Active Disturbance Rejection Control Design with Suppression of Sensor Noise Effects in Application to DC-DC Buck Power Converter
Krzysztof Łakomy, Rafal Madonski, Bin Dai, Jun Yang, Piotr Kicki, Maral Ansari, Shihua Li
AActive Disturbance Rejection Controlwith Sensor Noise Suppressing Observerfor DC-DC Buck Power Converters
Krzysztof Łakomy, Rafal Madonski, Bin Dai, Jun Yang,
Senior Member, IEEE , Piotr Kicki,Maral Ansari,
Student Member, IEEE , Shihua Li,
Fellow, IEEE
Abstract —The class of active disturbance rejection con-trol (ADRC) algorithms has been shown in the literature tobe an interesting alternative to standard control methods inpower electronics devices. However, their robustness andstability are often limited in practice by the high-frequencymeasurement noise, common in industrial applications. Inthis article, this problem is addressed by replacing theconventional high-gain extended state observer (ESO) witha new cascade observer structure. The presented exper-imental results, performed on a DC-DC buck power con-verter system, show that the new cascade ESO designhas increased estimation/control performance compared tothe standard approach, while effectively suppressing thedetrimental effect of sensor noise over-amplification.
Index Terms —noise suppression, power converter, high-gain observer, extended state observer, ADRC
I. I
NTRODUCTION T HE majority of modern power electronic circuits aresupplied with switching-mode power converters, likepulsewidth modulated DC-DC converters. A buck converter,for example, is responsible for stepping down voltage, whilestepping up current, from its input (supply) to its output (load).Its relatively high efficiency (often higher than 90%) and lowprice, makes a buck converter a choice of practitioners invarious power systems. With the constant pursuit of techno-logical improvement, finding more effective control methodsfor power converters is an active research topic.Practically appealing results on buck converter control usingthe idea of active disturbance rejection control (ADRC) wererecently reported in [1]–[3]. Interestingly, some motor controlcompanies attracted by the ADRC-based solutions (e.g. Texas
K. Łakomy is with the Institute of Automatic Control and Robotics,Poznan University of Technology, Piotrowo 3A, 60-965 Poznan, Poland,e-mail: [email protected]. Madonski is with the Energy Electricity Research Center, Interna-tional Energy College, Jinan University, 519070 Zhuhai, P. R. China, e-mail: [email protected]. Dai, J. Yang and S. Li are with the School of Automation,Southeast University, Key Laboratory of Measurement and Controlof CSE, Ministry of Education, 210096 Nanjing, P. R. China, e-mail: { bin 1994/j.yang84/lsh } @seu.edu.cnP. Kicki is with the Institute of Robotics and Machine Intelligence,Poznan University of Technology, Piotrowo 3A, 60-965 Poznan, Poland,e-mail: [email protected]. Ansari is with the Faculty of Engineering and Information Tech-nology, University of Technology Sydney, 81 Broadway, Sydney NSW,Australia, e-mail: [email protected] Instruments), have embedded this approach in their selectedcommercial products [4]. The key element in any ADRCscheme is the extended state observer (ESO), which is respon-sible for estimating the system state vector and reconstructingthe overall disturbance (also referred to as total disturbance )affecting the controlled variable [5].However, since the conventional form of ADRC uses a high-gain observer (HGO) to estimate selected signals, its capabil-ities are intrinsically limited by the presence and severity ofhigh-frequency sensor noise, as shown in [6]–[8]. The HGO-based ADRC design and tuning often comes down to a forcedcompromise between speed/accuracy of signals reconstructionand sensitivity to noise [9]. Same compromise can be seen inthe ADRC works for buck converters in which the measuredsystem output (voltage) is oftentimes corrupted with high-frequency noise [10]. Several solutions were proposed to solvethe problem of attenuating the effects of measurement noisein high-gain observers. They mainly address the problemby: employing nonlinear [11], [12] or adaptive techniques[13], redesigning the local behavior by combining differentobservers [14], employing low-power structures [15]–[17], ormodifying standard low-pass filters [18].Motivated by the above problem, a new cascade ESO-based ADRC solution is introduced. It is based on a virtualdecomposition of the total distubance present in the DC-DC buck converter system, allowing to design a cascadestructure of ESO, where each level of the observer cascadeis responsible for handling a particular type and frequencyrange of estimated signal. The proposed topology enhancesconventional state/disturbance estimation performance whileavoiding over-amplification of sensor noise. The user-definednumber of cascade levels allow to customize the overall controlsystem structure to meet certain disturbance rejection require-ments. Although a multi-level cascade observer is proposed,a straightforward design and implementation methodology isgiven, together with an intuitive tuning rules. The proposedADRC with cascade ESOs is validated in this work usinghardware experiments conducted on a DC-DC buck converterlaboratory testbed.
