#  SCIENCE CHINA Information Sciences, Volume 62 , Issue 5 : 052202(2019) https://doi.org/10.1007/s11432-018-9617-0

## Decentralized control for linear systems with multiple input channels More info
• ReceivedApr 25, 2018
• AcceptedSep 4, 2018
• PublishedApr 3, 2019
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### Abstract ### Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant Nos. 61403235, 61573221, 61633014, 61873332) and Qilu Youth Scholar Discipline Construction Funding from Shandong University.

### Supplement

Appendix

Proof of Theorem 3.2

We firstly prove the case of $N=2,$ i.e., under Assumption 3.1, the optimization problem $\min_{u_2}\min_{u_1}J$ s.t. (1) with $N=2$ has a unique solution if and only if ARE (4) has a solution such that $\Gamma_1>0$ and $\Gamma_2>0,$ and ARE \begin{equation}L=A'\Upsilon_{1}'\Upsilon_{2}'L[I+\Phi_1L]^{-1}\Upsilon_{2}\Upsilon_{1}A-A'\Upsilon_{1}'\Psi_1\Upsilon_{1}A \tag{27}\end{equation} has a solution such that the matrix $(I+\Phi_1L)^{-1}\Upsilon_2\Upsilon_1A$ is stable. In this case, the centralized optimal controllers are given by $u_1(k)=K_1x(k)$ and $~u_2(k)=K_2x(k)$ where \begin{align}&K_1=-\Gamma_1^{-1}B_1'\Upsilon_{2}'[P+L(I+\Phi_{1}L)^{-1}\Upsilon_{2}\Upsilon_{1}]A, \tag{28} \\ &K_2=-\Gamma_2^{-1}B_2'\Upsilon_{1}'[P+L(I+\Phi_{1}L)^{-1}\Upsilon_{2}\Upsilon_{1}]A. \tag{29} \end{align} “Necessity". The proof of the necessity mainly relies on the maximum principle, that is, the optimal controller satisfies $0=R_1u_1(k)+B_1'\lambda(k)$, where $\lambda(k)$ is the solution of the backward adjoint system \begin{equation}\lambda(k-1)=A'\lambda(k)+Qx(k). \tag{30}\end{equation}

The detailed proof is divided into four parts. Firstly, we consider the LQR problem with $u_2=0$ which shows that ARE (4) has a solution $P\geq0$ such that $\Gamma_1>0$. Secondly, the case of $u_2\neq0$ is discussed by introducing a new costate. Thirdly, the positive definiteness of $\Gamma_2$ is given. Lastly, we obtain the solvability of ARE (27) and establish the relationship between the new costate and the original state.

(i) The unique solvability of $\min_{u_2}\min_{u_1}J$ s.t. (1) implies that $\min_{u_1}J_1$ s.t. (1) with $u_2=0$ has a unique solution. Together with Assumption 3.1, there exists a solution $P>0$ to the ARE (4) such that $\Gamma_1>0$ and the matrix $A-\Gamma_1^{-1}B_1'PA$ is stable. In this case, it holds that $\lambda(k)=Px(k+1).$ The detailed proof is referred to literature 1).

(ii) In the case of $u_2\neq0$, the relationship becomes nonhomogeneous, that is, there exists $\zeta(k)$ such that $\lambda(k)=Px(k+1)+\zeta(k)$. Substituting the relationship into $0=R_1u(k)+B_1'\lambda(k)$ yields that $0=\Gamma_1u_1(k)+B_1'PAx(k)+B_1'PB_2u_2(k)+B_1'\zeta(k).$ Using the fact that $\Gamma_1>0$, one has \begin{equation}u_1(k)=-\Gamma_1^{-1}[B_1'PAx(k)+B_1'PB_2u_2(k)+B_1'\zeta(k)]. \tag{31}\end{equation}

Then the dynamic of the state becomes \begin{equation}x(k+1)=\Upsilon_1Ax(k)+\Upsilon_1B_2u_2(k)-B_1\Gamma_1^{-1}B_1'\zeta(k). \tag{32}\end{equation}

