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

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  • ReceivedApr 25, 2018
  • AcceptedSep 4, 2018
  • PublishedApr 3, 2019



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.



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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