Robust Fractional Control An overview of the research activity carried on at the University of Brescia

Università degli Studi di Brescia Dipartimento di Ingegneria dell Informazione Robust Fractional Control An overview of the research activity carried on at the University of Brescia Fabrizio Padula and Antonio Visioli www.ing.unibs.it/fabrizio.padula www.ing.unibs.it/visioli {fabrizio.padula,anotnio.visioli}@unibs.it Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 1 / 46

Fractional control for robust performance Why Fractional Control? extra degrees of freedom and capability of modeling a wider range of dynamics PUSH AHEAD THE FUNDAMENTAL ROBUSTNESS/PERFORMANCE TRADE-OFF! Warning! The design is much more complex Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 2 / 46

Outline 1 Fractional PID control Tuning rules 2 H Optimal Control A model-matching problem 3 H Model-matching controller design Optimal Controller Robust stability 4 Dynamic inversion of fractional systems Command signal design 5 Optimal feedback/feedforward control Combined feedback/feedforward design 6 Conclusions Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 3 / 46

Agenda 1 Fractional PID control Tuning rules 2 H Optimal Control A model-matching problem 3 H Model-matching controller design Optimal Controller Robust stability 4 Dynamic inversion of fractional systems Command signal design 5 Optimal feedback/feedforward control Combined feedback/feedforward design 6 Conclusions Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 4 / 46

Fractional-order proportional-integral-derivative controller Fractional-Order Proportional-Integral-Derivative (FOPID) controllers are the natural generalization of standard PID controllers FOPID controller C(s) = K p T i s λ + 1 T i s λ (T d s µ + 1) λ and µ are the non integer orders of the integral and derivative terms they have 5 parameters instead of three they are more flexible: through the exponents a continuous regulations of the slope is possible... Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 5 / 46

FOPID controller magnitude [db] 100 50 0 λ µ 10 5 10 0 10 5 frequency [Rad/s] 200 phase [ ] 100 0 100 λ µ 10 5 10 0 10 5 frequency [Rad/s] Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 6 / 46

Tuning rules: problem formulation d r e C P y Task set-point step following load disturbance step rejection Dynamics Integral Plus Dead Time (IPDT) process P(s) = K s e Ls First-Order Plus Dead Time (FOPDT) process P(s) = K Ts+1 e Ls Unstable First-Order Plus Dead Time (UFOPDT) process P(s) = K 1 Ts e Ls Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 7 / 46

Tuning rules: optimization function and constraints In order to get optimal tuning rules the integrated absolute error (IAE) has been minimized. IAE = e(t) dt = 0 0 r(t) y(t) dt, Maximum Sensitivity M s 1 M s = max ω [0,+ ) 1 + C(s)P(s) M s represents also the inverse of the minimum distance of the Nyquist plot from the critical point M s = 1.4 robust tuning M s = 2.0 aggressive tuning The optimization process has been numerically solved for different normalized dead time L T The results have been interpolated to obtain general tuning rules The optimal IAE have been interpolated to obtain performance assessment rules For the sake of comparison tuning rules have been developed also for integer PID controllers Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 8 / 46

Results for FOPDT IAE increment [%] IAE increment [%] 30 20 10 0 0 0.2 0.4 0.6 τ 25 20 15 10 5 0 0 0.2 0.4 0.6 τ IAE increment [%] IAE increment [%] 10 5 0 0 0.2 0.4 0.6 τ 60 40 20 0 0 0.2 0.4 0.6 τ Top-left: set-point with M s = 1.4. Top-right: set-point with M s = 2.0. Bottom-left: load disturbance with M s = 1.4. Bottom-right: load disturbance with M s = 2.0. Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 9 / 46

Results for IPDT and UFOPDT Integral M s 1.4 sp 2.0 sp 1.4 ld 2.0 ld IAE[%] 17.2 6.34 19.1 22.7 The optimization can be performed just once! Unstable 45 60 40 35 50 30 40 IAE [%] 25 20 IAE [%] 30 15 20 10 5 10 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 L/T L/T Left: set-point. Right: load disturbance Fractional controllers always perform better than their integer counterparts! Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 10 / 46

