Multivariable feedback control of electromagnetic reluctance force actuators

electromagnetic reluctance force actuators

NICLAS SUNDQVIST

Masters’ Degree Project

Stockholm, Sweden May 2009

XR-EE-RT 2009:009

This master’s thesis is part of a cooperation between BMW AG and Technische Universität München, in order to design a road simulator for Squeak and Rattle studies based on electromagnetic actuators.

The study covers the field of decoupling control and adaptive inverse control, applied to a multi-post road simulator with electromagnetic actuators. The objective is to derive a controller which is robust and stabilizes the system without disturbances, thereto it is important to strive for a suitable bandwidth.

At first the different control theory areas are covered and divided into the main problem groups, stabilization through state-feedback and bandwidth-adjustment measures through feedforward-filters. This is followed by the system modelling, where an actuator model is designed, with separate modelling steps for the electric, magnetic and mechanical parts.

A rigid body vehicle model is derived and connected to four actuators. This leads to the complete system model, which is balanced and analyzed in terms of stability, controllability and observability.

The model is a prerequisite for the design of the controller. Next the control theory pre- sented is applied to the derived model in terms of a decoupling controller and a reduced observer to stabilize the system. The stabilized system is attenuated with adaptive multi input multi output invserse control concepts, as well as static feedforward-filters. The finished control structure is simulated for a two actuator and a four actuator rig. Further- more, tests are made on an actual two-post road simulator with electromagnetic actuators, verifying the simulations.

The objectives stated in the project are achieved on the actual test rig, and proved to be robust enough for the purposes of this and future experiments. The design goals for the rig in the scope of this study were achieved with a combination of state-space decoupling controller, reduced observer, tracking error estimator and time waveform replication.

29.08.08, Garching Niclas Sundqvist

I

1 Introduction 1

1.1 Motivation . . . 1

1.2 Objectives . . . 2

1.3 Approach . . . 2

1.4 Outline . . . 3

2 Control theory 4 2.1 State feedback . . . 4

2.2 LQR . . . 5

2.3 Decoupling . . . 6

2.4 Luenberger observer . . . 7

2.5 Reduced state observer . . . 8

2.6 Tracking error estimator . . . 9

2.7 Adaptive inverse control . . . 10

2.7.1 Adaptive filters . . . 11

2.7.2 LMS algorithm . . . 12

2.7.3 RLS algorithm . . . 12

2.7.4 Transposed MIMO approach . . . 13

2.7.5 The Filtered-
approach . . . . 14

2.8 TWR . . . 15

3 Mathematical modelling 17 3.1 Actuator modelling: EMAc²-6 . . . 17

3.1.1 Electrical subsystem . . . 18

3.1.2 Magnetic subsystem . . . 19

3.1.3 Mechanical subsystem . . . 19

3.1.4 Complete actuator model . . . 19

3.2 Vehicle . . . 20

3.3 Two actuator half vehicle model . . . 20

3.4 Four actuator full vehicle model . . . 21

3.5 Complete plant model . . . 22

3.6 Model analysis . . . 24

3.6.1 Balancing and conditioning . . . 24

3.6.2 Controllability . . . 25

3.6.3 Observability . . . 25

3.6.4 System poles and zeros . . . 25

4 Implementation 27 4.1 Complete system overview . . . 27

4.2 Two shaker test-setup . . . 27

4.2.1 Amplifiers . . . 28

4.2.2 Sensors . . . 29

4.2.3 Filters . . . 29

II

4.2.4 Controller-board . . . 30

4.3 Controller implementation . . . 30

4.3.1 Overview . . . 30

4.3.2 Decoupling controller . . . 31

4.3.3 Observer . . . 32

4.3.4 Tracking error estimator . . . 33

4.3.5 Adaptive inverse control . . . 34

4.3.6 Time waveform replication . . . 36

5 Results 37 5.1 Simulations of a two actuator system . . . 37

5.1.1 Pole placement and optimal control . . . 37

5.1.2 Decoupling control . . . 38

5.1.3 Reduced observer . . . 40

5.1.4 Tracking error estimator . . . 41

5.2 Simulations of a four actuator system . . . 41

5.3 Comparing the actual system with simulations . . . 42

5.3.1 Model and actual system . . . 42

5.3.2 Reduced observer . . . 44

5.3.3 Tracking error estimator . . . 44

5.4 Adaptive inverse control . . . 45

5.4.1 Filtered-
. . . . 46

5.4.2 Transposed approach . . . 49

5.4.3 Static feedforward filter . . . 51

5.5 Tracking accuracy . . . 52

5.6 Disturbances without setpoint input . . . 55

6 Conclusions 56

7 Recommendations 58

2.1 State space model with the state feedback controller K. . . . 5

2.2 Luenberger observer. . . 8

2.3 Structure of the reduced observer. . . 9

2.4 Principle of error tracking combined with state feedback and a state estimator. 10 2.5 Scheme of the tracking error estimator. . . 10

