Adaptive Control of Power Converters Using Digital Network Analyzer ...
Adaptive Control of Power Converters Using Digital Network Analyzer Techniques Adam Barkley, Roger Dougal, Enrico Santi Department of Electrical Engineering University of South Carolina Columbia, SC USA [email protected], [email protected], [email protected] Abstract—We describe a new method for design of adaptive controls for switching power converters that leverages the information-rich frequency response data obtained using the correlation-based analysis dubbed Digital Network Analyzer Technique. Compared to existing single-frequency based adaptive control methods such as limit-cycle and relayfeedback based autotuning, this method provides a unified approach to target many distinct problems plaguing converter control designers. The major strength of this method is that a single adaptive controller structure is able to fix multiple converter problems with a generalized and online identification and adaptation procedure. Also, unlike the conservative nature of robust control, adaptive control attempts to maintain relatively high performance at each time instant whenever possible. A step-by-step procedure is given which explains how the control platform can use the converter to perturb the system, identify the non-parametric frequency response of the plant, fit the data to a parametric model, and synthesize a control which meets user specifications. Simulation results are provided for a comprehensive set of realistic scenarios, where each test case uniquely degrades the converter frequency response. In each case, the performance and stability degradation is mitigated through targeted control adaptation.
I.
INTRODUCTION
Electrical AC and DC power distribution systems are becoming increasingly complex due to a widening variety of power sources, loads, energy storage systems, reconfiguration options, and the incorporation of controlled power electronic converters. Ensuring robustness of these distribution systems is imperative to achieve high quality of service despite the time-varying nature of many sources and loads. Historically, the distribution-level control system has been highly centralized. Current research indicates an emerging trend towards hierarchical and decentralized control where more of the control decisions are made at lower levels in the hierarchy. This work introduces a new strategy for adaptive control that enables each converter in a system to adjust its own local control to accommodate changes in its environment.
This research was sponsored by the Electric Ship Research and Development Consortium (ESRDC) under grant N00014-08-1-0080 entitled “Power Electronics, Architecture, Controls, and Design Tools for Electric Ship Systems”
Figure 1. Generalized schematic of a switching converter, its embedded digital controller, and the proposed adaptive control structure
Recent advancements in digital control theory and the availability of low-cost embedded control platforms have enabled new functionality in power converters: higher control performance, and additional flexibility in control design [1]. However, tightly regulated multi-converter systems are subject to potentially destabilizing interactions according to the so-called Middlebrook impedance criterion [2]-[3]. Also, large parameter variations can negatively affect converter stability and performance. Much literature has been published in the area of converter modeling [3]-[5] and system identification [6]-[8] to address this problem. Previous work has demonstrated that a converter and its digital control platform, shown in Figure 1, can be used to perform online identification and system monitoring. In particular, important small-signal converter-level transfer functions [9]-[10] and impedances [11]-[13] can be monitored in real time. Several researchers have proposed adaptive control of switching converters [14]-[16], but they tend to target single problems around the crossover frequency by relying on existing or purposefully introduced
B. Weighted Least Squares Fitting The fitting problem is then to find coefficient vectors A and B such that the modeling error defined in (2) is minimized. As this error approaches zero for all frequencies of interest, Gcandidate(jω) is a good approximation of the real plant Guy(jω). ε fit ( jω k ) = Gcandidate ( s ) − Guy ( s ) s= jω
(2)
k
The method of least squares fitting is used to define a cost function as a metric to evaluate the quality of the fitting. This cost function, JWLS, is defined in (3) as the weighted sum of the square of the modeling error defined in (2) over the frequency indices of interest. Minimization of this cost function by adjusting coefficient vectors A and B can be accomplished by iterative numerical methods such as a Gauss-Newton algorithm. Figure 2. Flowchart for adaptive control design using digital network analyzer techniques
limit-cycling at that single frequency. The method proposed here and shown in Figure 2 uses the information-rich frequency response estimations obtained from correlation analysis to target a wide range of potential problems with a single adaptive controller structure. A model fitting procedure is described in section II, and one option for adaptive control synthesis is proposed in section III. A simulation testbed is presented in section IV along with results for a set of practical converter problems. II.
MODEL FITTING
This section describes a procedure to fit the nonparametric complex frequency response data obtained from the cross-correlation techniques of [9]-[10] to a parametric model of user-specified order. This fitted model is a more compact representation of the estimated model and is suitable for communication to a hierarchical or decentralized control and for the synthesis of a local converter-level control.
(
J WLS = ∑ W ( jω k ) ⋅ Gcandidate ( s ) s= jω − Guy ( jωk ) k
k
)
2
Recall that the correlation method of system identification yields the system’s complex gain vs. linearlyspaced discrete frequency indices as shown in Figure 3 for a typical converter control-to-output transfer function (notice the crowding of data points at high frequency). In fact, 90% of the data points lie in the upper decade of the frequency range of interest. The weighting function, W(jωk), is chosen to logarithmically de-emphasize the importance of fitting with increasing frequency in order to equally prioritize the fitting across the frequency range of interest. This choice of weighting function can be seen in the lower trace of Figure 3. The resulting fitted transfer function is shown in Figure 4 against the nonparametric source data. Notice that good matching was obtained across all frequencies of interest.
