Enhanced Finite Spectrum Assignment with disturbance compensation for LTI systems with input delay

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Journal of the Franklin Institute 355 (2018) 3541–3566 www.elsevier.com/locate/jfranklin

Enhanced Finite Spectrum Assignment with disturbance compensation for LTI systems with input delay Tito L.M. Santos∗, Taniel S. Franklin Department of Electrical and Computer Engineering (DEEC), Federal University of Bahia (UFBA), R. Aristides Novis, N. 02, Salvador, Bahia CEP 4210-630, Brazil Received 17 October 2017; received in revised form 23 February 2018; accepted 6 March 2018 Available online 13 March 2018

Abstract This paper presents a Finite Spectrum Assignment (FSA) with a generalized feedforward control for Linear Time-Invariant (LTI) systems with input delay and bounded unmeasured disturbances. A novel two-layer feedforward strategy is proposed in order to deal with matched and unmatched disturbances. The proposed control law is based on a filtered disturbance estimator and a generalized feedforward compensation which can be applied to any Artstein based predictor. An optimization design procedure is presented to improve disturbance attenuation properties in the presence of band-limited disturbances. The conditions to achieve disturbance rejection are also shown to deal with deterministic disturbance models. Furthermore, the proposed solution can be used to define either continuous-time or discrete-time control algorithms. Two case studies are presented to illustrate the benefits of the new approach. © 2018 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction Input-delay compensation (also known as dead-time compensation) has been investigated since Smith’s seminal article [28] due to the importance of time-delay effect in real control problems [19]. The reduction method is recognized as a natural way to perform dead-time ∗

Corresponding author. E-mail address: [email protected] (T.L.M. Santos).

https://doi.org/10.1016/j.jfranklin.2018.03.003 0016-0032/© 2018 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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compensation based on a state-space model [1]. In this context, Finite Spectrum Assignment (FSA) is known as the combination of the reduction method with a state-feedback control law [13,18]. Despite the generality of these dead-time compensation strategies, the main concept can be explored with different types of controllers [20] and disturbance compensation approaches [22]. Unmeasured disturbance attenuation is an important closed-loop requirement for time-delay systems [3,6,15,21,27]. Disturbance compensation ideas such as active disturbance rejection control [8] and extended state observer control [30] have received significant attention recently. Several strategies have been proposed to improve FSA with respect to disturbance attenuation properties as discussed in [6,14,22,27]. For discussion purposes, these modified approaches can be divided into two special classes by considering LTI (Linear Time-Invariant) systems with input delay and bounded disturbances: (i) compensators for systems with matched disturbances [6,11,27], and (ii) compensators for systems with arbitrary additive disturbances (matched and unmatched cases) [10,14,22]. This classification is useful since matched disturbances can be directly handled by the control action without any kind of gain gap. As a consequence, matched disturbance assumption greatly simplifies feedforward design problem. However, most of the real disturbances are not structurally matched with the control input. As previously pointed out, modified predictors have been proposed to improve disturbance rejection and/or attenuation properties. Optimal design of the output feedback problem with the FSA strategy was studied in [5]. The reduction method with a sliding mode controller was proposed in [20]. Sliding mode controller is commonly used to attenuate or reject disturbance effect [9,12]. Modified dead-time compensators have been proposed to attenuate disturbance effect in recent works [14,22,27]. Matched disturbances have been treated in [6,27] with a feedforward action. Feedforward control with probabilistic information of the time-varying delay was the main issue of concern in [34]. The problem of model uncertainty was considered in [17] with an adaptive approach. However, these works are based on specific dead-time compensation strategies. An interesting related work has recently proposed a discrete-time compensator to deal with output-feedback problem with a static feedforward action [7]. However, dynamic effect of the feedforward action was not considered and the general problem with matched and unmatched components was not analyzed due to its scope [7]. In this context, feedforward action has been used is related works, but a general feedforward framework has not been considered in the context of reduction method and FSA. A general framework would be useful since the main properties of the predictors may be preserved, but disturbance attenuation performance can be improved. This paper proposes a two-layer feedforward compensation strategy for LTI systems with bounded disturbances and single input-delay. The first layer is used to improve unmeasured matched disturbance attenuation (or rejection) while the second-layer is proposed to deal with the unmatched disturbances effect. This generalized solution can be applied to any Artsteinbased method (reduction method) which is one of the main contributions of this work since the discussion is not limited to a specific predictor. Ideal disturbance rejection is analyzed in order to define the desired design objective. Deterministic disturbance signals such as steplike, ramp-like, and sinusoidal disturbances can be rejected by considering this framework. An optimization-based design is proposed for disturbance attenuation of band-limited signals by following the ideas of [27]. Moreover, the discrete-time counterpart is also presented since implementation aspects have been treated as important issues in the context of real applications with long time-delays [6,32].

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The work is organized as follows: problem statement is presented in Section 2, the proposed two-layer feedforward action is proposed in Section 3, discrete-time case is discussed in Section 4, illustrative simulation examples are shown in Section 5, and the concluding remarks are drawn in Section 6. 2. Problem statement and preliminaries Consider a continuous-time LTI system with single-input delay, also known as dead-time, described as follows: x˙(t ) = Ax(t ) + Bu(t − h) + Bw w(t ),

(1)

u(t ) = φ(t ), t ∈ [−h, 0),

(2)

x(0) = x0 ,

(3)

where x0 and φ(t) for t ∈ [−h, 0) define the initial condition of the system, x(t ) ∈ Rn represents the state vector, u(t ) ∈ Rq is the control signal vector, and w(t ) ∈ Rm describes the unknown additive disturbance. In this kind of linear system, d (t ) = Bw w(t ) is called matched disturbance if BB+ d (t ) = d (t ) which implies that BB+ Bw = Bw where B+ denotes the Moore-Penrose pseudo-inverse. Otherwise, d (t ) = Bw w(t ) with BB+ Bw = Bw represents an unmatched (or mismatched) disturbance. If Bw = B as in [27], the matched disturbance condition directly holds which greatly simplifies disturbance attenuation problem. Assumption 1. A and B are constant and known matrices and the pair (A, B) is controllable. Assumption 2. The unknown disturbance signal is bounded by1 ||w(t )|| ≤ Wm , ∀t ≥ 0. Assumption 3. The unknown disturbance signal is locally integrable. Assumption 4. The input delay, namely h > 0, is constant and known. Remark 1. The assumption that the delay is known can be replaced by a constant unknown delay (hr ) with bounded control input. Note that x˙(t ) = Ax(t ) + Bu(t − hr ) + Bw wr (t ) can be rewritten as x˙(t ) = Ax(t ) + Bu(t − h) + w(t ), where w(t ) = Bw wr (t ) + B(u(t − hr ) − u(t − h)). In this case, also observe that BB+ w(t ) = BB+ Bw wr (t ) + B(u(t − hr ) − u(t − h)) because BB+ B = B by definition. Therefore, if ||wr (t )|| = 0, then BB+ w(t ) = w(t ) if and only if BB+ Bw = Bw . In summary, the disturbance nature is invariant with respect to the delay uncertainty, which means that w(t ) is unmatched if Bw wr (t ) is also an unmatched disturbance. Feedforward control strategies are commonly used to improved closed-loop performance in the presence of matched measured disturbances. Three control difficulties oppose the application of the standard feedforward strategy in this kind of time-delay problem: (i) disturbances are not measurable, (ii) control signal cannot be instantaneously used to counteract disturbance effect due to input delay effect, and (iii) unmatched disturbance imposes a gap between disturbance effect and control action. Challenges (i) and (ii) have been handled in [27] by using a filtered disturbance observer which approximately define a predicted disturbance in 1

In this formulation, the bound described by Wm is not required to be known.