2.6 Input-output model of adaptive filter. . . 11

2.7 Schematical view of a Finite Inpulse Response (FIR) filter. . . . 11

2.8 Identifying a SISO or MIMO plant. . . 13

2.9 Calculating the delayed inverse of a MIMO plant with the Transposed ap- proach. . . 14

2.10 The Filtered-
approach. . . . 15

3.1 The EMAc²-6 actuator. . . 17

3.2 Assembly drawing of the actuator showing simplified flux. . . 18

3.3 The comprising parts of the model. . . 18

3.4 Drawing used for two actuator modelling, side-view. . . 21

3.5 Drawing used for four shaker modelling. . . 21

3.6 Overview of the A-matrix from the complete state space model. . . 23

3.7 Pole-zero map describing the four actuator complete model, poles [x] zeros [0]. . . 26

4.1 The comprising parts of the system. . . 27

4.2 The test setup with two actuators, placed diagonally. . . 28

4.3 One of the four amplifiers, with the H-bridge removed and placed on top. . 28

4.4 Pole-zero map using the LQ-regulator with all weights set to one, poles [x] zeros [0]. . . 31

4.5 One of the transfer functions used for determining the decoupled system dynamics. . . 32

4.6 Simulink implementation of the reduced observer. . . 33

4.7 Simulink implementation of the tracking error estimator, with position ref- erence input. . . 34

4.8 Simulink implementation of the LMS adaptive MIMO filter. . . 34

4.9 Simulink implementation of the RLS adaptive MIMO filter. . . 35

4.10 Simulink implementation of the Transposed approach . . . 35

4.11 Simulink implementation of the Filtered-
method . . . 36

5.1 Bode-diagram using LQ-regulator for a two actuator plant. . . 38

5.2 Step-response using LQ-regulator for a two actuator plant. . . 39

5.3 Pole-zero map using the decoupling controller for a two actuator plant, poles [x] zeros [0]. . . 39

5.4 Bode-diagram using decoupling controller for a two actuator plant. . . 40

5.5 Step-response using decoupling controller for a two actuator plant . . . 41

IV

5.6 Bode-diagram indicating the difference between a system with [—] and with- out [- -] tracking error esimator. . . 42 5.7 Bode-diagram using decoupling controller for a four actuator plant, display-

ing only magnitude. . . 43 5.8 Step-response using decoupling controller for a four actuator plant. . . 43 5.9 Bode-diagram of the system model from input (voltage) to output (posi-

tion) and a corresponding measurement from the test rig, [—] model [- -]

measurement. . . 44 5.10 Verifying the Reduced observer, the measured current [gray] and the ob-

served [- -] current . . . 45 5.11 Falb Wolovich decoupling with [—] and without [- -] tracking error estimator 45 5.12 Using an RLS adaptive filter to indentify the inverse according to the

Filtered-
method. Simulating with 100 filter weights and a sample-rate of 1 kHz. System with filter output [- -] and input [gray]. Adapting algo- rithm turned on at t = 0.2. . . . 46 5.13 System identification with four FIR-filters with 200 taps each, Measurement

[—] Identification [- -]. . . 47 5.14 Inverse system identification with four FIR-filters with 200 taps each, iden-

tification [—] upsampled identification [- -] . . . 48 5.15 System with identified delayed inverse as feedforward filter, offline calculation 48 5.16 Impulse response of the inverse FIR filter, Identification 100 weights [—]

Upsampling 200 weights (spline interpolation) [- -] . . . 49 5.17 Using an RLS adaptive filter to indentify the plant. Simulating with 200

filter weights and a sample-rate of 1 kHz. Filter output [- -] and plant output [gray]. Adapting algorithm turned on at t = 0.1 . . . 50 5.18 Using the transposed approach to identify the plant and adapt an inverse.

Simulating with 200 filter weights and a sample-rate of 1 kHz. Filter connec- tion in series with plant, output [- -] and input [gray]. Adapting algorithm turned on at t = 0.5 . . . . 51 5.19 measured FRF matrix, with [- -] and without [—] offline identified feedfor-

ward filter . . . 52 5.20 Sine 10 Hz, (Upper) without offline identified feedforward filter, [—] target

positon, [- -] actual. (Lower) with offline identified feedforward filter, [—]

target positon, [-·-] filter output, [- -] actual. . . 53 5.21 PSD comparison between disturbed MIMO controller (actuator placed rear

right) with and without offline calculated filter, [- -] with [—] without [gray]

noise reference. . . 54 5.22 PSD of TWR 3 iteration and reference (Autobahn 120 km/h), [- -] TWR

MIMO output [gray] reference. . . 54 5.23 PSD comparison between the old SISO controllers and the derived MIMO

controller, [- -] MIMO [gray] SISO. . . 55

1.1 Shaker rig types overview . . . 2 4.1 Sensor overview. . . 29

VI

This chapter provides motivation for the study presented in this thesis. The project is part of a cooperation between BMW AG and Technische Universität München, in order to design a road simulator for Squeak and Rattle research based on electromagnetic actuators.

The chapter begins by explaining the need for such a road simulator, followed by objectives and approach. The chapter concludes with an outline of the thesis.

1.1 Motivation

The development of better noise damping measures from external sources and engine in modern vehicles has made vibration-induced sounds from components, and contact between components, increase in importance, as they are the main source of Squeak and Rattle in modern premium vehicles. In the quality assurance process in the automotive industry, the squeak and rattle perception is very important. On account of this, the efforts to reduce these problems have increased. Suppliers and OEMs approach this problem by means of extensive vibration-tests, test drives and complete-vehicle laboratory tests on Shaker rigs, also know as Road Simulators. A major trend in the industry is to utilize more indoor lab-based test equipment, as the testing in a laboratory environment allows for greater control of the experiment and reduces the time needed. In the acoustic comfort application, vehicle accelerations are measured on the test track and then reproduced in the laboratory. Test rigs yield a number of advantages, some of them are:

• Reproducibility: the tests can be accurately reproduced, with complete control of weather, and without influence from traffic and driver.

• New possibilities to look for problem sources, not only with measured test data, but also with synthetic signals and "compressed" signals to accelerate the process.

• Availability: the tests can be run 24 hours a day 7 days a week.

