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IEEE/CAA Journal of Automatica Sinica  2018, Vol. 5 Issue(6): 1089-1098 PDF
Stabilization of Networked Control Systems Using a Mixed-Mode Based Switched Delay System Method
Qing-Kui Li1, Xiaoli Li2, Jiuhe Wang1, Shengli Du2
1. School of Automation, Beijing Information Science and Technology University, Beijing 100192, China;
2. Faculty of Information Technology, Beijing University of Technology, Beijing 100124, China
Abstract: The phenomenon of mixed-mode is one of the most important characteristics of switched delay systems. If a networked control system (NCS) with network induced delays and packet dropouts (NIDs & PDs) is recast as a switched delay system, it is imperative to consider the effects of mixed-modes in the stability analysis for an NCS. In this paper, with the help of the interpolatory quadrature formula and the average dwell time method, stabilization of NCSs using a mixed-mode based switched delay system method is investigated based on a novel constructed Lyapunov-Krasovskii functional. With the Finsler's lemma, new exponential stabilizability conditions with less conservativeness are given for the NCS. Finally, an illustrative example is provided to verify the effectiveness of the developed results.
Key words: Interpolatory quadrature formula     mixed-mode     networked control system (NCS)     switched delay system
Ⅰ. INTRODUCTION

With the rapid advancements of network technology, in a number of control engineering applications, sensors and actuators connect with a remote controller over a multipurpose networks, allowing the networks to exchange information among distributed spatial system components. Such systems are well known as networked control system (NCSs) and have gained special interests due to their attractive features such as simple installation and maintenance, increased system flexibility, higher system testability and resource utilization, lower cost, and reduced weight and power [1]-[8]. Typical examples can be seen in a wide range of areas including industrial automation, satellite clusters, traffic control, and mobile robots [9]-[11]. Consequently, in the past decades, research on NCSs has attracted considerable attentions in control and network community, e.g., [12]-[20]. Despite the advantages of NCSs, the use of a shared network result in communication constrains due to the inherent limitation of network bandwidth. It is well recognized that network induced delays and packet dropouts (NIDs & PDs) are two important constrains which are the potential causes of performance deterioration or instability of the NCSs [21]. How NIDs & PDs affect the stability and performance of NCSs has thus become an important issue. Several methodologies have been developed for modeling, analyzing and synthezing for NCSs in the presence of NIDs & PDs. Some important categories are mentioned for examples as follows:

1) Time delay system methodology models packet dropouts as a kind of delays, which are lumped with network induced delays in the system. Then, the NCS is modeled as a time delay system with mixed delays [22], [23].

2) Stochastic system method holds the property of an NCS that intermittent packet dropouts and network induced delays are random, and is viewed as an important strategy to handle random phenomenon in NCS. One of the stochastic methods uses a linear function of stochastic variable to model random delays, e.g., [1], [24], or random packet dropouts, e.g., [21], [25]; the other one is to use the Markovian chains to model random delays, e.g., [26], [27], or random packet dropouts, e.g., [28], [29].

3) Switched system method provides a powerful tool for modeling of an NCS with time delays and packet dropouts due to the fact that, whether the transmission of a data packet is successful or not, the switching and delay characteristics of the NCS are obvious, and thus, a switched system model can naturally capture the properties of the NCS with time delays and packet dropouts, e.g., [22], [30]-[32].

In the aforementioned results concerning NCSs with NIDs & PDs, it should be pointed out that a time delay system method mixes the packet dropouts with the delay term, the effects of dropouts are not clearly formulated. On the other hand, a stochastic system method mainly focuses on the discrete system model of NCSs, and more importantly, it is usually assumed that the NIDs & PDs meet certain probability distributions. A switched system method, which has made great advances in the study of the theories and methods [33]-[36], has found a good application for NCSs [22], [30], [31]. However, to the best knowledge of authors, existing results concerning NCSs suffering from NIDs & PDs have mainly focused on either delay or switching behaviors [30], [37]. The coexistence and interaction of switchings and delays in an NCS due to NIDs & PDs, which leads to a more complicated stability analysis, have not been fully studied [38]. The importance of the study of this issue, arises from the natural property of the NCS, and the requirement of performance improvement of the system. This motivates the present study.

As far as a switched delay system is concerned, it usually means a switched system with time delays, where the delay may be contained in the system state, control input or switching signal [39]. Distinctive features of switched delay systems are the so called mixed-mode [40], i.e., when switching occurs, owing to the influence of time delays, both the present subsystem and the previous subsystems determine the state of the switched delay system. Consequently, it is natural to understand that the stability analysis problem for switched delay systems is not easy to handle based on the Lyapunov-Krasovskii functional theory, and there are still more space to improve existing results, e.g., [38], [40], especially for a switched time-varying delay system with unstable subsystems, which constitutes a major difficulty in dealing with the mixed-modes in time-varying delay case.

As is well known, switched delay systems can model a class of important systems that include switchings and delays, and an NCS can be seen as a typical example. Due to limited bandwidth, NIDs & PDs are viewed as important phenomena, and if we see packet dropouts occurring on an NCS as the switching behavior between the system with, and without packet dropouts, a switched delay system is regarded as a suitable model for real processes involving the NIDs & PDs, see [22], [30], [32], [41]. Therefore, stability analysis considering the effect of mixed-modes, from the viewpoint of the switched delay system, should be taken into account for an NCS. Unfortunately, the mixed-modes have not been considered in existing results for the NCSs with switched delay system models, see e.g. [22], [30], [41].

In this paper, stabilization of NCSs using a mixed-mode based switched delay system method is studied. The main contributions of this paper are as follows. Firstly, significantly capturing the properties of NCSs which lead to the controllers switching, an NCS with NIDs & PDs will be modeled as a continuous-time switched linear time-varying delay system. Secondly, when mixed-modes are considered in the stability analysis of NCSs based on the Lyapunov-Krasovskii theory, it is necessary for the corresponding subsystems to decompose the integral interval of the constructed Lyapunov-Krasovskii functional, and the estimation of the integral terms resulted by the processing of Lyapunov-Krasovskii functional is implemented by the interpolatory quadrature formula. Meanwhile, the average dwell time approach is used and the threshold of the delay upper bound which leads to the switching is selected by an optimization technique rather than a pre-specified constant, e.g. [41], [42]. Utilizing the Finsler's Lemma, new exponential stabilizability conditions for NCS with certain tolerated large delay and packet dropout rate are established. Distinctly different from exiting results that concerned with switched strategy for NCSs, e.g. [22], [30], the switched delay system model, and the stabilizability conditions allow the NCS to be unstabilizable during the period of the packet dropouts. Finally, the proposed method is demonstrated by an illustrative example.

Notations: $\mathbb{R}^{n}$ denotes the $n$ dimensional Euclidean space, and for given $\tau>0$, let $C_{n}=C([-\tau, 0], \mathbb{R}^{n})$ be the Banach space of continuous mapping from $([-\tau, 0], \mathbb{R}^{n})$ to $\mathbb{R}^{n}$ with topology of uniform convergence. Let $x_{t}\in C_{n}$ be defined by $x_{t}(\theta)=x(t+\theta), \theta \in[-\tau, 0]$. $\|\cdot\|$ denotes the usual 2-norm and $\|x_{t}\|_{cl}=\sup_{-\tau\leq \theta\leq0}\{\|x(t+\theta)\|, \|\dot{x}(t+\theta)\|\}$. The minimum eigenvalue of matrix $S$ is denoted by $\lambda_{\min}(S)$.

Ⅱ. MODELING OF AN NCS AS A SWITCHED DELAY SYSTEM

Consider the following linear continuous-time system described by

 \begin{align}\label{10-s1} \dot{x}(t)&=Ax(t)+Bu(t) \end{align} (1)

where $x(t)\in \mathbb{R}^{n}$, $u(t)\in \mathbb{R}^{p}$ are the state, and the control input, respectively; $A, B$ are constant matrices of appropriate dimensions, and $(A, B)$ is assumed to be stabilizable.

Noted that introducing the network usually leads to the NIDs & PDs [1], [23]. This paper assumes that a state feedback controller (realized by a zero-order-hold (ZOH)) with piecewise continuous function is applied to synthesize the NCS with NIDs & PDs. The following assumption is given to derive stabilizability conditions.

Assumption 1: The sensor is clock-driven, the controller and actuator are event-driven, and the data is transmitted with a single packet.

In the following, we will model the NCS with NIDs & PDs as a continuous-time switched delay system.

With the state feedback controller, the NCS (1) with NIDs & PDs can be depicted in Fig. 1.

 Download: larger image Fig. 1 Schematic of NCS with NIDs & PDs

The sampling period is assumed to be a positive constant $h$, and the sampling instants are denoted as $i_{k}h\, (k=1, 2, \ldots)$. It is assumed that the updating signal which is the signal successfully transmitted from the sampler to the controller and the ZOH, suffers the transmission delays $\eta_{k}=\eta_{k}^{sc}+\eta_{k}^{ca}$, where $\eta_{k}^{sc}$ and $\eta_{k}^{ca}$ are the delays from the sampler to the controller and, from the controller to the ZOH, respectively. Fig. 2 shows the timing diagram of signal transmitting in NCS with delays and dropouts.

 Download: larger image Fig. 2 Diagram of signal transmitting in NCS with delays and dropouts

Now, impose the piecewise linear continuous state feedback control via a network channel.

First, assume that at the sampling instant $i_{k}h$, the signal can be transmitted successfully from the sensor to the actuator. One can take the state feedback controller as the following form

 \begin{align*} u(t_{k})=Kx(i_{k}h), \;k=1, 2, \ldots \end{align*}

where $t_{k}$ is the instant when a packet is successfully transmitted from the sensor. If in the control input introducing the artificial time-varying delay $d(t)=t-i_{k}h$, taking into account the behavior of the ZOH gives

 \begin{align}\label{controller_1} &u(t)=Kx(t-d(t))\\ & t\in [i_{k}h+\eta_{k}, i_{k+1}h+\eta_{k+1}), \, k\in\{1, 2, \ldots\}. \end{align} (2)

The network induced delays $\eta_{k}$ are assumed to satisfy

 \begin{align}\label{delaybound_1} 0\leq\eta_{k}\leq\eta_{m}, \quad k=1, 2, \ldots \end{align} (3)

where $\eta_{m}$ is the upper bound of delays that ensures the stability of NCS with no packet dropout. If $\eta_{k}>\eta_{m}$, the NCS may be unstable. This case is viewed as the same of packet dropout.

Second, suppose that the sampling instant $i_{k}h$ is the instant when the signal fails to be transmitted successfully from the sensor to the actuator, and the latest instant when the signal is successful sampled is $i_{k}h-l(k)h$, where $l(k)\in\mathbb{Z}^{+}$ and $0\leq l(k)\leq L$ for a constant $L>0$. Take the state feedback controller as

 \begin{align*} u(t_{k})=Kx(i_{k}h-l(k)h), \;k=1, 2, \ldots \end{align*}

Furthermore, considering the behavior of the ZOH yields

 \begin{align}\label{controller_2} u(t)&=Kx(t-d(t)-l(k)h), t\in[i_{k}h+\eta_{m}, i_{k+j}h+\eta_{k+j}) \end{align} (4)

where $i_{k+j}h+\eta_{k+j}$ denotes the next update instant, $j = 1, 2, \ldots$.

