop transfer function of interest and W j (s) the associated performance weighting [3]. Considering different performance channels, a typical H∞ -synthesis consists in finding a stabilizing controller K(s) that satisfies ideally1 the constraints kW j (s)T j (s)k∞ < 1, j = 1, . . . , M
(10)
(6)
Typical control design requirements (bandwidth, roll-off, stability margins) are expressed as weighting functions and are cast as normalized H∞ constraints of the form kW j (s)T j (s)k∞ < 1
H(s) = Fl (P(s), diag(K(s), . . . , K(s))
(8)
(15)
The final controller must be then validated on the whole domain Θ. In the case when θ is time-varying, not much can be assessed from this approach, nevertheless it could provide good insights on the difficulty to control the plant. 3) Self-Scheduled Synthesis: If a single LTI controller K cannot stabilize the plant on the operating domain Θ, it could be scheduled as K(s, θ ) provided that θ is measurable in real-time. As suggested in (Eq. 3), one would choose an interpolation formula and would augment the outputs of the system with the scheduling variables (Eq. 5). For a finite set of values, the scheduled controller should satisfy
By denoting H j (s) = W j (s)T j (s), the normalized H∞ constraints kHi (s)k∞ < 1 can be aggregated in a single constraint kH(s)k∞ < 1 with
The aggregate transfer function H(s) to optimize is therefore:
H(s) := diag (H1 (s), . . . , HM (s))
H(s) = diag(. . . ,W1,θ j Sθ j ,W2,θ j Tθ j . . . )
1 Or
for H∞ -norm close to 1.
kW1,θ j Sθ j k∞ < 1, kW2,θ j Tθ j k∞ < 1 ( j = 1 . . . M)
(9)
The weighting functions could be a function of θ j . 1594
(16)
(17)
Three parametric cases are considered: a nominal case
III. L AUNCH V EHICLE M ODEL AND R EQUIREMENTS We mainly focus on the atmospheric ascent from 25 s to 60 s. For attitude control purpose, launch vehicle dynamics are then generally described by “short-period” equations of motion whose parameters are changing heavily due to mass variation, velocity and altitude. Therefore, the mathematical model for this time span is time dependent. The controller main task is to minimize the angle of attack α, although it is not available to feedback. Large values of α lead to large lateral forces and thus considerable stress that could lead to the vehicle breakdown [23], [24].
nom and two worst cases with high and low frequency rigid modes, resp. lf and hf. Morever, six flight instants
equally distributed between 25 s and 60 s are chosen along the trajectory: t1 = 25 s, t2 = 32 s, . . . , t6 = 60 s. A gridding of 18 rigid + flexible models are thus available for design purpose. Finally, for simulation and validation needs, an LFT model was developped from the gridding [27].
200 0.32
0.23
0.14
0.07
180
A. Launcher Model
175
160 0.44 150
Relative orientation of the vehicle in the aerodynamic context is defined by the angle of attack α while the launch vehicle orientation is defined by the attitude θ . Both of them are related by the equation: W Vr
(18)
θ¨ θ˙
θ α
= =
0 1 0 0
a12 0 1 1
θ˙ θ
θ˙ θ
+
+
0 d12
b13 0 W 0 β 0 β¨
b12 0 0 0
0.58 100
100 nd
2
Bending mode
80
75 0.74 50
b11 0
125
60 40
with W the wind input, Vr the launch vehicle speed and γ the path angle (0◦ for vertical trajectory). The rigid launcher model (Ar , Br ,Cr , Dr ) is then described by the LTV model [25], [26]:
4th order actuator
120 Imaginary Axis
α = θ +γ −
140
1st Bending mode
0.92
25
20
Rigid mode
0 −20 −80
−70
−60
−50
−40 −30 Real Axis
−20
−10
0
10
W β (19) β¨ (20)
with β the actual nozzle deflection. The coefficients ai j and bi j are time-varying and depend on the launch vehicle characteristics and flight conditions. For example, we have : QSref CNα (21) J where J is the launcher inertia (varying with ascent), Sref the reference surface, CNα the aerodynamic coefficient and Q = 1 2 2 ρVr the dynamic pressure (the density of air ρ decreases with altitude). In the following, we denote xr = [θ˙ θ ]T and u = [W β β¨ ]T . By including two bending modes (i.e. four extra state variable denoted by vector x f ), the flexible model is finally x˙r Ar M f xr Br = + u (22) x˙ f 0 Af xf Bf xr θ 0 1 = + Dr u (23) Cf α 0 1 xf
Fig. 2: Open-loop pole-zero map (case t1 ,nom) B. Requirements The available output for feedback is the pitch angle θ . However the objective is to minimize the angle of attack α in response to a worst-case wind profile. The requirements are imposed as follows: •
a12 =
The inputs β and β¨ are generated by a 4th -order actuator: β = Gact (s)βc (24) β¨ whose dynamics is defined by two pair of complex poles: (ζ1 , ω1 ) = (0.67, 102) and (ζ2 , ω2 ) = (0.2, 173). A typical pole/zero dispersion is depicted in Fig. 2.
