Abstract
A proper Hamilton-Jacobi-Isaacs (HJI) inequality must be solved in a nonlinear H∞ control problem. The sum of squares (SOS) method can now be used to solve an analytically unsolvable nonlinear problem. A HJI inequality suitable for SOS approach is derived in the paper. The SOS algorithm for solving the HJI inequality is also provided. Conservativeness of the SOS method is then discussed in the paper. The conservativeness of the SOS approach is caused by the method itself, because it is really a synthesis method over the entire state space. To reduce the conservativeness, a local H∞ design on a restricted state-space region is proposed. But the SOS approach for the local H∞ design also suffers from the conservativeness problem, because the S-procedure for solving the set-containment constraint provides only a sufficient condition. The above-mentioned sources of conservativeness are peculiar for the SOS approaches. So a proper approach must be carefully selected in the design process to get a reasonable result. A design example is also given in the paper.
Highlights
The control problems of complicated nonlinear system are always the research hotspots [1,2,3,4]
This article derives a suitable HJI inequality for sum of squares (SOS) method and turns it into a matrix inequality depending on the state variables similar to LMI and uses the functions in SOSTOOLS to solve the linear matrix inequality which is status-dependent before giving the nonlinear H∞ control law
We can know from the method of linear matrix inequality (LMI) that the solution P of the Riccati equation does not make up a convex problem, but if we use the inverse of the solution, just P−1, to present, we can get LMI [2]
Summary
The control problems of complicated nonlinear system are always the research hotspots [1,2,3,4]. In order to avoid solving this complicated HIJ inequality, researchers get L2 gain controller through constructed the Hamilton function generally [6, 8]. The SOS method [8,9,10], the abbreviation of the sum of squares, coming out recent years, has opened up a new way to solve the HJI inequality a numerical solution. Except for the nonlinearity of the object itself, if we want to use Lyapunov function higher than quadratic terms, or design a high order nonlinear control law, a polynomial in general form has to be studied. This article derives a suitable HJI inequality for SOS method and turns it into a matrix inequality depending on the state variables similar to LMI and uses the functions in SOSTOOLS to solve the linear matrix inequality which is status-dependent before giving the nonlinear H∞ control law. The article discusses the conservatism and the treatment countermeasures while using SOS to solve the HJI inequality
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