When $t\in [i_{k}h+\eta_{k}, i_{k+1}h+\eta_{k+1})$, if there is no packet dropout occurring, with the assumption $i_{k+1}=i_{k}+1$ one can get

 \begin{align}\label{eq_tau} \eta_{k}\leq t-i_{k}h Note that the admissible upper bound of network delays satisfy\eta_{k}\leq\eta_{m}$. There is $ \begin{align} d(t)

Remark 1: For an instant $\tau_{m}$, once a large delay $d(t)\geq \tau_{m}$, the stability of NCS cannot be guaranteed. In this case, no matter whether or not the packet dropout occurs, we all see that the packet dropouts occur on NCS. "Viewing the long delay as packet dropout" can be seen as an active packet dropout strategy, which is convenient to give a switched delay system model for NCS with NIDs & PDs (see in Fig. 2). Therefore, $\tau_{m}$ becomes the threshold of the switching between the subsystems with packet dropouts and without packet dropouts respectively, and it is the critical upper bound of the time-varying delay $d(t)$ which is closely related to the network induced delay $\eta_{k}$. It is clear that when delay $d(t)\geq \tau_{m}$, the packet dropout of NCS may occur but does not necessarily occur. In the following sections, "the packet dropout" includes both cases of packed dropout and "large delay".

By virtue of the technique of controller switching, imposing the controllers (2) and (4) to system (1) gives rise to the following system

 \begin{align}\label{10-s2} \dot{x}(t)=\left\{\begin{array}{ll} Ax(t)+&\!\!\!\!\!\!BKx(t-d(t)), \\ &t\in[i_{k}h+\eta_{k}, i_{k+1}h+\eta_{k+1})\\ &\mathit{\rm{or}}\, t\in[i_{k}h+\eta_{k}, i_{k+1}h+\eta_{m})\\ Ax(t)+&\!\!\!\!\!\!BKx(t-d(t)-l(k)h), \\ &t\in[i_{k+1}h+\eta_{m}, i_{k+j}h+\eta_{k+j}).\end{array} \right. \end{align} (7)

Introducing the following notions $d_{1}(t)=d(t)$ and $d_{2}(t)=d(t)+l(k)h$, the following switched delay system can recast system (7) as

 \begin{align}\label{10-s33} \dot{x}(t)=Ax(t)+BKx(t-d_{\sigma(t)}(t)) \end{align} (8)

where the right continuous function $\sigma(t) : [0, \infty)\rightarrow \mathcal{P}=\{1, 2\}$ is the switching signal. $\sigma(t)=1$ if $t\in[i_{k}h+\eta_{k}, i_{k+1}h+\eta_{k+1})$ or $t\in[i_{k}h+\eta_{k}, i_{k+1}h+\eta_{m})$ which corresponds to the NCS without packet dropout; $\sigma(t)=2$ if $t\in[i_{k+1}h+\eta_{m}, i_{k+j}h+\eta_{k+j})$ which corresponds to the NCS with packet dropouts. In the following, for convenience, denote $t_{k}= i_{k}h+\eta_{k}$.

For the development of the main results, the upper bound of $d_{2}(t)=d(t)+l(k)h$ is presented as

 \begin{align}d_{2}(t)=d(t)+l(k)h\leq\tau_{M}\end{align} (9)

and obviously, $\tau_{m}\leq\tau_{M}$.

Remark 2: The switched delay system model (8) is capturing the switching property of NCS when packet dropouts or long delays occur. It should be mentioned that in the recent references [41], [42], a switched delay system model in the continuous-time case for NCS has been studied based on an idea of control missing. It is obvious that there are distinct structure differences between the two switched delay system models, and importantly, the switching signal used for the NCS in this paper are generated by the real signal from the NIDs & PDs.

The following definitions are needed.

Definition 1 [43]: For any $T_{2}>T_{1}\geq 0$, let $N_{\sigma}(T_{1}, T_{2})$ denote the number of switching of $\sigma(t)$ over $(T_{1}, T_{2})$. If $N_{\sigma}(T_{1}, T_{2})\leq N_{0}+\frac{T_{2}-T_{1}}{T_{\alpha}}$ holds for $T_{\alpha}>0, N_{0}\geq 0$, then $T_{\alpha}$ is called average dwell time and $N_{0}$ the chatter bound.

Definition 2: For any $T_{2}>T_{1}$, let $T_{d}(T_{1}, \, T_{2})$ denote the total time interval of packet dropouts during $[T_{1}, T_{2})$, and the ratio $\triangle_{d}=\frac{T_{d}(T_{1}, \, T_{2})}{T_{2}-T_{1}}$ the packet dropouts rate.

Definition 3: System (1) is said to be exponentially stabilizable under the control law $u=u(t)$, if the solution $x(t)$ of system (1) through $(t_{0}, \phi)$ satisfies

 $\|x(t)\|\leq \kappa \|x_{t_{0}}\|_{cl}e^{-\lambda (t-t_{0})}, \;\forall t\geq t_{0}$

for some constants $\kappa \geq 0$ and $\lambda>0$.

To end this section, some useful lemmas are needed.

Lemma 1 (Finsler's Lemma) :Let $\xi\in\mathbb{R}^{n\times n}$, $\Xi=\Xi^{T}\in\mathbb{R}^{n\times n}$, and $\mathcal {H}\in\mathbb{R}^{m\times n}$ such that rank$(\mathcal {H})=r<n$, then the following statements are equivalent:

ⅰ) $\xi^{T}\Xi\xi<0$, for all $\xi\neq0, \mathcal {H}\xi=0$;

ⅱ) $\exists \mathcal {X}\in\mathbb{R}^{n\times m}$ such that $\Xi+\mathcal {X}\mathcal {H}+\mathcal {H}^{T}\mathcal {X}^{T} <0.$

Lemma 2: For any constant matrix $S\in \mathbb{R}^{n\times n}, \, S=S^{T}>0$, scalars $\alpha>0$, $0<d(t)\leq \tau$, and vector function $\dot{\omega}:[-\tau, 0]\rightarrow \mathbb{R}^{n}$ such that the integrations in the following are well defined, then

 \begin{align*} -&\int_{t-d(t)}^{t}\dot{\omega}^{T}(\xi)S\dot{\omega}(\xi)d\xi\leq-\upsilon^{T}(t)\tau^{-1}S\upsilon(t) \end{align*}

where $\upsilon(t)=\omega(t)-\omega(t-d(t))$.

Proof: Following the same line used in the proof of Lemma 1 [44] gives directly this lemma.

Ⅲ. MAIN RESULTS

In this section, a mixed-mode based switched delay system method is used for stabilizing controller design of the NCS with NIDs & PDs. To begin, how the mixed modes affect the stability of the switched delay system will be studied in the Lyapunov-Krasovskii sense.

In order to study the stability of NCS (8), consider the following form of Lyapunov-Krasovskii functional

 \begin{align}\label{10-LF0} V_{\sigma(t)}(x_{t})&=x^{T}(t)P_{\sigma(t)}x(t)\\ &~~~~\!+\!\int_{-\tau_{\sigma(t)}}^{0}\int_{t+\theta}^{t}\dot{x}^{T}(s)e^{\chi_{\sigma(t)}(t-s)}S\dot{x}(s)dsd\theta\!\!\!\! \end{align} (10)

where $\chi_{i}\, (i\in \mathcal{P})$ are constant coefficients of $(t-s)$, and $P_{i}$ and $S$ are the positive matrices to be designed. Without loss of generality, suppose that $\sigma(t)=i=1$, the NCS with no packet dropouts is active, thus $\chi_{i}=-\alpha, \, \chi_{j}=\beta$ ($\alpha$, $\beta$ are positive scalars) and $\tau_1=\tau_m, \, \tau_2=\tau_M$, i.e., $V_{1}(x_{t})$ corresponds to the NCS with no packet dropouts and, $V_{2}(x_{t})$ the NCS with packet dropouts. We have the following proposition.

Proposition 1: When system (1) suffers packet dropout or large delay, the manifold $\mathcal {U}(s)= \dot{x}^{T}(s)e^{\beta(t-s)}S\dot{x}(s)$ satisfies

 \begin{align}\label{10-e10} \mathcal {U}(\zeta)\leq q\mathcal {U}(t) \end{align} (11)

for $\zeta\in[t-\tau_{i}, t_{k}]\, (i=1, 2)$, if the following holds

 $$$\label{CON-pro0} A^{T}S+SA-\beta S\!\geq\! -q_{0}I$$$ (12)

where $q_{0}$ is an adjustable parameter, and $q=e^{\frac{q_{0}\tau_{i}}{\lambda_{\min}(S)}}$.

Proof: See Appendix A.

According to the average dwell time strategy, the switched system guarantees the stability (or the stabilizability) if the proportion rate is relatively small between the total activation time of the unstable (or unstabilizable) subsystems and the one of the stable (or stabilizable) subsystems. The following condition is needed.

The switching signals $\sigma(t)$ in (10) which are generated by the switchings between the NCS without packet dropouts and the NCS with packet dropouts, are supposed to be satisfied with the following average dropout rate:

 \begin{align}\label{dwt} \triangle_{d}=\frac{T_{d}(T_{1}, T_{2})}{T_{2}-T_{1}}<\frac{\alpha-\alpha^{\ast}}{\alpha+\beta} \end{align} (13)

where $\alpha^{\ast}\in(0, \alpha)$.

For the convenience of disposal of the mixed-modes for the NCS using a switched delay system mothod, it is helpful to exchange the integral order of the term $\int_{-\tau_{\sigma(t)}}^{0}\int_{t+\theta}^{t}\dot{x}^{T}(s)e^{\chi_{\sigma(t)}(t-s)}S\dot{x}(s)dsd\theta$ in (10) and one has

 \begin{align}\label{10-LF} V&_{\sigma(t)}(x_{t})=x^{T}(t)P_{\sigma(t)}x(t)\\ &~~~~~~~~~~~\!~\!\!+\!\!\int_{t\!-\!\tau_{\sigma(t)}}^{t}\!(s\!-\!t\!+\!\tau_{\sigma(t)})\dot{x}^{T}(s)e^{\chi_{\sigma(t)}(t\!-\!s)}S\dot{x}(s)ds.\!\! \end{align} (14)
A. Exponential Estimation of the Lyapunov-Krasovskii Functionals

Before developing the stability conditions for the mixed-modes based switched delay systems, a preliminary result is presented. The following lemma is based on the interpolatory quadrature formula (see, e.g. [45]).

Lemma 3: Let

 $W_{1}(t)=\int_{t-\tau_{m}}^{t}(s-t+\tau_{m})\dot{x}^{T}(s)e^{-\alpha(t-s)}S\dot{x}(s)ds$

and

 $W_{2}(t)=\int_{t-\tau_{M}}^{t}(s-t+\tau_{M})\dot{x}^{T}(s)e^{\beta(t-s)}S\dot{x}(s)ds$

when $t>t_{k}$, we have the following estimation inequalities

 $\begin{array}{l} \int_{t - {\tau _m}}^{{t_k}} {(s - t + {\tau _m}){\cal U}(s)} ds\\ \le q\frac{{{t_k} - t + {\tau _m}}}{{{\tau _m}}} \times \int_{t - {\tau _m}}^t {(s - t + {\tau _m}){\cal U}(s)} ds \end{array}$ (15)

$\mathit{\rm{where}} \; \; \mathcal {U}(s)= \dot{x}^{T}(s)e^{\beta(t-s)}S\dot{x}(s)$

 ${W_2}(t) \le {\left( {\frac{{{\tau _M}}}{{{\tau _m}}}} \right)^2} \times {e^{(\alpha + \beta ){\tau _M}}} \times {W_1}(t).$ (16)

Proof: See Appendix B.

With the help of Lemma 3, the following lemma facilitates us to give the exponential decay and increase estimation of Lyapunov-Krasovskii functionals corresponding to the NCS without packet dropouts and, the NCS with packet dropouts, respectively.

