The problem of simultaneous \(H^\infty\) control (Q1809016)

The author of this paper shows that the \(H^\infty\) control theory can be extended into the realm of multiple systems. He considers the problem of simultaneous \(H^\infty\) control of a finite collection of linear time-invariant systems defined on \([0,\infty)\), of the form: \[ \begin{cases} \dot x_i(t)=A_ix_i(t)+B_iw_i(t)+D_iu(t);\\ z_i(t)=K_ix_i(t)+G_iu(t);\\ y(t)=C_ix_i(t)+v_i(t),\quad i=1,2,\dots,k,\end{cases}\tag{1} \] where \(x_i(t)\in\mathbb{R}^{n_i}\) is the state, \(w_i(t)\in\mathbb{R}^{p_i}\) and \(v_i(t)\in\mathbb{R}^{l_i}\) are the disturbance inputs, \(u(t)\in\mathbb{R}^h\) is the control input, \(z_i(t)\in\mathbb{R}^{q_i}\) is the error output, and \(y(t)\in\mathbb{R}^l\) is the measured output. The following hypotheses are made: \(K_i' G_i=0\), \(G_i'G_i>0\), the pair \((A_i,K_i)\) is observable and the pair \((A_i,B_i)\) is controllable. The author considers the following Riccati algebraic equations: \[ A_i\widetilde P_i+\widetilde{P}_iA_i'+\widetilde{P}_i[(1+\varepsilon_0)K_i'K_i-C_i'C_i]\widetilde{P}_i+B_iB_i'=0,\quad i=1,\dots,k\quad (\varepsilon_0>0)\tag{2} \] and the matrices \[ \widehat{A}_i\triangleq A_i+\widetilde{P}_i[(1+\varepsilon_0)K_i'K_i-C_i'C_i]. \] The main result states that there exists a nonlinear digital output feedback controller defined by \[ \begin{aligned} & u(j\delta)=\bigcup[j\delta,y(\cdot)|j_0\delta],\quad u(t)=u(j\delta),\quad \forall t\in [j\delta,\;(j+1)\delta],\quad \forall j=0,1,2,\dots\\ & y(t)=0,\quad \forall t\in [0,j\delta]\Rightarrow u(j\delta)=0,\end{aligned}\tag{3} \] which solves the simultaneous \(H^\infty\) control problem for (1) if and only if there exist constants \(\varepsilon_0>0\) and \(\varepsilon_1>0\) and solutions \(\widetilde{P}_i=\widetilde{P}_i'>0\) to the Riccati algebraic equations (2) such that the matrices \(\widehat{A}_i\) are stable and the dynamic programming equation having the form \[ V(z_0)=\inf_{u^0\in\mathbb{R}^h}F(\varepsilon_1,z_0,u^0,V(\cdot))\tag{4} \] has a solution \(V(z_0)\) which satisfies a certain inequality for all \(z_0\in\mathbb{R}^n\), \(V(0)=0\), for \(z_0=0\) the infimum in (4) is achieved at \(u^0=0\). Moreover, if the simultaneous \(H^\infty\) control problem for the \(k\) systems (1) of orders \(n_1,n_2,\dots,n_k\) can be solved via a nonlinear digital output feedback controller (3), then this problem can be solved via a nonlinear digital output feedback dynamic controller of order \(n\triangleq n_1+n_2+\cdots+n_k+k\).