There are a number of different types of shaker rigs appropriate for this type of stud- ies. The two types normally used for complete vehicle interior noise analysis are either Hydraulic or Electrodynamic. These are however inherently cumbersome, due to size, power-consumption, cooling needs and complexity of the facility (a small comparison is presented in Table 1.1). An alternative to these are electromagnetic actuators. These have a large amount of power compared to the electrodynamic, and are very compact and lightweight. Construction of a new type of road simulator is under way, making use of electromagnetic actuators. The purpose of this study is to develop control strategies for this type of road simulator.

1

Type Model Power [kW] Stroke [mm] Force Actuator weight

Hydraulic MTS 270 400+ 55 kN not mobile

Electrodynamic LDS 40 20 1.5 kN 350 kg

Electromagnetic EMAc²-6 4 6 8 kN 25 kg

Table 1.1: Shaker rig types overview

1.2 Objectives

Electromagnetic actuators are compact and powerful [9], making them especially suitable for vibration-excitation of vehicle components and complete vehicles. Utilized as compo- nent shaker, the electromagnetic actuator EMAc²-5, has proved to be useful combined with a state feedback regulator extended with a PI-controller [7]. The need for these controllers is attributed to the unstable properties of the actuator. Using these Single Input Single Output (SISO) controllers on a road simulator with two or more actuators has lead to problems, due to the strong coupling in the system, as the actuators are directly clamped to the body of the vehicle.

Within the scope of this master’s thesis the SISO controllers are discarded, and different Multiple Input Multiple Output (MIMO) concepts are designed. Upon excitation of the vehicle, to be able to achieve good tracking accuracy of the measured vehicle responses¹, the interaction of the actuators through the body of the vehicle has to be handled. This is achieved through MIMO control combined with static and adaptive online and offline feedforward concepts.

The actual goals of the derived controller combined with different feedforward concepts are a bandwidth of 100 Hz, and a trajectory tracking accuracy with a maximum Root Mean Square (RMS) error of 5 percent.

1.3 Approach

To achieve these goals the following approach is taken. A mathematical model of the complete system is designed. Having constructed a model, it is used to derive a state esti- mator, reducing the need for sensors. When all signals are available, different controllers are developed – this to stabilize the actuators – keeping them in the desired position with- out disturbances. For the final step, achieving good tracking accuracy, different solutions are presented, mainly based on the online concept of Adaptive Inverse Control (AIC) and offline calculated filters. As reference the offline procedure called Time Waveform Repli- cation (TWR) is introduced, this method is however not in the scope of this thesis as it is a parallel ongoing research project.

All steps are accompanied by simulations and actual implementations on a rapid proto- typing system from dSpace®. Measurements verifying the results are presented as well.

1 Also referd to asdrive-files.

1.4 Outline

In the Second Chapter the control theory used in the study is introduced. The chapter presents different observer structures, state-feedback, adaptive inverse control and time waveform replication.

In the Third Chapter the electromagnetic actuator is presented and a linear actuator model is derived. The modelling of the actuator is followed by derivation of a linear model of the vehicle. This is put together to a complete system model and is analyzed in terms of stability, observability and controllability. Model balancing is also discussed.

In the Fourth Chapter the actual test rig is presented. The chapter contains hardware and software implementations, and different aspects of the controller implementations are discussed.

The Fifth Chapter contains results obtained from the model and the tests, based on the methods and algorithms presented in the previous chapters.

In the Sixth Chapter a few conclusions are drawn and the achieved results are anal- ysed.

In the Seventh Chapter recommendations for future work is presented.

The methods presented in this chapter are the different techniques and algorithms used on the shaker rig, and cover a wide spectrum of the control theory field. As the control objective is to, at first, stabilize the actuators and then follow a trajectory, a number of different approaches have been designed and implemented. The different techniques can be divided into four groups:

• Multivariable feedback control

• Estimators

• Adaptive inverse control

• Time Waveform Replication (TWR)

The part about multivariable feedback control covers two types of MIMO state feedback, the linear quadratic regulator, and decoupled control according to Falb-Wolovich [13].

Observer structures presented are reduced estimators and tracking error estimators. The adaptive inverse control part consists of two implementations, the Filtered-
approach and what in this thesis is called the Transposed approach¹ [10]. Finally a short introduction to time waveform replication is given. This will familiarize the reader with the control concepts used in this study.

2.1 State feedback

For the benefit of the comprehensibility of this master’s thesis a short introduction to complete state feedback is given. It is assumed that a differential equation written in state space is familiar. An overview can be found in [5].

In Figure 2.1 a standard state feedback is depicted, describing the state space model attenuated with a feedback of the state-vector x. The state space model can be written as

˙x = Ax + Bu (2.1)

y = Cx + Du. (2.2)

˙x = (A − BK)x + Bu (2.3)

y = Cx + Du (2.4)

Extending these equations with state feedback leads to the closed loop model in (2.3–2.4), as depicted in Figure 2.1, and enables adaptation of the system dynamics.

1 The method is presented in [10] by G. L. Plett, where no specific name is given, The nameTrans- posed approach is hence introduced for clarity.

4

Plant (in state space notation)

State Feedback A

B C

−K

˙x x y

u

Figure 2.1: State space model with the state feedback controller K.

The state feedback, or control law used in (2.3) is

u = −Kx. (2.5)

2.2 Linear Quadratic Regulator

The Linear Quadratic state feedback Regulator (LQR) for state space systems comes from the control theory area of optimal control [5]. It is optimal in the sense that its cost function is minimized. The cost function

J(u) =

∞

0

(x^TQx + u^TRu) dt (2.6)

consists of two parts, x² and u². The first parts main objective is to make sure that the transient from the starting position x₀ to x = 0 should be fast but without oscillations.

