不確定分數階系統(tǒng)的自適應控制研究
發(fā)布時間：2018-09-08 16:43
【摘要】：隨著工程技術的發(fā)展,越來越多的場合對控制提出了更高的要求。分數階微積分作為整數階微積分延伸和推廣,對于相當一部分復雜系統(tǒng),能夠描述得更為簡潔有效,進而降低對控制器魯棒性的要求；且分數階微積分的引入能夠增加控制器設計的自由度,改善控制品質。然而,隨著實際系統(tǒng)的運行,環(huán)境變化、元器件老化等原因使得所建模型不再能精確地描述實際系統(tǒng),所以研究不確定分數階系統(tǒng)的分數階控制問題有著重要的理論價值和實踐意義。 自適應方法在解決不確定整數階系統(tǒng)控制問題中,已經取得了豐碩的成果,然而,在將其拓展到分數階情形時,存在很多困難和挑戰(zhàn)。分數階微積分可以看作是整數階微積分的某種“連續(xù)過渡”。但從整數階到分數階,系統(tǒng)發(fā)生了某些本質的改變,如系統(tǒng)特征函數由單值變?yōu)槎嘀?狀態(tài)空間由有限維變?yōu)闊o窮維,使其相應的理論體系發(fā)生了本質變化,這也正是分數階自適應控制研究進展緩慢的原因之一。因此,本文將把握分數階系統(tǒng)的本質特性,借助間接李雅普諾夫方法,開展分數階自適應控制的研究,為不確定分數階系統(tǒng)的分數階控制問題提供有效解決方案。 首先,本文對分數階直接模型參考自適應控制方法進行了改進。對于階次0α≤1的SISO情形,為降低參數估計對跟蹤誤差的依賴性和避免不希望的跳變,本文在參數更新律中加入預測誤差項,得到了改進的控制策略,并將結果推廣至1α2的情形。對于MIMO情形,則利用右增益矩陣去除了與高頻增益矩陣相關的正定性的苛刻假設,提出了一個能適用于任意相對階對象的控制方案。同時,基于連續(xù)頻率分布模型和間接李雅普諾夫方法,本文證明了閉環(huán)控制系統(tǒng)的穩(wěn)定性、輸出跟蹤誤差以及參數估計誤差的漸近收斂特性。 其次,為獲得更好的參數收斂特性和控制效果,本文首次提出了分數階間接模型參考自適應控制方案,并進行了深入研究。首先討論了分數階系統(tǒng)的參數估計問題,針對系統(tǒng)參數有無約束的兩種情況,分別給出了參數估計方案。基于這一結果,分別針對SISO單變量分數階系統(tǒng)和SISO多變量分數階系統(tǒng)給出了分數階間接模型參考自適應控制器設計方法,解決了參考模型的選擇、控制器結構的構建和控制器參數的整定等問題。 然后,考慮到上述兩種方法通常只能適用于線性參數化模型,本文提出了可用于非線性系統(tǒng)的分數階自適應Backstepping控制方法。針對全狀態(tài)可測的情形,本文首先通過合適的坐標變換,將被控對象變換為歸一化的嚴格反饋系統(tǒng)；然后構造新的誤差變量,設計分數階自適應Backstepping狀態(tài)反饋控制器。針對部分狀態(tài)可測的情形,首先設計狀態(tài)觀測器,然后構造新的誤差變量。通過引入新穎的李雅普諾夫函數,解決了在觀測誤差漸近收斂的情況下閉環(huán)控制系統(tǒng)穩(wěn)定性證明的問題。并基于所提出的分數階跟蹤微分器,給出了分數階自適應Backstepping輸出反饋控制器的一般化設計流程和實現方法。 另外,巧妙地基于辨識的思想,從兩個角度研究了分數階算子的逼近問題：最高精度逼近和最低階次逼近。針對第一種情形,考慮到Oustaloup遞推逼算法是從不精確的幅頻特性出發(fā)得到的,且不能取復極點,所以并不是嚴格意義上的變極點方法,本文則基于矢量擬合方法,實現真正意義上的變極點有理逼近。針對第二種情形,提出了定極點逼近方法,將逼近問題轉化為一個線性最小二乘問題,并考慮了純積分環(huán)節(jié)的特性,給出了較優(yōu)的初始極點選擇方法。最后,在分數階積分算子逼近的基礎上,實現了對分數階系統(tǒng)的逼近,指出了分數階偽狀態(tài)空間模型與其頻率分布模型之間的關系,討論并解決了非零初始值的系統(tǒng)響應問題。上述相關工作為本文所提出的分數階自適應控制策略的驗證提供了有效的方法。
[Abstract]:With the development of Engineering technology, more and more occasions put forward higher requirements for control. Fractional calculus, as an extension and extension of integer-order calculus, can be described more concisely and effectively for a considerable number of complex systems, thereby reducing the requirements for controller robustness; and the introduction of fractional calculus can increase control. However, with the operation of the actual system, environmental changes, component aging and other reasons, the model can no longer accurately describe the actual system, so the study of Fractional-order Control of uncertain fractional-order systems has important theoretical value and practical significance.
Adaptive methods have achieved fruitful results in solving the control problems of uncertain integer-order systems. However, there are many difficulties and challenges in extending them to fractional-order systems. Qualitative changes, such as the change of system characteristic function from single-valued to multi-valued, and the change of state space from finite-dimensional to infinite-dimensional, make the corresponding theoretical system change essentially, which is one of the reasons why the research progress of fractional-order adaptive control is slow. The research on fractional-order adaptive control is carried out to provide an effective solution to the problem of Fractional-order Control for uncertain fractional-order systems.
