Stabilization of Unstable Aircraft Dynamics under Control Constraints
4 STABILITY SUBOPTIMAL CONTROLLER FOR TWO UNSTABLE EIGENVALUES
0
(xT(t)x(t) + uT(t)u(t))dt, (11)
has been synthesized for comparison purposes. The LQR stability region boundaries (dashed lines) have been computed by direct closed-loop system simulation and they are shown in Fig. 1 (see the top and left middle plots). One can see that the maximum stabi-lizer rejects the level of external disturbances approximately three times bigger than LQR.
In the bottom plots in Fig. 1 several trajectories of the closed-loop system with LQR and maximum stabilizer controllers are shown to illustrate the size of stability regions.
4 STABILITY SUBOPTIMAL CONTROLLER FOR TWO UNSTABLE EIGENVALUES
In this case the unstable subsystem is described by a second-order linear system
˙z(t) = Azz(t) + Bzu(t), whereAz∈ R2,Bz∈ R2×m,u(t) ∈ U and z(t) = QT2x(t).
4.1 Controllability Region Computation
A variational method for controllability region computation for an unstable system with only unstable eigenvalues has been developed by Formalsky [6].
The idea of this method is in maximization of a scalar linear functionf(z) = lTz considering points in the controllability region. The maximization process is performed toward the directionl, so the final solution coincides with the point zlon the boundary of controllability regionC having vector l normal to the boundary at this point (see the top left plot in Fig. 2):
zl= arg max
z∈C(lTz).
A solution of this task always exists, because the controllability region from Theo-rem 2 is totally bounded. The optimization task may be solved for different directionslk
and obtained points zlk can approximate the controllability region boundary as well, as necessary, because according to Theorem 1 the controllability region is convex.
To compute the controllability region boundary, let us consider a null-controllable setCtof all points in the state-space, from which the system can be moved to the origin in the presence of control constraints in a time less than or equal tot. Note that the boundaries of null-controllable sets approach the controllability region boundary ast → ∞.
The points in anull-controllable set Ctcan be expressed explicitly as initial points z(0) for the Cauchy problem through the applied control function u(τ) on the time interval [0, t]. The general solution of system (1),
z(t) = eAztz(0) +
t
0
eAz(t−τ)Bzu(τ)dτ, z(t) = 0,
can be used to describe null-controllable sets as follows:
Ct= {z|z = −
t
0
e−AzτBzu(τ)dτ, u(τ) ∈ U}.
The optimization task for maximizing a linear functionlTz on Cthas only one solu-tionzl:
zl= arg max
z∈CtlTz = −
m i=1
t
0
e−Azτbziu(i)maxsign(−lTe−Azτbzi)dτ, (12)
wherebziis a column of matrixBz(see the top left plot in Fig. 2).
Computing boundary pointszlkof anull-controllable set Ctfor a number of different directionslk,
lk = (cos ϕk, sin ϕk)T; ϕk =2πk
N , k = 1, N, we can approximateCtrather accurately.
Considering a sequence ofti= ti−1+ ∆ approaching infinity ti→ ∞ we will have Cti→ C. This convergence process can be stopped if
maxk ||zlk(t) − zlk(t + ∆)||2≤ ., where∆ is a small time increment and . is required accuracy.
4.2 Stability Suboptimal Linear Controller
The design objectives in this case are similar to the previous design with one real eigen-value. A saturated linear controlleru = sat(Kz) has to stabilize the closed-loop system in the absence of control saturation and prevent leaving the controllability region when sys-tem operates close to its boundary. We will consider anull-controllable set Ctwith large enought as an approximation of controllability region C.
In every boundary point ofCtthe control function following (12) is expressed in the form of relay law:
u(i)(0) = u(i)maxsign(−lTbzi), i = 1, m, (13) wherel is a vector normal to the boundary of Ctat this point.
Each component of control vector takes on the boundary ofCtat its positive or neg-ative extreme value and there are some points where it changes sign. Because any null-controllable set is symmetrical, the points on the boundary of Ct, where control changes its sign, are also symmetrical.
These points±z0ifor every component of control vectoru(i)are defined by the con-dition that the vectors±li, which are normal to the boundary ofCtat these points, are also normal to controllability vectorbzi, i.e.,liTbzi= 0 (see the top right plot in Fig. 2).
A relay control law
u(i)(z) = u(i)maxsign(kiTz), i = 1, m, (14)
112 M.G. Goman and M.N. Demenkov
where vectorski are normal to switch lines passing through±z0i, can produce at every boundary point ofCtthe same direction of control vector as in (13), and therefore prevent the system from moving outside of Ct. The direction of ki is defined from inequality kiTbzi < 0. In this case produced control vector bziu(i)(z) will “push” the system state toward the controllability region interior.
We will consider the linear control law withkivectors from relay control law (14):
u(i)= γkTi∗z, where ki∗ =kiu(i)max
||ki||2 . (15)
At the presence of control deflection constraints the control functions become nonlin-ear and can be expressed as
u(i)(z) = u(i)maxsat(γki∗Tz), i = 1, m.
The switching lines of relay control law (14) in the phase plane, defined bykiTz = 0, are transformed into “strips,” where the closed-loop system is free from control saturation and therefore linear.
