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A New Estimation Concept for Automotive Suspensions *
Proceedings of the 20th World The Federation of Automatic The International International Federation of Congress Automatic Control Control Proceedings of the 20th9-14, World The International Federation of Congress Automatic Control Toulouse, France, July 9-14, 2017 Toulouse, France, July 2017 Available online at www.sciencedirect.com The International of Automatic Control Toulouse, France,Federation July 9-14, 2017 Toulouse, France, July 9-14, 2017
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PapersOnLine 50-1 (2017) 15622–15626 NewIFAC Estimation Concept for Automotive New Estimation Concept ⋆for Automotive Suspensions New Estimation Concept for Automotive Suspensions ⋆⋆ Suspensions ∗∗ C. Ma ∗∗∗ ∗∗∗ A. Beckerman ∗∗ F. Assadian ∗∗
A. Beckerman ∗ F. Assadian ∗∗ C. Ma ∗∗∗ A. Beckerman F. Assadian C. Ma ∗ ∗∗ A. Beckerman F.Davis, Assadian C. USA Ma ∗∗∗ ∗ ∗ University of California, CA 95616 (e-mail: University of California, Davis, CA 95616 USA (e-mail: ∗ University of California, Davis, CA 95616 USA (e-mail: [email protected]). [email protected]). ∗ ∗∗ Davis, [email protected]). ∗∗University University of of California, California, Davis, CA CA 95616 95616 USA USA (e-mail: (e-mail: University of California, Davis, CA 95616 USA (e-mail: ∗∗ [email protected]). University of California, Davis, CA 95616 USA (e-mail: [email protected]). [email protected]). ∗∗ ∗∗∗ University California, Davis, CA 95616 USA (e-mail: [email protected]). ∗∗∗ Hyundai Motors,ofResearch Division, Division, Suspension Suspension and and Steering Steering Team Team ∗∗∗ Hyundai Motors, Research [email protected]). Hyundai Motors, Research Division, Suspension and Steering Team 2 (e-mail: [email protected]). 2 (e-mail: [email protected]). ∗∗∗ Hyundai Motors, [email protected]). Division, Suspension and Steering Team 2 (e-mail: 2 (e-mail: [email protected]). Abstract: Abstract: A A new new approach approach for for vehicle vehicle suspension suspension damping damping force force and and road road input input estimation estimation Abstract: approach for vehicle suspension damping force and roadController input estimation is proposed. proposed.AA Anew Multi-Input-Multi-Output Youla Parameterization Parameterization based Controller Output is Multi-Input-Multi-Output Youla based Output Abstract: for vehicle suspension damping force sprung and road input estimation is proposed. Anew Multi-Input-Multi-Output Youla Parameterization based Controller Output Observer is A used. Inapproach this scheme, scheme, two measurements measurements are required: required: and unsprung mass Observer is used. In this two are sprung and unsprung mass is proposed. A Multi-Input-Multi-Output Youla Parameterization based Controller Output Observer is used. In this scheme, two measurements are required: sprung and unsprung mass acceleration. This technique is beneficial in that no assumptions are made regarding damper acceleration. This technique is beneficial in that no assumptions are made regarding damper Observer is used. Indynamics. this scheme, two measurements areinput required:are sprung and unsprung mass acceleration. beneficial in that assumptions made regarding damper characteristics and both road content and characteristicsThis andtechnique dynamics.is Additionally, Additionally, both no road input frequency frequency content and discrete discrete acceleration. This technique is beneficial in that no assumptions are made regarding damper characteristics and dynamics. Additionally, both road input frequency content and discrete events are are estimated. estimated. Two Two simulation simulation examples examples are are used used to to illustrate illustrate the the technique: technique: Skyhook Skyhook events characteristics and dynamics. Additionally, both roughness road frequency content and Skyhook discrete events are estimated. Two force simulation usedinput to coefficient illustrate the technique: based loop control and estimation. based closed closed loop damping damping force controlexamples and road roadare roughness coefficient estimation. events are estimated. Two force simulation used to coefficient illustrate the technique: Skyhook based closed loop damping controlexamples and roadare roughness estimation. © 2017,closed IFAC loop (International of Automatic Control) Hosting by Elsevierestimation. Ltd. All rights reserved. based dampingFederation force control and road roughness coefficient Keywords: Estimation; Estimation; vehicles; vehicles; suspension; suspension; damping damping force; force; road road input; input; controller controller output output Keywords: Keywords: Estimation; vehicles; suspension; damping force; road input; controller output observer; Youla Youla observer; Keywords: Estimation; vehicles; suspension; damping force; road input; controller output observer; Youla observer; Youla 1. INTRODUCTION INTRODUCTION on model model based based damper damper estimation estimation (i.e. (i.e. Rajamani Rajamani et et al al 1. on 1. INTRODUCTION on modelthese based damperoften estimation al (1995)), these methods often make aa (i.e. greatRajamani number of ofetasas(1995)), methods make great number 1. INTRODUCTION on model based damper estimation (i.e. Rajamani et al (1995)), these methods often make a great number of assumptions regarding the damper and its properties. In this The proliferation proliferation of of active active and and semi-active semi-active suspensions suspensions sumptions regarding the damper and its properties. In this The (1995)), these methods often make a great number of asregarding theregarding damper and properties. In this The proliferation of active to and semi-active suspensions paper, the damper are provides new opportunities to improve rider comfort comfort and sumptions paper, no no assumptions assumptions regarding the its damper are needed. needed. provides new opportunities improve rider and sumptions regarding theregarding damper and its properties. In this The proliferation of active to and semi-active suspensions paper, no assumptions the damper are needed. provides new opportunities improve rider comfort and vehicle handling characteristics. For example, the damping vehicle handling characteristics. For example, the damping Road roughness is an important characteristic of road paper, no assumptions regarding the damper are needed. Road roughness is an important characteristic of road provides new opportunities to improve rider comfort and vehicle handling characteristics. For example, the damping force may may be be controlled controlled in in response response to to body body velocities velocities Road force roughness is an important characteristic of road profile. It is a single value representative of the frequency profile. It is a single value representative of the frequency vehicle handling characteristics. For example, the damping force may be controlled in response to body velocities or changing changing road road conditions. conditions. These These systems systems may may operate operate Road roughness is an important characteristic of road or profile. Itof isthe a single value representative of the coefficient frequency content of the road. A larger road roughness coefficient content road. A larger road roughness force may be controlled in response to body velocities or changing conditions. These systems may operate through the road control of damper damper valve position, magne- profile. Itofisthe a single value representative ofoccupants the coefficient frequency through the control of valve position, magnecontent road. A larger road roughness is typically typically more uncomfortable for vehicle vehicle and more uncomfortable for occupants and or changing road conditions. These systems may operate through the control of damper valve position, magnetorheological fluid, or hydraulic actuation to name few. is content oftothe road. A larger road roughness coefficient torheological fluid, or hydraulic actuation to name aa few. is typically more uncomfortable for vehicle occupants and may lead undesirable body motions. It is favorable to may lead to undesirable body motions. It is favorable to through the control of damper valve position, magnetorheological fluid, or hydraulic actuation to name a few. Such systems systems would would greatly greatly benefit benefit from from estimations estimations of of is typically more uncomfortable for vehicle occupants and Such may lead to undesirable body motions. It is favorable to adapt the suspension to changing road conditions, but adapt the suspension to changing road conditions, but torheological fluid, or hydraulic actuation to name a few. Such systems would greatly benefit from estimations of damping force force and and road road input. input. This This paper paper proposes proposes aa may lead undesirable motions. Itconditions, is favorable to damping adapt theto suspension tobody changing road but this often requires direct measurement through expensive this often requires direct measurement through expensive Such systems would greatly benefit from estimations of damping force and road input.both Thisdamping paper proposes simple strategy strategy for estimating estimating both damping force and anda adapt the suspension to changing road conditions, but simple for force this often requires direct measurement through expensive sensing equipment. equipment. While While an an approach approach was was