The second part tries to minimize the control input energy. Expanding this concept with individual weights for every state and input leads to the Quadratic cost function (2.6) Trying to solve this optimization problem yields the Riccati equation

A^TS + SA − (SB + N)R⁻¹(B^TS + N^T) +Q = 0. (2.7) Its derivation and solution is omitted here. The controllerK is derived from the solution S as

K = R⁻¹(B^TS + N^T). (2.8)

The controller gainK extracted from (2.8), can be used as depicted in the control-law

u = −Kx. (2.9)

Choosing the weights in the diagonal matrices Q and R is done by, when e.g. x_iis especially important to minimize, making the corresponding weight q_ii larger, relative to all other weights. The same concept can be used to minimize the energy needed by tuning the weights in R.

A high penalty on the inputs leads to a system that does not need much control energy, but instead has poor performance (the elements in R are larger than the ones in Q), and the

other way around. More performance with more energy has a side effect, an actual system always has limitations on the actuating variable, thus making the system less robust and in a worst case scenario unstable, when it goes into saturation.

2.3 Decoupling control

Decoupled control for MIMO systems means that a change in one reference signal does not influence other outputs, than the one assigned to the input. This area can be divided into two main groups, decentralized decoupled controllers and MIMO decoupling controllers.

Decentralized decoupling means integrating pre- and post-compensators on the inputs and outputs of the system, and then using SISO controllers. This method is however not discussed in this thesis. Here a MIMO decoupling control design is used, more precise a controller first introduced in [13], and will from here on be referred to as a Falb-Wolovich decoupling controller. The structure of such a controller is the same as presented in Figure 2.1, but attenuated with a prefilter on the input.

A standard state space model, (2.1–2.2) with D=0, with p outputs y_i i = (1,2,...,p), all have a class, meaning the time derivative y^(δ) where the input first appears, as

y^(δ_i ⁱ⁾ = c^T_i A^δix + c^Ti A^δi−1B

  

=0

u. (2.10)

Applying this to all outputs leads to

⎛

⎜⎜

⎝ y^δ₁¹

... y^δp^p

⎞

⎟⎟

⎠=

⎛

⎜⎜

⎝ c^T₁A^δ1

... c^T_pA^δp

⎞

⎟⎟

⎠

  

A^∗

x +

⎛

⎜⎜

⎝

c^T₁A^δ1−1B ... c^T_pA^δp−1B

⎞

⎟⎟

⎠

  

D^∗

u=^! ¯u. (2.11)

If D∗ is non-singular (det(D^∗) = 0), the system is decouplable, and the integrator-chain (2.11) can be written as

u = (D^∗)⁻¹(−(A^∗)x + ¯u). (2.12)

Considering (2.12) as control law leads to the decoupled system dynamics (p number of input/output channels)

y^(δ_i ⁱ⁾ =¯ui, i = 1,...,p. (2.13)

Stabilization of this integrator-chain is done by chosing a stable transfer function for every input/output channel, as

y^(δ_i ⁱ⁾+ M_δⁱ_i−1y_i^(δi−1)+ . . . + M₁ⁱy_i+ M₀ⁱ = k_iw_i. (2.14) Comparing a transfer function (2.14) with the decoupling for every channel delivers the expression

y_i^(δi) =¯ui =−Mδⁱi−1y_i^(δi−1)− . . . − M₁ⁱy_i− M₀ⁱ+ k_iw_i. (2.15) Substituting y^(k)_i = c^T_i A^k results in the actual state feedbacku = −Rx + F w, shown here

in detail,

R = (D^∗)⁻¹

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

c^T₁A^δ1 +

δ₁−1 v=0

M_1vc^T₁A^v ...

c^T_pA^δp+

δp−1

v=0

M_4vc^T_pA^v

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠

(2.16)

F = (D^∗)⁻¹

⎛

⎜⎝

k₁ · · · 0 ... . . . ...

0 · · · kp

⎞

⎟⎠. (2.17)

To give the controller the right transfer properties, choosing k_i = M₀ⁱ leads to the transfer function,

yi= M₀ⁱ

s^δi + M_δⁱ_i−1s^(δi−1)+ . . . + M₁ⁱs + M₀ⁱwi (2.18) which then for t→ ∞ gives yi = w_i.

Having chosen a transfer function for every input/output channel, the system is decoupled.

A necessary and sufficient condition for decoupling is det(D^∗) = 0, meaning that this matrix must be non-singular. The class of every output determines the transfer function order to be chosen [13]. This stabilizes and decouples the system as long as the individual transfer functions are stable.

2.4 Luenberger observer

Considering a state-space description of a system with n states (2.1–2.2), as starting point, with a complete state feedback controlleru = −Kx, it needs all n states as measurements, or estimates of the actual system states to be able to close the loop. It happens that some states are not measurable, and/or expensive to measure. A state observer can give access to these states. If the plant is observable (Section 3.6.3), it is possible to estimate all states needed, given measurements of the plants inputs and outputs. The state observer is a mathematical system model of the real system, providing an estimate of its internal states.

The observer model of the system is derived from (2.1–2.2), by adding additional terms to ensure that the observer model states converge to the actual system states. In particular, for a so called Luenberger observer, taking the output of the observer and subtracting it from the output of the plant and then multiplying it with a matrix L (Figure 2.2) ensures convergence. Adding this to (2.1) gives the expressions

˙ˆx = Aˆx + L(y − ˆy) + Bu, (2.19)

ˆy = C ˆx + Du. (2.20)

Assuming that D = 0 leads to the closed loop equation

Observer A A

B B

C C

L

˙x x y

u

ˆx

˙ˆx ˆx ˆy

Figure 2.2: Luenberger observer.

