Piecewise Linear Control Systems: A Computational Approach by Mikael K.-J. Johansson

By Mikael K.-J. Johansson

2. Piecewise Linear Modeling . . . . . . . . . . . . . . . . . . . . . nine 2. 1 version illustration . . . . . . . . . . . . . . . . . . . . . nine 2. 2 resolution techniques . . . . . . . . . . . . . . . . . . . . . . . 2. three Uncertainty types . . . . . . . . . . . . . . . . . . . . . . 2. four Modularity and Interconnections . . . . . . . . . . . . . . 26 2. five Piecewise Linear functionality Representations . . . . . . . . . 28 2. 6 reviews and References . . . . . . . . . . . . . . . . . . 30 three. Structural research . . . . . . . . . . . . . . . . . . . . . . . . . . 32 three. 1 Equilibrium issues and the regular kingdom attribute . . 32 three. 2 Constraint Verification and Invariance . . . . . . . . . . . 35 three. three Detecting beautiful Sliding Modes on mobilephone barriers 37 three. four reviews and References . . . . . . . . . . . . . . . . . . 39 four. Lyapunov balance . . . . . . . . . . . . . . . . . . . . . . . . . . forty-one four. 1 Exponential balance . . . . . . . . . . . . . . . . . . . . . . forty-one four. 2 Quadratic balance . . . . . . . . . . . . . . . . . . . . . . . forty two four. three Conservatism of Quadratic balance . . . . . . . . . . . . . forty six four. four From Quadratic to Piecewise Quadratic . . . . . . . . . . . forty eight four. five Interlude: Describing Partition homes . . . . . . . . . fifty one four. 6 Piecewise Quadratic Lyapunov capabilities . . . . . . . . . fifty five four. 7 research of Piecewise Linear Differential Inclusions . . . . sixty one four. eight research of platforms with beautiful Sliding Modes . . . . sixty three four. nine enhancing Computational potency . . . . . . . . . . . . sixty six four. 10 Piecewise Linear Lyapunov services . . . . . . . . . . . seventy two four. eleven A Unifying View . . . . . . . . . . . . . . . . . . . . . . . . seventy seven four. 12 reviews and References . . . . . . . . . . . . . . . . . . eighty two five. Dissipativity research . . . . . . . . . . . . . . . . . . . . . . . . eighty five five. 1 Dissipativity research through Convex Optimization . . . . . . 86 21 14 Contents Contents five. 2 Computation of £2 triggered achieve . . . . . . . . . . . . . . 88 five. three Estimation of temporary strength . . . . . . . . . . . . . . . . 89 five. four Dissipative platforms with Quadratic provide premiums . . . . . ninety one five. five reviews and References . . . . . . . . . . . . . . . . . . ninety five Controller layout . . . . . . . . . . . . . . . . . . . . . . . . . . . ninety six 6. 1 Quadratic Stabilization of Piecewise Linear" platforms . . . ninety seven 6. 2 Controller Synthesis in line with Piecewise Quadratics . . . ninety eight 6. three reviews and References . . . . . . . . . . . . . . . . . . one hundred and five 7. chosen themes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7. 1 Estimation of areas of allure . . . . . . . . . . . . .

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43 Chapter 4. 1 to verify exponential stability of the origin. 1. 1. 1 Linear (white) and saturated (shaded) regions in the state space. 3) w i t h / / I. 1 guarantees a much larger region of attraction than the Lyapunov equation approach, the result is still very weak. In fact, the sat urated system is globally asymptotically stable (we will prove this in Chap ter 7, where we also develop specialized methods for" estimating regions of attraction for" linear" systems with saturation). One of the reasons for" the disappointing performance is that we have only analyzed the behaviour in the unsaturated region.

Piecewise linear system with non traversal sliding. 3 Uncertainty Models Uncertainty and robustness are central themes in modeling and analysis of feedback systems. One of the most important reasons for" using feedback is to guarantee that system specifications are met despite variations in sys tern components and exogenous disturbances. Furthermore, since there is always a mismatch between the model used in the mathematical analysis and the actual physical system, it is important to account for" this uncer tainty to ensure that the results derived from the model will hold in reality.

The following result is central [83, 30]. 4). 6) i 1, . . , L has a solution, then the origin is globally exponentially stable. 2. 2 only gives sufficient conditions for stability. 2 is a convex optimiza tion problem a solution P can always be found if it exists. The conservatism comes from the fact that quadratic Lyapunov functions are only sufficient for establishing stability of linear differential inclusions [142]. 2 is that the search for a quadratic Lyapunov function has been formulated as a convex optimization problem.

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