Sum Of Squares Approach Research Articles

Stability analysis of aperiodic impulsive delay systems using sum of squares approach

AbstractThe stability problem of aperiodic impulsive delay systems with unstable continuous and discrete dynamics is addressed in this article. New delay‐dependent stability conditions are proposed by employing a clock‐dependent Lyapunov–Krasovskii‐like functional composed of a Lyapunov–Krasovskii functional and a high‐order polynomial matrix function. The stability conditions are then solved using sum of squares programming techniques, yielding less conservative results than the looped‐functionals approach. Finally, the simulation results demonstrate the effectiveness of the results.

Observer‐based control for nonlinear Hadamard fractional‐order systems via SOS approach

AbstractPractical stability refers to the notion that the origin is not an equilibrium point (EP) and that the system states tend to converge toward a sphere centered at the origin. The first goal of this paper is to analyze the concept of “practical stability” in Caputo–Hadamard fractional‐order derivative (CHFOD) systems. Then, using the Lyapunov approach, a polynomial fuzzy (PF) observer‐based controller for stabilizing CHFOD PF systems is created. The observer‐based control is innovative since it was created and proven using the sum‐of‐squares (SOS) method. In conclusion, a numerical illustration is provided to corroborate the theoretical findings.

State/fault Estimation and Fault-tolerant Control by SOS approaches under Unmeasurable States

This paper constructs a novel state/fault estimation (SFE) observer and integrated fault-tolerant controller for a series of tunnel diode circuits based on SOS approaches. The unmeasurable states are taken into consideration, expanding the application scope of the method, while a reduced-order observer is proposed to lighten the computational burden. The fault-tolerant control (FTC) unit is co-designed considering the estimation error, and a decoupling technique is employed to reduce computational complexity. For lower conservativeness, stable constraints ensuring the stability of the augmented system under H ∞ performance γ are given in the format of sum-of-squares (SOS). Finally, the validity of the proposed method is verified through a simulation experiment of a tunnel diode circuit.

Design of Experiments for Evaluating the Relevance of Change in Test Method for Kinematic Viscosity of Opaque Oils

Viscosity is a physicochemical property that evaluates the resistance that fuel offers to flow, influencing the engine’s operation and combustion process. Its control is aimed at good fuel atomization and the preservation of lubricating characteristics. Changes in viscosity can lead to wear on various parts of the engine. Viscometers typically measure the viscosity of fuels in the oil and gas industry. These instruments can measure the time it takes for a fluid to move a given distance through a pipe or the time it takes for an object of a given size and density to pass through the liquid. The traditional test method, ASTM D445, differentiates the procedure for opaque liquids from transparent ones; that is, it requires a warm-up of the sample between 60 °C and 65 °C for 1 h. This additional step can overload laboratory routines, although it is not guaranteed to have a metrologically significant effect on the final result. Thus, this study evaluated the relevance of complying with this step in the test method for the kinematic viscosity of opaque liquids using a 32 factorial experimental design. Based on the F test, p-value, confidence intervals, and percentage contribution of the sum of squares approaches concerning the regression analysis, one concluded that the warm-up time was not a relevant factor in the kinematic viscosity, specifically of very low sulphur fuel oil, Brazilian fuel oil, and atmospheric residue diluted with diesel oil, which are fluids at room temperature.

On Solving MAX-SAT Using Sum of Squares

We consider semidefinite programming (SDP) approaches for solving the maximum satisfiability (MAX-SAT) problem and weighted partial MAX-SAT. It is widely known that SDP is well-suited to approximate (MAX-)2-SAT. Our work shows the potential of SDP also for other satisfiability problems by being competitive with some of the best solvers in the yearly MAX-SAT competition. Our solver combines sum of squares (SOS)–based SDP bounds and an efficient parser within a branch-and-bound scheme. On the theoretical side, we propose a family of semidefinite feasibility problems and show that a member of this family provides the rank-two guarantee. We also provide a parametric family of semidefinite relaxations for MAX-SAT and derive several properties of monomial bases used in the SOS approach. We connect two well-known SDP approaches for (MAX)-SAT in an elegant way. Moreover, we relate our SOS-SDP relaxations for partial MAX-SAT to the known SAT relaxations. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms – Discrete. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2023.0036 .