˙ˆx = (A − LC)ˆx + Bu + Ly. (2.21)

The Luenberger observer is therefore asymptotically stable when the matrix A - LC has all the eigen-values with strictly negative real part. To use the output from the observer in the feedbacku = −K ˆx the observer needs to be faster then the actual plant, meaning the closed-loop poles of the observer A− LC are chosen to converge several times faster than the closed-loop poles of the system A− BK [5].

2.5 Reduced state observer

When high quality measurements are available, it is not necessary to use a complete state observer like the Luenberger observer. Presented here is the so called Reduced observer, estimating only those states that are not available or available with poor quality (Signal to Noise Ratio (SNR)). The basic idea is the same as for the complete state observer. A good overview of this can be found in [8]. To make the design process easier it is important to first reorganize rows and columns in the state space description of the system, so that all measured states come first in thex-vector and the states to be observed subsequently after. This leads to the state space model

˙y

˙r



=

A₁₁ A₁₂ A₂₁ A₂₂

 y r

 +

B₁ B₂



u, (2.22)

where r is the vector containing reduced system states to be estimated. Extracting the reduced system yields the equation

˙r = A22r + A21y + B2u. (2.23)

With the reduced output

˙y − A₁₁y − B₁u = A₁₂r. (2.24)

Substituting, r → x A22→ A

A21y + B1uB → u y − A₁₁r − B₂u → y

A12→ C^T (2.25)

into the Luenberger observer equation (2.21) results in the reduced observer. This together with ρ = ˆr − Ly, to remove the derivative of y in the equation, reveales the reduced observer system (Figure 2.3)

˙ρ = (A₂₂− LA₁₂)ρ + [(A₂₂− LA₁₂)L + A₂₁− LA₁₁]y + (B₂− LB₁)u (2.26)

ˆr = ρ + Ly. (2.27)

Reduced observer

L y

u

B₂− LB1

(A₂₂− LA₁₂)L + A₂₁− LA₁₁

A₂₂− LA₁₂

Figure 2.3: Structure of the reduced observer.

2.6 Tracking error estimator

In systems where tracking accuracy is the main goal, controlling the outputs such that they follow a given reference trajectory, a normal state feedback is not adequate, mainly due to the damping and stabilizing properties of such a regulator. As a state feedback is not error based, it drives all states and hence all outputs to zero. There are however techniques and modifications that can be used to make the closed loop system follow a given trajectory. The basic idea is to let the states follow given reference states xr. To generate the reference states a reference state estimator can be used with the system equation

˙xr = Axr+ But+ Ld(r − Cxr). (2.28) In (2.28) L_d has the same function as L in a normal state observer (Section 2.4). Taking the reference states and subtracting the estimated states from the observer, and using this as input to the control law (Figure 2.4), results in the control law

− +

− +

Observer Reference state estimator

Control law

Plant

Ld L

−K ˆ G

G⁻¹

u y ut

r

xd ˆx

Figure 2.4: Principle of error tracking combined with state feedback and a state estimator.

u = ut− K(ˆx − xr). (2.29)

The reference state observer input ut is not available without an inverse system model, making this whole algorithm cumbersome and hard to implement on a real system. A solutions to this problem is to take L_d = L reducing the control scheme to what in literature [1] [2] is referred to as the Tracking Error Estimator (TEE), shown in Figure 2.5.

Observer Control Law

Plant

−K L

G y

u

e r

ut

uf b

xe

Figure 2.5: Scheme of the tracking error estimator.

When using y − r as tracking error, the observer produces the output xe = ˆx − xr. In accordance with Figure 2.5 this can then be used as feedback. This basically means moving the coordinate system to follow the trajectory.

2.7 Adaptive inverse MIMO control

In the previous sections the theory behind the concept of state-feedback control was de- scribed. This theory is needed to stabilize and give the system suitable properties. To achieve the second objective, good tracking accuracy, a few other concepts are needed.

The following section discusses feedforward concepts and not classical feedback control.

The purpose of this section is to show the basic idea of adaptive inverse control for MIMO systems. Adaptive inverse control means adaptation of a plant inverse in order to obtain a system where the output is exactly that of the input. This then implies decoupling

properties, as an input can be choosen to only influence one output. The adaptivness of the algorithms means that the inverse is calculated from measurements in real-time and no modelling is required. An overview of the algorithms available are found in [3].

A short introduction to adaptive filters is given, together with a brief description of two algorithms for adapting filters. For the actual online adaptive inverse control for MIMO systems two methods are presented, The Transposed approach [10], which is completely online, and the Filtered-
approach [3], which is partly online. The Transposed approach can also be used offline to calculate a static feedforward filter.

2.7.1 Adaptive filters

− +

xk yk

dk

ek

Figure 2.6: Input-output model of adaptive filter.

The fundamental building block of adaptive signal processing and control is an adaptive filter. The adaptive filter, seen as a block, has an input xk, an output yk and a special input d_k called desired response. This signal is the desired filter response or output, and is used internally to create an error e_k (Figure 2.6) and, through adaptation of the filter weights, minimizing it. This concept is applicable in the single input single output case as well as for MIMO applications. There are a number of different adaptive filter types, in this thesis however, the type of filter is limited to Finite Impulse Response (FIR) filters.

+

w_0k w_1k w_2k wN k

xk xk−1 xk−2 xk−N

yk

z⁻¹ z⁻¹

z⁻¹

Figure 2.7: Schematical view of a Finite Inpulse Response (FIR) filter.