•
Angle of attack α ≤ 3◦ in steady-state and minimize overshoot in response to a worst-case wind input W Dominant dynamics well damped 1) ζ ≥ 0.5 for nom case 2) ζ ≥ 0.15 for hf and lf cases
•
Minimum gain margin 1) GM ≥ 3 dB for nom and hf cases 2) GM ≥ 1 dB for lf case
•
Minimum delay margin DM > 40 ms for every case
Although the delay margin should be greater than one control period of 27 ms, a larger margin (40 ms) is desirable. The deflection angle β of the nozzle should remain less than 5◦ . However, in this work, no saturations are considered. IV. A PPLICATION TO THE LAUNCHER PROBLEM Structured H∞ synthesis was previously applied with success in [20]. Six LTI controllers were obtained for each ti on the nom case; robustness assessment and gain-scheduling w.r.t. ascent time ta were done a posteriori. In this section, we take advantage of new features of hinfstruct to do it all at once, or at least to alleviate the design procedure.
1595
W
A. Controller structure
K.c.Free = false; K.d.Value = zeros(1,2); K.d.Free = false;
c
The control architecture is depicted in Fig. 3 and consists of two main components: • a controller K(s,ta ) scheduled ascentc time ta to K ( s, tw.r.t. a) control the unstable rigid mode; • a notch filter F(s) to handle the bending modes. W
c
Controller
Notch filter
K ( s, ta )
F ( s)
c
Actuator
Launcher
Gact ( s )
G(s)
ta
The transfer function poles are both initialized to −30 to help convergence. The bending modes (especially the first one) tend to destabilize in closed loop if no careful attention is paid. The passive approach consists in using notch filters centered on bending mode frequencies. Previous study [20] actually showed that only the first bending mode needs to be cared of while the second bending mode is easily handled with roll-off. The filter structure is as follows:
n
F(s) =
Fig. 3: Control architecture [20] showed that a PID-like controller was sufficient to fulfil the requirements on the rigid model; this controller was piecewise affinely scheduled w.r.t. ta . We first consider the following structure : 1 Ki (ta ) Kd (ta )s + (25) K(s,ta ) = K p (ta ) + s τ1 s + 1 τ2 s + 1
(26)
1st order filter with time constant τ1 to limit the bandwidth of the derivative action; • 1st order filter with time constant τ2 to impose roll-off. Because of the scheduling, two PIDs must be optimized. To our knowledge, it is not possible to impose the same parameter τ1 in both of them if the function ltiblock.pid is used. Moreover by imposing two real poles −1/τ1 and −1/τ2 , we lose the possibility of having a complex pair. Finally the controller is sought as a pure PID + 2nd order transfer function, i.e. a0 Ki (ta ) + Kd (ta )s 2 (27) K(s,ta ) = K p (ta ) + s s + a1 s + a0 •
The controller can then be rewritten as a state-space model: 0 0 0 Ki0 Ki1 θ x˙ = 1 0 −a0 x + a0 K p0 K p1 (28) ta θ 0 1 −a1 Kd0 Kd1 θ 0 0 0 0 1 uc = x+ (29) ta θ This highly structured state-space model is specified as: K = ltiblock.ss(’K’,3,1,2); K.a.Value = [0 0 0;1 0 -900;0 1 -60]; K.a.Free(:,1:2) = false; K.a.Free(1,3) = false; % integral pole K.c.Value = [0 0 1];
(30)
The filter has unitary static gain and pure imaginary zeros that can be tuned with the parameter b2 . The filter is not scheduled; indeed we aim at finding a robust single filter that will handle the first bending mode variation along the ascent phase and for the different cases. With a0 = ωn2 and a1 = 2ζ ωn , this filter is specified by the M ATLAB commands2 : a0 a1 b2 F
= = = =
realp(’a0’,500); realp(’a1’,10); realp(’b2’,1); tf([b2 0 a0],[1 a1 a0]);
B. Weighting functions
where the tunable elements are: • scheduled PID gains K p (ta ) = K p0 + ta .K p1 Ki (ta ) = Ki0 + ta .Ki1 Kd (ta ) = Kd0 + ta .Kd1
b2 s2 + ωn2 s2 + 2ζ ωn s + ωn2
The selection of the weighting functions is a key factor for the synthesis, and is also probably the most tedious step. Indeed, it requires several iterations to find adequate shaping functions. For traditional H∞ synthesis, the order of the weighting function is usually chosen as low as possible to limit the controller order. For structured H∞ synthesis, this does not apply, but weighting function will still be chosen as simple as possible to simplify the designer work (fewer parameters to tune). Similarly, the number of weighted transfers minimised by the H∞ synthesis should be the lowest as long as the requirements are met. The weighting function applied on the sensitivity function T1 (s) between the setpoint θc and the tracking error ε is chosen as: s/M1 + ω1 W1 (s) := (31) s + ω1 ε1 where ω1 is approximately the 0 dB frequency (system bandwidth), ε1 enforces high gain at low frequencies (precision) and 1/M1 is the system minimum modulus margin. A powerful use of the multi model approach is that the weighting functions can be different according to the considered model (nom, lf or hf). For example, the gain margin requirement is not the same for the different cases and can then be adjusted with M1 (Tab. I). The closed-loop transfer T2 (s) from the noise input nθ to the command signal βc is weighted by W2 (s) :=
10s s + 104
(32)
2 The variable change avoids unnecessary parameter repetition in the LFT parameterization of F(s) [21].
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TABLE I: Parameters of the weighting function W1 nom
lf
hf
1.75 1e−3 0.5
1 1e−3 0.15
1 1e−3 0.5
Singular Values (dB)
Parameters ω1 ε1 1/M1
Singular Values 50 1/W1 0
−50
−100 −4 10
This weighting is not very constraining but can somewhat limit the control signal.