Notation.
Within this article, we treat R as a set of realnumbers, R + = { x ∈ R ∶ x > } as a set of positive realnumbers, Z as a set of integers, λ min ( AAA ) and λ max ( AAA ) arerespectively the minimal and maximal eigenvalues of matrix AAA , while
AAA ≻ means that matrix AAA is positive definite.Function f ( x ) ∶ R → R belongs to class K when it is strictly a r X i v : . [ ee ss . S Y ] S e p ig. 1: Semiconductor realization of the considered DC-DCbuck power converter, with diode V D and control switch
V T .increasing and f ( ) = . The expression ls ∞ ∶= lim sup t →∞ was used for the sake of notation compactness. II. P
RELIMINARIES
A. Simplified plant model and control objective
Following [3], an average dynamic model of a DC-DC buckconverter, depicted in Fig. 1, can be written as: ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ dv o ( t ) dt = C i L ( t ) − CR v o ( t ) , di L ( t ) dt = V in L [ µ ( t ) + d ( t )] − L v o ( t ) ,y o ( t ) = v o ( t ) + n ( t ) , (1)where µ ∈ [ , ] is the duty ratio, y o [V] is the measured systemoutput that consists of the average capacitor voltage v o [V] andthe sensor noise n [V], i L [A] is the average inductor current, R [ Ω ] is the load resistance of the circuit, L [H] is the filterinductance, C [F] is the filter capacitance, V in [V] is the inputvoltage source, and d ( t ) represents the unknown (possiblytime-varying and nonlinear) external disturbance.The control objective is to force v o ( t ) to follow a referencecapacitor output voltage trajectory v r ( t ) [V] by manipulating µ ( t ) with following assumptions applying. Assumption 1:
Following the limitations resulting from thephysical properties of the considered electronic circuit, we mayassume that the values of voltage and current are bounded, andbelong to some compact set such that sup t ≥ ∣ i L ( t )∣ < r i L and sup t ≥ ∣ v o ( t )∣ < r v o for r i L , r v o > . Assumption 2:
Output voltage v o ( t ) is the only measur-able signal and is additionally corrupted by bounded, high-frequency measurement noise sup t ≥ ∣ n ( t )∣ < r n . Assumption 3:
The unknown external disturbance signal sup t ≥ ∣ d ( t )∣ < r d is bounded and has bounded first time-derivative sup t ≥ ∣ ˙ d ( t )∣ < r ˙ d . Assumption 4:
There exists a positive constant r v r such thatthe reference signal and its specific time-derivatives satisfyinequality sup t ≤ {∣ v ( i ) r ( t )∣} ≤ r v r , for i ∈ { , , , } . B. Application of the ADRC principle
Following the standard ADRC design, system (1) need to bereformulated, emphasizing the system input-output relation: d v o ( t ) dt = − CR † a dv o ( t ) dt − CL † a v o ( t ) + V in CL – b [ µ ( t ) + d ( t )] . (2) Combing the uncertain (or unknown) terms of (2), includingthe imperfect identification of the input gain, results in afollowing form of the output voltage dynamics: ¨ v o = a v o + a ˙ v o + bµ − ˆ bµ + bd ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ F ( t, ˙ v o ,v o ,µ,d ) + ˆ bµ = F (⋅) + ˆ bµ, (3)where ˆ b ≠ is an estimate of the input gain b from (2) and F (⋅) represents the matched total disturbance of system (3).Since v r ( t ) and its derivatives may not be known a priori ,which may lead to possible inability of constructing the feed-forward term in the controller µ , let us reformulate dynamics(3) into an error-domain as ¨ e = ¨ v r − ¨ v o = ¨ v r − F (⋅)·„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„¶ F ∗ (⋅ , ¨ v r ) − ˆ bµ, (4)where e ( t ) = v r ( t )− v o ( t ) is the control error signal and F ∗ (⋅) is the total disturbance in the error-domain. In this article, weutilize a standard form of the ADRC controller µ = ˆ b − ( ˆ F ∗ + µ ) , (5)which is constructed to simultaneously compensate the in-fluence of disturbance using the estimated value of totaldisturbance ˆ F ∗ and to stabilize system (4) in a close vicinityof the equilibrium point e = using the output-feedbackstabilizing controller µ . Assumption 5:
Stabilizing controller µ has a structure thatguarantees the boundedness of µ (⋅) and ˙ µ (⋅) . Althoughthis assumption may seem conservative, it is relaxed with thepreviously introduced Assumptions 1, 3, and 4.We will first put the focus on precise and on-line estimationof perturbing term F ∗ (⋅) , crucial for proper active disturbancerejection. To calculate ˆ F ∗ , we first need to define the extendedstate z = [ z z z ] ⊺ ≜ [ e ˙ e F ∗ ] ⊺ ∈ D z , where D z ≜ { xxx ∈ R ∶∥ x ∥ < r z } for some r z ∈ R + . The dynamics of state vector z can be expressed, upon (4), as a state-space model ⎧⎪⎪⎨⎪⎪⎩ ˙ z = AAA z − ddd ˆ bµ + bbb ˙ F ∗ ,y = e − n = ccc ⊺ z − n, (6)where AAA ≜ [ × III × × ] , ddd ≜ [ ] ⊺ , ccc ≜ [ ] ⊺ , and bbb ≜ [ ] ⊺ . Given (6), the output of this system y correspondsto the control error e which, according to Assumption 2, isinfluenced by the measurement noise n . Remark 1:
Control error e , together with its derivative ˙ e arebounded according to the Assumptions 1, 3, and 4, and thespecific form of system dynamics (1). Remark 2:
Under the Assumptions 1, 3 and 4, func-tion F ∗ ( t ) is continuously differentiable, and thus, thereexist bounded continuous functions Ψ F ∗ , Ψ ˙ F ∗ such that sup t ≥ ∣ F ∗ ( t )∣ < Ψ F ∗ ( e, ˙ e, v r , ˙ v r , ¨ v r , µ ) , sup t ≥ ∣ ˙ F ∗ ( t )∣ < Ψ ˙ F ∗ ( e, ˙ e, v r , ˙ v r , ¨ v r , ... v r , µ, ˙ µ ) , for all [ e ˙ e ] ⊺ ∈ R . Both prac-tical and theoretical justifications of lumping selected com-ponents as parts of F ∗ (⋅) , including control signal an state-dependent variables, has been thoroughly discussed in [5]. II. M
AIN RESULT : PROPOSED CASCADE
ESO ADRC
To calculate the estimated value of extended state vector z ,let us now introduce a novel p -level structure of a cascadeobserver ( p ∈ Z and p ≥ ) in a following form ˙ ξξξ ( t ) = AAAξξξ ( t ) − ddd ˆ bµ ( t ) + lll [ y ( t ) − ccc ⊺ ξξξ ( t )] ˙ ξξξ i ( t ) = AAAξξξ i ( t ) + ddd ⎛⎝− ˆ bµ ( t ) + bbb ⊺ i − ∑ j = ξξξ j ( t )⎞⎠+ lll i ccc ⊺ [ ξξξ i − ( t ) − ξξξ i ( t )] i ∈ { , ..., p } , (7)where ξξξ j ≜ [ ξ j, ξ j, ξ j, ] ⊺ ∈ R is the state of particularobserver cascade level, lll j ≜ [ ω oj ω oj ω oj ] ⊺ ∈ R is theobserver gain vector with design parameter ω oj ≜ α j − ω o ∈ R + for α > , ω o ∈ R + , and j ∈ { , ..., p } . The estimate of z ,resulting from the observer (7) can be expressed as ˆ z = [ ˆ z ˆ z ˆ z ] ⊺ ≜ ξξξ p + bbbbbb ⊺ p − ∑ j = ξξξ j ∈ R . (8) Remark 3:
It is worth noting, that if we reduce the observerto a single level ( p = ), we would obtain a standard formof a linear high-gain ESO, as seen in [19]. An introductionof the subsequent cascade levels allows us to keep the sameobservation quality with smaller values of ω o , resulting in thedecrease of the measurement noise amplification, see (7). Thiseffect will become visible in the upcoming experiments. Remark 4:
The idea of the cascade observer structureproposed in (7) is based on a specific choice of the firstlevel observer bandwidth ω o , which should be large enoughto guarantee a precise estimation of the first element of theextended state vector z , and low enough to filter out themeasurement noise. The latter elements of the extended statevector, i.e. z and z , usually have faster transients, andthus, are not estimated precisely with the first level observer.The consecutive observer levels are introduced to improvethe estimation quality of z and z using higher observerbandwidths ω oi for i > . The following observer levels areusing the state vectors of previous observer levels instead ofa measured signal, and thus, should result in lower noiseamplification than a single-level ESO with high bandwidthvalue.Having ˆ z , the application of control action (5) to the system(4) results in following second-order error dynamics ¨ e = F ∗ − ˆ F ∗ ·„„„„„„„„„„„„„‚„„„„„„„„„„„„„¶ ˜ F ∗ − µ , (9)where ˜ F ∗ ∈ R is the residue of the total disturbance resultingfrom the imperfect observation of F ∗ by observer (7). Theintroduction of consecutive cascade level of the observer canbe interpreted as an attempt to estimate the total disturbanceresidue, based on the signals returned on the output of previouscascade level, and its inclusion in the overall estimate of theextended state vector (8).A block diagram of the proposed ADRC with cascade ESOfor the DC-DC buck power converter is shown in Fig. 2. ESO1st level ()ESO2nd levelDC-DCconverter (1) μ p -th level c ascade ESO (7) ξ ξ Controller(5) v o v r μ - z ESO1st levelState selector (8) p = ESO p- th level ξ p y n - p = :.:. ...... (design parameter) d .. Fig. 2: Proposed ADRC with sensor noise suppression viacascade ESO structure for the DC-DC buck power converter.
Theorem 1:
Under Assumptions 2-5, and by taking a stabi-lizing proportional derivative controller µ ≜ k p ( e − n )·„„„„„„„„‚„„„„„„„„¶ y + k d ˆ z , k p , k d > , (10)the observation errors of the extended state obtained with the p -level cascade observer, defined as ˜ z p ≜ z − ˆ z , = z − ξξξ p − bbbbbb ⊺ p − ∑ j = ξξξ j ∈ R , (11)together with the control error e , described with the dynamics(9), are bounded. In other words: ∀ ω o ,k > ∃ δ ˜ z ,δ e > ∶ ls ∞ ∥ ˜ z p ( t )∥ < δ ˜ z ∧ ls ∞ ∣ e ( t )∣ < δ e . (12) Remark 5:
To keep the notational conciseness of the fol-lowing theoretical analysis, we propose to tune the stabilizingcontroller (10) with a single parameter k > , setting thevalues of proportional and derivative gains, respectively, as k p = k and k d = k . Chosen tuning procedure places the polesof control error dynamics (9) at the value of −√ k . Similarcontroller parametrization was used in [19]). Proof of Theorem 1.