The FIR-filter output is the weighted sum of the current and N previous samples (Figure 2.7), described by

y_k= W X_k, (2.30)

Xk= [x^T_k, x^T_k−1· · · x^T_k−N]^T. (2.31) In (2.30) W are the filter-weights. Given a SISO FIR-filter, W is a row vector, containing the impulse response of the filter. A MIMO filter can have an arbitrary number of inputs

and outputs; giving W multiple rows containing impulse responses from all inputs to all outputs². Adapting W can be done in a number of different ways. The following two are presented: Least Mean Squares (LMS) and Recursive Least Squares (RLS) algorithms.

2.7.2 LMS algorithm

The Least Mean Squares (LMS) algorithm is a steepest-descent method for finding the minimum value of a function and can be used for filter weight adaptation. In Figure 2.6 the expression for the error can be seen combined with the FIR-filter. This is the instantaneous error

e_k= d_k− yk= d_k− WkX_k. (2.32)

from (2.32) it is possible to construct a quadratic cost function which is the instantaneous squared error

e_k² = d²_k+ X_k^TW_kW_k^TX_k− 2dkW_k^TX_k. (2.33) Taking the gradient of the cost function leads to the expression

∇ = ∂e²_k

∂W_k =

⎡

⎢⎢

⎢⎣

∂e²_k

∂w_1k

...

∂e²_k

∂wNk

⎤

⎥⎥

⎥⎦= 2e_k

⎡

⎢⎢

⎣

∂ek

∂w_1k

...

∂ek

∂wNk

⎤

⎥⎥

⎦=−2ekX_k. (2.34)

The resulting gradient is attenuated with a small learning constant or step size, µ, as in

W_k+1= W_k− µ∇ = Wk+ 2µe_kX_k. (2.35)

Shown in (2.35) is the complete SISO LMS algorithm. The derivation for the MIMO case is more or less exactly the same and is well described in [6]. Due to the simplicity of the algorithm it is well suited for online filter adaptation, but compared to other, more complex algorithms, it lacks in convergence speed. This problem can not be solved by making the learning constant bigger as this can lead to instability.

2.7.3 RLS algorithm

The Recursive Least Squares (RLS) algorithm is a very fast [6] algorithm for linear filter weight adaptation. The algorithm presented below is a Matrix-RLS algorithm, but is derived in the same way as a scalar RLS algorithm. The derivation is omitted here, but can be found in [6]. The algorithm is

π_k= X_k^Tφ⁻¹_xx(k− 1) r_k= 1

λ + πkXk

Kk = rkπk

2 In this thesis a MIMO filter of orderN refers to one filter in the filter matrix; hence the number of weights is actually 4N.

ς_k = d_k− Wk−1X_k Wk=Wk−1+ ς_kK_k φ⁻¹_xx(k) = 1

λtri



φ⁻¹_xx(k− 1) − πk^TKk



. (2.36)

In (2.36), λ is a forgetting constant, and should be set slightly less than one. The tri- operator takes the upper triangular part of the matrix and replaces the lower triangular part.

According to Haykin, RLS converges in two times as many iterations as there are taps in the filter [6]. That is faster than most other algorithms. As an example a filter with 200 taps would converge within 400 samples, and with a sample rate of 0.2 ms it converges in 80 ms, which is exceptionally good. However the complexity of the algorithm makes it harder (compared to matrix-LMS) to implement, needing a lot more computing power, especially for a MIMO plant.

2.7.4 Transposed MIMO approach

To be able to explain this control method a short introduction to linear SISO adaptive inverse control is needed. In Figure 2.8, the way to identify a plant is shown; this is analogue in the SISO and MIMO case. But the problem at hand is to be able to adapt an inverse controller. In the SISO case, this is very simple due to the commutability of transfer functions of linear SISO systems (C(z)P_plant(z) = P_plant(z)C(z)), meaning that interchanging input and output on the adaptive filter, will adapt an inverse controller.

There are however a few limitations; as an inverse of real system is usualy non-causal the adapted controller would have to be non-causal, which is not possible, the adaptive algortihm can not converge. To solve this problem it is necessary to filter the input signal, with a filter M (z), to ensure that the resulting controller is causal and stable, also making it possible to give the system special response properties. A FIR-filter is always stable [6], but the adapting algorithms can become unstable.

− + P (z)

plant

P (z)ˆ

ˆyk

ek

yk

Figure 2.8: Identifying a SISO or MIMO plant.

Considering an adaptation of a feedforward controller C(z). The objective is to adapt the controller to give the system the transfer function properties of M (z), where M (z) is a user defined static filter, containing the desired system dynamics, from here on called a reference model. The system input signal u is filtered through M (z) creating the desired system response dk. The difference between the desired system response and the measured system output is used to create a system error e_k = d_k− yk. Minimizing this error with e.g. the LMS algorithm will (in theory) deliver a system transfer function M (z) = P_plant(z)C(z).

M (z) can be chosen to give the system a desired response according to a wide variety of

design criteria, it is however often practical to use a delay corresponding to the transport delay of the plant, making the resulting controller causal, thus adapting the feedforward filter to the delayed system inverse C(z) = P_∆⁻¹(z).

In the MIMO case however, as MIMO systems have, in contrast to SISO systems, not only transfer functions but matrices with transfer functions, the same approach can not be adopted. As a matrix multiplication in general is not commutative, one can not use the simple method to adapt an inverse MIMO controller described for the SISO case. In the MIMO case, what is true is that P_plant(z)C(z) = C^T(z)P_plant^T (z) (as C(z) is considered to be the inverse of P_plant), making MIMO adaptation possible according to the block diagram in Figure 2.9.

− + C(z)

copy

P (z)

plant

P^T(z)

copy

Cˆ^T(z)

Mˆ^T(z) dk

ek

xk yk

Figure 2.9: Calculating the delayed inverse of a MIMO plant with the Transposed approach.