Lemma 4: For given positive constants $\alpha, \, \beta$, the derivative of the Lyapunov-Krasovskii functional (14) along the trajectory of system (8) satisfies

 $$$\label{eqLF1} V_{1}(x_{t})\leq e^{-\alpha(t-t_{0})}V_{1}(x_{t_{0}})$$$ (17)

when $\sigma(t)=1$ and satisfies

 $$$\label{eqLF2} V_{2}(x_{t})\leq e^{\beta(t-t_{0})}V_{2}(x_{t_{0}})$$$ (18)

when $\sigma(t)=2$, if there exist symmetric positive definite matrices $P_{i}\; \textrm{and}\; S$, matrix $K$, any free weighting matrices $X_{i}, Y_{i}, Z_{i}\, (i=1, 2)$ with appropriate dimensions, and positive constants $\tau_{m}, \, \tau_{M}$ such that the following optimization problem:

 \begin{align}\label{CON-pro} &\max \quad \tau_{m}, \;\tau_{M} \\ &\begin{array}{ll} \mathit{\rm{ s.t.}}&\!\!\Theta_{i}:=\left[\begin{array}{ccc} \varphi_{11}^{i}\;&\;\varphi_{12}^{i}\;&\;\varphi_{13}^{i}\\ *&\varphi_{22}^{i}&\varphi_{23}^{i}\\ *&*&\varphi_{33}^{i} \end{array}\right]<0, \quad i=1, 2 \end{array} \end{align} (19)

is solvable, where

 \begin{align*} \varphi_{11}^{i}&=A^{T}P_{i}\!+\!P_{i}A\!+\!K^{T}B^{T}P_{i}\!+\!P_{i}BK\!+\!\chi_{i} P_{i}\!+\!X_{i}\!+\!X_{i}^{T}\\ \varphi_{12}^{i}&=-P_{1}BK-X_{i}+Y_{i}^{T}\\ \varphi_{13}^{i}&=(A+BK)^{T}S-X_{i}+Z_{i}^{T}\\ \varphi_{22}^{1}&=-\left[\tau_{m}^{-1}e^{-\alpha\tau_{m}}-q(\alpha-\alpha^{\ast})e^{\beta\tau_{m}}\right]S -Y_{1}-Y_{1}^{T}\\ \varphi_{22}^{2}&=-\tau_{M}^{-1}e^{-\alpha\tau_{M}}S-Y_{1}-Y_{1}^{T}\\ \varphi_{23}^{i}&=-K^{T}B^{T}S-Y_{i}-Z_{i}^{T}\\ \varphi_{33}^{i}&=-\tau_{i}^{-1}S-Z_{i}-Z_{i}^{T}\quad(i=1, 2). \end{align*}

Proof: See Appendix C.

Remark 3: In order to deal with the mixed-modes based stability analysis for NCS with NIDs & PDs, the interpolatory quadrature formula is key to deriving the exponential decay or increase estimation conditions for the Lyapunov-Krasovskii functionals. It should be noted that simple bounding techniques may not give the available solution of the problem addressed.

Remark 4: By the introduction of the Finsler's lemma, the solvability problem of $\dot{V}_{i}(x_{t})+\alpha V_{i}(x_{t})\leq\xi^{T}(t)\Xi_{i}\xi(t)$ $<0$ is converted to the solvability problem of inequalities (19). The merit of this scheme lies in that the matrices in the Lyapunov functional and the control gains which are to be determined are decoupled. On the other hand, since some free weighting matrices can be selected, the flexibility of solving inequalities (19) is increased, and thus the conservativeness of the conditions (19) is decreased accordingly.

We are now in a position to give the stabilization conditions for the NCS with NIDs & PDs.

Theorem 1: For given positive constants $\alpha, \, \beta, \, q$, if there exist symmetric positive definite matrices $P_{i}\, \, \textrm{and}\, \, S$, matrices $K$, any free weighting matrices $X_{i}, Y_{i}, Z_{i}, T_{i}, M_{i}, N_{i}$ $(i=1, 2)$ with appropriate dimensions, and positive constants $\tau_{m}, \, \tau_{M}$ such that (12) and (9) are satisfied, the system (1) with NIDs & PDs is exponentially stabilizable under the switching signal $\sigma(t)$ which orchestrates the switchings between the system without packet dropout and with packet dropouts, and satisfies the following conditions:

ⅰ) the average dwell time is satisfied with

 \begin{align}\label{eq26} \frac{\ln\mu\varepsilon}{\alpha}<\frac{\ln\mu\varepsilon}{\alpha^{*}}=T_{\alpha}^{*} where\varepsilon=e^{\frac{2(\ln \tau_{M}-\ln \tau_{m})+(\alpha+\beta)\tau_{M}}{2}}$$ {\rm{and}}\;\mu = \mathop {\inf }\limits_\nu \left\{ {\nu \left| {\begin{array}{*{20}{c}} {{P_i} \le \nu {P_j}, \;\forall i, j \in {\cal P}} \end{array}} \right.} \right\} $(21) in which$P_{1}$and$P_{2}$are defined in Lemma 4; ⅱ) the packet dropouts rate meets (3). Proof: The proof is similar to the one in [38], and thus is omitted. Remark 5: By using the switched delay system method, an NCS with NIDs & PDs can be modeled as a switched delay system with both stable and unstable subsystems, which extremely extends the delay upper bound for an NCS. Note that the large delay case has been viewed as packet dropout, so the packet dropouts rate can be depicted as the occurrence frequency of the unstable subsystem, i.e., we can obtain the packet dropouts rate by (13) and the optimal solutions of (12) and (19). Now, we present the procedure to get the maximum packet dropouts rate$\Delta_{d}$. Step 1: Preset the values of positive constants$\alpha, \, \beta$, solve inequality (19) to get the feasible solutions of$\tau_{m}$and$\tau_{M}$. Step 2: Bidirectionally adjust the constants$\alpha, \, \beta$to obtain the optimal solution of$q_{0}$according to (12), and then to get the optimal solutions of$\tau_{m}$and$\tau_{M}$. Step 3: Select an appropriate parameter$\alpha^{*}$to get the maximum packet dropouts rate according to (13), such that the NCS can be stabilizable. Remark 6: The delay upper bounds$\tau_{m}, \, \tau_{M}$, which we obtained by the optimal techniques, are different from existing results which are given as constants for the NCSs, see [23], [41], [42] for instance. Most importantly, the key point$\tau_{m}$we derived is the threshold of the switching between the NCS with packet dropouts and that without packet dropout. Remark 7: Literally, the main results obtained in this paper are similar to those of [19] due to that the switchings, networked delays, the mixed" properties and the average switching techniques are studied in both papers. However, compared with [19], stability analysis of NCSs considering of the coexistence and interaction of switchings and delays, i.e., the effect of mixed-mode, is the obvious the main task of this paper. Ⅳ. AN ILLUSTRATIVE EXAMPLE In this section, a numerical example is introduced to demonstrate the mixed-mode based switched delay system method for NCS with NIDs & PDs. Consider the same NCS (1) as in [38], where $ \begin{equation*} A=\left[\begin{array}{cc}-0.8&-0.01\\1&0.1\end{array}\right], \quad B=\left[\begin{array}{c}0.4\\0.1\end{array}\right]. \end{equation*} $It is obvious that the open-loop system is not stable. With the networked control scheme, we can model the following switched delay system $ \begin{align*}\dot{x}(t)=Ax(t)+BKx(t-d_{\sigma(t)}(t)). \end{align*} $By the mixed-mode based switching delay system method proposed in this paper, we can obtain the critical delay bound$\tau_{m}=1.03$and the admissible upper bounds of the delay$\tau_{M}=1.66$. The control gain $ K=\left[\begin{array}{cc}-7.2893&-5.6621\end{array}\right] $and the Lyapunov-Krasovskii functional (14) with $ P_{1}=\left[ \begin{array}{cc} 11.1070& 6.2535\\ 6.2535 & 8.4610 \end{array} \right], \quad P_{2}=\left[ \begin{array}{cc} 10.2466& 3.9710\\ 3.9710 & 7.2438 \end{array} \right] $and $ S=\left[ \begin{array}{cc} 153.6669& 13.6497\\ 13.6497& 132.9459 \end{array} \right] $are calculated with the MATLAB LMI control Toolbox based on conditions (19). Note that, though the effect of the mixed-mode of the delays and dropouts is considered, by introducing the Finsler's lemma to deal with the network induced delays and dropouts, we can still establish the conditions of less conservativeness, which enables us to get larger critical delay bound$\tau_{m}$and admissible delay bound$\tau_{M}$than existing results. Table Ⅰ presents the relative results by comparison. Table Ⅰ THE CRITICAL DELAY BOUND$\tau_{m}$AND THE ADMISSIBLE DELAY BOUND$\tau_{M}$OBTAINED BY DIFFERENT METHODS By selecting the parameters$\alpha=0.4320, \, \beta=0.6201$, we can get$\mu= 1.3553$and$T_{\alpha}^{*}=4.3312$according to condition (ⅱ) of Theorem 1 with the optimization algorithm, and the maximum packet dropouts ratio$\triangle_{d}\rightarrow0.0475\, (T_{\alpha}\rightarrow\infty)$can be calculated. Due to Finsler's lemma being adopted, more free weighting matrices facilitate us to get less conservative results than existing ones, e.g. [38], [41]. Table Ⅱ gives the comparison of the influences of the delay bounds$\tau_{m}, \, \tau_{M}$to$\mu$and the dropouts rate$\triangle_{d}$. Table Ⅱ COMPARISON OF INFLUENCES OF CRITICAL DELAY BOUND$\tau_{m}$AND THE ADMISSIBLE DELAY BOUND$\tau_{M}$TO$\mu$, THE DROPOUTS RATE$\triangle_{d}$We simulate the network environment such that the NIDs & PDs satisfy the conditions of Theorem 1. By selecting the initial state as$x_{1}(0)=-0.1$and$x_{2}(0)=0.1$, Fig. 3 presents the switching signal generated by the delays and packet dropouts. Satisfactory simulation results of the control input and the state response are illustrated in Figs. 4 and 5, respectively.  Download: larger image Fig. 3 The switching signal resulted by the packet dropouts  Download: larger image Fig. 4 The control input  Download: larger image Fig. 5 The state response of NCS with NIDs & PDs Ⅴ. CONCLUSION This paper addressed the modeling of NCS with network induced delays and packet dropouts as a continuous-time switched linear delay system. The effect of mixed-mode to the stability analysis, from the viewpoint of a switched delay system, was investigated for NCSs based on a novel constructed Lyapunov-Krasovskii functional, and Newton-Cotes quadrature formula was taken into account for the decay rate estimation of the switched Lyapunov Krasovskii functional. With the help of the Finsler's lemma, new exponential stabilizability conditions with less conservativeness were given for NCSs. Finally, an example was given to illustrate the effectiveness of the developed results. Issues of the stabilization problem for NCSs using the mixed-mode based switching delay system method with more general Lyapunov functionals, or for NCSs with multiple packets transmission and other communication constraints, are interesting and challenging, and thus deserve further investigation in the future. APPENDIX A PROOF OF PROPOSITION 1 Consider the derivative of$\mathcal {U}(s)$with respect to$s$. For this purpose, let$z(s)=\dot{x}(s)$, and then$\mathcal {U}(s)=z^{T}(s)e^{\beta(t-s)}Sz(s)$. For$t\in[i_{k+1}h+\eta_{m}, i_{k+j}h+\eta_{k+j})$which corresponds to the packet dropout, differentiating both sides of (8) yields $ \begin{align}\!\dot{z}(s)\!=\!Az(s)\!+\!(1\!-\!\dot{d}_{2}(s))BKz(s\!-\!d_{2}(s)).\end{align} $(A1) Note that the sawtooth structure of the artificial delays, i.e.,$d_{2}(t)=t-i_{k}h+l(k)h$and$\dot{d}_{2}(t)=1$seen almost anywhere. System (A1) reduces to $ \begin{align}\label{unsys1}\dot{z}(s)=Az(s).\end{align} $(A2) Fixing$t$as a parameter, and taking derivative of$\mathcal {U}(s)$with respect to$s$along system (A1) yield $ $$\dot{\mathcal {U}}(s)=z^{T}(s)e^{\beta(t-s)}\left[A^{T}S+SA-\beta S\right]z(s).$$ $(A3) The conditions of (12) help us to get$\dot{\mathcal {U}}(s)\geq-q_{0}\parallel z(s)\parallel^{2}$. Noticing that$\parallel z(s)\parallel^{2}\leq\frac{\mathcal {\mathcal {U}}(s)}{\lambda_{\min}(S)}$, we can further obtain $ \begin{align}\dot{\mathcal {U}}(s)\geq-\frac{q_{0}}{\lambda_{\min}(S)}\mathcal {U}(s).\end{align} $(A4) Integrating (A4) from$\zeta\in[t-\tau_{i}, t_{k}]$to$t$gives$\ln\frac{U(t)}{U(\zeta)}\geq-\frac{q_{0}}{\lambda_{\min}(S)}(t-\zeta)\geq-\frac{q_{0}\tau_i}{\lambda_{\min}(S)} =\ln\frac{1}{q}$, which leads to $ $$\mathcal {U}(\zeta)\leq q \mathcal {U}(t).$$ $(A5) APPENDIX B PROOF OF LEMMA 3 Consider the Newton-Cotes quadrature formula [45]. ⅰ) Note that inequality (15) is equivalent to the following inequality $ \begin{align} &\frac{q (t_{k}-t)+(q-1)\tau}{\tau}\int_{t-\tau}^{t_{k}}(s-t+\tau)\mathcal {U}(s)ds\\ &\quad\quad+\frac{q (t_{k}-t)+q\tau}{\tau} \int_{t_{k}}^{t}(s-t+\tau)\mathcal {U}(s)ds\geq0. \end{align} $(A6) If (A6) can be guaranteed, the proof is finished. Now, consider the left hand of (A6). According to the integral mean value theory, there exist$\xi_{1}\in[t-\tau, t_{k}]$and$\xi_{2}\in[t_{k}, t]$, such that $ \begin{align} &\frac{q (t_{k}-t)+(q-1)\tau}{\tau}\int_{t-\tau}^{t_{k}}(s-t+\tau)\mathcal {U}(s)ds\\ &\;\, =\frac{q (t_{k}-t)\!+\!(q-1)\tau}{\tau} (t_{k}\!-t\!+\!\tau)[(\xi_{1}\!-t\!