Parameterγ, which is the same for all control components, allows one to vary the width of these “strips” and thus to find the compromise between the local stability charac-teristics expressed in terms of eigenvalues location and the size of the closed-loop system stability region. Of course, the stability region of this saturated linear control law is less than the controllability region, but in practice it can approach it very closely.
Example 2. As a design example, consider the longitudinal dynamics of an aircraft trimmed at altitude H = 25000 ft and Mach number M = 0.9 [1, 15]. The linearized system of motion equations has the following stability and controllability matrices:
[A|B] =
−0.0226 −36.6170 −18.8970 −32.0900 3.2509 −0.7626 0 0 0.0001 −1.8997 0.9831 −0.0007 −0.1708 −0.0050 0 0 0.0123 11.7200 −2.6316 0.0009 −31.6040 22.3960 0 0
0 0 1 0 0 0 0 0
The first part of the state vector consists of the aircraft’s basic rigid body variables:
perturbation in the forward velocity δV , the angle of attack α, the pitch rate q, and the pitch attitude angleθ. Two first-order lags x(5)andx(6) are appended to the state vector to represent actuators dynamics. The control variablesδeandδcare signals sent to elevon and canard actuators. In the control law design the following constraints are considered:
|δe| ≤ 0.035, |δc| ≤ 0.035. Note that the angles and angular rates are expressed in radians and radians per second, respectively.
The system due to aircraft aerodynamic static instability has an oscillatory unstable mode in the phugoid frequency range, which can be separated off via the Schur decompo-sition:
[Az|Bz] =
0.7067 2.6227 5.1766 −3.6583
−0.0237 0.6729 6.7815 −4.6968
,
z(t) = QT2x(t), QT2
=
0.0098 −0.0377 −0.1852 −0.9589 0.1726 −0.1219
−0.0036 −0.9269 −0.2146 0.1385 0.2260 −0.1566
.
Using the Formalsky method, thenull-controllable sets Cthave been computed until convergence to the boundary of controllability regionC. In this example, the convergence process has been stopped att = 10 sec. In the middle left plot in Fig. 2 the null-controllable sets are plotted with time increment∆ = 0.2 sec in the interval t = [0, 10] sec. The null-controllable sets were approximated by 120 points.
To design linear control law (15), the vectorsli, which are normal to the boundary of C10at the points where control functions change the sign (lTi bzi = 0, i = 1, 2), have been defined:
l1= (b(2)z1, −b(1)z1)T = (6.7815, −5.1766)T, l2= (b(2)z2, −b(1)z2)T = (−4.6968, 3.6583)T.
Now the control switching points on the boundary of controllability region, approxi-mated byC10, can be computed:
z01= arg max
z∈C10lT1z
= −0.035
2 i=1
10
0
e−Azτbzisign(−lT1e−Azτbzi)dτ = (1.5703, −0.5421)T,
z02= arg max
z∈C10lT2z
= −0.035
2 i=1
10
0
e−Azτbzisign(−lT2e−Azτbzi)dτ = (−1.5703, 0.5421)T.
In our case there are two switching lines passing through points±z01and±z02 on the controllability region boundary, and these lines are numerically the same for each of two controls. These switching lines are described by the following equations:
kT1z = 0, kT2z = 0,
wherekiare defined as
k1= (z01(2), −z01(1))T = (−0.5421, −1.5703)T, k2= (z02(2), −z02(1))T = (0.5421, 1.5703)T.
Since kT1bz1< 0, k2Tbz2< 0, these kialready have correct signs.
In the middle right plot in Fig. 2 the root-locus for the closed-loop system with control law (15) is shown forγ = 0 ÷ 10. Parameter γ = 6 is taken as a candidate for the “stability suboptimal” control law.
114 M.G. Goman and M.N. Demenkov
Figure 2 The Formalsky method (top left plot), relay law computation (top right plot), null-controllable sets (mid-dle left plot), closed-loop system root locus (mid(mid-dle right plot), stability and controllability regions for unstable subsystem (bottom left plot), cross-sections of stability regions in the plane (α, θ) (bottom right plot).
In the state-space of the original system this suboptimal control law has the form u(x) = sat(Ksubx), where
Ksub=
6||k0.0351||2kT1QT2 6||k0.0352||2kT2QT2
=
0 0.186 0.0552 0.0382 −0.0566 0.0394 0 −0.186 −0.0552 −0.0382 0.0566 −0.0394
.
The comparison of the stability region for the designed “stability suboptimal” control law and the open-loop system controllability region is presented in the bottom left and right plots in Fig. 2 in the plane of unstable variables and in physical plane (α , θ). It is clearly seen that the closed-loop system stability region with designed linear control is rather close to the controllability region.
In the two bottom plots in Fig. 2 one can see also the cross-sections of stability regions for two linear quadratic regulators designed for comparison purposes. Linear quadratic regulators have been computed by minimizing the performance index (11): one considering only unstable subspace LQR1 and the other for the original system including stable and unstable subspaces LQR2.
One can see that the stability region for LQR1is bigger than the region for the LQR2, because LQR1stabilizes only the unstable subsystem, while LQR2allocates some control resourses for the stable subspace and leaves less for the unstable one. Meanwhile the designed stability suboptimal control law provides a much larger stability region than both linear quadratic regulators and its size is very close to that of the controllability region.