Output Redefinition-Based Dynamic Inversion Control For A Nonminimum .
Output Redefinition-Based Dynamic InversionControl for a Nonminimum Phase HypersonicVehicleAbstract—Output redefinition based dynamic inversion(ORDI) control is proposed for a nonminimum phasehypersonic vehicle. When velocity and altitude are selectedas control outputs, a hypersonic vehicle exhibitsnonminimum phase behavior, preventing the application ofstandard dynamic inversion due to the unstable zerodynamics. This problem is solved by the ORDI controlarchitecture, where output redefinition is utilized at first torender the modified zero dynamics stable, and thendynamic inversion is used to stabilize the new externaldynamics. Three kinds of ORDI controllers with differentchoices of new control output are investigated. The firsttakes the internal variable as the control output, whichexhibits good robustness but with restricted performance.The second utilizes a synthetic output, which is a linearcombination of the system output and internal variable,making the zero dynamics adjustable, and thus improvesthe tracking performance. The third adds an integral item tothe synthetic output, and thus ensures zero steady-stateerror even with model uncertainties. A systematic way isproposed to determine the combination coefficient toachieve zero dynamics assignment by using the root locusmethod. The efficiency of the method is illustrated bynumerical simulations.Index Terms—Nonminimum phase, output redefinition,dynamic inversion, hypersonic vehicle, zero dynamicsassignmentHI. INTRODUCTIONYPERSONIC vehicles (HSVs) refer to a vehicle thattravels at velocity greater than Mach 5. This is regarded asone of the most promising technology for achieving costeffective and reliable access to space. One of the most difficultchallenges encountered in designing flight control systems forHSVs is the nonminimum phase problem due to elevator-to-liftcoupling [1]. When the nonlinear control method, dynamicinversion, is straightforwardly applied to nonminimum phasesystems, it results in exact tracking but the unstable zerodynamics remains an unstable part in the closed-loop system.Therefore, the nonminimum phase character of an HSVprevents the application of standard dynamic inversion and allof its invariants, bringing great challenges to nonlinearcontroller design for these vehicles.The nonminimum phase problem of HSV can be avoided byadding a canard. Since the elevator-to-lift coupling is canceledby the canard, the nonminimum phase behavior is removed.Many nonlinear methods are applied to the canard configuredHSV, such as sliding mode control [2-4], dynamic surfacecontrol [5-7], and feedback linearization control [8-9], to namejust a few. Although a canard is beneficial to avoid thenonminimum phase problem, it is a problem for the vehiclestructure since the canard must withstand a large thermal stressat hypersonic speeds. Therefore, it is of great significance toinvestigate the control problem of an HSV without a canard,which means that a controller must be designed directly basedon the nonminimum phase HSV model. This issue has receivedmore and more attention in recent years but only a few newcontrol methods have been proposed [1,10-15].Though the nonminimum phase character of an HSV limitsthe application of classical nonlinear control methods, linearcontrol methods are still available. In [10], a linear controller isdeveloped for an HSV by a pole assignment method. As animprovement, a stable inversion approach [16-17] was appliedto an HSV in [11]. It achieves exact tracking by imbedding theideal internal dynamics into a linear feedback controller.However, this method is noncausal and greatly depends onexact model knowledge.In addition, some nonlinear control methods are proposed fora nonminimum phase HSV. One typical method is approximatefeedback linearization [12-13]. By strategically ignoring theelevator-to-lift coupling and resorting to dynamic extension atthe input side, an