developed developed in in damping force and road input. Thisdamping paperwhile proposes simple for estimating both force anda sensing vertical strategy road velocity profile simultaneously making this often requires direct measurement through expensive vertical road velocity profile simultaneously while making sensing equipment. While an approach was developed in Gonzlez et al. (2008), it requires 12 (compared with four Gonzlez et al. (2008), it requires 12 (compared with four simple strategy for estimating both damping force and vertical road velocity profile simultaneously while making no assumptions assumptions regarding regarding damper damper or or road. road. This This estimation estimation sensing equipment. While an approach was developed in no Gonzlez et al. (2008), it requires 12 (compared with four in this paper) parameters and assumes linear damping this paper) parameters and assumes linear damping -vertical road simultaneously while making in no assumptions regarding damper or road. This estimation technique is velocity based onprofile the Controller Output Observer Gonzlez et al. (2008), it requires 12 (compared with four technique is based on the Controller Output Observer in this paper) parameters and assumes linear damping this paper paper makes makes no no assumptions assumptions on on linearity linearity or or damper damperthis no assumptions regarding damper or road. This estimation technique isfirst based on the Observer framework first presented in Ozkan al. and this paper) parameters and assumes linearasor damping framework presented inController Ozkan et etOutput al. (2008) (2008) and in this paper makes no assumptions on linearity damperconfiguration. This fact is very important, dampers This fact is very important, as dampers technique isfirst based on the Controller Observer framework etOutput al. (2008) and configuration. applied again again in Varnhagen al. for this paper makes no fact assumptions on region. linearity ordampers damper applied in presented Varnhageninet et Ozkan al. (2014) (2014) for longitudinal longitudinal configuration. This is very important, as often behave well outside their linear Additionally, behave well outside their linear region. Additionally, framework first presented inet Ozkan et al. (2008) and often applied again Varnhagen al.the (2014) for longitudinal dynamics. As proof of concept, concept, the technique is applied applied configuration. This fact is very important, as dampers dynamics. As aainproof of technique is often behave well outside their linear region. Additionally, the technique technique in in Gonzlez Gonzlez et et al. al. (2008) (2008) does does not not estimate estimate applied Varnhagen et al.the (2014) for longitudinal dynamics. As ainproof concept, technique is applied through again simulation foroftwo two applications: Skyhook based the often behave well outside their region. Additionally, through simulation for applications: aa Skyhook based the technique in Gonzlez et al.linear (2008) does not estimate discrete road events like speed bumps and potholes. discrete road events like speed bumps and potholes. dynamics. As a proof of concept, the technique is applied through simulation for force two applications: Skyhook based the closed loop loop damping control roughness technique in Gonzlez et al.bumps (2008)and does not estimate closed damping force control and and aroad road roughness road events like speed potholes. through simulation for force two applications: Skyhook based discrete closed loop damping control and aroad roughness coefficient estimator. discrete road events like2. speed bumps and potholes. coefficient estimator. 2. MODEL MODEL closed loopestimator. damping force control and road roughness coefficient 2. MODEL Skyhook based based controllers are are common common in in the the automoautomocoefficient estimator. Skyhook controllers 2. MODELin this estimation frameIt will be shown in Section Skyhook based controllers are common in the automowill be shown in Section 33 that, that, in this estimation frametive industry industry (Karnopp (Karnopp et tive et al. al. (1974)). (1974)). To To summarize, summarize, aa It It will be shown in Section 3 that, in thison frameSkyhook based controllers are commonwhile the automo-a work, work, unknown system inputs acting onestimation the real real system system tive industry (Karnopp (1974)). Toin summarize, unknown system inputs acting the reference damping force etis is al. calculated low-level reference damping force calculated while aa low-level will be shown in Section 3inputs that, in this framework, unknown system inputs acting onestimation the real system tive industry (Karnopp etis al. (1974)). To summarize, a It are treated as controllable to the estimator. These reference damping force calculated while a low-level are treated as controllable inputs to the estimator. These controller attempts to apply it to the vehicle. In practice, controller attempts to apply it to the vehicle. In practice, work, unknown system inputs acting on the real system are treated as controllable inputs to the estimator. These reference damping force is calculated while a low-level inputs are not measurable, so a controller is developed that controller attempts to apply it to the vehicle. In practice, inputs are not measurable, so a controller is developed that damping force is often estimated through lookup tables damping force is often estimated through lookup tables are treated as measurable, controllable inputs to the estimator. These inputs are not so a controller is developed that controller attempts to apply it to the vehicle. In practice, drives the error between some measured and estimated damping force is often estimated through lookup tables the error between some measured and estimated that may may be be functions functions of of current current valve valve position position and and relative relative drives that inputs are not measurable, so a controller isunknown developed that drives the error between some measured and estimated damping force is often estimated through lookup tables signals to zero. It is then assumed that the inputs that may be functions of current valve position and relative signals to zero. It is then assumed that the unknown inputs velocity across across the the damper. damper. This This practice practice suffers suffers from from drives the error between some measured and estimated velocity signals to zero. It is then assumed that the unknown inputs that may be functions of current valve position and relative acting on on the the real real system system are are equivalent equivalent to to the the controlcontrolvelocity across the damper. Thisare practice suffers various shortcomings: shortcomings: all dampers dampers are different, tablesfrom are acting various all different, tables are signals to zero. It is then assumed that the unknown inputs acting on the real system are controller. equivalent to the controlvelocity across theof damper. Thisare practice suffers from lable inputs determined by the This framework various shortcomings: all dampers different, tables are lable inputs determined by the controller. This framework not representative dynamic systems, damper properties not representative of dynamic systems, damper properties acting ona the realwith system are equivalent to and the controllable inputs determined by the controller. This framework various shortcomings: all dampers are different, tables are requires model the required inputs outputs not representative of dynamic systems, damper properties a model with the required inputs and outputs change with with operating operating conditions conditions and and time, time, and and tables tables requires change inputs determined by the controller. This framework requires a model with the required and outputs not representative of While dynamic systems, properties representing the unknown unknown inputs andinputs measured signals, change with operating conditions anddamper time,been and tables lable representing the inputs and measured signals, include limited data. many papers have been written include limited data. While many papers have written requires a model with the required inputs and outputs representing the unknown inputs and measured signals, change with operating conditions and time, and tables respectively. Here, Here, the the unknown unknown inputs inputs are are damping damping force include limited data. While many papers have been written respectively. force representing the unknown inputs and measured signals, respectively. Here, the unknown inputs are damping force ⋆ Research include limited data. While many papers have been written and vertical road input velocity, and the measured ⋆ supported by by the the Hyundai Hyundai Center Center of of Excellence Excellence in in Vehicle Vehicle and vertical road input velocity, and the measured signals signals Research supported respectively. Here, the unknown inputs are dampingsignals force ⋆ and vertical road input velocity, and the measured Research supported by the Hyundai Center of Excellence in Vehicle are sprung and unsprung mass acceleration. Dynamic Systems & Control. are sprung and unsprung mass acceleration. Dynamic Systems & Control. ⋆ Research and verticaland road input velocity, and the measured signals supported by the Hyundai Center of Excellence in Vehicle are sprung unsprung mass acceleration. Dynamic Systems & Control. are sprung and unsprung mass acceleration. Dynamic Systems & Control.
