The continuous two-mass model from Eq. (3.12) with matrices (3.44) and three-mass model from Eq. (3.27) with matrices (3.48) and (3.49) are discretized such that the closed loop dynamics of a digital control system can be analyzed. Subsequently, the discretized systems are augmented by input and output dead times.
5.2.1 Discretized Control Models
The general transformation of continuous-time systems to discrete-time models, as for instance given in [FranklinPowellWorkman98] or [Levine10], is derived. Consider an nth-order discrete system described by the equations
xk+1 =Adxk+Bduk, yk =Cdxk+Dkuk, xk ∈Rn, (5.12) whereby, discrete matrices and states are denoted by the index d.
The discrete system matrix Ad can be calculated by
Ad =eAT0, (5.13)
where A is the continuous system matrix and T0 the sampling time. Further, due to the zero-order hold, it is assumed that the control input is piecewise constant over the sample time. Furthermore, the discrete input matrix is given as
Bd= Z T0
0
eAqAdq=A−1(Ad−I)B, (5.14) for Anonsingular and with identity matrix I. Moreover, it is
Cd =C, Dd =D. (5.15)
Equations (5.13) and (5.14) are now applied to the two-mass system (3.12) and three-mass control system (3.27).
5.2.1.1 Discrete Two-Mass Control Model
The discrete two-mass control model with discrete states xk= [∆ϕk,∆ωk]T can be derived analytically by applying Cayley–Hamilton theorem. The transition matrix can be rewritten as a power series
eAT0 =µ0I+µ1A+µ2A2+. . .+µn−1An−1. (5.16) The coefficients µ0, µ1, . . . , µn−1 can be calculated using the Cayley–Hamilton theorem.
The theorem stated that the following n equations
eλit=µ0 +µ1λi+µ2λ2i +. . .+µn−1λn−1i , (5.17)
5.2 Powertrain System with Time Delay 99
with λi as the eigenvalue of Afor i= 1,2. . . , nare valid, see for instance [Unbehauen07].
For the two-mass system (3.12) we have a pair of complex conjugate eigenvalues
λ1 =−a0+ω0i, λ2 =−a0−ωoi. (5.18) Thus, the following linear system of equations has to be solved
"
The equations can be reformulated to µ1 = 1
Using (5.16), then the discrete system matrix is analytically given as A2,d =µ0I+µ1A2 =
and the discrete input matrix reads
B2,d=A−12 (A2,d−I)B2 =
Once the discrete system is derived, the corresponding sampling times can be calculated using Eq. (5.11) for the two-mass control models of conventional powertrain, and battery electric powertrain, respectively. The closed-loop bandwidth ωb is calculated with output controller from Eq. (3.83) and critical proportional gain from Eq. (3.90). The critical gain kp is chosen in order to calculate the closed loop with the fastest system response.
Closed-loop bandwidth ωb of the system, appropriate sampling frequency ω0 = 10ωb and appropriate sampling time T0 are summarized in Tab. 5.1. For further investigations the sampling time T0 = 5 msis chosen, which is appropriate for all three two-mass control models.
kp ωb ω0 T0
conv. powertrain closed clutch 296.4 39.6 rad/s 396 rad/s 15.9 ms conv. powertrain open clutch 74.5 112 rad/s 1120 rad/s 5.6 ms battery electric powertrain 65.6 91rad/s 910 rad/s 6.9 ms
Table 5.1: Calculated parameters for two-mass models.
5.2.1.2 Discrete Three-Mass Control Model
The analytical derivation of the discrete three-mass control model with discrete states xk= [∆ϕ1,k,∆ϕ2,k,∆ω1,k,∆ω2,k]T is more difficult, since the system order is higher than of the two-mass model. However, the discrete system matrix A3,d and input matrices B3,u1,d,B3,u2,d can be calculated numerically using Eq. (5.13) and Eq. (5.14). Thereby, the continuous matrices from Eq. (3.34), Eq. (3.48), and Eq. (3.49) are applied with an appropriate sampling time T0. The appropriate sampling time for the three-mass model is calculate analogous to the two-mass control models. For feedback control the critical control gain kp = 260 is applied to the output feedback controller from Eq. (3.91) using the second input u2. The bandwidth of this closed loop system is calculated and is used to derive an appropriate sampling time. Table 5.2 shows the values with minimum appropriate sampling time T0 = 13.1 ms. As in the case of the two-mass control models a sampling time of T0 = 5 msis chosen.
kp ωb ω0 T0
hybrid electric powertrain 260 48 rad/s 480 rad/s 13.1 ms
Table 5.2: Calculated parameters for the hybrid electric three-mass model.
5.2.2 Augmentation of Time Delay
Actuator dead time τact and measurement dead time τms are part of the control loop. In the following, it is assumed that actuator dead time τact and measurement dead time τms
are a multiple of the sampling time T0. Then, it is valid nact = τact
T0
, nms = τms
T0
, with nact, nms ∈N, T0 6= 0 (5.26) and the discrete signals are delayed by nact and nms steps, respectively. Figure 5.5 shows the digital control loop with continuous and discrete delayed signals.
The discrete system plant from Eq. (5.12) has to be augmented by the delayed states in order to analyze the closed loop stability and design compensation methods. The amount of delayed steps ntotal depends on the total dead time. It is
ntotal=nact+nms. (5.27)
5.2 Powertrain System with Time Delay 101
Gc(z) A/D
+ D/A Gp(s) y(t)
y(k)
Gp,τact(z)
τms
r(k−nms) e(k−nms)
y(k−nms)
uc(k−nms) uc(t−τms) τact
uc(t−τms−τact)
Figure 5.5: Digital control loop with discretized plant Gp,τact(z), actuator dead time τact
and measurement dead time τms.
Then, the augmented system of the nth-order discrete system from Eq. (5.12) has
na =n+next (5.28)
states with the number of extended states
next=n·ntotal. (5.29)
The augmented state vector reads
xa,k = [xk, xk−1, . . . , xk−ntotal]T ∈Rna, (5.30) where xk−i denote delayed states at time step (k−i) with i= 1,2, . . . ntotal.
The general augmented discrete system matrix is given as Ad,a=
"
Ad 0n,next Inext,next 0next,n
#
∈Rna×na, (5.31)
where 0n,next denotes then×next zero matrix and0next,n denotes thenext×n zero matrix, respectively. Moreover, Inext,next denotes the next×next identity matrix.
The augmented discrete input matrix reads Bd,a=
"
Bd 0next,1
#
∈Rna×na. (5.32)
Two-Mass System
Applying the augmentation to the discrete two-mass control system, the augmented state vector
x2,a,k = [∆ϕk,∆ωk,∆ϕk−1,∆ωk−1, . . . ,∆ϕk−ntotal,∆ωk−ntotal]T (5.33) results.
The augmented discrete system matrix is given as
and the augmented discrete input matrix reads
B2,d,a =
The discrete three-mass control system can also be augmented by delayed states and the augmented state vector is given as
x3,a,k = [∆ϕ1,k,∆ϕ2,k,∆ω1,k,∆ω2,k,∆ϕ1,k−1, . . . ,∆ω2,k−ntotal]T. (5.36) The augmented discrete system matrix reads
A3,d,a =
with 4×4 identity matrix I and the augmented discrete input matrix is given as
B2,d,a =