approximate model with full vector relativedegree is obtained. Then standard dynamic inversion can beapplied to the approximate model, resulting in approximatelinearization of the original model. Other methods can also beused to the approximate model, such as backstepping [14]. Thismethod works mainly because the approximate model hashigher relative degree so that the internal variables are includedin the control loop. However, this method only works when thecoupling is weak enough, i.e., a “slightly” nonminimum phasesystem [18]. The control law designed by the approximatemodel will result in instability when applied to the model withstronger nonminimum phase behavior [13].Another nonlinear control method focuses on the redefinitionof the zero dynamics. In [1], a preliminary feedbacktransformation is used to convert the model into theinterconnection of systems with feedforward and feedback form,respectively. Then the original output is converted into a statetrajectory of new zero dynamics. Hence, additional controleffort is not required for stabilizing the internal dynamics. In[15], with the definition of two separate nested zero dynamicssubsystems, the elevator is treated as the primary effector tocontrol the regulated output and the stabilization of the internaldynamics as a secondary objective.The idea of the two aforementioned papers is very similar tothe output redefinition method [19], whose main concept is: (1)to perform an output redefinition such that the zero dynamicswith respect to the new output are acceptable; and (2) to definea modified desired trajectory for the new output to track suchthat the original output tracks the original desired trajectoryasymptotically. Since the second step can be realized by stableinversion [16-17], the main difficulty lies in how to find a
minimum phase output. In [19], the new output is constructedthrough the B-I norm form. But it is nontrivial to implement inpractice since the B-I norm form of a complex system is usuallydifficult to obtain. In [20], the flatness-based approach isproposed, where a variable with full relative degree is selectedas the control output. This variable is called by the flat output,and there is no zero dynamics corresponding to it. However, nosystemic way is provided to find such a flat output, which limitsthe application of this method. Another method is staticallyequivalent output [21-22], where the new output is computedon the basis of the solution of a singular partial differentialequation to induce the prescribed zero dynamics.Inspired by [1,19], an output redefinition-based dynamicinversion control (ORDI) method is developed to achieve stabletracking control for a nonminimum phase HSV. The ORDImethod combines the advantage of linear and nonlinear control.In the first step, the zero dynamics are stabilized by constructinga synthetic output which is a linear combination of the systemoutput, an internal variable and integral tracking error, whoseeffect is very similar to PI control. In the second step, theexternal dynamics are stabilized by dynamic inversion, whichtakes advantage of nonlinear control. As a result, the closedloop system becomes an asymptotic stable linear systemcascaded with a locally stable nonlinear zero dynamics. Basedon the ORDI control architecture, three kinds of ORDIcontrollers are developed for the HSV with different choices ofthe new output definitions.The main contributions of this paper are twofold. Firstly, forthe nonminimum phase system control theory, a systematic wayis proposed to construct a minimum phase output. By selectingthe new output as a linear combination of the system output, aninternal variable and integral tracking error, and using the rootlocus method to determine the combination coefficient, aneffective way is proposed to achieve zero dynamics assignment.Compared to the methods in the aforementioned references [1922], the proposed method is based on the original coordinateand the classical root locus