+\tau)\mathcal {U}(\xi_{1})] \end{align} $(A7) and $ \begin{align} &\frac{q (t_{k}-t)+q\tau}{\tau}\int_{t_{k}}^{t}(s-t+\tau)\mathcal {U}(s)ds\\ &\qquad=\frac{q (t_{k}-t)+q\tau}{\tau}(t-t_{k})[(\xi_{2}-t+\tau)\mathcal {U}(\xi_{2})]. \end{align} $(A8) Noticing that$t-\tau\leq\xi_{1}\leq t_{k}\leq\xi_{2}\leq t$, and considering condition (A5) gives $ \begin{align} &\frac{q (t_{k}-t)+(q-1)\tau}{\tau}\int_{t-\tau}^{t_{k}}(s-t+\tau)\mathcal {U}(s)ds\\ &\qquad+\frac{q (t_{k}-t)+q\tau}{\tau} \int_{t_{k}}^{t}(s-t+\tau)\mathcal {U}(s)ds\\ &\quad=\frac{q (t_{k}-t)+(q-1)\tau}{\tau} (t_{k}-t+\tau)[(\xi_{1}-t+\tau)\mathcal {U}(\xi_{1})]\\ &\qquad+\frac{q (t_{k}-t)+q\tau}{\tau}(t-t_{k})[(\xi_{2}-t+\tau)\mathcal {U}(\xi_{2})]\\ &\quad\geq\frac{q (t_{k}-t)+(q-1)\tau}{\tau} (t_{k}-t+\tau)[(\xi_{1}-t+\tau)\mathcal {U}(\xi_{1})]\\ &\qquad+\frac{q (t_{k}-t)+q\tau}{\tau}(t-t_{k})[(\xi_{1}-t+\tau)\mathcal {U}(\xi_{1})]\frac{1}{q} \end{align} $(A9) noticing that$q=e^{\frac{q_{0}\tau_{i}}{\lambda_{\min}(S)}}\geq1$and$0 <\frac{1}{q}\leq1$one can easily obtain that (A9) is greater than or equal to$0$. The proof is completed. ⅱ) Considering the integral transformation for$W_{2}(t)$with$s=\frac{\tau_{M}}{\tau_{m}}(x-t+\frac{\tau_{m}}{2})+(t-\frac{\tau_{M}}{2})$, one can obtain $ \begin{align*}W_{2}(t)&=\int_{t-\tau_{M}}^{t}(s-t+\tau_{M})U_{2}(s)ds\\ &=\left(\frac{\tau_{M}}{\tau_{m}}\right)^{2} \int_{t-\tau_{m}}^{t}(x-t+\tau_{m})\\ &~~~~\times U_{2}\left(\frac{\tau_{M}}{\tau_{m}}(x-t+\frac{\tau_{m}}{2})+(t-\frac{\tau_{M}}{2})\right)dx. \end{align*} $Note that when$x\in[t-\tau_{m}, t], \, 0 <\tau_{m}<\tau_{M}$, there has $ \begin{align*}&\frac{\tau_{M}}{\tau_{m}}\left(x-t+\frac{\tau_{m}}{2}\right)+(t-\frac{\tau_{M}}{2})-x\\ =&\left(\frac{\tau_{M}}{\tau_{m}}-1\right)(x-t)\leq0. \end{align*} $Note that$U_{2}(s)\leq e^{(\alpha+\beta)\tau_{M}}U_{1}(s)$, inequality (A7) holds. APPENDIX C PROOF OF LEMMA 4 Firstly, we consider the following decomposition of the Lyapunov-Krasovskii functional (14). Suppose that system (8) is in the$i$th mode for$t\in[t_{k}, t_{k+1})$, and the$i$th subsystem was activated from the$j$th subsystem at$t=t_{k}$, it is also assumed that$(t-\tau_{i})\in [t_{k-1}, t_{k})$. When$t\in[t_{k}, t_{k}+\tau_{i})$it follows from (14) that $ \begin{align} V_{i}(x_{t})=&x^{T}(t)P_{i}x(t)\\ &+\int_{t-\tau_{i}}^{t_{k}}(s-t+\tau_{i})\dot{x}^{T}(s)e^{\chi_{j}(t-s)}S\dot{x}(s)ds\\ &+\int_{t_{k}}^{t}(s-t+\tau_{i})\dot{x}^{T}(s)e^{\chi_{i}(t-s)}S\dot{x}(s)ds\dot{x}(s) \end{align} $(A10) and when$t\in [t_{k}+\tau_{i}, t_{k+1})$$ $$V_{i}(x_{t})=x^{T}(t)P_{i}x(t) +\int_{t-\tau_{i}}^{t}(s-t+\tau_{i})\dot{x}^{T}(s)e^{\chi_{i}(t-s)}S\dot{x}(s)ds.$$ $(A11) We will discuss the derivatives of the Lyapunov-Krasovskii functionals in two cases, i.e.,$t_{k+1}-t_{k}\leq\tau_{i}$and$t_{k+1}-t_{k}>\tau_{i}$. Case 1:$t_{k+1}-t_{k}\leq\tau_{i}$. Note that in this case, the mixed-mode may cover the whole switching interval$[t_{k}, t_{k+1})$. Thus, for$t\in[t_{k}, t_{k+1})$, differentiating the Lyapunov-Krasovskii functional (14) along the trajectory of system (8) gives $ \begin{align} \dot{V}_{i}(x_{t})=&\, 2x^{T}(t)P_{i}\dot{x}(t)+\tau_{i} \dot{x}^{T}(t)S\dot{x}(t)\nonumber\\ &-\int_{t-\tau_{i}}^{t_{k}}\dot{x}^{T}(\theta)e^{\chi_{j}(t-\theta)}S\dot{x}(\theta)d\theta\nonumber\\ &-\int_{t_{k}}^{t}\dot{x}^{T}(\theta)e^{\chi_{i}(t-\theta)}S\dot{x}(\theta)d\theta\nonumber\\ &+\chi_{j}\int_{t-\tau_{i}}^{t_{k}}(s-t+\tau_{i})\dot{x}^{T}(\theta)e^{\chi_{j}(t-\theta)}S\dot{x}(\theta)d\theta\nonumber\\ &+\chi_{i}\int_{t_{k}}^{t}(s-t+\tau_{i})\dot{x}^{T}(\theta)e^{\chi_{i}(t-\theta)}S\dot{x}(\theta)d\theta. \end{align} $(A12) If it is assumed that$\sigma(t)=i=1$, thus$\chi_{i}=-\alpha, \, \chi_{j}=\beta$and$\tau_1=\tau_m, \, \tau_2=\tau_M$. Therefore $ \begin{align} \dot{V}_{1}(x_{t})&+\alpha V_{1}(x_{t})\\=&\, 2x^{T}(t)P_{1}\dot{x}(t)+\tau_{m} \dot{x}^{T}(t)S\dot{x}(t)+x^{T}(t)\alpha P_{1}x(t)\nonumber\\ &-\int_{t-\tau_{m}}^{t_{k}}\dot{x}^{T}(\theta)e^{\beta(t-\theta)}S\dot{x}(\theta)d\theta\nonumber\\ &-\int_{t_{k}}^{t}\dot{x}^{T}(\theta)e^{-\alpha(t-\theta)}S\dot{x}(\theta)d\theta\nonumber\\ &+\beta\int_{t-\tau_{m}}^{t_{k}}(s-t+\tau_{m})\dot{x}^{T}(\theta)e^{\beta(t-\theta)}S\dot{x}(\theta)d\theta\nonumber\\ &-\alpha\int_{t_{k}}^{t}(s-t+\tau_{m})\dot{x}^{T}(\theta)e^{-\alpha(t-\theta)}S\dot{x}(\theta)d\theta\nonumber\\ &+\alpha\int_{t-\tau_{m}}^{t_{k}}(s-t+\tau_{m})\dot{x}^{T}(\theta)e^{\beta(t-\theta)}S\dot{x}(\theta)d\theta\nonumber\\ &+\alpha\int_{t_{k}}^{t}(s-t+\tau_{m})\dot{x}^{T}(\theta)e^{-\alpha(t-\theta)}S\dot{x}(\theta)d\theta.\ \end{align} $(A13) Considering the effect of the mixed-mode, during$[t-\tau_{m}, \, t_{k})$, the 2nd subsystem is active. With the estimation inequality (15) in Lemma 3 and the average activation rate (13), we have $ \begin{align} &(\alpha+\beta)\int_{t-\tau_{m}}^{t_{k}}(s-t+\tau_{m})\dot{x}^{T}(\theta)e^{\beta(t-\theta)}S\dot{x}(\theta)d\theta\\ &\leq(\alpha+\beta)\cdot\frac{t_{k}-t+\tau_{m}}{\tau_{m}}\cdot q\tau_{m} \int_{t-\tau_{m}}^{t}\dot{x}^{T}(\theta)e^{\beta\tau_{m}}S\dot{x}(\theta)d\theta\\ &\leq(\alpha+\beta)\cdot\frac{\alpha-\alpha^{\ast}}{\alpha+\beta}\cdot q\tau_{m} \int_{t-\tau_{m}}^{t}\dot{x}^{T}(\theta)e^{\beta\tau_{m}}S\dot{x}(\theta)d\theta\\ &\leq q(\alpha-\alpha^{\ast})\cdot\tau_{m} \int_{t-\tau_{m}}^{t}\dot{x}^{T}(\theta)e^{\beta\tau_{m}}S\dot{x}(\theta)d\theta \end{align} $(A14) and it is obvious that $ \begin{align} &-\int_{t-\tau_{m}}^{t_{k}}\dot{x}^{T}(\theta)e^{\beta(t-\theta)}S\dot{x}(\theta)d\theta -\int_{t_{k}}^{t}\dot{x}^{T}(\theta)e^{-\alpha(t-\theta)}S\dot{x}(\theta)d\theta\nonumber\\ &\leq-\int_{t-\tau_{m}}^{t}\dot{x}^{T}(\theta)e^{-\alpha\tau_{m}}S\dot{x}(\theta)d\theta. \end{align} $(A15) Substituting (A14) and (A15) into (A13) gives rise to $ \begin{align} \dot{V}_{1}&(x_{t})+\alpha V_{1}(x_{t})\nonumber\\ \leq\, &2x^{T}(t)P_{1}[Ax(t)+BKx(t-d_{1}(t))]\nonumber\\ &\!+\!\tau_{m} [Ax(t)\!+\!BKx(t\!-\!d_{1}(t))]^{T}S[Ax(t)\!+\!BKx(t\!-\!d_{1}(t))]\\ &\!+\!x^{T}(t)\alpha P_{1}x(t)-\int_{t-\tau_{m}}^{t}\dot{x}^{T}(s)e^{-\alpha\tau_{m}}S\dot{x}(s)ds\nonumber\\ &\!+\!q(\alpha-\alpha^{\ast})\tau_{m} \int_{t-\tau_{m}}^{t}\dot{x}^{T}(\theta)e^{\beta\tau_{m}}S\dot{x}(\theta)d\theta.\\ \leq\, &2x^{T}(t)P_{1}\dot{x}(t)+\tau_{m} \dot{x}^{T}(t)S\dot{x}(t)+x^{T}(t)\alpha P_{1}x(t)\nonumber\\ &-\left[e^{-\alpha\tau_{m}}-q(\alpha-\alpha^{\ast})\tau_{m} e^{\beta\tau_{m}}\right] \int_{t-d_{1}(t)}^{t}\dot{x}^{T}(\theta)S\dot{x}(\theta)d\theta. \end{align} $(A16) Note that$0<d_{1}(t)\leq \tau_{m}$. Lemma 2 facilitates us to obtain $ \begin{align} -&\int_{t-d_{1}(t)}^{t}\dot{x}^{T}(\theta)S\dot{x}(\theta)d\theta\leq-\upsilon^{T}(t)\tau_{m}^{-1}S\upsilon(t) \end{align} $(A17) where$\upsilon(t)=\int_{t-d_{1}(t)}^{t}\dot{x}^{T}(\theta)d\theta=x(t)-x(t-d_{1}(t))$. Let$\xi(t)=[\begin{array}{ccc}x^{T}(t), &x^{T}(t-d_{1}(t)), &\upsilon^{T}(t)\end{array}]^{T}$, and then it can be concluded that $ \begin{align} \dot{V}_{1}(x_{t})+\alpha V_{1}(x_{t})\leq\xi^{T}(t)\Xi_{1}\xi(t) \end{align} $(A18) where $ \begin{equation*} \Xi_{1}:=\left[\begin{array}{ccc} \phi_{11}^{1}\;&\;\phi_{12}^{1}\;&\;\phi_{13}^{1}\\ *&\phi_{22}^{1}&\;\phi_{23}^{1}\\ *&*&\;\phi_{33}^{1} \end{array}\right]\end{equation*} $with $ \begin{align*} \phi_{11}^{1}&=A^{T}P_{1}+P_{1}A+K^{T}B^{T}P_{1}+P_{1}BK+\alpha P_{1}\\ \phi_{12}^{1}&=-P_{1}BK\\ \phi_{13}^{1}&=(A+BK)^{T}S\\ \phi_{22}^{1}&=-\left[\tau_{m}^{-1}e^{-\alpha\tau_{m}}-q(\alpha-\alpha^{\ast})e^{\beta\tau_{m}}\right]S\\ \phi_{23}^{1}&=-K^{T}B^{T}S\\ \phi_{33}^{1}&=-\tau_{m}^{-1}S. \end{align*} $Note that, on the other hand, when$\sigma(t)=1$, inequality (19) can be rewritten as follows $ \begin{align} \Xi_{1}+\mathcal {X}_{1}\mathcal {H}+\mathcal {H}^{T}\mathcal {X}_{1}^{T}<0 \end{align} $(A19) where $ \begin{align*} \mathcal {X}_{1}&=\left[ \begin{array}{cc} X_{1} \\ Y_{1} \\ Z_{1} \end{array} \right]\\ \mathcal {H}&=\left[ \begin{array}{ccc} I &-I &-I \\ \end{array} \right]. \end{align*} $Obviously,$\mathcal {H}\xi(t)=0$. According to Finsler's lemma, there is$\xi^{T}(t)\Xi_{1}\xi(t) <0$, which implies$\dot{V}_{1}(x_{t})+\alpha V_{1}(x_{t})\leq0$and $ \begin{align}\tag{A20} V_{1}(x_{t})\leq e^{-\alpha(t-t_{0})}V_{1}(x_{t_{0}}) \end{align} $holds apparently. Case 2:$t_{k+1}-t_{k}>\tau_{i}$. In this case, two subintervals:$[t_{k}, \, t_{k}+\tau_{i})$and$[t_{k}+\tau_{i}, \, t)$are considered. For$t\in[t_{k}, \, t_{k}+\tau_{i})$, the analysis is the same as in Case 1. For$t\in[t_{k}+\tau_{i}, \, t)$,$V_{i}(x_{t})$is given by (32). Note that no time interval decomposition of the integral is needed. When$i=1$, condition (19) must be satisfied with this case, so the proof is omitted. Similarly, it is easy to obtain$V_{2}(x_{t})\leq e^{\beta(t-t_{0})}V_{2}(x_{t_{0}})\$ with the same procedure above.