−2
−1
10
0
1
10 10 Frequency (rad/s)
2
10
3
10
10
Singular Values (dB)
200
C. Synthesis For each case c (i.e. nom, hf, lf,) and flight ascent ti , we can build the diagonal structure: Hc,ti (s) = diag(W1,c T1,c,ti ,W2 T2,c,ti )
−3
10
1/W2 100 0 −100
(33) −200 −2 10
where each closed-loop transfer function T j,c,ti (s) can be written in Standard Form as T j,c,ti (s) = Fl (Pj,c,ti ,C) with C(s) := diag(K, F)). The plants Pj,c,ti are obtained from a Simulink diagram (using linlft command) and the closedloop transfers T j,c,ti (s) are computed with the lft command. As presented in Section 2, the plant output is augmented with the output ti θ that provides the scheduling part of the controller. This corresponds to the following command lines:
−1
10
0
10
1
10
2
10 Frequency (rad/s)
3
4
10
10
5
10
6
10
Fig. 4: H∞ constraints on nom case Stability Margins Gain Margin (dB)
10
C = blkdiag(K,F); Hi = blkdiag(W1*lft(Pi1,C),W2*lft(Pi2,C));
5
0 25
30
35
40
45
50
55
60
45
50
55
60
45
50
55
60
Time (s) Phase Margin (deg)
All diagonal structures Hc,ti (s) are concatenated in a single diagonal structure H(s) to be optimized with hinfstruct. Ideally we are looking for a stabilizing controller C(s) such that kH(s)k∞ < 1 + δ with δ as small as possible to satisfy the constraints.
40 35 30 25 25
hf nom lf 30
35
40 Time (s)
0.3272s2 + 444.3 (35) s2 + 8.558s + 444.3 The global maximum H∞ norm of H(s) is 1.2, meaning that requirements are mostly satisfied for each of the 18 models (Fig. 4 for nom case). The stability margins are all satisfied for the specified times (Fig. 5) and the damping constraint on the rigid mode is also satisfied (Fig. 6). Concerning the time-responses to a wind gust (Fig. 7), the overshoot is quite low (3.73◦ ) for the worst case hf and the violation of the requirement (α ≤ 3◦ ) does not last more than 0.47 s for the worst case lf. Overall, the results have been improved since [20]. F(s) =
3 The computational time was 15 mn on an iMac 2012 with 2.7 GHz quad-core Intel Core i5, 16GB RAM and M ATLAB R2012b
0.2 Delay Margin (s)
D. Results The more models (case, flight instant) we aggregate in H(s), the more demanding and time-consuming the optimization is. First tests are conducted with models at time t1 , t3 and t6 to fully understand the process and to guide our weighting choice. We finally run the optimization on the 18 full order models at once3 . The optimization yields the controller: 0.034 2.73 + 1.4s − ta 0.087 + + 0.018s K = 6.32 + s s 897 × 2 (34) s + 22.67s + 897
0.15 0.1 0.05 0 25
30
35
40 Time [s]
Fig. 5: Stability margins
V. C ONCLUSION The application of self-scheduled and structured H∞ synthesis to a realistic launch vehicle attitude control problem has been presented. This work successfully took advantage of new features of the hinfstruct command. Instead of doing a posteriori gain-scheduling, the authors were able to cast the problem of gain-scheduling and parametric robustness into the structured H∞ framework; this provided a more efficient controller than previous studies. Although the computational effort is higher, it avoids the iterative procedure of traditional gain-scheduling and allows the user to choose the interpolation formula a priori. ACKNOWLEDGMENTS The authors would like to thank ASTRIUM-ST who allowed the launch vehicle model to be used for this article.
1597
100
100 0.046
0.032
0.02
0.046
0.032
0.02
0.009
0.046
90 80
80
80
50 0.095
30 20
20
40
40
0.15
30
0.15 20
0
−3
−2 Real Axis
−1
0
1
−10 −5
0.15 20
20 0.3 10
0
−4
40
40
0.3 10
−10 −5
50 0.095
30
20
0.3 10
60
60 Imaginary Axis
Imaginary Axis
40
40
80
70 60
60
50 0.095
0.009
0.064
70 60
60
0.02
80
0.064
70
0.032
90
80
0.064
Imaginary Axis
100
0.009
90
0
−4
−3
(a) nom case
−2 Real Axis
−1
(b) lf case
0
1
−10 −5
−4
−3
−2 Real Axis
−1
0
1
(c) hf case
Fig. 6: Closed-loop poles
α (deg) 5
0
−5 25
30
35
40
45
50
55
60
Time (s) θ (deg) 2 hf nom lf
1 0 −1 25
30
35
40
45
50
55
60
45
50
55
60
Time (s) βc (deg) 0
−5
−10 25
30
35
40 Time (s)
Fig. 7: Time-responses to a worst-case wind profile
R EFERENCES [1] G. Zames, “Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximations,” IEEE Transactions on Automatic Control, vol. 26, no. 2, pp. 301–320, 1981. [2] J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, “Statespace solutions to standard H2 and H∞ control problems,” IEEE Transactions on Automatic Control, vol. 34, no. 8, pp. 831–847, 1989. [3] K. Zhou and J. C. Doyle, Essentials of robust control. Upper Saddle River, N.J.: Prentice Hall, 1998. [4] P. Gahinet and P. Apkarian, “A linear matrix inequality approach to H∞ control,” International Journal of Robust and Nonlinear Control, vol. 4, pp. 421–448, 1994. [5] M ATLAB, Robust Control Toolbox. Natick, MA: The MathWorks Inc., 2012. [6] J. V. Burke, D. Henrion, A. S. Lewis, and M. L. Overton, “HIFOO - a Matlab package for fixed-order controller design and H∞ optimization,” in Proc. 5th IFAC Symposium on Robust Control Design, Toulouse, France, July 2006. [7] P. Gahinet and P. Apkarian, “Decentralized and fixed-structure H∞ control in M ATLAB,” in 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, USA, 2011, pp. 8205–8210. [8] P. Apkarian and D. Noll, “Nonsmooth H∞ synthesis,” IEEE Transactions on Automatic Control, vol. 51, no. 1, pp. 71–86, 2006. [9] ——, “Nonsmooth optimization for multiband frequency domain control design,” Automatica, vol. 43, no. 3, pp. 724–731, 2007.
