March  2022, 12(1): 135-157. doi: 10.3934/naco.2021056

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

1. 

School of Electrical Engineering and Automation, Hefei University of Technology, Hefei 230009, China

2. 

School of Foreign Studies, Hefei University of Technology, Hefei 230009, China

3. 

School of Mathematical Sciences, Sunway University, Malaysia, Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin, China

* Corresponding author: Canghua Jiang

Received  February 2021 Revised  April 2021 Published  March 2022 Early access  November 2021

Fund Project: 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

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.

Citation: Canghua Jiang, Dongming Zhang, Chi Yuan, Kok Ley Teo. An active set solver for constrained $ H_\infty $ optimal control problems with state and input constraints. Numerical Algebra, Control and Optimization, 2022, 12 (1) : 135-157. doi: 10.3934/naco.2021056
References:
[1]

A. BemporadM. MorariV. Dua and E. N. Pistikopoulos, The explicit linear quadratic regulator for constrained systems, Automatica, 38 (2002), 3-20.  doi: 10.1016/S0005-1098(01)00174-1.

[2]

F. Borrelli, Discrete Time Constrained Optimal Control, Ph.D. thesis, Swiss Federal Institute of Technology (ETH), Zurich, 2002.

[3]

A. E. Bryson and Y.-C. Ho, Applied Optimal Control: Optimization, Estimation, and Control, Hemisphere, Washington, 1975.

[4]

J. Buerger, M. Cannon and B. Kouvaritakis, An active set solver for min-max robust control, in Proceedings of the 2013 American Control Conference, Washington, USA, (2013), 4227–4233.

[5]

J. BuergerM. Cannon and B. Kouvaritakis, An active set solver for input-constrained robust receding horizon control, Automatica, 50 (2014), 155-161.  doi: 10.1016/j.automatica.2013.09.032.

[6]

J. BuergerM. Cannon and B. Kouvaritakis, Active set solver for min-max robust control with state and input constraints, International Journal of Robust and Nonlinear Control, 26 (2016), 3209-3231.  doi: 10.1002/rnc.3501.

[7]

M. CannonW. Liao and B. Kouvaritakis, Efficient MPC optimization using Pontryagin's minimum principle, International Journal of Robust and Nonlinear Control, 18 (2008), 831-844.  doi: 10.1002/rnc.1247.

[8]

R. GhaemiJ. Sun and I. V. Kolmanovsky, Neighboring extremal solution for nonlinear discrete-time optimal control problems with state inequality constraints, IEEE Transactions on Automatic Control, 54 (2009), 2674-2679.  doi: 10.1109/TAC.2009.2031576.

[9]

P. J. GoulartE. C. Kerrigan and T. Alamo, Control of constrained discrete-time systems with bounded $l_2$ gain, IEEE Transactions on Automatic Control, 54 (2009), 1105-1111.  doi: 10.1109/TAC.2009.2013002.

[10]

M. Green and D. J. N. Limebeer, Linear Robust Control, Prentice-Hall, Englewood Cliffs, NJ, 1995.

[11]

I. Kolmanovsky and E. G. Gilbert, Theory and computation of disturbance invariant sets for discrete-time linear systems, Mathematical Problems in Engineering, 4 (1998), 317-367. 

[12]

M. V. KothareV. Balakrishnan and M. Morari, Robust constrained model predictive control using linear matrix inequalities, Automatica, 32 (1996), 1361-1379.  doi: 10.1016/0005-1098(96)00063-5.

[13]

D. Q. MayneS. V. RakovićR. B. Vinter and E. C. Kerrigan, Characterization of the solution to a constrained ${H}_{\infty}$ optimal control problem, Automatica, 42 (2006), 371-382.  doi: 10.1016/j.automatica.2005.10.015.

[14]

J. Nocedal and S. J. Wright, Numerical Optimization, Springer, New York, 2006.

[15]

S. V. RakovićB. KouvaritakisM. CannonC. Panos and R. Findeisen, Parameterized tube model predictive control, IEEE Transactions on Automatic Control, 57 (2012), 2746-2761.  doi: 10.1109/TAC.2012.2191174.

[16]

P. O. M. Scokaert and D. Q. Mayne, Min-max feedback model predictive control for constrained linear systems, IEEE Transactions on Automatic Control, 43 (1998), 1136-1142.  doi: 10.1109/9.704989.

