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ECE/MAE7360. Robust and Optimal Control.
Electrical and Computer Engineering, Utah State University
Project #2: Space-shuttle robustness analysis (stability and
performance)
Submit via e-mail only.
1. Project Objectives: Become expert in doing robustness
(stability/performance) analysis using MATLAB mu-synthesis Toolbox.
2. System model and sample analysis (see the attached.)
3. Required Investigations.
3.1. Assuming no delay in the actuator delay (refer to page 7-48, the rudder
model), repeat all the steps in the space-shuttle robustness analysis (page 7-
44 to page 7-74) using the existing given data set. (Note, assume here no
delay in both rudder and elevon!)
3.2. Assuming that the natural frequency and the damping factor for BOTH
rudder and elevon are all scaled down to 90% of its original value (i.e.,
ξ ele =0.720.90, ωele =140.90 rad/sec. and ξrud =0.750.90, ωrud =210.
rad/sec.), repeat all the steps in the space-shuttle robustness analysis (page
7-44 to page 7-74) using the existing given data set. (Note, we need to
include the delay models of both rudder and elevon in this case!)
3.3. Compare three sets of analysis results (original/sample demo, delay
removed, and actuator natural frequency/damping factor down-scaled) and
make your comments and possible suggestions.
Note:
Project report format:
(1) A suitable title.
(2) Introduction of the robustness analysis problem.
(3) Effect of actuator delay (with vs. without).
(4) Effect of changes in actuator's damping factor and natural frequency.
(5) Summary and comments.
(6) References
Bonus: Make your Project Report self-containing using MS Word or LaTeX.
Attached: A sample robustness analysis. (Extracted from mu-Toolbox User's Guide)
7 Robust Control Examples
Space Shuttle Robustness Analysis
This section outlines a robust stability and robust performance analysis of the Space Shuttle lateral axis flight control system during re-entry. It serves as a general illustration of the usefulness of the real and complex μ analysis methods. The system is a simplified model of the Space Shuttle, in the final stages of landing, as it transitions from supersonic to subsonic speeds. The material in this chapter is based on the paper:
Doyle, J., K. Lenz, and A. Packard, “Design Examples Using μ Synthesis: Space Shuttle Lateral Axis FCS During Re-entry,” NATO ASI Series, Modelling, Robustness, and Sensitivity Reduction in Control Systems , vol. 34, Springer-Verlag, 1987. The analysis procedure involves several steps:
1 Build uncertain model of plant.
2 Define performance specifications and uncertainty bounds.
3 Construct open-loop interconnection.
4 Close feedback loop with controller.
5 Perform a variety of real and complex μ analysis tests on the closed-loop system, and explore the impact of the uncertainty model (real vs. complex) on the robust stability and robust performance requirements.
6 Construct worst-case perturbations, and see their effect on the closed-loop system in the frequency and time domain.
7 Robust Control Examples
All variables in y are measured with inertial devices (gyroscopes and accelerometers) whose individual noise characteristics are discussed later.
Aircraft Model: Aerodynamic Uncertainty
The major source of uncertainty in the aircraft model (AC) is in the aerodynamic coefficients. These are standard aerodynamic parameters which express incremental forces and torques generated by incremental changes in sideslip, elevon, and rudder angles. This is a linear relationship, expressed as
The coefficients c •• are typically estimated based on theoretical predictions, numerical calculations, experiments in wind tunnels, and flight tests. At Mach 0.9, the shuttle is in a transonic regime involving a combination of subsonic and supersonic flows. Theoretical, computational, and wind tunnel techniques are inaccurate at this flight condition, so with extremely limited flight data (early in the shuttle program), the coefficient uncertainty for the shuttle model is unusually large.
Uncertainty in these coefficients is modeled as a nominal value, plus a perturbation.
where the values of the r •• are
side force yawing moment rolling moment
c (^) y β c (^) ya c (^) yr c ηβ c η a c η r c (^) l β c (^) la c (^) lr
β θele θrud
c (^) y β c (^) ya c (^) yr c ηβ c η a c η r c (^) l β c (^) la c (^) lr
c (^) y β c (^) ya c (^) yr c ηβ c η a c η r c (^) l β c (^) la c (^) lr
r (^) y β δ y β r (^) ya δ ya r (^) yr δ yr r ηβ δ (^) ηβ r η a δ (^) η a r η r δ (^) η r r (^) l β δ l β r (^) la δ la r (^) lr δ lr
Space Shuttle Robustness Analysis
and the perturbations δ•• are assumed to be fixed, unknown, real parameters, with each satisfying |δ••| ≤ 1. We use the notation r ••. * δ•• to denote the 3 × 3 perturbation matrix in the model for the aero coefficients, c ••.