A. Candidate Transfer Function To begin the model fitting process, a candidate transfer function is defined in (1). The numerator order, N, and denominator order, M, must be chosen large enough to capture all salient features of the model under the worst-case set of expected parameter variations. Vectors A and B are the vectors of numerator and denominator coefficients of the candidate transfer function. Gcandidate ( s) =
AN s N + AN −1s N −1 + " + A0 BM s M + BM −1s M −1 + " + B0
(1)
(3)
Figure 3. A typical Bode plot of nonparametric complex frequency response data and the weighting function
Figure 6. Internal Model Control (IMC) structure
Figure 4. A Bode plot of the fitted parametric model overlaid onto the nonparametric data
III.
ADAPTIVE CONTROL
This section proposes an adaptive control structure using correlation-based system identification model fitting techniques described in previous sections and depicted in Figure 1. Conventional time-domain adaptive control structures such as Model Reference Adaptive Control (MRAC) rely on persistent excitation and a gradual convergence to the desired closed-loop system dynamics [18]-[19]. In contrast, the correlation-based system identification actively perturbs the converter and yields a regularly updated model estimate. In this sense, many of the conventional control design methodologies can be used online to synthesize a control at each snapshot in time. If this procedure is repeated as in the flowchart of Figure 2, the controller is updated in response to changes in the converter system.
Perturb
Post Process
Synthesize Control
Perturb
Post Process Detect
Perturb
Post Process
Perturb
Post Process
One possible choice of timing is sketched in Figure 5, where the perturbation is repeated at regular time intervals. Note that one complete identification event consists of a perturbation interval and post processing interval. This identification takes a finite amount of time, TIDENT, determined primarily by the frequency range of interest. The complete response time duration from a system parameter change to successful control adaptation is defined as TTOTAL and is a function of both the polling interval and TIDENT.
Figure 5. Timing overview of the system identification based adaptive control
A. Internal Model Control One option for a generalized control synthesis algorithm is Internal Model Control (IMC), which is commonly used in industrial process control [20]-[22]. The block diagram is shown in Figure 6, where the plant to be controlled is modeled as a generic standalone-stable transfer function, g(s). It is assumed that an estimation of the plant, g~( s) , is available. In this paper, the plant estimation is exactly the fitted parametric model from section II. The concept of the IMC control is that there is an internal model estimate running within the control (in parallel with the actual plant). Because the same control input, u, is applied to both the actual plant and the internal model, both outputs should be equal (after, possibly, an initial transient due to different initial conditions). The feedback signal is the difference between the actual measured output and the internal model’s output such that there is no feedback signal if the estimation is perfect. That is, when g~( s) = g ( s) . With no information in the feedback path, one can guarantee BIBO stability if and only if both c(s) and g(s) are standalone stable [21]. To design the IMC compensator c(s), the plant estimate is first separated into an invertible part, g~( s) − , and noninvertible part, g~( s) + , as in (4). The noninvertible part may include elements such as RHP zeros and time delays. RHP zeroes cannot be inverted because they become unstable RHP poles. Time delays cannot be inverted in causal systems. g~( s) = g~ ( s ) + g~( s) −
(4)
At least two methods of handling RHP zeros are commonly presented in the literature, depending on the objective function the user seeks to minimize [20]-[21]. Factoring the noninvertible part according to (5) is Integralof-Absolute-Error-optimal for step setpoint and output disturbances and is equivalent to leaving the RHP zeros unaffected. g~ ( s) + = e −τs ∏ (− τ n s + 1) , Re(τ n ) > 0 n
(5)
Factoring according to (6) is Integral-of-Squared-Erroroptimal (ISE-optimal) for the same step setpoint and output disturbances and is equivalent to adding additional LHP zeros (mirrored about the imaginary axis) to the invertible
part of the plant [20]-[21]. The ISE-optimal method of (6) is used exclusively in this work.
(− τ n s + 1) , Re(τ ) > 0 g~ ( s ) + = e −τs ∏ n (τ n s + 1) n
(6)
The IMC compensator, c(s), serves two purposes: to invert the invertible portions of g~( s) , and to insert the desired closed-loop system dynamics. The dynamics, f(s), are typically chosen as a collection of n repeated real poles at λ [rad/s] to make c(s) strictly proper. The design equation for the IMC compensator is (7). 1 c( s) = ~ f ( s) , with g ( s) −
f ( s) =
1
(λs + 1)n
(7)
With algebraic block diagram manipulation, the IMC structure of Figure 6 can be transformed into the equivalent conventional negative feedback form shown in Figure 7. In this form, the equivalent conventional compensator, gc(s) is found to be (8). c( s ) c( s) g c (s ) = = ~ 1 − c( s) g (s ) 1 − f ( s) g~ ( s) +
(8)
Since most of the basic converter topologies feature a second order control-to-output transfer function, the result is generally a PID-like compensator. Even if the conventional feedback control form is implemented, the IMC design methodology is appealing for two reasons. First, there is only one design parameter to adjust (λ). Secondly, when perfect estimation is available, stability analysis is reduced to inspection of the open-loop IMC compensator c(s). This method assumes that a perfect estimation is available from the most recent system identification. However as shown in Figure 5, there exists a time, TTOTAL, between a potential system parameter change and a new identification and control synthesis cycle. In this interim time, stability is determined by both g(s) and the aggressiveness of c(s). For this reason, the choice of the IMC filter parameter λ is a trade-off between closed-loop performance and robustness to parameter variation during the time interval TTOTAL. Additionally, care should be taken to avoid direct application of the IMC design to systems with lightly-damped poles and zeros [18] and open-loop unstable systems [23]. These references propose modifications to the standard IMC structure to deal with these two problem cases.
