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Stability Analysis for a Class of Discrete-Time Switched Systems With Partial Unstable Subsystems Abstract: This brief is concerned with the stability problem for a class of discrete-time switched systems with unstable subsystems. By constructing a quasi-time-dependent Lyapunov function, stability analysis criterion for nonlinear switched systems is developed under a designed switching rule in which the fast and slow switching techniques are adopted for unstable and stable subsystems, respectively.
Then, stability criteria for linear switched systems are presented. It is shown that the obtained results are more general and less conservative than existing ones. Some comparative simulations are provided to demonstrate the effectiveness of the proposed method.
Article :. Date of Publication: 01 February DOI: Need Help?N2 - In this paper, we study the stability analysis of discrete-time interconnected positive systems by means of a weighted l1-induced norm, where the weighted l1-induced norm is computed from the l1 norm of the input and output signals evaluated with given weighting vectors. In the literature,it was shown that the weighted L1-induced norm of continuous-time LTI positive systems can be characterized by linear programming problem LP.
Moreover, the weighted L1-induced norm was proved to be useful for the stability analysis of interconnected positive systems. On the basis of these results, we first show that the weighted l1-induced norm of discrete-time LTI positive systems can be again characterized by LP. From this preliminary result, we can construct a transformation from a given discrete-time positive system to a continuous-time positive system, where the weighted L1-induced norm of the resulting system is equivalent to the weighted l1-induced norm of the original system.
By means of this key transformation, we can readily extend the continuous-time case results to the stability analysis of discrete-time interconnected positive systems. AB - In this paper, we study the stability analysis of discrete-time interconnected positive systems by means of a weighted l1-induced norm, where the weighted l1-induced norm is computed from the l1 norm of the input and output signals evaluated with given weighting vectors.
Section of Measurement and Control Engineering. Access to Document U2 - Documentation Help Center. Stability is a standard requirement for control systems to avoid loss of control and damage to equipment.
For linear feedback systems, stability can be assessed by looking at the poles of the closed-loop transfer function. Gain and phase margins measure how much gain or phase variation at the gain crossover frequency will cause a loss of stability. Together, these two quantities give an estimate of the safety margin for closed-loop stability.
The smaller the stability margins, the more fragile stability is. Examine the pole and zero locations of dynamic systems graphically and numerically.
Examine the effect of stability margins on closed-loop response characteristics of a control system. Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select:.
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Search MathWorks. Open Mobile Search. Off-Canvas Navigation Menu Toggle. Stability Analysis Gain and phase margins, pole and zero locations. Functions expand all Pole and Zero Locations.
Pole and Zero Locations. Open Live Script. Assessing Gain and Phase Margins. Open Script.
5.7: 5.7 Linear Stability Analysis of Discrete-Time Nonlinear Dynamical Systems
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Email Address. Sign In. Stability analysis of discrete-time fuzzy dynamic systems based on piecewise Lyapunov functions Abstract: This paper presents a stability analysis method for discrete-time Takagi-Sugeno fuzzy dynamic systems based on a piecewise smooth Lyapunov function.
It is shown that the stability of the fuzzy dynamic system can be established if a piecewise Lyapunov function can be constructed, and moreover, the function can be obtained by solving a set of linear matrix inequalities that is numerically feasible with commercially available software.
It is also demonstrated via numerical examples that the stability result based on the piecewise quadratic Lyapunov functions is less conservative than that based on the common quadratic Lyapunov functions.
Article :. Date of Publication: 19 February DOI: Need Help?All of the discussions above about eigenvalues and eigenvectors are for linear dynamical systems. Can we apply the same methodology to study the asymptotic behavior of nonlinear systems?
Unfortunately, the answer is a depressing no. Asymptotic behaviors of nonlinear systems can be very complex, and there is no general methodology to systematically analyze and predict them. We will revisit this issue later.
Having said that, we can still use eigenvalues and eigenvectors to conduct a linear stability analysis of nonlinear systems, which is an analytical method to determine the stability of the system at or near its equilibrium point by approximating its dynamics around that point as a linear dynamical system linearization.
The basic idea of linear stability analysis is to rewrite the dynamics of the system in terms of a small perturbation added to the equilibrium point of your interest. This operation is what linearization is all about. Here is how linear stability analysis works. The right hand side of the equation above is still a nonlinear function. Together with this result and Eq. Interestingly, this conclusion has some connection to the cobweb plot we discussed before. If the slope is too steep, either positively or negatively, trajectories will diverge away from the equilibrium point.
If the slope is less steep than 1, trajectories will converge to the point. You may have noticed such characteristics when you drew the cobweb plots. Linear stability analysis offers a mathematical explanation of that. Using variable replacement similar to Eq.
However, the assumption that they are extremely small helps simplify the analysis here. This linear approximation allows us to rewrite Eqs. It is a linear approximation of the nonlinear function. Note that the orders of rows and columns of a Jacobian matrix must match. Look at how simple it can get! Sometimes, an unstable equilibrium point may come with other eigenvalues that show stability. Such equilibrium points are called saddle pointswhere nearby trajectories are attracted to the equilibrium point in some directions but are repelled in other directions.
But actually, proving that the point is truly neutral requires more advanced nonlinear analysis, which is beyond the scope of this textbook. Finally, if the eigenvalues are complex conjugates, oscillatory dynamics are going on around the equilibrium points. Such equilibrium points are called a stable or unstable spiral focus or a neutral centerdepending on their stabilities. Figure 5. Find an equilibrium point of the system you are interested in.
Calculate the Jacobian matrix of the system at the equilibrium point. Calculate the eigenvalues of the Jacobian matrix. If the absolute value of the dominant eigenvalue is:.Thank you for the wonderful experience. I will share our trip with everyone and encourage them to use your company to book their travel. When we travel to the North again we will definitely use Nordic Visitor.
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Stability Analysis of Discrete-time Interconnected Positive Systems using Weighted l1-induced Norm
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