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a band-limited frequency range. This strategy provides an enhanced disturbance attenuation, but the solution was limited to systems with matched disturbances due to the compensation strategy which is a significant limitation for most of the real applications. Moreover, discretetime counterpart has not been discussed is most of the related works, and the dead-time compensation strategies are related to particular prediction approaches [27], [22], and [14]. In this work, the following alternative description is considered from (1) without any loss of generality: x˙(t ) = Ax(t ) + Bu(t − h) + d (t ),

(4)

y(t ) = Cx(t ),

(5)

u(t ) = φ(t ), t ∈ [−h, 0),

(6)

x(0) = x0 ,

(7)

where d (t ) = Bw w(t ) in order that Bw is not explicitly required and y(t ) ∈ R is an arbitrary linear combination of the states. This output vector should be used to define set-point tracking priority in the presence of unmatched disturbances. Objective. The main objective of this paper is to generalize the feedforward idea used [6,27] in order to achieve the following properties: i) its application directly holds for any Artstein based predictor, ii) matched and unmatched disturbances can be considered, iii) disturbance attenuation/rejection performance can be directly defined as a filter design problem [22,25] and iv) either continuous-time or discrete-time approach can be used. q

2.1. Preliminaries This section briefly presents the main ideas that relate Artstein based dead-time compensation control and Finite Spectrum Assignment with the feedforward action. 2.2. Artstein predictor It is well known that delay effect could be theoretically removed from the control loop by using the exact prediction given by:  t x(t + h) = eAh x(t ) + eA(t−τ ) [Bu(τ ) + d (τ + h)]dτ. (8) t−h

However, d (τ + h) with τ ∈ [t − h, t] is not available for control purposes due to causality condition. The Artstein predictor is an approximated solution which can be obtained by neglecting the effect of disturbance signal as follows:  t xˆ(t + h) = eAh x(t ) + eA(t−τ ) Bu(τ )dτ. (9) t−h

In this case, xˆ(t + h) = x(t + h) because d (τ + h) = 0n,1 in practice.2 Then, due to the reduction method, the following equivalent description can be considered to take disturbances 2

The notation 0n, m describes a null matrix with n rows and m columns.

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into account: x˙ˆp (t ) = Axˆp (t ) + Bu(t ) + d p (t ),

(10)

x(t ) = xˆp (t − h) + e p (t ).

(11)

where ep (t) is the prediction error, dp (t) is the equivalent disturbance and x˙ˆp (t ) is a prediction for x(t + h). In the reduction method, dp (t) disturbs closed-loop evolution since u(t) is computed from the predicted signal defined by xˆp (t ). On the other hand, ep (t) disturbs state evolution and reference tracking performance, as a consequence, since prediction error describes the difference between real and predicted evolutions. More details can be found in [22] for instance. Artstein predictor implicitly provides the following relationship between d(t), dp (t), and ep (t): d p (t ) = eAh d (t ),  e p (t ) =

t−h

(12)

eA(t−h−τ ) d (τ + h)dτ.

(13)

t−2h

The main objective of the Artstein based predictor is to modify the definition of xˆp (t ) in order to reduce the disturbance effect over dp (t) and ep (t). For example, null prediction error in the presence of constant disturbances was ensured in [14], null prediction error in the presence of deterministic disturbances was treated in [22], and a feedforward compensation was used in [27] for systems with matched disturbances. Moreover, band-limited disturbance attenuation was considered in [27] by using a Taylor series-based polynomial approximation of the delay inverse. However, these strategies do not consider the general feedforward case. 2.3. Equivalent description This equivalent model described by Eqs. (10) and (11) can be used to analyze any of the modified Artstein predictor [22]. The only difference comes from the definition of dp (t) and ep (t) with respect to d(t). In this work, a general framework is analyzed. Thus, no specific predictor is required, but four mild assumptions should be considered for the sake of analysis purposes: ||d p (t )|| ≤ sup α(||d (τ )|| ), ∀t ≥ 0,

(14)

||e p (t )|| ≤ sup β(||d (τ )|| ), ∀t ≥ 0,

(15)

lim ||d (t )|| = 0 ⇒ lim ||d p (t )|| = 0,

(16)

lim ||d (t )|| = 0 ⇒ lim ||e p (t )|| = 0,

(17)

τ ∈(t0 ,t]

τ ∈(t0 ,t]

t→∞

t→∞

t→∞

t→∞

where t0 ∈ (−∞, t ) is a fixed initial instant, and α( · ) and β( · ) are K-functions.3 These assumption are easily verified by considering any recent Artstein based predictor such as 3

A K-function is defined as α(·) : R+ → R strictly increasing with α(0) = 0.

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[14,22,27]. Roughly speaking, these conditions guarantee: (i) ||dp (t)|| and ||ep (t)|| are bounded by an increasing function of the worst case of d(τ ) with τ ∈ (t0 , t], and ii) steady-state disturbance effect is null in the presence of vanishing disturbances, i.e limt→∞ ||d (t )|| = 0. Note that limt→∞ ||d (t )|| = 0 is not a required assumption. The required prediction property is defined such that if limt→∞ ||d (t )|| = 0 holds, then limt→∞ ||d p (t )|| = 0 and limt→∞ ||e p (t )|| = 0 should be ensured by the Artstein based predictor. Actually, if limt → ∞ ||d(t)|| = 0, then nothing can be stated a priori about the general convergence of ||dp (t)|| and ||ep (t)|| based on these mild assumptions. Anyway, specific convergence properties in the presence of non-vanishing disturbances can be derived independently for each Artstein based strategy as discussed in [14,22,27] for instance. For illustration purposes, consider standard Artstein predictor case as follows: ||d p (t )|| = ||eAh d (t )||

(18)

≤ σ ||d (t )|| ≤ sup σ ||d (τ )||

(19) (20)

≤ sup σ ||d (τ )||, ∀t ≥ 0,

(21)

τ ∈(t −,t ] τ ∈(,t]

where σ = ||eAh || and  > 0 is any arbitrary small and positive constant. The same holds for the prediction error given by:  ||e p (t )|| = ||

t−h

eA(t−h−τ ) [d (τ + h)]dτ ||

(22)

t−2h

≤ hλ sup ||d (τ )||

(23)

≤

τ ∈[t −h,t ]

≤

sup

hλ||d (τ )||

(24)

sup

hλ||d (τ )||, ∀t ≥ 0,

(25)

τ ∈(t −h−,t ] τ ∈(−h−,t]

where λ = maxτ ∈[0,h] ||eAτ || and t0 = min (−, −h − ) = −h − . The same idea can be applied to [14,22,27] for instance. This general description is useful since performance discussion is not related to a specific Artstein based prediction strategy. 2.3.1. Finite Spectrum Assignment with feedforward control In this work, the Finite Spectrum Assignment (FSA) is used with two-layer feedforward concept. This disturbance compensation provides the additional degree of freedom in order to enhance closed-loop performance. Set-point tracking is presented instead of regulation to the origin by virtue of presentation completeness. A nominal steady-state target is defined by the pair (x , u) where Ax + Bu = 0n,1 . Observe that z = [x  u ] can be any vector of the following nullspace N ([A B]). Then, for a given nominal target, the FSA control law without feedforward action is defined by: u(t ) = K (xˆp (t ) − x ) + u, where xˆp (t ) is an Artstein based prediction computed with x(t) and u(t).

