16.30 Topic 5: Introduction to state-space models

Topic #5 Feedback Control Systems state-space Systems What are state-space models ? Why should we use them? How are they related to the transfer functions used in classical control design and how do we develop a state-space model ? What are the basic properties of a state-space model , and how do we analyze these? Fall 2010 5 2 SS Introduction State space model : a representation of the dynamics of an Nth order system as a first order differential equation in an N-vector, which is called the state. Convert the Nth order differential equation that governs the dy namics into N first-order differential equations Classic example: second order mass-spring system mp + cp + kp = F Let x1 = p, then x2 = p = x 1, and x 2 = p =(F cp kp)/m =(F cx2 kx1)/m p 01 p 0 = + u p k/m c/m p 1/m Let u = F and introduce the state x1 p x == x = Ax + Bu x2 p If the measured output of the system is the position, then we have that p x1 y = p =10 =10 = Cx p x2 September 21, 2010 Fall 2010 5 3 Most general continuous- time linear dynamical system has form x (t) = A(t)x(t) + B(t)u(t) y(t) = C(t)x(t)+ D(t)u(t) where: t R denotes time x(t) Rn is the state (vector) u(t) Rm is the input or control y(t) Rp is the output A(t) Rn n is the dynamics matrix B(t) Rn m is the input matrix C(t) Rp n is the output or sensor matrix D(t) Rp m is

Sep 21, 2010 · • A system is said to be time-invariant if the relationship between the ... • Can develop linear models from nonlinear system dynamics September 21, 2010 ... • Can then develop the linearized equations by using the Taylor series expansion of f(·, ·) about x e and u e.

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  Series, Model, Time, Nonlinear

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