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The model chosen, a quarter car, is simple and commonly used in the automotive industry - see Figure 1. The two masses represents the tire (unsprung mass, mus ) and onequarter of the body (sprung mass, ms ). The linear spring elements include suspension (ks ) and tire (kt ) stiffness. The controllable inputs to the estimator (as required for estimating the real unknown inputs) are a force (Fa ) and velocity (vi ) actuator representative of the damping force and vertical road input velocity, respectively. A linear damper (bs ) is included to stabilize the plant and make for easier control of Fa and vi . It should be noted that the total damping force estimate will be the combination of the passive damping force and Fa . Fig. 1. Quarter Car Schematic. The suspension stiffness can be obtained from the Ride Rate, or relationship between vertical wheel stroke (qs ) and spring force, and is therefore represented by a vertical spring. The damping force (Fd ), however, is modified by a lever ratio. That is, the deflection is approximately ∂q ′ proportional to (by ∂qss ), but not equal to, the wheel stroke. This is shown in Figure 2. Fig. 2. Damper configuration with lever ratio. In future steps the combined effects of the passive damping force and controllable force will be converted to the total damping force at this new position through Equation 1, where Fa is the controllable input and bs (ˆ vus − vˆs ) is the passive damping force. Hats (ˆ a) above variables denote estimated quantities. ′
∂q Fˆd = ( s )−1 (Fˆa + bs (ˆ vus − vˆs )) ∂qs
(1)
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T
y = [v˙ s v˙ us ] T ˆ = Fˆa vˆi u
Youla Parameterization was chosen as the control methodology for its simplicity and robustness. This technique allows for the shaping of the closed loop response directly. The block diagram in Figure 3 is reorganized and shown in Figure 4. These systems are equivalent. Fig. 4. Block diagram with feedback. It is now apparent that the boxed region in Figure 4 ˆ p ) can be replaced by an equivalent (containing Gc and G transfer function matrix, Y . This is shown in Figure 5. Fig. 5. Block diagram without feedback. Y is therefore equivalent to the closed loop response from ˆ and given by Equation 4. ym to u ˆ p Gc )−1 Gc Y = (I + G
In this paper, the measured signals are sprung (v˙ s ) and unsprung (v˙ us ) mass acceleration (Equation 2). The estimated unknown inputs are the additional damping force (Fˆa as described in Section 2) and road input velocity (vi ) (Equation 3).
(4)
It is easily shown that the closed loop response from ym to yˆ (or complimentary sensitivity function, T ) is given by Equation 5. ˆ pY T =G
Fig. 3. Reorganized block diagram.
(3)
ˆ p are If the input-output dynamic characteristics of G representative of the actual plant, and yˆ follows ym , then ˆ must be representative of the actual input - this assumes u no other major inputs act on the system. However, if the ˆ p is not representative of the real system, u ˆ behavior of G will compensate to ensure yˆ and ym are similar, but do ˆ Therefore, so at the expense of an accurate estimate of u. the plant input-output relationships must be considered carefully. This paper makes use of the quarter car, a model commonly used in practice.
3. YOULA CONTROLLER OUTPUT OBSERVER DESIGN In a Youla Parameterization based Controller Output Observer, unknown system inputs (u), states (x), and outputs (y) may be estimated. Youla based control techniques are well documented in Youla et al. (1976), Skogestad et al. (1976), and Assadian (2015), while Controller Output Observers were first presented in Ozkan et al. (2008). Unknown system inputs are treated as controllable ˆ to the estimator. Using Youla parameterization inputs (u) techniques, a controller (Gc ) is used to drive the error ˆ output between certain measured (ym ) and estimated (y) signals to zero. The estimated outputs signals come from a ˆ p ) that is propagated forward in time model of the plant (G ˆ Any additional output signal can be reconstructed by u. ˆ p . This control structure is from the internal states of G shown in Figure 3.
(2)
(5)
Next, consider the sensitivity function, or S, as defined as I − T . This represents the closed loop response between output disturbances and estimated outputs. Good performance and robustness are achieved when certain conditions on T , S, and Y are met. This is accomplished by shaping Y alone. For good tracking and disturbance rejection, T should be stable and close to identity and very small at low and high frequencies, respectively. This determines, from the constraint that S = I − T , that S should be very small and identity at low and high frequencies, respectively. Additionally, Y must be proper and stable. Finally, the desired controller, Gc , is obtained from Equation 6. Gc = S −1 Y
(6)
In summary, the unknown system inputs and outputs are estimated through Equations 7 and 8, respectively, where Y and T are defined in Equations 4 and 5, respectively.
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ˆ = Y ym u yˆ = T ym
(7) (8)
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3.1 Smith McMillan Form
250
ˆ p is a transfer function matrix (i.e. has MultipleWhen G Inputs and Multiple-Outputs, MIMO), the system is decoupled using the Smith McMillan transformation. A diagonalized version of Y is then designed so diagonalized versions of T and S have desired properties. Once complete, the transformation is inversed and MIMO transfer function matrices obtained (Assadian (2015)).
200
Magnitude, dB
150
ˆ p through The diagonalized plant (Gsm ) is related to G two unimodular matrices (Ul , Ur ) as shown in Equation 9.
100 50 0 -50 -100 -150
Gp =
UL−1 Gsm UR−1
(9)
Ul and Ur are defined by Equations 10 ans 11, where Np and Nsm are the numerators, and Dp and Dsm the ˆ p and Gsm , respectively. denominators, of G −1 −1 UL = (Np Nsm ) −1 UR = Dp Dsm
(10) (11)
The diagonal elements of Gsm are shown in Equations 12 and 13, where G∗sm1 and G∗sm2 represent the stable poles and zeros of Gsm1 and Gsm2 , respectively. Note that seperation of the stable and unstable components is required for control design. This will be shown in Section 3.2. Gsm1 = sG∗sm1 Gsm2 = s2 G∗sm2
(12) (13)
T S Y
10 -5
10 0
10 4
10 -5
10 0
Frequency, Hz
10 4
Frequency, Hz
Fig. 6. (Left) Bode plot of T1 , S1 and Y1 . (Right) Bode plot of T2 , S2 , Y2 the magnitude of T1 and T2 are approximately one for the majority of the frequency range as desired. T , S and Y are obtained from Equations 16 - 18, where UR and UL are the unimodular matrices from Equations 10 and 11 and MY is a diagonal matrix whose elements are Y1 and Y2 from Equations 14 and 15. (16) (17) (18)
Y = UR MY UL T = Gp Y S =I −T
The Bode plots of these new transfer function matrices are shown in Figure 7.
k1 (G∗sm1 )−1 2 )(s + p ) (s + p1 )(s2 + 2ζ1 ωn1 s + ωn1 2 ∗ −1 k2 (Gsm2 ) Y2 = 2 2 )(s2 + 2ζ ω s + ω 2 ) (s + 2ζ2 ωn2 s + ωn2 3 n3 n3 Y1 =
(14)
200 0 -200 200
Magnitude, dB
The requirements on T , S, and Y discussed in Section 3 now apply to their diagonal elements (Ti , Si , and Yi ). That is, Ti should be stable and close to one and zero at low and high frequencies, respectively, while the inverse is true for Si . Additionally, the diagonal elements of Yi must be proper and stable. Y1 and Y2 are shown in Equations 14 and 15, where G∗sm1 and G∗sm2 are defined in Equation 12 and 13.