Figure 8. Simulink implementation of the 4th order IMC compensator in conventional feedback form
B. Compensator Implementation The compensator designed in (8) requires implementation within the digital control platform. There are many common choices of control structure [24], but any structure compatible with the proposed adaptive control structure of Figure 1 must satisfy two additional constraints: •
Adjustment of any transfer function coefficient gain must not cause a disturbance in the compensator output, and
•
The compensator must be smoothly enabled and disabled without wind-up when disabled
To meet these constraints, all transfer function coefficient gains must appear before any integrators, and all integrator states must be held when the compensator is disabled. A generic 4th order compensator which satisfies these constraints is presented in Figure 8. The transfer function’s numerator and denominator coefficients are shown in yellow, integrators are shown in gray, and the summing junction is shown in blue. An enable flag (green) switches the input to all integrators to zero and performs a hold operation on the compensator output. IV.
VERIFICATION
A. Simulation Testbed To verify the proposed system-identification based adaptive control, a simulation testbed was constructed in Matlab/Simulink. A basic schematic of the converter system is displayed in Figure 9. Each test case involves a variation of one of the converter parameters in the red dashed circles: output bus capacitance, output capacitor ESR, downstream constant power load, and undamped LC input filter. Table I lists all nominal and off-nominal component values. Table II lists all cases investigated. The actual Simulink implementation of the system using the SimPowerSystem toolbox is shown in Figure 10. A full switching model of the converter is used, and the switching frequency is 10 kHz for all test cases.
Figure 7. Conventional feedback control form
Figure 9. Simulation testbed schematic with variable parameters circled.
TABLE I.
NOMINAL AND OFF-NOMINAL COMPONENT VALUES
Component Name
Nominal Value
Off-nominal Value
Vg
60 [V]
60 [V]
Lout
1.0 [mH]
1.0 [mH]
Cout
90 [µF]
300 [µF]
RC_ESR
0.1 [Ω]
1.0 [mΩ]
PCPL
0 [W]
175 [W]
Lfilt
Not connected
8.0 [mH]
Cfilt
Not connected
80 [µF]
Rload
3.0 [Ω]
3.0 [Ω]
TABLE II.
fitting algorithm is able to successfully fit the low-order plant to a higher-order model, pushing the excess poles and zeros to very high frequencies. The IMC control synthesized after both identifications closely matches the control synthesized from the analytic model. The proposed implementation of a 4th order generic compensator block is able to handle the case of a lower order plant by setting the unused coefficients equal to zero. Finally, since this case does not involve a parameter change, the step response after the control adaptation is identical to the step response before the control adaptation as shown in Figure 14. In summary, this case shows the repeatability and reliability of the proposed identification and control synthesis operations.
TEST CASES INVESTIGATED
Case Number
Parameter Change at t = 0.245 seconds
0
No parameter changes
1
Step increase of Cout
2
Step decrease of RC_ESR
3
Step increase of PCPL
4
Switch Lfilt and Cfilt into the circuit
B. Internal Model Control Results 1) Test Case 0: Nominal Parameters This first “test case” does not involve any converter parameter changes, but merely serves to show proper nominal operation of the following events: converter startup, three output-voltage-reference steps, two successful identification and control synthesis cycles, and a converter shutdown. An overview of the time-domain results for this nominal case is shown in Figure 11. Note that for all cases, the IMC filter pole location, ωf = 1/ λ, was set to 2*π*2kHz. This aggressive specification is equivalent to placing the desired closed-loop bandwidth at one fifth of the 10kHz switching frequency.
Figure 11. Time-domain overview of the entire Case 0 simulation with adaptive IMC control
There are a few important points to note about these results. First, the response to a step change of the output voltage reference (Figure 13) is well-behaved with the nominal plant parameters and using the nominal controller. Second, the identified nominal plant of Figure 12 matches the analytic expectation of the plant. Thirdly, the parametric Figure 12. Estimated frequency response of the nominal control-to-output transfer function and fitted parametric model
Figure 10. Simulink implementation of the testbed converter system described in Figure 9.