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# Using A While Loop Write A Simple Coin Flip Game

This is an introduction to Pygame for people who already know Python. Line 7 is a while loop that has a condition of simply the value True. There are a lot of different sorts of game loops you can use. ar Thu Jul 27 09:36:18 ART 2000. For example, instead of good job you can write “James is 6” or you can add a line thanking friends for celebrating with you. Packet Updated: July 2018: With summer month options. It is a bit of harmless fun, our residents. In the last instruction , you have been asked to use equals() method and keep in mind that you have a local variable called "choice" with you holding the user input. Sorry for the basic question but I'm new to Java. Whether your child is just starting their educational journey in preschool or taking the leap into 5th grade, our collection of games will help your child practice the skills they need to succeed, while making learning a positive experience they're excited to continue. We have created a program that can toss a coin over and over until it comes up head 10 times. Let us know a little bit about you so we can give you the best information possible. In this guide, we will learn for loop and the other two loops are covered in the separate tutorials. An (unnamed) turtle object is automatically created whenever any of the functions derived from a Turtle method is called. 1+ (nx/1!) - (n(n-1)x^2/2!) Write a C++ program to 1. If you guess right, pass the coin to your right. Five Reasons to Use Games in the Classroom Thanks to its partnership with publisher Eye on Education, Education World is pleased to present this blog post by Rebekah Stathakis, author of A Good Start:147 Warm-Up Activities for Spanish Class. What we want is a repeatable sequence of seemingly random numbers that satisfy certain properties, such as the average value of a list of random numbers between say, 0 and 1000, should be 500. If the player guesses wrong, then the prompt appears again until the guess is correct. cls( 1 ) pyxel. Turn a cardboard box and some laundry baskets into this simple skee ball game! See the details here: Laundry Basket Skee Ball. Limit the number of flips per run from 1 - 100,000 4. Words, Vocabulary. You need to use loops. It’s similar in concept to the Advanced Vocabulary Builder below, but it’s been created specifically for the little ones (but don’t let that deter you!). The outcome of the first coin flip determines the encryption rule just as in the previous example. The show function displays the screen and waits until the ESC key is pressed. For example, C programs uses constructs such as if-else, for-loop, array, function, pointer, which are low-level and hard to abstract real problems such as a Customer Relationship Management (CRM) system or a computer soccer game. Although still moderated, SE definitely represents the Wild. Follow along to learn more about how Bitcoin works and the best places to buy Bitcoin. Script writing is the main part of any Episode Interactive story. Check out 10-year-old Ayesha’s online reviews of children's books. Create-A-Kid Students flip a pair of coins to determine what genotype and what phenotype their child will have for 29 given traits. You should also measure the slope of each. So if your child is gliding, you will use an /r/ word. The program prompts the user to enter a guess and reports whether the guess is correct or incorrect. Some have pictures and drawings, while others are just plain. The AI's smarts for playing Tic Tac Toe will follow a simple algorithm. Your coin flip function is kind of odd. What is the use of break and continue in Python? In Python, break and continue statements can alter the flow of a normal loop. In simple words: The java programs can be distributed on more than one systems that are connected to each other using internet connection. Staples EasyClose Security Tinted #10 Business Envelopes, 4 1/8" x 9 1/2", White, 500/Box (50312) Get it tomorrow, Jan 15. Some arcade games are about fighting, while others focus on dodging obstacles, scoring points and getting power-ups to help along the way. Help your students build a growth mindset using social-emotional learning and ScienceWorld magazine. 3 V / 5 V (better) GND -> GND. let step = 0 step = 0 Detecting a step. It returns a string "Heads" if the boolean variable "sideShowing" is TRUE. ¢ is the symbol used to denote cents. The output looks something like this: Welcome to the coin. Guides that also cover topics like business, distribution and marketing. We do not consider the ﬁrst two tosses to constitute a run, since the third toss has the same value as the ﬁrst two. In this tutorial we’ll explain how these two methods work, and give some practical examples. Unfortunately, I do not believe I was successful in explaining to Kent why my figures were correct. org is the very first 2 player games portal in the world and has the largest games achieve in its field. The first activity, Sorting, helps children to recognise the different coins. Keep in the loop. Office Party Game Ideas. Its simple-to-use interface is winning the hearts and minds of everyone. A fake coin weighs 1 gram less than a real coin. 099999 This just compounds. COM's practical food and fitness tools, expert resources and an engaged community. Let’s Play! 9 Awesome Online Games to Learn Spanish Word Toss. In case the. Cover the waxed paper with another piece of waxed paper. The While loop in R Programming is used to repeat a block of statements for a given number of times until the specified expression is False. import pyxel pyxel. Second is the authenticity and originality of the coin. Do While Loop. Taking The Piss: messing or screwing around. Arcade is a Python library for creating 2D video games that is easy to start using, and very capable as you gain experience. Experimental Probability When asked about the probability of a coin landing on heads, you would probably answer that the chance is ½ or 50%. The max-loop differs from the search-loop since the max-loop must look at every element, while the search-loop can exit as soon as it finds a match. you need either a for loop or a while loop around the part you want to loop. I need to write a python program that will flip a coin 100 times and then tell how many times tails and heads were flipped. (optional) - this value is added to the sum of items of the iterable. Basically, the loop keeps cycling through until a logical test is failed (returns a false). Find the probability of rolling even numbers three times using a six-sided die numbered from 1 to 6. DocuSign enables people to electronically sign agreements from almost anywhere. GetAxis or Input. Let me write this, the probability of exactly two heads, I'll say H's there for short. The subscript here indicates that this is the PMF of the random variable. Volunteer-led clubs. 5 then it's Heads or otherwise Tails. Give the prayer to the person who is chosen to be "it," and have the rest of the players hide their eyes while that person hides with the prayer. You need to use loops. The while loop’s main responsibility is. Write a few sizable non-graphical programs in python -- parse some text files, write a guessing game or a journal-entry program or something. For example, the pattern [^abc] will match any single character except for the letters a, b, or c. That was flip number 130,659,178 Flip again? Color The Coin!. Weighing only once and using the minimum number of coins, how can you find the bags containing the fake coins?. It returns a string "Heads" if the boolean variable "sideShowing" is TRUE. Create-A-Kid Students flip a pair of coins to determine what genotype and what phenotype their child will have for 29 given traits. Toss results can be viewed as a list of individual outcomes, ratios, or table. Learn more: Gallykids. Then, the test expression (i <= 10) evaluates to false and while loop terminates. getNumberOfTosses: This function asks the user how many times the. Write a program Permutations. The break is used to break out of the case statements. We are proud to be the #1 G2 industry leader in user research and software testing. 00 The while loop is used for the game loop. As long as my students were learning to write and recognize their names, I. Java exercises. Rehearse and record slide timings When you choose the following show types: Presented by a speaker (full screen) and Browsed at a kiosk (full screen), you’ll want to rehearse and record timings for effects and slides. Program should display "H" for heads and "T" for tails for each individual toss off the coin 6. In the C# language we can use break, return and goto to change control flow in a while-loop. The latest reviewed version was checked on 21 May 2019. Double for loop. This is an introduction to Pygame for people who already know Python. Hint: use a do-while loop to generate random integers until you get one that indexes a male student. Here is a variable that is used for iterating over a. The largest free online library about the art of magic and the amazing feats of magicians! Want to know about the World's Most Dangerous Magic Trick (more than 15 magicians have died performing it!) Or learn the real names of famous magicians, the true record holders in magic, the magician who helped win a World War- and more! Our magic shop's. Learn new concepts by solving fun challenges in 25+ languages addressing all the hot programming topics. For example, when you are working with single-dimensional Array, you can use Java For Loop to iterate from. You can do programming in C#, Java, or Boo, a language similar to Python. As long as my students were learning to write and recognize their names, I. This can help students decode new. PIPS=6 # A die has 6 pips. Icons, clipart, logotypes and more. Obv: IMP CAES VESPAS AVG P M TR P P P COS III; Head of Vespasian, laureate, r. It was the first competition added to the Olympics that was not a footrace. 5 /* Coin toss. Your customizable and curated collection of the best in trusted news plus coverage of sports, entertainment, money, weather, travel, health and lifestyle, combined with Outlook/Hotmail, Facebook. Finally, label each item with the fraction name. There are 3 types of loops in Java: for loop, // Infinite while loop while (true) { // body of while loop }. In order to pick participants, they might choose people using a technique that is the statistical equivalent of a coin toss. play interesting!. If you need a simple choice between one thing or another, here's a function that generates a random number between 0 and 1 and returns it. But you also control a moving box. Games: Mobile gaming has become a big contender in the world of entertainment. (In this photo, children were counting sets of 2. Tickety-Boo: phrase for when everything’s going great (Example: “All is tickety-boo in my world. The consumables can be found in-game right now. Searchable site of thousands of quality teaching resources, interactive resources, homework, exam and revision help. Use on an IWB, PC or mac. The answer is simple. Double for loop. To represent this, we use a similar expression that excludes specific characters using the square brackets and the ^ ( hat ). 15 Simple Drinking Games Every Fresher Should Know. The basic idea with the max-loop is to keep a local variable that stores max-so-far as we go through the loop, and each element is compared to the max-so-far. */ int coinFace = rand() % 2 + 1. Notice the symmetry between "number" and "reversed", inside the loop. Bank notes start with the one dollar note while coins are less than one dollar. The syntax of the sum () function is: The sum () function adds start and items of the given iterable from left to right. By following the step by step guide, students can create a letter and then print it. Write a program to represent the tossing of a coin. This is a loop that will run for the entire game, where we handle events and updates the screen and positions of our objects. Tickety-Boo: phrase for when everything’s going great (Example: “All is tickety-boo in my world. Weighing only once and using the minimum number of coins, how can you find the bags containing the fake coins?. A simple game of chance. (e) Determine the probability of exactly three tails. On every iteration it takes the next. 2D Animation is a technique used to create the illusion of movement using static images. Boost your team’s productivity using an interactive whiteboard. Current Customer? Just Browsing?. Help your students build a growth mindset using social-emotional learning and ScienceWorld magazine. Students need to come up to the whiteboard, say the time and then flip a coin. 14 in 2015 dollars. You must remember to close the file or risk having some data not be written and saved to the file. Learn how to play volleyball games with these volleyball training tips and volleyball rules and drills at Volleyball. available() to read all characters in file. Free online maths games for kids. The outcome of the toss should be printed and the result should be return to the main program. IXL is the world's most popular subscription-based learning site for K–12. Java Coin Flip. ” and its size has been set to a width of 300 pixels and height of 100 pixels. This article contains a step by step guide on building an animation using the python tkinter package. Show Hide all comments. Let’s Play! 9 Awesome Online Games to Learn Spanish Word Toss. We can think of control flow (how statements are encountered) like a river. The game is simple. There are precedents for getting rid of coins that are too small to use. Please Sign up or sign in to vote. Smashing Magazine — for web designers and developers. Creating the Game Loop. The while loop will run as long as the variable counter is less or equal with 100. For example reverse of 839 is 938. PIPS=6 # A die has 6 pips. Can you collect all the stickers on Noah's adventure? Clifford The Big Red Dog. Your program must have the following functions: 1. (optional) - this value is added to the sum of items of the iterable. A single control can affect several properties at once. The while loop will run as long as the variable counter is less or equal with 100. Players line up on either side of the clothesline that can be taped or tied to walls or pillars. These are all of the different ways that I could flip three coins. TNW is one of the world’s largest online publications that delivers an international perspective on the latest news about Internet technology, business and culture. And don't worry, no singing skills are. MAXTHROWS=600 # Increase this if you have nothing better to do with your time. A Do While loop is probably the lesser known of the looping methods out there, but it can be very useful in certain situations. All the games are free and only use a few items that you probably already have laying around the house. The while loop will check for a value of 'Y' or 'N'. This loop will execute the code block once, before checking if the condition is true, then it will repeat the loop as long as the condition is true. Using the os. I use different things, because some work better than others depending on where you are in the room and what the activity is. 2 (Coin Tossing) As we have noted, our intuition suggests that the probability of obtaining a head on a single toss of a coin is 1/2. COM offers diet, nutrition and fitness tips for a healthier lifestyle. This game is simple but fun and keeps their attention while you help them discriminate between correct and incorrect productions. (the timing mechanism). Otherwise, the expression is false. - Duration: 4:51. You'll build a roller coaster track for marbles using foam pipe insulation and masking tape, and see how much of an initial drop is required to get the marble to "loop the loop. Packet Updated: July 2018: With summer month options. Accurately track hurricanes, thunderstorms, lightning strikes, and other severe weather with the help of the top live storm tracker apps on the market today. You need to use loops. The Slack interface showing a shared channel, where two companies have a conference call and share files to reach a decision together. #include #include using namespace std; int main() { cout << "This is a very simple number guessing game. choice([-1, 1]) Your while loop is also odd. We have a sample space of 2³ = 8 possible outcomes { (HHH), (HHT), (HTH. Code::Blocks IDE is used to write programs, most of these will work with GCC and Dev C++ compilers. Turn a game of sardines into a group prayer session by writing a prayer on an index card or other piece of paper. math random function javascript binary. while expression, statements, end evaluates an expression , and repeats the execution of a group of statements in a loop while the expression is true. Come up with a fictional problem that your team must solve. I was trying to find a way to allocate more memory, but I couldn't find a way. Writing a Simple Game. We will count the number of heads and tails. Note: **While supplies last. The wrestling competition was held in the. I assume the following is true: assuming a fair coin, getting 10 heads in a row whilst tossing a coin does not increase the chance of the next coin toss being a tail, no matter what amount of probability and/or statistical jargon is tossed around (excuse the puns). It belongs in this article thanks to its dedicated client which allows you to use its services quickly and. A For( loop is basically a specialized form of a While loop, with the main differences being that it is executed a specific number of times and it has the. Hope this helps! Below is the output of the sample menu that we will be creating: Sample Menu Here we have a function print_menu() which is used only to print the menu and the options available. Combination generator. Determining the best writing journals and diaries Writing journals have evolved from simple plain books to modern ones that are of different materials and shapes. The game goes like this: you need to give the team as many balls as possible in two minutes. For example, we might be able to justify independence by looking at the way the random experiment is performed. There's eight possible outcomes. Upgrade them to suit your race style and gain advantages. These are all of the different ways that I could flip three coins. Help Martha fix these funny pictures! Noah's Adventure. Below is the syntax highlighted version of Flip. Our kids games online are proof that learning can be super entertaining and fun. # #Example file for working with loops # x=0 #define a while loop # while (x <4): # print x # x = x+1 #Define a. ; Examples and usage in Python. java using a programming text editor (such as Sublime Text, Atom, Notepad++, Textpad, gEdit) or an IDE (such as Eclipse or NetBeans). The code and assets for this example and others can be found in the gosu-examples Ruby gem. Multiply Matrices 4. Make your own images with our Meme Generator or Animated GIF Maker. Used by over 10,000,000 students, IXL provides personalized learning in more than 8,000 topics, covering math, language arts, science, social studies, and Spanish. English Language Arts, Writing, Back to School. Loop can be used to iterate over a list, data frame, vector, matrix or any other object. Permutations. It is not always easy to decide what is heads and tails on a given coin. Verilog code for Full Adder 20. The first activity, Sorting, helps children to recognise the different coins. 