Disturbance decoupling by measurement feedback

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1 Preprints of the 19th Word Congress The Internationa Federation of Automatic Contro Disturbance decouping by measurement feedback Arvo Kadmäe, Üe Kotta Institute of Cybernetics at TUT, Akadeemia tee 21, 12618, Tainn, Estonia (e-mai: IRCCyN, Ecoe Centrae de Nantes, 1 rue de a Noe, Nantes Cedex 3, France Abstract: The paper addresses the disturbance decouping probem for MIMO discretetime noninear systems. A sufficient conditions are derived to sove the probem by dynamic measurement feedback, i.e. the feedback that depends on measurabe outputs ony. The soution to the disturbance decouping probem, described in this paper, is based on the input-output inearization, which is used to inearize certain functions. Two exampes are added to iustrate the resuts. 1. INTRODUCTION The disturbance decouping probem (DDP is one of the fundamenta probems in contro theory. There are a ot of papers, that sove the probem by state feedback, see Aranda-Bricaire and Kotta [2001, 2004], Fiegner and Nimeier [1994], Grizze [1985], Monaco and Normand-Cyrot [1984] for noninear discrete-time systems and Conte et a. [2007], Isidori [1995], Nimeier and van der Schaft [1990] for noninear continuous-time systems. For output or measurement feedback, the probem acks the fu soution. The first paper that appied measurement feedback to sove the DDP was Isidori et a. [1981], where sufficient sovabiity conditions were given for continuous-time systems, and the feedback that was used was restricted to the so-caed pure dynamic measurement feedback. In Kadmäe et a. [2013], simiar resuts as in Isidori et a. [1981] were given for discrete-time systems (though, more genera feedback was used, using agebraic approach (attice theory, that is abe to address aso certain type of non-smooth systems. A more genera feedback, where the state of the compensator is not a function of the state of the system, but can be chosen independenty of it, was used in Xia and Moog [1999] and Kadmäe and Kotta [2012b], where sufficient conditions for the sovabiity of the probem by dynamic measurement feedback were given for continuous- and discrete-time SISO systems, respectivey. For static measurement feedback soutions see Pothin et a. [2002] and Kadmäe and Kotta [2012a]. In this paper, we extend the resuts of Kadmäe and Kotta [2012b] for MIMO discrete-time systems 1. However, the extension is not direct since we reax certain integrabiity conditions. The resut of this paper depends heaviy on the soution of the input-output inearization probem, see Kadmäe and Kotta [2014]. We show that a feedback This work was supported by the European Union through the European Regiona Deveopment Fund, by the ETF grant nr and by the Estonian Research Counci, persona research funding grant PUT Note that there are no soutions for MIMO continuous-time systems. that inearizes certain functions aso soves the disturbance decouping probem. It is our conecture that our resuts can be generaized directy for continuous-time systems, though the computations are different because the differentia operator and forward-shift operator act differenty on the set of functions. 