Figure 13. Output-voltage-reference step response (30V to 32V to 30V) for Case 0 before control adaptation
Figure 14. Output-voltage-reference step response (30V to 32V to 30V) for Case 0 after control adaptation
Figure 15. Output-voltage-reference step response for Case 1 before control adaptation
Figure 16. Estimated frequency response of the Case 1 off-nominal controlto-output transfer function and fitted parametric model
2) Test Case 1: Step Change of Output Capacitance All of the following test cases use the same simulation testbed, event timing, model fitting settings, and IMC filter parameter setting as test case 0. To evaluate the improvement of the control adaptation, two output-voltagereference steps are performed: one beginning at t = 0.26 s (off-nominal parameter, outdated control) and a second one beginning at t=0.475 s (off-nominal parameter, after second identification and control adaptation). Case 1 introduces a step increase in output capacitance at time t = 0.245s. The resulting time-domain overviews for all test cases are similar to Figure 11 and are omitted here due to space constraints. The response to a step change of the output-voltagereference before control adaptation is displayed in Figure 15, where a significantly less-damped step response is observed. The identification and fitting of the off-nominal control-tooutput frequency response is shown in Figure 16 and features a lower output filter corner frequency and increased Q when compared with the nominal case of Figure 12. After a new control is synthesized and re-engaged, the improved step response of Figure 17 results.
Figure 17. Output-voltage-reference step response for Case 1 after control adaptation
3) Test Case 2: Step Change of Output Capacitor ESR It is well known that the electrolytic capacitors commonly used in the output filter of switching converters
feature a large equivalent series resistance (ESR). This resistance introduces an additional LHP high-frequency zero in the control-to-output transfer function. Generally, the phase lead associated with this ESR zero helps to stabilize the closed-loop system, but a capacitor’s ESR is notoriously variable with temperature, age, and other factors. Test case 2 introduces a step decrease in the output capacitor’s equivalent series resistance (ESR) at time t = 0.245s. As a result of the step change in capacitor ESR, the control-to-output transfer function is modified from Figure 18 to Figure 12. Notice the decrease of the high-frequency phase asymptote from -90° in Figure 18 to nearly -180° in Figure 12. This unexpected removal of the stabilizing highfrequency phase lead results in increased overshoot and highfrequency ringing of the off-nominal output-voltagereference step response of Figure 19, where a drastically decreased damping ratio is observed. After the second identification and control adaptation, the improved outputvoltage-reference step response is shown in Figure 20, where it can be seen that the adapted closed-loop system dynamics return to the nominal (desired) behavior. Note that the highfrequency oscillation of the output voltage in Figures 19 and 20 occurs at 10 kHz and it is switching ripple due mostly to the output capacitor ESR; therefore, it cannot be eliminated through control.
Figure 19. Zoom of output-voltage-reference step response for Case 2 before control adaptation
Figure 20. Zoom of output-voltage-reference step response for Case 2 after control adaptation
4) Test Case 3: Step Change of Constant Power Load
Figure 18. Estimated frequency response of the Case 2 nominal control-tooutput transfer function and fitted parametric model
Test case 3 involves the connection of a constant power load to the output bus at time t = 0.245s through t = 0.55s. This load is representative of a downstream converter under perfect feedback control. The constant power load presents a negative incremental resistance to the converter’s load which combines with the nominal load resistance to ultimately reduce the damping of the buck converter’s LCR output filter. The net effect is an increased Q in the control-tooutput transfer function, which is clearly seen in the frequency response of Figure 21, and the output-voltagereference step response of Figure 22. The improved step response obtained after the control adaptation is shown in Figure 23.
transfer function by two and it also introduces two complexconjugate RHP zeroes which cannot be completely cancelled in the IMC control. The phase shift associated with the complex-conjugate RHP zeros is clearly seen in Figure 24 and makes the closed-loop system (using the nominal compensator) unstable. This unstable closed-loop behavior is seen in the step response of Figure 25 and fixed by the adaptive controller in Figure 26. Notice that since the complex-conjugate RHP pair of zeros in the plant cannot be completely cancelled the step response exhibits lower closed-loop performance than previous cases. Also, the step response exhibits an initial oscillation as typical of systems with even numbers of RHP zeros [25]. Figure 21. Estimated frequency response of the Case 3 off-nominal controlto-output transfer function and fitted parametric model
Figure 24. Estimated frequency response of the Case 4 off-nominal controlto-output transfer function and fitted parametric model
Figure 22. Zoom of output-voltage-reference step response for Case 3 before control adaptation
Figure 25. Output-voltage-reference step response for Case 4 before control adaptation
Figure 23. Output-voltage-reference step response for Case 3 after control adaptation
5) Test Case 4: Introduction of an LC Input Filter Test case 4 represents one of the most severe events that may occur in a converter system: the sudden introduction of a lightly damped LC input filter at time t=0.245s. It is an extremely challenging case for two reasons: the additional input filter increases the order of the control-to-output
[8]
[9]
[10]
[11] Figure 26. Zoom of output-voltage-reference step response for Case 4 after control adaptation [12]
V.
CONCLUSION
This work described a new adaptive control structure based on the online cross-correlation method of system identification techniques. Compared to traditional adaptive control structures, this new method does not rely on a persistent excitation and gradual convergence to a desired closed-loop behavior. Instead, a control algorithm is synthesized based on a regularly-updated and fitted parametric model. The concept of Internal Model Control was applied to synthesize a control. Finally, a simulation testbed using a full switching model for the converter was constructed in Simulink and proper adaptive control behavior was verified for several unique and challenging test cases.