(26)

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2.3.2. Two-layer feedforward control The proposed feedforward strategy has two different components: (i) the matched disturbance one, which is not used for prediction purposes, and (ii) the unmatched disturbance action which appears in the second compensation layer. These feedforward actions can be represented as follows: u f f (t ) = uu (t ) + um (t ),

(27)

where um (t) is used to deal with matched disturbances, uu (t) is defined to handle the undesired effect of the unmatched disturbances, and uff (t) is the overall feedforward signal. A virtual control signal (v(t )) is defined for prediction purpose as follows: v(t ) = u(t ) − um (t ).

(28)

The overall control action (u(t)) is effectively applied, but its effect is removed from v(t ). In this case, the original system can be rewritten as follows: x˙(t ) = Ax(t ) + Bv(t − h) + Bum (t − h) + d (t ),

(29)

y(t ) = Cx(t ),

(30)

u(t ) = φ(t ), t ∈ [−h, 0),

(31)

x(0) = x0 .

(32)

The main idea is to use Bum (t − h) to counter-act matched additive disturbances, but the standard single-layer feedforward strategy can be easily recovered with um (t) ≡ 0q, 1 . However, the two-layer is useful in the presence of matched disturbances because Bum (t − h) can be directly used to mitigate or reject d(t) before prediction stage. Now, the new prediction can be computed by using x(t) and v(t ). From now on, Xˆ p (t ) is used to emphasize the prediction obtained with the two-layer approach. For example, if Arstein predictor is used with this feedforward approach, the prediction is defined by:  t Xˆ p (t + h) = eAh x(t ) + eA(t−τ ) Bv(τ )dτ. (33) t−h

Note that if um (t) ≡ 0q, 1 , then Xˆ p (t + h) = xˆp (t + h). Thus, this approach can be analyzed as an extension of the standard solution. Now, complete FSA control law with feedforward action can be directly defined by u(t ) = K (Xˆ p (t ) − x ) + u + uu (t ) + um (t ),

(34)

where Xˆ p (t ) is a prediction computed from x(t) and v(t ), um (t) is the matched disturbance compensation term and uu (t) is the unmatched feedforward action. If the two-layer approach is used, the equivalent model is given by X˙ˆ p (t ) = AXˆ p (t ) + Bv(t ) + d p (t ), (35) x(t ) = Xˆ p (t − h) + e p (t ),

(36)

where dp (t), ep (t) are computed with respect to Buu (t − h) + d (t ) instead of d(t). For example, if the original Artstein prediction is used, then: d p (t ) = eAh (Buu (t − h) + d (t )),

(37)

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 e p (t ) =

t−h

eA(t−h−τ ) (Buu (τ ) + d (τ + h))dτ.

(38)

t−2h

As unmatched disturbances cannot be rejected by the first layer (um (t)), their undesired propagation effects with respect to the desired system response may be attenuated by using the action of the second layer (uu (t)). Disturbance estimator, generalized compensation, and tuning strategies can be presented to improve disturbance attenuation performance by considering the complete control law definition. 3. Main result The additional feedforward action can be used to attenuate the effect of the additive disturbances and the prediction error. As a consequence of the two-layer approach, three main signals should be considered: (i) d(t), (ii) dp (t), and (iii) ep (t). Note that d(t) is useful to define the matched disturbance compensation action while dp (t) and ep (t) can be treated with the unmatched layer. The proposed feedforward action is based on disturbance observers as presented in [35] and used in [27] and [22]. Three estimation filters are introduced in order to define the observers: (i) F1 (s) is a stable, strictly proper and SISO filter, (ii) F2 (s) is a stable, strictly proper and MIMO filter and (iii) F3 (s) is any stable, proper and MIMO filter.4 The filtered disturbances and the filtered output prediction error are given by: d˜(t ) = L−1 {F1 (s)[(sI − A )L{x(t )} − e−sh BL{u(t )}]}, (39) δ˜p (t ) = L−1 {F2 (s)[(sI − A )L{Xˆ p (t )} − BL{v(t )}]},

(40)

e˜y,p (t ) = L−1 {F3 (s)[L{Cx(t )} − e−sh L{C Xˆ p (t )}]},

(41)

where d˜(t ) ∈ Rn , δ˜p (t ) ∈ R p , e˜y,p (t ) ∈ R p , L denotes the unilateral Laplace transform, s represent Laplace complex variable and null initial conditions are assumed for implementation simplicity. In this case, F2 (s) is described by a q × n transfer matrix and F3 (s) is represented by a q × q filter.5 Note that Eqs. (39)–(41) are used for implementation purposes, but these signals can be rewritten as follows: d˜(t ) = L−1 {F1 (s)L{d (t )}}, (42) δ˜p (t ) = L−1 {F2 (s)L{d p (t )}},

(43)

e˜y,p (t ) = L−1 {F3 (s)L{Ce p (t )}}.

(44)

If the system is not initially at rest, non-null initial conditions can be considered as unmeasured disturbance to simplify the implementation of the observer. This result can be verified from the unilateral Laplace transform and the definition of Eqs. (4), (35) and (36) given by: d (t ) = L−1 {sL{x(t )} − AL{x(t )} − e−sh BL{u(t )} − x(0)},

(45)

F3 (s) is a SISO filter if q = 1. Note that L{x(t )}, L{u(t )}, L{v(t )}, and L{Xˆ p (t )} can be respectively represented by X(s), U(s), V(s), and Xˆ p (s). However, the inverse Laplace transform was used to emphasize the difference between a signal and a system (signals are represented with time notation). 4 5

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d p (t ) = L−1 {sL{xˆp (t )} − A{xˆp (t )} − B{v(t )} − xˆp (0)},

(46)

ey,p (t ) = C(x(t ) − xˆp (t − h)).

(47)

Anyway, the initial condition effect can be easily computed because F1 (s) and F2 (s) are known filters. If desired, this effect can be handled as a measured disturbance, which recovers the generality for the proposed estimator. Remark 2. The signals δ˜p (t ) and e˜y,p (t ) can be directly obtained from d(t) (or dm (t ) = d (t ) + Bum (t − h)), but these relationships depend on the the prediction strategy and d(t) needs to be estimated anyway. Therefore, d˜(t ), δ˜p (t ) and e˜y,p (t ) are assumed to be directly obtained from their respective estimators for discussion generality. Then, matched and unmatched feedforward compensation are respectively defined by: um (t ) = −B+ d˜(t ),

(48)

uu (t ) = −δ˜p (t ) − e˜y,p (t ).