[10] W. J. Rugh and J. S. Shamma, “Research on gain-scheduling,” Automatica, vol. 36, no. 10, pp. 1401–1425, 2000. [11] W. J. Rugh, “Analytical framework for gain scheduling,” IEEE Control Systems Magazine, vol. 11, no. 1, pp. 79–84, January 1991. [12] D. A. Lawrence and W. J. Rugh, “Gain scheduling dynamic linear controllers for a nonlinear plant,” Automatica, vol. 31, no. 3, pp. 381– 390, March 1995. [13] J. S. Shamma and M. Athans, “Analysis of gain-scheduled control for nonlinear plants,” IEEE Transactions on Automatic Control, vol. 35, no. 8, pp. 898–907, 1990. [14] D. J. Stilwell and W. J. Rugh, “Stability preserving interpolation methods for the synthesis of gain-scheduled controllers,” Automatica, vol. 36, no. 5, pp. 665–671, 2000. [15] B. Clément and G. Duc, “An interpolation method for gainscheduling,” in Proceedings of the 40th Conference on Decision and Control, vol. 2. IEEE, 2001, pp. 1310–1315. [16] D. J. Leith, W. E. Leithead, D. A. Lawrence, and W. J. Rugh, “Comments on gain scheduling dynamic linear controllers for a non linear plant,” Automatica, vol. 34, no. 8, pp. 1041–1043, 1998. [17] C. S. Mehendale and K. M. Grigoriadis, “A new approach to lpv gainscheduling design and implementation,” in Proceedings of the 43rd IEEE Conference on Decision and Control, Atlantis, Paradise Island, Bahamas, vol. 3, December 2004, pp. 2942–2947. [18] J. F. Magni, Y. Le Gorrec, and C. Chiappa, “A multimodel-based approach to robust and self-scheduled control design,” in Proceedings of the 37th IEEE Conference on Decision and Control, Tampa, Florida. IEEE, 1998, pp. 3009–3014. [19] C. Döll, Y. Le Gorrec, G. Ferreres, and J. F. Magni, “A robust selfscheduled missile autopilot: Design by multi-model eigenstructure assignment,” Control Engineering Practice, vol. 9, no. 10, pp. 1067– 1078, October 2001. [20] M. Knoblauch, D. Saussié, and C. Bérard, “Structured h∞ control for a launch vehicle,” in American Control Conference (ACC), Montréal, Canada, 2012. [21] P. Gahinet and P. Apkarian, “Structured H∞ synthesis in M ATLAB,” in Proceedings of the 18th IFAC World Congress, Milan, Italy, vol. 18, no. 1, 2011, pp. 1435–1440. [22] J. F. Magni, “Multimodel eigenstructure assignment in flight control design,” Aerospace Sciences and Technologies, vol. 3, no. 3, pp. 141– 151, 1999. [23] M. Hernandez, “Launcher study case: modal self-scheduling control,” Master’s thesis, ONERA-France, 2008. [24] D. Saussié, C. Bérard, L. Saydy, and O. Akhrif, “Robust scheduled control of longitudinal flight with handling quality satisfaction,” The Aeronautical Journal, vol. 115, no. 1165, pp. 163–174, 2011. [25] L. Greensite, Analysis and design of space vehicle control system. New York: Spartan Books, 1970. [26] L. Mangiacasale, Veicoli aerospaziali: dinamica, stabilita, controllo: flessibilita strutturale, sloshing, traiettorie. Roma: Esagrafica, 2003. [27] C. Bérard, J. Biannic, C. Döll, and D. Saussié, “Modélisation LFT et commande auto-séquencée - application au pilotage d’un lanceur,” Journal Européen des Systèmes Automatisés, 2012.
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Self-Scheduled and Structured H∞ Synthesis : a Launch Vehicle Application. David Saussié1 , Quentin Barbès2 and Caroline Bérard2 Abstract— This paper presents a new application of structured H∞ synthesis to tune self-scheduled controllers. Newly available M ATLAB-based tools allow to tune fixed-structure linear controllers while satisfying H∞ constraints. Moreover multi-model synthesis capabilities can extend their application to self-scheduled controllers. This technique is successfullyw1 applied to the attitude control of a launch vehicle in atmospheric ascent phase.