The aircraft model acnom has the nominal aerodynamic coefficients absorbed into the state-space data. In addition to the inputs μ and outputs y described earlier, acnom has three fictitious inputs and outputs such that the uncertain behavior of the aircraft AC is given by the linear fractional transformation in Figure 7-33.
The state-space model for acnom is created by the M-file mk_acnom. A listing of state-space model acnom is given in “Shuttle Rigid Body Model” at the end of this section.
Figure 7-33: Uncertain Aircraft Model
r (^) y β r (^) ya r (^) yr r ηβ r η a r η r r (^) l β r (^) la r (^) lr
Space Shuttle Robustness Analysis
deflections, rates, and accelerations of the control surfaces, the state-space models created in mk_act each have three outputs, as shown below.
Exogenous Disturbances, Noises, and Commands
There are three sources of exogenous signals:
- Wind gusts - Sensor noise - Pilot bank-angle command
In the H ∞ framework, all time domain signals are modeled as the unit ball in
L 2 , filtered by problem dependent weighting functions which reflect typically
occurring signals in the application. In addition to the L 2 gain, the H ∞ norm
also has an interpretation in terms of gain from sinusoids to sinusoids. Now, suppose h represents one of the exogenous signals, and W (^) h is the associated stable weighting function. Then, the signal h is assumed to be any signal from the set
h ∈ { W (^) h η h : ||η h || 2 ≤ 1}
By choosing the form of W (^) h ( s ), the spectral content of such signals h can be shaped.
- Lateral Wind Gusts: The set of lateral wind gusts is modeled as
The set on the right-hand side of the equation models the typical wind gusts that the shuttle will encounter at this flight condition.
actrud
urud
rud ^ _rud ^ rud
actele
uele
ele ^ _ele ^ ele
d gust W gustηgust : W gust = 30 1 + s ⁄ 2 1 + s -------------------, ηgust 2 ≤ 1
7 Robust Control Examples
- Sensor Noise: Each measurement is corrupted with sensor noise which becomes more severe with increasing frequency. Since p and r are measured with comparable gyroscopes, their sensor noise weights are identical,
These weighting functions imply a low frequency measurement error in p and r of 0.0003 rads/sec, and a high frequency error of 0.015 rads/sec. The model of the measured value of p , denoted p meas , is given by p meas = p + W (^) p η p where η p is an arbitrary signal, with ||η p || 2 ≤ 1. This type of weighted, additive
L 2 sensor noise is assumed for each of the 4 measured variables.
The measurement of φ is obtained from a navigation package at a reduced sample rate, so its weight is chosen to be
which is relatively large in the mid-to-high frequency range. The sensor noise weight on the n (^) y accelerometer is
For the variables r , φ, and n (^) y , we have
- Pilot Bank-Angle Command: In this problem, the pilot (or autopilot) takes the shuttle through a series of sweeping “S” turns to slow the vehicle down.
W (^) P W (^) r 0.0003^1 + s /0. 1 + s /0.
W φ 0. 1 + s /0. 1 + s /
W (^) n (^) y 0. 1 + s /0. 1 + s /
r meas = r + W (^) r η r φmeas =φ + W φ η (^) φ n (^) y meas
= n (^) y + W (^) n (^) y η n (^) y
7 Robust Control Examples
This performance specification can be loosely interpreted as a requirement that the closed-loop system should, under the excitation of the modeled exogenous signals, maintain θele to below 0.25 radians, to below 1 rad/ sec, to below 200 rads/sec 2 , and so on for the rudder variables. For notational purposes, let W act be the 6 × 6 constant matrix so that
- Performance variables: - The ideal bank angle response (φideal) of the shuttle to a bank-angle command (φcmd ) is
where ω = 1.2 rad/sec, and ξ = 0.7. The bank-angle tracking error is defined as φ – φideal.
