外文翻译--双速率数据采样系统的仿真

中文 2630 字 Simulation of dual-rate sampled-data system Abstract: The simulation problem of a dual-rate system is studied by applying discrete lifting technology, quick sampling operator and quick hold operator. The method can achieve the result that is close to the simulation of continuous-time signal. The concrete simulation is steped and programmed with a real example under MATLAB environment. Key words: Dual-rate sampled-data system; Discrete lifting technology; Quick sampling operator; Quick hold operator 1. Introduction Sampling control system refers to the object controller for the continuous and digital systems. At present, most control systems are continuously charged by the object under the control of the computer realization of discrete sampling control system. With the continuous improvement of the system requirements, single-rate sampled-data systems can not meet the requirements, so multi-rate sampled-data systems in place. Multi-rate sampling control system works in practice with the prospect of a wide range of practical, this is because: 1) In the complex multi-variable control system, requires that all physical signals in the same sampling frequency is not realistic. 2) sampling and to maintain the higher frequency, the better the performance of the system, but the fast A / D and D / A conversion means that the cost is. So for different signal bandwidth, you should use a different A / D and D / A conversion rate, in order to achieve performance and the best compromise between price. 3) multi-rate controller is generally time-varying controller, it has a single-rate controller can not compare the merits. Such as increasing the system gain margin, consistent with the stability of the system to facilitate the realization of decentralized control. A relatively simple multi-rate sampled-data control system is dual-rate sampled-data systems, virtual box as shown in Figure 1. Simulation of the system is defined as: for a given input signal w, simulation of its continuous output signal z process. Figure 1 dual-rate sampled-data systems wth a virtual sampler and holder Literature 4 is given a single-rate sampling of high-precision control system simulation. In single-rate sampled-data systems exist in only a single sampling period, thus only the application of the simulation process of some of the more sophisticated theory, such as the continuous transfer function of a single rate discrete. Dual-rate sampled-data systems, because of the existence of two types of sampling period, and the controller too variable controller, thus increasing the difficulty of the simulation. In this paper, discrete technological upgrading, the system in two different sampling period organically linked to the controller into a time-varying time-invariant controller. At the same time, the use of rapid sampling and rapid operator to maintain, given the dual-rate sampling control system simulation method. 2. Prior knowledge Figure 1 sampler sampling period T1 = ph, sampling operator S: y (k) = Syc (t) = yc (kph), holder of the sampling period T2 = qh, maintain operator H: uc (kqh + r) = Hu (k), 0 0, Dr space for the continuous delay operator, that is, Druc (t) = uc (tT); U space for the discrete step lag operator; U2 for the discrete space operator step ahead. 