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Adaptive-Gain Second Order Sliding Mode Observer Design for Switching Power Converters
Jianxing Liu, Salah Laghrouche, M. Harmouche, Maxime Wack
TL;DR
Multicell converters need capacitor-voltage information, yet voltage sensing adds cost and complexity and standard rank-based observability can be insufficient. The paper uses Z(T_N)-observability and proposes an adaptive-gain SOSML observer using load current and switching inputs. Under suitable switching sequences, capacitor voltages become observable, and simulations show greater robustness than a Luenberger switched observer under noise and load variation.
Problem
Capacitor-voltage sensors increase converter cost and complexity, while the system is only partially observable under the standard observability-matrix rank condition.
Method
The paper analyzes Z(T_N)-observability under switching sequences and designs an adaptive-gain SOSML observer using load current and known switching inputs.
Results
Under suitable switching sequences, capacitor voltages become observable, and simulations show the proposed observer is more robust than a Luenberger switched observer under output noise and load variations up to 50%.
Takeaways & Limitations
Capacitor voltages can be estimated without direct voltage sensors when the switching-input conditions required for hybrid observability hold.
Takeaways & Limitations
Some switching modes make capacitor voltages completely unobservable, so observability depends on the switching sequence.
Abstract
from arXiv · showhide
In this paper, a novel adaptive-gain Second Order Sliding Mode (SOSM) observer is proposed for multicell converters by considering it as a class of hybrid systems. The aim is to reduce the number of voltage sensors by estimating the capacitor voltages only from the measurement of load current. The proposed observer is proven to be robust in the presence of perturbations with \emph{unknown} boundary. However, the states of the system are only partially observable in the sense of observability rank condition. Due to its switching behavior, a recent concept of $Z(T_N)$ observability is used to analysis its hybrid observability, since its observability depends upon the switching control signals. Under certain condition of the switching sequences, the voltage across each capacitor becomes observable. Simulation results and comparisons with Luenberger switched observer highlight the effectiveness and robustness of the proposed observer with respect to output measurement noise and system uncertainties (load variations).
1. INTRODUCTION
Multicell converters require capacitor-voltage information for many control methods, but adding voltage sensors increases system cost and complexity. The paper therefore develops an adaptive-gain SOSM observer and analyzes observability under switching.
- Many multicell-converter control methods require capacitor-voltage measurements, increasing sensor count, cost, and system complexity.
- Because the system is only partially observable by the standard observability-rank condition, hybrid observability must account for switching control signals.
- The paper analyzes Z(T_N)-observability under switching sequences and designs an adaptive-gain SOSM observer for capacitor-voltage estimation.
- The proposed observer addresses perturbations whose first-derivative boundaries are unknown and combines nonlinear and linear SOSM terms through adaptive gain scaling.
- Simulations compare the proposed observer with a Luenberger switched observer under disturbances.
2. MODELING OF the MULTI-CELL CONVERTER
The paper models a multicell converter as a hybrid system with continuous electrical states and discrete switching variables. Using load-current measurements and known switching inputs, it formulates an observer-oriented switched-affine representation for estimating capacitor voltages.
- A multicell converter combines cells containing energy-storage elements and commutators, producing hybrid behavior from continuous states and discrete switching logic.
- The converter is connected to an inductive load, with current flowing from the source through the converter switches to the output.
- Each commutation cell uses a binary signal S_j, where S_j=1 and S_j=0 select complementary upper- and lower-switch states.
- Assuming only load current is measured, the system is represented as a switched-affine model x_dot=f(x,u)=A(u)x+B(u), y=h(x,u)=Cx.
- The observer estimates capacitor voltages from load current and the associated switching control input, which is assumed known.
3. HYBRID OBSERVABILITY ANALYSIS
The converter is not fully observable under the standard rank condition, so hybrid observability must account for switching trajectories and input sequences. Z(T_N)-observability provides conditions under which capacitor-voltage states become observable across switching intervals.
- Switching-mode analysis: Some switching modes make capacitor voltages unobservable, but these modes correspond to cells that are not switching and do not occur for all control sequences.For a three-cell converter, the analysis examines all eight switching configurations and applies projection conditions across selected intervals.
- Standard observability: The observability matrix has rank 2, which is less than p, so continuous states are not observable from load-current measurements alone.This motivates an observability analysis beyond the standard rank condition.