Magnitude, dB
400
3.2 Controller Design
0 -100 -200
(15)
Unfortunately, the unstable zeros shown in Equations 12 and 13 can’t be canceled without introducing unstable poles into Y1 and Y2 . A compromise is made by introducing a pole or poles extremely close to the origin to essentially cancel the unstable zeros. This results in the magnitude of T1 and T2 equaling something smaller than one at the lowest frequencies and the inability for steady state tracking. The frequency response of the first and second diagonal element of T , S, and Y are shown in the left and right pane of Figure 6, respectively. It is apparent that
T S Y
100
10 -5
10 0
Frequency, Hz
10 4 10 -5
10 0
10 4
Frequency, Hz
Fig. 7. MIMO transfer function frequency response.
4. APPLICATIONS The utility of this estimator is made apparent by two example simulation applications: Skyhook based damping force estimation for closed loop control in active/semiactive suspensions, and road roughness coefficient estimation.
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4.1 Skyhook Based Control
Fd,SH = bSH vs
200
Damping Force, N
The control methodology chosen is Skyhook control, a common strategy with variants deployed on many vehicle brands. It was proposed that an optimal suspension is one in which the sprung mass is connected through a linear damper to some inertial reference. Due to the inability to build such a contraption, this ideal damping force is instead applied within the suspension between sprung and unsprung masses through either force actuator or control of the damper. Although sub-optimal, this strategy has proven highly effective. The ideal damping force is given by Equation 19, where bSH is a tuning parameter and vs is the sprung mass vertical velocity.
The control structure in Figure 8 is proposed and described in this paragraph. A simple controller (Gc,SH , different from that in Section 3) is developed that drives the effective damping force towards the ideal Skyhook damping force (Fd,SH ) using the controlled input force (Fi ). This effective damping force is the combination of both passive damping and the controllable damping force. The estimate of the damping force (Fˆd ) will inherently contain both these forces. Note that u and Est represents additional system inputs and the estimator, respectively. Fig. 8. Damping force feedback control structure. Success of the estimator is measured by the level of degradation in root-mean-square (RMS) heave acceleration as this value is strongly correlated with ride comfort. A comparison is made between uncontrolled, controlled with no estimator, and controlled with estimator. This is done in two scenarios: (1) driving over a motorway at 100 KPH and (2) driving over a large bump (0.1 m tall) at 40 KPH. The results are summarized in Table 1 where RMS heave acceleration is shown for the uncontrolled case and percent change for cases with control. Both control scenarios are shown: with direct damping force feedback, and with estimated damping force feedback. It is apparent that little degradation in RMS heave acceleration exists due to the inclusion of the estimator versus direct measurement feedback. Additionally, Figure 9 shows the estimated versus actual damping force for motorway conditions using control.
100 0 -100 -200 -300
(19)
A fully active Skyhook is applied to a high order vehicle model (Beckerman et al (April 2016), Beckerman et al (July 2016)). Each corner suspension is modeled as a nonlinear spring and damper (lookup tables) with an active element capable of applying ideal forces. As damping force is used in feedback with the controller, either a damping force measurement or estimate is required. In this paper, results are obtained for both scenarios and compared. It will be shown that the influence of the estimator does little to inhibit the performance of the controller.
Actual Estimate
300
14
14.5
15
15.5
16
16.5
Time, s
Fig. 9. Actual versus estimated damping force. measure the road directly, but the method proposed in this paper needs only sprung and unsprung mass accelerations. A road roughness coefficient (G0 ) is a parameter that fits a special curve as shown in Equation 20 to a Power-SpectralDensity (PSD) plot of the road - v is the spatial frequency and v0 is the cutoff spatial frequency, both in rad/m, and Gz is the road PSD in m2 /(rad/m) (Assadian (2015)). Such a plot is shown in Figure 10 where results are shown for a motorway (bottom) and farm road (top). Shown in gray, blue, and red are the PSD given the actual road data, the best fit of the actual road PSD, and a best fit of the estimated road PSD, respectively. The general shape of this plot is a result of decreasing road variation over smaller distances. Gz (v) = G0 [1 + (v0 /v)2 ]/(2πv)2
(20)
It is apparent from Figure 10 that the best fit of both the actual and estimated road data are very similar over a large frequency range. Additionally, the actual road roughness coefficient G0 varies by 3.3 and 2.4 percent for the motorway and farm road, respectively. 5. CONCLUSION The method proposed in this paper allows for the simple estimation of vehicle suspension damping force and road input velocity. Estimation of these two quantities is very useful in active or semi-active suspension configurations. A Youla Parameterization based Controller Output Observer was utilized while only requiring that sprung and unsprung mass accelerations be measured. To validate the concept, two tests were performed. The first test consisted of a Skyhook based control law for an active suspension
4.2 Road Roughness Estimation Road roughness estimation is important for suspension parameter adaptation. Expensive sensors may be used to 16195
Table 1. Heave acceleration results for the uncontrolled and controlled case, with and without the estimator Test No control [RMS, m/s2 ] Control without estimator [% change] Control with estimator [% change]
Motorway 0.0279 -52.3176 -46.5230
Bump 0.1192 -30.2342 -22.0213
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troller output observer for disturbance rejection.” SAE International Journal of Passenger Cars-Mechanical Systems 7.2014-01-0125 (2014): 65-72. Youla, D., H. Jabr, and J. Bongiorno. ”Modern WienerHopf Design of Optimal Controllers–Part II: The Multivariable Case.” IEEE Transactions on Automatic Control 21.3 (1976): 319-38. Web.