[13]
[14]
[15]
[16]
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[2]
[3]
[4]
[5]
[6] [7]
Maksimovic, D.; Zane, R.; Erickson, R., "Impact of digital control in power electronics," Power Semiconductor Devices and ICs, 2004. Proceedings. ISPSD '04. The 16th International Symposium on, vol., no., pp. 13-22, 24-27 May 2004 R. D. Middlebrook, “Input filter considerations in design and application of switching regulators,” in Proc. IEEE Industry Applications Soc. Annu. Meeting, 1976, pp. 366-382. Byungcho Choi; Dongsoo Kim; Donggyu Lee; Seungwon Choi; Jian Sun, "Analysis of Input Filter Interactions in Switching Power Converters," Power Electronics, IEEE Transactions on , vol.22, no.2, pp.452-460, March 2007 Davoudi, A.; Jatskevich, J.; De Rybel, T., "Numerical state-space average-value modeling of PWM DC-DC converters operating in DCM and CCM," Power Electronics, IEEE Transactions on , vol.21, no.4, pp. 1003-1012, July 2006 Alonge, F.; D'Ippolito, F.; Raimondi, F. M.; Tumminaro, S., "Nonlinear Modeling of DC/DC Converters Using the Hammerstein's Approach," Power Electronics, IEEE Transactions on , vol.22, no.4, pp.1210-1221, July 2007 L. Ljung, System Identification: Theory for the User, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1999. B. Johansson, M. Lenells, "Possibilities of obtaining small-signal models of DC-to-DC power converters by means of system
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I.
INTRODUCTION
Electrical AC and DC power distribution systems are becoming increasingly complex due to a widening variety of power sources, loads, energy storage systems, reconfiguration options, and the incorporation of controlled power electronic converters. Ensuring robustness of these distribution systems is imperative to achieve high quality of service despite the time-varying nature of many sources and loads. Historically, the distribution-level control system has been highly centralized. Current research indicates an emerging trend towards hierarchical and decentralized control where more of the control decisions are made at lower levels in the hierarchy. This work introduces a new strategy for adaptive control that enables each converter in a system to adjust its own local control to accommodate changes in its environment.
This research was sponsored by the Electric Ship Research and Development Consortium (ESRDC) under grant N00014-08-1-0080 entitled “Power Electronics, Architecture, Controls, and Design Tools for Electric Ship Systems”
Figure 1. Generalized schematic of a switching converter, its embedded digital controller, and the proposed adaptive control structure
Recent advancements in digital control theory and the availability of low-cost embedded control platforms have enabled new functionality in power converters: higher control performance, and additional flexibility in control design [1]. However, tightly regulated multi-converter systems are subject to potentially destabilizing interactions according to the so-called Middlebrook impedance criterion [2]-[3]. Also, large parameter variations can negatively affect converter stability and performance. Much literature has been published in the area of converter modeling [3]-[5] and system identification [6]-[8] to address this problem. Previous work has demonstrated that a converter and its digital control platform, shown in Figure 1, can be used to perform online identification and system monitoring. In particular, important small-signal converter-level transfer functions [9]-[10] and impedances [11]-[13] can be monitored in real time. Several researchers have proposed adaptive control of switching converters [14]-[16], but they tend to target single problems around the crossover frequency by relying on existing or purposefully introduced
B. Weighted Least Squares Fitting The fitting problem is then to find coefficient vectors A and B such that the modeling error defined in (2) is minimized. As this error approaches zero for all frequencies of interest, Gcandidate(jω) is a good approximation of the real plant Guy(jω). ε fit ( jω k ) = Gcandidate ( s ) − Guy ( s ) s= jω
(2)
k
The method of least squares fitting is used to define a cost function as a metric to evaluate the quality of the fitting. This cost function, JWLS, is defined in (3) as the weighted sum of the square of the modeling error defined in (2) over the frequency indices of interest. Minimization of this cost function by adjusting coefficient vectors A and B can be accomplished by iterative numerical methods such as a Gauss-Newton algorithm. Figure 2. Flowchart for adaptive control design using digital network analyzer techniques
limit-cycling at that single frequency. The method proposed here and shown in Figure 2 uses the information-rich frequency response estimations obtained from correlation analysis to target a wide range of potential problems with a single adaptive controller structure. A model fitting procedure is described in section II, and one option for adaptive control synthesis is proposed in section III. A simulation testbed is presented in section IV along with results for a set of practical converter problems. II.
MODEL FITTING
This section describes a procedure to fit the nonparametric complex frequency response data obtained from the cross-correlation techniques of [9]-[10] to a parametric model of user-specified order. This fitted model is a more compact representation of the estimated model and is suitable for communication to a hierarchical or decentralized control and for the synthesis of a local converter-level control.
(
J WLS = ∑ W ( jω k ) ⋅ Gcandidate ( s ) s= jω − Guy ( jωk ) k
k
)
2
Recall that the correlation method of system identification yields the system’s complex gain vs. linearlyspaced discrete frequency indices as shown in Figure 3 for a typical converter control-to-output transfer function (notice the crowding of data points at high frequency). In fact, 90% of the data points lie in the upper decade of the frequency range of interest. The weighting function, W(jωk), is chosen to logarithmically de-emphasize the importance of fitting with increasing frequency in order to equally prioritize the fitting across the frequency range of interest. This choice of weighting function can be seen in the lower trace of Figure 3. The resulting fitted transfer function is shown in Figure 4 against the nonparametric source data. Notice that good matching was obtained across all frequencies of interest.
