Linear Time-Invariant subsystem decoupling

This section investigates the decoupling of the unstable, symmetric and asymmetric flutter modes from the rigid body dynamics at a single 60 m/s airspeed. The results presented here, are building on the decoupling algorithm developed in Chapter 3. For highlighting the benefits of the proposed input and output transformations, three sample stabilizing control laws are synthesized: static output-feedback, static state-feedback and dynamic output-feedback.

Flutter decoupling

The aim is to synthesize input and output transformations for the aircraft model at 60 m/s airspeed, which maximize the transfer through the flutter modes, while they are limiting the interactions with the rigid body dynamics. For this purpose the model is brought to the (3.2) subsystem form, where Gc(s) contains the flutter modes, while the rigid body dynamics are collected in Gd(s).

A necessary condition for static output-feedback design is that, the blended system satisfies n_u,b ×n_y,b ≥ n_x [135]. In order to satisfy this constraint, K_u ∈ R^nu×nu,b and K_y ∈ R^ny×ny,b blend matrices were synthesized, where n_u,b and n_y,b denote the input and output dimensions after the blend matrices are applied to the system. The synthesis of such blend matrices has

0.01 0.1 1 10 100 1000 10000

−200

−100 0

frequency [rad/s]

magnitude[dB]

K_uGc(s)K_y^T K_uGd(s)K_y^T

Figure 6.4: LTI example: singular value ranges of the blended subsystems

been discussed in Chapter 3. In terms of K_u, this modification means that, after computing an initial K_u0 blend matrix, its largest n_u−n_u,b singular vectors have to be retained. Then the alternating projection sequences have to be evaluated only nu−n_u,b times, which actually reduces the number of necessary evaluations compared to the k_u blend vector design, where n_u−1 projection sequences are needed. The same applies for theK_y output blend matrix. Note that the selection ofn_u,b and n_y,b is problem dependent. The frequency range where decoupling should be achieved is characterized by the [0, ω_n] interval, where ω_n has been selected as the maximum natural frequency of the Gc(s) controlled modes.

For the decoupling of the two complex flutter modes (nx = 4), n_u,b = 4 and n_y,b = 5 have been selected. The available inputs were the 8 ailerons, and the four ruddervators. They_L3,y_L6, y_R3,y_R6IMUs provided theω_x,i,ω_y,i,ω_z,iangular rate measurements on the wings, and thep,q, r signals were measured at the center of gravity. The resulting Ku and Ky blend matrices were applied to the subsystems, yielding the result shown in Figure 6.4. Note that, each subsystem has four singular value curves, which lie in the shaded intervals. It is obvious that, a high level of decoupling is achieved, since the largest singular value ofKuGd(s)Ky is significantly less than the maximum corresponding to the controlled subsystem. Furthermore, the maximum gain of the decoupled subsystems is only slightly higher that the minimum amplification ofK_uGc(s)K_y. Static output-feedback

The objective of the static output-feedback control is the stabilization of the unstable flutter modes, and so drive any nonzero initial conditions to zero. For this a partial pole placement technique is proposed, which only affects the targeted poles and leaves the remaining ones untouched. In this example the proposedK_u and K_y blend matrices assure that, the rigid body poles are not affected by the feedback law. The static output-feedback design problem is defined next.

Problem 5.: The static output-feedback problem [69]. Given the system

˙

x=Ax+Bu,

y=Cx, (6.1)

withx∈R^nx,u∈R^nu, andy∈R^ny. Find the feedback coefficient matrixF ∈R^nu×ny such that, with

u=−F y, (6.2)

the closed-loop system is stable, and it satisfies performance objectives prescribed by the J = 1

2 Z∞ 0

x^TQx+u^TRu

dt, (6.3)

quadratic cost function, whereQ and R are symmetric positive semidefinite weighting ma-trices ^a.

aThe relative magnitudes ofQandRrepresent design tradeoffs between the magnitude of the states and the necessary control energy. Larger elements inR will penalize the control energy and so it will decrease.

In order to achieve faster convergence of the states to zero, one may select largerQ.