- Turn coordination: in an ideal turn, the bank angle, and the yaw rate are related. For this aircraft, a turn coordination error is defined as r (^) p := r – 0.037φ - In a turn, it is desired that the pilot feel very little lateral acceleration, hence, the lateral acceleration variable, n (^) y , is an error. These error signals are weighted by frequency dependent weights to give a performance error vector as
θ^ ·^ ele θele
..
e act W act
θele
θ^ ·^ ele θele θrud
θ^ ·^ rud θrud
..
..
φideal :=
1 2 ξ(s/ω ) (s/ω ) 2
---------------------------------------------------------φcmd
Space Shuttle Robustness Analysis
For notational purposes, let W (^) p erf be a 3 × 5 transfer function matrix so that
The error weight on the lateral acceleration indicates a tolerance for low frequency accelerations of 1.25 ft/sec^2 , which is relaxed at high frequency, allowing accelerations up to 12.5 ft/sec^2. Again, these specifications correspond to n (^) y errors produced by the exogenous signal set (wind gusts, measurement noises, and bank angle commands). Similar interpretation is given to the other performance variables.
LFT Aero-Coefficient Uncertainty
The perturbations in the aero-coefficients can be written as an LFT (linear fractional transformation) on a structured uncertainty matrix. Define constant matrices W (^) L ∈ R^3 ×^9 and W (^) R ∈ R^9 ×^3 such that
for all δ••. This is easily done with the permutation matrices WL and W (^) R shown below.
e perf :=
0.8 1 ---------------------- +^1 + s /0.1 s^0
0 500 1 ------------------------- +^1 s^ +/0.01 s - 0
0 0 250 1 ------------------------- +^1 s^ +/0.01 s -
n (^) y r – 0.037φ φ – φideal
e perf W perf
p r n (^) y φ φideal
W (^) L ⋅ diag[δ y β,δ (^) ηβ,δ l β,δ ya ,…,δ lr ] ⋅ W (^) R
r (^) y β δ y β r (^) ya δ ya r (^) yr δ yr r ηβ δ (^) ηβ r η a δ (^) η a r η r δ (^) η r r (^) l β δ l β r (^) la δ la r (^) lr δ lr
Space Shuttle Robustness Analysis
Figure 7-34: Shuttle Interconnection Structure
The M-file mk_olic uses the sysic command to create a SYSTEM matrix description of the open-loop interconnection structure. In the workspace, the open-loop system is denoted by olic, and has 23 states, 23 outputs, and 17 inputs.
mk_olic; minfo(olic)
A schematic diagram, with the specific input/output ordering for olic, is shown in Figure 7-35.
Ideal bank angle response model
cmdcmd - W cmd - (^) cmd
noisy(p,r ,ny , ) rudder cmd
p r ny
Wp erf (^)
ep erf (p,r^ ,ny^ , )
acnom
pertoutf-1-9g
Wr (^) Wl
pertinf1-9g
gust Wgust
actrud
^ rud
ele^ actele
Wact
eact
7 Robust Control Examples
Figure 7-35: Schematic Diagram of Space Shuttle olic
olic
pertinf 1 g pertinf 2 g pertinf 3 g pertinf 4 g pertinf 5 g pertinf 6 g pertinf 7 g pertinf 8 g pertinf 9 g
exogenous disturbances
p r ny gust cmd elevon cmd u rudder cmd
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17
pertoutf 1 g pertoutf 2 g pertoutf 3 g pertoutf 4 g pertoutf 5 g pertoutf 6 g pertoutf 7 g pertoutf 8 g pertoutf 9 g
ep erf
weighted ny weighted r weighted er r
eact
weighted elevon acc weighted elevon rate weighted elevon pos weighted rudder acc weighted rudder rate weighted rudder pos
y
cmd noisy p noisy r noisy ny noisy
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19 20 21 22 23
7 Robust Control Examples
The two other controllers have already been designed and stored in the file shutcont.mat. load shutcont minfo(k_x) minfo(k_mu)
Nominal Frequency Responses
The closed-loop system is constructed using the star product command starp.
In the closed-loop system, there are six exogenous signals (the six η signals: four sensor noises, wind gust, bank angle command) and nine errors (weighted performance error vector and the weighted actuator error vector). The nominal performance objective is that this multivariable transfer function matrix
should have an H ∞ norm less than 1. Using μ-Tools, it is easy to evaluate this
performance criterion. Simply form the closed-loop system, calculate its frequency response, and plot the norm of the appropriate transfer function versus frequency.
