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By Chen C.-T.

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Among them, two are from the original integrator plant, and one is from the augmentation of the plant model. 1. The objective of this tutorial is to demonstrate how to obtain a discrete-time state-space model from a continuous-time state-space model, and form the augmented discrete-time state-space model. Consider a continuoustime system has the state-space model: ⎡ ⎡ ⎤ ⎤ 010 1 x˙ m (t) = ⎣ 3 0 1 ⎦ xm (t) + ⎣ 1 ⎦ u(t) 010 3 y(t) = 0 1 0 xm (t). 9) Step by Step 1. m. We form a continuous-time state variable model; then this continuous-time model is discretized using MATLAB function ‘c2dm’ with specified sampling interval Δt.

47) Then we can use this model to calculate the state variable x ˆm (k), k = 1, 2, . . , with an initial state condition x ˆm (0) and input signal u(k) as x ˆm (k + 1) = Am x ˆm (k) + Bm u(k). 48) This approach, in fact, would work after some transient time, if the plant model is stable and our guess of the initial condition is nearly correct. What could be the problems with this type of approach? Basically, it is an open-loop ˆm (k) satisfies the difference equation: prediction. The error x ˜m (k) = xm (k)− x x ˜m (k + 1) = Am (xm (k) − xˆm (k)) ˜m (k).

Stable modes here means that the corresponding eigenvalues are strictly inside the unit circle. 24 1 Discrete-time MPC for Beginners specifically stated, we require the model to be both controllable and observable in order to achieve desired closed-loop performance. 6 to illustrate the importance of observability for the design of observer. Because the augmented model introduced additional integral modes, we need to examine under what conditions these additional modes become controllable. The simplest way for the investigation is based on the assumption of minimal realization of the plant model.

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