1st toss is 100% either head or tails. Please Sign up or sign in to vote. Random sentences can also spur creativity in other types of projects being done. Java Coin Flip. This Java program is used to toss a coin using Java random class. You do not know the bias of the coin. Where, var: var reads each element from the list starting from the first element. This article will teach you the steps to building a simple game that has the player dodging bouncing balls. function coinFlip() { return Math. In Python we have three types of loops for, while and do-while. use the function rbinom () to draw a number from a Bernoulli distribution: theta <- 0. java that take an integer command-line argument n and prints all n! permutations of the n letters starting at a (assume that n is no. A coin should always have two sides. Nice and Simple code. We do not consider the ﬁrst two tosses to constitute a run, since the third toss has the same value as the ﬁrst two. Please Sign up or sign in to vote. For example, if an int. And aside from being free, it's really not programming at all — if you can write a story, you can make a Twine game. Our kids games online are proof that learning can be super entertaining and fun. Coin Toss: Simulation of a coin toss allowing the user to input the number of flips. Talk to a representative. From Amazon's development blog, learn the GameMaker basics from. Math Game Time’s free and fun math videos appeal to children. It’s similar in concept to the Advanced Vocabulary Builder below, but it’s been created specifically for the little ones (but don’t let that deter you!). But unlike while loop which depends on condition true or false. To its right, you will see main. The loop retrieves (posts) content from database in templates, so it s Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. import pyxel pyxel. Interactive questions, awards, and certificates keep kids motivated as they master skills. This example will count the number of heads and tails without using if statements. Wait, what's this? if, else, repeat, while, for, case - it's Verilog that looks exactly like C (and probably whatever other language you're used to program in)! Even though the functionality appears to be the same as in C, Verilog is an HDL, so the descriptions should translate to hardware. Finally you will have video which needs to be fixed and you are in the right place. Background research is necessary so that you know how to design and understand your experiment. You never know, one of your pennies could be worth the price of a nice dinner — or a whole lot more. It continues to be a reliable source for new puzzles and insights in decision theory. Thus, the PMF is a probability measure that gives us probabilities of the possible values for a random variable. How to Flip a Coin. Printable board games help kids learn in a fun and challenging way. There's a good reference for implementing card shuffling correctly on Wikipedia. The Tic Tac Toe AI's algorithm will compute the best move to make, as shown in Figure 10-4. Note: **While supplies last. This is what I have so far but I keep getting errors. While that file is open, you will see a Graphical Layout tab on the bottom of that file. Use all four fingers to pick up the string noose—the near hand string goes between the thumb and forefinger. Moving Animated Walking, Running, Crawling Creature and Animal Animations Walking animations at netanimations. All the pins will get their mode set to OUTPUTs in this manner. Upgrade them to suit your race style and gain advantages. Share using Email. Rather than always iterating over an arithmetic progression of numbers (like in Pascal), or giving the user the ability to define both the iteration step and halting condition (as C), Python’s for statement iterates over the items of any sequence (a list or a string), in the order. With pygame imported and initialized, the display set, and the game surface being updated, we can start working on our main game loop. com is not affiliated with The Girl Scouts of America. Some teachers write out the list on the board, or print it out on a worksheet - and while those methods can sometimes work, there’s something about flashcards that seems to make memorization a lot more intuitive. You can use it as much as you like at no cost. We can think of control flow (how statements are encountered) like a river. There is only one way to get two heads, out of four. So if your child is gliding, you will use an /r/ word. If it lands on a name, that player drinks. Nice and Simple code. C Program to print Fibonacci series in a given range 6. I’m working on prototyping a top-down 2D game (in 3D) and wanted to start off with a bare bones/boilerplate player controller. 6 Suppose a coin is tossed 9 times, with the result HHHTTTTHT. Offer Expires on 01/18/2020. Output: Value of x:1 Value of x:2 Value of x:3 Value of x:4 for loop: for loop provides a concise way of writing the loop structure. This can help students decode new. The default value of. The first flip was a tail, and after that flip the relative frequency (rf) of heads is 0. A coin should always have two sides. The above-mentioned games will not only entertain you but also help you earn a handsome amount of money. Most Current Blog Posts. Here is a variable that is used for iterating over a. Hot Wheels Iconic Scooby Doo! The Mystery Machine Die-Cast Vehicle. Guides that also cover topics like business, distribution and marketing. So even if you know Python and PyGame but you need ideas, you could give this book a skim. The MCO VISA Card allows you to spend anywhere at perfect interbank exchange rates with crypto cashback. Find the probability of rolling even numbers three times using a six-sided die numbered from 1 to 6. If you are creating a game, most of what you are looking for may already be included in the many PythonGameLibraries that are available. Share your experience. Offers In-App Purchases. Word Toss is the perfect game for beginners who want to learn new words in Spanish while being a little silly. By doing this you can stimulate a coin flip in Python easily. ) and "hide" it in your hand. Below is an example of using LabVIEW to program a counter on an M Series device to create a digital pulse train signal using NI-DAQmx. Hello World Program in C 2. The subscript here indicates that this is the PMF of the random variable. I'm trying to make a simple program that simulates the flipping of a coin and has the user guess whether the coin landed on heads or tails. Random Word Generator is the perfect tool to help you do this. The winner of the game is the one who has the most cards at the end of the game. The program asks the user to guess the coin toss and then compares the value with the actual coin toss result. Penalty: For failure to comply: Loss of coin-toss option for both halves and overtime, and loss of 15 yards from the spot of the kickoff for the first half only. If you're behind a web filter, please make sure that the domains *. Just check that part. The completed program should look like the following: Please use the code I attached as it's the template i'm using. flip-card:hover. Now channels can help you work as closely with external partners and clients as you do with teams down the hall. Volunteer-led clubs. Numismatics (the scientific study of money) defines the obverse and. On every iteration it takes the next. Making statements based on opinion; back them up with references or personal experience. Coin toss probability is explored here with simulation. But if I ask you whether that coin is heads or tails while it’s spinning, you might say the answer is both. And aside from being free, it's really not programming at all — if you can write a story, you can make a Twine game. Learning something as potentially complicated as graphics programming will be a real chore if you're also unfamiliar with the language you're using. Write a program that simulates coin tossing. The coin of the highest value, less than the remaining change owed, is the local optimum. Speed down on a randomized slope. Click Play, play the game for a bit then save. Whether you’re approving a purchase, closing a sale, or signing an agreement, it’s easy with DocuSign—reliable and trusted worldwide for electronic signatures. OmmWriter opens up in full-screen mode, so nothing disrupts your creative process. This example will count the number of heads and tails without using if statements. You could also set up a simple guessing game and award a prize to the person who. In R programming, while loops are used to loop until a specific condition is met. The object of the game is to remember where similar cards are located and to pick up as many pairs as possible. This video use a while loop to simulate a series of coin tosses. If the value of i has a remainder of zero when divided by 2 (that’s why we use the modulus operand %%) we don’t enter the if statement, execute the print function and loop back. SplashLearn’s online money games capture children’s attention through virtual coins they can tap on and flip over. When i is between 1 and 10 we enter the loop and if not the loop stops. SysFont ( "Comic Sans MS", 20) myfont2 = pygame. It costs a dollar to play. randrange(2), and use an if-else statement to print Heads when the result is 0, and Tails otherwise. - Duration: 4:51. This game is a fast-paced and exciting way to work on problem-solving. You will gain the respect of your co-workers for being organized and inventive, and all your hard work will pay off when your office party games get the whole room rolling with laughter!. Hello World Program in C 2. It feels as though each word flows from you naturally. The body of for loop is separated from the rest of the code using indentation. ” and its size has been set to a width of 300 pixels and height of 100 pixels. What I am trying to do is make a coin that plays one animation when the game starts (I have no problem with this part) and when I collect that coin, it plays a little animation of a few little twinkly stars fading out. Tossing a Biased Coin Michael Mitzenmacher When we talk about a coin toss, we think of it as unbiased: with probability one-half it comes up heads, and with probability one-half it comes up tails. You could create. Use this generator to generate a trully random, cryptographically safe number. It's just a simple example showing the importance of loop in computer programming. There is simple rule that you can use to tell if you should use a while-loop. FREE Coin Flipper is unlocked when you download the app. Notice how most. A coin should always have two sides. People also post their games and projects on the forums, so it's a nice way of exploring what LÖVE can do. With R we can play games of chance - say, rolling a die or flipping a coin. When you are working with multi-layered data, then use this Java Nested For loops to extract the layered data, but please be careful while using it. It won't do anything except print some text to the screen but it will demonstrate the first step that all our Android game projects will require. Close your hand into a fist, but leave a tiny gap between your index and middle fingers. A correct guess pays $2. After clicking Publish, a new link will appear in the window. Earlier, we show you a few possibilities to adapt this function so you can apply a different VAT. The function should have no input and no return value. The Making Games book uses game examples from the PyGame library to teach you how to make similar games. Beginner allows students to count money using any amount of coins and. For example, instead of good job you can write “James is 6” or you can add a line thanking friends for celebrating with you. We welcome you to Hscripts. As a covered wagon party of pioneers, you head out west from Independence, Missouri to the Willamette River and valley in. Currently we have 18980 for you to play online for free. Choose a word or phrase and draw dashes on the board for each letter. We will count the number of heads and tails. In this tutorial, we will learn, For Loop Syntax and Examples. This game program illustrates the use of Random class and enumerators in Java. You can use a loop instead of repeating yourself with die 1 die 2 and die 3. Because when you are not relying on, or reading from, a word-for-word text you are free to interact with your audience. We will use of the java. Based in the USA, Coinbase is available in over 30 countries worldwide. It costs a dollar to play. We can think of control flow (how statements are encountered) like a river. 1st toss is 100% either head or tails. This is what i have so farI need to add a function named coin to simulate a coin toss where heads is represented by a 1 and tails a 2. How to write a C Program to Create Simple Calculator using Switch case, Functions, and Else If Statement. Many classics like Mario and Sonic the Hedgehog fall in this group. 3 has been released. You must remember to close the file or risk having some data not be written and saved to the file. Greedy algorithms determine minimum number of coins to give while making change. Copyright © 2000-2017, Robert Sedgewick and Kevin Wayne. (Game: heads or tails) Write a program that lets the user guess whether the flip of a coin results in heads or tails. It is primarily used in its most literal sense by businesses seeking to account for unpaid loan obligations, unpaid receivables, or losses on stored inventory. BPA and phthalate-free, dishwasher safe. All variables are created this way. Tiki Toss (Hook and Ring Game): What You’ll Need: Tiki Toss game (or a large metal ring, hook, eye hook and string to make your own). Throw dough, sift flour, and cook with Skits! Dogs, Vocabulary. In our draw() loop, we send whatever we want over the serial port by using the write method from the Processing Serial library. counter = 0 while counter = 100: print counter counter + 2 Nested Loops In some script you may want to use nested loops. In R programming, while loops are used to loop until a specific condition is met. Login to reply the answers. // code block to be executed. Share using Email. Tossing a Biased Coin Michael Mitzenmacher When we talk about a coin toss, we think of it as unbiased: with probability one-half it comes up heads, and with probability one-half it comes up tails. I want the simulation to end when I get a certain amount of money. C programs with output showing usage of operators, loops, functions, arrays, performing operations on strings, files, pointers. You can totally adjust these games for whatever age group you have or however many players you have. (the timing mechanism). We show two ways of solving this problem. Use MathJax to format equations. Mike Gold Apr 05, 2019. Tosh: total bullshit, nonsense or rubbish. (optional) - this value is added to the sum of items of the iterable. In our draw() loop, we send whatever we want over the serial port by using the write method from the Processing Serial library. Bank notes start with the one dollar note while coins are less than one dollar. This pocket-sized device is built to last and easy to hold. We offer easy solution for developers to share their games. Alphabet, Spelling, Letters, Vocabulary, Literacy. In Python we have three types of loops for, while and do-while. This video use a while loop to simulate a series of coin tosses. Find the probability of rolling even numbers three times using a six-sided die numbered from 1 to 6. We will count the number of heads and tails. Have the participants start at the first tee and attempt to throw their beanbag ball into the bucket with as few throws as possible. Definition of flipper in the Idioms Dictionary. For example reverse of 839 is 938. I’m using just this approach in an upcoming simple HML5 game built in. You have the condition twice, once on the while loop and once on the if statement for the break. We offer easy solution for developers to share their games. By Teach With Me. MakingFriends. Your program must have the following functions: 1. 3 Example- Converting A For Loop to a While Essentially every for loop can be written as a while loop. init( 120 , 120 ) pyxel. That way several tutors can. In C++, Using a while loop write a simple coin flip game. Flip through memes, gifs, and other funny images. 2 (Coin Tossing) As we have noted, our intuition suggests that the probability of obtaining a head on a single toss of a coin is 1/2. By Kathleen Charlson. The steps of performing a bubble sort are: Compare the first and the second element of the list and swap them if they are in wrong order. Example 2: Simple For Loop with flush. Prior to this version, it was. Passers are suppose to cross the lines from the starting point and back. The game should continue until the user indicates they do not want to play anymore. When children play games, they can apply their math skills to new and unique scenarios while building logic and critical thinking skills. So, if you have 2 free cells empty you can move 3 cards together. NEW games added every week. Animation desk is a sleek and very powerful app for animation on the iPad. // Begin File for Assignment 8 // Coin Flip Game, Random Heads or Tails Betting Game #include. I am just learning Python on class so I am really at the basic. If our for loop began for x = 1:2:15, we must state that x = 1 initially, before our while loop begins. Program should display each run as a series of tosses 7. Here is what I have programmed: #include #include #include using namespace std; int main() { int userGuess; /*Takes input from user, user guesses whether a coin will land on heads or tails. use the function rbinom () to draw a number from a Bernoulli distribution: theta <- 0. Here is a simple example of a for loop. Do While Loop. flip-card-inner { transform: rotateY(180deg);} but we cannot warrant full correctness of all content. Then the while loop is iterated until. 3 V / 5 V (better) GND -> GND. A coin worth 5 cents is called a nickel, while a coin worth 10 cents is a dime. Froot Loop words “Boo!” card game 4-in a row for oo words Worksheet pack for oo words Roll & read ee words Worksheet pack for ee words Board games for ee words 4-in-a-row for ee words Long & Short Vowel Sorting Mats Roll & Write long vowel words Card game for ai/ay words ai / ay path game. What is the use of break and continue in Python? In Python, break and continue statements can alter the flow of a normal loop. This is what I have so far but I keep getting errors. Use features like bookmarks, note taking and highlighting while reading Happiness, Anxiety, and Writing: Using Your Creativity To Live A Calmer, Happier Life (Writing As A Second Career Book 4). This is what i have so farI need to add a function named coin to simulate a coin toss where heads is represented by a 1 and tails a 2. As long as my students were learning to write and recognize their names, I. It exits out of the loop, when the condition becomes true. Here is a simple example of a for loop. Here is a snapshot if you don't mind:. Which of the following require using arrays. Minute to Win Party Games. Pivot query help us to generate an interactive table that quickly combines and compares large amounts of data. For each, the input comes from standard input and consists of N real numbers between 0. In Alice you can easily build interactive elements for game programs and explore the art of game design. This game is like a simple hide and clap game but you need to play this game in absolute darkness. io game in which you need to expand your territory on the map! When the game starts, you own a very small block. The output looks something like this: Welcome to the coin. If you are designing a website, then you can find all the resources you need for webmasters and web developers such as free scripts, web tools, programming tutorials, web design and applications, clipart images, web icons etc. There is simple rule that you can use to tell if you should use a while-loop. MAXTHROWS=600 # Increase this if you have nothing better to do with your time. We have created a program that can toss a coin over and over until it comes up head 10 times. Clicking “Present” will show the Slides without auto-advance or looping. Assuming you attach the micro:bit to your foot or ankle, it will get shaken when you take a step. In each iteration of the loop, the remainder when n is divided by 10 is calculated and the value of n is reduced by 10 times. It costs a dollar to play. Even young kids can start to understand the difference between a need and a want. An algorithm is a finite series of instructions to compute a result. Then the while loop is used until n != 0 is false (0). Solve interactive problems on setting time on an analog clock, reading time, and telling the same time in different. The first activity, Sorting, helps children to recognise the different coins. Bon appétit! Learn more: E is for Explore. By Deborah Stewart. Write a program Permutations. It is a detailed and easily understandable description of steps of algorithms or a program, which does not use any programming concepts, rather uses natural language. There is nothing wrong with wanting things and understanding how to save for those wants will help kids learn about money. One output may be connected to multiple inputs, however. Lisa Dziuba is one of the co-founders of Flawless App, a tool for iOS developers to compare the expected design with the real app inside the dev environment. The while loop will run as long as the variable counter is less or equal with 100. Let Overstock. // Begin File for Assignment 8 // Coin Flip Game, Random Heads or Tails Betting Game #include. The joy of playing Pictionary and Pictionary Air is that the rules of the game are simple, but executing them is a challenge and inevitably produces a lot of fun and laughter. IXL is the world's most popular subscription-based learning site for K–12. 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Verilog code for counter with testbench 21. Occasionally chess games do not end with a winner, but with a draw. In case we enter the loop, we need to check if the value of i is uneven. Loops iterate over a block of code until the test expression is false, but sometimes we wish to terminate the current iteration or even the whole loop without checking test expression. If you need a simple choice between one thing or another, here's a function that generates a random number between 0 and 1 and returns it. In conventional computer processor a transistor is either up or down, heads or tails. At the end we need to close the file. To use multiple turtles on a screen one has to use the object-oriented interface. (Imagine using assembly codes, which is a very low level code, to write a computer soccer game. For each, the input comes from standard input and consists of N real numbers between 0. The game loop. Then the while loop is iterated until. 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# What limits the modulation of digital signals?