Selected publications F. Padula and A. Visioli. Tuning rules for optimal PID and fractional-order PID controllers. Journal of Process Control, 21(1):69-81, 2011. F. Padula and A. Visioli. Optimal tuning rules for proportional-integral-derivative and fractional-order proportional-integral-derivative controllers for integral and unstable processes. IET Control Theory and Applications, 6(6):776-786, 2012. F. Padula and A. Visioli. Set-point weight tuning rules for fractional-order PID controllers. Asian Journal of Control, 15(4):1-13, 2013. Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 11 / 46

The standard H control problem Let C be the set of stabilizing controllers Problem 1 where min K C Tzw T zw = G 11 + G 12 K(1 G 22 K) 1 G 21 Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 13 / 46

A Model-matching problem Theorem: Youla parametrization The set C of all stabilizing controllers K is: { X + MQ C = Y NQ : Q H } where P = MN 1 and M, N, X, Y satisfy the Bezout identity NX + MY = 1 using the previous result z = (T 1 QT 2 )w Problem 2 Find Q H such that the model-matching error T 1 QT 2 is minimized, where both T 1 and T 2 are in FRH using inner-outer factorization where T 1 T 2 Q = R X R = FRL, X = H Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 14 / 46

Nehari s theorem There exists a closest H -matrix X to a given L -matrix R, and R X = Γ R, where Γ R is the Hankel operator with symbol R R can be factorized as R = R 1 + R 2 with R 1 RL (integer!) unstable and analytic in the left half plane (antistable) and R 2 H and it holds that Γ R = Γ R1 R 1 is integer, thus Γ R has finite rank and can be computed by means of known techniques It can be shown that the optimal model-matching error is integer and real-rational......this is a Nevanlinna-Pick optimal interpolation problem! Each RHP zero of T 2 plays the role of an interpolation constraint to avoid internal instability: Q H, no zero/pole cancelations in the RHP Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 15 / 46

An optimal interpolation problem Theorem Consider the model-matching problem, the optimal model matching error is an all-pass in RH whose coefficients are completely determined by the interpolation constraints E o (z i ) = T 1 (z i ) i = 1,...,n d k E o (s) = ds s=zi dk T 1 (s) k = 1,...,m k ds s=zi k i 1; i = 1,...,n being m i the multiplicity of the ith RHP zero of T 2 and E o the optimal model-matching error The optimal interpolation error is integer and real-rational! The optimal Youla parameter Q is FRH The optimal controller is a fractional real-rational function Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 16 / 46

Selected publications F. Padula, S. Alcantara, R. Vilanova, and A. Visioli. H control of fractional linear systems. Automatica 49(17):2276-2280, 2013. Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 17 / 46

Weighted model-matching problem Using the IMC controller C(s) the closed loop transfer function has a very simple expression: T(s) = G n(s)c(s) the equivalent feedback controller can be easily recovered by means of: K(s) = C(s) 1 C(s)G n(s) a) feedback configuration, b) IMC configuration Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 19 / 46

Weighted model-matching problem By means of the IMC controller we can set up a weighted model-matching problem: C o (s) := min C(s) W(s)(M(s) C(s)Gn(s)) Find C(s) such as the -norm of the weighted difference between the nominal closed-loop transfer function and the desired closed-loop transfer function is minimized The role of the weighting function is of main concern: it allows the user to give more importance to certain frequency ranges (typically low frequencies) and less importance to other frequency ranges (typically high frequency). Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 20 / 46

Weighted model-matching problem 50 40 30 20 10 0 10 20 30 target plant nominal closed loop plant weighting function 0.14 0.12 0.1 0.08 0.06 40 50 60 10 4 10 2 10 0 nominal closed-loop frequency response (red), target frequency response (green), weighting function (blue) ω 0.04 0.02 0 10 4 10 2 10 0 10 2 10 4 ω model-matching error The model mismatch is bigger at those frequencies where the weighting function is smaller Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 21 / 46

