Linear Port-Hamiltonian Systems on Infinite-dimensional by Birgit Jacob, Hans J. Zwart

By Birgit Jacob, Hans J. Zwart

This e-book presents a self-contained creation to the speculation of infinite-dimensional structures thought and its functions to port-Hamiltonian platforms. The textbook starts off with straight forward recognized effects, then progresses easily to complex themes in present research.

Many actual structures might be formulated utilizing a Hamiltonian framework, resulting in types defined via traditional or partial differential equations. For the aim of keep watch over and for the interconnection of 2 or extra Hamiltonian platforms it truly is necessary to bear in mind this interplay with the surroundings. This booklet is the 1st textbook on infinite-dimensional port-Hamiltonian platforms. An summary sensible analytical process is mixed with the actual method of Hamiltonian platforms. This mixed technique ends up in simply verifiable stipulations for well-posedness and stability.

The booklet is available to graduate engineers and mathematicians with a minimum history in sensible research. in addition, the idea is illustrated via many worked-out examples.

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Then the following equality holds: d x(t) 2H = Re (y(t)∗ u(t)) . 30) dt Proof. 31) 2 where we have used that J is skew-adjoint and H is self-adjoint. 30), we can conclude several facts. If u ≡ 0, then the Hamiltonian is constant. Thus the solutions lie on isoclines, and the energy remains constant when applying no control. , how to steer the state to zero. The input u(t) = −ky(t), k ≥ 0, makes the energy non-increasing. This line of research will be further developed in Chapter 4. 1. 2, then ⎤ ⎤ ⎡ ⎡ 1 0 0 L1 0 0 L1 C 1 ⎦ , H = ⎣ 0 L2 0 ⎦ .

This motivates the notion “exponentially stable”. 1) which do not tend to zero. 2) and we try to find a suitable input function u such that the corresponding solution x converges to zero for t → ∞. 3. Let A ∈ Kn×n and B ∈ Kn×m . 2) converges to zero for t → ∞. Clearly, controllable systems Σ(A, B) are stabilizable. 1. 2 The pole placement problem It is the aim of the following two sections to characterize stabilizability and to show that the stabilizing control function u can be obtained via a feedback law u(t) = F x(t).

8. 2. Here V denotes the voltage source, L1 , L2 denote the inductance of the inductors, and C denotes the capacitance of the capacitor. 4), we obtain the system equations dIL1 = VL1 = VC − V, dt dIL2 = VL2 = VC − V, L2 and dt dVC = IC = −IL1 − IL2 . 10) and the control u(t) = V (t), we receive ⎤ − L11 ⎦ x(t) + ⎣ − 1 ⎦ u(t). 2. Normal forms 33 The controllability matrix is given by ⎡ 0 − L11 ⎢ −1 0 R(A, B) = ⎣ L2 1 0 + CL1 1 CL2 1 CL21 1 CL22 ⎤ + CL11 L2 + CL11 L2 ⎥ ⎦. 12) Since L1 times the first row equals L2 times the second row, we obtain rk R(A, B) = 2 and thus the system Σ(A, B) is not controllable.

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