Effect of Corporate governance on Financial Distress among the Firms Listed at Nairobi Securities Exchange

This study delves into the intricate effect of corporate governance on financial distress in Kenya among the firms listed at the Nairobi Securities Exchange. The recent economic growth and development in Kenya have attracted increased attention to the effectiveness of corporate governance practices adopted by businesses. The study examines the effect of various corporate governance factors, such as Board Size, Board Independence, Audit Committee Independence, and Firm Size, on financial distress in listed companies on the Nairobi Securities Exchange. To evaluate the significance of each factor in influencing financial distress, the analysis employs a Type I Sum of Squares approach using XLSTAT software. The findings reveal that Board Size and Firm Size exhibit highly significant relationships with financial distress, indicating their crucial roles in influencing a company's financial stability. Furthermore, Board Independence demonstrates a statistically significant relationship with financial distress, underscoring the importance of independent directors in mitigating financial risks. Although Audit Committee Independence exhibits borderline significance, it provides indications of a potential relationship with financial distress. Overall, the study offers valuable insights into the factors affecting financial distress in Kenyan companies and emphasizes the significance of robust corporate governance practices in ensuring financial stability and resilience.

SOS Approach for Practical Stabilization of Tempered Fractional-Order Power System

Fractional systems have been widely utilized in various fields, such as mathematics, physics and finance, providing a versatile framework for precise measurements and calculations involving partial quantities. This paper aims to develop a novel polynomial controller for a power system (PS) with fractional-order (FO) dynamics. It begins by studying the practical stability of a general class of tempered fractional-order (TFO) nonlinear systems, with broad applicability and potential for expanding its applications. Afterward, a polynomial controller is designed to guarantee the practical stability of the PS, encompassing the standard constant controller as a specific instance. The design conditions for this controller are resolved using the sum of squares (SOS) approach, a powerful technique for guaranteeing stability and control design. To showcase the practical value of the analytical findings, simulations of the PS are conducted utilizing SOSTOOLS.

Design of Polynomial Observer-Based Fault-Tolerant Controller for Polynomial Systems with State Delay: A Sum of Squares Approach

Design of Polynomial Observer-Based Fault-Tolerant Controller for Polynomial Systems with State Delay: A Sum of Squares Approach

Lyapunov Function computation for Periodic Linear Hybrid Systems via Handelman, Polya and SoS approaches: A comparative study

We propose a method for the stability analysis of linear hybrid systems with periodic jumps. The method relies on the solution to polynomial inequalities based on the Handelman decomposition. Compared to existing approaches, such as sum-of-squares (SoS) and Polya's theorem, the proposed method reduces the computation time to obtain stability certificates.

Continuous Stability TS Fuzzy Systems Novel Frame Controlled by a Discrete Approach and Based on SOS Methodology

Generally, the continuous and discrete TS fuzzy systems’ control is studied independently. Unlike the discrete systems, stability results for the continuous systems suffer from conservatism because it is still quite difficult to apply non-quadratic Lyapunov functions, something which is much easier for the discrete systems. In this paper and in order to obtain new results for the continuous case, we proposed to connect the continuous with the discrete cases and then check the stability of the continuous TS fuzzy systems by means of the discrete design approach. To this end, a novel frame was proposed using the sum of square approach (SOS) to check the stability of the continuous Takagi Sugeno (TS) fuzzy models based on the discrete controller. Indeed, the control of the continuous TS fuzzy models is ensured by the discrete gains obtained from the Euler discrete form and based on the non-quadratic Lyapunov function. The simulation examples applied for various models, by modifying the order of the Euler discrete fuzzy system, are presented to show the effectiveness of the proposed methodology.

Modeling and Control of Electrical Power and Energy Produced by a Synchronous Generator Using Polynomial Fuzzy Systems and Sum-Of-Squares Approach

The synchronous generator, as the main component of power systems, plays a key role in these system’s stability. Therefore, utilizing the most effective control strategy for modeling and control the synchronous generator results in the best outcomes in power systems’ performances. The advantage of using a powerful controller is to have the synchronous generator modeled and controlled as well as its main task i.e., stabilizing power systems. Since the synchronous generator is known as a complicated nonlinear system, modeling and control of it is a difficult task. This paper presents a sum of squares (SOS) approach to modeling and control the synchronous generator using polynomial fuzzy systems. This method as an efficacious control strategy has numerous superiorities to the well-known T-S fuzzy controller, due to the control framework is a polynomial fuzzy model, which is more general and effectual than the well-known T-S fuzzy model. In this case, a polynomial Lyapunov function is used for analyzing the stability of the polynomial fuzzy system. Then, the number of rules in a polynomial fuzzy model is less than in a T-S fuzzy model. Besides, derived stability conditions are represented in terms of the SOS approach, which can be numerically solved via the recently developed SOSTOOLS. This approach avoids the difficulty of solving LMI (Linear Matrix Inequality). The Effectiveness of the proposed control strategy is verified by using the third-part Matlab toolbox, SOSTOOLS.