method, making it much easier to becarried out, especially for complex systems. Secondly, for theHSV control problem, the ORDI method is successfully appliedto solve the nonminimum phase problem. The proposed methodhas advantages over existing ones [1,10-15]. Compared to thelinear methods [10-11], the developed method is nonlinearwhich takes advantage of dynamic inversion control. Comparedto [12-14], this method is able to deal with strongernonminimum phase behavior due to the inclusion of the zerodynamics assignment process. Finally, compared to [1,15], thismethod transfers the high-order control problem into twocontrol problems of lower order, which greatly simplifies thecontrol design.The remainder of the paper is organized as follows: InSection II, the HSV model and its zero dynamics analysis arepresented. The main idea of ORDI control is provided inSection III. Then, the ORDI controllers for an HSV will bedeveloped in Section IV and Section V. Next, simulations anddiscussions are given in Section VI. Finally, the conclusions aresummarized in Section VII.II. HYPERSONIC VEHICLE MODELAND ZERO DYNAMICS ANALYSISA. Nonminimum Phase HSV ModelThe model considered in this paper is the rigid-bodylongitudinal model of an air-breathing HSV, which isdeveloped in [1] to verify the control algorithm for HSV withnonminimum phase characteristics. Following [1], the model iswritten as V (T cos α D mg sin γ ) / mh V sin γγ ( L T sin α mg cos γ ) / ( mV )(1)θ QQ M / I yyThe expressions for thrust T , lift L , drag D , and the pitchingmoment M are given by T qS CT (α ) CT φ (α ) φ L qS CL (α ) CLδ e δ e D qSCD (α )(2)M zT T q cS CM (α ) CMδ e δ e whereCT (α ) CT3α 3 CT2α 2 CT1 α CT0CT φ (α ) CTφα α 3 CTφα α 2 CTφα α CTφ32CL (α ) CLα α CL0(3)C D (α ) CDα α 2 CDα α CD02C M (α ) CMα α 2 CMα α CM02TABLE IVEHICLE PARAMETERSMeaningNotationValuemVehicle mass147.9 slug/ftgAcceleration due to gravity32.17 ft/s2I yyMoment of inertia86722.5 slug·ft2/radSReference area17 ft2cMean aerodynamic chord17 ftzTThrust-to-moment coupling coefficient8.36 ftρ0Air density at nominal altitude6.7429 10-5 slugs/ft3hsInverse of the air density exponential decay rate21358.8 fth0Nominal altitude85000 ftThis model comprises five state variables x [V , h, γ , θ , Q ]Trepresenting velocity, altitude, flight path angle, pitch angle andpitch rate, respectively. There are two control inputsu [φ , δ e ] , representing fuel to air ratio and elevatorTdeflection, respectively. The dynamic pressure q in (2) is
calculated by q ρ ( h ) V 2 / 2 with ρ ( h ) ρ0 e ( h h0 ) / hsbeingthe atmospheric density. All the parameter values in the modelare shown in Table I and Table II [1,23].αCL5.9598 rad-1δeCL0.73408 radCL0-0.024377CDα2CDαC0D-1α2CMCMα0M6.8888 ft-15.1390 ftφαCT-10.199043TCTCT20.97141 rad-1-0.074020 rad-1 C φα 3 -14.038 rad-3T-0.019880 rad-2 C φα 2 -1.5839 rad-1CT10.037275 rad-1Ξ y :[V , h]T0TC-0.021635and the admissible flight range is{7500 V 11000 ( ft / s ) , 70000 h 135000 ( ft )}.Thecontrol objective is to design a control law u [φ , δ e ] suchTthat the system outputs track the given constant commandsV * , h* asymptotically in the admissible flight range Ξ y eedbacksHSVIn standard dynamic inversion, the system outputs [V , h ]Tare employed as control outputs. According to [24], zerodynamics are the remaining dynamics when the outputs areidentically zero. Denote the tracking errors aseV V V * , eh h h* . Since the commands V * , h* arenonzero, the tracking errors [ eV , eh ] are used as regulatedToutputs to analyze the zero dynamics.For the convenience of zero dynamics analysis and controldesign, the HSV model (1) is written in an affine form as f g φV VV h V sin γQ f q g qφ φ g qδ δ ewhere(6) g hφ gV sin γ gγφV cos γ(7)g hδ gγδ V cos γWhen the regulated outputs are identically zero, the inputscan be derived by setting the right side of (6) to zero, which are 1 φ 0 gV 0 fV 0 δ e g hφ g hδ f h Substituting (8) into the θ ,Q dynamics yieldsθ Q(8)(9)This is the zero dynamics