As seen in Figure 2.9 the first step is to find the "transposed" system plant, this is done by normal system identification (Figure 2.8) and then taking the "transpose" of the identified system. Transpose in the sense of transposing the filters, not a mathematical transpose.

This makes adaptation of the C^T(z) inverse controller possible. Taking the transpose of the found filter and copying all filter coefficients to the feedforward filter, results in an MIMO plant inverse. This method was first presented in [10].

2.7.5 The Filtered-
approach

Another way of solving the Adaptive Inverse MIMO Control problem is by making use of the Filtered-
method. As already mentioned in Section 2.7.4, SISO methods for adaptive inverse control require impermissible commutation, and can on account of this not directly be transferred to the MIMO case, except for the SISO Filtered-
, where the signal flow occurs in such a way that it can be used in the MIMO case. A block diagram Filtered-

adaptation scheme is shown in Figure 2.10. This method (presented for the first time in [3]) tries to minimize the error between the optimal inverse controller C(z) and the adapted controller ˆC(z). The difference between the outputs from these two is called
^ (Figure 2.10), an ideal error. This is however unavailable. Considering the overall system error vector, meaning difference between output from the reference system and the actual plant, and filtering the vector through a delayed inverse model of the system ( ˆP_∆⁻¹ ) produces the vector
_k, the filtered error or Filtered-
. This error can be used to adapt a controller, instead of
^ [3]. The first problem to be solved in this approach is finding an appropriate delayed system inverse. This problem can be solved offline and online, in this thesis the delayed inverse is identified offline, by means of the Transposed approach (Section 2.7.4),

which works just as well offline, using measurements. The offline identification would then be done at the outset, initializing the online adaptation.

Offline adaptation is a more simple solution to this problem and can also be used in its own right; more on this in Chapters 4 and 5.

− +

− C(z) +

C(z)ˆ

copy

P (z)

plant

C(z)ˆ

Pˆ⁻¹(z)

delayed

M (z)

delay

’

u y

Figure 2.10: The Filtered-
approach.

The blocks in Figure 2.10 are:

• P (z) the plant

• M (z) is a reference model, the desired system response

• ˆP_∆⁻¹ is a delayed inverse plant model

• ˆC(z) the adapted controller

2.8 Time waveform replication

The current industrial standard for achieving good tracking accuracy, such that the mea- sured signals from the test drives are accurately reproduced on the vehicle in the road simulator, is an offline procedure called Time Waveform Replication. TWR is an offline iterative feedforward procedure, used to calculate new trajectories based on the output er- ror (difference between measurements from test drives and the output delivered using only the normal controller) [12]. This is done by measuring the Frequency Response Function (FRF) matrix of the road simulator and the tracking error from the previous iteration, an

applying this to

u^(j+1)_{f f} (ω) = u^(j)_{f f}(ω) + (G^m(ω))⁻¹Q^(j)(r(ω)− y^(j)(ω)). (2.37) The elements of (2.37) are:

• j is the iteration number

• u^(j)_{f f} the current feedforward signal

• G^m(ω) the measured FRF matrix

• Q^(j) is a diagonal gain matrix with specific gains for each output

• r(ω) the vector containing the reference signals

• y^(j)(ω) the vector of outputs measured in iteration j

This is the basic implementation of TWR, and in practice it is often attenuated by gains is both the time and frequency domain, where the gains are iteration dependent, according to the situation to improve convergence speed [12].

Being an offline procedure, the TWR algorithm has a few advantages over AIC, as it uses the inverse measured FRF matrix, this can exactly as in the AIC case be a non-causal transfer function matrix, but as the procedure is done completely offline this poses no problem, eliminating the need for delays.

This chapter is divided into four main parts, actuator modelling, car modelling, complete system modelling and model analysis. The goal is a four actuator full car model suitable for control design. To achieve this a linear actuator model is derived, as well as a mass- spring-damper model of the car. The derived complete model is then analyzed in terms of observability, controllability and stability. The chapter also covers the balancing of the model. The verification of the finished model can be found in the results Section 5.3.1.

3.1 Actuator modelling: EMAc

-6

Figure 3.1: The EMAc²-6 actuator.

The EMAc²-6 is an actuator producing a linear motion and is used for shaker rigs. The actuator is of Electromagnetic type, using the reluctance force. A schematic configuration of the actuator can be seen in Figure 3.2. It consists of two armatures, permanent magnets, a magnetic core, two coils and membrane springs. The armatures are connected with a shaft, and together they make up the actuating body for the magnetic power transmission.

The shaft is held in place by the membrane guidance springs, as they are radially stiff but (in comparison) axially flexible. The magnetic force is controlled by

Fm = dWm

ds . (3.1)

The magnetic reluctance force Fm appears in the boundary layer between materials with different permeability, and are governed by (3.1), whereWm is the magnetic field energy and s can be seen as the air gap between magnet and armature [9]. A magnet only produces attracting forces (on ferromagnetic materials); in this case however, as the armatures are on both sides of the magnet, the attracting force on one side is a repelling force on the other side, thus making both possible. The attracting forces are nonlinear, and this property is what makes the actuator unstable, as the forces only balance each other in the middle (due to symmetry). To be able to control the position, and making it into an actuator,

17

+

+

−

− Membrane

Membrane Coil

Coil

Magnetic core

Axle

Armature Armature

Permanent magnets

Figure 3.2: Assembly drawing of the actuator showing simplified flux.

two coils are built in. A current through the coils induces a magnetomotive force in the magnetic core, making control of the reluctance force possible. An in-depth analysis of the working principle of the actuator can be found in [9].

u i F z

Gel Gmag Gmech

Figure 3.3: The comprising parts of the model.