2
PSD, m /(rad/m)
10 -5
10 -10 Raw - Actual Fit - Actual Fit - Estimate
10 -1
10 0
10 1
10 2
Spatial frequency, rad/m
Fig. 10. Power Spectral Density of road profile for a motorway (bottom) and farm road (top). where damping force feedback was utilized. Minimal degradation in performance due to estimation was witnessed. The second test consisted of road roughness coefficient estimation. It was shown that values between actual and estimated coefficients differed by no more than 3.3 percent. Additionally, this method for road input estimation applies to discrete events in addition to the frequency analysis performed in this paper. REFERENCES Assadian, F., MAE 272 - Theory and Design of Control Systems Class Notes. 2015, University of California, Davis: Davis, CA. Beckerman, Alex, and Francis Assadian, Dr. Generalized Template for Suspension Dynamic Modeling With the Use of Bond Graphs. Proc. of Spring Simulation MultiConference 2016, Pasadena, CA. 2016. Beckerman, Alex K., and Francis Assadian, Dr. ”ICBGM.” Society for Modeling & Simulation. Proc. of Summer Simulation Symposium, Montreal, Quebec, Canada. N.p.:n.p., n.d. N. pag. Print. Karnopp, D., M. J. Crosby, and R. A. Harwood. ”Vibration Control Using Semi-Active Force Generators.” Journal of Engineering for Industry 96.2 (1974): 619. Web. Gonzlez, A., E.j. O’brien, Y.-Y. Li, and K. Cashell. ”The Use of Vehicle Acceleration Measurements to Estimate Road Roughness.” Vehicle System Dynamics 46.6 (2008): 483-99. Web. Ozkan, Basar, Donald Margolis, and Marco Pengov. ”The controller output observer: Estimation of vehicle tire cornering and normal forces.” Journal of Dynamic Systems, Measurement, and Control 130.6 (2008): 061002. Rajamani, R., and J.k. Hedrick. ”Adaptive Observers for Active Automotive Suspensions: Theory and Experiment.” IEEE Transactions on Control Systems Technology 3.1 (1995): 86-93. Web. Skogestad, Sigurd, and Ian Postlethwaite. Multivariable Feedback Control: Analysis and Design. Chichester: Wiley, 1996. Print. Varnhagen, Scott, and Donald Margolis. ”Longitudinal slip ratio control of electric powertrains using a con16196
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PapersOnLine 50-1 (2017) 15622–15626 NewIFAC Estimation Concept for Automotive New Estimation Concept ⋆for Automotive Suspensions New Estimation Concept for Automotive Suspensions ⋆⋆ Suspensions ∗∗ C. Ma ∗∗∗ ∗∗∗ A. Beckerman ∗∗ F. Assadian ∗∗
A. Beckerman ∗ F. Assadian ∗∗ C. Ma ∗∗∗ A. Beckerman F. Assadian C. Ma ∗ ∗∗ A. Beckerman F.Davis, Assadian C. USA Ma ∗∗∗ ∗ ∗ University of California, CA 95616 (e-mail: University of California, Davis, CA 95616 USA (e-mail: ∗ University of California, Davis, CA 95616 USA (e-mail: [email protected]). [email protected]). ∗ ∗∗ Davis, [email protected]). ∗∗University University of of California, California, Davis, CA CA 95616 95616 USA USA (e-mail: (e-mail: University of California, Davis, CA 95616 USA (e-mail: ∗∗ [email protected]). University of California, Davis, CA 95616 USA (e-mail: [email protected]). [email protected]). ∗∗ ∗∗∗ University California, Davis, CA 95616 USA (e-mail: [email protected]). ∗∗∗ Hyundai Motors,ofResearch Division, Division, Suspension Suspension and and Steering Steering Team Team ∗∗∗ Hyundai Motors, Research [email protected]). Hyundai Motors, Research Division, Suspension and Steering Team 2 (e-mail: [email protected]). 2 (e-mail: [email protected]). ∗∗∗ Hyundai Motors, [email protected]). Division, Suspension and Steering Team 2 (e-mail: 2 (e-mail: [email protected]). Abstract: Abstract: A A new new approach approach for for vehicle vehicle suspension suspension damping damping force force and and road road input input estimation estimation Abstract: approach for vehicle suspension damping force and roadController input estimation is proposed. proposed.AA Anew Multi-Input-Multi-Output Youla Parameterization Parameterization based Controller Output is Multi-Input-Multi-Output Youla based Output Abstract: for vehicle suspension damping force sprung and road input estimation is proposed. Anew Multi-Input-Multi-Output Youla Parameterization based Controller Output Observer is A used. Inapproach this scheme, scheme, two measurements measurements are required: required: and unsprung mass Observer is used. In this two are sprung and unsprung mass is proposed. A Multi-Input-Multi-Output Youla Parameterization based Controller Output Observer is used. In this scheme, two measurements are required: sprung and unsprung mass acceleration. This technique is beneficial in that no assumptions are made regarding damper acceleration. This technique is beneficial in that no assumptions are made regarding damper Observer is used. Indynamics. this scheme, two measurements areinput required:are sprung and unsprung mass acceleration. beneficial in that assumptions made regarding damper characteristics and both road content and characteristicsThis andtechnique dynamics.is Additionally, Additionally, both no road input frequency frequency content and discrete discrete acceleration. This technique is beneficial in that no assumptions are made regarding damper characteristics and dynamics. Additionally, both road input frequency content and discrete events are are estimated. estimated. Two Two simulation simulation examples examples are are used used to to illustrate illustrate the the technique: technique: Skyhook Skyhook events characteristics and dynamics. Additionally, both roughness road frequency content and Skyhook discrete events are estimated. Two force simulation usedinput to coefficient illustrate the technique: based loop control and estimation. based closed closed loop damping damping force controlexamples and road roadare roughness coefficient estimation. events are estimated. Two force simulation used to coefficient illustrate the technique: Skyhook based closed loop damping controlexamples and roadare roughness estimation. © 2017,closed IFAC loop (International of Automatic Control) Hosting by Elsevierestimation. Ltd. All rights reserved. based dampingFederation force control and road roughness coefficient Keywords: Estimation; Estimation; vehicles; vehicles; suspension; suspension; damping damping force; force; road road input; input; controller controller output output Keywords: Keywords: Estimation; vehicles; suspension; damping force; road input; controller output observer; Youla Youla observer; Keywords: Estimation; vehicles; suspension; damping force; road input; controller output observer; Youla observer; Youla 1. INTRODUCTION INTRODUCTION on model model based based damper damper estimation estimation (i.e. (i.e. Rajamani Rajamani et et al al 1. on 1. INTRODUCTION on modelthese based damperoften estimation al (1995)), these methods often make aa (i.e. greatRajamani number of ofetasas(1995)), methods make great number 1. INTRODUCTION on model based damper estimation (i.e. Rajamani et al (1995)), these methods often make a great number of assumptions regarding the damper and its properties. In this The proliferation proliferation of of active active and and semi-active semi-active suspensions suspensions sumptions regarding the damper and its properties. In this The (1995)), these methods often make a great number of asregarding theregarding damper and properties. In this The proliferation of active to and semi-active suspensions paper, the damper are provides new opportunities to improve rider comfort comfort and sumptions paper, no no assumptions assumptions regarding the its damper are needed. needed. provides new opportunities improve rider and sumptions regarding theregarding damper and its properties. In this The proliferation of active to and semi-active suspensions paper, no assumptions the damper are needed. provides new opportunities improve rider comfort and vehicle handling characteristics. For example, the damping vehicle handling characteristics. For example, the damping Road roughness is an important characteristic of road paper, no assumptions regarding the damper are needed. Road roughness is an important characteristic of road provides new opportunities to improve rider comfort and vehicle handling characteristics. For example, the damping force may may be be controlled controlled in in response response to to body body velocities velocities Road force roughness is an important characteristic of road profile. It is a single value representative of the frequency profile. It is a single value representative of the frequency vehicle handling characteristics. For example, the damping force may be controlled in response to body velocities or changing changing road road conditions. conditions. These These systems systems may may operate operate Road roughness is an important characteristic of road or profile. Itof isthe a single value representative of the coefficient frequency content of the road. A larger road roughness coefficient content road. A larger road roughness force may be controlled in response to body velocities or changing conditions. These systems may operate through the road control of damper damper valve position, magne- profile. Itofisthe a single value representative ofoccupants the coefficient frequency through the control of valve position, magnecontent road. A larger road roughness is typically typically more uncomfortable for vehicle vehicle and more uncomfortable for occupants and or changing road conditions. These systems may operate through the control of damper valve position, magnetorheological fluid, or hydraulic actuation to name few. is content oftothe road. A larger road roughness coefficient torheological fluid, or hydraulic actuation to name aa few. is typically more uncomfortable for vehicle occupants and may lead undesirable body motions. It is favorable to may lead to undesirable body motions. It is favorable to through the control of damper valve position, magnetorheological fluid, or hydraulic actuation to name a few. Such systems systems would would greatly greatly benefit benefit from from estimations estimations of of is typically more uncomfortable for vehicle occupants and Such may lead to undesirable body motions. It is favorable to adapt the suspension to changing road conditions, but adapt the suspension to changing road conditions, but torheological fluid, or hydraulic actuation to name a