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- In control theory, optimal projection equations constitute necessary and sufficient conditions for a locally optimal reduced-order LQG controller. The linear-quadratic-Gaussian (LQG) control problem is one of the most fundamental optimal control problems. It concerns uncertain linear systems disturbed by additive white Gaussian noise, incomplete state information (i.e. not all the state variables are measured and available for feedback) also disturbed by additive white Gaussian noise and quadratic costs. Moreover, the solution is unique and constitutes a linear dynamic feedback control law that is easily computed and implemented. Finally the LQG controller is also fundamental to the optimal perturbation control of non-linear systems. The LQG controller itself is a dynamic system like the system it controls. Both systems have the same state dimension. Therefore, implementing the LQG controller may be problematic if the dimension of the system state is large. The reduced-order LQG problem (fixed-order LQG problem) overcomes this by fixing a-priori the number of states of the LQG controller. This problem is more difficult to solve because it is no longer separable. Also the solution is no longer unique. Despite these facts numerical algorithms are available to solve the associated optimal projection equations. (en)
- 最佳投影方程（optimal projection equations）是控制理论中，建構局部最佳降階LQG控制器的充分必要条件。 LQG控制（線性二次高斯控制）問題是最优控制領域中最基礎的問題之一，這問題包括了存在不確定性的線性系統，受到加性高斯白噪声的影響，沒有完整的狀態資訊（無法量測到所有的狀態變數，也無法透過回授得知），對應二次的成本泛函。不過存在唯一解，而且可以建構線性動態回授的控制律，易於計算以及實現。而LQG控制器也是非線性系統中最佳擾動控制的基礎。 LQG控制器的架構會類似要控制的系統，兩者會有相同的維度。因此若系統本身就是高維度，要實現（全階）LQG控制器會很困難。降階LQG問題（固定階LQG問題）事先固定LQG控制器的階數，因此克服了這個困難。不過在全階LQG控制器中適用的分離原理，在降階LQG問題中已無法適用，因此這方面會更困難，而且其解也不唯一。不過可以找到數值分析的演算法來求解對應的最佳投影方程。 (zh)
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- 最佳投影方程（optimal projection equations）是控制理论中，建構局部最佳降階LQG控制器的充分必要条件。 LQG控制（線性二次高斯控制）問題是最优控制領域中最基礎的問題之一，這問題包括了存在不確定性的線性系統，受到加性高斯白噪声的影響，沒有完整的狀態資訊（無法量測到所有的狀態變數，也無法透過回授得知），對應二次的成本泛函。不過存在唯一解，而且可以建構線性動態回授的控制律，易於計算以及實現。而LQG控制器也是非線性系統中最佳擾動控制的基礎。 LQG控制器的架構會類似要控制的系統，兩者會有相同的維度。因此若系統本身就是高維度，要實現（全階）LQG控制器會很困難。降階LQG問題（固定階LQG問題）事先固定LQG控制器的階數，因此克服了這個困難。不過在全階LQG控制器中適用的分離原理，在降階LQG問題中已無法適用，因此這方面會更困難，而且其解也不唯一。不過可以找到數值分析的演算法來求解對應的最佳投影方程。 (zh)
- In control theory, optimal projection equations constitute necessary and sufficient conditions for a locally optimal reduced-order LQG controller. The linear-quadratic-Gaussian (LQG) control problem is one of the most fundamental optimal control problems. It concerns uncertain linear systems disturbed by additive white Gaussian noise, incomplete state information (i.e. not all the state variables are measured and available for feedback) also disturbed by additive white Gaussian noise and quadratic costs. Moreover, the solution is unique and constitutes a linear dynamic feedback control law that is easily computed and implemented. Finally the LQG controller is also fundamental to the optimal perturbation control of non-linear systems. (en)
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- Optimal projection equations (en)
- 最佳投影方程 (zh)
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