Piston position control

The first one of the proposed methods for position control is the linear quadratic (LQ) method.

LQ optimal techniques are widely studied and applied since the 1960s. The advantages of this controller are simple construction, easy implementation and low computing cost. Besides, it provides an optimal feedback rule for a given cost function of the output and input energy. The detailed treatment of LQ optimal control can be found in the following textbooks [57, 58]. In the last decades the LQ control was introduced into several applications [59,60,61,62,63].

In order to achieve a nominal performance and meet robust stability specifications a H∞

synthesis is examined to take the unmodelled dynamics, effects of disturbances and parametric uncertainty of the plant into consideration. This way, the controller can be designed to provide the track of the predefined reference signal, which reduces the effects of the disturbances and the uncertainties on performances. The robust and optimal control theory are discussed in [64,65].

In the last decades theH∞control was applied several applications, too [66,67,68,69,70].

The third examined control design method is the nonlinear type sliding mode control (SMC), which could achieve higher disturbance rejection performance and wider stability margin versus the linear methods. Besides the disturbance rejection and stability, it provides a well defined dynamic behavior on a sliding surface, which is better suited to achieve the predefined dynamic requirements. The detailed treatment of sliding mode control can be found in [43, 71, 72].

Moreover, several papers have been published on the topic of sliding mode control of electro-pneumatic actuators [73,74,75,76,77,78,79].

The aim of the position control design is to find the most suitable approach, which provides satisfying position tracking performance, good disturbance rejection, wide stability margin and appropriate to embedded application.

5.3.1 Linear quadratic approach

The first proposed feedback control algorithm is the LQ servo control method, an extended version of the LQ control, which has full state feedback and can follow a reference signal [80,81].

The LQ provides an optimal feedback rule for a given cost function of the output and input energy, so the input energy can be considered for the control design. The LQ servo includes an additional artificial state with an integrator, which can ensure the tracking capability of the control. For LQ servo control design, the system should be given in LTI form. The linearization can be seen inSection B.1 ofAppendix B.

To achieve the optimal feedback rule, the associated performance index is written as follows:

J = Z ^∞

0

x^TQx+u^TRu

dt, (5.1)

whereQ=Q^T ≥0and R=R^T >0.

The goal is to construct a stabilizing linear state-feedback controller of the form: u=−Kx that minimizes the performance indexJ.

Besides stabilizing the actuator in a steady state, it should track a reference signal. In consequence there is an error signal e, which is the deviation from the desired and current actuator state. This error could be decreased to zero using a new state variable as follows:

˙z=e=x_ref −x_p=x_ref −C_px, (5.2) whereC_p selects the controlled states.

The steady state of the new state equation is zero, which impliese= 0. Thus the augmented system can be given by the following form:

x˙

˙z

=

Aˆ

z }| { A 0

−C_p 0 x z

+

Bˆ1

z }| { B

0

u+

Bˆ2

z }| { 0

I_p

x_ref. (5.3)

Suppose that the augmented system

A,ˆ Bˆ₁

is stabilizable and the complete state of the augmented plant can be accurately measured or estimated at all times and is available for feed-back. In this way the associated optimal control can be obtained as:

u=−[Kr Kz] x

z

; [Kr Kz] =R⁻¹Bˆ^T₁P, (5.4) whereP≥0is the unique solution of the Control Algebraic Ricatti Equation (CARE) of

P ˆA+Aˆ^TP−P ˆB₁R⁻¹Bˆ^T₁P+Q= 0. (5.5) Thus the closed-loop augmented system in state space form is obtained as follows:

x˙

˙z

=

A−BK_r BK_z

−C_p 0

x z

+

0 I

xref. (5.6)

The LQ servo block structure can be seen inFig. 5.2, whereG=C(sI−A)⁻¹Bcorresponds to the linearized model.

R Kz G

Kr

Cp

xref z˙ z u x

− −

Figure 5.2: LQ servo block structure

The LQ servo control hasRandQmatrices, which are tunable parameters with appropriate dimensions. For the position control of the clutch actuator the reference signal is the piston position (x_pst), hence the error e is the deviation from the desired and actual piston position.