few. Such systems would greatly benefit from estimations of damping force force and and road road input. input. This This paper paper proposes proposes aa may lead undesirable motions. Itconditions, is favorable to damping adapt theto suspension tobody changing road but this often requires direct measurement through expensive this often requires direct measurement through expensive Such systems would greatly benefit from estimations of damping force and road input.both Thisdamping paper proposes simple strategy strategy for estimating estimating both damping force and anda adapt the suspension to changing road conditions, but simple for force this often requires direct measurement through expensive sensing equipment. equipment. While While an an approach approach was was developed developed in in damping force and road input. Thisdamping paperwhile proposes simple for estimating both force anda sensing vertical strategy road velocity profile simultaneously making this often requires direct measurement through expensive vertical road velocity profile simultaneously while making sensing equipment. While an approach was developed in Gonzlez et al. (2008), it requires 12 (compared with four Gonzlez et al. (2008), it requires 12 (compared with four simple strategy for estimating both damping force and vertical road velocity profile simultaneously while making no assumptions assumptions regarding regarding damper damper or or road. road. This This estimation estimation sensing equipment. While an approach was developed in no Gonzlez et al. (2008), it requires 12 (compared with four in this paper) parameters and assumes linear damping this paper) parameters and assumes linear damping -vertical road simultaneously while making in no assumptions regarding damper or road. This estimation technique is velocity based onprofile the Controller Output Observer Gonzlez et al. (2008), it requires 12 (compared with four technique is based on the Controller Output Observer in this paper) parameters and assumes linear damping this paper paper makes makes no no assumptions assumptions on on linearity linearity or or damper damperthis no assumptions regarding damper or road. This estimation technique isfirst based on the Observer framework first presented in Ozkan al. and this paper) parameters and assumes linearasor damping framework presented inController Ozkan et etOutput al. (2008) (2008) and in this paper makes no assumptions on linearity damperconfiguration. This fact is very important, dampers This fact is very important, as dampers technique isfirst based on the Controller Observer framework etOutput al. (2008) and configuration. applied again again in Varnhagen al. for this paper makes no fact assumptions on region. linearity ordampers damper applied in presented Varnhageninet et Ozkan al. (2014) (2014) for longitudinal longitudinal configuration. This is very important, as often behave well outside their linear Additionally, behave well outside their linear region. Additionally, framework first presented inet Ozkan et al. (2008) and often applied again Varnhagen al.the (2014) for longitudinal dynamics. As proof of concept, concept, the technique is applied applied configuration. This fact is very important, as dampers dynamics. As aainproof of technique is often behave well outside their linear region. Additionally, the technique technique in in Gonzlez Gonzlez et et al. al. (2008) (2008) does does not not estimate estimate applied Varnhagen et al.the (2014) for longitudinal dynamics. As ainproof concept, technique is applied through again simulation foroftwo two applications: Skyhook based the often behave well outside their region. Additionally, through simulation for applications: aa Skyhook based the technique in Gonzlez et al.linear (2008) does not estimate discrete road events like speed bumps and potholes. discrete road events like speed bumps and potholes. dynamics. As a proof of concept, the technique is applied through simulation for force two applications: Skyhook based the closed loop loop damping control roughness technique in Gonzlez et al.bumps (2008)and does not estimate closed damping force control and and aroad road roughness road events like speed potholes. through simulation for force two applications: Skyhook based discrete closed loop damping control and aroad roughness coefficient estimator. discrete road events like2. speed bumps and potholes. coefficient estimator. 2. MODEL MODEL closed loopestimator. damping force control and road roughness coefficient 2. MODEL Skyhook based based controllers are are common common in in the the automoautomocoefficient estimator. Skyhook controllers 2. MODELin this estimation frameIt will be shown in Section Skyhook based controllers are common in the automowill be shown in Section 33 that, that, in this estimation frametive industry industry (Karnopp (Karnopp et tive et al. al. (1974)). (1974)). To To summarize, summarize, aa It It will be shown in Section 3 that, in thison frameSkyhook based controllers are commonwhile the automo-a work, work, unknown system inputs acting onestimation the real real system system tive industry (Karnopp (1974)). Toin summarize, unknown system inputs acting the reference damping force etis is al. calculated low-level reference damping force calculated while aa low-level will be shown in Section 3inputs that, in this framework, unknown system inputs acting onestimation the real system tive industry (Karnopp etis al. (1974)). To summarize, a It are treated as controllable to the estimator. These reference damping force calculated while a low-level are treated as controllable inputs to the estimator. These controller attempts to apply it to the vehicle. In practice, controller attempts to apply it to the vehicle. In practice, work, unknown system inputs acting on the real system are treated as controllable inputs to the estimator. These reference damping force is calculated while a low-level inputs are not measurable, so a controller is developed that controller attempts to apply it to the vehicle. In practice, inputs are not measurable, so a controller is developed that damping force is often estimated through lookup tables damping force is often estimated through lookup tables are treated as measurable, controllable inputs to the estimator. These inputs are not so a controller is developed that controller attempts to apply it to the vehicle. In practice, drives the error between some measured and estimated damping force is often estimated through lookup tables the error between some measured and estimated that may may be be functions functions of of current current valve valve position position and and relative relative drives that inputs are not measurable, so a controller isunknown developed that drives the error between some measured and estimated damping force is often estimated through lookup tables signals to zero. It is then assumed that the inputs that may be functions of current valve position and relative signals to zero. It is then assumed that the unknown inputs velocity across across the the damper. damper. This This practice practice suffers suffers from from drives the error between some measured and estimated velocity signals to zero. It is then assumed that the unknown inputs that may be functions of current valve position and relative acting on on the the real real system system are are equivalent equivalent to to the the controlcontrolvelocity across the damper. Thisare practice suffers various shortcomings: shortcomings: all dampers dampers are different, tablesfrom are acting various all different, tables are signals to zero. It is then assumed that the unknown inputs acting on the real system are controller. equivalent to the controlvelocity across theof damper. Thisare practice suffers from lable inputs determined by the This framework various shortcomings: all dampers different, tables are lable inputs determined by the controller. This framework not representative dynamic systems, damper properties not representative of dynamic systems, damper properties acting ona the realwith system are equivalent to and the controllable inputs determined by the controller. This framework various shortcomings: all dampers are different, tables are requires model the required inputs outputs not representative of dynamic systems, damper properties a model with the required inputs and outputs change with with operating operating conditions conditions and and time, time, and and tables tables requires change inputs determined by the controller. This framework requires a model with the required and outputs not representative of While dynamic systems, properties representing the unknown unknown inputs andinputs measured signals, change with operating conditions anddamper time,been and tables lable representing the inputs and measured signals, include limited data. many papers have been written include limited data. While many papers have written requires a model with the required inputs and outputs representing the unknown inputs and measured signals, change with operating conditions and time, and tables respectively. Here, Here, the the unknown unknown inputs inputs are are damping damping force include limited data. While many papers have been written respectively. force representing the unknown inputs and measured signals, respectively. Here, the unknown inputs are damping force ⋆ Research include limited data. While many papers have been written and vertical road input velocity, and the measured ⋆ supported by by the the Hyundai Hyundai Center Center of of Excellence Excellence in in Vehicle Vehicle and vertical road input velocity, and the measured signals signals Research supported respectively. Here, the unknown inputs are dampingsignals force ⋆ and vertical road input velocity, and the measured Research supported by the Hyundai Center of Excellence in Vehicle are sprung and unsprung mass acceleration. Dynamic Systems & Control. are sprung and unsprung mass acceleration. Dynamic Systems & Control. ⋆ Research and verticaland road input velocity, and the measured signals supported by the Hyundai Center of Excellence in Vehicle are sprung unsprung mass acceleration. Dynamic Systems & Control. are sprung and unsprung mass acceleration. Dynamic Systems & Control.
