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An active set solver for constrained $ H_\infty $ optimal control problems with state and input constraints

  • * Corresponding author: Canghua Jiang

    * Corresponding author: Canghua Jiang 

This paper is dedicated to the memory of Professor Han-Chin Lai. The first author is supported by the Natural Science Foundation of Anhui Province, China, under Grant 2008085MF216

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  • This paper proposes an active set solver for $ H_\infty $ min-max optimal control problems involving linear discrete-time systems with linearly constrained states, controls and additive disturbances. The proposed solver combines Riccati recursion with dynamic programming. To deal with possible degeneracy (i.e. violations of the linear independence constraint qualification), constraint transformations are introduced that allow the surplus equality constraints on the state at each stage to be moved to the previous stage together with their Lagrange multipliers. In this way, degeneracy for a feasible active set can be determined by checking whether there exists an equality constraint on the initial state over the prediction horizon. For situations when the active set is degenerate and all active constraints indexed by it are non-redundant, a vertex exploration strategy is developed to seek a non-degenerate active set. If the sampled state resides in a robust control invariant set and certain second-order sufficient conditions are satisfied at each stage, then a bounded $ l_2 $ gain from the disturbance to controlled output can be guaranteed for the closed-loop system under some standard assumptions. Theoretical analysis and numerical simulations show that the computational complexity per iteration of the proposed solver depends linearly on the prediction horizon.

    Mathematics Subject Classification: Primary: 93B45, 93B36; Secondary: 90C31.


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  • Figure 1.  The CPU time per iteration and the maximum number of iterations

    Figure 2.  The controlled pitch dynamics

    Table 1.  Computational complexity of Algorithm 1

    Step Number of operations
    2) $ \begin{array}{l}O((\frac{4}{3}(n_{A_w}^2n_w+n_{A_u}^2n_u)\\+(n_wn_{A_w}+n_{A_w}^2+n_un_{A_u}+n_{A_u}^2)n_x+n_{A_u}n_x^2)N)\end{array} $
    3) $ \begin{array}{l}O((a_1(n_{A_w}n_w^2+n_{A_u}n_u^2)+a_2(n_w^3+n_u^3)+a_3(n_w^2+n_u^2)n_x\\+(n_w+n_{A_w}+n_u+n_{A_u})n_x^2+2n_x^3)N)\end{array} $
    4) $ \begin{array}{l}O\left((b_1n_x^2+b_2(o_k^d)^3)N\right)\end{array} $
    8) $ \begin{array}{l}O(n_k^d(\frac{4}{3}(n_{A_w}^2n_w+n_{A_u}^2n_u)\\+(n_wn_{A_w}+n_{A_w}^2+n_un_{A_u}+n_{A_u}^2)n_x+n_{A_u}n_x^2)N)\end{array} $
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