1 If the definition of (U2) q1KUp1 = K to set up, said K for the (p, q) - discrete controller cycle. 2 If the definition of G for the system to meet the DrG = GDr, said the G for the T-cycle for time-varying systems. 3 Simulation Algorithm 2.1 Simulation of the expression K is a known theorem (p, q) - cycle of discrete controllers, operator and maintenance of sampling operator as mentioned above, the HKS for the T-cycle for time-varying systems. See Figure 1 to prove the relationship between the signal, there are established under the style qhrH K S DH K SDtyH K S DtH K S yDtyH K S DTtH K S yphptH K S ypkH K ykyH K U pkKyH U qkuH U qqkHurqhqkuqhqtuTtutuDtH K S yDTTcTcTcTccccccTcT0)()()()()()()()()()()()()()()(11111111we can see from the definition 2,HKS for the T-cycle for time-varying systems. HKS is a cycle as a result of T, so the case with the single rate is similar to Figure 1 in the relationship between input and output systems can be expressed as wH K S GH K S GIGGz )( 211221211 Or wH K S GHK S GIHGGz )( 211221211 Dual-rate sampling control system input and output channels, by adding a virtual sampler and maintain fast, and as shown in Figure 1, the virtual fast sampler and holder of the sampling period T / n. Wd is the w to T / n for the sampling period of the sampling signal, when the input signal mph time for the simulation, there Wd = w (kT / n), k = 0,1, ., mn/p1 zd and the relationship between z Ibid. Clearly, when n when, wd = w, zd = z. To make the number of discrete-time sequence for positive integer, n as the integer multiple of l. Study shown in Figure 1 of the simulation system, virtual box can be dual-rate sampling control system input and output signals for the simulation results. Figure 1 zd = Snz, w = Hnwd, it is by the type (2) dnnnnndnnndwGK S HHK S GIHSGGwHK S GHK S GIHGGSzSz)()(211221211211221211Which G11n, G12n, G21n to correspond to the cycle of T / n of the discretization. Formula (4) is dual-rate sampling control system simulation expression. 2.2 Simulation of the calculation of expression Expression of desire (4), first obtained G11n, G12n, G21n, SnH, SHn, (I-KSG22H)-1K, etc. value. Which G11n, G12n, G21n continuous transfer function of G11, G12, G21 single-cycle T / n of the discretization are easy to calculate. Discussed below SnH, SHn, (I-KSG22H)-1K calculations. (1) SnH calculation Figure 2 Expressiong for Input and Output of SHn Figure 2 of the cycle in Hn for T / n = lh / n, S the cycle ph, while x2 (0) = x1 (0), x2 (1) = x1 (n/p1), ., x2 (m -1) = x1 (m-1) n/p1), It is SHn = pmnpmpmpm/010000000100000001(2) SnH calculation Similarly available expression SnH qnHSmn /21000100000010001 (3) (I-KSG22H)-1K calculation By discrete sampling and the discrete operator to maintain the definition of operator, there are (k) = (kp)Sp2l l, = Sp (kq + r) = (k)Hq2l l = Hq R = 0,1, ., q-1 SG22H = SrSAG22HhHq = SpG22dHq (5) G22d which can be separated by a single rate process h been. For (I-KSpG22dHq)-1K is still