Refined Algorithms for Adaptive Optimal Output Regulation and Adaptive Optimal Cooperative Output Regulation Problems

Given a linear unknown system with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> inputs, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$p$</tex-math></inline-formula> outputs, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula>-dimensional state vector, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-dimensional exosystem, the problem of the adaptive optimal output regulation of this system boils down to iteratively solving a set of linear equations and each of these equations contains <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\frac{n (n+1)}{2} + (m+q)n$</tex-math></inline-formula> unknown variables. A problem with a moderate size may entail forming and then solving a few hundreds of such linear equations. Thus, the computational cost of solving such a problem can be formidable. In this article, we first improve the existing algorithm by decoupling each of these linear equations into two lower dimensional linear equations. The first one contains <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$nq$</tex-math></inline-formula> unknown variables, and the second one contains <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\frac{n (n+1)}{2} + mn$</tex-math></inline-formula> unknown variables. Thus, this improved algorithm reduces a linear equation with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\frac{n (n+1)}{2} + (m+q)n$</tex-math></inline-formula> unknown variables to a linear equation with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$nq$</tex-math></inline-formula> unknown variables and a linear equation with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\frac{n (n+1)}{2} + mn$</tex-math></inline-formula> unknown variables. As a result, not only the computational cost of the problem is drastically reduced but also the solvability conditions for these equations are significantly weakened. Moreover, we also apply this improved algorithm to the adaptive cooperative optimal output regulation of linear multiagent unknown systems and reduce the computational cost of the problem more saliently.