For the input weighting matrixRthe identity matrix is used and theQmatrix contains the state variable’s weights on the main diagonal and zeros elsewhere. These weights target to achieve the desired system dynamic behavior. Since the control signals are saturated, the error could become high and the integrator winds up easily. In order the control signal remains in its nominal physical range, the values of theQmatrix should be chosen carefully [82] or an appropriate anti wind up should be used (see details later).

5.3.2 H^∞ approach with exact linearization

The second proposed control method is the H∞ approach. In this case exact linearization via state feedback is used to take the nonlinearity of the system into consideration. The process of the exact linearization can be seen inSection B.3 ofAppendix B.

Consider the closed-loop system which includes the feedback structure of the linearized model G and controller K, and elements associated with the uncertainty models and performance objectives (seeFig. 5.3).

In the diagram,ris the reference,vis the control input,yis the output,nis the measurement noise andz_eis the deviation of the output from the desired one. The structure of the controllerK may be partitioned into two parts: K = [K_r K_y], whereK_y is the feedback part of the controller and K_r is the pre-filter part [83].

W_cmd K_r G W_e

W_n W_m

∆_m Try

Ky

r v

y

ze

n

−

e d

˜ r

−

Figure 5.3: Closed loop interconnection structure

The required transfer function T_ry from the reference command in physical units to y is used to introduce a time domain specification into the design process. The W_cmd describes the magnitude and the frequency dependence of the reference command generated by the normalized reference signal r. The model error is represented with multiplicative uncertainty at the plant input byWm and∆m.

It is assumed that the transfer function W_m is known, and it reflects the uncertainty in the model introduced by the unmodelled dynamics, the linearization and the error of the feedforward part. The transfer function∆m is assumed to be stable and unknown with the norm condition, k∆_mk<1. In the diagram, eis the input of the perturbation and dis its output.

The weighting functionW_ereflects the relative importance of the different frequency domains in terms of tracking error. The weighting function Wn represents the impact of the different frequency domains in terms of sensor noisen.

Let the required transfer function from the reference to the piston position be the following second-order system: T_ry = _τ2s²+2ζτ s+1¹ . The reference tracking should ideally be decoupled at the output channels and must fulfill the requirements determined in the time domain. Thus the parameters of theT_ry transfer function are as follows: τ = 0.01 andζ = 1.7, these yield an over damped second-order behavior.

In order to meet the requirements for the tracking error, aW_e weighting function is applied, which reduces the steady state error below 2%: We= 5·10^3s/10_s+1²⁺¹. It follows from the condition that the transfer function from the reference signal to the position must be less than 1/W_e in theH∞norm sense i.e. less than 2·10⁻⁴m in steady state.

In general, the sensors are often accurate at low frequency and in steady state, but respond poorly to frequency increase. Hence, it is assumed that the sensors noise is 1% in the frequency domain of the actuator and increases above it, thus the weighting function of the sensors noise are represented by: W_n= 10^−5s/10_s/10²4⁺¹+1.

Let the frequency weighting functions of the unmodelled dynamics are as follows: W_m = 5^s/10_s/10²3⁺¹+1. These result that in the low frequency domain, the uncertainties are about 10% and, in the upper frequency domain they are up to 100%.

The reference signalrin case of manual transmission comes from the clutch pedal and in case of automated mechanical transmission (AMT) comes from the transmission control unit. The driver can generate stick input reference commands with the clutch pedal up to a bandwidth of about 10Hz, but the transmission control unit generates unit steps as well. The magnitude of the reference in case of central release bearing is10mmhence, theW_cmd= 10⁻².

In order to derive the robust performance of a closed-loop system the uncertain structure depicted in Fig. 5.3 has to be reformulated in LFT form to get the so called ∆ P K structure (see Fig. 5.4).