2.1 Agebraic toos 2. PRELIMINARIES In this paper, x stands for x(t and for k 1, x [k] stands for kth-step forward time shift of x, defined by x [k] := x(t+ k. Simiar notations are used for the backward shift and the other variabes. Consider a noninear system, described by the equations = f(x, u, w y = h (x (1 z = h(x, where x X R n is the state, u U R m is the controed input, w W R ι is the disturbance input, y Y R p is the controed output and z Z R q is the measured output. It is assumed that the functions f, h and h are meromorphic. Aso, we assume, that the system (1 is submersive, meaning that genericay, i.e. everywhere except on a set of measure zero, [ f ] rank = n. (2 (x(t, u(t Aso, throughout the paper it is assumed that i = 1,..., p. Let K denote the fied of meromorphic functions which depend on finite number of variabes from the set {x, u [k], w [k] ; k 0}. Introduce the forward-shift operator δ : K K, defined by the equations (1; in particuar δx := f(x, u, w and for k 0, δu [k] := u [k+1], δw [k] := w [k+1]. Moreover, Copyright 2014 IFAC 7735

2 19th IFAC Word Congress δφ(x, u, w,..., u [k], w [s] := φ(f(x, u, w, u [1], w [1],..., u [k+1], w [s+1] for φ K. Under the submersivity assumption (2, the pair (K, δ is a difference fied. In genera, this difference fied is not inversive, i.e. the operator δ is not inversive in K. However, one can aways find an overfied K of K, caed the inversive cosure of K, which is inversive. See Aranda-Bricaire et a. [1996], Aranda-Bricaire and Kotta [2004] for detais how to compute K. From now on, we assume that difference fied (K, δ is inversive and denote it by K. Note that then there exists an operator δ 1, which is caed backward-shift operator. By δ k and δ k we denote the k-fod appication of operators δ and δ 1, respectivey. Define the vector space of one-forms as E = span K {dφ φ K}. Aso, define X := span K {dx}, W := span K {dw [k], k 0}. The operators δ and δ 1 are extended to E by the rues δ ( a dφ = δ(a d(δφ δ 1( a dφ = δ 1 (a d(δ 1 φ, where a, φ K. A one-form ω is caed exact, if it is a differentia of some function ξ K, i.e ω = dξ. Let y = (y 1,..., y p be the controed output vector of the system (1. The reative degree r i of an output y i with respect to input u is defined by r i := min{k N dy [k] i / X + W}. If there does not exist such integer k, then set r i :=. In genera, a one-form ω is a inear combination over K of finite number of standard basis eements of E, i.e. {dx, du [k], dw [k] ; k 0}. However, it is often possibe to find a ineary independent set of exact one-forms with ess eements than those basis eements of E in terms of which ω can be expressed. Definition 1. A number γ N is caed the rank of a oneform ω, if γ is minima number of ineary independent exact one-forms necessary to express a one-form ω. The set of these exact one-forms is caed the basis of ω. Next we define two subspaces Ω and Ω u of X in the foowing way: and Ω = {ω X k N : (3 δ k ω span K {dx, dy [ri] i,..., dy [ri+k 1] Ω u = {ω X k N : δ k ω span K {dx, du, (4..., du [k 1], dy [ri] i,..., dy [ri+k 1] By definitions, Ω Ω u. For SISO systems Ω = Ω u, since du can be written as a inear combination of dx and dy [r], where r is the reative degree of output y with respect to input u. Foowing emmas give procedures for computing subspaces Ω and Ω u. Lemma 1. Kadmäe and Kotta [2012a] The subspace Ω may be computed as the imit of the foowing agorithm: Ω 0 = X (5 Ω k+1 = {ω Ω k δω Ω k + span K {dy [ri] Lemma 2. The subspace Ω u may be computed as the imit of the foowing agorithm: Ω 0 = X (6 Ω k+1 = {ω Ω k δω Ω k + span K {du, dy [r i] Suppose Ω = span K {dθ 1,..., dθ s }. Next define the k- time forward-shift of subspace Ω eementwise by Ω [k] = span K {dθ [k] 1,..., dθ[k] s } for k Probem statement The DDP by