By combining with $\lambda(k)=Px(k+1)+\zeta(k)$ and (30), we have \begin{align}\lambda(k-1) &=A'P\Upsilon_1Ax(k)+A'P\Upsilon_1B_2u_2(k)-A'PB_1\Gamma_1^{-1}B_1'\zeta(k)+A'\zeta(k)+Qx(k) \\ &=Px(k)+A'\Upsilon_1'\zeta(k)+A'P\Upsilon_1B_2u_2(k), \end{align} where $P$ satisfying ARE (4) has been used in the derivation of the last equality. Accordingly, $\lambda(k)=Px(k+1)+\zeta(k)$ holds where the dynamic of $\zeta$ is given by \begin{equation}\zeta(k-1)=A'\Upsilon_1'\zeta(k)+A'P\Upsilon_1B_2u_2(k). \tag{33}\end{equation}

We now calculate the cost function. In view of (1) and (30), it yields that \begin{align}x'(k)\lambda(k-1)-x'(k+1)\lambda(k) &=x'(k)Qx(k)-u_1'(k)B_1'\lambda(k)-u_2'(k)B_2'\lambda(k) \\ &=x'(k)Qx(k)+u_1'(k)R_1u_1(k)-u_2'(k)B_2'\lambda(k), \end{align} where $0=R_1u(k)+B_1'\lambda(k)$ has been inserted in the last equality. Taking summation from $0$ to $N$ and letting $N$ tend to $\infty$ yields that $x'(0)\lambda(-1)=\sum_{k=0}^{\infty}[x'(k)Qx(k)+u_1'(k)R_1u_1(k)-u_2'(k)B_2'\lambda(k)].$ The cost function (3) is then reformulated as \begin{equation}J=x'(0)\lambda(-1)+\sum_{k=0}^{\infty}[u_2'(k)R_2u_2(k)+u_2'(k)B_2'\lambda(k)]. \tag{34}\end{equation}

(iii) Consider the problem $\min_{u_2}J$ s.t. (32) and (33) where $J$ is given in (34). Using again the maximum principle, the optimal controller $u_2$ satisfies that \begin{equation}0=(R_2+B_2'P\Upsilon_1B_2)u_2(k)+B_2'P\Upsilon_1Ax(k)+B_2'\Upsilon_1'\zeta(k). \tag{35}\end{equation}

It is now shown that $\Gamma_2=R_2+B_2'P\Upsilon_1B_2$ is invertible. Let $u_2(k)=0$, $k\geq0$; from (32) and (33), one has $\zeta(k-1)=0$ and $x(k+1)=\Upsilon_1Ax(k)$ for $k\geq~0$. Noting that the matrix $\Upsilon_1A$ is stable, the zero controller $u_2(k)=0$ is stabilizing. Now consider the case of $x(0)=0$; the optimal controller must be $u_2(k)=0$, $k\geq~0$ with the corresponding optimal cost of $0$. Selecting $u_2(s)=0$, $s>0$ and $u_2(0)\neq0$ which is arbitrarily chosen to be stabilizing, then $\zeta(k)=0$, $k\geq0$ and the optimal cost can be rewritten from (34) as $J=u_2'(0)\Gamma_2u_2(0)$ which is strictly positive. This implies that $\Gamma_2>0.$ Accordingly, from (35), we have \begin{equation}u_2(k)=-\Gamma_2^{-1}[B_2'P\Upsilon_1Ax(k)+B_2'\Upsilon_1'\zeta(k)]. \tag{36}\end{equation}

Substituting (36) into (32) and (33) yields the Hamiltonian-Jacobi system \begin{align}&x(k+1)=\Upsilon_2\Upsilon_1Ax(k)-\Phi_1\zeta(k), \tag{37} \\ &\zeta(k-1)=A'\Upsilon_1'\Upsilon_2'\zeta(k)-A'\Upsilon_1'\Psi_1\Upsilon_1Ax(k). \tag{38} \end{align} (iv) From the existence and uniqueness of the optimal solution to problem $\min_{u_2}\min_{u_1}J$ s.t. (1), it holds that the system (37) and (38) has a unique solution. In view of the stability of the matrix $\Upsilon_1A$ and the admissible set of $u_2$, we have $x(k)\in~l_2$. Thus, it holds that $\lim_{k\rightarrow\infty}\zeta(k)=L\lim_{k\rightarrow\infty}x(k+1)=0$ for any matrix $L$. Using the induction technique, we assume that there exists a constant matrix $L$ such that $\zeta(k)=Lx(k+1)$ holds. Substituting it into (37) yields that $(I+\Phi_1L)x(k+1)=\Upsilon_2\Upsilon_1Ax(k).$ By combining with the uniqueness of solution to (37) and (38), one has that $I+\Phi_1L$ is invertible and \begin{equation}x(k+1)=(I+\Phi_1L)^{-1}\Upsilon_2\Upsilon_1Ax(k). \tag{39}\end{equation}