P

K

∆

r n d e

v

z_e

y

˜ r w

Figure 5.4: The ∆-P-K structure

The generalized plant model P with inputs [d, w, v], where w = [r, n] and outputs [e, z_e, ˜r, y], respectively, can be formalized as follows:

P =







0 0 0 W_m

W_eG −W_eT_ryW_cmd 0 W_eG

0 W_cmd 0 0

−G 0 −W_n −G





. (5.7)

Using the weighting functions of the nominal performance and the robust stability specifica-tions, the optimal H∞ controller is designed. To find a controller that minimizes the transfer function from w to z_e, the hinfsyn MATLAB function is used. With the LMI-based synthesis approach of the controllerγ = 0.34 and controller of 9th order are resulted.

In order to analyze the performance and robustness requirements, the closed loop system is expressed by the lower linear fractional transformation:

M =F_l(P, K) =

M₁₁ M₁₂ M21 M22

. (5.8)

Then the robust stability (RS) is equivalent to: kM₁₁k_∞ < 1, the nominal performance (NP) is achieved if the performance objective: kM₂₂k_∞ < 1 is satisfied. Finally, the robust performance (PR) is equivalent: kF_u(M,∆)k_∞<1.

The M11 and M22 transfer functions associated with transfer from dto e and transfer from w = [r, n]to z_e can be evaluated separately. The controlled system achieves RS, NP and RP as well. These results are represented with the singular value plots of the closed-loop system in Fig. 5.5.

10⁻¹ 10⁰ 10¹ 10² 10³

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Frequency [rad/s]

Magnitude

RS NP RP

Figure 5.5: Singular values with full order controller regarding RS, NP and RP

Since the order of the resulted controller is high, the Hankel singular value based order reduction procedure [64] is applied. With reduce MATLAB function a 5th and a 4th order controller are derived. The singular value plots of the closed loop system with 9th order controller and with reduced order controllers are compared. Then it found that, the 5th order one may fulfill the requirements since, the characteristics of the singular values deviate only in low frequency domain (seeFig. 5.6).

10⁻¹ 10⁰ 10¹ 10² 10³

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Frequency [rad/s]

Magnitude

9^th order 5^th order 4^th order

Figure 5.6: Singular values with full and reduced order controllers

In order to handle the saturation a high-gain anti-windup [84] is applied. The anti-windup compensation is provided by subtracting the difference between the actual and the saturated

physical control signals through a high-gain matrixX from the controller input. The implemen-tation is shown in Fig. 5.7.

K u uˆ EPC

X

r e y

−

−

Figure 5.7: High-gain anti-windup structure

The limits of the control signal u, i.e. the available maximum and minimum mass flow rates, are dependent on the pressure ratio between the chamber and the supply or ambient pressure.

Hence, the saturation limits are calculated according to Eq. (2.28). In case of the upper limit, A_in corresponds to the cross sections of the load valves,p_in corresponds to the supply pressure andp_out corresponds to the chamber pressure. While in case of the lower limit,A_incorresponds to the cross sections of the exhaust valves, p_in corresponds to the chamber pressure and p_out corresponds to the ambient pressure [85].

5.3.3 Sliding mode control approach

Consider a single-input dynamic system of normal form:

x⁽ⁿ⁾=f(x) +g(x)u, (5.9)

where the scalar x is the output of interest, the scalar u is the control input and x= [x,x, . . . , x˙ ⁿ⁻¹]^T is the state vector. In order to rewrite the EPC model into normal form a coordinate transformation is applied (see details inSection B.2 of Appendix B). The f(x) and g(x) coordinate functions are derived fromEq. (B.19)and Eq. (B.20)respectively. In Eq. (5.9) the functionf(x)is not exactly known, but the extent of the inaccuracy onf(x)is upper bounded by a known continuous function of x; similarly, the control input function g(x) is not exactly known, but is of known sign and is bounded by a known, continuous functions of x.

The control problem is to get the state x to track a specific time varying state x_d = [x_d,x˙_d, . . . , xⁿ_d⁻¹]^T in the presence of model imprecision on f(x) and g(x). For the tracking task to achieve a finite controlu, the initial desired state x_d(0)must be such that

x_d(0) =x(0). (5.10)

Letx˜=x−x_d be the tracking error in variablex, and let

˜

x=x−x_d= [ ˜x x . . .˙˜ x˜⁽ⁿ⁻¹⁾ ]^T (5.11) be the tracking error vector. Furthermore, let us define a time-varying surfaceS(t) in the state-spaceRⁿ by the scalar equations(x, t) = 0, where

s(x, t) = d

dt +λ (n−1)

˜

x (5.12)

and λis a strictly positive constant.

