# Robust model predictive control for discrete-time fractional-order systems

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1. Robust MPC for fractional-order discrete-time systems P. Sopasakis,, S. Ntouskas and H. Sarimveis Institute for Advanced Studies Lucca, Italy, National Technical University of Athens, Greece. June 17, 2015 2. Fractional-order systems and nite-dimensional approximations 3. Fractional derivatives The Grunwald-Letnikov derivative1 of order > 0 is dened as D x(t) = lim h0+ hx(t) h (1) where h is the fractional dierence operator of order with step size h > 0 given by hx(t) = j=0 (1)j j c j x(t kh), (2) where j = j1 i=0 i i + 1 . (3) 1 S. Samko, A. Kilbas, O. Marinhev, Fractional integral and derivatives, Gordon & Breach Science Publishers, 1993; Section 20. 1 / 18 4. Fractional derivatives in action A few examples: Pharmacokinetics (Dokumetzidis et al. 2010, Magin et al. 2004) Semi-innite power transmission lines (Clarke et al., 2004) Viscoelastic polymers (Hilfer 2000) Anomalous diusion (Magin 2010, Pereira 2010) Electromagnetic theory (Schafer and Kruger 2006, Zelenyi and Milovanov 2004) Statistical mechanics (Tarasov 2005) and many other... 2 / 18 5. Fractional systems Fractional-order systems l i=1 AiDi x(t) = Bu(t) (4) Euler-type discretisation [Replace Di hi i h ] l i=1 Aii h xk+1 = Buk (5) Notice that i h is an innite-dimensional operator! 3 / 18 6. Finite-dimensional approximation h can be written as h = h, + R h,, (6) where h,x(t) = j=0 c j x(t kh), (7) and R h, is a bounded operator. If we plug (6) into the original system we get l i=1 Aii h,xk+1 Finite dimensional + l i=1 AiRi h,xk+1 Bounded term = Buk (8) 4 / 18 7. LTI approximation We can now write the system as an LTI with a bounded disturbance term dk as follows xk+1 = Axk + Buk + Gdk, (9) with state xk = (xk, xk1, . . . , xk+1) and dk D Rn. Assuming xk X where X is a balanced compact set, we have D = l i=1 A1 0 Ai(i)X, (10) where () = j=0 |c j | and A0 = l i=1 Aici j . 5 / 18 8. () is quickly decreasing with 50 100 150 200 250 300 () 10 -2 10 -1 10 0 =0.7 =0.5 =0.4 ...and as a result, D can become arbitrarily small for adequately large ! 6 / 18 9. Model Predictive Control using the approximate system model 10. Why MPC? Because... Optimisation-based control Accounts for state/input constraints xk X, uk U. 7 / 18 11. Tube-based MPC Nominal System Tube-based MPC + - + + Fractional System v u Ke K e x z Concept: The control action u is calculated as uk = vk + Kek, where ek = xk zk and vk is computed by an MPC controller. See: D.Q. Mayne, M.M. Seron, S.V. Rakovic, Robust model predictive control of constrained linear systems with bounded disturbances, Automatica 41(2), 219224, 2005. 8 / 18 12. Tube-based MPC The set S = i=0 Ai KGD (11) is robustly positive invariant for the deviation dynamics ek+1 = AKek + Gdk (12) Choosing z0 = x0 we have xk {zk} S, (13) so the constraints will be satised if zk X S, (14a) vk U KS. (14b) See: J. B. Rawlings and D. Q. Mayne, Model Predictive Control: Theory and Design. Nob Hill Publishing, 2009. 9 / 18 13. This leads to the following MPC formulation: PN :VN (zk) = min vkVN (zk) VN (zk, vk), (15) The terminal set Xf and cost function Vf are selected according to the stabilising conditions in D. Mayne, J. Rawlings, C. Rao, and P. Scokaert, Constrained model predictive control: Stability and optimality, Automatica 36 (6), 789814, 2000. 10 / 18 14. This leads to the following MPC formulation: PN :VN (zk) = min vkVN (zk) VN (zk, vk), (15) where VN (zk|k, vk)= Vf (zk+N|k) Terminal cost + N1 i=0 (zk+i|k, vk+i|k), (16) The terminal set Xf and cost function Vf are selected according to the stabilising conditions in D. Mayne, J. Rawlings, C. Rao, and P. Scokaert, Constrained model predictive control: Stability and optimality, Automatica 36 (6), 789814, 2000. 10 / 18 15. This leads to the following MPC formulation: PN :VN (zk) = min vkVN (zk) VN (zk, vk), (15) where VN (zk|k, vk)= Vf (zk+N|k) Terminal cost + N1 i=0 (zk+i|k, vk+i|k), (16) and for some S S VN (zk)= v zk+i+1|k=Azk+i|k+Bvk+i|k, iN[0,N1] zk|k = zk zk+i|k X S, iN[1,N] vk+i|k U KS, iN[0,N1] zk+N|k Xf Terminal constraints (17) The terminal set Xf and cost function Vf are selected according to the stabilising conditions in D. Mayne, J. Rawlings, C. Rao, and P. Scokaert, Constrained model predictive control: Stability and optimality, Automatica 36 (6), 789814, 2000. 10 / 18 16. Closed-loop properties The solution of PN is {v0(zk), v1(zk), . . . , vN1(zk)} and the control law is N (zk) = v0(zk), (18) the input applied to the system is (zk, xk) = N (zk) + K(xk zk) and the closed-loop system (in terms of both zk and xk) becomes xk+1 = Axk + B(zk, xk) + Gdk, (19a) zk+1 = Azk + BN (zk). (19b) Stability: The set S {0} is exponentially stable for (19). 11 / 18

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