I. I NTRODUCTION
w2
Initiated by Zames [1] and further developed by Doyle [2], H∞ synthesis has ever since offered an efficient frequencydomain framework to design SISO and MIMO controllers [3]. In its most basic form, H∞ control boils down to a stabilization and disturbance rejection problem. Classical frequency-domain techniques such as SISO loop-shaping and gain/phase margin analysis are thus easily extended to the MIMO case. In this framework, design requirements areu expressed through H∞ gain constraints (as weighting filters). Efficient algorithms based upon a state-space solution [2], or an LMI characterization [4], have been proposed and implemented in the M ATLAB Robust Control Toolbox [5]. In spite of great qualities, H∞ synthesis still suffers from several practical limitations. The order of H∞ controllers is always as high as the order of the augmented system; thus controller reduction has to be performed to lower the complexity. Moreover no particular structure arises from the controller and physical interpretation by engineers are not easy. This led to techniques that tried to directly tune arbitrary control architectures while satisfying H∞ constraints. The most noticeable contributions in the field are the HIFOO toolbox [6] and the hinfstruct M ATLAB function [7], which is part of the Robust Control Toolbox [5]; the latter function is based upon nonsmooth optimizers presented in [8], [9]. Depending on the system, an unique controller might not be sufficient to ensure performance on the whole domain of evolution; control engineers have therefore resorted to gain-scheduling [10] to overcome this problem. Controller parameters are then updated accordingly to scheduling variables representative of the operating point. Classical gainscheduling consists in designing a set of LTI controllers corresponding to a given set of linearized models around equilibrium points [11]–[13] which are then interpolated 1 D. Saussié is with Electrical Engineering Departement, Polytechnique Montréal, QC H3T 1J4, Canada. [email protected] 2 Q. Barbès and C. Bérard are with the Mathematics, Computer Science, Automatics Departement, ISAE 31055 Cedex 4 Toulouse, France.
{quentin.barbes, caroline.berard}@isae.fr 978-1-4799-0176-0/$31.00 ©2013 AACC
versus scheduling variables which may be exogenous and/or endogenous variables. A posteriori interpolation can however become a challenging question when dynamic compensators are considered [14], [15] and implementation problems can occur [16], [17]. To avoid such problems, a self-scheduling P ( s ) technique was proposed in [18], [19] to choose a priori the scheduling formula. This paper extends the work done in [20]. Taking advanW3multi-model ( s) tage of new capabilities of hinfstruct, we propose to use structured H∞ synthesis to design directly a self-scheduled controller. The paper is organized as follows. W2 ( s ) Section 2 discusses the formulation of self-scheduling in the structured H∞ synthesis framework. Section 3 depicts the launch vehicle problem (mathematical model and requireW1 ( s ) ments). Finally, Section 4 shows the application of our selfscheduling technique.
( s ) S TRUCTURED H II. S ELF -S CHEDULING INGTHE ∞ F RAMEWORK We explain here how to design self-scheduled structured controller in the structured H∞ framework
K ( s)
A. Standard Form
As suggested in [21], we use the Standard Form as a standardized representation of linear systems (Fig. 1). W case where the plant and Moreover we consider the general the controller depend on parameters (θ , θ 0 ). The Standard Form consists then of two main components: c model P(s, θ ) with parameter vector θ whose • A linear Gact ( s ) G(s) components are time-varying, uncertain or a mixture of both. ref vector • A structured controller K(s, θ 0 ) scheduled w.r.t. F ( s) K ( s) θ 0 known in real-time and possibly a subvector of θ .
w u
z
P ( s, )
y
K ( s, ') Fig. 1: Standard Form for Structured H∞ Synthesis Exogenous inputs are gathered in w and regulated outputs in z, u is the command signal and y the available measure-
1593
z2 z1
y
ments. P(s, θ ) is then partitioned as: z P11 (s, θ ) P12 (s, θ ) w = y P21 (s, θ ) P22 (s, θ ) u
Moreover each H j (s) can be rewritten as H j (s) := Fl (Pi (s), K(s)) and the standard form for H(s) looks like (1)
Finally, the linear-fractional transformation (LFT) yields the closed-loop transfer function from w to z: Tzw (s) = Fl (P, K) = P11 + P12 K(I − P22 K)−1 P21
(2)
The scheduled controller K(s, θ 0 ) evolves w.r.t. to θ 0 . As traditionally done in gain-scheduling [10], different LTI controllers are designed for several models linearized around equilibrium points and are scheduled a posteriori w.r.t. some scheduling variables. As presented in [19], we plan to choose a priori the scheduling formula so it will be taken into account during the synthesis process. This choice can be oriented according to physical considerations, open-loop studies or previous work. As an example, consider a scheduling w.r.t. to parameter θ1 and an interpolation formula: K(s, θ1 ) = K0 (s) + θ1 K1 (s) + θ12 K2 (s)
(3)
Inspired by Proposition 2 from [22], the synthesis of this controller would be equivalent to the synthesis of a nonscheduled structured controller: Keq (s) = K0 (s) K1 (s) K2 (s) (4) w.r.t. the new augmented system: P11 (s, θ ) P12 (s, θ ) w P21 (s, θ ) P22 (s, θ ) z = θ1 P21 (s, θ ) θ1 P22 (s, θ ) u yeq θ12 P21 (s, θ ) θ12 P22 (s, θ ) (5) B. Structured H∞ -synthesis The H∞ -synthesis consists in finding a stabilizing controller K(s) that minimizes the H∞ -norm of the transfer function Tzw where kTzw (P, K)k∞ := max ω6=0
kzk2 = max σ¯ (Tzw ) kwk2 ω6=0
