Laboratory of Adaptive Control Institute of Automation and Robotics Poznań University of Technology IAR-PUT Maciej M. Michałek E6 Active/Adaptive Disturbance Rejection Control(ADRC) The exercise is devoted to the control design problem in the ADRC(Active/Adaptive Disturbance Rejection Control) scheme for the exemplary plant and to verification of the designed control system in the Matlab-Simulink environment. In the ADRC control scheme, the adaptation process results from the appropriate on-line update of an additional control signal which is responsible for compensation of the so-called total disturbance encompassing all the unknown and uncertain terms of the plant model. 1 Description of the plant Let us consider the process of rolling motion for the delta-wing aircraft presented in Fig. 1. Thankstothedifferentialdeflectionδ a (expressedin[rad])ofaileronsontheleftandright Figure1:Aviewofthedelta-wingaircraftintherollingmotionwithangularvelocity β(based on E. Lavretsky, K. A. Wise: Robust and Adaptive Control with Aerospace Applications, Springer, London, 2013) sides of the aircraft body, it is possible to change the aircraft roll-angle β expressed in[rad]. One can approximate the roll-angle dynamics by the following nonlinear differential equation β=θ 10 β+θ 20 β+θ60 (δ a +d)+(θ 30 β +θ 40 β ) β+θ 50 β 3, (1)
IAR-PUT: Laboratory of Adaptive Control E6 2 whereθ 10,θ 20,...,θ 60 arethetrueparametersoftheplant,whilethetermdrepresentssome external disturbance which affects the plant through the input channel. From now on, we assume what follows: A1.theonlymeasurablesignalsarethecontrolinputu δ a andtheplantoutput,thatis, therollangle:y β, A2.thevaluesofparametersθ 10 = 0.018andθ 20 =0.015areperfectlyknownuponthea priori knowledge, A3.upontheaprioriknowledgeitisknownthattheinputgainθ 60 [0.05,0.80], A4.valuesofparametersθ 30,θ 40,andθ 50 areunknown, A5.theexternaldisturbancedisboundedbutitisunmeasurable,unknown,andcanbetime varying. Notethatvaluesofcontrolinputu=δ a mustbeinherentlyconstrainedtoa(subsetof)range [ π;π]radduetophysicalinterpretationofδ a. 2 Control performance requirements We are interested in designing the ADRC control system for the roll-angle dynamics represented by equation(1) which, under assumptions A1 to A5, guarantees satisfaction of the following prescribed performance requirements: R1.signaly r (t)=β r (t)isaboundedtime-varyingreferencetrajectoryfortheaircraftroll anglesuchthatẏ r (t)andÿ r (t)existandarebounded, R2.trackingerrore(t) y r (t) y(t)convergeswithnoovershoottoanarbitrarysmall vicinityofzero,thatis e(t) ǫfort,whereǫ 0isasufficientlysmallconstant, R3.settlingtimeT s1% oftheclosed-loopsystemsatisfiest s1% αforα>0expressedin[s]. 3 Control system design 3.1 Step 1: description of the plant model in the extended-state space The roll-angle dynamics(1) represents a highly uncertain system which can be rewritten as where β=θ 10 β+θ 20 β+ˆθ6 δ a +F(β, β,δ a,d), (2) F(β, β,δ a,d)=(θ 30 β +θ 40 β ) β+θ 50 β 3 +θ 60 d+(θ 60 ˆθ 6 )δ a will be called the total disturbance since it aggregates all the unknown or uncertain terms of the plantmodel.inordertodesigntheadrccontrolsystem,weneedtowritetheplantdynamics (2) in the form of an extended-state space model. Let us introduce the extended state x 1 β x= x 2 β (3) x 3 F where the third state variable is equal to the total disturbance. The expression extended state meansthatthenaturalstateofthesecondorderdynamics(2)isextendedbyanadditional (artificial) state variable equal to the total disturbance. Justification of this extension will be
IAR-PUT: Laboratory of Adaptive Control E6 3 providedinsections3.2and3.3.uponequation(2),andtakingu δ a (seeassumptiona1), one derives the extended-state space model in the following form 0 1 0 x 1 0 0 ẋ= θ 10 θ 20 1 x 2 + ˆθ 6 u+ 0 F (4) 0 0 0 x 3 0 1 A B which shall be complemented by the output equation(according to assumption A1) [ ] y= 1 0 0 x=x 1. (5) C 3.2 Step 2: design of the controller structure A nominal controller in the ADRC scheme consists of two loops: the total disturbance compensationlooprepresentedbycomponentu cn,andtheouterfeedback-feedforwardlooprepresented bycomponentu.thenominaladrccontrolsignalisdefinedasfollows u n u +u cn ˆθ 6, u cn F, (6) wherethetermu willbedesignedinsection3.4.thegeneralideacharacteristicforthecontrol law(6)reliesonacompensationfortheunknowntermfindynamics(2),andthenondesigning theouter-loopcontrolu forthecompensatedplantmodel.apracticalefficiencyoftheadrc controller substantially depends on the effectiveness of the total disturbance compensation process. Since the total disturbance F is unknown and unmeasurable, the nominal control law (6) cannot be applied in practice. Therefore, one proposes a practical version of ADRC control lawintheform u u +u c ˆθ 6, u c ˆF, (7) whereˆfisanestimateofthetotaldisturbance.sincefisathirdvariableoftheextendedstate (3),itispossibletodesignastateobserverwhichwillallowestimatingFandnextutilizingit inthecontrollaw(7). 