Note that by substituting (6.2) to (6.3), it yields J = 1

2 Z∞ 0

x^T

Q+C^TF^TRF C x

dt, (6.4)

which is not linear in the feedback matrix F. It has been shown that, for finding such real F feedback matrix, a necessary condition is n_u×n_y ≥n_x [135]. The solution of Problem 5 leads to a set of coupled nonlinear matrix equations. A detailed overview of suitable algorithms for finding anF output-feedback matrix is given in [121]. The applied iterative method, which finds a sub-optimal F starting from an initial F₀ is discussed in Chapter 8 of [69].

For the synthesis of the Ku and Ky blend matrices, the system is already brought to the subsystem form in (3.2), and suppose that D= 0, which is a necessary condition according to (6.1). Then, by applying the F feedback, the closed-loop state matrix is given as

A_c 0 0 A_d

− B_c

B_d

F

C_c C_d

=

A_c−B_cF C_c −B_cF C_d

−B_dF Cc A_d−B_dF C_d

. (6.5)

Note that, a generalF output-feedback matrix will both affect the controlled and the decoupled subsystems as well. Furthermore, from the off-diagonal terms it is obvious that the subsystems are coupled. By applying the synthesizedK_u and K_y^T blend matrices, the closed-loop dynamics yields

A_c 0 0 A_d

− B_c

B_d

KuF K_y^T

Cc C_d

=







A_c−B_cK_uF K_y^TC_c −B_cK_uF

≈0

z }| { K_y^TC_d

−B_dKu

| {z }

≈0

F K_y^TCc A_d−B_dKu

| {z }

≈0

F K_y^TC_d

| {z }

≈0







≈

A_c−B_cK_uF K_y^TC_c 0

0 A_d

.

(6.6)

The blend matrices are designed such that, they are minimizing the transfer through the de-coupled subsystem. By the assumptions B_dK_u ≈ 0 and K_y^TC_d ≈ 0, (6.6) is approximately blockdiagonal, where theA_d block is not affected by the feedback. However, note that the cross

terms on the off-diagonal are such that, theB_cK_uandK_y^TC_care maximizing the transfer through the targeted subsystem. This shows that, a successful decoupling necessitates ||B_dK_u||2 ≪ 1 and ||K_y^TC_d||2 ≪1, and in the extreme case ofK_u ∈ker(B_d),K_y ∈ker(C_d^T), the approach may lead to perfect decoupling. In light of the before mentioned arguments, partial pole placement is not guaranteed by the proposed method, but it is often achieved depending on the actual parameters of the underlying system.

Numerical result

The aim is to find a static output-feedback controller, which by using minimum control energy stabilizes the unstable flexible modes, while do not interact with the rigid body dynamics. For system stabilization, minimum control energy corresponds to the case, when the unstable poles are mirrored to the imaginary axis [67]. This is achieved, when the weighting matrices are selected asQ= 0 and R=I in (6.3).

Two output-feedback approaches were synthesized. The first one uses all available inputs and outputs, without any transformation. As intended it has mirrored the flexible poles, and left the rigid body poles unattained. The other approach uses theK_u ∈R^nu×nu,b andK_y ∈R^ny×ny,b input and output transformations to eliminate the coupling between the subsystems, withn_u,b = 4 and n_y,b= 5. The resultingF ∈R^4×5 output-feedback matrix mirrors the poles of the flutter modes, while the decoupled dynamics are unaltered. Note that, due to the decoupling, this output-feedback matrix solely depends on the controlled subsystem, i.e., it can be designed by substitutingA_c,B_c and C_cinto (6.1).

Figure 6.5 shows time domain simulation results of the closed-loops, for the two output-feedback controllers. In both cases, the subsystems were started from x^T_c0 =

20 20 20 20T

andx_d0= 0 initial conditions. This selection allows to quantify the level of interactions between the two subsystems, since if no coupling is present, x_d(t) = 0 for the whole simulation. For better visualization, all subfigure show the two most dominant signals, but the remaining ones stay in the shaded areas as well. The left column in Figure 6.5 shows the output-feedback stabilization without the decoupling transformations. In this case due to the cross couplings in (6.5) the Gd(s) subsystem is pushed out from the x_d0 = 0 initial condition. However, when the input and output blend matrices are applied, the couplings are significantly reduced, according to (6.6). This is shown in the right column of Figure 6.5.