measurement feedback can be stated as foows. Find a dynamic measurement feedback of the form η [1] = F (η, z, v (7 u = H(η, z, v, where η R ρ and v R m, such that controed outputs y i of the cosed-oop system do not depend on disturbance w at any time instant, i.e. dy [k] i span K {dx, dη} k < r i dy [k] i span K {dx, dη, dv,..., dv [k r i] } k r i, where r i is the reative degree of output y i of the cosed oop system with respect to u. Lemma 3. If the reative degrees r i of outputs y i with respect to u are finite then system (1 is disturbance decouped if and ony if dy [r i] i Ω u + span K {du}. (8 Proof: Necessity. Since r i is the reative degree of output y i with respect to input u, m dy [ri] i = ω 0 + b i, du, =1 where b i, K and ω 0 span K {dx}. We show that ω 0 Ω u. Assume contrary that ω 0 / Ω u. Then there exists k N such that δ k ω 0 / span K {dx, du,..., du [k 1] }. This means that one-form ω 0 is not disturbance decouped and thus y i aso is not disturbance decouped. This is a contradiction and thus ω 0 Ω u. Sufficiency. If (8 is true, then by Lemma 2 Ω [1] u Ω u + span K {du}. Thus, Ω u is invariant with respect to the system dynamics and since dy Ω u, the system is disturbance decouped. 3. MAIN RESULTS 3.1 Input-output inearization Since our soution of the DDP depends on the soution of the input-output (i/o inearization probem, we start with the statement of the i/o inearization probem. For 7736

3 19th IFAC Word Congress more information, see Kadmäe and Kotta [2014]. In this section, et = 1,..., q. Consider a discrete-time muti-input muti-output (MIMO noninear system, described by the difference equations z [n ] = Φ (z τ,..., z [n τ ] τ, u,..., u [n 1] (9 for τ = 1,..., q, = 1,..., m, where Φ are supposed to be meromorphic functions of their arguments and the indices in (9 satisfy the reations n 1 n 2 n q, n τ < n τ n τ < n, τ (10 n τ n, τ >. Aso, we assume, that system (9 is submersive, i.e. the map Φ = (Φ 1,..., Φ q T satisfies genericay the condition [ Φ ] rank = q, (z, u where z = (z 1,..., z q and u = (u 1,..., u m. In this section, et K be the fied of meromorphic functions in variabes z, u and a finite number of their independent forward shifts, i.e. variabes from the set, u [k] ; k 0}. Aso, et E k :=, du,..., du [k 1] } for any k N and r denotes the reative degree of the output z with respect to the input u. C = {z,..., z [n 1] span K {dz,..., dz [k 1] Given a discrete-time MIMO noninear contro system of the form (9, we say that system (9 is i/o inearized by feedback (7, if the differentias of the input-output equations of the cosed-oop system satisfy the reations dz [n ] span R {dz [n τ ] τ,..., dz τ, dv} (11 for τ = 1,..., q. In case when span R {dv}, system (9 is said to be stricty i/o inearized. dz [n ] We say that functions φ (z,..., z [s 1], u,..., u [s 1] are inearizabe (stricty inearizabe if the system z [s] = φ (z,..., z [s 1], u,..., u [s 1] is i/o inearizabe (stricty i/o inearizabe. Let ω := dz [n ] mod span R {dz [n τ ] τ,..., dz τ }, where τ = 1,..., q. 2 For sovabiity of the i/o inearization probem, it is necessary that 3 ω E n r +1, (12 since otherwise noninearities appear before the input u starts to affect the output y i. First, et ω, = 1,..., q, be the basis eements of span R { ω }. In the rest of this section assume that, τ = 1,..., q and = 1,..., m. Let σ be such that ω E σ. Next, define the one-forms 2 In the case of strict inearizabiity, one has to take ω := dz [n ]. 