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The model chosen, a quarter car, is simple and commonly used in the automotive industry - see Figure 1. The two masses represents the tire (unsprung mass, mus ) and onequarter of the body (sprung mass, ms ). The linear spring elements include suspension (ks ) and tire (kt ) stiffness. The controllable inputs to the estimator (as required for estimating the real unknown inputs) are a force (Fa ) and velocity (vi ) actuator representative of the damping force and vertical road input velocity, respectively. A linear damper (bs ) is included to stabilize the plant and make for easier control of Fa and vi . It should be noted that the total damping force estimate will be the combination of the passive damping force and Fa . Fig. 1. Quarter Car Schematic. The suspension stiffness can be obtained from the Ride Rate, or relationship between vertical wheel stroke (qs ) and spring force, and is therefore represented by a vertical spring. The damping force (Fd ), however, is modified by a lever ratio. That is, the deflection is approximately ∂q ′ proportional to (by ∂qss ), but not equal to, the wheel stroke. This is shown in Figure 2. Fig. 2. Damper configuration with lever ratio. In future steps the combined effects of the passive damping force and controllable force will be converted to the total damping force at this new position through Equation 1, where Fa is the controllable input and bs (ˆ vus − vˆs ) is the passive damping force. Hats (ˆ a) above variables denote estimated quantities. ′
∂q Fˆd = ( s )−1 (Fˆa + bs (ˆ vus − vˆs )) ∂qs
(1)
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T
y = [v˙ s v˙ us ] T ˆ = Fˆa vˆi u
Youla Parameterization was chosen as the control methodology for its simplicity and robustness. This technique allows for the shaping of the closed loop response directly. The block diagram in Figure 3 is reorganized and shown in Figure 4. These systems are equivalent. Fig. 4. Block diagram with feedback. It is now apparent that the boxed region in Figure 4 ˆ p ) can be replaced by an equivalent (containing Gc and G transfer function matrix, Y . This is shown in Figure 5. Fig. 5. Block diagram without feedback. Y is therefore equivalent to the closed loop response from ˆ and given by Equation 4. ym to u ˆ p Gc )−1 Gc Y = (I + G
In this paper, the measured signals are sprung (v˙ s ) and unsprung (v˙ us ) mass acceleration (Equation 2). The estimated unknown inputs are the additional damping force (Fˆa as described in Section 2) and road input velocity (vi ) (Equation 3).
(4)
It is easily shown that the closed loop response from ym to yˆ (or complimentary sensitivity function, T ) is given by Equation 5. ˆ pY T =G
Fig. 3. Reorganized block diagram.
(3)
ˆ p are If the input-output dynamic characteristics of G representative of the actual plant, and yˆ follows ym , then ˆ must be representative of the actual input - this assumes u no other major inputs act on the system. However, if the ˆ p is not representative of the real system, u ˆ behavior of G will compensate to ensure yˆ and ym are similar, but do ˆ Therefore, so at the expense of an accurate estimate of u. the plant input-output relationships must be considered carefully. This paper makes use of the quarter car, a model commonly used in practice.
3. YOULA CONTROLLER OUTPUT OBSERVER DESIGN In a Youla Parameterization based Controller Output Observer, unknown system inputs (u), states (x), and outputs (y) may be estimated. Youla based control techniques are well documented in Youla et al. (1976), Skogestad et al. (1976), and Assadian (2015), while Controller Output Observers were first presented in Ozkan et al. (2008). Unknown system inputs are treated as controllable ˆ to the estimator. Using Youla parameterization inputs (u) techniques, a controller (Gc ) is used to drive the error ˆ output between certain measured (ym ) and estimated (y) signals to zero. The estimated outputs signals come from a ˆ p ) that is propagated forward in time model of the plant (G ˆ Any additional output signal can be reconstructed by u. ˆ p . This control structure is from the internal states of G shown in Figure 3.
(2)
(5)
Next, consider the sensitivity function, or S, as defined as I − T . This represents the closed loop response between output disturbances and estimated outputs. Good performance and robustness are achieved when certain conditions on T , S, and Y are met. This is accomplished by shaping Y alone. For good tracking and disturbance rejection, T should be stable and close to identity and very small at low and high frequencies, respectively. This determines, from the constraint that S = I − T , that S should be very small and identity at low and high frequencies, respectively. Additionally, Y must be proper and stable. Finally, the desired controller, Gc , is obtained from Equation 6. Gc = S −1 Y
(6)
In summary, the unknown system inputs and outputs are estimated through Equations 7 and 8, respectively, where Y and T are defined in Equations 4 and 5, respectively.
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ˆ = Y ym u yˆ = T ym
(7) (8)
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3.1 Smith McMillan Form
250
ˆ p is a transfer function matrix (i.e. has MultipleWhen G Inputs and Multiple-Outputs, MIMO), the system is decoupled using the Smith McMillan transformation. A diagonalized version of Y is then designed so diagonalized versions of T and S have desired properties. Once complete, the transformation is inversed and MIMO transfer function matrices obtained (Assadian (2015)).
200
Magnitude, dB
150
ˆ p through The diagonalized plant (Gsm ) is related to G two unimodular matrices (Ul , Ur ) as shown in Equation 9.
100 50 0 -50 -100 -150
Gp =
UL−1 Gsm UR−1
(9)
Ul and Ur are defined by Equations 10 ans 11, where Np and Nsm are the numerators, and Dp and Dsm the ˆ p and Gsm , respectively. denominators, of G −1 −1 UL = (Np Nsm ) −1 UR = Dp Dsm
(10) (11)
The diagonal elements of Gsm are shown in Equations 12 and 13, where G∗sm1 and G∗sm2 represent the stable poles and zeros of Gsm1 and Gsm2 , respectively. Note that seperation of the stable and unstable components is required for control design. This will be shown in Section 3.2. Gsm1 = sG∗sm1 Gsm2 = s2 G∗sm2
(12) (13)
T S Y
10 -5
10 0
10 4
10 -5
10 0
Frequency, Hz
10 4
Frequency, Hz
Fig. 6. (Left) Bode plot of T1 , S1 and Y1 . (Right) Bode plot of T2 , S2 , Y2 the magnitude of T1 and T2 are approximately one for the majority of the frequency range as desired. T , S and Y are obtained from Equations 16 - 18, where UR and UL are the unimodular matrices from Equations 10 and 11 and MY is a diagonal matrix whose elements are Y1 and Y2 from Equations 14 and 15. (16) (17) (18)
Y = UR MY UL T = Gp Y S =I −T
The Bode plots of these new transfer function matrices are shown in Figure 7.
k1 (G∗sm1 )−1 2 )(s + p ) (s + p1 )(s2 + 2ζ1 ωn1 s + ωn1 2 ∗ −1 k2 (Gsm2 ) Y2 = 2 2 )(s2 + 2ζ ω s + ω 2 ) (s + 2ζ2 ωn2 s + ωn2 3 n3 n3 Y1 =
(14)
200 0 -200 200
Magnitude, dB
The requirements on T , S, and Y discussed in Section 3 now apply to their diagonal elements (Ti , Si , and Yi ). That is, Ti should be stable and close to one and zero at low and high frequencies, respectively, while the inverse is true for Si . Additionally, the diagonal elements of Yi must be proper and stable. Y1 and Y2 are shown in Equations 14 and 15, where G∗sm1 and G∗sm2 are defined in Equation 12 and 13.
