Robust Stability of Time-Varying Markov Jump Linear Systems with Respect to a Class of Structured, Stochastic, Nonlinear Parametric Uncertainties
Abstract
:1. Introduction
- (i)
- An estimation of the lower bound of the stability radius is obtained for a class of continuous-time Markovian jump linear systems subject to block-diagonal stochastic parameter perturbations. The considered parametric uncertainties are of multiplicative white noise type with unknown intensity;
- (ii)
- Scaling techniques have been used in order to effectively address the multi-perturbations case. This allows us to provide a lower bound of the stability radius in terms of the unique bounded and positive semidefinite solutions of adequately defined parameterized backward Lyapunov differential equations;
- (iii)
- A second characterization of a lower bound of the stability radius is given in terms of the existence of positive solutions of adequately defined parameterized backward Lyapunov differential inequalities. This second formulation allows us to state and solve a robust stabilization problem as a convex optimization problem.
2. Problem Formulation
2.1. Model Description
- (a)
- is almost surely continuous in any ;
- (b)
- for any , ;
- (c)
- .
2.2. Robust Stability: Stability Radius
- (a)
- globally exponentially mean square stable with conditioning (GESMS–C) if there exist , with the property:, , and any initial probability distribution of the Markov process;
- (b)
- globally stochastically stable with conditioning (GSS–C) if there exists with the property:, , and any initial probability distribution of the Markov process.
- (a)
- (b)
- Since the nominal system (4) is a special case of a system of type (1) (with ), it follows that the previous definition is also applicable in the case of the nominal system. It is worth noting that the system (4) is a linear system and this is why the "global" epithet of the stability is redundant. At the same time, in the linear case, the stability property is not related to a solution, it is a property of the whole system. Therefore, we shall say that the nominal system is exponentially stable in mean square with conditioning (ESMS–C) if its solutions have a behavior like that described by (14).
- (c)
- Applying Theorem 8.3.7 from [4], we deduce that the zero solution of a system is GESMS–C if and only if it is GSS–C.
3. Several Preliminary Issues
3.1. The Lyapunov Type Operators and Lyapunov Differential Equations
3.2. The Scaling of the Uncertainties
- (a)
- The assumption is fulfilled;
- (b)
- The nominal system (4) is ESMS–C;
4. The Main Results
4.1. A Lower Bound of the Stability Radius
- (a)
- The assumptions and hold true;
- (b)
- The nominal system (4) is ESMS–C.
- (i)
- (ii)
- Is obtained by subtracting (25) from (40) and using Proposition 2 to conclude that .
- (a)
- Assume that the assumptions of Theorem 4 are fulfilled. Let be given. If there exists a vector of scaling parameters and a bounded solution of the corresponding BLDI (40), satisfying the conditionthen
- (b)
- Assume that the assumptions of Hypotheses 1a–1c is fulfilled. If there exists a vector of scaling parameters and a bounded and uniform positive on solution of the BLDI (41) satisfying a condition of type (42), then
- (i)
- the nominal system (4) is ESMS–C;
- (ii)
- (a)
- (b)
- The fact that the nominal system is ESMS–C is obtained from Lemma 6 (i). The part (ii) is obtained in the same way as in the proof of (a) from above. Thus the proof ends.
4.2. Robust Stabilization via a State Feedback
- (a)
- There exist -matrix valued functions , which are bounded with a bounded derivative;
- (b)
- There exist bounded and continuous matrix valued functions ;
- (c)
- There exist positive scalars , , , , , satisfying the following system of linear matrix inequalities (LMIs):, , , where we denoted, .
- (a)
- It is worth mentioning that if the matrix valued functions, which are involved as coefficients of (47), are periodic functions, and if (47) has a solution and with , , then (47) also has a solution , , which is periodic with the same period as the coefficients of (47);
- (b)
- Since the constant functions can be regarded as periodic functions of an arbitrary period, one obtains that, if the coefficients of (47) do not depend upon t, then (47) has a constant solution and with , if it is solvable. Hence, without loss of generality, in the periodic case to test the solvability of (47) we look for a periodic solution with the same period as the coefficients. Moreover, if the coefficients of (47) do not depend upon t, then we test its solvability, looking for constant solutions.
5. Numerical Experiments
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Dragan, V.; Aberkane, S. Robust Stability of Time-Varying Markov Jump Linear Systems with Respect to a Class of Structured, Stochastic, Nonlinear Parametric Uncertainties. Axioms 2021, 10, 148. https://doi.org/10.3390/axioms10030148
Dragan V, Aberkane S. Robust Stability of Time-Varying Markov Jump Linear Systems with Respect to a Class of Structured, Stochastic, Nonlinear Parametric Uncertainties. Axioms. 2021; 10(3):148. https://doi.org/10.3390/axioms10030148
Chicago/Turabian StyleDragan, Vasile, and Samir Aberkane. 2021. "Robust Stability of Time-Varying Markov Jump Linear Systems with Respect to a Class of Structured, Stochastic, Nonlinear Parametric Uncertainties" Axioms 10, no. 3: 148. https://doi.org/10.3390/axioms10030148
APA StyleDragan, V., & Aberkane, S. (2021). Robust Stability of Time-Varying Markov Jump Linear Systems with Respect to a Class of Structured, Stochastic, Nonlinear Parametric Uncertainties. Axioms, 10(3), 148. https://doi.org/10.3390/axioms10030148