Polynomial Fuzzy Observer-Based Integrated Fault Estimation and Fault-Tolerant Control With Uncertainty and Disturbance

This article studies the integrated fault-tolerant control (FTC) and fault estimation (FE) problem using polynomial fuzzy model (PFM) based on the sum of squares (SOS) approach. The integrated approach not only considers the bidirectional interaction between fault diagnosis and control units but also ensures an optimal response. The transient management in the fault occurrence situation and the alleviation of bidirectional interaction between FE and FTC are the motivational challenges. However, the integrated approach increases the complexity and problem's dimension. In this situation, PFM can reduce the problem's dimensions and increase modeling accuracy. The additive actuator faults, input disturbance, and structured uncertainty are considered to bring the problem closer to practical applications. A polynomial fuzzy observer with unknown input is designed to estimate the time-varying faults that is used to remove the effects of faults on the control input. Thus, the proposed approach benefits from PFM in FE and FTC synthesis to reduce the design dimensions, have a more precise model, and make the synthesis less conservative. Moreover, the PFM is more robust against intense model uncertainties and faults. To evaluate the proposed approach, the FTC of an inverted-pendulum system is simulated. The fault-tolerant and fault estimation performances of the proposed approach are compared with those of the linear matrix inequality (LMI) approach. The simulated results show that the proposed SOS approach outperforms the LMI approach.

Interval Observers Design for Polynomial Fuzzy Singular Systems by Utilizing Sum-of-Squares Program

This paper designs interval observers for polynomial fuzzy singular systems, in which the external disturbances and unknown parameters are included in the systems. The polynomial fuzzy modelings are proposed, which are more practical than Takagi–Sugeno fuzzy modelings. Then, the interval observers are designed for nonnegative and generic systems. The stability conditions are proposed by sum-of-squares (SOS) approach via SOSTOOLS. Compared to the linear matrix inequality-based approaches, the SOS approach is more relaxed and general. Estimation accuracy is analyzed via ${L_{1}/L_{2}}$ framework. The efficiency of the interval observers are demonstrated through two simulation examples.

An SOS Observer-Based Sensorless Control for PMLSM Drive System

Sum-of-squares (SOS) approach is an effective technique for analysis and design of nonlinear systems. Due to the advantage, this study presents an SOS observer-based control approach for sensorless position servo of permanent magnet linear synchronous motor. To apply the SOS approach, a nonlinear polynomial model is firstly established to describe motor dynamics. Based on this model, a full-order Luenberger observer is developed to estimate the system state. The observer gain that ensures the convergence of state estimation is deduced via the stability analysis and solved by using the third-party toolbox SOSTOOLS. Compared with earlier studies, the advantage of the proposed observer is that the exact motor dynamics is used. Therefore, no further hypothesis or adaptive mechanism on the mover speed is required, which is usually utilized to solve the problem of speed variation with the state observation. According to the principle of backstepping control, the controller is systematically designed to guarantee the asymptotic stability of the closed-loop system. Since both the mechanical and electrical dynamics have been considered, the control signal is directly presented as a voltage. This is different from the conventional current control strategy, which applies the proportional–integral controller to produce a voltage command. The experiment results are provided to validate the proposed control scheme. Moreover, a recent work that is based on the sensorless linear quadratic regulator optimal control is adopted as a comparison study.

Sum of squares approach for ground vehicle lateral control under tire saturation forces

This work presents the stability analysis and design of a lateral controller for a nonlinear ground vehicle applying the concept of polynomial sum of squares relaxations. The system is approximated by a polynomial vector field that describes the lateral vehicle dynamics. The resulting polynomial system falls on a class of non-affine in input system, which makes the control synthesis more involved. This issue is circumvented by an input-affine approximation, simplifying the stability analysis and the design procedure of a polynomial state-feedback controller able to enlarge the region of attraction (RoA). We also compare the estimated region of attraction with the standard LQR optimal controller.