3 Note that if r = 1, then the condition (12 is aways satisfied. ω,λ span K {dz [σ λ],..., dz [σ 1], du [σ λ],..., du [σ 1] }, where λ = 1,..., σ 1, such that ω ω,λ E σ λ (13 and ω,σ := ω. (14 It means that the one-forms ω,λ depend on the (σ λth and higher order terms of the one-forms ω. Let γ,λ be the rank of a one-form ω,λ for λ = 1,..., σ. Then there exist γ,λ functions ϕ k,λ (z[σ λ],..., z [σ 1], u [σ λ],..., u [σ 1] such that ω,λ span K {d ϕ 1,λ,..., d ϕ γ,λ,λ }. Finay, define the function ϕ k,λ as a (σ λ step backward shift of the function ϕ k,λ, i.e. ϕ k,λ := (δ 1 σ λ ϕk,λ = δ λ σ ϕk,λ for λ = 1,..., σ and k = 1,..., γ,λ. Theorem 1. Kadmäe and Kotta [2014] Under the assumption (12 the system (9 is input-output inearizabe by dynamic output feedback of the form (7 if and ony if ϕ k,λ dim(span K {dϕ k,λ} = rank K (u, δϕ k,λ, (15 for λ = 1,..., σ, λ = 1,..., σ 1, k = 1,..., γ,λ and functions ϕ 1,σ are independent from a the other functions. 3.2 Sufficient conditions for sovabiity of the DDP The theorem beow gives sufficient sovabiity conditions of the DDP by dynamic measurement feedback. Theorem 2. Under the assumption that a the reative degrees r i of outputs y i with respect to u are finite, the DDP by dynamic measurement feedback is sovabe for system (1, if (i there exist one-forms ω i span K {dz,..., dz [s 1], du,..., du [s 1] } with rank ω i =: γ i such that ω i Ω + + Ω [s 1] for some s 1; (ii for ω i = γ i =1 β i,dα i, (z,..., z [s 1], u,..., u [s 1] from (i, the functions α i, are stricty inearizabe by dynamic measurement feedback. dy [r i+s 1] i Proof: We show that the feedback that inearizes stricty the functions α i, in (ii, soves the disturbance decouping probem. Note that the reative degree of y i with respect to input v is r i = r i + s 1. Since for the cosed-oop system ω i span K {dv}, one gets from (i that dy [ ri] i Ω + + Ω [s 1] + span K {dv}. Next, we show that Ω = Ω + + Ω [s 1], where Ω is the subspace Ω for the cosed-oop system. From the definition of the subspace Ω, Ω + + Ω [s 1] span K {dx, dy [ri] i,..., dy [ri+s 2] i }. 7737

4 19th IFAC Word Congress Since r i = r i + s 1, then in the cosed-oop system Thus, Ω + + Ω [s 1] span K {dx, dη}. Ω + + Ω [s 1] = { ω span K {dx, dη} k N : ω [k] span K {dx, dη, dy [ri+s 1] i,..., dy [ri+s k 2] i }} = Ω. The ast equaity comes from the definition (3 of the subspace Ω. Since Ω Ω u, then by Lemma 3, system (1 is disturbance decouped. Coroary 1. For SISO systems, the conditions of Theorem 2 are necessary and sufficient. Proof: It remains to prove the necessity. By Lemma 3, since the cosed-oop system is disturbance decouped, dy [ r] Ω u + span K {dv}, (16 where r is the reative degree of y in the cosed-oop system with respect to the new input v and Ω u is the subspace Ω u for the cosed-oop system. We choose s 1 such that r = r + s 1. Since for singe input systems Ω = Ω u, one can show, as in the proof of Theorem 2, that Ω u = Ω + + Ω [s 1]. Now, one can find the one-form ω span K {dv}, with rank 1, such that we get from (16 dy [r+s 1] ω Ω + + Ω [s 1]. Assume that ω = βdα for some functions β, α K. Ceary, the feedback that soves the disturbance decouping probem, aso inearizes stricty function α, since for the cosed-oop system ω span K {dv}. Thus conditions (i and (ii of Theorem 2 are satisfied. Note that if we take s = 1 in Theorem 2, we get sovabiity conditions for DDP by static measurement feedback. In this case the strict inearizabiity of functions α i, means that system of equations α i, (z, u = v µ, µ = 1,..., m, is sovabe