Both cascade and overall MPC structures have their pros and cons, which are worth of more detailed discussions and comparisons.
6.4.1 Sampling Time
Firstly, the cascade structure requires the sampling time of inner-loop cur- rent control to be several times faster than that of the outer-loop speed control and the ZOH is assumed to interface the two systems with differ- ent sampling rates. This configuration allows the response of the inner-loop current control to be enormously fast and independent of the outer-loop con- figurations. The benefit of super-fast current control is particularly demon- strated by the quick response in the regulation of d-axis current to zero.
However, for the overall MPC structure, a single sampling time has to be utilized. It implies that the current control and speed control share the same sampling time. In the implementation, the extra computation cost due to the increase of the system dimension and complexity of QP algorithm with the overall structure requires a much slower sampling time to be employed than that of the inner-loop current control in the cascade structure. As a result, the d-axis response will be inevitably slowed due to the slower sampling time employed in the overall structure.
6.4.2 Modelling Error
Secondly, the challenge of applying RPC compared to MPC is the degree of the accuracy of the system model in the high frequency bands when the closed-loop bandwidth is stretched by the inclusion of an extra frequency modes. In the cascade structure, the inner-loop is equipped with a MPC controller which offers more robustness to the modelling error due to the
linearization. Moreover, with a relatively fast inner-loop current response, the model used for the outer-loop RPC design is reduced to a second or- der linear system considering the compensation of the inner-loop dynamics. Thus, the effect of the nonlinearity is overcome by the inner-loop MPC and a low order linear system is resulted for the outer-loop design. Since the cascade structure prevents inner-loop nonlinearity from affecting the outer- loop system, the RPC could readily applied in the outer-loop system for disturbance rejection, speed and position tracking.
In the overall structure, the modelling error due to the linearization could not be ignored for the design of RPC due to its higher sensitivity to modelling errors than MPC. In addition, the increase in the system order adds the risks to apply the RPC in the overall structure. Those are the two key reasons why the RPC has not yet been applied in the overall structure.
6.4.3 Disturbance Rejection
Finally, the main disturbances in a PMSM system are the load torque distur- bance and the equivalent pulsing toque disturbance arising from the current sensor offsets. In the cascade structure, both disturbances could be treated as the input disturbances occurring on the q-axis current command i∗q, as shown in Figure 5.2 and their impact is eventually rejected by the outer- loop RPC controller. The analysis of disturbance rejection in the cascade structure is quite straightforward by the employment of input sensitivity function.
In the overall structure, the load torque disturbance TL is rejected be-
MPC Elec. Model Mech. Model ω∗ e ωe i∗ d i0d i0q id iq vd vq δid δiq ωe + + − Tl/Kt Machine Model
Figure 6.6: Disturbance rejection with overall MPC structure.
state variable iq, which makes the situation become much more complicated
than the case in the cascade structure. Except for the sensitivity issues with RPC in overall structure, it is another reason why not to apply the RPC in the overall structure.
6.4.4 Constraints on iq
In the overall structure, the constraint on iqis imposed via the one step pre-
diction based on the augmented system model and hence the performance of the algorithm would depend on the accuracy of the prediction. In com- mon practice, this type of constraints is categorized into the group of soft constraints [6, 99] since it can not guarantee that the constraints are satis- fied in face of modelling errors. However, the iq constraint, in the cascade
structure, is imposed via its reference signal i∗q that is assumed to be equal to iq. The constraint on the i∗q itself is the input constraint that belongs to
the type of hard constraints [6, 99]. As its name indicates, hard constraints are guaranteed regardless of any circumstance. Figures 6.7 and 6.8 demon- strate the comparison between the two cases. As shown in Figure 6.8(a), the constraint on i∗
q is guaranteed but is violated slightly by iq due to the
is normally small and only happens during transient responses when an in- tegrator is applied for the inner-loop, the resulting short time over-current is acceptable for the servo drive systems. However, the formulation of state constraints with the overall structure is more vulnerable to modelling errors and system noises that could deteriorate the control performance, even at the steady-state. 0 0.5 1 1.5 2 2.5 3 −10 −8 −6 −4 −2 0 2 4 6 8 10 i d i q
(a) d-q axis current (id, iq)
0 0.5 1 1.5 2 2.5 3 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 ω e * ω e (b) electrical speed ωe Figure 6.7: iq constrained
with overall structure
0 0.5 1 1.5 2 2.5 3 −10 −5 0 5 10 i d i q i q *
(a) d-q axis current (id, iq)
0 0.5 1 1.5 2 2.5 3 0 500 1000 1500 2000 ω e * ω e (b) electrical speed ωe Figure 6.8: iq constrained
with cascade structure
6.4.5 Sum of Comparisons
In sum, the discussion above indicates that the cascade structure still offers some advantages in terms of the existence of the inner-loop modelling error and disturbance rejection.
6.5
Conclusions
This chapter focuses on the development of a single MPC for the combined speed and current control of a PMSM. The design is carried out based on the assumption of the linearization and the employment of gain scheduled control. In practice, only few, for example two or three, operating points is actually required for the gain scheduled control, since the MPC with an embedded integrator possesses a certain degree of tolerance to the model mismatch due to the linearization. It also reveals that it is difficult to apply RPC to the overall structure due to the model mismatch and increase of system order.
The constrained MPC design in the overall structure differs from the cas- cade structure in that the constraint on iqoccurs on the state variable of the
overall model. Thus, the formulation of state constraints adds more com- plexities to solve the active Lagrange multipliers on-line using Hildreth’s procedure. Special consideration is given to the situation where the con- flict of active constraints happens. The problem of conflicts is solved by giving priorities to constraints according to their significance in the appli- cation. The less important constraints are relaxed when conflicting in order to guarantee that more important constraints are satisfied. The experimen- tal results given in this chapter prove the feasibility of the overall control structure. In addition, the choice of tuning parameters and their effects on velocity response and noise attenuation have been discussed and supported by experimental results.
Last but not the least, through the comparison with the cascade struc- ture, it shows that the cascade structure still offers some advantages in face of the selection of control structures.
Chapter 7
Comparison with Cascade PI
The objective of this chapter is to compare the two MPC structures: cascade MPC and overall MPC, with the classical cascade PI structure. Section 7.1 introduces the design of the three PI controllers using pole-assignment technique and it provides the benchmark for the comparison with the other two MPC structures. Section 7.2 illustrates the comparison between the two MPC structures with the benchmark in terms of the ability of constraints and the performance under parameter variations. Section 7.3 summarizes the findings of the comparison and concludes the chapter.