The dynamics of the observation errordefined for a particular cascade level, i.e. ˜ z i ≜ z − ξξξ i − bbbbbb ⊺ ∑ i − j = ξξξ j ∈ R for i ∈ { , ..., p } , can be expressed (aftersome algebraic transformations) as ˙˜ z = ( AAA − lll ccc ⊺ ) ˜ z − lll n + bbb ˙ F ∗ , ˙˜ z i = ( AAA − lll i ccc ⊺ ) ˜ z i + ( lll i ccc ⊺ − bbbbbb ⊺ lll i − ccc ⊺ ) ˜ z i − − bbbbbb ⊺ lll n + bbb ˙ F ∗ − bbbbbb ⊺ i − ∑ j = ( lll j ccc ⊺ − lll j + ccc ⊺ ) ˜ z j , for i ∈ { , ..., p } . (13)Equations (13) allow us to write the dynamics of the aggre-gated observation error ˜ ζ ≜ [ ˜ z ⊺ ... ˜ z ⊺ p ] ⊺ ∈ R p in a form ˙˜ ζ = H ζ ˜ ζ + δ ˙ F ∗ + γ n, (14)where matrix H ζ is lower triangular and its eigenvalues λ i ∈ {− ω o , − αω o , ..., − α p ω o } for i ∈ { , ..., p } , vector = [ bbb ⊺ ... bbb ⊺ ·„„„„„„„„„„„„‚„„„„„„„„„„„¶ p times ] ⊺ , and γ = [ lll ⊺ bbbbbb ⊺ ... lll ⊺ bbbbbb ⊺ ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ p times ] ⊺ . Introducing thetransformation ˜ ζ = ΛΛΛ χ χ for ΛΛΛ χ ≜ blkdiag { LLL , ..., LLL p } ∈ R p × p where LLL i ≜ diag {( α i − ω o ) − , ( α i − ω o ) − , } ∈ R × for i ∈ { , ..., p } ∈ R p × p , we can rewrite (14) to a form ˙ χ = ΛΛΛ − χ H ζ ΛΛΛ χ χ + ΛΛΛ − χ δ ˙ F ∗ + ΛΛΛ − χ γ n = ω o HHH χ χ + δ ˙ F ∗ + γ n, (15)where HHH χ is dependent only on parameter α and its eigenval-ues λ i ∈ {− , − α, ..., − α p } for i ∈ { , ..., p } . To conduct astability analysis of the observation subsystem, let us introducea Lyapunov function candidate V χ = χ ⊺ PPP χ χ ∶ R p → R lim-ited by λ min ( PPP χ ) ∥ χ ∥ ≤ V χ ≤ λ max ( PPP χ ) ∥ χ ∥ , where PPP χ ≻ is the solution of Lyapunov equation HHH χ ⊺ PPP χ + PPP χ HHH χ = − III.
The derivative of V χ , based on (15), can be written down as ˙ V χ = − ω o χ ⊺ χ + χ ⊺ PPP χ ( δ ˙ F ∗ + γ n )≤ − ω o ∥ χ ∥ + ∥ χ ∥ λ max ( PPP χ )√ p (∣ ˙ F ∗ ∣ + ω o ∣ n ∣) (16)and holds ˙ V χ ≤ −( − ν χ ) ω o ∥ χ ∥ for ∥ χ ∥ ≥ λ max ( PPP χ )√ pω o ν χ ∣ ˙ F ∗ ∣ + λ max ( PPP χ )√ pω o ν χ ∣ n ∣ , (17)where ν χ ∈ ( , ) is a chosen majorization constant. Thelower bound of ∥ χ ∥ is a class K function with respect to theperturturbations ∣ ˙ F ∗ ∣ and ∣ n ∣ , so according to the Remark 2and Assumption 2, system (15) is input-to-state stable (ISS),and according to [20], satisfiesls ∞ ∥ χ ( t )∥ ≤ ρ χ λ max ( PPP χ )√ pω o ν χ Ψ ˙ F ∗ (⋅)+ ρ χ λ max ( PPP χ )√ pω o ν χ r n , (18)for ρ χ = √ λ max ( PPP χ )/ λ min ( PPP χ ) . Since λ max ( ΛΛΛ χ ) = max { , ( ν p − χ ω o ) − } and ˜ z p is a subvector of ˜ ζ , we maywrite down that ∥ ˜ z p ∥ ≤ ∥ ˜ ζ ∥ ≤ λ max ( ΛΛΛ χ ) ∥ χ ∥ and thus that theasymptotic relationls ∞ ∥ ˜ z p ( t )∥ ≤ λ max ( ΛΛΛ χ ) ls ∞ ∥ χ ( t )∥ =∶ δ ˜ z , (19)which completes the proof of the observer part of (12). Remark 6:
Upon the result (18), we can see that in thenominal conditions, when n ( t ) ≡ , the asymptotic relationls ∞ ∥ χ ( t )∥ → as ω o → ∞ resulting in the possibility ofgetting an arbitrarily small value of δ ˜ z .Let us define control error vector (cid:15)(cid:15)(cid:15) = [ e ˙ e ] ⊺ ∈ R . Theapplication of feedback controller (10) to dynamics (9) gives ˙ (cid:15)(cid:15)(cid:15) = [ − k − k ]·„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ KKK (cid:15)(cid:15)(cid:15) + [ k ]·„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„¶ ZZZ ˜ z p − [ k ]– κκκ n, (20) which can be transformed with substitution (cid:15)(cid:15)(cid:15) = ΛΛΛ ε εεε , where ΛΛΛ ε ≜ diag { k − , } , into ˙ εεε = ΛΛΛ − ε KKK
ΛΛΛ ε εεε + ΛΛΛ − ε ZZZ ˜ z p − ΛΛΛ − ε κκκn = k [ − − ]·„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„¶ HHH ε εεε + ZZZ ˜ z p − κκκn. (21)Let us now introduce a Lyapunov function candidate V ε = εεε ⊺ PPP ε εεε ∶ R → R limited by λ min ( PPP ε ) ∥ εεε ∥ ≤ V ε ( εεε ) ≤ λ max ( PPP ε ) ∥ εεε ∥ , where PPP ε ≻ is the solution of Lyapunovequation HHH ⊺ ε PPP ε + PPP ε HHH ε = − III.