3.3 Step 3: design of the extended-state observer(leso) In order to estimate the total disturbance term, one introduces the Linear Extended-State Observer(LESO), which corresponds to the classical Luenberger observer defined for the extended state(3), that is, ˆx=Aˆx+Bu+L(y Cˆx) =(A LC) ˆx+Bu+Ly, (8) Γ wherematricesa,b,andcaretakenfrom(4)and(5),respectively,while L= [ l 1 l 2 l 3 ] (9) istheobservergainvectorwhichhastobedesignedyet.thematrix l 1 1 0 Γ=A LC= θ 10 l 2 θ 20 1 (10) l 3 0 0
IAR-PUT: Laboratory of Adaptive Control E6 4 must have all the eigenvalues in the left-half complex plane. Determination of the observer gains is possible by appropriate locating the eigenvalues of matrix Γ using the design condition det(λi Γ)=W o (λ), W o (λ) (λ+ω o ) 3 =λ 3 +3ω o λ 2 +3ω 2 oλ+ω 3 o (11) wherew o (λ)isthedesiredpolynomialguaranteeinglocationofatriplerealeigenvalueat λ= ω o forsomeprescribedfrequencyω o >0.Since det(λi Γ)=λ 3 +(l 1 θ 20 )λ 2 +(l 2 θ 10 l 1 θ 20 )λ+l 3, (12) thusbycomparingthecorrespondingcoefficientsofw o (λ)andpolynomial(12)onegetsthe solution: l 1 =3ω o +θ 20, (13) l 2 =3ω 2 o+θ 10 +l 1 θ 20, (14) l 3 =ω 3 o. (15) Equation(8) determines LESO for the continuous time domain. To obtain a discrete time version of LESO, one can discretize equation(8) using, e.g., the Euler-Forward method which yields ˆx(n)=(I+T a Γ) ˆx(n 1)+T a Bu(n 1)+T a Ly(n 1), (16) Γ a B a L a wheret a >0isasamplingtimeusedfortheestimationprocess. 3.4 Step4:designoftheouter-loopcontrolcomponentu Sincethetotaldisturbancecanbeon-lineestimatedbyLESO,onemayreplaceˆFindefinition (7)withtheestimateˆx 3 providedbyasolutionof(8)or(16).aftersubstituting(7)into(4) one gets the dynamics ẋ 2 =θ 10 x 1 +θ 20 x 2 +x 3 +ˆθ 6 u ˆx 3 ˆθ 6 =θ 10 x 1 +θ 20 x 2 +u +(x 3 ˆx 3 ) (17) Designoftheouter-loopcontrolu willbeconductedforthecaseofperfectcompensation (cancellation)ofthetotaldisturbance,thatis,forx 3 ˆx 3 0.Intheseidealconditionsholds ẋ 2 =θ 10 x 1 +θ 20 x 2 +u β=θ 10 β+θ 20 β+u. (18) Bydefinitionofthetrackingerrorwecanwrite e=β r β β=β r e (19) ė= β r β β= β r ė (20) ë= β r β β= β r ë (21) which allow rewriting the compensated dynamics(18) in the form or, after simple reordering the particular terms, as β r ë=θ 10 (β r e)+θ 20 ( β r ė)+u (22) ë θ 20 ė θ 10 e= β r θ 20 βr θ 10 β r u. (23) Let us define the outer-loop control in a way which ensures that the closed-loop error dynamics is bounded and asymptotically convergent to zero. To this aim, we propose a definition u β r θ 20 βr θ 10 β r feedforward +k d ė+k p e PD feedback (24)
IAR-PUT: Laboratory of Adaptive Control E6 5 which consists of the feedforward term(depending on the reference signals) and the PD feedbacktermwithtwocontrollergainsk p,k d >0whichhavetobedesignedyet.Bysubstituting the control law(24) into(23) yields the closed-loop error dynamics ë+(k d θ 20 )ė+(k p θ 10 )e=0 (25) whichisasymptoticallystableifthetwocoefficientsnearthetermseandėarepositive.in order to satisfy the performance requirements R2 and R3, one proposes to select the controller gainsk p andk d accordingtothefollowingdesigncondition s 2 +(k d θ 20 )s+(k p θ 10 )=W c (s), W c (s) (s+ω c ) 2 =s 2 +2ω c s+ω 2 c (26) wherew c (s)isthedesiredpolynomialguaranteeinglocationofadoublerealpoleats= ω c forsomeprescribedfrequencyω c >0.Bycomparingthecorrespondingcoefficientsofthe polynomials in(26) one gets the synthesis equations k p =ω 2 c+θ 10, (27) k d =2ω c +θ 20. (28) Notethatfrequencyω c shouldbeappropriatelyrelatedtothefrequencyω o usedintheleso design(see(11)).ageneralruleis ω c ω o (29) to assure that the convergence of estimation error ˆx(t) x(t) will be substantially faster than the convergence of tracking error e(t). Additionally, upon requirement R3 one shall select ω c 2π α because T s1% 2π ω