The modelling of the actuator is done in three separate steps; electrical, magnetic and mechanical. The connections between the models are illustrated in Figure 3.3, where u is voltage, i current, F force and z position.

3.1.1 Electrical subsystem

The electrical subsystem consists of mainly two parts, amplifiers and coils, two coils in every actuator and one amplifier per coil. The amplifiers are there to convert the output from the controller to a voltage and subsequently a current, more on this in Section 4.2.1.

The current produces a magnetic flux in the coil, according to u = Ri + ˙ψ = Ri + (L(z,i) + i∂L(z,i)

∂i )di

dt + i∂L(z,i)

∂z dz

dt. (3.2)

In (3.2):

• u, is the voltage from the amplifier [V].

• R, is the resistance in the R-L circuit [ohm].

• i, is the current [A].

• ˙ψ, is, according to Faraday’s law of induction, the induced electromotive force. It is proportional to the change in the magnetic flux.

The change in the magnetic flux ˙ψ can be expanded according to Equation 3.2 [7], and shows the self-induction due to the current and the external induction from the magnets.

The inductivity L(z,i) is dependent on position and current. The decision is made to make

this equation linear and time-invariant by choosing L constant, thus completely neglecting the external induction and the current dependency. After these simplifications the model is reduced to the somewhat more familiar expression

u = Ri + Ldi

dt. (3.3)

3.1.2 Magnetic subsystem

A large amount of work has been put into the modelling of the magnetic properties during the development of the transducer [4] [7]. There are however a few system properties that lead to complexity problems, mainly due to Eddy-currents and hysteresis of the magnetic core. The modelling of the dynamic properties together with verifying measurements have shown that the properties can be modelled with the first order system

Tmag ˙F + F = k_{f i}. (3.4)

In (3.4) the first order system is written in the time-domain, where:

• k_{f i} is the current to force constant

• T_mag is the time constant of the system

3.1.3 Mechanical subsystem

When only considering the actuator, the actuating body in the transducer is the only moving mass. This can be seen as a single degree of freedom system with the equation

m_sh¨z + cz = F − mshg. (3.5)

In (3.5), m_sh is the mass of armatures and shaft and c = c_s+ c_m as the spring of the system is a combination of the membrane guidance springs csand cmthe negative magnetic stiffness of the magnetic core. The static load term m_shg is from here on neglected, as it is inert from a dynamic point of view.

3.1.4 Complete actuator model

To be able to combine electrical, magnetic and mechanical models (3.3, 3.4 and 3.5) into a state-space model, the states need to be identified. Performing a Laplace transform on (3.4) and (3.5) and replacing the force in the mechanics-equation with the expression in

the magnetic-equation, gives

(m_shs²+ c)z(s) = F (s) (3.6)

(Tmags + 1)F (s) = k_{f i}. (3.7)

Putting (3.6) and (3.7) together and returning to the time domain gives the equation mshTmag...

z =−msh¨z + kf ii− cTmag˙z − cz. (3.8)

In the approximation of the magnetic circuit, the term ˙F leads to, according to (3.8), that the highest time derivative in the system is the jerk, making the acceleration a state. Writing equations (3.3) and (3.8) on state space form leads to the state vector x =



z ˙z ¨z i ^. The system equation is

⎡

⎢⎢

⎢⎣

¨z˙z ...z

˙i

⎤

⎥⎥

⎥⎦=

⎡

⎢⎢

⎢⎣

0 1 0 0

0 0 1 0

−_Tmag¹ _m^c_sh −_m^c_{sh Tmag}¹ _k¹_{f iTmag}¹

0 0 0 −^R_L

⎤

⎥⎥

⎥⎦

⎡

⎢⎢

⎢⎣ z

¨z˙z i

⎤

⎥⎥

⎥⎦+

⎡

⎢⎢

⎢⎣ 0 0 0 1/L

⎤

⎥⎥

⎥⎦u. (3.9)

3.2 Vehicle

In this thesis, the mechanical modelling of the car is done with a homogenous rigid body connected to four spring-damper-units as a model of the suspension. This system is then connected through another two/four spring-damper-units to the actuators. This gives the car three degrees of freedom (DOF); heave as well as roll and pitch. The clamping of the actuators on the car is carried out at the jacking positions. As these are designed for static loads they are not especially stiff, which has been verified through measurements, hence the spring-damper modelling of them as well. On the actual shaker rig, the actuators are bolted to a seismic mass, to isolate the system from the building. The fact that the actuators are bolted means that the movement in the x-y-plane (perpendicular to the z-axis or heave) is only due to the flexibility in the connecting shaft, an undesirable side-effect of the actual construction, therefore these DOFs are not modelled.

3.3 Two actuator half vehicle model

Approaching the modelling problem with two actuator models combined with a half vehicle model (Figure 3.4) reduces the DOFs of the car to only heave and pitch. This model can be used for half vehicle shaker rigs, when only one side of the car is connected to shakers.

The rig currently under development is however a complete vehicle rig, which nevertheless can be used with only two actuators. When using only two, these are normally placed diagonally, rendering this model obsolete. The model designed in Section 3.4 is furthermore scalable, and can be reduced to exactly this model. On account of this the modelling steps for the two actuator model are omitted.

In Figure 3.4 the rigid body modelling of the two actuator half vehicle model is shown. The tires are not modelled at all, the complete suspension is replaced with a spring-damper- unit. The mechanical models of the actuators are shown as well with the connecting spring-damper-units modelling the jacking points. This figure could be seen as a side- view of the four actuator full-car model in Section 3.4. In Figure 3.4 z_sxx are vehicle body positions, z_shxxactuator positions and d_xx/c_xx are the damper and spring constants respectively.