Static state-feedback

When the states of the targeted subsystem are available for feedback, then it is possible to further reduce the couplings in (6.6). Along with the previous discussion, the goal is to synthesize an F static state-feedback matrix which stabilizes the unstable flutter modes, while not interacting with the rigid body dynamics. The static state-feedback problem is given by Problem 5, if (6.2) is replaced by u=−F x. The readily availablelqr command in MATLAB provides F, which is computed according to the Qand R weight matrices.

For a partial pole placement approach, partition F such that F =

Fc F_d

, and select F_d= 0. Then the closed-loop dynamics, with aK_u input blend matrix will have the form

A_cl=

A_c 0 0 A_d

− B_c

B_d

K_u F_c 0

=



A_c−B_cK_uF_c 0

−B_dKuFc

| {z }

≈0

A_d



≈

A_c−B_cK_uK_c 0

0 A_d

. (6.7) The cross coupling term in the upper right corner of A_cl is zero due to the F_d = 0 selection.

Furthermore, since BdKu ≈0 for a successful decoupling, the effect of the second off-diagonal term is also suppressed, leading to an approximate partial pole placement. Similarly to the output-feedback case, the achievable reduction of the coupling is problem dependent.

−40

−20 0 20 40

xc(t)

Stabilize without blend

−40

−20 0 20 40

Stabilize with blend

0 0.5 1 1.5 2

−5 0 5

time [s]

xd(t)

0 0.5 1 1.5 2

0.1 0.05 0

−0.05

time [s]

Figure 6.5: LTI example: numerical simulation with static output-feedback (with for con-trolled and for decoupled subsystems)

Numerical result

For the example the same aircraft model is used as in the static output-feedback case, and mini-mum energy stabilizing state-feedback controllers were synthesized by MATLAB’slqr command, with Q= 0 andR =I. Two state-feedback gain matrices were designed, with and without the K_u blend matrix. The time domain simulation is carried out in the same vein as in the previous example, and its result is shown in Figure 6.6. Note that for the unblended case, the oscillations in the decoupled subsystem have reduced by 50 % compared to the output-feedback design.

Since K_u andK_y have already achieved a high level of decoupling in the output-feedback prob-lem, no significant improvements are observable when the blend matrices are applied. However, as (6.7) shows, state-feedback technique has clear benefits over the output-feedback, when the level of output decoupling is not satisfactory (K_y^TC_d̸≈0).

Dynamic output-feedback

Up to this point it has been shown how to synthesize a state-feedback gain, which only interacts with the targeted dynamics. However, when the states are not available, their values have to be reconstructed by dynamic observers. Due to the fact that, the feedback gain only interacts with a subset of the states, in the forthcoming the special case is investigated, when only these states are recovered. This leads to a smaller order observer, which on the other hand provides guarantees on the worst-case estimation error, and the rate of convergence. Together with the state-feedback gain, this observer constitutes a dynamic controller, which has the same order as the controlled subsystem.

Next, the design of an estimator is discussed, which minimizes the effects of the decoupled subsystem on the estimation error. Before proceeding, the definition of α stability is intro-duced.

−40

−20 0 20 40

xc(t)

Stabilize without blend

−40

−20 0 20 40

Stabilize with blend

0 0.5 1 1.5 2

−2 0 2

time [s]

xd(t)

0 0.5 1 1.5 2

0.1 0.05 0

−0.05

time [s]

Figure 6.6: LTI example: numerical simulation with state-feedback (with for controlled and for decoupled subsystems)

Lemma 18.: The α stability [47]. Let A be a given square matrix and α be a given positive scalar. Then the following statements are equivalent.

a, The system ˙x(t) =Ax(t) isα stable.

b, There exists a matrix X≻0 such that (A+αI)^T X+X(A+αI)≺0.