c. (30) Combination of rule(29) and condition(30) gives the proposed synthesis equations ω c 2π α, ω o= ω c, 0<µ 1 (31) µ where µ is a sufficiently small scaling factor. Worth to emphasize here that the synthesis equations(27)-(28) and tuning rules(31) guarantee satisfaction of requirement R3 with noovershooting response only in the perfect compensation conditions. Thus, if the compensation ofthetotaldisturbanceisnotperfect(apracticalcase),therequirementr3willbemetonly approximately. Summarizing, the ADRC control law is a combination of definition(7) with(24), and takes the final form ] u= [ βr θ 20 βr θ 10 β r +k d ( β r ˆx 2 )+k p (β r β) ˆx 3 /ˆθ 6, (32) whereˆx 2 andˆx 3 aretakenfromleso.theterm β r θ 20 βr θ 10 β r isresponsibleforthe feedforwardcontrol,thetermk d ( β r ˆx 2 )+k p (β r β)isthepdfeedbackcontrol,whileˆx 3 is responsible for a compensation of the total disturbance. According to assumption A1, the D partofthepdcontrolcomponenthasbeenrewrittenbyusingtheestimateˆx 2,becausesignal β is not measurable and it cannot be used in the controller. Moreover, in practical applications it is recommended to post-process the control signal(32) with the saturation function Sat(u,u m ) min{ u,u m } sign(u), (33) whereu m >0isaprescribedsaturationlevelofthecontrolsignal(inthecaseofsystem(1), oneshalltakeu m π).aresultantschemeoftheadrccontrolsystemcorrespondingtothe control law(32) followed by the saturation block is presented in Figure 2.
IAR-PUT: Laboratory of Adaptive Control E6 6 Figure2:BlockschemeoftheADRCsystemfortheaero-plant;theblocksandarrowshighlighted in gray constitute the adaptive loop in the control system while the black ones correspond to the conventional part of the control system(rsg = Reference Signals Generator) 3.1 Reference trajectory tracking in the ADRC system. Open the file DeltaWingPlantADRC.mdl which contains the plant described by dynamics(1) and the reference signal generator(rsg). The RSG block produces twotypesofthereferencetrajectoryy r (t)anditstimederivativesẏ r (t),ÿ r (t): TYPE1: y r (t) Y r sin(ω r t), (34) TYPE2: y r (t) H(s)[Y r (t)rect(ω r t)], (35) whererect(ω r t)representsarectangularsignalwithunitamplitudeandfrequency ω r rad/s,h(s)isaprescribedthirdorderlinearfilter,whiley r (t)isatime-varying amplitude. On the scheme in file DeltaWingPlantADRC.mdl implement the estimator LESO in the continuous-time-domain form determined by(8), the outer-loop PD+FF controller(24), and close the loop according to the control law(32) followed by thesaturationblock.selectthefollowingparameters:u m =π,α=2.0s,ˆθ 6 from amiddlepointoftheknownrange[0.05,0.8],andµ=0.2. RuntheADRCsystemfortheexternaldisturbanced(t) 0andtheplantinitial conditionsβ(0)=0.4rad, β(0)=0(tothelatteraim,oneneedstoappropriately initialize two global variables beta0 and betap0). Analyze the resultant control qualityforbothtypesofareferencetrajectorygeneratedbythersgblock see (34)-(35) usingthedefaultparameters(readthemoutfrominsideofthersg blocks). Important: for the analysis purposes check the time plots of the tracking error e(t)aswellasthecontrolsignalu(t),andcomparestatex(t)withitsestimate ˆx(t)computedbyLESO;comparethereferencesignaly r (t)withtheplantoutput onthesameplot;checkalsobehavioroftheestimateˆf(t). Repeat and analyze simulations for µ {0.5;0.1;0.03}. (36) Does the system satisfy performance requirements R2 and R3 in all the cases? How does the decreasing value of µ influence the control performance and the control signal?whateffectsdoyouexpectafteraddingastochasticnoisetosignaly?
IAR-PUT: Laboratory of Adaptive Control E6 7 Compare the control performance of the full ADRC controller with the performance obtained when the compensation(adaptation) loop is open(by forcing ˆx 3 (t) 0). Turnontheexternaldisturbanced(t),whichcanbeselectedinsideablockofthe plant. Check the control performance of the ADRC system in these conditions. Repeat the simulations for µ taken from the set determined by(36). Compare the control performance when the compensation(adaptation) loop is open. Replace the continuous-time LESO with its discrete-time counterpart represented by(16).checktheinfluenceofthesamplingtimet a usedforcomputationsofthe discrete-time LESO on the overall tracking performance for the following set of sampling time values: T a {0.04;0.03;0.01}s. (37) Note: the above simulations conduct using µ = 0.1.