The derivative ˙ V ε = − kεεε ⊺ εεε + εεε ⊺ PPP ε ZZZ ˜ z p − εεε ⊺ PPP ε κκκn ≤ − k ∥ εεε ∥ + ∥ εεε ∥ λ max ( PPP ε ) [ m Z ∥ ˜ z p ∥ + k ∣ n ∣] , (22)where m Z = max { , k } , holds ˙ V ε ≤ −( − ν ε ) k ∥ εεε ∥ for ∥ εεε ∥ ≥ λ max ( PPP ε ) ν ε k [ m Z ∥ ˜ z p ∥ + k ∣ n ∣] (23)The lower boundary of ∥ εεε ∥ is class K with respect to argu-ments ∥ ˜ z p ∥ and ∣ n ∣ . According to the Remark 2, Assumption 2and result (17), system (21) is ISS and satisfiesls ∞ ∥ εεε ( t )∥ ≤ ρ ε λ max ( PPP ε ) ν ε k [ m Z ls ∞ ∥ ˜ z p ( t )∥ + k r n ]≤ ρ ε λ max ( PPP ε ) ν ε k ⎡⎢⎢⎢⎢⎣ ρ χ m Z λ max ( PPP χ )√ pω o ν χ Ψ ˙ F ∗ (⋅)+ ( ρ χ m Z λ max ( PPP χ )√ pω o ν χ + k ) r n ⎤⎥⎥⎥⎥⎦ , (24)where ρ ε = √ λ max ( PPP ε )/ λ min ( PPP ε ) . According to transforma-tion between original control error vector (cid:15)(cid:15)(cid:15) and the trans-formed εεε , we write ∥ ˙ (cid:15)(cid:15)(cid:15) ∥ ≤ max { k − , } ∥ εεε ∥ =∶ m k ∥ εεε ∥ and thusls ∞ ∥ (cid:15)(cid:15)(cid:15) ( t )∥ ≤ m k ρ ε λ max ( PPP ε ) ν ε k ⎡⎢⎢⎢⎢⎣ ρ χ λ max ( PPP χ )√ pω o ν χ Ψ ˙ F ∗ (⋅)+ ( ρ χ λ max ( PPP χ )√ pω o ν χ + max { k − , } k ) r n ⎤⎥⎥⎥⎥⎦ =∶ δ e , (25)which completes the proof of Theorem 1. Remark 7:
Similarly to the comment made in Remark 6, inthe case of n ( t ) ≡ and upon the result (25), we can say thatls ∞ ∥ (cid:15)(cid:15)(cid:15) ( t )∥ → as ω o → ∞ ∨ k → ∞ , making it possible toget an arbitrarily small value of δ e . Remark 8:
Upon the result (25), we may observe that theincreasing gains of both observer and controller are amplify-ing measurement noise, thus, it is not recommended to useextremely high values of ω o and k in practice. IV. H
ARDWARE EXPERIMENT
A. Testbed description
The experimental setup used for the study is seen in Fig. 3.The output voltage was measured by a Hall effect-based sensorand converted through a 16-bit A/D converter in the dSPACEplatform. The output was recorded by a digital oscilloscopeig. 3: Laboratory setup, with a - buck converter, b - dSPACEcontroller, c - input voltage, d - oscilloscope, e - voltage sensor,f - A/D converters, and g - PC with control software.and dedicated PC-based software. The sampling period wasset to T s = Hz. The physical parameters of the DC-DCconverter, described with (1), were V in = V, L = . H, C = . F, and R = . This allowed to straightforwardlycalculate the system gain in (3) as ˆ b = V in /( CL ) = × .The tested control algorithm was first implemented in aMatlab/Simulink-based model, from which a C code programwas generated and run on the dSPACE controller in real-time.Considering the above parameters of the utilized testbed andthe controller/observer structures introduced in (5), (10), and(7), we can derive the transfer-function-based relation U ( jω ) = G uy ( jω ) [ E ( jω ) − N ( jω )]·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ Y ( jω ) , (26)where U ( jω ) , E ( jω ) , N ( jω ) , and Y ( jω ) correspond re-spectively to signals µ ( t ) , e ( t ) , n ( t ) , and y ( t ) after Laplacetransformation. The amplitude Bode diagram of G uy ( jω ) obtained for the observer levels p ∈ { , , } and tuned withthe nominal parameters utilized in the experiment is presentedin Fig. 4. The vertical dashed lines represent the chosencontroller bandwidth k , which is the range we expect theclosed-loop system to operate in, and the experiment samplingfrequency ω s . The green area represents the frequency range,where CESO ( p = and p = ) should react more rapidlythan standard ESO, and red area is the