With initial condition Eq. (5.10), the problem of tracking x ≡ x_d is equivalent to that of remaining on the surfaceS(t) for allt >0; indeeds= 0 represents a linear differential equation whose unique solution is ˜x ≡ 0. Thus, the problem of tracking the n-dimensional vector x_d can be reduced to that of keeping the scalar quantity s at zero. More precisely, the problem of tracking the n-dimensional vector x_d can be effectively replaced by a 1^st-order stabilization problem in s. Indeed, since from Eq. (5.12) the expression of s contains x˜⁽ⁿ⁻¹⁾, it is needed only to differentiatesonce for the input uto appear. Furthermore, bounds onscan be directly translated into bounds on the tracking error vector ˜x, and therefore the scalar s represents a true measure of tracking performance. Specifically, assuming that˜x(0) = 0, then

∀t≥0, |s(t)| ≤Φ=⇒ ∀t≥0, x˜⁽ⁱ⁾(t)

≤(2λ)ⁱǫ

i= 0, . . . , n−1, (5.13) whereǫ=Φ/λⁿ⁻¹.

In the case of x_d(0)6=x(0), bounds of Eq. (5.13)are obtained asymptotically, i.e., within a short time-constant (n−1)/λ. Thus, an n^th-order tracking problem is replaced by a 1^st-order stabilization problem, and is quantified with Eq. (5.13) the corresponding transformations of performance measures. The simplified,1^st-order problem of keeping the scalarsat zero can now be achieved by choosing the control lawu of Eq. (5.9)such that outside of S(t)

1 2

d

dts² ≤ −η|s|, (5.14)

whereη is a strictly positive constant [43].

Consider the model of the EPC in an appropriate coordinate system which guarantees that the model is in form of Eq. (5.9). The system dynamics is not exactly known, but the model error is assumed to be bounded by some known functionF =F(x) as follows:

|f −f_real| ≤F. (5.15)

In order for the system to track x(t) ≡ x_d(t), a sliding surface (n = 3) s = 0 is defined according toEq. (5.12), namely:

s= d

dt +λ (2)

˜

x= ¨x˜+ 2λx˙˜+λ²x.˜ (5.16) Then

˙

s = ...

˜

x + 2λx¨˜+λ²x˙˜=

= ...

x −...

x_d+ 2λx¨˜+λ²x˙˜=

= f(x) +g(x)u−...

x_d+ 2λx¨˜+λ²x.˙˜ (5.17) The best approximation u_eq of a continuous control law that would achieve s˙= 0 is thus

u_eq= −f(x) +...

x_d−2λx¨˜−λ²x˙˜

g(x) . (5.18)

In order to satisfy the sliding condition (Eq. (5.14)) despite uncertainty on the dynamics f, an additional term (udr) is added to ueq, which is discontinuous across the surface s= 0 as:

u_{SM C} =u_eq+u_dr =u_eq−k sgn(s), (5.19)

where u_dr is the disturbance rejection part of the control. Smoothing of control discontinuity is essential to avoid chattering effect thus in Eq. (5.19) the sgn(s) is replaced bysat(s), where sat(.)is a linear saturated function to±1with gradient of one. This achieves a trade-off between tracking precision and robustness to unmodeled dynamics.

Finally, substituting Eq. (B.19),Eq. (B.20)and Eq. (5.16) intoEq. (5.19)the SMC law can be obtained and the controlk andλparameters are tuned for the controlled EPC system.

In order to decrease the load of the computing device the reduction of the equivalent control part to zero (u_eq = 0) is examined. In this wayu =−k sat(s) is the control rule, which can be considered as a simplified SMC. This reduction can be done since in static state the equivalent control part is equal to zero. Otherwise the u = −k sat(s) part could ensure the s= 0 sliding condition and through these the prescribed dynamics [86].

In document Observer based feedforward/feedback control of electro-pneumatic clutch systems (Pldal 71-78)