外文翻譯--雙速率數(shù)據(jù)采樣系統(tǒng)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. 雙速率數(shù)據(jù)采樣系統(tǒng)的仿真 摘要 ：雙速率系統(tǒng)的仿真問題是采用離散提升技術(shù)、快速采樣算子和快速保持算子來研究的。該模型實(shí)現(xiàn)的結(jié)果與連續(xù)信號非常相近。最后給出具體地仿真步驟，并結(jié)合實(shí)例在 MATLAB 環(huán)境下編程實(shí)現(xiàn)。 關(guān)鍵詞 ：雙速率數(shù)據(jù)采樣系統(tǒng)，離散提升技術(shù)，快速采樣算子，快速保持算子 1.簡介 采樣控制系統(tǒng)是指連續(xù)和數(shù)字系統(tǒng)的對象控制器。目前，大多數(shù)的控制系統(tǒng)是繼續(xù)的由計(jì)算機(jī)實(shí)現(xiàn)的采樣控制系統(tǒng)控制器實(shí)現(xiàn)的。隨著對系統(tǒng)要求的不斷提高，單速率的采樣控制系統(tǒng)變得不能滿足應(yīng)用的要求，因此其地位被混合采樣速率的 采樣控制系統(tǒng)所替代。混合采樣速率控制系統(tǒng)在實(shí)際應(yīng)用中能夠滿足于很廣泛的應(yīng)用場合，這是因?yàn)椋?1）在復(fù)雜的多變量控制系統(tǒng)中，要求所有的物理量在被采樣的時(shí)候都具備相同的采樣速率是不現(xiàn)實(shí)的事情。 2）在對信號進(jìn)行采樣的工程中，采樣的頻率越高，系統(tǒng)對信號的復(fù)現(xiàn)性能就越好，但是快速的 A/D 和 D/A 轉(zhuǎn)換器意味著更高的花費(fèi)。因此，對于不同的信號帶寬，你應(yīng)該使用不同速率的 A/D 及 D/A 轉(zhuǎn)換器，進(jìn)而是的系統(tǒng)的功能達(dá)到一個(gè)較高的水平的同時(shí)，又不致使系統(tǒng)的花費(fèi)太大。 3）多速率控制器一般而言是采樣時(shí)間可變的控制器，這是但速率采 樣控制器不能與之相較的優(yōu)點(diǎn)。如增加系統(tǒng)增益裕度，則就要保持系統(tǒng)的穩(wěn)定性從而保證系統(tǒng)離散控制功能的實(shí)現(xiàn)。 雙速率采樣控制系統(tǒng)是一個(gè)相對簡單的多速率采樣控制系統(tǒng)，其系統(tǒng)的框圖如圖1 所示?？刂葡到y(tǒng)仿真被定義為：對于一個(gè)給定的輸入 W，對系統(tǒng)的輸出信號 Z進(jìn)行模擬的過程。 圖 1 帶虛擬采樣器和保持器的雙速率采樣控制系統(tǒng) 文獻(xiàn) 4中給出了一個(gè)高精度的單速率采樣控制系統(tǒng)仿真的樣本。在單速率采樣控制系統(tǒng)中僅存在一種采樣周期，這樣因而其仿真過程只需應(yīng)用一些較成熟的理論。例如單速率連續(xù)傳遞函數(shù)的離散化。對于雙速率 采樣控制系統(tǒng)而言，由于系統(tǒng)中存在兩種不同的采樣周期，并且控制器為時(shí)變控制器，這樣就增加了仿真的難度。 本文采用離散提升技術(shù)，將系統(tǒng)中兩種不同的采樣周期有機(jī)地聯(lián)系起來，把時(shí)變控制器變?yōu)闀r(shí)不變控制器。同時(shí)采用快速采樣算子和快速保持算子，給出了雙速率采樣控制系統(tǒng)的仿真方法 2.知識背景 圖 1 采樣器的采樣周期 T1=ph，采樣控制器 S： y(k)=Syc(t)=yc(kph)，保持器的采樣周期 T2=qh，保持器算子： uc=(kqh+r)=Hu(k),00， Dr 為連續(xù)空間上的延遲算子， Druc (t) = uc (tT);U 為離散空間上的一步滯后算子； U2 為離散空間上的一步超前算子。 定義 1 如果（ U2） q1KUp1=K 成立 ，則稱 K 為（ p,q） -周期離散控制器。 定義 2 如果連續(xù)系統(tǒng) G 滿足 DrG=GDr，則稱 G 為 T-周期連續(xù)時(shí)變系統(tǒng)。 3.仿真算法 3.1 仿真表達(dá)式 K 是一個(gè)已知的定義（ p,q） -周期的離散控制器，采樣算子和保持算子如上所述，則 HKS 以 T 為周期的時(shí)變系統(tǒng)。如圖 1 即可證明信號之間的關(guān)系，在已知既定的條件下下式成立： 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我們可以由定義 2 看到， HKS 為 T 周期的時(shí)變系統(tǒng)。由于 HKS 的周期是 T，因此同單速率系統(tǒng)類似，圖 1 中輸出與輸入的關(guān)系可以表示為： wH K S GH K S GIGGz )( 211221211 或者是 wH K S GHK S GIHGGz )( 211221211 在雙速率采樣控制系統(tǒng)輸出與輸入通道中，通過增加一個(gè)可見的采樣器且保持快速，像在圖 1 中顯示的一樣，這個(gè)可見快速采樣器及保持器的采樣周期均為T/n。 Wd 是 w 以 T/n 為采樣周期的采樣信號，當(dāng)輸入信號的仿真時(shí)間為 mph 時(shí)，有： Wd=w(kT/n)， k=0,1,mn/p1 zd與 z 的關(guān)系同上。顯然，當(dāng) n 時(shí)， wd=w， zd=z。為使離散時(shí)間序列的個(gè)數(shù)為正整 數(shù)， n 選為 l 的整數(shù)倍。研究圖 1 所示系統(tǒng)的仿真，便可得到虛框中雙速率采樣控制系統(tǒng)連續(xù)輸入輸出信號的仿真結(jié)果。 圖 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 為對應(yīng)于周期 T/n 的離散化。式 (4)即為雙速率采樣控制系統(tǒng)的仿真表達(dá)式。 3.2 仿真表達(dá)式的計(jì)算 欲求表達(dá)式（ 4），首先要得到 G11n， G12n,， G21n,， SnH,， SHn，以及(I-KSG22H)-1K 等等變量 ， G11n, G12n, G21n 分別是連續(xù)傳遞函數(shù) G11, G12, G21 以 T 為采樣周期采樣后的離散傳遞函數(shù)，均以計(jì)算。 下面討論SnH,SHn,(I-KSG22H)-1K 的計(jì)算。 5. 計(jì)算 SnH 圖 2 SHn 的輸入與輸出框圖 圖 2 中 Hn的周期為 T/n=lh/n， S 的周期為 ph，當(dāng) x2 (0) = x1 (0), x2 (1) = x1 (n/p1), ., x2 (m -1) = x1 (m-1) n/p1)， SHn = pmnpmpmpm/0100000001000000016. 計(jì)算 SnH 同理可得 到 SnH 的表達(dá)式： qnHSmn /21000100000010001 7. 計(jì)算 (I-KSG22H)-1K 由離散采樣以及離散算子的定義，有： (k) = (kp)Sp2l l, = Sp (kq