Output blend calculation

Algorithm 1 Input blend calculation with alternating projection

1: Given: The subsystemsGc and Gd are given in a form as shown in Figure 3.2.

2: Initialization: Solve the following optimization problem, forβ²,γ²,X_d,X_c,Y,K_u: minimize

Xd, Ku, Xc, Y, β², γ²−β²+γ² s.t.:

X_dA^T_d +A_dX_d+B_dK_uB_d^T X_dI_d^T

I_dX_d −γ²I

⪯0, A^T_c I^T_c

I 0 T

Ξ

A^T_c I^T_c I 0

+

B_c^T 0

0 I

T

Π

B_c^T 0

0 I

≺0, Ku ∈S^nu,0⪯Ku ⪯I, Y ≻0.

3: Seti= 0

4: fork= 1 to nu−1 do

5: K_u^⋆_i =P_Γ^n−k

rankK_u

6: forj=1 to maximum number of iterations do

7: Solve the following optimization problem forX_c,Y,X_d,S,K_u: minimize

X_d, Ku, Xc, Y, Strace (S) s.t.:

X_dA^T_d +A_dX_d+B_dK_uB^T_d X_dI_d^T

I_dX_d −γ²I

⪯0, A^T_c I^T_c

I 0 T

Ξ

A^T_c I^T_c I 0

+

B_c^T 0

0 I

T

Π

B_c^T 0

0 I

≺0, Ku∈S^nu,0⪯Ku⪯I, Y ≻0,

S K_u−K_u^⋆_i K_u−K_u^⋆_i I

⪰0.

8: K_u^⋆_i+1 =P_Γ^n−k

rankK_u

9: if j >1 then

10: if ^|K

ui+1⋆ −K^⋆_ui|

|K_ui+1^⋆ | >thresholdthen

11: i=i+ 1

12: break for loop

13: end if

14: end if

15: i=i+ 1

16: end for

17: end for

18: From the K_u^⋆_i = U SV^T singular value decomposition find ku as the left singular vector corresponding to the largest singular value.

Gc(s)

Gd(s)

B_ck_u R A_c

C_c k_y

+

+ ¯yc

B_dk_u R A_d

C_d k_y

+

+ ¯y_d

¯u

Figure 3.4: Problem layout for output blend calculation controlled and to be decoupled are given by

Ac Bcku

I 0

T

Ξ

Ac Bcku

I 0

+

Cc 0 0 I

T

Π

Cc 0 0 I

≺0, (3.10)

and

A^T_dX_d+X_dA_d+C_d^TK_yC_d X_dB_dk_u k_u^TB_d^TX_d −γ²I

⪯0, (3.11)

respectively, where Π =

−Ky 0 0 β²I

. Note the introduced K_y = k_yk_y^T output blend matrix.

The optimization problem to be solved is given in Proposition 2 with variables X_c,Y,X_d,K_y, β²,γ².

Proposition 2 .: The output blend design. The optimal output blend vector k_y for the system given in (3.9) can be calculated as the left singular vector corresponding to the largest singular value of the blend matrix K_y, whereK_y satisfies the following optimization problem

minimize

Xd, Ky, Xc, Y, β², γ² −β²+γ²

subject to (3.10), (3.11),X_d∈S^nxd, X_d⪰0, X_c∈S^nxc, Y ∈H^nxc, Y ≻0,

K_y ∈S^ny,0⪯K_y ⪯I,and rank (K_y) = 1.

(3.12)

The rank one solution for the blend matrixKycan be achieved by a similar alternating projection algorithm as in the case of the input blend. This is summarized in Algorithm 2. By applying the output blend to each of the subsystems, they will have the from (with direct feedthrough)

˙

x_{c,d}(t) =A_{c,d}x_{c,d}(t) +B_{c,d}k_u

¯u(t),

¯y_{c,d}(t) =k^T_yC_{c,d}x_{c,d}(t) +k^T_yDku

¯u(t). (3.13)

Note that the direct feedthrough term was not involved into the optimization process which means that the optimal transformation of Dby the blend vectors ku and k_y^T is not guaranteed.

However, since theDterm is the same for each modes, it is possible to modify the overall control scheme presented in Figure 3.1, by introducing a k^T_yDk_u feedforward term as shown in Figure 3.5. Note that the proposed decoupling algorithm is extendable to discrete time systems as well by modifying the corresponding LMI conditions. Such an extension is discussed in [TB3].