range where onlyCESO p = should provide quicker response with respectto control errors. The points at the intersection of ω s andobservers graphs indicate the amplification factors of highfrequency signals (e.g. measurement noise) within signal µ ( t ) .Consequently, in the following experiments, we can expectthe measurement noise to be least amplified in CESO p = ,followed by CESO p = and finally standard ESO.TABLE I: Used bandwidth parameterization of CESOs. Bandwidth Cascade level p = p = p = ω o ) λ λα λα ω o ) – λ λα ω o ) – – λ -2 Fig. 4: Bode diagram representing the module of G uy ( jω ) . B. Test methodology and results
Two following sets of experiments were conducted to testthe ADRC scheme with the proposed cascade ESO (CESO):E1: Comparison with standard ESO (i.e. CESO with p = ).E2: Influence of parameters ω o (E2a), k (E2b), and α (E2c). Remark 9:
Please note that a standard, single ESO issynonymous with the CESO with cascade level p = .The control objective in both tests was to track a smoothvoltage trajectory despite the presence of a varying input-additive external disturbance shown in Fig. 5. Referencetrajectory was designed as a filtered and biased square signalwith the bias equal to 7V, the amplitude of square signal equalto 6V, and period 1s. The filtering transfer function applied tothe square signal was G f ( s ) = . s + . s + .The results of E1 are gathered in Fig. 6. The observerbandwidth for the standard ESO ( p = ) was ω o = rad/s,which was close to the maximum that could be obtained for a kHz sampling without observing any undesirable effects.For the comparison, only CESOs with p = and p = levels were utilized to maintain legibility of the results whilenot loosing their generality. In order to provide a systematictuning methodology across tested observers, bandwidths of theCESOs were parameterized and set according to Table I with α = and λ = rad/s. The controller gains from (10) wereset to k p = and k d = in each case, which correspondsto controller bandwidth k = , introduced in Remark 5.One can notice from Fig. 6 that, with the applied tuningmethodology, all the tested controllers have realized the giventask, however the standard ESO ( p = ) provided the worstperformance in terms of tracking accuracy and noise sup-pression. On the other hand, with the increase of cascadelevel p in CESO, better performance was achieved. Thisobservation is supported with the calculated integral qualityindices in Table II. Besides the improvement of control errorperformance, the transfer of sensor noise into the control signalhas decreased with the increase of parameter p thanks to theTABLE II: Integral quality criteria for experiment E1. Observer type Criterion ∫ ∣ e ( t )∣ dt ∫ ∣ u ( t )∣ dt ∫ ∣ ˙ u ( t )∣ dt Standard ESO ( p = ) 0.2310 0.5368 315.58Cascade ESO ( p = ) 0.0467 0.5496 113.23Cascade ESO ( p = ) 0.0381 0.5545 29.11 Fig. 5: External disturbance applied in all of the experiments.lower values of ω o related to the first level of CESO. Thisresult is supported with the values of ∫ ∣ ˙ u ( t )∣ dt criterion inTable II, which represents the impact of rapid fluctuations ofcontrol signal, mostly caused by the amplified noise.The initial premises formulated upon Fig 4 have beenconfirmed with the results presented in Fig. 6. As expected, thecontrol signal with the lowest content of measurement noisewas obtained for CESO p = , then CESO p = , and finallythe standard ESO.Next, in order to provide potential CESO users with guide-lines for its construction and tuning, the influence of its designparameters was investigated. To this effect, the results of E2are seen in Fig. 7-9. In contrast