[17]

P. TøndelT. A. Johansen and A. Bemporad, An algorithm for multi-parametric quadratic programming and explicit MPC solutions, Automatica, 39 (2003), 489-497.  doi: 10.1016/S0005-1098(02)00250-9.

[18]

Y. Wang and S. Boyd, Fast model predictive control using online optimization, IEEE Transactions on Control Systems Technology, 18 (2010), 267-278. 

[19]

G. M. Ziegler, Lectures on Polytopes, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4613-8431-1.

show all references

References:
[1]

A. BemporadM. MorariV. Dua and E. N. Pistikopoulos, The explicit linear quadratic regulator for constrained systems, Automatica, 38 (2002), 3-20.  doi: 10.1016/S0005-1098(01)00174-1.

[2]

F. Borrelli, Discrete Time Constrained Optimal Control, Ph.D. thesis, Swiss Federal Institute of Technology (ETH), Zurich, 2002.

[3]

A. E. Bryson and Y.-C. Ho, Applied Optimal Control: Optimization, Estimation, and Control, Hemisphere, Washington, 1975.

[4]

J. Buerger, M. Cannon and B. Kouvaritakis, An active set solver for min-max robust control, in Proceedings of the 2013 American Control Conference, Washington, USA, (2013), 4227–4233.

[5]

J. BuergerM. Cannon and B. Kouvaritakis, An active set solver for input-constrained robust receding horizon control, Automatica, 50 (2014), 155-161.  doi: 10.1016/j.automatica.2013.09.032.

[6]

J. BuergerM. Cannon and B. Kouvaritakis, Active set solver for min-max robust control with state and input constraints, International Journal of Robust and Nonlinear Control, 26 (2016), 3209-3231.  doi: 10.1002/rnc.3501.

[7]

M. CannonW. Liao and B. Kouvaritakis, Efficient MPC optimization using Pontryagin's minimum principle, International Journal of Robust and Nonlinear Control, 18 (2008), 831-844.  doi: 10.1002/rnc.1247.

[8]

R. GhaemiJ. Sun and I. V. Kolmanovsky, Neighboring extremal solution for nonlinear discrete-time optimal control problems with state inequality constraints, IEEE Transactions on Automatic Control, 54 (2009), 2674-2679.  doi: 10.1109/TAC.2009.2031576.

[9]

P. J. GoulartE. C. Kerrigan and T. Alamo, Control of constrained discrete-time systems with bounded $l_2$ gain, IEEE Transactions on Automatic Control, 54 (2009), 1105-1111.  doi: 10.1109/TAC.2009.2013002.

[10]

M. Green and D. J. N. Limebeer, Linear Robust Control, Prentice-Hall, Englewood Cliffs, NJ, 1995.

[11]

I. Kolmanovsky and E. G. Gilbert, Theory and computation of disturbance invariant sets for discrete-time linear systems, Mathematical Problems in Engineering, 4 (1998), 317-367. 

[12]

M. V. KothareV. Balakrishnan and M. Morari, Robust constrained model predictive control using linear matrix inequalities, Automatica, 32 (1996), 1361-1379.  doi: 10.1016/0005-1098(96)00063-5.

[13]

D. Q. MayneS. V. RakovićR. B. Vinter and E. C. Kerrigan, Characterization of the solution to a constrained ${H}_{\infty}$ optimal control problem, Automatica, 42 (2006), 371-382.  doi: 10.1016/j.automatica.2005.10.015.

[14]

J. Nocedal and S. J. Wright, Numerical Optimization, Springer, New York, 2006.

[15]

S. V. RakovićB. KouvaritakisM. CannonC. Panos and R. Findeisen, Parameterized tube model predictive control, IEEE Transactions on Automatic Control, 57 (2012), 2746-2761.  doi: 10.1109/TAC.2012.2191174.

[16]

P. O. M. Scokaert and D. Q. Mayne, Min-max feedback model predictive control for constrained linear systems, IEEE Transactions on Automatic Control, 43 (1998), 1136-1142.  doi: 10.1109/9.704989.

[17]

P. TøndelT. A. Johansen and A. Bemporad, An algorithm for multi-parametric quadratic programming and explicit MPC solutions, Automatica, 39 (2003), 489-497.  doi: 10.1016/S0005-1098(02)00250-9.

[18]

Y. Wang and S. Boyd, Fast model predictive control using online optimization, IEEE Transactions on Control Systems Technology, 18 (2010), 267-278. 

[19]

G. M. Ziegler, Lectures on Polytopes, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4613-8431-1.

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} $
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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