Remark 4. It is possible to incorporate the Ddirect feedthrough term into the optimization problem, by slight modifications of the algorithm. Instead of using the modal outputs in (3.5) and (3.6), one may apply a k_y0 ∈ R^ny initial output blend vector to the subsystems. This

G(s) ku

Gc(s) Gd(s)

k^T_y

−λ_c(s) k^T_yDk_u

u ⁺

+

y +

¯y

¯u

-Figure 3.5: Modified closed-loop control scheme with input and output blending

would replace I_c and I_d by k^T_y0Cc and k^T_y0C_d, respectively, and retain the k^T_y0Dc, k_y0^T D_d terms at their corresponding locations. The k_y0 initial output blend vector has to be selected such that it does not turn any subsystems unobservable. Note that selecting k_y0 as a column of ones usually suffices. If not, initial ky0 can be found based on the Popov-Belevitch-Hautus (PBH) observability test [61]. It states that the system is observable if rank

k_y0^T C sI−A

=nx. In other words for all p eigenvectors of A, the k_y0^T Cp ̸= 0 relationship must hold. By collecting all eigenvectors to a P matrix, k_y0 can be found as a solution to k^T_y0CP = I, where I is a vector of ones with appropriate dimension. For complex eigenvalues the real and imaginary parts of a corresponding eigenvector have to be substituted.

Until this point we have considered thek_u and k_y input and output transformations as vec-tors. This is desirable because it significantly simplifies the control problem, by turning the plant into a SISO one. However, these transformations may also be matrices and sometimes this is necessary to achieve proper decoupling. In these cases there are 1 or more (n) singular values of K_u0 which have comparable magnitude to the largest one. This means that there are n possible input directions which yield acceptable decoupling. On the contrary, the singular vectors corresponding to the smallest singular values ofK_u0 are denoting input directions which excite the decoupled dynamics. For keeping n input directions, one needs to run the alternat-ing projection sequences nu −n times, where at the beginning of each sequences the smallest nonzero singular value of K_u is zeroed out (removing directions from K_u corresponding to the decoupled subsystem). For details see the alternating projection algorithm in Appendix D. Note that finding the desired value of n is problem dependent and may involve certain engineering judgment.

The application of input and output blends may introduce unstable zeros to the open loop SISO plant. Some of them are directly connected to the suppression of the undesired dynamics, and in the transfer function form may lead to pole-zero cancellations. On the other hand the appearance of additional zeros can be avoided when necessary by the fact that non-square systems rarely have transmission zeros [113, 116]. This means that the open loop system might be converted to a non-square system by suitable input and (or) output transformation matrices.

The computation of the input and output blend vectors require an initial output and input blend vector, respectively. At the same time, their joint calculation would lead to a bilinear optimization problem, whose solution is often obtained in an iterative manner with one variable fixed in each step. A similar computation is suggested for the k_u and k_y blend vectors. Starting from an initial k_y0 an optimal solution fork_u is calculated based on Section 3.3.1 with a corre-sponding ky relying on Section 3.3.2. Then by selecting ky0 =ky this iteration continues until convergence defined by pre-set thresholds and the final values are then considered as optimal ones. According to numerical studies, better decoupling can be achieved through the iterative

Algorithm 2 Output blend calculation with alternating projection

1: Initialization: Solve the following optimization problem, forβ²,γ²,X_d,X_c,Y,K_y: minimize

Xd, Ky, Xc, Y, β², γ²−β²+γ² s.t.

A^T_dXd+XdAd+C_d^TKyCd XdBdku

k^T_uB_d^TX_d −γ²I

⪯0, Ac Bcku

I 0

T

Ξ

Ac Bcku

I 0

+

Cc 0 0 I

T

Π

Cc 0 0 I

≺0, K_y ∈S^ny,0⪯K_y ⪯I, Y ≻0.