corresponding to the regulatedToutputs y [ eV , eh ] , which represents the remaining dynamicsVehicle sensorsB. Zero Dynamics Analysis for Standard DynamicInversionθ Qdynamics are e V fV gV 0 φ e h f h g hφ g hδ δ e where f h fV sin γ fγ V cos γQ f q g qφ φ 0 g qδ δ e0Fig. 1. The control system of an HSVγ fγ gγφ φ gγδ δ eqSCT (α ) q cSCM (α ) ) / I yyWith the regulated outputs y [ eV , eh ] , the externalThe control system of an HSV is shown in Fig. 1. We willonly focus on the controller design in this paper. The systemoutputs areTT-1.3642 rad-1T(z(5) g qφ z g qδ q cSCMδ e / I yyT qSCT φ (α ) / I yy ,φCCMδ eDfγ ( qSCL (α ) qSCT (α ) sin α mg cos γ ) / ( mV ) fqValue1.0929 rad-37.9641 rad-2TgV qSCT φ (α ) cos α / m0.69341 rad-10.16277C( qSC (α ) cos α qSC (α ) mg sin γ ) / m gγφ qSC gγδ qSCLδ e / ( mV )T φ (α ) sin α / ( mV ) ,TABLE IIAERODYNAMIC PARAMETERSValueNotation Value NotationNotation fV(4)when eV 0, eh 0 . It can be observed that the zero dynamicsis a second-order nonlinear dynamic equation associated withV * , h* . The stability can be analyzed by Jacobian linearization.Each time a pair of constant commands V * , h* are selectedfrom the range Ξ y , then the zero dynamics are linearized andthe roots are calculated. When the whole range Ξ y is covered,the root map of the linearized zero dynamics is obtained asshown in Fig. 2. It can be observed from Fig. 2 that thelinearized zero dynamics have a positive real root, indicatingTthat the zero dynamics are unstable with respect to y [ eV , eh ] .10.50-0.5-1-30-20-100102030Fig. 2. Root map of the linearized zero dynamics for yUsing standard dynamic inversion means obtaining theTcontrol law [φ , δ e ] by inversion of the external dynamics (6).As a result, when the goal of exact tracking is achieved, i.e., eV 0, eh 0 , the remaining dynamics θ ,Q becomeequivalent to the zero dynamics (9) which are unstable.
Therefore, standard dynamic inversion is not applicable.III. THE MAIN CONCEPT OF ORDIIn order to avoid confusion, it is necessary to state thedifference between system outputs and control outputs. Systemoutputs refer to the system states which are actually desired tofollow some predefined reference commands, while controloutputs refer to the variables used as outputs for the purpose ofcontroller design. For HSV problems, velocity and altitude aresystem outputs, and are used as control outputs in standarddynamic inversion. Pitch angle and pitch rate, which cannot beexpressed as the combination of the system outputs and theirderivatives, are internal variables.The control scheme of ORDI includes two steps as shown inFig. 3.Step 1: Zero dynamics assignment by output redefinition.By constructing a new control output, the zero dynamics canbe modified to make the modified zero dynamics stable.Meanwhile, a proper command should be designed for the newcontrol output to track such that the system output can track thepredefined reference trajectory asymptotically, which will becalled the equal convergence principle.System outputsInternal variablesOutputredefinitionNew external dynamicsDynamicinversionStable zero dynamicsFig. 3. Schematic diagram of ORDIcoefficient at one possible value in the guiding range, the rootmap of the linearized zero dynamics at all equilibriums isobtained. Finally, the feasible values for the coefficient withstable zero dynamics can be selected from the root map.(3) Synthetic output with integralWhen there are model uncertainties, the command θ *calculated from the nominal model will no longer satisfy theequal convergence principle. Then the replaced output, i.e., thealtitude, will have steady-state tracking error for the twomethods above. This problem can be solved by adding anintegral item of the tracking error to the synthetic output, andthen zero steady-state error can be guaranteed in the case ofmodel uncertainties for the replaced output. In this case, theTnew output is chosen as y c [ eV , eh λc1eθ λc 2σ h ] , whereσ h eh and λc1 , λc 2 are