(49)

The proposed FSA control law is directly given by: u(t ) = K (Xˆ p (t ) − x ) + u − δ˜p (t ) − e˜y,p (t ) − B+ d˜(t ).

(50)

Note that F1 (s) = 0, F2 (s) = 0q,n , and F3 (s) = 0q,q , can be respectively used to neglect the matched disturbance compensation term (first layer), unmatched disturbance compensation action (second-layer), and the prediction error compensation term (second layer). These filters provide the flexibility of the proposed approach since the role of each contribution is clearly identified. In summary, Xˆ p (t ) stands for the prediction based on the first-layer feedforward compensation, dp (t) is the equivalent additive disturbance, ep (t) is the prediction error, v(t ) is the virtual control signal used for prediction purposes, u(t) is the effective control signal, and the notation s˜(t ) represents a filtered version of a signal s(t). The filtered signals are used to implement the feedforward action where δ˜p (t ) maps d˜p (t ) onto the control dimension (Rq ). Note that this transformation is directly obtained because F2 (s) is a q × n transfer matrix. This generalized disturbance compensation concept does not guarantee performance improvement by itself. Now, ideal disturbance rejection is discussed to show benefits and limitations of the proposed approach. 3.1. Ideal disturbance rejection Ideal dynamic compensation is a key concept in time-delay system problems [19, Chapter 5]. The theoretical perfect disturbance rejection is able to show limitations and to point out feasible solutions for practical implementation purposes. In this generalized two-layer compensation strategy, perfect disturbance rejections from d(t) to x(t) (with matched disturbances) and from d(t) to y(t) (with unmatched disturbances) are achieved if the following conditions hold: ||F1 (s) − esh || = 0,

(51)

||F2 (s) − [C(sI − A − BK )−1 B]−1C(sI − A − BK )|| = 0,

(52)

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||F3 (s) − esh [C(sI − A − BK )−1 B]−1 || = 0.

(53)

For presentation simplicity, proof is presented in Appendix A. However, the main idea is simple. Note that conditions (i) ||F1 (s) − esh || = 0 and (ii) d (t ) = BB+ d (t ) implies that x˙(t ) = Ax(t ) + Bv(t − h). Then, ep (t) ≡ 0n, 1 , dp (t) ≡ 0n, 1 , Xˆ p (t ) = x(t + h), and x(t ) − x = e(A+BK )(t−h) (x(0) − x ) which is the nominal autonomous evolution (matched case). If unmatched disturbances are considered, the following ideal theoretical conditions are considered: i) ||F2 (s) − [C(sI − A − BK )−1 B]−1C(sI − A − BK )|| = 0, and ii) ||F3 (s) − esh [C(sI − A − BK )−1 B]−1 || = 0 which implies that y(t ) − y = C e(A+BK )(t−h) (Xˆ p (t ) − x ) as shown in Appendix A.2. This ideal discussion is useful since design guidelines can be obtained from Eqs. (51)–(53). Note that F1 (s), F2 (s) and F3 (s) are stable, causal and rational filters. Thus, these conditions cannot be achieved since (i) esh and [C(sI − A − BK )−1 B]−1 are non-causal, and (ii) esh is not rational. In this context, the conceptual contribution of Sanz et al. [27] can be very helpful since these ideal conditions can be approximately obtained in the desired frequency range. Remark 3. In this new feedforward strategy, disturbance estimators and feedforward compensators are integrated into a single stage defined by F1 (s), F2 (s) and F3 (s). This lumped solution simplifies the overall design since disturbance rejection improvement depends on both stages (estimator and compensator) which can be optimized taken into account a given performance criterion. Anyway, the two-layer concept can be also used with a disturbance estimator (extended state observer – for instance) and the standard feedforward compensator designed in separated stages. However, the solutions with multiple stages unnecessarily increase tuning complexity and overall implementation order. Differently from [27], the proposed feedforward action can be used to attenuate both matched and unmatched disturbances. In [27], a polynomial approximation was considered as the design framework since only F1 (s) was implicitly used. Now, the ideal disturbance rejection paradigm is applied in order to increase flexibility with respect to the definition of F1 (s), F2 (s) and F3 (s). 3.1.1. Estimator design In this section, an approximated solution is proposed based on the ideal compensation result in order to improve disturbance attenuation. Now, consider the following transfer matrices: H2 (s) =[C(sI − A − BK )−1 B]−1C(sI − A − BK ),

(54)

H3 (s) =[C(sI − A − BK )−1 B]−1 ,

(55)

which comes from the rational part (without delay) of Eqs. (52) and (53). Assume that θ 1 , θ 2 and θ 3 are the respective free design parameters of F1 (s), F2 (s) and F3 (s). These filter can be defined by the arguments that solve the following optimization problems: min ||F1 (s) − esh ||, s.t.: s = jω, ∀ω ∈ (0, ω],

(56)

min ||F2 (s) − H2 (s)||, s.t.: s = jω, ∀ω ∈ (0, ω],

(57)

min ||F3 (s) − esh H3 (s)||, s.t.: s = jω, ∀ω ∈ (0, ω],

(58)

θ1

θ2

θ3

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where ω freely defines the desired frequency range. For obvious reason, the quality of the approximation depends on the filter order (number of free tuning parameters). As can be expected, a larger bandwidth requires a higher filter order in order to keep disturbance rejection performance. The disturbance attenuation performance and the filter complexity imposes a natural trade-off due to the infinite dimensional nature of the delay. As shown in [27], a causal approximation based on Taylor series has provided good results for F1 (s) even with low-order filters. Constant and sinusoidal disturbance rejection can be achieved by considering the following condition: F1 (s)|s=0,± jωi = esh |s=0,± jωi ,

(59)

F2 (s)|s=0,± jωi = H2 (s)|s=0,± jωi ,

(60)

F3 (s)|s=0,± jωi = esh H3 (s)|s=0,± jωi ,

(61)

where ωi , i = 1, . . . , q define the undesired frequencies. The parameters of F1 (s), F2 (s) and F3 (s) can be computed as discussed in [25]. The same idea can be applied to reject deterministic disturbances with repeated poles such as piecewise ramp-like and parabolic-like disturbances. In this case, the derivative of Eqs. (59)–(61) should be considered [22]. Also note that unmatched disturbance rejection condition implicitly requires that the system has no transmission zeros at the disturbance frequency, i.e. det (C(sI − A − BK )−1 B)|s=0,± jωi = 0, but this condition is not imposed to the matched disturbance rejection problem. This assumption is expected as discussed in [25]. This analysis illustrates the usefulness of the first layer action in the presence of matched disturbances for LTI systems with transmission zeros. Remark 4. The benefits of the proposed approach can be directly evaluated by comparing disturbance attenuation with and without filters. This analysis can be performed from the singular values of M1 (s) = 1 − F1 (s)e−sL , M2 (s) = C(sI − A − BK )−1 (I − BF2 (s)) and M3 (s) = [I − e−shC(sI − A − BK )−1 BF3 (s)]. 3.2. Constant disturbance problem As shown in Appendix A.1, if d(t) is a matched disturbance, then the delayed system with matched feedforward action can be rewritten as follows: x˙(t ) = Ax(t ) + Bv(t − h) + dm (t ),

(62)

where dm (t ) = L−1 {(1 − e−sh F1 (s))L{d (t )}}.