A. Candidate Transfer Function To begin the model fitting process, a candidate transfer function is defined in (1). The numerator order, N, and denominator order, M, must be chosen large enough to capture all salient features of the model under the worst-case set of expected parameter variations. Vectors A and B are the vectors of numerator and denominator coefficients of the candidate transfer function. Gcandidate ( s) =
AN s N + AN −1s N −1 + " + A0 BM s M + BM −1s M −1 + " + B0
(1)
(3)
Figure 3. A typical Bode plot of nonparametric complex frequency response data and the weighting function
Figure 6. Internal Model Control (IMC) structure
Figure 4. A Bode plot of the fitted parametric model overlaid onto the nonparametric data
III.
ADAPTIVE CONTROL
This section proposes an adaptive control structure using correlation-based system identification model fitting techniques described in previous sections and depicted in Figure 1. Conventional time-domain adaptive control structures such as Model Reference Adaptive Control (MRAC) rely on persistent excitation and a gradual convergence to the desired closed-loop system dynamics [18]-[19]. In contrast, the correlation-based system identification actively perturbs the converter and yields a regularly updated model estimate. In this sense, many of the conventional control design methodologies can be used online to synthesize a control at each snapshot in time. If this procedure is repeated as in the flowchart of Figure 2, the controller is updated in response to changes in the converter system.
Perturb
Post Process
Synthesize Control
Perturb
Post Process Detect
Perturb
Post Process
Perturb
Post Process
One possible choice of timing is sketched in Figure 5, where the perturbation is repeated at regular time intervals. Note that one complete identification event consists of a perturbation interval and post processing interval. This identification takes a finite amount of time, TIDENT, determined primarily by the frequency range of interest. The complete response time duration from a system parameter change to successful control adaptation is defined as TTOTAL and is a function of both the polling interval and TIDENT.
Figure 5. Timing overview of the system identification based adaptive control
A. Internal Model Control One option for a generalized control synthesis algorithm is Internal Model Control (IMC), which is commonly used in industrial process control [20]-[22]. The block diagram is shown in Figure 6, where the plant to be controlled is modeled as a generic standalone-stable transfer function, g(s). It is assumed that an estimation of the plant, g~( s) , is available. In this paper, the plant estimation is exactly the fitted parametric model from section II. The concept of the IMC control is that there is an internal model estimate running within the control (in parallel with the actual plant). Because the same control input, u, is applied to both the actual plant and the internal model, both outputs should be equal (after, possibly, an initial transient due to different initial conditions). The feedback signal is the difference between the actual measured output and the internal model’s output such that there is no feedback signal if the estimation is perfect. That is, when g~( s) = g ( s) . With no information in the feedback path, one can guarantee BIBO stability if and only if both c(s) and g(s) are standalone stable [21]. To design the IMC compensator c(s), the plant estimate is first separated into an invertible part, g~( s) − , and noninvertible part, g~( s) + , as in (4). The noninvertible part may include elements such as RHP zeros and time delays. RHP zeroes cannot be inverted because they become unstable RHP poles. Time delays cannot be inverted in causal systems. g~( s) = g~ ( s ) + g~( s) −
(4)
At least two methods of handling RHP zeros are commonly presented in the literature, depending on the objective function the user seeks to minimize [20]-[21]. Factoring the noninvertible part according to (5) is Integralof-Absolute-Error-optimal for step setpoint and output disturbances and is equivalent to leaving the RHP zeros unaffected. g~ ( s) + = e −τs ∏ (− τ n s + 1) , Re(τ n ) > 0 n
(5)
Factoring according to (6) is Integral-of-Squared-Erroroptimal (ISE-optimal) for the same step setpoint and output disturbances and is equivalent to adding additional LHP zeros (mirrored about the imaginary axis) to the invertible
part of the plant [20]-[21]. The ISE-optimal method of (6) is used exclusively in this work.
(− τ n s + 1) , Re(τ ) > 0 g~ ( s ) + = e −τs ∏ n (τ n s + 1) n
(6)
The IMC compensator, c(s), serves two purposes: to invert the invertible portions of g~( s) , and to insert the desired closed-loop system dynamics. The dynamics, f(s), are typically chosen as a collection of n repeated real poles at λ [rad/s] to make c(s) strictly proper. The design equation for the IMC compensator is (7). 1 c( s) = ~ f ( s) , with g ( s) −
f ( s) =
1
(λs + 1)n
(7)
With algebraic block diagram manipulation, the IMC structure of Figure 6 can be transformed into the equivalent conventional negative feedback form shown in Figure 7. In this form, the equivalent conventional compensator, gc(s) is found to be (8). c( s ) c( s) g c (s ) = = ~ 1 − c( s) g (s ) 1 − f ( s) g~ ( s) +
(8)
Since most of the basic converter topologies feature a second order control-to-output transfer function, the result is generally a PID-like compensator. Even if the conventional feedback control form is implemented, the IMC design methodology is appealing for two reasons. First, there is only one design parameter to adjust (λ). Secondly, when perfect estimation is available, stability analysis is reduced to inspection of the open-loop IMC compensator c(s). This method assumes that a perfect estimation is available from the most recent system identification. However as shown in Figure 5, there exists a time, TTOTAL, between a potential system parameter change and a new identification and control synthesis cycle. In this interim time, stability is determined by both g(s) and the aggressiveness of c(s). For this reason, the choice of the IMC filter parameter λ is a trade-off between closed-loop performance and robustness to parameter variation during the time interval TTOTAL. Additionally, care should be taken to avoid direct application of the IMC design to systems with lightly-damped poles and zeros [18] and open-loop unstable systems [23]. These references propose modifications to the standard IMC structure to deal with these two problem cases.