I am trying to understand why there are different speed on different frequency bands. I see that e.g. ZigBee can do 250kb/s for both 2.4GHz and 868/915MHz, but WiFi are also 2.4GHz with much larger speeds, why are they different? How come 2.4GHz is so slow for ZigBee and so fast for WiFi when they operate in the same frequency band?

And is it possible to obtain larger speeds on lower frequency bands just by modulating the signal the same as WiFi does? Is modulation impaired by lower frequencies or is it only dependent on how good the modulating hardware is?

Thank you very much in advance!

The maximum useful datarate is a fraction of the carrier frequency. However, the difference between ZigBee and WiFi has nothing to do with that. 2.4 GHz is so high, that it is not even close to the limiting factor for data rate.

Generally, there are two tradeoffs with higher data rate over the same frequency carrier: RF bandwidth and power.

A perfect sine wave, like a unmodulated carrier, has a bandwidth of zero. As soon as you start messing with that, the bandwidth becomes non-zero. All modulation schemes by definition must modulate the carrier in some way, so all cause the bandwidth of the transmitted signal to be non-zero. This means that signal will use up some finite amount of the RF spectrum. There are legal and practical reasons why you want to keep the amount of RF spectrum space to a minimum. Different modulation schemes have different characteristics, but one way or the other you're going to pay for high data rate with a high spectrum width, or bandwidth.

The other issue is power. This is the main reason ZigBee is much lower data rate than WiFi. I won't go into all the reasons why since it seems off topic to the question and the answer could be a whole book, but it takes more power to transmit a higher data rate at the same signal to noise ratio as a lower one. In some sense, you can trade off data rate with signal to noise ratio at the same power. There are also power and dollar costs in the electronics that have to handle the higher data rate independent of the RF power. Since ZigBee is intended for small low power battery operated devices, a tradeoff was made between data rate, power requirements, distance, and usefulness. Because battery operation was so important, the data rate was made lower. WiFi, on the other hand, had little constraint on the power required for the electronics. Its intended use also demands higher data rate, so a different tradeoff was made.

• +1 - nice answer, thought I'd mention there are also practical reasons why you may sometimes want to widen the bandwidth (e.g. spread spectrum) – Oli Glaser Oct 7 '12 at 22:30
• @Olin Actual numbers comparing bandwidth and rf power for wifi vs zigbee would be useful and would support the claims made. – apalopohapa Oct 8 '12 at 1:59
• Great answer, thank you. But when talking about the tradeoff between speed and spectrum width, could you provide a small example? Thank you – chwi Oct 8 '12 at 5:00
• @Wilhelmsen: Look up the frequency analysis of AM modulation. Briefly, the spectrum of a AM transmission is the carrier with the modulated signal on either side of it. A 1 kHz signal on a 1 MHz carrier therefore has frequencies at 999 kHz, 1.000 MHz, and 1.001 MHz. The spread would be wider with a higher modulation frequency. That is just one simplified example. There is a lot written about that out there. – Olin Lathrop Oct 8 '12 at 12:13
• Simplified or not, it was exactly what I wanted so thank you very much! Excellent! – chwi Oct 8 '12 at 17:45


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WARNING: This function will be removed soon; see oce-defunct.

findInOrdered(x, f)

Arguments

x Ignored, since this function is defunct. Ignored, since this function is defunct.


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AdaOpt is a probabilistic classifier based on a mix of multivariable optimization and a nearest neighbors algorithm. More details about it are found in this paper. When reading the paper, keep in mind that the algorithm is still very new; only time will allow to fully appreciate all of its features. Plus, its performance on this dataset is not an indicator of its future performance, on other datasets.

Currently, the package containing AdaOpt, mlsauce, can be installed from the command line as:

pip install git+https://github.com/Techtonique/mlsauce.git

In this post, we’ll use mlsauce’s AdaOpt on a handwritten digits dataset from UCI Machine Learning repository.

The model is firstly trained on a set of digits – to distinguish between a “3”, or a”6”, etc.:

from time import time
from tqdm import tqdm
import mlsauce as ms
import numpy as np
from sklearn.metrics import classification_report
from sklearn.model_selection import train_test_split
from sklearn.datasets import load_digits

Z = digits.data
t = digits.target

# Split data in training and testing sets
np.random.seed(2395)
X_train, X_test, y_train, y_test = train_test_split(Z, t,
test_size=0.2)

learning_rate=0.3,
reg_lambda=0.1,
reg_alpha=0.5,
eta=0.01,
gamma=0.01,
tolerance=1e-4,
row_sample=1,
k=3)

# Teaching AdaOpt to recognize digits
start = time()
obj.fit(X_train, y_train)
print(time()-start)

0.03549695014953613

Then, AdaOpt is tasked to recognize new, unseen digits (X_test, y_test), based on what it has seen on the training set (X_train, y_train):

start = time()
print(obj.score(X_test, y_test))
print(time()-start)

0.9944444444444445
0.19525575637817383

The accuracy is high on this dataset. Additional error metrics are presented in the following table:

preds = obj.predict(X_test)
print(classification_report(preds, y_test))

              precision    recall  f1-score   support

0       1.00      1.00      1.00        31
1       1.00      0.97      0.99        40
2       1.00      1.00      1.00        36
3       1.00      1.00      1.00        45
4       1.00      1.00      1.00        37
5       0.97      1.00      0.98        29
6       1.00      0.98      0.99        42
7       1.00      1.00      1.00        35
8       0.97      1.00      0.99        33
9       1.00      1.00      1.00        32

accuracy                           0.99       360
macro avg       0.99      1.00      0.99       360
weighted avg       0.99      0.99      0.99       360

Ad here is a confusion matrix:

At test time, AdaOpt uses a nearest neighbors algorithm. Which means, a task with quadratic complexity (a large number of operations). But there are a few tricks implemented in mlsauce’s AdaOpt to alleviate the potential burden on very large datasets, such as: instead of comparing the testing set to the whole training set, comparing it to a stratified subsample of the training set.

row_sample == 0.1 for example in the next figure, means that 1/10 of the training set is used in the nearest neighbors procedure at test time. The figure represents a distribution of test set accuracy:

We also have the following timings in seconds (current, could be faster in the future) for training+prediction, as a function of row_sample:

The paper contains a more detailed discussion of how these figures are obtained, and a description of AdaOpt.

Note: I am currently looking for a gig. You can hire me on Malt or send me an email: thierry dot moudiki at pm dot me. I can do descriptive statistics, data preparation, feature engineering, model calibration, training and validation, and model outputs’ interpretation. I am fluent in Python, R, SQL, Microsoft Excel, Visual Basic (among others) and French. My résumé? Here!