2: Seti= 0

3: fork= 1 to n_y−1 do

4: K_y^⋆_i =P_Γ^n−k

rankK_y

5: forj=1 to maximum number of iterations do

6: Solve the following optimization problem forXc,Y,X_d,S,Ky: minimize

Xd, Ky, Xc, Y, Strace (S) s.t.:

X_dA^T_d +A_dX_d+B_dK_yB^T_d X_dI_d^T

I_dX_d −γ²I

⪯0, A^T_c I^T_c

I 0 T

Ξ

A^T_c I^T_c I 0

+

B_c^T 0

0 I

T

Π

B^T_c 0

0 I

≺0, K_y ∈S^ny,0⪯K_y ⪯I, Y ≻0,

S K_y−K_y^⋆_i K_y−K_y^⋆_i I

⪰0.

7: K_y^⋆_i+1 =P_Γ^n−k

rankK_y

8: if j >1 then

9: if ^|K

yi+1⋆ −K^⋆_yi|

|K_yi+1^⋆ | >thresholdthen

10: i=i+ 1

11: break for loop

12: end if

13: end if

14: i=i+ 1

15: end for

16: end for

17: From the K_y^⋆_i = U SV^T singular value decomposition find k_y as the left singular vector corresponding to the largest singular value.

process than by separate design of the blend vectors. However, in many cases the improvement provided by this iteration is marginal compared to the increased computational time, and so upon satisfactory results after the first step further iterations could be neglected.

Remark 5. It might be desirable to identify some metrics which provide information about whether the decoupling is possible before calculating the actual blend vectors. According to [49] the magnitude of |q_i^Tb_j|=|q_i||b_j|cos(θ_ij) is an indication of controllability of the i^th mode from thej^thinput, whereqiis the left eigenvector corresponding to the i^th mode,bj is the input vector corresponding to the j^th input, and cos(θ_ij) is the angle between the two vectors. In the applied modal form this reduces to the following criteria for the i^th mode. In order to be controllable from thej^thinput, the input vector bj should contain non-zero elements in the rows

corresponding to thei^th mode. The magnitude of these elements are measures of controllability.

In case of the blend calculation problem this means that the blended input matrix Bku should contain non-zero values at the locations corresponding to the targeted mode, while it’s other elements should be small, possibly zero. This is clearly achievable if the row vectors of B_d are far from the subspace spanned by the rows of Bc. Similar reasoning corresponds to the output decoupling, based on observability and the columns ofC.

Remark 6. The case of unstable subsystems needs some further discussion. The minimum sensitivity characterized by theH−index can be calculated for unstable subsystems as well. On the other hand theH∞ norm is defined only for stable systems (i.e., poles having negative real part). At the same time, unstable systems that have no poles on the imaginary axis have an L∞ norm (also known as the peak gain). This peak gain can be computed using (2.27) after mirroring the unstable poles over the imaginary axis [140]. This modification is used in the decoupling algorithm for the unstable subsystems.

Remark 7. The proposed decoupling method is an appealing tool for subsystem identification.

In this case based on a preliminary model specific input and output transformations are synthe-sized to excite only a certain part of the dynamics and to reduce the identification problem to a SISO one. This way the approach scales down the complexity of the identification problem, by reducing the dimension of the unknown parameter vector. The effect of the neglected dynam-ics can be characterized by an additional disturbance term. Preliminary results of decoupled parameter identification are discussed in [TB91]. For less accurate preliminary models robust decoupling transformations can be synthesized, following the methods in Chapter 4.

It is important to review the achievable theoretical bounds on the decoupling performance.

Consider the maximization of the minimal sensitivity for the transfer corresponding to the controlled subsystem. The decoupling is carried out by one-norm blend vectors ku and k^T_y. By applying matrix norm identities on an arbitrary transfer function matrix Gny×nu, one gets:

√1n_u||G||i∞≤σ(¯ G), (3.14) where||·||i∞is the maximal row sum. This shows that with a normalized blending, the maximum achievable singular value is bounded from above. Consequently by applying one-norm input and output blends, no higher H− index can be achieved than the maximum singular value of the given subsystem.

Concerning the suppression of the subsystem to be decoupled, the highest suppression rate can be achieved if the blended inputs or outputs are as close as possible to the corresponding null spaces. A blending approach proposed by [98] carries out decoupling based on a null space transformation. This approach is certainly valid, however, the existence of null space and its sensitivity for model parameters and changes might hinder its application for real problems.

In document Optimal Decoupling of Dynamical Systems (Pldal 46-50)