in u. 4. EXAMPLES Exampe 1. Consider the system 1 = u 1 2 = x 3u 3 + x 2 x 4 u 2 x 1 3 = u 2 4 = x 1w (17 5 = u 1u 2 x 4 + x 2 y 1 = x 2 y 2 = x 5 z = x 4. First, note that the reative degrees r 1 and r 2 of outputs y 1 and y 2 with respect to u are both 1. One can aso computes subspaces Ω = span K {dx 2, dx 5 } and Ω u = span K {dx 1, dx 2, dx 3, dx 5 }. Ceary, dy i / Ω u + span K {du} for i = 1, 2. Therefore, system (17 is not disturbance decouped. To find the one-forms ω i, defined in (i of Theorem 2, we cacuate dy [ri+si 1] i for s i = 1, 2,..., unti dy [r i+s i 1] i Ω + + Ω [s i 1] + span K {dz,..., dz [si 1], du,..., du [si 1] }. For system (17, we cacuate dy [1] 1 = u 3dx 3 dx 1 + zu 2 dx 2 + x 3 du 3 + x 2 d(zu 2 Ω + span K {du, dz} dy [1] 2 = dx 2 + d(u 1 u 2 z Ω + span K {du, dz}. Thus, s 2 = 1. Compute Ω + Ω [1] = span K {dx 2, dx 5, d d 5 }. Now, dy [2] 1 = d(u[1] 3 u 2 u 1 + z [1] u [1] 2 dx[1] d(z[1] u [1] 2 Ω + Ω [1] + span K {du, du [1], dz, dz [1] }, meaning that s 1 = 2. Next, we can choose the one-forms ω i as ω 1 = d(u [1] 3 u 2 u d(z[1] u [1] 2 ω 2 = d(u 1 u 2 z. Obviousy, rank ω 1 = 2 and rank ω 2 = 1. It remains to check whether the functions α 1,1 = u [1] 3 u 2 u 1, α 1,2 = z [1] u [1] 2 and α 2,1 = u 1 u 2 z are inearizabe. One can find, that the dynamic feedback η [1] 1 = z(η 2v 1 + v 3 η 2 2 η [1] 2 = v 2 2, u 1 = v 3 η 2 (18 u 2 = η 2 z u 3 = η 1, inearizes functions α 1,1, α 1,2, α 2,1 and aso decoupes disturbances from the controed outputs y 1 and y 2. Reay, in the cosed-oop system y [2] 1 = v v 2 y [1] 2 = v 3 + x 2 and since Ω u = span K {dx 1, dx 2, dx 5, d 2, dη 2}, the conditions of Lemma 3 are satisfied. This means that the cosed-oop system is disturbance decouped. Exampe 2. The next exampe is taken from Kadmäe et a. [2013]. The system in Figure 1 is a typica subsystem in many appications and consists of inear subsystems d W 1 = k 1 /(1 + T 1 dt, W d 2 = k 2 /(1 + T 2 dt, W 3 = d k 3 T 3 dt /(1 + T 3 d dt, W 4 = k 4 / d dt and saturation operation, { x, if x x0 σ(x = x 0 sign x, if x > x

5 19th IFAC Word Congress u y k 5 W 1 σ W 2 W z W 3 Fig. 1. System with saturation operation. that corresponds to the ampifier. Here k 1,..., k 5, are rea coefficients, T 1, T 2 are certain time constants and T 3 may be considered as unknown function of disturbance w because of the unexpected changes in the feedback oop. After the Euer discretization, one gets a system described by the equations: 1 = k 4x 2 + x 1 2 = k 2 T 2 σ(x 3 + x 2 (1 1 T 2 3 = 1 T 1 (k 1 k 5 (u x 1 k 1 k 3 (x 2 x 4 + x 3 (1 1 T 1 4 = 1 T 3 (w x 2 + x 4 (1 1 T 3 (w (19 y = x 1 z = k 3 (x 2 x 4. In Kadmäe et a. [2013], a dynamic measurement feedback is found that soves the DDP for system (19. However, note that the probem statement of Kadmäe et a. [2013] is somewhat different from that in this paper. Namey, in Kadmäe et a. [2013] the state η of a compensator is assumed to be a function of state x, i.e. η = ϕ(x. Beow we sove the DDP for system (19 using the method described in this paper. Since our method assumes a functions to be meromorphic, we take σ(x 3 = x 3 in (19, i.e. x 3 x 3,0 for some x 3,0 R. Note that if x 3 > x 3,0, one can show by Lemma 3 that the system (19 is aready disturbance decouped. The reative degree of output y with respect to input u is r = 3. Next, we have to find, by Lemma 1, the subspace Ω. Compute Ω = Ω 1 = span K {dx 1, dx 2, dx 3 }. Since y [3] = ( 1 k 1k 2 k 4 k 5 x1 + ( 3k 4 3k 4 + k 4 T 1 T 2 T 2 T2 2 x2 + ( 3k 2 k 4 k 2k 4 