(63)

This result is important since the first layer can previously reduce the prediction error caused by matched disturbances. Note that if F1 (0) = 1 and B+ d (t ) = d (t ), then limt→∞ ||dm (t )|| = 0, limt→∞ ||d p (t )|| = 0, limt→∞ ||e p (t )|| = 0 and limt→∞ (x(t ) − x ) = 0 as consequence. The second layer generalizes the main idea of Sanz et al. [27]. In this case, the reachable steady-state set in the presence of constant disturbance is given by Ax d + Bud = −Bd which depends on d . Thus, the prioritized set-point tracking concept is used to define the desired linear combination of the states which can be tracked in the presence of constant unmatched disturbances. In summary, limt→∞ C(x(t ) − x ) = 0 if F2 (0) = I , F3 (0) = I , limt→∞ d (t ) = d in the presence of either matched or unmatched disturbances.

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This is an important result since constant disturbance rejection is a relevant control specification and one of the main drawbacks of the original Artstein predictor [14]. Moreover, no integral action is explicitly required which is a useful property with respect to tuning simplicity. It was recently shown that explicit integral action is not necessary to achieve setpoint tracking in Smith predictor based strategies [26,29]. Thus, this result closes this gap between these recent simplified filtered Smith predictors and the Artstein based predictors in the presence of convergent disturbances. Moreover, other feedforward tuning rules (based on transfer function methods) can be used to define F1 (s), F2 (s) and F3 (s) in the case with constant disturbances. Remark 5. Smith-predictor strategies are based on input-output models so that internal stability should be ensured with additional design constraints if the open-loop system is not Bounded-Input Bounded-Output stable [19]. Artstein based approaches have no additional internal stability requirement since any (stable) estimation filters can be used. However, filtered Smith predictor tracking properties depend only on the prediction error due to the input-output description while equivalent additive disturbances should be considered with the Artstein based predictors. This discussion shows how to deal with Finite Spectrum Assignment tracking properties in state-space context based on a two-layer feedforward concept. 4. Computer-based control Discrete-time problem is relevant in practice due to the importance of digital control. In real computer-based control applications, the FSA numerical integration may require small steps to avoid significant integration errors. As pointed out in [6], the approximation error of this integral may even cause instability. On the other hands, discrete-time implementation avoids approximation error at sampling instants and hidden inter-sample oscillations can be avoided with a standard sampling period choice [2]. In this context, the discrete-time counterpart may be directly used to reduce computational requirements. Thus the discrete-time model obtained with a sample-hold discretization method is given by: x[k + 1] = Ad x[k] + Bd v[k − hd ] + qd [k],

(64)

y[k] = Cx[k],

(65)

u[k] = φ[k], k ∈ [−hd , 0], k ∈ Z

(66)

x[0] = x0 ,

(67)

where Z describes the sets of integers, x[k] = x(k Ts ), y[k ] = y(k Ts ), u[k ] = u(k Ts ), v[k ] = v(kTs ), Ts is the sampling period, hd describes the discrete-time input delay, qd [k] = Bd um [k − hd ] + dd [k], and dd [k] is the discrete-time unmeasured disturbance. As usual, the nominal steady-state target, which is defined by (Ad − I )x + Bd u = 0n,1 , can obtained from N ([(Ad − I ) Bd ]). Similarly to continuous-time case, the disturbance estimators are given by: d˜[k] = Z −1 {F1 (z)[(zI − Ad )Z{x[k]} − z−hd Bd Z{u[k]}]},

(68)

δ˜p [k] = Z −1 {F2 (z)[(zI − Ad )Z{Xˆ p [k]} − Bd Z{v[k]}]},

(69)

T.L.M. Santos, T.S. Franklin / Journal of the Franklin Institute 355 (2018) 3541–3566

e˜y,p [k] = Z −1 {F3 (z)[Z{x[k]} − z−hd Z{Xˆ p [k]}]}.

3553

(70)

Similarly to continuous-time case, the notation Z{·} is used to emphasize the Z-transform of a signal. The proposed control law is similar to the continuous-time problem: u[k] = K (Xˆ p [k] − x ) + u − δ˜p [k] − e˜y,p [k] − B+ d˜[k] , (71) where Xˆ p [k] is obtained with a discrete-time Artstein based predictor from x[k] and v[k] = u[k] + B+ d˜[k]. As expected, ideal disturbance rejection is theoretically achieved if the following conditions hold: ||F1 (z) − z−hd || = 0,

(72)

||F2 (z) − [Cd (zI − Ad − Bd Kd )−1 B]−1Cd (zI − Ad − Bd Kd )|| = 0,

(73)

||F3 (z) − z−hd [Cd (zI − Ad − Bd Kd )−1 Bd ]−1 || = 0.

(74)

Proof is omitted since arguments are basically the same of the continuous-time case. Note that similar discussion applies, but frequency range is analyzed with z = e j , = [0, ], < π ; constant disturbance is handled with = 0 (z = 1); and sinusoidal disturbance rejection is achieved by considering j = ω j Ts , where ωj is the undesired frequency in continuous-time and Ts is the sampling period. Despite the similarity of continuous-time and discrete-time problems, these aspects should be clarified in order correctly achieve the expected results. Actually, most of related FSA results are presented in continuous-time, but the discrete-time counterpart also holds which is important with respect to implementation simplicity. 5. Case study This section presents two case studies. The first one is based on the open-loop system presented in [14] and reproduced in [22,27]. Multiple disturbance scenarios are considered to emphasize the generality of the proposed method. The main difference with respect to previous works comes from the detailed discussion with respect to unmatched disturbance effect. The second case is a non-linear DC-DC boost converter with input delay. The feedforward effect is analyzed in the presence of model uncertainty caused by non-linear dynamics and external disturbances. 5.1. Case 1 – Benchmark problem 

Consider the open-loop unstable system presented in [14]:         x˙1 (t ) 1 x1 (t ) 0 0 1 = + (u(t − h) + wm (t )) + w (t ), x˙2 (t ) −9 3 x2 (t ) 1 1 u

(75)

where h = 0.5, wm (t ) describes the matched disturbance and wu (t ) represents the unmatched disturbance. It is supposed that the system is initially at rest with x(0) = 0 and u(t ) = 0 for t ∈ [−0.5, 0). The control law is defined by u(t ) = −[45 18]Xˆ p (t ) in all cases, but two prediction strategies are considered in order to compute Xˆ p (t ): i) the original Artstein predictor, and ii) the Artstein based prediction proposed in [14]. The output y(t ) = [1 0]x(t ) is chosen to define a target priority in the presence of unmatched disturbances and yt = 1 is the desired target.

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Reference ( Lechappe et al., 2015 ) Artstein Modified Artstein

x1(t)

2

1

0 0

10

20

10

20

30

40

50

30

40

50

4 x2(t)

2 0 −2 −4 0

Time (s) Fig. 1. Output response comparison: case with constant disturbance.