parameters to be designed. Comparedto y b , the additional integral item in y c can help to eliminatethe altitude tracking error in steady state when there are modeluncertainties. The effect is very similar to PI control.Step 2: Dynamic inversion control for the new externaldynamics.When the new output is decided, the new external dynamicscan be obtained by taking derivatives of the new output untilthe input appears. Then dynamic inversion control can beutilized to stabilize the new external dynamics.After these two steps, the new external dynamics is fullylinearized while the whole system is partially linearized. As aresult, the closed-loop system becomes a cascade system:e ξ Aeξ(10)e η q ( eξ ,e η )Three ways are provided here to construct a minimum phasenew output:(1) Internal variable as new outputThis way just simply selects the internal variable as a newoutput, thus it only works for certain systems whose internalvariable leads to stable zero dynamics. For an HSV, the internalvariable θ can replace the system output V or h as the output.With θ being an output, its corresponding command θ * isdetermined by the equilibrium manifold [21] to satisfy the equalconvergence principle. Denote eθ θ θ * . Fortunately, it iswhere the external dynamics eξ is an independent asymptoticthe modified zero dynamics are stable. However, in this way thezero dynamics are fixed and will restrict the trackingperformance for the altitude.(2) Static synthetic outputTo make the zero dynamics adjustable, the control output canbe selected as a linear combination of the system output and theinternal variable. In this way more flexibility is available toimprove the tracking performance of the replaced output. ForIn the next two sections, the ORDI controller will bedesigned for an HSV with different choices of new output.found that when y a [ eV , eθ ] are chosen as the control outputs,yban HSV, the new output is chosen as [eV , eh λb eθ ]T,where λb is a parameter to be designed.A systematic way is proposed to determine the combinationcoefficient. First, the root locus of the linearized zero dynamicsat one equilibrium is obtained when the coefficient changes. Aguiding range of the coefficient is derived such that the rootlocates in the left-hand-plane (LHP). Then, by fixing thestable linear system, and the internal dynamics e η is a locallystable nonlinear system influenced by the external states. TheJacobian linearization of (10) is e ξ A 0 eξ (11) e η B C e η where B q / eξ and C q / e η . The stability of (11) isdetermined by a block lower triangular matrix. Since A andC are both stable, system (11) is stable.IV. ORDI WITH INTERNAL VARIABLE AS OUTPUTIn this section, the pitch angle θ is used as an output toreplace the altitude h . In order to satisfy the equal convergenceprinciple, the command θ * should be chosen as the equilibriumcorresponding to V * , h* , i.e., θ * is decided by the equilibriummanifold [21]. For model (4), assume the equilibrium isTTx* V * , h* , 0, θ * , 0 , with u* φ * , δ e* . From V 0 , γ 0and Q 0 , the following equations can be constructed:
(12)*2 h h/h1( 0) s *where q * ρ0 e(V ) is the dynamic pressure2corresponding to V * , h* . In (12), there are three equations withfour unknowns q* , θ * , φ * , δ e* . Once V * , h* are given, then q*V * 8000 ft / s and h* 100000 ft in the region of interest{70000 h 135000 ( ft ) , 5 γ 5 ( deg )} is shown in Fig. 5.It can be observed that any initial value in this region will beattracted to the desired equilibrium point. Similar results can beobtained for other commands.64γ (deg)q * S CT φ (θ * ) φ * CT (θ * ) cos θ * q * SCD (θ * ) 0q * S CL (θ * ) CLδ e δ e* q * S CT φ (θ * ) φ * CT (θ * ) sin θ * mg 0δe ******zT q S CT φ (θ ) φ CT (θ ) q cS CM (θ ) CM δ e 0is determined, and θ * can be solved from (12). Forconvenience, denote the solution as follows(13)θ * qθ ( q * )This is the equilibrium manifold [21]. Therefore the commandθ * for the new output θ can be calculated from (13) for givencommands V * , h* beforehand.A. Zero Dynamics AnalysisWith the regulated outputs y a [ eV , eθ ] , the externalTdynamics are0 φ e V fV gV(14) e θ f q g qφ g qδ δ e When the regulated outputs are identically zero, then the inputscan be derived by setting the right side of (14) to zero, whichare 10 fV (15) g qδ f q Substituting (15) into the h, γ dynamics, the zero dynamicsare obtained as follows:e h V sin γ(16)γ fγ gγφ φ a gγδ δ ea φ a gV a δ e g qφFig. 