the cycle of change SpG22dHq and K, this paper discrete operator to upgrade to turn it into time-invariant systems, the specific process as shown in Figure 3. Simulation of expression at this time (4) can be expressed as Figure 4. Enhanced by the discrete, periodic time-varying link SpG22dHq and K into the time-invariant Lp1SpG22dHqL-1 q1 and Lq1K L-1 q1, calculated as follows: (1) Lq1K L-1 q1 calculation If the dual-rate controller of the state equation for K 1,1,0)()()()()()1(1101,111011qijkpyDkxCikqujkpyBkAxkxpjjikipjjkk，While Lq1K L-1 q1 state equation can be expressed as )()()()()1(kyDkxCkukyBkxAkxkkk Among which Figure 3 (I-KSG22H)-1K to upgrade the discrete signal Figure 4 (4) simulation indicate , 1 yLy p uLu q1 AA 110 1 pBBBB 1101qCCCC ,1,10,11,00,01111pqqPDDDDD (2) Lp1SpG22dHqL-1 q1 calculation Lemma 1 for P for the state variables x, the state model for A, B, C, D, m, n and s meet the following relationship is positive integer. The system state variables for the discrete sampling operator can be expressed as a state model. Which AA 10 1 11 nr j njr rjrj BABAB 1,10,11,10,11,00,0,nmmnmmjmDDDDDDDCACACC Among which )()( ,01)1( 1)1(, rBCAimDD immjjmrrimmjjmji Characteristics function X qrprqp ,0,1)(,Take11,1 , lsqnpmqnpm , Conclusions from the Appeal, G22d obtained from Lp1SpG22dHqL-1 q1 of the state space model. Integrated on the system, we can see in Figure 4 for the simulation process: mph input signal period, then )/()( nkTwkw d nnnn GGGG 22211211 , 122 11 qqdpp LHGSL 111 pq KLL 12211 , xSHxwGx ndn 23 1 xLx p 3122141111)(xLHGSLKLLIxqqdpppq 415 1 xLx q 56 HxSx n dnn wGyxGy 1116122 , 12 yyz d 3. Simulation example Figure 1 for the generalized plant G 00100001004.0014.000004.0014.0A 0 0 4.00 0 4.0000 0 4.00000000,00001001CDBAnd controller K is 2 3 7 7 0.52 1 3 0 7.9 5 52 5 8 6 9.9 5 301,019 5 4 3 8.00 4 5 6 2.0ijijDCBASampling period T1 = 2s, T2 = 3s, p = 2, q = 3, h = 1, p1 = 3, q1 = 2, l = 6, T = 6. So that m = 6, n, respectively, for 4800,7200, 9600, wd for unit step input signal. Using MATLAB programming language, and the system simulation, the results shown in Figure 5. 4. Conclusion In this paper, dual-rate sampling control system of the characteristics of discrete applications to upgrade their skills, rapid sampling and rapid operator to maintain operator to study the dual-rate sampling control system simulation methods, and gives concrete examples of simulation steps and guidelines. Dual-rate controller as a result of changing the controller too, so the dual-rate sampled-data control system simulation to verify the accuracy of the problem to be further studied. Sampling control system technology has undergone more than a decade of development, but there is a fundamental problem. Especially since the use of upgraded technology, sampling control theory has entered a new stage of development. Because it can take into account the performance between the sampling moment, therefore seems to enhance the transformation has become a sampling control system analysis and design of the only correct way, and their use is also expanding, but in the real design was brought out higher requirements. Upgrade