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Discretization with Asynchronous Actuation and Sensing

Hiroyasu Tsukamoto

Let ii, jj, and kk be the sampling indices for the control, attack, and sensing, respectively, tft_{f} be the terminal time, ti=iΔtut_{i}=i\Delta t_{u}, tj=jΔtat_{j}=j\Delta t_{a}, tk=kΔtyt_{k}=k\Delta t_{y}, I=tf/ΔtuI=\lfloor t_{f}/\Delta t_{u}\rfloor, J=tf/ΔtaJ=\lfloor t_{f}/\Delta t_{a}\rfloor, K=tf/ΔtyK=\lfloor t_{f}/\Delta t_{y}\rfloor, ui=u(ti)u_{i}=u(t_{i}), aj=a(tj)a_{j}=a(t_{j}), xk=x(tk)x_{k}=x(t_{k}), Ak=A(ν(tk))A_{k}=A(\nu(t_{k})), Bu,k=Bu(ν(tk))B_{u,k}=B_{u}(\nu(t_{k})), Ba,k=Ba(ν(tk))B_{a,k}=B_{a}(\nu(t_{k})), Ck=C(ν(tk))C_{k}=C(\nu(t_{k})), Dk=D(ν(tk))D_{k}=D(\nu(t_{k})),

u(t)\displaystyle u(t) =i=0i=Ihu(tti)ui\displaystyle=\sum_{i=0}^{i=I}h_{u}(t-t_{i})u_{i} (1)
a(t)\displaystyle a(t) =j=0j=Jha(ttj)aj\displaystyle=\sum_{j=0}^{j=J}h_{a}(t-t_{j})a_{j} (2)

and

hu(t)={1t[0,Δtu)0otherwise\displaystyle h_{u}(t)=\begin{cases}1&t\in[0,\Delta t_{u})\\ 0&\text{otherwise}\end{cases} (4)
ha(t)={1t[0,Δta)0otherwise\displaystyle h_{a}(t)=\begin{cases}1&t\in[0,\Delta t_{a})\\ 0&\text{otherwise}\end{cases} (5)

Discretization (t=tk=kΔtyt=t_{k}=k\Delta t_{y}) assuming the topology ν(t)\nu(t) is fixed during t[tk,tk+Δty],kt\in[t_{k},t_{k}+\Delta t_{y}],~{}\forall k\in\mathbb{N}:

xk+1=\displaystyle x_{k+1}= eAkΔtyxk+i=0i=I(tktk+ΔtyeAk(tk+Δtyτ)Bu,khu(τti)𝑑τ)ui+j=0j=J(tktk+ΔtyeAk(tk+Δtyτ)Ba,kha(τtj)𝑑τ)aj\displaystyle e^{A_{k}\Delta t_{y}}x_{k}+\sum_{i=0}^{i=I}\left(\int_{t_{k}}^{t_{k}+\Delta t_{y}}e^{A_{k}(t_{k}+\Delta t_{y}-\tau)}B_{u,k}h_{u}(\tau-t_{i})d\tau\right)u_{i}+\sum_{j=0}^{j=J}\left(\int_{t_{k}}^{t_{k}+\Delta t_{y}}e^{A_{k}(t_{k}+\Delta t_{y}-\tau)}B_{a,k}h_{a}(\tau-t_{j})d\tau\right)a_{j} (6)
=\displaystyle= eAkΔtyxk+i=0i=I(tktk+ΔtyeAk(tk+Δtyτ)hu(τti)𝑑τ)Bu,kui+j=0j=J(tktk+ΔtyeAk(tk+Δtyτ)ha(τtj)𝑑τ)Ba,kaj\displaystyle e^{A_{k}\Delta t_{y}}x_{k}+\sum_{i=0}^{i=I}\left(\int_{t_{k}}^{t_{k}+\Delta t_{y}}e^{A_{k}(t_{k}+\Delta t_{y}-\tau)}h_{u}(\tau-t_{i})d\tau\right)B_{u,k}u_{i}+\sum_{j=0}^{j=J}\left(\int_{t_{k}}^{t_{k}+\Delta t_{y}}e^{A_{k}(t_{k}+\Delta t_{y}-\tau)}h_{a}(\tau-t_{j})d\tau\right)B_{a,k}a_{j} (7)
yk=\displaystyle y_{k}= Ckxk+Dkj=0j=Jha(tktj)aj\displaystyle C_{k}x_{k}+D_{k}\sum_{j=0}^{j=J}h_{a}(t_{k}-t_{j})a_{j} (8)

where hgh_{g} is the impulse response of the hold device. Note that since the impulse response at time tt as a result of the discrete-time control ukΔ(tk)u_{k}\Delta(t_{k}) at time tkt_{k} is given by hg(ttk)ukh_{g}(t-t_{k})u_{k} by definition, and thus (1) is just a superposition of these impulse responses. In particular, when we have a constant time interval Δt=Δtk\Delta t=\Delta t_{k} and

hu(t)={1t[0,Δtu]0otherwise\displaystyle h_{u}(t)=\begin{cases}1&t\in[0,\Delta t_{u}]\\ 0&\text{otherwise}\end{cases} (9)
ha(t)={1t[0,Δta]0otherwise\displaystyle h_{a}(t)=\begin{cases}1&t\in[0,\Delta t_{a}]\\ 0&\text{otherwise}\end{cases} (10)