4 shows the root map of the linearized zero dynamics forall V * , h* in the range Ξ y , from which it can be seen that allthe roots stay in the LHP. So y a is a minimum phase outputand can be used for ORDI control. When the new outputsTy a [ eV , eθ ] are driven to zero, the altitude tracking error isdetermined by the modified zero dynamics (16) and will alsoconverge to zero. However, in this case the modified zerodynamics cannot be adjusted; thus the altitude trackingperformance is restricted.0.04-2-4-60.60.81h (ft)1.21.45x 10Fig. 5. Phase portrait of zero dynamics for yaB. Controller DesignAccording to the new external dynamics (14), design thecontrol inputs as follows: 10 ka11eV ka12σ V fV φ gV (17) δ g e qφ g qδ ka 21e θ ka 22 eθ ka 23σ θ f q where ka11 , ka12 , ka 21 , ka 22 , ka 23 are positive gains to be designed,and σ V , σ θ are integral of the new outputs to cancel modeluncertainties:σ V eV(18)σ θ eθthen the closed-loop system becomes e V ka11eV ka12σ V (19) eθ ka 21e θ ka 22 eθ ka 23σ θ Finally, the velocity tracking error eV is determined by thelinear equation (19), which can be adjusted almost arbitrarily.However, the altitude tracking error eh is determined by themodified zero dynamics (16), which is fixed and will restrictthe altitude tracking performance.V. ORDI WITH SYNTHETIC OUTPUTIn this section, a synthetic output is constructed as a linearcombination of the system states. First, a static synthetic outputis investigated, and then synthetic output with an integral itemis investigated.A. Static Synthetic Output1) Zero Dynamics AssignmentybFirst, the new output is chosen as 0.020-0.02-0.04-0.820-0.6-0.4-0.20Fig. 4. Root map of linearized zero dynamics for yaRemark 1: Root map is employed here to analyze the stabilityof the zero dynamics. However, it is not a global property,which only guarantees stability when the system is close to theequilibrium point. For further investigation, a phase portrait for[eV , eh λb eθ ]T,where λb is a parameter to be designed. The adjustableparameter in the new output gives the opportunity to adjust theeh λb eθ . The newmodified zero dynamics. Denote e bexternal dynamics corresponding to the new output are e V fV gV 0 φ (20) e b f b gbφ gbδ δ e where
f b fV sin γ fγ V cos γ λb f qgbφ gV sin γ gγφV cos γ λb g qφ(21) gbδ gγδ V cos γ λb g qδThe inputs to keep zero outputs are 1 φ b gV 0 fV (22) b δ e gbφ gbδ f b By substituting (22) into the h, γ dynamics, the modified zerodynamics are as follows:e h V sin γ(23)γ fγ gγφ φ b gγδ δ ebThe combination coefficient λb will affect the zerodynamics (23) through φ b , δ eb since λb appears in f b , gbφ , gbδ(see (21)) and then affects φ b , δ eb (see (22)). Therefore thevalue of λb will have a direct bearing on the stability of themodified zero dynamics. A root locus approach will be appliedto determine the feasible value of λb which renders stablezero dynamics.Fig. 6 shows the root locus of the linearized zero dynamicsTat the equilibrium x* [8000,100000, 0,1.324, 0] when λbranges from 1000 to 1000. The roots are divided into twoparts: when 1000 λb 18 , there are two real roots (onepositive, one negative) which go away from the origin as λbincreases; when 19 λb 1000 , there are two LHP complexroots which go close to the origin as λb increases. So theguiding range is λb 19 .50dynamics (23) and will also converge to zero. However, if thereexists model uncertainties, the command θ * calculated fromthe nominal model no longer satisfies the equal convergenceprinciple, thus eh will have