Magnitude, dB
400
3.2 Controller Design
0 -100 -200
(15)
Unfortunately, the unstable zeros shown in Equations 12 and 13 can’t be canceled without introducing unstable poles into Y1 and Y2 . A compromise is made by introducing a pole or poles extremely close to the origin to essentially cancel the unstable zeros. This results in the magnitude of T1 and T2 equaling something smaller than one at the lowest frequencies and the inability for steady state tracking. The frequency response of the first and second diagonal element of T , S, and Y are shown in the left and right pane of Figure 6, respectively. It is apparent that
T S Y
100
10 -5
10 0
Frequency, Hz
10 4 10 -5
10 0
10 4
Frequency, Hz
Fig. 7. MIMO transfer function frequency response.
4. APPLICATIONS The utility of this estimator is made apparent by two example simulation applications: Skyhook based damping force estimation for closed loop control in active/semiactive suspensions, and road roughness coefficient estimation.
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4.1 Skyhook Based Control
Fd,SH = bSH vs
200
Damping Force, N
The control methodology chosen is Skyhook control, a common strategy with variants deployed on many vehicle brands. It was proposed that an optimal suspension is one in which the sprung mass is connected through a linear damper to some inertial reference. Due to the inability to build such a contraption, this ideal damping force is instead applied within the suspension between sprung and unsprung masses through either force actuator or control of the damper. Although sub-optimal, this strategy has proven highly effective. The ideal damping force is given by Equation 19, where bSH is a tuning parameter and vs is the sprung mass vertical velocity.
The control structure in Figure 8 is proposed and described in this paragraph. A simple controller (Gc,SH , different from that in Section 3) is developed that drives the effective damping force towards the ideal Skyhook damping force (Fd,SH ) using the controlled input force (Fi ). This effective damping force is the combination of both passive damping and the controllable damping force. The estimate of the damping force (Fˆd ) will inherently contain both these forces. Note that u and Est represents additional system inputs and the estimator, respectively. Fig. 8. Damping force feedback control structure. Success of the estimator is measured by the level of degradation in root-mean-square (RMS) heave acceleration as this value is strongly correlated with ride comfort. A comparison is made between uncontrolled, controlled with no estimator, and controlled with estimator. This is done in two scenarios: (1) driving over a motorway at 100 KPH and (2) driving over a large bump (0.1 m tall) at 40 KPH. The results are summarized in Table 1 where RMS heave acceleration is shown for the uncontrolled case and percent change for cases with control. Both control scenarios are shown: with direct damping force feedback, and with estimated damping force feedback. It is apparent that little degradation in RMS heave acceleration exists due to the inclusion of the estimator versus direct measurement feedback. Additionally, Figure 9 shows the estimated versus actual damping force for motorway conditions using control.
100 0 -100 -200 -300
(19)
A fully active Skyhook is applied to a high order vehicle model (Beckerman et al (April 2016), Beckerman et al (July 2016)). Each corner suspension is modeled as a nonlinear spring and damper (lookup tables) with an active element capable of applying ideal forces. As damping force is used in feedback with the controller, either a damping force measurement or estimate is required. In this paper, results are obtained for both scenarios and compared. It will be shown that the influence of the estimator does little to inhibit the performance of the controller.
Actual Estimate
300
14
14.5
15
15.5
16
16.5
Time, s
Fig. 9. Actual versus estimated damping force. measure the road directly, but the method proposed in this paper needs only sprung and unsprung mass accelerations. A road roughness coefficient (G0 ) is a parameter that fits a special curve as shown in Equation 20 to a Power-SpectralDensity (PSD) plot of the road - v is the spatial frequency and v0 is the cutoff spatial frequency, both in rad/m, and Gz is the road PSD in m2 /(rad/m) (Assadian (2015)). Such a plot is shown in Figure 10 where results are shown for a motorway (bottom) and farm road (top). Shown in gray, blue, and red are the PSD given the actual road data, the best fit of the actual road PSD, and a best fit of the estimated road PSD, respectively. The general shape of this plot is a result of decreasing road variation over smaller distances. Gz (v) = G0 [1 + (v0 /v)2 ]/(2πv)2
(20)
It is apparent from Figure 10 that the best fit of both the actual and estimated road data are very similar over a large frequency range. Additionally, the actual road roughness coefficient G0 varies by 3.3 and 2.4 percent for the motorway and farm road, respectively. 5. CONCLUSION The method proposed in this paper allows for the simple estimation of vehicle suspension damping force and road input velocity. Estimation of these two quantities is very useful in active or semi-active suspension configurations. A Youla Parameterization based Controller Output Observer was utilized while only requiring that sprung and unsprung mass accelerations be measured. To validate the concept, two tests were performed. The first test consisted of a Skyhook based control law for an active suspension
4.2 Road Roughness Estimation Road roughness estimation is important for suspension parameter adaptation. Expensive sensors may be used to 16195
Table 1. Heave acceleration results for the uncontrolled and controlled case, with and without the estimator Test No control [RMS, m/s2 ] Control without estimator [% change] Control with estimator [% change]
Motorway 0.0279 -52.3176 -46.5230
Bump 0.1192 -30.2342 -22.0213
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troller output observer for disturbance rejection.” SAE International Journal of Passenger Cars-Mechanical Systems 7.2014-01-0125 (2014): 65-72. Youla, D., H. Jabr, and J. Bongiorno. ”Modern WienerHopf Design of Optimal Controllers–Part II: The Multivariable Case.” IEEE Transactions on Automatic Control 21.3 (1976): 319-38. Web.
2
PSD, m /(rad/m)
10 -5
10 -10 Raw - Actual Fit - Actual Fit - Estimate
10 -1
10 0
10 1
10 2
Spatial frequency, rad/m
Fig. 10. Power Spectral Density of road profile for a motorway (bottom) and farm road (top). where damping force feedback was utilized. Minimal degradation in performance due to estimation was witnessed. The second test consisted of road roughness coefficient estimation. It was shown that values between actual and estimated coefficients differed by no more than 3.3 percent. Additionally, this method for road input estimation applies to discrete events in addition to the frequency analysis performed in this paper. REFERENCES Assadian, F., MAE 272 - Theory and Design of Control Systems Class Notes. 2015, University of California, Davis: Davis, CA. Beckerman, Alex, and Francis Assadian, Dr. Generalized Template for Suspension Dynamic Modeling With the Use of Bond Graphs. Proc. of Spring Simulation MultiConference 2016, Pasadena, CA. 2016. Beckerman, Alex K., and Francis Assadian, Dr. ”ICBGM.” Society for Modeling & Simulation. Proc. of Summer Simulation Symposium, Montreal, Quebec, Canada. N.p.:n.p., n.d. N. pag. Print. Karnopp, D., M. J. Crosby, and R. A. Harwood. ”Vibration Control Using Semi-Active Force Generators.” Journal of Engineering for Industry 96.2 (1974): 619. Web. Gonzlez, A., E.j. O’brien, Y.-Y. Li, and K. Cashell. ”The Use of Vehicle Acceleration Measurements to Estimate Road Roughness.” Vehicle System Dynamics 46.6 (2008): 483-99. Web. Ozkan, Basar, Donald Margolis, and Marco Pengov. ”The controller output observer: Estimation of vehicle tire cornering and normal forces.” Journal of Dynamic Systems, Measurement, and Control 130.6 (2008): 061002. Rajamani, R., and J.k. Hedrick. ”Adaptive Observers for Active Automotive Suspensions: Theory and Experiment.” IEEE Transactions on Control Systems Technology 3.1 (1995): 86-93. Web. Skogestad, Sigurd, and Ian Postlethwaite. Multivariable Feedback Control: Analysis and Design. Chichester: Wiley, 1996. Print. Varnhagen, Scott, and Donald Margolis. ”Longitudinal slip ratio control of electric powertrains using a con16196