to E1, here only control errorsand control signals are presented to save space. However, theestimated total disturbance is part of the control signal (see (5))so its influence is explicitly visible in the control signal.The results of E2a are depicted in Fig. 7. In case of thestandard ESO ( p = ), a well-known relation known fromhigh-gain observers, discussed in the Introduction, can beobserved. Namely, with the increase of observer bandwidth ω o , significant noise amplification occurs in the control signal.At the same time, a slight improvement of the control errorwas obtained. In case of the proposed CESO ( p = and p = ),with the increase of ω o , the amplitude of the control signalincreases but no visible improvement in the control accuracycan be observed. In other words, in engineering practice, atsome point, due to multiple factors like maximum samplingfrequency and noise characteristics, increasing the observerbandwidth ω o will no longer provide better performance. Wecan conclude, that with CESO one can achieve better controlperformance for the wide range of ω o values, comparing tothe results of standard ESO ( p = ) in Fig. 7(a).The results of E2b are depicted in Fig. 8. In the caseof the standard ESO ( p = ), it is clear that increasing thecontroller bandwidth k improves the control accuracy whilekeeping a significant, undesired level of control signal andnoise therein. In case of the proposed CESO ( p = and p = ),increasing the controller bandwidth k results in comparablecontrol errors retaining similar level of the control signal. Dueto the characteristics of CESO, it is possible to obtain bettercontrol performance for the wide range of k values, comparingto the results obtained for standard ESO in Fig. 8(a).The results of E2c are depicted in Fig. 9. In case of theCESO ( p = ), increasing α improves both the trackingaccuracy and noise suppression in the control signal. However,in case of the CESO ( p = ), increasing α keeps improvingthe noise suppression in the control signal but at some point Output
Reference
Control error [ - ] Control signal Total disturbance
Fig. 6: Results of experiment E1.deterioration in the tracking accuracy can be spotted. It resultsfrom a fact that, in this case, observer bandwidth ω o is set toosmall, which makes the observer not providing a fast-enoughand accurate-enough estimate of the first state variable of theextended state vector (see Remark 4). V. C
ONCLUSION
An active disturbance rejection control with a novel cascadeextended state observer (CESO) for DC-DC buck convertershas been proposed. The validity of the new approach hasbeen shown through a dedicated stability analysis and aset of hardware experiments. The comparison between theproposed cascade ESO-based ADRC and a standard singleESO-based ADRC showed that the former has stronger capa-bilities of sensor noise suppression and provides better controlperformance (understood as tracking accuracy and energyefficiency). The structure of the proposed ADRC is bulkierthan the conventional one but in return provides an additionaland practically appealing degree of freedom in shaping theinfluence of measurement noise on the observer/controller part.
VI. A
CKNOWLEDGEMENT
The article was created thanks to participation in programPROM of the Polish National Agency for Academic Ex-change. The program is co-financed from the European SocialFund within the Operational Program Knowledge EducationDevelopment, non-competitive project entitled International
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Control error [ - ] Control signal (a) CESO p = Control error [ - ] Control signal (b) CESO p =3