Weighted model-matching problem The functions involved in the model-matching problem are chosen as follows: 1 Process model: 2 Nominal process transfer function G nt (s) = 3 Target closed-loop transfer function 4 Weighting function K 1+Ts α e Ls G n(s) = K 1 Ls 1+Ts α M(s) = 1 1+T ms λ W(s) = 1+zsµ s µ where T m, λ, z and µ are parameters to be selected Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 22 / 46

Suboptimal FOPID controller the equivalent feedback controller is K o (s) = 1 K ρ 1+Ts α γ sµ + T ms λ + T m(z + ρ γ ( )sλ+µ (1+ ρtm γl µ sµ ) 1+ n 1 k=m L k k n s n n 1 k=0 L k n s k n + m 1 k=n L k n s k n it has the same low-frequency behavior of a filtered FOPID controller, neglecting the last term (that for low frequencies tends to one) a suboptimal controller is obtained: L µ +z 1 (1+Ts α )(1+T m L K(s) = µ +T m s µ ) K( ρ + Tm) ρ γ s µ (1+T m γ +z ρ γ +Tm sµ ) It is a filtered FOPID controller in series form ) When µ = 1, K(s) = K o (s) In any case the suboptimal controller stabilizes the nominal system! Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 23 / 46

Robust stability Assume that the process belongs to a family F defined as: F = {G(s) = G nt (s)(1+ m(s)) : m(jω) < Γ(jω) } m(s) = (G(s) G nt (s))/g nt (s) is the uncertainty description Γ(jω) is a frequency dependent function that upper bounds the system uncertainty. Robust stability condition: Γ(s)T n(s) < 1 where T n(s) is the nominal closed-loop transfer function. Sufficient condition: T n(jω) < 1/Γ(jω) The right hand side of this inequality is usually a low-pass transfer function. It defines a robust stability boundary. Based on robustness and desired bandwidth tuning guidelines have been provided. Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 24 / 46

Example G nt,f(s) = 1 G(s) = s 2 + 0.4s+1 e 0.6s 1 1 1.05s 1.702 + 1 e 0.74s G nt,i(s) = 0.566s+1 e 0.9s 40 30 20 magnitude [db] 10 0 10 20 30 10 1 10 0 10 1 ω T m has been fixed to 1.5 (µ = 1) In the fractional case z can be reduced to 0 preserving robust stability. The selection of z can be done just to speed up or slow down the system response In the integer case it is necessary to set z = 10 to achieve robust stability Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 25 / 46

Example Step responses 1.5 process variable 1 0.5 0 0 10 20 30 40 50 time 1.5 control variable 1 0.5 0 0 10 20 30 40 50 time Step-responses with T m = 1.5L and µ = 1: integer model (dotted line z = 10) and the fractional model for different values of z (dash-dot line z = 10, dashed line z = 1 and solid line z = 0.1) Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 26 / 46

Selected publications F. Padula, R. Vilanova, and A. Visioli. H optimization based fractional-order PID controllers design. International journal of Robust and Nonlinear Control (in press, available online). Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 27 / 46

Command signal design where G(s) = Ḡ(s)e Ls and T(s) = K(s)G(s) 1+K(s)G(s) Find a command signal such that a smooth transition of the output between 0 and 1 is obtained within in finite amount of time τ satisfying a set of constraints on the control variable and its derivatives Given a sufficiently smooth desired output ȳ( ; τ) find the command signal r( ; τ) such that, for the τ parameterized couple r( ; τ), ȳ( ; τ), it holds that L[ȳ(t L;τ)] = T(s)L[r(t;τ))] Moreover, satisfy D i u(t;τ) < u i M, t > 0, i = 0, 1,...,l Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 29 / 46

Desired output Desired output,τ-parameterized transition polynomial ȳ(t;τ) C (n) 0 if t < 0 (2n+1)! ȳ(t;τ) := n ( 1) n r τ r t 2n r+1 n!τ 2n+1 r=0 if 0 t τ r!(n r)!(2n r+1) 1 if t > τ smooth, through the parameter n the regularity of the transition polynomial can be arbitrarily selected; monotonic; finite time transition. Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 30 / 46

