在實務上，系統鑑別所面對的是具不確定性與限制的系統，包括程序本身及其所處環境的多種未知與限制因素。譬如程序初始狀態、模型的結構和參數、負載擾動和雜訊皆可能是未知的，另外鑑別實驗之輸入訊號改變須受到實際裝置的限制。面對這些客觀存在的諸多不確定性與限制，如何設計實際可行的鑑別實驗和發展估測演算法以獲得適當的程序與擾動模型，就是系統鑑別研究的主要課題。
本論文發展三個線性系統鑑別方法，用於解決現場鑑別測試常遭遇的實際困難，以獲得正確的微差分方程或轉移函數模型。這些方法具備以下優點：首先，鑑別實驗較不受輸入測試訊號和實際操作條件的限制；其次，能夠直接從一組輸出入實驗數據同時估測出程序結構以及對應的模型參數，無須冗長或重覆之實驗操作過程；最後，能夠處理實際操作狀況，包括未知初始狀態、未知負載擾動以及雜訊干擾。
第一個鑑別方法應用多重積分轉換來處理連續時間單一輸入/單一輸出程序，其特點是採用先繁後簡的兩段式測試訊號和兩階段順序式最小平方演算，將複雜的鑑別問題分解成兩個較簡單的參數估測問題。在第一階段利用後段測試數據來獲得程序轉移函數的分母多項式、未知擾動與初始狀態的估測，然後在第二階段利用前段測試數據來完成整個程序模型參數的估測。模擬和實驗範例的結果顯示，此法能夠有效地去除靜態、漂移或週期性負載擾動及初始狀態對程序鑑別的影響並且由於多重積分的使用對雜訊存在與模型結構不吻合的情況具有相當的強韌性。此法也提供簡易方式來獲得正確的模型結構並在非零初始狀態和擾動存在下達到模型預測和輸出量測相吻合的結果。
第二個鑑別方法適用於離散時間多重輸入/多重輸出程序，針對每一個多重輸入/單一輸出子程序，使用一個含有可調時間縮放因子的Laguerre模型建立一條回歸方程式，以進行最小平方參數估測，其特點是可處理在未知時間進入鑑別測試的未知動態負載擾動。當負載擾動的動態與程序動態不同時，此回歸式中Laguerre模型的階次被擴增來包含程序、初始狀態、負載擾動及隨機性擾動。所得之Laguerre係數最小平方估測器配合三個誤差標準的最小化可用以有效和正確地建構程序與擾動模型。對於決定性負載擾動，兩個誤差標準被用來選擇適當的時間縮放因子、擾動進入時間與程序時延，可以確保程序模型的正確性，而對於隨機性擾動，另一個誤差標準被用來選擇適當的時間縮放因子與程序時延，以去除擾動所造成的負面效應。模擬研究證實此法能夠處理各種各樣決定性和隨機性擾動，並且對取樣時間的選擇不敏感，這對離散系統的鑑別尤其重要。此法的缺點則為Laguerre模型的擴增後階次必須大於真實程序階次，否則無法有效地描述未知擾動，因此難以獲得優良的降階程序模型。
最後一個方法應用雙Laguerre模型於離散時間多重輸入/多重輸出程序的鑑別，其原理為在同一組測試數據下，針對程序和未知負載擾動採用兩個含有不同時間縮放因子的Laguerre模型來分開處理。其優點為允許針對程序之Laguerre模型階次小於真實的程序階次，因此可以獲得優良的降階程序模型，而其缺點是必須能夠適當地選出兩個不同的時間縮放因子來區分出並準確描述程序和擾動對程序輸出的影響。此法已被證實為可信賴的，能夠根據所提誤差標準來得到優良的降階程序模型。

關鍵字: 多重積分、順序演算法、MIMO系統鑑別、Laguerre模型、模型降階、未知擾動

In practice, system identification is faced with uncertain and constrained systems, including various unknown and constrained factors in the process itself and its surroundings. For instance, initial states of the process, model structure and parameters, load disturbances and noise can be unknown. In addition, changes in the input signal to an identification test may be restricted by practical devices. Under such uncertainties and constraints existing objectively, the major role of the identification studies is to design feasible identification experiments and to develop estimation algorithms that can acquire adequate process and disturbance models.
This dissertation presents three identification methods to resolve various difficulties often encountered in field identification testing and hence obtain accurate differential (difference) equations or transfer function models. These methods possess the following advantages. First, their identification tests are less restricted to the input test signal and practical operating conditions. Second, they can estimate the model structure and the corresponding model parameters from a single set of test data, without lengthy or repetitive experimental procedures. Last, they can cope with practical operating conditions, such as unknown initial states, unknown load disturbances and noises.
The first identification method applies the multiple integration transform to deal with the identification of continuous-time single-input/single-output processes. The feature of the method is that the two segments of a test signal, first complicated and then simple, and two-stage sequential least-squares algorithms are employed to divide the complicated identification problem into two simpler parameter estimation problems. At the first stage of estimation based on the latter segment of the test data, the algorithms can yield the estimation of the denominator polynomial of the transfer function model as well as unknown disturbances and initial states. At the second stage of estimation based on the former segment of the test data, the algorithms give the complete model parameter estimates of the process. Simulation and experimental studies reveal that the method can reject the effects of unknown initial states as well as static, drifting and periodic disturbances on identification results, and is rather robust with respect to noise and model structure mismatch due to the use of multiple integration. It also provides convenient ways to find the model structure and to fit model predictions to the output measurements in the face of unknown initial states and disturbances.
The second identification method is suited to discrete-time multiple–input/multiple–output processes. For each multiple–input/ single–output subprocess, a regression equation is established based on a Laguerre model involving an adjustable time-scaling factor for least-squares parameter estimation. The feature of the method is that it could deal with unpredicted load disturbances entering the identification test at any unknown time. As the load disturbances possess dynamics distinct from the process dynamics, the order of the Laguerre model in the regression equation is augmented to incorporate terms regarding the process, load disturbances and stochastic disturbances. The resulting least–squares estimator of the Laguerre coefficients in conjunction with the minimization of three developed error criteria is thus employed to construct process and disturbance models efficiently and accurately. For deterministic disturbances, two error criteria are used to infer proper values of the time-scaling factor, the load entering time, and process time delays. In the presence of a stochastic disturbance, another error criterion is employed to determine process time delays together with the time–scaling factor that rejects the most deteriorating effect of the stochastic disturbance on parameter estimation. Simulation studies demonstrate that the proposed method is reliable against multifarious characteristics of deterministic and stochastic disturbances, and is insensitive to the selection of the sample time, which is particularly useful for discrete identification. The disadvantage of the method is that the assumed order of the Laguerre model must be greater than the actual process order, lest the unknown disturbances should become indescribable. This implies a good reduced-order model is not obtainable by this method.
The last method applies double Laguerre models to discrete-time multiple–input/multiple–output processes. The key is to describe the process and disturbance dynamics separately by two Laguerre models involving different time-scaling factors under the same set of test data. Its advantage is to allow the Laguerre model for the process to be smaller than the actual process order, so that a good reduced-order model is obtainable. The disadvantage is that two different time-scaling factors must be properly selected such that the effects of the process and disturbance on the process output can be distinguished and accurately described. It is demonstrated that the method is reliable in that it can obtain a good reduced-order process model based on the proposed error criteria.

Keywords: Multiple Integration, Sequential Algorithms, MIMO System Identification, Laguerre Model, Model Reduction, Unknown Disturbances

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