Plugging (39) into (38), it is obtained that \begin{equation}\zeta(k-1)=[A'\Upsilon_1'\Upsilon_2'L(I+\Phi_1L)^{-1}\Upsilon_2\Upsilon_1A-A'\Upsilon_1'\Psi_1\Upsilon_1A]x(k).\end{equation}

Thus, $\zeta(k-1)=Lx(k)$ holds where $L$ satisfies (27). This implies that ARE (27) has a solution. Note that $x(k)\in~l_2$, and then the matrix $(I+\Phi_1L)^{-1}\Upsilon_2\Upsilon_1A$ is stable.

“Sufficiency". Assume that (4) has a positive semi-definite solution such that $\Gamma_1>0,\Gamma_2>0$, and then the matrix $A'\Upsilon_1'$ is stable under Assumption 3.1. The detailed proof of the sufficiency is consisting of two steps. First, we obtain the optimal controller $u_1(k)$ by completing the square. Second, based on the optimization of the controller $u_1$ and the corresponding state trajectory and cost function, we derive the optimal controller $u_2$ by the sufficient maximum principle.

(i) First, we derive the optimal controller $u_1$. To this end, we introduce a new variable $\zeta$ with the following dynamic: \begin{equation}\zeta(k-1)=A'\Upsilon_1'\zeta(k)+A'\Upsilon_1'PB_2u_2(k). \tag{40}\end{equation}

Noting that $u_2\in~l_2,$ one has $\lim_{k\rightarrow\infty}\zeta(k)=0$. Using (1), it yields that \begin{align}x'(k+1)Px(k+1)-x'(k)Px(k) =& x'(k)(A'PA-P)x(k)+2u_1'(k)B_1'PAx(k)+2u_2'(k)B_2'PAx(k) \\ &+u_1'(k)B_1'PB_1u_1(k)+2u_1'(k)B_1'PB_2u_2(k)+u_2'(k)B_2'PB_2u_2(k). \end{align} From (1) and (40), one has \begin{align}2x'(k+1)\zeta(k)-2x'(k)\zeta(k-1) =& 2u_1'(k)B_1'\zeta(k)+2u_2'(k)B_2'\zeta(k) \\ &+2x'(k)A'PB_1\Gamma_1^{-1}B_1'\zeta(k) -2x'(k)A'\Upsilon_1'PB_2u_2(k). \end{align} Then, it yields by simple calculation that \begin{align}&x'(k+1)Px(k+1)-x'(k)Px(k)+2x'(k+1)\zeta(k)-2x'(k)\zeta(k-1) \\ & =-x'(k)Qx(k)-u_1'(k)R_1u_1(k)+[u_1(k)+\Gamma_1^{-1}B_1'PAx(k)+\Gamma_1^{-1}B_1'PB_2u_2(k)+\Gamma_1^{-1}B_1'\zeta(k)]' \\ & \times\Gamma_1[u_1(k)+\Gamma_1^{-1}B_1'PAx(k)+\Gamma_1^{-1}B_1'PB_2u_2(k)+\Gamma_1^{-1}B_1'\zeta(k)]+u_2'(k)B_2'P\Upsilon_1B_2u_2(k) \\ & +2u_2'(k)B_2'\Upsilon_1'\zeta(k)-\zeta'(k)B_1\Gamma_1^{-1}B_1'\zeta(k). \end{align} Accordingly, Eq. (3) can be reformulated as \begin{align}J=& x'(0)Px(0)+2x'(0)\zeta(-1)+\sum_{k=0}^{\infty}\Big([u_1(k)+\Gamma_1^{-1}B_1'PAx(k)+\Gamma_1^{-1}B_1'PB_2u_2(k) \\ &+\Gamma_1^{-1}B_1'\zeta(k)]'\Gamma_1 [u_1(k)+\Gamma_1^{-1}B_1'PAx(k)+\Gamma_1^{-1}B_1'PB_2u_2(k)+\Gamma_1^{-1}B_1'\zeta(k)] \\ &+u_2'(k)\Gamma_2u_2(k)+2u_2'(k)B_2'\Upsilon_1'\zeta(k)-\zeta'(k)B_1\Gamma_1^{-1}B_1'\zeta(k)\Big). \end{align} In view of the fact that $\Gamma_1>0$, the optimal controller of $u_1$ is given by \begin{equation}u_1(k)=-\Gamma_1^{-1}B_1'PAx(k)-\Gamma_1^{-1}B_1'PB_2u_2(k)-\Gamma_1^{-1}B_1'\zeta(k). \tag{41}\end{equation}