Note that α stability guarantees that the free response of the system decays to zero faster than e^−αt [47].

Suppose that the Ky output transformation matrix is already calculated, and the system is given in the subsystem form (3.2). Then a linear observer for the controlled subsystem is characterized by the state-space equation^*

ˆ˙

x_c(t) =Exˆ_c(t) +Hu(t) +Gy(t). (6.8) The estimation error is defined asec=xc−xˆc, and the error dynamics (with blended input) is

˙

e_c=A_cx_c+B_cK_uu−Exˆ−Hu−GK_y^TC_cx_c−GK_y^TC_dx_d. (6.9) Selecting H =BcKu and E = Ac−GK_y^TCc, the LTI system describing the error dynamics is

given as

˙ e_c e_c

=

Ac−GK_y^TCc

−GK_y^TC_d

I 0

e_c x_d

. (6.10)

The substitution of (6.10) into the 2×2 block form of the Bounded Real Lemma (see Lemma 8), tells that, the error dynamics is stable and the maximum error is less than γ if there exists P ≻0 such that

P A_c−P GK_y^TC_c+A^T_cP−C_c^TK_yG^TP+I −P GK_y^TC_d

−C_d^TK_yG^TP −γ²I

≺0. (6.11)

*The time dependency is omitted in the rest of the section to ease the notation.

0 0.5 1 1.5 2

−40

−20 0 20 40

time [s]

xc(t)

0 0.5 1 1.5 2

0.1 0.05 0

−0.05

−0.1

time [s]

xd(t)

Figure 6.7: LTI example: numerical simulation with dynamic output-feedback (with for controlled and for decoupled subsystems)

Note that this is a bilinear matrix inequality which can be linearized by selecting Y =P G, to

give

P Ac−Y K_y^TCc+A^T_cP −C_c^TKyY^T +I −Y K_y^TC_d

−C_d^TK_yY^T −γ²I

≺0. (6.12)

According to Lemma 18, the error dynamics is α stable, and the peak estimation error is less thanγ if the

P Ac−Y K_y^TCc+A^T_cP−C_c^TKyY^T +I+ 2αP −Y K_y^TC_d

−C_d^TK_yY^T −γ²I

≺0 (6.13)

inequality holds. After solving for the minimal γ, the observer gain matrix is found as G = P⁻¹Y. Note that K_y may be neglected (merged into the G optimization variable) and so one may look for the observer gain over all available measurements. By closing the feedback loop through theF state-feedback gain and the (6.8) dynamic observer, the approach directly allows to synthesize dynamic controllers for the targeted subsystem, which has order equal to the controlled subsystem, and reduces interaction with the decoupled dynamics.

Remark 15. Note that this dynamic controller design is fairly straightforward. However, by having optimal K_u and K_y blend matrices, more sophisticated observers and controllers can be designed relying on Kalman filtering or H∞ control results. In these cases one needs to carry out the synthesis for the targeted subsystem with applied input and output blends. TheK_u and K_y matrices will then assure the limited interaction with the decoupled subsystems.

Numerical result

For the time domain simulation, a dynamic output-feedback controller is composed in the form ˆ˙

x_c(t) =Exˆ_c(t) +Hu(t) +Gy(t),

u(t) =KuFxˆc. (6.14)

The dynamic observer is designed according to (6.13), where the K_y output blend has been neglected, and α = 10. The same F matrix had been used as for the state-feedback example.

Figure 6.7 shows the results with dynamic output-feedback controller. The controller stabi-lizes the flexible subsystem, but the initial oscillations have slightly larger amplitudes for both subsystems, than for the state-feedback design.

The estimation error of the controlled states are shown in Figure 6.8. The two most significant estimation errors are plotted, the remaining two also stay in the shaded interval, which has a decay rate of e^−10t. Note that this has been a design specification, by selectingα= 10.

0 0.07 0.15 0.23 0.3

−20 0 20

time [s]

ec(t)

Figure 6.8: LTI example: estimation error dynamics

In document Optimal Decoupling of Dynamical Systems (Pldal 106-113)