Figure 8. Simulink implementation of the 4th order IMC compensator in conventional feedback form
B. Compensator Implementation The compensator designed in (8) requires implementation within the digital control platform. There are many common choices of control structure [24], but any structure compatible with the proposed adaptive control structure of Figure 1 must satisfy two additional constraints: •
Adjustment of any transfer function coefficient gain must not cause a disturbance in the compensator output, and
•
The compensator must be smoothly enabled and disabled without wind-up when disabled
To meet these constraints, all transfer function coefficient gains must appear before any integrators, and all integrator states must be held when the compensator is disabled. A generic 4th order compensator which satisfies these constraints is presented in Figure 8. The transfer function’s numerator and denominator coefficients are shown in yellow, integrators are shown in gray, and the summing junction is shown in blue. An enable flag (green) switches the input to all integrators to zero and performs a hold operation on the compensator output. IV.
VERIFICATION
A. Simulation Testbed To verify the proposed system-identification based adaptive control, a simulation testbed was constructed in Matlab/Simulink. A basic schematic of the converter system is displayed in Figure 9. Each test case involves a variation of one of the converter parameters in the red dashed circles: output bus capacitance, output capacitor ESR, downstream constant power load, and undamped LC input filter. Table I lists all nominal and off-nominal component values. Table II lists all cases investigated. The actual Simulink implementation of the system using the SimPowerSystem toolbox is shown in Figure 10. A full switching model of the converter is used, and the switching frequency is 10 kHz for all test cases.
Figure 7. Conventional feedback control form
Figure 9. Simulation testbed schematic with variable parameters circled.
TABLE I.
NOMINAL AND OFF-NOMINAL COMPONENT VALUES
Component Name
Nominal Value
Off-nominal Value
Vg
60 [V]
60 [V]
Lout
1.0 [mH]
1.0 [mH]
Cout
90 [µF]
300 [µF]
RC_ESR
0.1 [Ω]
1.0 [mΩ]
PCPL
0 [W]
175 [W]
Lfilt
Not connected
8.0 [mH]
Cfilt
Not connected
80 [µF]
Rload
3.0 [Ω]
3.0 [Ω]
TABLE II.
fitting algorithm is able to successfully fit the low-order plant to a higher-order model, pushing the excess poles and zeros to very high frequencies. The IMC control synthesized after both identifications closely matches the control synthesized from the analytic model. The proposed implementation of a 4th order generic compensator block is able to handle the case of a lower order plant by setting the unused coefficients equal to zero. Finally, since this case does not involve a parameter change, the step response after the control adaptation is identical to the step response before the control adaptation as shown in Figure 14. In summary, this case shows the repeatability and reliability of the proposed identification and control synthesis operations.
TEST CASES INVESTIGATED
Case Number
Parameter Change at t = 0.245 seconds
0
No parameter changes
1
Step increase of Cout
2
Step decrease of RC_ESR
3
Step increase of PCPL
4
Switch Lfilt and Cfilt into the circuit
B. Internal Model Control Results 1) Test Case 0: Nominal Parameters This first “test case” does not involve any converter parameter changes, but merely serves to show proper nominal operation of the following events: converter startup, three output-voltage-reference steps, two successful identification and control synthesis cycles, and a converter shutdown. An overview of the time-domain results for this nominal case is shown in Figure 11. Note that for all cases, the IMC filter pole location, ωf = 1/ λ, was set to 2*π*2kHz. This aggressive specification is equivalent to placing the desired closed-loop bandwidth at one fifth of the 10kHz switching frequency.
Figure 11. Time-domain overview of the entire Case 0 simulation with adaptive IMC control
There are a few important points to note about these results. First, the response to a step change of the output voltage reference (Figure 13) is well-behaved with the nominal plant parameters and using the nominal controller. Second, the identified nominal plant of Figure 12 matches the analytic expectation of the plant. Thirdly, the parametric Figure 12. Estimated frequency response of the nominal control-to-output transfer function and fitted parametric model
Figure 10. Simulink implementation of the testbed converter system described in Figure 9.