its technology was originally designed for the needs of related, but not limited to the actual situation in many areas of the individual. This is the special nature of sampled-data systems, especially in its structure on the signal path. Sampling control system signal channel constituted by two parts, a continuous channel, and the other is sampling channel. Sampling control system upgrade, its norm is not entirely equivalent. Taking into account the characteristics of the two-channel frequency response method proposed can also be given the systems frequency response induced by the true norm, will be sampled-data control systems analysis and design the right way. 双速率数据采样系统的仿真 摘要 ：双速率系统的仿真问题是采用离散提升技术、快速采样算子和快速保持算子来研究的。该模型实现的结果与连续信号非常相近。最后给出具体地仿真步骤，并结合实例在 MATLAB 环境下编程实现。 关键词 ：双速率数据采样系统，离散提升技术，快速采样算子，快速保持算子 1.简介 采样控制系统是指连续和数字系统的对象控制器。目前，大多数的控制系统是继续的由计算机实现的采样控制系统控制器实现的。随着对系统要求的不断提高，单速率的采样控制系统变得不能满足应用的要求，因此其地位被混合采样速率的 采样控制系统所替代。混合采样速率控制系统在实际应用中能够满足于很广泛的应用场合，这是因为： 1）在复杂的多变量控制系统中，要求所有的物理量在被采样的时候都具备相同的采样速率是不现实的事情。 2）在对信号进行采样的工程中，采样的频率越高，系统对信号的复现性能就越好，但是快速的 A/D 和 D/A 转换器意味着更高的花费。因此，对于不同的信号带宽，你应该使用不同速率的 A/D 及 D/A 转换器，进而是的系统的功能达到一个较高的水平的同时，又不致使系统的花费太大。 3）多速率控制器一般而言是采样时间可变的控制器，这是但速率采 样控制器不能与之相较的优点。如增加系统增益裕度，则就要保持系统的稳定性从而保证系统离散控制功能的实现。 双速率采样控制系统是一个相对简单的多速率采样控制系统，其系统的框图如图1 所示。控制系统仿真被定义为：对于一个给定的输入 W，对系统的输出信号 Z进行模拟的过程。 图 1 带虚拟采样器和保持器的双速率采样控制系统 文献 4中给出了一个高精度的单速率采样控制系统仿真的样本。在单速率采样控制系统中仅存在一种采样周期，这样因而其仿真过程只需应用一些较成熟的理论。例如单速率连续传递函数的离散化。对于双速率 采样控制系统而言，由于系统中存在两种不同的采样周期，并且控制器为时变控制器，这样就增加了仿真的难度。 本文采用离散提升技术，将系统中两种不同的采样周期有机地联系起来，把时变控制器变为时不变控制器。同时采用快速采样算子和快速保持算子，给出了双速率采样控制系统的仿真方法 2.知识背景 图 1 采样器的采样周期 T1=ph，采样控制器 S： y(k)=Syc(t)=yc(kph)，保持器的采样周期 T2=qh，保持器算子： uc=(kqh+r)=Hu(k),00， Dr 为连续空间上的延迟算子， Druc (t) = uc (tT);U 为离散空间上的一步滞后算子； U2 为离散空间上的一步超前算子。 定义 1 如果（ U2） q1KUp1=K 成立 ，则称 K 为（ p,q） -周期离散控制器。 定义 2 如果连续系统 G 满足 DrG=GDr，则称 G 为 T-周期连续时变系统。 3.仿真算法 3.1 仿真表达式 K 是一个已知的定义（ p,q） -周期的离散控制器，采样算子和保持算子如上所述，则 HKS 以 T 为周期的时变系统。如图 1 即可证明信号之间的关系，在已知既定的条件下下式成立： qhrH K S DH K SDtyH K S DtH K S yDtyH K S DTtH K S yphptH K S ypkH K ykyH K U pkKyH U qkuH U qqkHurqhqkuqhqtuTtutuDtH K S yDTTcTcTcTccccccTcT0)()()()()()()()()()()()()()()(11111111我们可以由定义 2 看到， HKS 为 T 周期的时变系统。由于 HKS 的周期是 T，因此同单速率系统类似，图 1 中输出与输入的关系可以表示为： wH K S GH K S GIGGz )( 211221211 或者是 wH K S GHK S GIHGGz )( 211221211 在双速率采样控制系统输出与输入通道中，通过增加一个可见的采样器且保持快速，像在图 1 中显示的一样，这个可见快速采样器及保持器的采样周期均为T/n。 Wd 是 w 以 T/n 为采样周期的采样信号，当输入信号的仿真时间为 mph 时，有： Wd=w(kT/n)， k=0,1,mn/p1 zd与 z 的关系同上。显然，当 n 时， wd=w， zd=z。为使离散时间序列的个数为正整 数， n 选为 l 的整数倍。研究图 1 所示系统的仿真，便可得到虚框中双速率采样控制系统连续输入输出信号的仿真结果。 图 1 中的 zd=Snz,w=Hnwd，故由式（ 2）得 dnnnnndnnndwGK S HHK S GIHSGGwHK S GHK S GIHGGSzSz)()(211221211211221211其中 G11n,G12n,G21n 为对应于周期 T/n 的离散化。式 (4)即为双速率采样控制系统的仿真表达式。 3.2 仿真表达式的计算 欲求表达式（ 4），首先要得到 G11n， G12n,， G21n,， SnH,， SHn，以及(I-KSG22H)-1K 等等变量 ， G11n, G12n, G21n 分别是连续传递函数 G11, G12, G21 以 T 为采样周期采样后的离散传递函数，均以计算。 下面讨论SnH,SHn,(I-KSG22H)-1K 的计算。 5. 计算 SnH 图 2 SHn 的输入与输出框图 图 2 中 Hn的周期为 T/n=lh/n， S 的周期为 ph，当 x2 (0) = x1 (0), x2 (1) = x1 (n/p1), ., x2 (m -1) = x1 (m-1) n/p1)， SHn = pmnpmpmpm/0100000001000000016. 计算 SnH 同理可得 到 SnH 的表达式： qnHSmn /21000100000010001 7. 计算 (I-KSG22H)-1K 由离散采样以及离散算子的定义，有： (k) = (kp)Sp2l l, = Sp (kq