Dynamic inversion Consider the transfer function H(s) of Σ: H(s) = b(s) a(s) = where ν is the commensurate order. ρ = (p m)ν the relative order of Σ. m k=0 b ks kν s pν + p 1 k=0 a ks kν Define the set of all the cause/effect pairs associated with Σ: { B := (u( ), y( )) P c P c : m k=0 b kd kν u = D pν y + } p 1 k=0 a kd kν y Consider the system H(s), given the desired transition polynomial (i.e, C (k) for some k N) ȳ(t;τ), find the input u(t;τ) such as the τ parameterized couple (u( ;τ), ȳ( ;τ)) B, ȳ(0;τ) = 0 and ȳ(t;τ) = 1 t τ, moreover satisfy D i u(t;τ ) < u i M, t > 0 i = 0, 1,...,l Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 31 / 46

Dynamic inversion Using Laplace transform, the inversion is algebraic in the frequency domain: 1 U(s;τ) = H (s)ȳ(s;τ) Σ is assumed to be commensurate here thus the following techniques apply: Polynomial division Partial fraction expansion and the inverse system can be decomposed as follows H 1 (s) = γ n ms ρ +γ n m 1 s ρ ν + +γ 1 s ν +γ 0 + H 0 (s) H 0 (s), zero dynamics of Σ, strictly proper m g i H 0 (s) = (s ν λ i ) k i+1 Inverse transforming... i=1 Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 32 / 46

Dynamic inversion...the zero dynamics is the summation of Mittag-Leffler functions m g i m η 0 (t) = k i! ε g i k i (t,λ i ;ν,ν) = k i! t k iν+ν 1 di k E d(λ i t ν ) k i ν,ν(λt ν ) Proposition i=1 If n > [ρ]+1+l, for τ sufficiently large then i=1 u(t;τ) = γ n md ρ ȳ(t;τ)+γ n m 1 D ρ ν ȳ(t;τ)+ +γ 1 D ν ȳ(t;τ)+γ 0 ȳ(t;τ)+ t 0 η 0(t ξ)ȳ(ξ;τ)dξ The convolution integral becomes t η 0 0(t ξ)y(ξ;τ)dξ = m g i i=1 k i! [ε ki (t,λ i ;ν, 2n r + 2+ν) + [ (2n+1)! n!τ 2n+1 n r=0 ( 1) n r τ r (2n r + 1)! r!(n r)!(2n r+1) 0 ( ) if 0 t τ 2n r+1 2n r + 1 j=0 (2n r + 1 j)!τ j j { ε ki (t τ,λ i ;ν, 2n r + 2 j +ν) ] if t > τ 0 if 0 t τ ε ki (t τ,λ i ;ν, 1+ν) if t > τ Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 33 / 46

Command signal synthesis, again The open loop system is first inverted obtaining r ol (t;τ) via dynamic inversion of the delay-free open loop transfer function K(s)Ḡ(s) a delayed correction term is then added to avoid the delayed feedback effect finally, the command signal is computed r c(t;τ) = ȳ(t L;τ) r(t;τ) = r ol (t;τ)+r c(t;τ) under the existence condition n [ρ Ḡ ]+1+l Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 34 / 46

Selected publications F. Padula and A. Visioli. Inversion-based feedforward and reference signal design for fractional constrained control systems. Automatica. 50(8):2169-2178, 2014. Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 35 / 46

Problem formulation So far we have introduced two results: robust controller design command signal synthesis now we want to put the previous results together. Consider the family of plants { K F = G(s) = e Ls : T K [K min, K max], s λ + 1 } T [T min, T max], λ [λ min,λ max], L [L min, L max] and the nominal system G(s) = K Ts λ + 1 e Ls whose parameters are the mean values of the corresponding uncertainty intervals Define extremal system G i (s) i = 1,...,16 for the family F each system obtained with any possible combination of the extremal values of the uncertainty intervals. Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 37 / 46