Enhancing Transient Stability of DC Microgrid by Enlarging the Region of Attraction Through Nonlinear Polynomial Droop Control

A methodology for enlarging the region of attraction (ROA) of a DC microgrid with constant power loads (CPLs) is proposed. The enlargement is achieved through the optimal design of a polynomial droop controller. The design of this nonlinear controller is done by solving a sum of squares (SOS) program. The proposed SOS program allows finding a Lyapunov function that serves to estimate the ROA. Therefore, the coefficients of the polynomial droop controller are optimized to maximize that estimate. Using the SOS approach, the estimate of the ROA exceeds the performance previously attained with alternative methods. It is illustrated how this nonlinear droop control approach is able to enlarge the ROA compared with a linear droop technique. Numerical simulations confirm that the proposed polynomial controller makes the system more robust against large disturbances and so enhances the transient stability.

Relaxed Sum-of-Squares Based Stabilization Conditions for Polynomial Fuzzy-Model-Based Control Systems

This paper presents a relaxed stabilization condition for polynomial-fuzzy-model-based control system by using sum-of-squares (SOS) approach. Full-block S-procedure is utilized to equivalently decouple the fuzzy weighting constraint into two constraints, which can be computed directly with the SOS approach. Besides, a free full-block multiplier imposed by this procedure also contributes to reducing the conservativeness of stability analysis. The copositivity verification relaxation performed on the membership-functions-related constraint can both obtain more relaxed results and reduce the computation complexity. The concept of polynomial Lyapunov function (PLF) with square matrical representation, which simplifies the stabilization condition by avoiding partial derivatives of a polynomial matrix, is also introduced to generalize the existing expression of PLFs. Then, the derived stabilization condition, which is represented in terms of SOS can be numerically solved. Three simulation examples are conducted to illustrate the effectiveness and applicability of the proposed method. These examples also demonstrate the advantages of the proposed method over the existing convex SOS approach.

Ellipsoidal state estimation based on sum of squares for non‐linear systems with unknown but bounded noise

This brief concerns the set-theoretic recursive state estimation of non-linear systems with polynomial dynamics, making use of ellipsoidal bounds. The sum of squares approach is used to construct an optimal ellipsoid containing the intersection of sets obtained with the use of non-linear system equations and measurement equations. The properties of the proposed filter are illustrated with two examples, one concerning the simultaneous estimation of parameters and state of a linear system, and the other on the Euler equations of the rigid body rotational motion.

Sum of Squares Approach for Nonlinear H∞ Control

A proper Hamilton-Jacobi-Isaacs (HJI) inequality must be solved in a nonlinear H∞ control problem. The sum of squares (SOS) method can now be used to solve an analytically unsolvable nonlinear problem. A HJI inequality suitable for SOS approach is derived in the paper. The SOS algorithm for solving the HJI inequality is also provided. Conservativeness of the SOS method is then discussed in the paper. The conservativeness of the SOS approach is caused by the method itself, because it is really a synthesis method over the entire state space. To reduce the conservativeness, a local H∞ design on a restricted state-space region is proposed. But the SOS approach for the local H∞ design also suffers from the conservativeness problem, because the S-procedure for solving the set-containment constraint provides only a sufficient condition. The above-mentioned sources of conservativeness are peculiar for the SOS approaches. So a proper approach must be carefully selected in the design process to get a reasonable result. A design example is also given in the paper.

Polynomial control design for polynomial systems: A non-iterative sum of squares approach

This paper proposes a non-iterative state feedback design approach for polynomial systems using polynomial Lyapunov function based on the sum of squares (SOS) decomposition. The polynomial Lyapunov matrix consists of states of the system leading to the non-convex problem. A lower bound on the time derivative of the Lyapunov matrix is considered to turn the non-convex problem into a convex one; and hence, the solutions are computed through semi-definite programming methods in a non-iterative fashion. Furthermore, we show that the proposed approach can be applied to a wide range of practical and industrial systems that their controller design is challenging, such as different chaotic systems, chemical continuous stirred tank reactor, and power permanent magnet synchronous machine. Finally, software-in-the-loop (SiL) real-time simulations are presented to prove the practical application of the proposed approach.