5.1.1. Constant disturbance – continuous-time Asymptotically constant disturbances are commonly found in real control problems, but the original Artstein compensator is not able to achieve null prediction error in the presence of this type of disturbance. The constant disturbances are represented by wm = 1(t − 10) and wu = 1(t − 30), where 1(t ) = 1, t ≥ 0 and 1(t ) = 0, t < 0. The modified Artstein strategy is obtained with the proposed disturbance compensation where the filters are given by: F1 (s) =

1 , τs + 1 

F2 (s) =

F3 (s) =

 15 1 , τs + 1 τs + 1

54 , τs + 1

(76)

(77)

(78)

τ = 0.5 is the only additional tuning parameter. This simplicity is an important requirement in several industrial applications [16]. Note that the matched disturbance settling time is approximately defined by ts,2% = 4τ + h since 4τ determines the filter settling time with constant disturbances and h is the disturbance propagation time with respect to the prediction error (Artstein predictor). Set-point tracking response is shown in Fig. 1. As expected, the original Artstein strategy has offset when either matched or unmatched constant disturbances are considered. The

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( Lechappe et al., 2015 ) Artstein Modified Artstein

|C ep(t)|

1

0.5

0 0

10

20

10

20

30

40

50

30

40

50

p

||d (t)||

2

10

5

0 0

Time (s) Fig. 2. Prediction error and equivalent disturbance comparison: case with constant disturbance.

offset is not significant with the approach of [14] in the case with matched uncertainty, but disturbance effect was not effectively rejected in the presence of unmatched disturbance. As expected, the Modified Artstein with generalized feedforward action is able to reject matched and unmatched constant disturbances. Note from Fig. 2 that the first layer was useful to achieve null prediction error and equivalent disturbance in the presence of matched disturbance, but the second layer is necessary to correct disturbance effect and prediction error in the presence of unmatched disturbance. Also observe that the prediction error is effectively smaller with the predictor of [14], but the second layer would be necessary to correct the effect of dp (t). Thus, the proposed idea can be easily used with [14] to correct this source of offset due to the generality of the new feedforward strategy. 5.1.2. Single sinusoidal disturbance – continuous-time The sinusoidal disturbance case in analyzed with wm ≡ 0 and wu = (0.5 + sin (t ))1(t − 10). In this problem, the first layer was not considered since only matched disturbance is an issue of concern from now on (F1 (s) = 0). Second layer filters are defined by   α1,2 s2 + α1,1 s + 15 α2,1 s2 + α2,1 s + 1 , F2 (s) = (79) (0.5s + 1)3 (0.5s + 1)3   β2 s2 + β1 s + 54 hs 1+ , F3 (s) = (0.5s + 1)3 0.1s + 1

(80)

where α i, j and β j , i = 1, 2, j = 1, 2 are tuning parameters used to define disturbance attenuation properties. Note that a causal first-order Taylor series approximation was used to consider

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0

−50

−100 −1 10

Signular Values (dB)

Signular Values (dB)

3556

0

10

1

2

10

10

0 Standard Exact Optimal

−20 −40 −60 −1 10

0

1

10 10 Frequency (rad/s)

2

10

Fig. 3. Singular value (σ ( · )) comparison: σ (M2 (jω)) – top, σ (M3 (jω)) – bottom.

  the delay inverse, i.e. esh ≈ 1 + 0.1hss+1 . The tuning parameters are computed by using two design procedure: i) optimization based approach6 with ω ∈ [0, 1] and ii) exact disturbance rejection method7 in order to eliminate sinusoidal disturbances with frequency 1 rad/s. Note that the delay approximation could be considered as an optimization variable, but the same decision variables have been used with exact and approximation approaches in order to perform a clear comparison. Optimization problem was solved with a simple Nelder-Mead algorithm.8 The comparisons of the singular value of M2 (jω) and M3 (jω) (Remark 4) are shown in Fig. 3 where standard denotes the case without feedforward action. The benefits of the proposed approach can be verified by the significant attenuation observed in particular for M2 (jω). As expected, the exact solution has a notch filter effect at 1 rad/s and the optimization based approach has a smaller worst-case gain in the desired frequency range (ω ∈ [0, 1]). However, the difference is not so significant since both strategies provide a significant disturbance attenuation at the considered frequency range. Also note that M3 (s) exhibits an amplification at midrange frequencies (from 1 to 10 rad/s) which is mainly caused by the inverse delay approximation. The midrange frequencies gain amplify transitory disturbance effect, but the high frequency region is not a problem since the filters are strictly proper and the proposed solution becomes similar to the case without feedforward action. Moreover, any undesired amplification can be reduced by using high-order (filters) approximations. The set-point tracking response is presented in Fig. 4. Exact and optimal solutions are applied with the standard Artstein predictor and compared with the approach proposed in [14]. As expected from the frequency response analysis, the sinusoidal disturbance is completely eliminated with the exact approach, but the optimization based solution also provides a good 6 7 8

Optimal solution: α1,2 = 12.64, α1,1 = 21.65, α2,2 = 0.75, α2,1 = 1.41, β2 = 65.04, β1 = 73.93. Exact solution: α1,2 = 12.62, α1,1 = 20.88, α2,2 = 0.75, α2,1 = 1.375, β2 = 64.25, β1 = 65.54. Matlab function “fminsearch” can be used to directly implement the Nelder-Mead optimization algorithm.

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3557

Reference ( Lechappe et al., 2015 ) Modified Artstein (Exact) Modified Artstein (Optimal)

x1(t)

2

1

0 0

10

20

10

20

30

40

50

30

40

50

4 x2(t)

2 0 −2 −4 0

Time (s) Fig. 4. Output response comparison: case with sinusoidal disturbance.

result if compared with the state of art without feedforward action. Note that an adaptive filtering approach can be used to reject sinusoidal disturbances with unknown frequencies [21,25]. 5.1.3. Multiple sinusoidal disturbances – continuous-time For completeness purposes, a case with multiples frequencies is analyzed with wu = (0.5 + sin (0.8t ) + sin (0.6t ) + sin (0.4t ))1(t − 10). As shown in Fig. 5, the optimization based feedforward control is better than the exact case, but the difference is not significant. This result can be expected from the frequency response analysis which clearly indicates the benefits to attenuate band-limited disturbance signals. Moreover, the exact solution seems to be a promising alternative due to its design simplicity. 5.1.4. Ramp disturbance – discrete-time Finally, the discrete-time case with ramp disturbance is considered to emphasize the generality of the proposed approach. In this problem, Ts = 0.01 defines the sampling time, Kd = [−40.3782 − 16.9267] is the discrete time gain in order to map continuous-time to discrete-time closed-loop eigenvalues, and the filters are:   351z − 357 17.86z − 16.94 , F2 (z) = (z − 0.9)2 (z − 0.9)2 48.54z − 48.04 F3 (z) = . (z − 0.9)2

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Reference ( Lechappe et al., 2015 ) Modified Artstein (Exact) Modified Artstein (Optimal)

x1(t)

2 1 0 0

20

40

20

40

60

80

100

60

80

100

x2(t)

5 0 −5 0

Time (s) Fig. 5. Output response comparison: case with multiple sinusoidal disturbance.