Figure 13. Output-voltage-reference step response (30V to 32V to 30V) for Case 0 before control adaptation
Figure 14. Output-voltage-reference step response (30V to 32V to 30V) for Case 0 after control adaptation
Figure 15. Output-voltage-reference step response for Case 1 before control adaptation
Figure 16. Estimated frequency response of the Case 1 off-nominal controlto-output transfer function and fitted parametric model
2) Test Case 1: Step Change of Output Capacitance All of the following test cases use the same simulation testbed, event timing, model fitting settings, and IMC filter parameter setting as test case 0. To evaluate the improvement of the control adaptation, two output-voltagereference steps are performed: one beginning at t = 0.26 s (off-nominal parameter, outdated control) and a second one beginning at t=0.475 s (off-nominal parameter, after second identification and control adaptation). Case 1 introduces a step increase in output capacitance at time t = 0.245s. The resulting time-domain overviews for all test cases are similar to Figure 11 and are omitted here due to space constraints. The response to a step change of the output-voltagereference before control adaptation is displayed in Figure 15, where a significantly less-damped step response is observed. The identification and fitting of the off-nominal control-tooutput frequency response is shown in Figure 16 and features a lower output filter corner frequency and increased Q when compared with the nominal case of Figure 12. After a new control is synthesized and re-engaged, the improved step response of Figure 17 results.
Figure 17. Output-voltage-reference step response for Case 1 after control adaptation
3) Test Case 2: Step Change of Output Capacitor ESR It is well known that the electrolytic capacitors commonly used in the output filter of switching converters
feature a large equivalent series resistance (ESR). This resistance introduces an additional LHP high-frequency zero in the control-to-output transfer function. Generally, the phase lead associated with this ESR zero helps to stabilize the closed-loop system, but a capacitor’s ESR is notoriously variable with temperature, age, and other factors. Test case 2 introduces a step decrease in the output capacitor’s equivalent series resistance (ESR) at time t = 0.245s. As a result of the step change in capacitor ESR, the control-to-output transfer function is modified from Figure 18 to Figure 12. Notice the decrease of the high-frequency phase asymptote from -90° in Figure 18 to nearly -180° in Figure 12. This unexpected removal of the stabilizing highfrequency phase lead results in increased overshoot and highfrequency ringing of the off-nominal output-voltagereference step response of Figure 19, where a drastically decreased damping ratio is observed. After the second identification and control adaptation, the improved outputvoltage-reference step response is shown in Figure 20, where it can be seen that the adapted closed-loop system dynamics return to the nominal (desired) behavior. Note that the highfrequency oscillation of the output voltage in Figures 19 and 20 occurs at 10 kHz and it is switching ripple due mostly to the output capacitor ESR; therefore, it cannot be eliminated through control.
Figure 19. Zoom of output-voltage-reference step response for Case 2 before control adaptation
Figure 20. Zoom of output-voltage-reference step response for Case 2 after control adaptation
4) Test Case 3: Step Change of Constant Power Load
Figure 18. Estimated frequency response of the Case 2 nominal control-tooutput transfer function and fitted parametric model
Test case 3 involves the connection of a constant power load to the output bus at time t = 0.245s through t = 0.55s. This load is representative of a downstream converter under perfect feedback control. The constant power load presents a negative incremental resistance to the converter’s load which combines with the nominal load resistance to ultimately reduce the damping of the buck converter’s LCR output filter. The net effect is an increased Q in the control-tooutput transfer function, which is clearly seen in the frequency response of Figure 21, and the output-voltagereference step response of Figure 22. The improved step response obtained after the control adaptation is shown in Figure 23.
transfer function by two and it also introduces two complexconjugate RHP zeroes which cannot be completely cancelled in the IMC control. The phase shift associated with the complex-conjugate RHP zeros is clearly seen in Figure 24 and makes the closed-loop system (using the nominal compensator) unstable. This unstable closed-loop behavior is seen in the step response of Figure 25 and fixed by the adaptive controller in Figure 26. Notice that since the complex-conjugate RHP pair of zeros in the plant cannot be completely cancelled the step response exhibits lower closed-loop performance than previous cases. Also, the step response exhibits an initial oscillation as typical of systems with even numbers of RHP zeros [25]. Figure 21. Estimated frequency response of the Case 3 off-nominal controlto-output transfer function and fitted parametric model
Figure 24. Estimated frequency response of the Case 4 off-nominal controlto-output transfer function and fitted parametric model
Figure 22. Zoom of output-voltage-reference step response for Case 3 before control adaptation
Figure 25. Output-voltage-reference step response for Case 4 before control adaptation
Figure 23. Output-voltage-reference step response for Case 3 after control adaptation
5) Test Case 4: Introduction of an LC Input Filter Test case 4 represents one of the most severe events that may occur in a converter system: the sudden introduction of a lightly damped LC input filter at time t=0.245s. It is an extremely challenging case for two reasons: the additional input filter increases the order of the control-to-output
[8]
[9]
[10]
[11] Figure 26. Zoom of output-voltage-reference step response for Case 4 after control adaptation [12]
V.
CONCLUSION
This work described a new adaptive control structure based on the online cross-correlation method of system identification techniques. Compared to traditional adaptive control structures, this new method does not rely on a persistent excitation and gradual convergence to a desired closed-loop behavior. Instead, a control algorithm is synthesized based on a regularly-updated and fitted parametric model. The concept of Internal Model Control was applied to synthesize a control. Finally, a simulation testbed using a full switching model for the converter was constructed in Simulink and proper adaptive control behavior was verified for several unique and challenging test cases.
[13]
[14]
[15]
[16]
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