Control systems for power converters typically must satisfy several specifications and requirements, while dealing with uncertainty or operating point dependence at the same time. Since worst-case models may not exist or be different for each specification, the conventional industry standard approaches, such as the ones based on voltage-mode [1
] and current-mode [2
] controllers, rely on expert knowledge, simulation and iteration in order to find an appropriate controller.
As an alternative to this manual iteration, the automatic synthesis of controllers for switched-mode power converters has been one active topic of research in the last decade. These approaches are of interest because they can take into account the requirements together with the uncertainty or the nonlinearities of the converter to provide robust stability and performance, and they can do all that by imposing conditions beforehand.
Methods based on linear matrix inequalities (LMIs) have been some of the most successful approaches to the synthesis of robust controllers for power converters. The first attempts [5
] demonstrated how uncertainty could be modeled and how the transient and frequency domain specifications could be taken into account. More recently, the efforts have been focused on approaches that do not require full state feedback [8
], that improve the robustness [9
] or the performance properties [10
]. Although these papers employ averaged models of the converters, other approaches have also tackled the problem from a hybrid system perspective [11
One of the open problems in the topic is the fact that the results may be conservative. The synthesized controller may not offer the best possible performance, when compared with conventionally tuned controllers, such as current-mode controllers. One possible solution to this conservativeness was shown in [13
], where excellent robustness and tight regulation were achieved simultaneously, at the expense of control complexity.
One of the causes behind the conservativeness of LMI methods in [7
] is the fact that the stability of the system is ensured no matter how large the derivative of the uncertain parameters may be. Specifically, when the uncertainty is characterized by being norm bounded, time varying and evolving in a set of polyhedral vertices, one difficulty remains: how to find an adequate mathematical representation for it, as well as for its rate of variation [14
]. Nonetheless, several ways for representing both the derivative of the time-dependent parameter and the parameter itself have been proposed in the literature [14
]. Different approaches to control these uncertain systems have been reported, such as state feedback gain-scheduling control [18
], output feedback [15
], linear quadratic Gaussian (LQG) or linear quadratic regulator (LQR) controllers [19
] and gain-scheduled linear quadratic regulators (LQRs) [22
In this paper, we propose a new method to synthesize robust LQR controllers for pulse-width modulated (PWM) converters, with the objective to improve the LQR synthesis that was proposed in [7
]. The method is based on the results introduced in [14
], such that the proposed approach can consider the time derivative of the uncertain parameters. As a consequence, the new LQR formulation can obtain less conservative results. This reduced conservativeness can be seen as a new degree of freedom. With this method, practicing engineers can synthesize controllers for larger sets of uncertainty (i.e., with improved robustness) or controllers that provide tighter regulation (i.e., improved performance) when compared with the previous method. The approach has been verified with the synthesis of a controller for a boost converter, such that a direct comparison with [7
] has been carried out. Note that the proposed method could also be used in other switched-mode power converters, such as the buck converter (which was also treated in [7
This paper is organized as follows. Section 2
briefly reviews the modeling of the boost converter and the LQR state feedback proposed in [7
]. Then, Section 3
proposes a new formulation of the LQR problem, such that novel LMI conditions are given. In Section 4
, the proposed synthesis method is employed in the boost converter, using the original model and other alternatives that allow us to obtain improved robustness or improved performance. The appropriateness of the approach is verified with simulations in Section 5
. Finally, conclusions are given in Section 6
5. Simulation Results
This section illustrates the properties of the different controllers
. We have performed a set of PSIM [26
] simulations of the switched DC–DC boost converter, according to Figure 1
. The first set of simulations is useful to establish the performance of the controllers, by analyzing the response of the converter with respect to changes in the output current. The second set aims to establish the robustness of the different controllers when there is a change in the operating point, by modifying the supply voltage.
First, the waveforms of the simulations with changes in the load are grouped in Figure 5
. The top waveforms in each subfigure correspond to the output voltage
, whereas the bottom waveform represents the output current
. In all simulations, the converter load is initially the nominal value
. At time t
= 1 ms, the load changes to
, which is the maximum load allowed by design in all polytopes. The load returns to
= 6 ms.
As a baseline for the comparison, Figure 5
a shows the performance of controller K
, as in [7
], and the performance of controller
obtained with the proposed method and the same polytope used in [7
. It can be seen that the disturbance rejection properties and the settling time are nearly identical. In contrast, Figure 5
b shows a comparison with controller
, which exhibits a tight regulation of the output voltage, such that the maximum error of
and its settling time are reduced to approximately one half of what is achieved with K
. As expected, the robust controller
presents loose regulation and a slower response, as shown in Figure 5
c, when compared to K
Note that Figure 5
a–c shows the response at the nominal operating point, when
and D’ = 0.5. In order to evaluate the performance at a different operating point, Figure 5
d shows the response of K
under an input voltage variation of −40%, such that the operating point is now D’ = 0.3. Again,
is the controller that achieves excellent regulation properties, maintaining its robustness in the expected region of operation.
is the controller that demonstrates that the proposed method can be used to improved regulation while maintaining the same robustness properties,
is the controller that demonstrates that the method can also be employed to enlarge the stability region. Figure 6
shows the waveforms of the simulations in which the input voltage is stepped, such that the operating point of the converter is modified in time. Figure 6
a shows a voltage step of −40%, which corresponds to a step in the duty cycle from D’ = 0.5 to D’ = 0.3 (D = 0.7). Since all polytopes considered such a region, the three controllers maintain the stability, with
exhibiting the best regulation performance. Figure 6
b shows a similar step in the input voltage, but now the input voltage decreases down to
, such that the operating point duty cycle moves from D’ = 0.3 to D’ = 0.1. The method proposed in [7
] did not allow us to consider such a large range of operating point uncertainty, whereas the proposed method resulted in controller
. As can be seen in the figure,
is the only controller that successfully maintains stability under those conditions, exhibiting excellent stability properties.
It is worth noting that the transient shown in Figure 6
b shows the saturation of the duty cycle at 100% with the unstable controllers. Although the modeling of that nonlinearity is out of the scope of this paper, this aspect has been treated in the specific context of switched-mode power converters in [27
Finally, Figure 7
depicts the waveforms of the converter startup, with the three controllers K
. The input voltage is Vg = 12 V and the voltage reference ramps up from 12 V to 24 V at t
= 0, with a rate of change of 2400 V/s. It can be observed that the three controllers operate inside the expected range of operation and stabilize the converter.
The numerical synthesis of robust LQR controllers for PWM DC–DC converters by means of LMIs has suffered from the conservativeness of the methods based on quadratic stability, since a single Lyapunov function is employed for the entire uncertainty region and because the uncertain parameters are assumed to change arbitrarily fast. This paper proposes a new method to synthesize robust LQR controllers. The method employs parameter-dependent Lyapunov functions and allows us to consider the rate of change of the uncertain parameters.
The method has been employed to synthesize LQR controllers for a PWM DC–DC boost converter. With that aim, the paper has reviewed two uncertainty models of the boost converter that were proposed in the past. In addition, it has introduced an enlarged version of one of them, with the objective to obtain stability for a very large region of uncertain parameters. While the conventional synthesis methods fail to obtain feasible solutions with these uncertainty models, the proposed method has been demonstrated to be useful in achieving better regulation performance or improved robustness.