steady-state error.2) Controller DesignAccording to the new external dynamics (20), design thecontrol inputs as follows: 1 φ gV 0 kb11eV kb12σ V fV(24) δ g e bφ gbδ kb 21e b kb 22 eb kb 23σ b f b where σ b eb , and kb11 , kb12 , kb 21 , kb 22 , kb 23 are positive gains tobe designed. Then the closed-loop system becomes e V kb11eV kb12σ V (25) eb kb 21e b kb 22 eb kb 23σ b Finally, the velocity tracking error decided by (25) can achievea prescribed tracking performance, while the altitude trackingerror, decided by the modified zero dynamics (23), can also beadjusted to some extent by tuning λb .B. Synthetic Output with Integral1) Zero Dynamics Assignmentyb On the basis of the synthetic outputintegral item can be added to deal with model uncertainties. SoTthe new output is chosen as y c [ eV , eh λc1eθ λc 2σ h ] ,where λc1 , λc 2 are parameters to be designed. Denoteec eh λc1eθ λc 2σ h . The new externalcorresponding to the new output are0 φ e V fV gV gg cδ δ e ec f c cφwhere0[eV , eh λb eθ ] , andynamics(26)f c fV sin γ fγ V cos γ λc1 f q λc 2V sin γg cφ gV sin γ gγφV cos γ λc1 g qφ(27) g cδ gγδ V cos γ λc1 g qδ-50-500The inputs to keep zero outputs are50Fig. 6. Root locus of linearized zero dynamics for yb 1Fig. 7 shows the root map of the linearized zero dynamics forall V * , h* in the range Ξ y with λb 20 and λb 1000 ,respectively. It can be seen that all the roots stay in the LHP. Soy b is a minimum phase output for these two values of λb , andcan be used for ORDI control. The performance for these twovalues will be compared in the simulations.1004λ 20b5000-50-2-100-6-5-4-3-2-1λ 10000γ fγ gγφ φ c gγδ δ ecb20 fV φ c gV(28) c δ e g cφ g cδ f c Due to the introduction of integral, σ h becomes an additionalinternal variable. Substituting (28) into the σ h , θ , Q dynamics,the modified zero dynamics are as follows:σ h ehe h V sin γ(29)In (29), the added zero dynamics σ h eh always have an-4-0.4-0.3-0.2-0.10Fig. 7. Root map of linearized zero dynamics for ybWhen the new outputs y b [ eV , eb ] are driven to zero, theTaltitude tracking error eh is determined by the modified zeroequilibrium eh 0 . Therefore the altitude tracking error eh canconverge to zero as long as (29) is (locally) asymptoticallystable, regardless of model uncertainties.Fixing λc1 1000 , Fig. 8 shows the root locus of thelinearized zero dynamics when λc 2 ranges from 1 to 1. It canbe seen there is one real root and two complex roots. The real
root goes from right to left as λc 2 increases, and lies in LHPwhen λc 2 0 . The complex roots go from left to right as λc 2increases, and lie in the LHP when λc 2 0.14 . So the guidingrange is 0 λc 2 0.14 .21-1modified zero dynamics (29) will converge to its equilibrium.Since eh 0 is always an equilibrium of (29), then the altitudetracking error eh will converge to zero even if there are modeluncertainties.4λ0-50-2-1uncertainties, as shown in Fig. 10, the regulated outputs eV , eθconverge to zero rapidly. Then eh converges to zero slowlyunder the effect of stable zero dynamics. Therefore both systemoutputs have no steady-state tracking error.100 100-10000-4-0.40.504000200-1.540020000.50-0.5-1 -1-1.5-0.5-50-1000.50500-2002000210t ( s) 1000, λ c2 0.02c1400-400t ( s)4002000t ( s)Fig. 10. Nominal model simulation results for ya20-2TIn the case with model uncertainties, as shown in Fig. 11, theregulated outputs [ eV , eθ ] still converge to zero rapidly. Then-0.3-0.2-0.10Fig. 9. Root map of linearized zero dynamics for yc2) Controller DesignAccording to the new external dynamics (26), the controlinputs are designed as follows: 10 kc11eV kc12σ V fV φ gV (30) δ gg kekekfσ cδ e cφc c 21 c c 22 c c 23 cwhere σ c ec , and kc11 , kc12 , kc 21 , kc 22 , kc 23 are positive gains tobe designed. Then the closed-loop system becomes e V kc11eV kc12σ V (31) ec kc 21e c kc 22 ec kc 23σ c Finally, the velocity tracking error is determined by the linearequation (31), and the altitude tracking error is determined bythe modified zero