Firstly, the fractional order direct model reference adaptive control method is improved. For SISO with order 0 alpha < 1, in order to reduce the dependence of parameter estimation on tracking error and avoid undesirable jump, the predictive error term is added to the parameter update law, and the improved control strategy is obtained, and the result is extended to 1 alpha2. In the case of MIMO, the righthand gain matrix is used to remove the rigorous assumption of positive definiteness associated with the high frequency gain matrix, and a control scheme for any relative order plant is proposed. Based on the continuous frequency distribution model and the indirect Lyapunov method, the stability of the closed-loop control system is proved. Asymptotic convergence properties of tracking error and parameter estimation error.
Secondly, in order to obtain better parameter convergence characteristics and control effect, the fractional-order indirect model reference adaptive control scheme is proposed for the first time and studied deeply in this paper. Firstly, the problem of parameter estimation for fractional-order systems is discussed. Results The design methods of fractional-order indirect model reference adaptive controller for SISO single-variable fractional-order systems and SISO multivariable fractional-order systems are presented respectively. The problems such as the selection of reference model, the construction of controller structure and the tuning of controller parameters are solved.
Then, considering that the above two methods can only be applied to linear parameterized models, a fractional-order adaptive backstepping control method for nonlinear systems is proposed. A new error variable is constructed and a fractional-order adaptive backstepping state feedback controller is designed. In the case of partial state measurability, a state observer is first designed, and then a new error variable is constructed. Based on the proposed fractional order tracking differentiator, the general design flow and implementation method of the fractional order adaptive backstepping output feedback controller are given.
In addition, based on the idea of identification, the approximation problem of fractional operators is studied from two aspects: the highest precision approximation and the lowest order approximation. For the second case, a fixed pole approximation method is proposed, which transforms the approximation problem into a linear least squares problem. Considering the characteristics of the pure integral link, an optimal initial pole selection method is given. On the basis of integral operator approximation, the approximation of fractional-order systems is realized. The relationship between fractional pseudo-state space model and its frequency distribution model is pointed out. The system response problem with non-zero initial value is discussed and solved. Method.
【學位授予單位】：中國科學技術大學
【學位級別】：博士
【學位授予年份】：2015
【分類號】：TP13
本文編號：2231093
[Abstract]:With the development of Engineering technology, more and more occasions put forward higher requirements for control. Fractional calculus, as an extension and extension of integer-order calculus, can be described more concisely and effectively for a considerable number of complex systems, thereby reducing the requirements for controller robustness; and the introduction of fractional calculus can increase control. However, with the operation of the actual system, environmental changes, component aging and other reasons, the model can no longer accurately describe the actual system, so the study of Fractional-order Control of uncertain fractional-order systems has important theoretical value and practical significance.
Adaptive methods have achieved fruitful results in solving the control problems of uncertain integer-order systems. However, there are many difficulties and challenges in extending them to fractional-order systems. Qualitative changes, such as the change of system characteristic function from single-valued to multi-valued, and the change of state space from finite-dimensional to infinite-dimensional, make the corresponding theoretical system change essentially, which is one of the reasons why the research progress of fractional-order adaptive control is slow. The research on fractional-order adaptive control is carried out to provide an effective solution to the problem of Fractional-order Control for uncertain fractional-order systems.
Firstly, the fractional order direct model reference adaptive control method is improved. For SISO with order 0 alpha < 1, in order to reduce the dependence of parameter estimation on tracking error and avoid undesirable jump, the predictive error term is added to the parameter update law, and the improved control strategy is obtained, and the result is extended to 1 alpha2. In the case of MIMO, the righthand gain matrix is used to remove the rigorous assumption of positive definiteness associated with the high frequency gain matrix, and a control scheme for any relative order plant is proposed. Based on the continuous frequency distribution model and the indirect Lyapunov method, the stability of the closed-loop control system is proved. Asymptotic convergence properties of tracking error and parameter estimation error.
Secondly, in order to obtain better parameter convergence characteristics and control effect, the fractional-order indirect model reference adaptive control scheme is proposed for the first time and studied deeply in this paper. Firstly, the problem of parameter estimation for fractional-order systems is discussed. Results The design methods of fractional-order indirect model reference adaptive controller for SISO single-variable fractional-order systems and SISO multivariable fractional-order systems are presented respectively. The problems such as the selection of reference model, the construction of controller structure and the tuning of controller parameters are solved.
Then, considering that the above two methods can only be applied to linear parameterized models, a fractional-order adaptive backstepping control method for nonlinear systems is proposed. A new error variable is constructed and a fractional-order adaptive backstepping state feedback controller is designed. In the case of partial state measurability, a state observer is first designed, and then a new error variable is constructed. Based on the proposed fractional order tracking differentiator, the general design flow and implementation method of the fractional order adaptive backstepping output feedback controller are given.
In addition, based on the idea of identification, the approximation problem of fractional operators is studied from two aspects: the highest precision approximation and the lowest order approximation. For the second case, a fixed pole approximation method is proposed, which transforms the approximation problem into a linear least squares problem. Considering the characteristics of the pure integral link, an optimal initial pole selection method is given. On the basis of integral operator approximation, the approximation of fractional-order systems is realized. The relationship between fractional pseudo-state space model and its frequency distribution model is pointed out. The system response problem with non-zero initial value is discussed and solved. Method.
【學位授予單位】：中國科學技術大學
【學位級別】：博士
【學位授予年份】：2015
【分類號】：TP13
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