These filters were obtained by considering Eqs. (73) and (74) and their derivatives with z = 1 and the poles 0.9 are defined as free tuning parameters. Details of the parameters computation can be found in [22] and [25]. The piecewise ramp disturbance (bounded) is described by wu = 0.5t[1(t − 10) − 1(t − 15)]. Discrete-time responses with ramp disturbance are presented in Fig. 6. Similar to the continuous-time case with constant disturbance, the predictor proposed in [14] has a better performance if compared with the original Artstein predictor, but both of them present a divergent tracking error during ramp disturbance effect. The proposed feedforward action is able to track the desired set-point in the presence of ramp disturbance because the generalized solution was defined to achieve this loop requirement. The same result could be achieved for both x1 (t) and x2 (t) simultaneously in the presence of matched by using the first layer feedforward action. The main contribution of this solution comes from the fact that this idea can be applied to other Artstein based strategies as [22], [27], and [14]. 5.2. Case 2 – DC-DC boost converter (discrete-time) A DC-DC boost converter is used in several applications in order to provide a fixed power despite load changes and external voltage variations. The schematic description is depicted in Fig. 7, where R represents the load, Sa denotes a mosfet used for switching purposes, and Vs is the external voltage source. The output voltage (Vo ) is the regulated variable while the duty-cycle is the control variable. Duty-cycle is converted into an effective control action by using a Pulse Width Modulation (PWM) switching command. As discussed in [31], this kind

T.L.M. Santos, T.S. Franklin / Journal of the Franklin Institute 355 (2018) 3541–3566

3559

Reference ( Lechappe et al., 2015 ) Artstein Modified Artstein

x1(t)

2

1

0 0

5

10

15

20

25

30

5

10

15 Time (s)

20

25

30

x2(t)

5 0 −5 0

Fig. 6. Output response comparison with discrete-time control: case with ramp disturbance.

L

D

iL Vs

sa

C

R

Vo

Fig. 7. Schematic representation of a DC-DC boost converter.

of control strategy is used to regulate the power of fuel cells in automobile applications for instance. It is assumed that the input time-delay comes from a CAN-based network control system strategy [33] with a worst-case triggered action. This trigger based on the worst-case delay can be used to convert a network-induced time-varying delay into a constant delay. This approach can be easily applied to CAN networks because worst-case delay can be analytically derived [4] or obtained from the so-called hyper-period analysis [33]. It is assumed a constant delay of 10 ms based on typical values obtained in automobile applications [33]. An average state-space model can be obtained with ξ (t ) = [I (t ) Vo (t )] , where I(t) is the inductor current, Vo (t) represents the capacitor voltage, and u(t) is the duty-cycle. The average non-linear description is given by:     −(1 − u(t − h)) ξ2L(t ) + VsL(t ) ξ˙1 (t ) = , (81) ξ˙2 (t ) (1 − u(t − h)) ξ1 (t ) − ξ2 (t ) C

RC

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where L = 500 × 10−6 H, C = 200 × 10−6 F, R = (24)2 /100 , Vs = 20 V, as proposed in [31]. Sampling period is given by Ts = 1 ms and PWM frequency rate is 30 KHz as defined in [31]. The linearized model is obtained by defining x1 (t ) = ξ1 (t ) − I ∗ and x2 (t ) = ξ2 (t ) − Vo∗ , where Vo∗ = 24 V is the desired equilibrium output voltage, I ∗ = (Vo∗ )2 /(Vs∗ R), and u∗ = 1 − (Vo∗ /I ∗ )/(R) as a consequence. Note that I∗ and u∗ depends on the correct knowledge of R and the steady-state value of Vs which may vary due to voltage source changes (fuel cell variations for instance). In this case, the linearized model is given by the following description: ∗       Vo∗  0 − 1−u x˙1 (t ) (t ) x 1 L = 1−u∗ + LI ∗ (u(t − h) − u∗ ), (82) 1 x˙2 (t ) x2 (t ) − −C C RC where h = 10 ms as previously discussed. Note that x = [0 0] and u − u∗ = 0 in this linear description. Digital control approach is used since control law is assumed to be remotely computed in a network-based automobile application. Discrete-time controller (Kd = [−0.0004 0.0128]) is obtained as the state-feedback Linear Quadratic Regulator solution based  ∗  ∗ on the following cost function: J = ∞ j=0 [x [ j] x [ j] + (u[ j] − u ) (u[ j] − u )]. Moreover, the following filters are used to eliminate steady-state effect of model uncertainty caused by non-linearities and voltage source variations:   0.0003566 0.001036 , F2 (z) = z − 0.95 z − 0.95 0.001103z F3 (z) = , z − 0.95 with F1 (z) = 0. Observe that the discrete-time pole given by 0.95 is the only additional design parameter which defines standard closed-loop trade-offs (robustness against disturbance rejection performance). The first simulation is based on the non-linear average model where the voltage source is varied from 20 to 16 V as in [31] at 0.1 s. Output performance is depicted in Fig. 8. Note that only the Modified Artstein strategy is able to regulate the output voltage at 24 V since the variation of Vs is an unmatched disturbance. Also observe that all closed-loop response are exactly the same between 0.1 and 0.11 s due to the delay effect, but set-point tracking responses become different after this interval. This is an expected consequence of the undesired delay effect. The duty-cycle response is shown in Fig. 9. Disturbance effect is not significant with respect to control action variation if the original Artstein predictor is used because d p [k] = Ahdd d[k] ≈ 0n,1 in this case study.9 Moreover, prediction error does not modify control action as previously discussed since u[k] is computed from Xˆ p [k] which does not depend on ep [k]. This attenuation property of the FSA for open-loop stable systems was previously analyzed in [24, Section 3.4]. Also note that the strategy proposed in [14] is not able to deal with this kind of disturbance (voltage source variation) because the effect of d p [k] = d[k] + Ahdd (d[k] − d[k − hd ]) ≈ d[k] cannot be directly rejected with the standard state-feedback. Hence, the new feedforward strategy can be used to enhance disturbance rejection properties by preserving control law simplicity which is expected from the theoretical analysis. 9

h

h

Also note that Add = eATs hd = eAh . In this case, ||Add ||2 = 0.02.

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Reference ( Lechappe et al., 2015 ) Artstein Modified Artstein

1

(t)

10

5

0 0

0.05

0.1

0

0.05

0.1

0.15

0.2

0.25

0.15

0.2

0.25

2

(t)

25 20 15 Time (s) Fig. 8. Responses obtained with the DC-DC boost converter based on the average non-linear model.

0.5 0.4

u(t)

0.3 0.2 0.1 0 0

0.05

0.1

0.15

0.2

0.25

Time (s) Fig. 9. Control responses obtained with the DC-DC boost converter based on the average non-linear model.

Now, a similar scenario is considered, but Matlab/Simulink components are used to simulate inductor, capacitor, diode, mosfet and voltage source. The main objective is to evaluate other sources of uncertainties such as diode and mosfet losses. Moreover, this new simulation avoids the average model description so that switching effect can be considered. The new results are shown in Figs. 10 and 11. It is remarkable the fact that overall results are similar, but only

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Reference ( Lechappe et al., 2015 ) Artstein Modified Artstein

1

(t)

10

5

0 0

0.05

0.1

0

0.05

0.1

0.15

0.2

0.25

0.15

0.2

0.25

2

(t)

25 20 15 Time (s) Fig. 10. Responses obtained with the DC-DC boost converter based on the non-linear switching model.