- Hybrid observability: Observability depends on the switching behavior, making Z(T_N)-observability appropriate for the converter's hybrid system.The analysis uses the measured load current and known switching-input sequence.
- Z(T_N)-observability: Z(T_N)-observability evaluates whether a state transformation is uniquely determined along a specified hybrid time trajectory and associated input sequence.Its conditions combine interval-wise observability, a full-rank projection collection, and constancy of unobservable components within intervals.
- Z(T_N)-observability: The hybrid trajectory and ordered switching inputs influence observability similarly to an input, while unobservable components must remain constant during the relevant interval.This condition allows information from multiple switching intervals to establish observability.
4. ADAPTIVE-GAIN SOSML OBSERVER DESIGN
The paper designs an adaptive-gain SOSML observer for multicell converters that estimates capacitor voltages while handling load-variation perturbations with unknown derivative bounds. Its error dynamics are proven to converge in finite time, and the reduced-order error system is exponentially stable under switching-related conditions.
- Observer motivation: The observer estimates capacitor voltages using load-current measurements, reducing reliance on additional voltage sensors whose noise and cost affect converter implementations.The design targets state observation without directly measuring capacitor voltages.
- Adaptive-gain design: The adaptive-gain SOSML design addresses load variations whose first-derivative bounds are unknown and does not require their a-priori knowledge.A time-scaling approach uses a novel adaptive law with one tuning parameter.
- Scope: The observer applies to hybrid switched-affine multicell converter systems represented by the paper’s system class.The stated applicability extends beyond the specific three-cell design example.
- Adaptive-gain design: The observer combines the SOSML algorithm with adaptive gains and defines observation-error dynamics for the current and capacitor-voltage estimates.The observer formulation uses the SOSML injection and tunable gains λ(t), α(t), kλ(t), kα(t), k1, and k2.
- Convergence analysis: Under the stated BIBS and observability assumptions, the error-system trajectories converge to zero in finite time despite perturbations bounded by an unknown positive constant.The theorem specifies adaptive-gain conditions and uses χ1 as an unknown perturbation-bound constant.
- Convergence analysis: After sliding motion, the reduced-order error system has exponentially convergent trajectories when its boundedness and switching-period conditions are satisfied.The switching signals allow the relevant interval parameters to be selected over one switching-sequence period.
5. SIMULATION RESULTS
Simulations evaluate the adaptive-gain SOSML observer against a Luenberger switched observer under nominal and perturbed conditions. The proposed observer remains robust to output noise and load variations, with modest additional computational requirements.
- Simulation setup: The simulation compares the adaptive-gain SOSML observer with a Luenberger switched observer using the model parameters listed in Table 2.The switched observer is formulated through a switched error system with constant gains selected using a positive-matrix condition.
- Nominal conditions: Under noise-free operation without load variation, both observers estimate capacitor voltages Vc1 and Vc2 with desired performance.Figure 2 reports the voltage estimates and their errors for both observers.
- Perturbed conditions: Under output noise and load-resistance variations up to 50%, the adaptive-gain SOSML observer remains robust, while the Luenberger switched observer is more sensitive.The output noise is included specifically to test robustness, and the load resistance is varied by up to 50%.
- Perturbed conditions: The adaptive-gain SOSML observer shows better performance because SOSM acts as a robust exact differentiator.The adaptive law is also reported to remain effective under load variations.
- Implementation: The adaptive-gain SOSML calculations are slightly more intensive than those of the Luenberger observer, but their additional real-time burden is described as low.The correction term and two design parameters support implementation with increased digital-computer capabilities.
6. CONCLUSIONS
The paper concludes that adaptive-gain SOSML can estimate multicell-converter capacitor voltages despite rank-condition limitations, provided switching sequences satisfy the required observability condition. Compared with a Luenberger switched observer, it is more robust to load-resistance variations and output measurement noise.
- Observability: Z(T_N)-observability makes capacitor voltages observable after suitable switching sequences even though the observability matrix never has full rank.The conclusion applies this hybrid observability concept under a certain condition of the input sequences.
- Robustness: The adaptive-gain SOSML observer is more robust than the Luenberger switched observer under load-resistance variations and output measurement noise.This comparison is reported as a principal conclusion of the paper.
- Advantages: The method requires tuning only one parameter k and does not require a-priori knowledge of perturbation bounds.These are identified as the two main advantages of the proposed method.