Problem formulation Given a unity feedback loop We want to design a controller K(s) and a command signal r(t) to satisfy: robustness control variable limitation overshoot limitation settling time minimization for the whole family of plantsf Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 38 / 46

Optimal feedback/feedforward control define the worst-case settling time (at a given percentage) {τ, T m, z} is a set of tuning parameter t s,wc(τ, T m, z) := max i=1,...,16 t s,i(τ, T m, z), Min-max problem subject to 1 (Robust stability) Γ(s)T n(s) < 1; min ts,wc(τ, Tm, z) τ,t m,z 2 (Maximum overshoot) max y i (t;τ, T m, z) < y f (1+O max), i = 1,...,16; 3 (Maximum control variable) max u i (t;τ, T m, z) < U max, i = i,...,16; where (u i ( ), y i ( )) is the input-output couple for the ith extremal system, y f is the set-point value, O max > is the maximum allowable overshoot and U max > 0 the maximum acceptable control variable. Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 39 / 46

Optimal feedback/feedforward control only a constraints on the control variable is imposed (l = 0) the existence condition for both control signal and command signal reduces to moreover it can be shown Lemma n [ρ K Ḡ ]+1 There always exists a couple of parameters T m, z such that the optimal controller K o (s) stabilizes the family F provided that the parametric uncertainty over the process dc-gain K is lower than 1, i.e., K max K K < 1 Theorem The min-max problem is solvable provided that and K max K K < 1 U max > y f G i (0), i = 1,...,16 Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 40 / 46

Simulation results G(s) = 1 s 1.5 + 1 e s 1 uncertainty of ±10% over the plant s parameters 2 settling time at 2% 3 unitary set-point value y f = 1 4 maximum control variable of U max = 1.5 5 maximum overshoot of O max = 0.2 Solving procedure Numerically compute the robust stability boundary by gridding the process uncertainty Obtain Γ(jω) by upper bounding the computed uncertainties for each frequency Select β = max[1,λ] = 1.5 and a transition polynomial with regularity n = 3 to satisfy the existence condition Numerically (genetic algorithm) solve the min-max problem Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 41 / 46

Example 20 10 2 magnitude [db] 0 10 20 1.5 1 0.5 30 0 40 10 2 10 1 10 0 10 1 10 2 frequency 0 2 4 6 8 10 12 14 16 time process variable 1 0.5 0 0 2 4 6 8 10 12 14 16 time The obtained optimal parameters are T m = 3.5469, z = 3.4155 and τ = 6.3122 control variable 1 0.5 0 0 2 4 6 8 10 12 14 16 time the optimal worst-case settling time is t s,wc = 13.42 Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 42 / 46

Simulation results Results optimizing the settling time but using a step command signal... magnitude [db] 20 15 10 5 0 5 10 15 20 10 2 10 1 10 0 10 1 10 2 frequency process variable process variable 1 0.5 0 0 2 4 6 8 10 12 14 16 time 1 0.8 0.6 0.4 0 2 4 6 8 10 12 14 16 time The obtained optimal parameters are T m = 4.8747, z = 5.6705 and τ = 9.7502 the optimal worst-case settling time is t s,wc = 18.11...the combined feedback/feedforward optimization performance improvements is 26 %! Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 43 / 46

Conclusions: ROBUST PERMFORMANCE a complete set of constrained optimal time-scale invariant tuning rules for PID and FOPID controllers, for stable unstable and integral processes the solution of the scalar standard H control problem for fractional SISO LTI systems an H model-matching robust design methodology suitable for both monotonic and nonmonotonic dynamics the solution of a constrained optimal input-output dynamic inversion problem for fractional LTI systems an inversion-based feedforward signal design for fractional control loops a combined feedback/feedforward control design technique to cope with uncertainty in an effective way... Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 45 / 46

Fabrizio Padula (University of Brescia) Robust Fractional Control Cape Town, 23/8/2014 46 / 46