0.5 0.4

u(t)

0.3 0.2 0.1 0 0

0.05

0.1

0.15

0.2

0.25

Time (s) Fig. 11. Control responses obtained with the DC-DC boost converter based on the non-linear switching model.

the modified approach was effective to regulate the output voltage before 0.1 s. During this initial interval, the voltage source is 20 V (nominal value), but the non-linear model is not perfect due to mosfet and diode losses as can be verified from the fact that V∗ , I∗ , and u∗ do not define an equilibrium. Once more, the modified strategy can be used to handle this kind of uncertainty since the feedforward action is able to compensate these additional sources of uncertainties.

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Finally, this result is important to show that the proposed feedforward compensation is useful in the presence of model uncertainty. Observe that if u(t) (or u[k]) and x(t) (or x[k]) are constrained, then model uncertainty effect can be described as an unmatched and bounded additive disturbance.10 Moreover, tracking was achieved without an augmented model with integral action which can be used to reject other types of disturbances such as ramp and sinusoidal unmeasured signals. In summary, the proposed approach is able to provide tracking properties to the FSA technique which is an important requirement in practical dead-time compensation problems. 6. Conclusions A novel two-layer feedforward concept has been proposed to deal with matched and unmatched bounded additive disturbances with the FSA approach. The generalized solution can be applied to any Artstein based predictor in order to improve disturbance attenuation or to achieve disturbance rejection with respect to usual deterministic disturbance types. An optimization based design solution is proposed to provide a systematic tuning procedure. The discrete-time counterpart is analyzed in order to simplify computer-based implementation and to reduce computational requirements. The proposed strategy is a complementary solution since disturbance rejection or attenuation properties can be included regardless of Artstein based predictor attributes. Simulation examples are presented based on a benchmark problem in order to illustrate the benefits of the proposed approach. The two-layer concept may be combined with the output feedback idea presented in [7] in order to improve the dynamic feedforward compensation without full-state knowledge. Also note that the performance limitation imposed by state and input constraints [23] may be explored as an interesting topic for future works. Acknowledgments Financial support from the Brazilian funding agencies CNPq (Grant Number 308245/20156) and CAPES (Grant Number 88881.119983/2016-01) are gratefully acknowledge. The authors especially acknowledge the helpful suggestions provided by the anonymous reviewers and the editor. Appendix A. Disturbance analysis A.1. First layer analysis – matched case Initially suppose that the disturbance can be rewritten in the matched form, i.e. d (t ) = Bw w(t ) with BB+ Bw = Bw . The as previously discussed, Eq. (29) can be alternatively de10 Note that x˙(t ) = A x(t ) + B u(t ), where A and B are unknown matrices can be described by nominal matrices as r r r r follows: x˙(t ) = Ax(t ) + Bu(t ) + d (t ) with d (t ) = (Ar − A )x(t ) + (Br − B)u(t ). In this case, d(t) is bounded if x(t), u(t) are constrained. The same transformation holds for discrete-time problems. Thus, bounded disturbance assumption can be achieved indirectly as in this example or by using robust design strategies for constrained systems as presented in [23] for instance. This kind of disturbance is typically unmatched which is handled by the second feedforward layer in the proposed framework. Also observe that the two-layer feedforward concept is a key contribution of this work if compared with the related literature.

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scribed by x˙(t ) = Ax(t ) + Bv(t − h) + dm (t ),

(A.1)

where dm (t ) = −BB+ d˜(t − h) + d (t ). Since F1 (s) is a scalar filter and BB+ Bw = Bw with d (t ) = Bw w(t ), the modified disturbance can be expressed as follows: dm (t ) = L−1 {(I − BB+ F1 (s)e−sh )L{d (t )}}, = L−1 {(1 − F1 (s)e−sh )L{BB+ d (t )}}, = L−1 {(1 − F1 (s)e−sh )L{d (t )}}.

(A.2) (A.3) (A.4)

In this case, matched disturbance attenuation is defined by M1 (s) = 1 − F1 (s)e−sh . From the ideal disturbance rejection point of view, ||M1 (s)|| = 0 or alternatively ||F1 (s) − esh || = 0 implies that ||dm (t )|| = 0 if only matched disturbances are considered. In the presence of unmatched disturbance, the effect of the filter is directly defined by (I − F1 (s)BB+ e−sh ), but its undesired effect is treated by the second layer. A2. Second layer analysis – unmatched case In the second layer analysis problem, consider that all the eigenvalues of the matrix  = A + BK are strictly inside the left-half plan. Then, Eq. (10) can be rewritten as follows by considering the two-layer approach: X˙ˆ p (t ) = AXˆ p (t ) + Bv(t ) + d p (t ),

(A.5)

x(t ) = Xˆ p (t − h) + e p (t ),

(A.6)

where Xˆ p (t ) is obtained from v(t ) and x(t). Alternatively, the description can be translated with respect to any desired nominal equilibrium z = [x  u ] : X˙ˆ p (t ) = A[Xˆ p (t ) − x ] + B[v(t ) − u] + d p (t ) = [Xˆ p (t ) − x ] + Bu p (t ) + d p (t ) = [Xˆ p (t ) − x ] + q(t )

(A.7) (A.8) (A.9)

where q(t ) = Bu p (t ) + d p (t ) for presentation simplicity. Thus, the predicted output can be expressed by: Xˆ p (t ) − x = et (Xˆ p (0) − x ) + L−1 {(sI − )L{q(t )}}.

(A.10)

Now, the Eq. (A.10) can be rewritten by multiplying matrix C, considering a time-delay h and adding the prediction error Cep (t) to both sides of the identity as follows: C(Xˆ p (t − h) − x ) + Ce p (t ) = L−1 {e−shC(sI − )−1 L{q(t )} + L{Ce p (t )}} + Ce(t−h) (Xˆ p (0) − x )

(A.11)

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Then, by taking into account that y(t ) = C Xˆ p (t − h) + C e p (t ), y = C x , q(t ) = Bu p (t ) + d p (t ) = −Bδ˜p (t ) − BC e˜ p (t ) + d p (t ), the following tracking error holds: y(t ) − y =L−1 {e−shC(sI − )−1 (I − BF2 (s))L{d p (t )} + [I − e−shC(sI − )BF3 (s)]L{Ce p (t )}} + Ce(t−h) (Xˆ p (0) − x ) = L−1 {M2 (s)L{d p (t )} + M3 (s)L{Ce p (t )}} + Ce(t−h) (Xˆ p (0) − x ),

(A.12) (A.13)

where M2 (s) = e−shC(sI − A − BK )−1 (I − BF2 (s)) and M3 (s) = [I − e−shC(sI − A − BK )BF3 (s)] describe disturbance and prediction error effect, and Ce(t−h) (Xˆ p (0) − x ) is the nominal evolution. Note that ideal disturbance rejection would be obtained with ||M2 (s)|| = 0 and ||M3 (s)|| = 0. These condition may be rewritten in terms of the filter as: ||F2 (s) − [C(sI − A − BK )−1 B]−1C(sI − A − BK )|| = 0,

(A.14)

||F3 (s) − esh [C(sI − A − BK )−1 B]−1 || = 0,

(A.15)

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