dynamics (29).VI. SIMULATIONS AND DISCUSSIONTo validate the effectiveness of the proposed method, twocases will be considered with Monte Carlo simulations.Case 1: Nominal model simulation. The proposed methodsare applied to the nominal model (4) without uncertainties. Thecommands are given as V * 8000 ft/s and h* 100000 ft .The initial values of the outputs are assumed on an randomeh becomes stable slowly under the effect of stable zerodynamics, but will not converge to zero. Therefore velocity hasno steady-state error while altitude has large steady-state 0510020002540-0.20100200t ( s)300-0.4eh ( f t ) 20, λ c2 0.02c1For y a [ eV , eθ ] , in the ideal case without modeleh ( f t )combination coefficients and can be used for ORDI control.TWhen the new outputs y c [ eV , ec ] are driven to zero, then thek k k 10, k k k k k k 1,a 21b 21c 21a12b12c12a 23b 23c 2325, λ 0.02 .k k k a 22b 22c 22c2θ1e ( deg)0.5eθ ( deg)0λc 2 0.02 . It can be observed that all the roots stay in the LHP.So y c is a minimum phase output for thes
When the nonlinear control method, dynamic inversion, is straightforwardly applied to nonminimum phase systems, it results in exact tracking but the unstable zero dynamics remains an unstable part in the -loop system. closed Therefore, the nonminimum phase of an HSV character prevents the application of standard dynamic inversion and all
A Nonlinear Dynamic Inversion Predictor-Based Model Reference Adaptive Controller for a Generic Transport Model Stefan F. Campbell and John T. Kaneshige T. A non-linear dynamic inversion control law is then: . applied to the fast dynamic inversion illustrated in Fig. 2.
Abstract Dynamic rupture inversion is a powerful tool for learning why and how faults fail, but much more work has been done in developing inversion methods than evaluating how well these methods work. This study examines how well a nonlinear rupture inversion method recovers a set of known dynamic rupture parameters on a synthetic fault based .
by dynamic model inversion is well known and has been applied to the control of high angle of attack ghter aircra [ , ]. e primary drawback of dynamic inversion for aircra ight control is the need for high- delity nonlinear model which must be inverted in real time. However, it is di cult to obtain the exact aircra dynamic model in practice. e .
Adaptive Incremental Nonlinear Dynamic Inversion for Attitude Control of Micro Air Vehicles Ewoud J. J. Smeur, Qiping Chu,† and Guido C. H. E. de Croon‡ Delft University of Technology, 2629 HS Delft, The Netherlands DOI: 10.2514/1.G001490 Incremental nonlinear dynamic inversion is a sensor-based control approach that promises to provide .
cients are treated as unknowns. A collection of nonlinear controllers are studied, specifically an ideal dynamic inversion controller and two switching dynamic inversion controllers. A dynamic inversion controller is modified with an adaptive term that learns the aerodynamic e ects on the equation of motion.
of adaptive Neural Nets to add robustness to the nonlinear dynamic inversion control law [9, 10, 11]. In this paper, the artificial neural network using Radial Basis Function [RBF] is applied in order to improve the robustness of the classical dynamic inversion. This can be done by incorporating this
Kalman Filter (SDKF), is used to transform the matrix inversion into an iterative scalar inversion. The proposed design acts as . ing matrix inversion is the modified Gram-Schmidt algorithm, which is based on QR decomposition. Irturk et al. took this approach in [10] and performed matrix inversion for up to an
Anaesthetic Machine Anatomy O 2 flow-meter N 2 O flow-meter Link 22. Clinical Skills: 27 28 Vaporisers: This is situated on the back bar of the anaesthetic machine downstream of the flowmeter It contains the volatile liquid anaesthetic agent (e.g. isoflurane, sevoflurane). Gas is passed from the flowmeter through the vaporiser. The gas picks up vapour from the vaporiser to deliver to the .