(ii) We now aim to obtain the optimal $u_2.$ Considering (41), the corresponding cost becomes \begin{equation}J=x'(0)Px(0)+2x'(0)\zeta(-1)+\sum_{k=0}^{\infty}\Big(u_2'(k)\Gamma_2u_2(k)+2u_2'(k)B_2'\Upsilon_1'\zeta(k)-\zeta'(k)B_1\Gamma_1^{-1}B_1'\zeta(k)\Big). \tag{42}\end{equation}

Substituting (41) into (1), we can rewrite the states as follows: \begin{equation}x(k+1)=\Upsilon_1Ax(k)+\Upsilon_1B_2u_2(k)-B_1\Gamma_1^{-1}B_1'\zeta(k). \tag{43}\end{equation}

Combining with (40) yields that \begin{equation}x'(k+1)\zeta(k)-x'(k)\zeta(k-1) =u_2'(k)B_2'\Upsilon_1'\zeta(k)-\zeta'(k)B_1\Gamma_1^{-1}B_1'\zeta(k)-x'(k)A'\Upsilon_1'PB_2u_2(k).\end{equation}

Together with the fact that $\lim_{k\rightarrow\infty}\zeta(k)=0$ and $\lim_{k\rightarrow\infty}x(k)=0,$ it is further obtained that \begin{equation}x'(0)\zeta(-1)=-\sum_{k=0}^\infty\Big(u_2'(k)B_2'\Upsilon_1'\zeta(k)-\zeta'(k)B_1\Gamma_1^{-1}B_1'\zeta(k)-x'(k)A'\Upsilon_1'PB_2u_2(k)\Big). \tag{44}\end{equation}

Plugging (44) into (42), one has \begin{equation}J =x'(0)Px(0)+x'(0)\zeta(-1)+\sum_{k=0}^{\infty}\Big(u_2'(k)[\Gamma_2u_2(k)+B_2'\Upsilon_1'\zeta(k)+B_2'P\Upsilon_1Ax(k)]\Big). \tag{45}\end{equation}

We then apply the sufficiency of the maximum principle and obtain that the optimal controller satisfies $0=\Gamma_2u_2(k)+B_2'\Upsilon_1'\zeta(k)+B_2'P\Upsilon_1Ax(k).$ Combining with the assumption that $\Gamma_2>0$, the optimal controller of $u_2$ must be \begin{equation}u_2(k)=-\Gamma_2^{-1}[B_2'\Upsilon_1'\zeta(k)+B_2'P\Upsilon_1Ax(k)]. \tag{46}\end{equation}

Provided that (27) has a stabilizing solution, then it holds that $\zeta(k)=Lx(k+1)$. Together with (41) and (46), the optimal controllers can be reformulated as $u_1(k)=K_1x(k)$, $u_2(k)=K_2x(k)$ where $K_1$, $K_2$ are defined by (28) and (29).

For the general case of $N>2,$ the fact of $\Gamma_i>0$ and the derivation of $L$ in (5) can be derived similarly to that of $\Gamma_2>0$ in (3) and (4), respectively. The sufficiency also follows similarly to the case of $N=2$. This completes the proof.

Tadmor G, Mirkin L. $H_\infty$ control and estimation with preview-part II: fixed-size ARE solutions in discrete time. IEEE Trans Autom Control, 2005, 50: 29–40.

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• Figure 1

(Color online) The state trajectories with suboptimal distributed controller.