T 2 T2 2 k 2k 4 x3 + k 1k 2 k 4 ( k5 u z, T 1 T 2 T 1 T 2 one can choose ω = k 5 du dz. Then condition (i of Theorem 2 is satisfied for s = 1. The rank of the one-form ω is obviousy 1 and α = k 5 u z. By taking v = k 5 u z, one gets u = 1 k 5 (v + z. This static measurement feedback soves the DDP for system (19. The reason, why we get static soution in this paper, but dynamic soution in Kadmäe et a. [2013], is that the seection of one-form ω, in Theorem 2, is more restricted, than the seection of certain function, based on which the soution is computed, in Kadmäe et a. [2013]. In the atter case the choice of a function that eads to static soution is not obvious. 5. CONCLUSION This paper addressed the DDP by dynamic measurement feedback. Using agebraic methods, sufficient sovabiity conditions were given. For SISO systems, the conditions are aso necessary. The key point of the soution is inearization of certain functions by measurement feedback. It is shown that this feedback aso soves the disturbance decouping probem. The future work wi incude finding necessary and sufficient sovabiity conditions for MIMO systems. Two exampes were given to iustrate the theory. REFERENCES E. Aranda-Bricaire and Ü. Kotta. Generaized controed invariance for discrete-time noninear systems with appication to the dynamic disturbance probem. IEEE Trans. Autom. Contro, 46: , E. Aranda-Bricaire and Ü. Kotta. A geometric soution to the dynamic disturbance decouping for discrete-time noninear systems. Kybernetika, 49: , E. Aranda-Bricaire, Ü. Kotta, and C. H. Moog. Linearization of discrete-time systems. SIAM J. Contro and Optimization, 34(6: , G. Conte, C.H. Moog, and A.M. Perdon. Agebraic Methods for Noninear Contro Systems. Theory and Appications. Springer, T. Fiegner and H. Nimeier. Dynamic disturbance decouping of noninear discrete-time systems. In Proc. of the 33rd IEEE Conf. on Decision and Contro, voume 2, pages , J.W. Grizze. Controed invariance for discrete-time noninear systems with an appication to the disturbance decouping probem. IEEE Trans. Autom. Contro, 30: , A. Isidori. Noninear contro systems. Springer, London, A. Isidori, A.J. Krener, C. Gori-Giorgi, and S. Monaco. Noninear decouping via feedback: A differentia gemetric approach. IEEE Trans. Autom. Contro, 26: , A. Kadmäe and Ü. Kotta. Disturbance decouping of muti-input muti-output discrete-time noninear systems by static measurement feedback. Proc. of the Estonian Academy of Sciences, 61(2:77 88, 2012a. A. Kadmäe and Ü. Kotta. Dynamic measurement feedback in discrete-time noninear contro systems. In Proc. of the 2012 American Contro Conference: Fairmont The Queen Eizabeth, Montrea, Canada, June 27-29, 2012, pages Montrea, 2012b. A. Kadmäe and Ü. Kotta. Input-output inearization of discrete-time systems by dynamic output feedback. European Journa of Contro, 20(2:73 78, A. Kadmäe, Ü. Kotta, A. Shumsky, and A. Zhirabok. Measurement feedback disturbance decouping in discrete-time noninear systems. Automatica, 49(9: , S. Monaco and D. Normand-Cyrot. Invariant distributions for discrete-time noninear systems. Systems and Contro Letters, 5: , H. Nimeier and A.J. van der Schaft. Noninear dynamica contro systems. Springer, New York,

6 19th IFAC Word Congress R. Pothin, C.H. Moog, and X. Xia. Disturbance decouping of noninear miso systems by static measurement feedback. Kybernetika, 38: , X. Xia and C.H. Moog. Disturbance decouping by measurement feedback for siso noninear systems. IEEE Trans. Autom. Contro, 44: ,