where P(s) can be inferred by input/output rearrangements of diag(P1 (s), . . . , PM (s)). As stated by [7], [21], standard H∞ algorithms cannot handle the resulting block-diagonal controller structure (Eq. 10), but, thanks to new features, hinfstruct can. This is definitely an advantage over standard H∞ synthesis; only the useful transfers are optimized over a common structured controller. C. Suggested strategies We present here different strategies depending on the nature of the problem. The hinfstruct function and the associated commands are definitely a well-suited set of tools to handle these problems. 1) Single Model Synthesis: As in [21], consider the design of a feedback controller K for the plant G under the H∞ constraints kW1 Sk∞ < 1, kW2 T k∞ < 1 (11) where S = 1/(1 + L), T = L/(1 + L) and L = GK. The diagonal structure to optimize is therefore: H(s) = diag(W1 S,W2 T )
(12)
2) Robust/Multi-Model Synthesis: Let the plant G(s, θ ), θ ∈ Θ to be uncertain. A robust controller K should then satisfy ∀θ ∈ Θ,
kW1 S(s, θ )k∞ < 1, kW2 T (s, θ )k∞ < 1
(13)
As this problem is not yet tractable, it could be approximated on a finite set of values, i.e. kW1 Sθ j k∞ < 1, kW2 Tθ j k∞ < 1 ( j = 1 . . . M)
(14)
The aggregate transfer function H(s) to optimize is therefore: H(s) = diag(. . . ,W1 Sθ j ,W2 Tθ j . . . )
(7)
where T j (s) is a closed-loop transfer function of interest and W j (s) the associated performance weighting [3]. Considering different performance channels, a typical H∞ -synthesis consists in finding a stabilizing controller K(s) that satisfies ideally1 the constraints kW j (s)T j (s)k∞ < 1, j = 1, . . . , M
(10)
(6)
Typical control design requirements (bandwidth, roll-off, stability margins) are expressed as weighting functions and are cast as normalized H∞ constraints of the form kW j (s)T j (s)k∞ < 1
H(s) = Fl (P(s), diag(K(s), . . . , K(s))
(8)
(15)
The final controller must be then validated on the whole domain Θ. In the case when θ is time-varying, not much can be assessed from this approach, nevertheless it could provide good insights on the difficulty to control the plant. 3) Self-Scheduled Synthesis: If a single LTI controller K cannot stabilize the plant on the operating domain Θ, it could be scheduled as K(s, θ ) provided that θ is measurable in real-time. As suggested in (Eq. 3), one would choose an interpolation formula and would augment the outputs of the system with the scheduling variables (Eq. 5). For a finite set of values, the scheduled controller should satisfy
By denoting H j (s) = W j (s)T j (s), the normalized H∞ constraints kHi (s)k∞ < 1 can be aggregated in a single constraint kH(s)k∞ < 1 with
The aggregate transfer function H(s) to optimize is therefore:
H(s) := diag (H1 (s), . . . , HM (s))
H(s) = diag(. . . ,W1,θ j Sθ j ,W2,θ j Tθ j . . . )
1 Or
for H∞ -norm close to 1.
kW1,θ j Sθ j k∞ < 1, kW2,θ j Tθ j k∞ < 1 ( j = 1 . . . M)
(9)
The weighting functions could be a function of θ j . 1594
(16)
(17)
Three parametric cases are considered: a nominal case
III. L AUNCH V EHICLE M ODEL AND R EQUIREMENTS We mainly focus on the atmospheric ascent from 25 s to 60 s. For attitude control purpose, launch vehicle dynamics are then generally described by “short-period” equations of motion whose parameters are changing heavily due to mass variation, velocity and altitude. Therefore, the mathematical model for this time span is time dependent. The controller main task is to minimize the angle of attack α, although it is not available to feedback. Large values of α lead to large lateral forces and thus considerable stress that could lead to the vehicle breakdown [23], [24].
nom and two worst cases with high and low frequency rigid modes, resp. lf and hf. Morever, six flight instants
equally distributed between 25 s and 60 s are chosen along the trajectory: t1 = 25 s, t2 = 32 s, . . . , t6 = 60 s. A gridding of 18 rigid + flexible models are thus available for design purpose. Finally, for simulation and validation needs, an LFT model was developped from the gridding [27].
200 0.32
0.23
0.14
0.07
180
A. Launcher Model
175
160 0.44 150
Relative orientation of the vehicle in the aerodynamic context is defined by the angle of attack α while the launch vehicle orientation is defined by the attitude θ . Both of them are related by the equation: W Vr
(18)
θ¨ θ˙
θ α
= =
0 1 0 0
a12 0 1 1
θ˙ θ
θ˙ θ
+
+
0 d12
b13 0 W 0 β 0 β¨
b12 0 0 0
0.58 100
100 nd
2
Bending mode
80
75 0.74 50
b11 0
125
60 40
with W the wind input, Vr the launch vehicle speed and γ the path angle (0◦ for vertical trajectory). The rigid launcher model (Ar , Br ,Cr , Dr ) is then described by the LTV model [25], [26]:
4th order actuator
120 Imaginary Axis
α = θ +γ −
140
1st Bending mode
0.92
25
20
Rigid mode
0 −20 −80
−70
−60
−50
−40 −30 Real Axis
−20
−10
0
10
W β (19) β¨ (20)
with β the actual nozzle deflection. The coefficients ai j and bi j are time-varying and depend on the launch vehicle characteristics and flight conditions. For example, we have : QSref CNα (21) J where J is the launcher inertia (varying with ascent), Sref the reference surface, CNα the aerodynamic coefficient and Q = 1 2 2 ρVr the dynamic pressure (the density of air ρ decreases with altitude). In the following, we denote xr = [θ˙ θ ]T and u = [W β β¨ ]T . By including two bending modes (i.e. four extra state variable denoted by vector x f ), the flexible model is finally x˙r Ar M f xr Br = + u (22) x˙ f 0 Af xf Bf xr θ 0 1 = + Dr u (23) Cf α 0 1 xf
Fig. 2: Open-loop pole-zero map (case t1 ,nom) B. Requirements The available output for feedback is the pitch angle θ . However the objective is to minimize the angle of attack α in response to a worst-case wind profile. The requirements are imposed as follows: •
a12 =
The inputs β and β¨ are generated by a 4th -order actuator: β = Gact (s)βc (24) β¨ whose dynamics is defined by two pair of complex poles: (ζ1 , ω1 ) = (0.67, 102) and (ζ2 , ω2 ) = (0.2, 173). A typical pole/zero dispersion is depicted in Fig. 2.