Space Shuttle Robustness Analysis
omega = logspace(-2,3,30); clp_h = starp(olic,k_h,5,2); clp_hg = frsp(clp_h,omega); clp_x = starp(olic,k_x,5,2); minfo(clp_x) clp_xg = frsp(clp_x,omega); minfo(clp_xg) clp_mu = starp(olic,k_mu,5,2); clp_mug = frsp(clp_mu,omega);
Note that the closed-loop systems have additional inputs and outputs from the nine aero-perturbation channels. The relevant exogenous signals and errors are selected (using sel) before calculating the maximum singular value (vnorm).
np_hg = sel(clp_hg,[10:18],[10:15]); np_xg = sel(clp_xg,[10:18],[10:15]); np_mug = sel(clp_mug,[10:18],[10:15]); vplot('liv,m',vnorm(sel(clp_hg,10:18,10:15)),... vnorm(sel(clp_xg,10:18,10:15)),... vnorm(sel(clp_mug,10:18,10:15))) title('NOMINAL PERFORMANCE: ALL CONTROLLERS')
Figure 7-36: Nominal Performance of k_h , k_x , and k_mu
10 −2^10 −1^100 101 102 0
1 NOMINAL PERFORMANCE: ALL CONTROLLERS
FREQUENCY (RAD/SEC)
H−INFINITY NORM
k_h − SOLID k_xk_mu − DOTTED − DASHED
Space Shuttle Robustness Analysis
Figure 7-37: Schematic Design of clp_RS
clp_hgRS = sel(clp_hg,1:9,1:9); clp_xgRS = sel(clp_xg,1:9,1:9); clp_mugRS = sel(clp_mug,1:9,1:9);
Calculate μ across frequency, and look at μ plots. Start with the complex uncertainty structure.
[bnds_h,dv_h,sens_h,rp_h]=mu(clp_hgRS,delsetrs_C); [bnds_x,dv_x,sens_x,rp_x]=mu(clp_xgRS,delsetrs_C); [bnds_mu,dv_mu,sens_mu,rp_mu]=mu(clp_mugRS,delsetrs_C); vplot('liv,d',bnds_h,'-',bnds_x,'--',bnds_mu,'-.') title('ROBUST STABILITY OF CLOSED-LOOP: COMPLEX')
clp RS
pertinf 1 g pertinf 2 g pertinf 3 g pertinf 4 g pertinf 5 g pertinf 6 g pertinf 7 g pertinf 8 g pertinf 9 g
1 2 3 4 5 6 7 8 9 pertoutf 1 g pertoutf 2 g pertoutf 3 g pertoutf 4 g pertoutf 5 g pertoutf 6 g pertoutf 7 g pertoutf 8 g pertoutf 9 g
1 2 3 4 5 6 7 8 9
7 Robust Control Examples
Figure 7-38: Complex Robust Stability μ Analysis of k_h , k_x , and k_mu
According to Figure 7-38, the k_mu controller has the best robust stability properties when the perturbations are treated as complex (dynamic). The peak of the lower bound, 0.9, implies that there is a diagonal complex perturbation of size, , that causes instability. The peak of the upper bound, approximately 0.99, implies that for diagonal perturbations smaller than , the closed-loop system remains stable. The gap between the upper and lower bound can be reduced by using the “c” option in the mu command. Without this option, the upper bound from mu is a computational approximation to
that can be refined (option “c”) at the expense of slower execution. Using the “c” option reduces the upper bound peak to 0.9, so that the complex μ analysis gives a tight estimate on the size of the smallest destabilizing perturbation. Similar interpretations are possible for the closed-loop systems with controllers k_h and k_x, though, since the μ plots have larger peaks, the bound on allowable perturbations is smaller. Hence, the closed-loop system with the
0
1
2
3
10 -2^10 -1^10 0 10 1 10
ROBUST STABILITY OF CLOSED-LOOP: COMPLEX
FREQUENCY (RAD/SEC)
MU
k_h - SOLID k_x - DASHED k_mu - DOTTED
k_h (solid line) k_x (dashed line) k_mu (dotted line)
1 0.9^ -------- 1 0.99^ -----------
inf σ DMD