•
Angle of attack α ≤ 3◦ in steady-state and minimize overshoot in response to a worst-case wind input W Dominant dynamics well damped 1) ζ ≥ 0.5 for nom case 2) ζ ≥ 0.15 for hf and lf cases
•
Minimum gain margin 1) GM ≥ 3 dB for nom and hf cases 2) GM ≥ 1 dB for lf case
•
Minimum delay margin DM > 40 ms for every case
Although the delay margin should be greater than one control period of 27 ms, a larger margin (40 ms) is desirable. The deflection angle β of the nozzle should remain less than 5◦ . However, in this work, no saturations are considered. IV. A PPLICATION TO THE LAUNCHER PROBLEM Structured H∞ synthesis was previously applied with success in [20]. Six LTI controllers were obtained for each ti on the nom case; robustness assessment and gain-scheduling w.r.t. ascent time ta were done a posteriori. In this section, we take advantage of new features of hinfstruct to do it all at once, or at least to alleviate the design procedure.
1595
W
A. Controller structure
K.c.Free = false; K.d.Value = zeros(1,2); K.d.Free = false;
c
The control architecture is depicted in Fig. 3 and consists of two main components: • a controller K(s,ta ) scheduled ascentc time ta to K ( s, tw.r.t. a) control the unstable rigid mode; • a notch filter F(s) to handle the bending modes. W
c
Controller
Notch filter
K ( s, ta )
F ( s)
c
Actuator
Launcher
Gact ( s )
G(s)
ta
The transfer function poles are both initialized to −30 to help convergence. The bending modes (especially the first one) tend to destabilize in closed loop if no careful attention is paid. The passive approach consists in using notch filters centered on bending mode frequencies. Previous study [20] actually showed that only the first bending mode needs to be cared of while the second bending mode is easily handled with roll-off. The filter structure is as follows:
n
F(s) =
Fig. 3: Control architecture [20] showed that a PID-like controller was sufficient to fulfil the requirements on the rigid model; this controller was piecewise affinely scheduled w.r.t. ta . We first consider the following structure : 1 Ki (ta ) Kd (ta )s + (25) K(s,ta ) = K p (ta ) + s τ1 s + 1 τ2 s + 1
(26)
1st order filter with time constant τ1 to limit the bandwidth of the derivative action; • 1st order filter with time constant τ2 to impose roll-off. Because of the scheduling, two PIDs must be optimized. To our knowledge, it is not possible to impose the same parameter τ1 in both of them if the function ltiblock.pid is used. Moreover by imposing two real poles −1/τ1 and −1/τ2 , we lose the possibility of having a complex pair. Finally the controller is sought as a pure PID + 2nd order transfer function, i.e. a0 Ki (ta ) + Kd (ta )s 2 (27) K(s,ta ) = K p (ta ) + s s + a1 s + a0 •
The controller can then be rewritten as a state-space model: 0 0 0 Ki0 Ki1 θ x˙ = 1 0 −a0 x + a0 K p0 K p1 (28) ta θ 0 1 −a1 Kd0 Kd1 θ 0 0 0 0 1 uc = x+ (29) ta θ This highly structured state-space model is specified as: K = ltiblock.ss(’K’,3,1,2); K.a.Value = [0 0 0;1 0 -900;0 1 -60]; K.a.Free(:,1:2) = false; K.a.Free(1,3) = false; % integral pole K.c.Value = [0 0 1];
(30)
The filter has unitary static gain and pure imaginary zeros that can be tuned with the parameter b2 . The filter is not scheduled; indeed we aim at finding a robust single filter that will handle the first bending mode variation along the ascent phase and for the different cases. With a0 = ωn2 and a1 = 2ζ ωn , this filter is specified by the M ATLAB commands2 : a0 a1 b2 F
= = = =
realp(’a0’,500); realp(’a1’,10); realp(’b2’,1); tf([b2 0 a0],[1 a1 a0]);
B. Weighting functions
where the tunable elements are: • scheduled PID gains K p (ta ) = K p0 + ta .K p1 Ki (ta ) = Ki0 + ta .Ki1 Kd (ta ) = Kd0 + ta .Kd1
b2 s2 + ωn2 s2 + 2ζ ωn s + ωn2
The selection of the weighting functions is a key factor for the synthesis, and is also probably the most tedious step. Indeed, it requires several iterations to find adequate shaping functions. For traditional H∞ synthesis, the order of the weighting function is usually chosen as low as possible to limit the controller order. For structured H∞ synthesis, this does not apply, but weighting function will still be chosen as simple as possible to simplify the designer work (fewer parameters to tune). Similarly, the number of weighted transfers minimised by the H∞ synthesis should be the lowest as long as the requirements are met. The weighting function applied on the sensitivity function T1 (s) between the setpoint θc and the tracking error ε is chosen as: s/M1 + ω1 W1 (s) := (31) s + ω1 ε1 where ω1 is approximately the 0 dB frequency (system bandwidth), ε1 enforces high gain at low frequencies (precision) and 1/M1 is the system minimum modulus margin. A powerful use of the multi model approach is that the weighting functions can be different according to the considered model (nom, lf or hf). For example, the gain margin requirement is not the same for the different cases and can then be adjusted with M1 (Tab. I). The closed-loop transfer T2 (s) from the noise input nθ to the command signal βc is weighted by W2 (s) :=
10s s + 104
(32)
2 The variable change avoids unnecessary parameter repetition in the LFT parameterization of F(s) [21].
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TABLE I: Parameters of the weighting function W1 nom
lf
hf
1.75 1e−3 0.5
1 1e−3 0.15
1 1e−3 0.5
Singular Values (dB)
Parameters ω1 ε1 1/M1
Singular Values 50 1/W1 0
−50
−100 −4 10
This weighting is not very constraining but can somewhat limit the control signal.
−2
−1
10
0
1
10 10 Frequency (rad/s)
2
10
3
10
10
Singular Values (dB)
200
C. Synthesis For each case c (i.e. nom, hf, lf,) and flight ascent ti , we can build the diagonal structure: Hc,ti (s) = diag(W1,c T1,c,ti ,W2 T2,c,ti )
−3
10
1/W2 100 0 −100
(33) −200 −2 10
where each closed-loop transfer function T j,c,ti (s) can be written in Standard Form as T j,c,ti (s) = Fl (Pj,c,ti ,C) with C(s) := diag(K, F)). The plants Pj,c,ti are obtained from a Simulink diagram (using linlft command) and the closedloop transfers T j,c,ti (s) are computed with the lft command. As presented in Section 2, the plant output is augmented with the output ti θ that provides the scheduling part of the controller. This corresponds to the following command lines:
−1
10
0
10
1
10
2
10 Frequency (rad/s)
3
4
10
10
5
10
6
10
Fig. 4: H∞ constraints on nom case Stability Margins Gain Margin (dB)
10
C = blkdiag(K,F); Hi = blkdiag(W1*lft(Pi1,C),W2*lft(Pi2,C));
5
0 25
30
35
40
45
50
55
60
45
50
55
60
45
50
55
60
Time (s) Phase Margin (deg)
All diagonal structures Hc,ti (s) are concatenated in a single diagonal structure H(s) to be optimized with hinfstruct. Ideally we are looking for a stabilizing controller C(s) such that kH(s)k∞ < 1 + δ with δ as small as possible to satisfy the constraints.
40 35 30 25 25
hf nom lf 30
35
40 Time (s)
0.3272s2 + 444.3 (35) s2 + 8.558s + 444.3 The global maximum H∞ norm of H(s) is 1.2, meaning that requirements are mostly satisfied for each of the 18 models (Fig. 4 for nom case). The stability margins are all satisfied for the specified times (Fig. 5) and the damping constraint on the rigid mode is also satisfied (Fig. 6). Concerning the time-responses to a wind gust (Fig. 7), the overshoot is quite low (3.73◦ ) for the worst case hf and the violation of the requirement (α ≤ 3◦ ) does not last more than 0.47 s for the worst case lf. Overall, the results have been improved since [20]. F(s) =
3 The computational time was 15 mn on an iMac 2012 with 2.7 GHz quad-core Intel Core i5, 16GB RAM and M ATLAB R2012b
0.2 Delay Margin (s)
D. Results The more models (case, flight instant) we aggregate in H(s), the more demanding and time-consuming the optimization is. First tests are conducted with models at time t1 , t3 and t6 to fully understand the process and to guide our weighting choice. We finally run the optimization on the 18 full order models at once3 . The optimization yields the controller: 0.034 2.73 + 1.4s − ta 0.087 + + 0.018s K = 6.32 + s s 897 × 2 (34) s + 22.67s + 897
0.15 0.1 0.05 0 25
30
35
40 Time [s]
Fig. 5: Stability margins
V. C ONCLUSION The application of self-scheduled and structured H∞ synthesis to a realistic launch vehicle attitude control problem has been presented. This work successfully took advantage of new features of the hinfstruct command. Instead of doing a posteriori gain-scheduling, the authors were able to cast the problem of gain-scheduling and parametric robustness into the structured H∞ framework; this provided a more efficient controller than previous studies. Although the computational effort is higher, it avoids the iterative procedure of traditional gain-scheduling and allows the user to choose the interpolation formula a priori. ACKNOWLEDGMENTS The authors would like to thank ASTRIUM-ST who allowed the launch vehicle model to be used for this article.
1597
100
100 0.046
0.032
0.02
0.046
0.032
0.02
0.009
0.046
90 80
80
80
50 0.095
30 20
20
40
40
0.15
30
0.15 20
0
−3
−2 Real Axis
−1
0
1
−10 −5
0.15 20
20 0.3 10
0
−4
40
40
0.3 10
−10 −5
50 0.095
30
20
0.3 10
60
60 Imaginary Axis
Imaginary Axis
40
40
80
70 60
60
50 0.095
0.009
0.064
70 60
60
0.02
80
0.064
70
0.032
90
80
0.064
Imaginary Axis
100
0.009
90
0
−4
−3
(a) nom case
−2 Real Axis
−1
(b) lf case
0
1
−10 −5
−4
−3
−2 Real Axis
−1
0
1
(c) hf case
Fig. 6: Closed-loop poles
α (deg) 5
0
−5 25
30
35
40
45
50
55
60
Time (s) θ (deg) 2 hf nom lf
1 0 −1 25
30
35
40
45
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55
60
45
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Time (s) βc (deg) 0
−5
−10 25
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40 Time (s)
Fig. 7: Time-responses to a worst-case wind profile
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