unsw ceic3000 w4 process modeling and analysis

unsw ceic3000 w4 process modeling and analysis - Process...

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Process Modelling and Analysis Week 4: Nonlinear Analysis - Part One Session 1, 2012 Lecturer: A/Prof. J. Bao Contents 1 Stability Analysis for Linear Systems 1 2 Multiplicity 3 2.1 Output multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Input multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Stability Analysis 4 4 Phase-plane Analysis 7 4.1 Second Order Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1 Stability Analysis for Linear Systems When we have a process model, we may ask the question, if the states are perturbed from the steady-state values, what will be the dynamic behaviour of this system? Will the state converge to a some steady-state or will it go to infinity? This is stability analysis concerned with. If the state converges to its original equilibrium or to a new steady-state, this system is said to be stable, otherwise, it is said to be unstable. If we have bounded input u s , we re-define input variable as u = u u s = 0 . x = x x s . A new model will be obtained with the new coordinate. As long as the input is bounded, such variable transform will not change the boundness of the model. Therefore we can define stability based on the zero-input form and for the equilibrium of zero. Mathematically, for a system with zero input ˙x = f ( x , 0 ) , x ( t 0 ) = x 0 (1) Then this system is (asymptotically) stable with the equilibrium of zero if lim t →∞ x ( t ) = 0 (2) Otherwise it is unstable. There are several stability definitions. Here we just adopt one common definition. We will learn other stability definitions such as BIBO stability in next session’s process control subject. 1
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An unstable system may be very sensitive to disturbances. If we find a process is unstable, we need take extra care, because any disturbance may cause this process to run away. If the state variable is temperature, unstable system implies the temperature may go to infinity, at least theoretically, it may cause fire or exposition. So it is very important to analyse the stability of models. However, it is hard to determine the stability of the process from the definition. So we need stability criteria. For a single variable linear system ˙ x = ax (3) has the solution: x ( t ) = e at x (0) (4) When t approaches infinity, x ( t ) will converge to 0 if a < 0; x ( t ) will stay at x (0) if a = 0; and be unbounded if a > 0 . Therefore the system is stable if and only if a < 0 . In a similar fashion, for a linear system ˙x = A x + B u (5) with x (0) = x 0 . The “zero-input” form of the state space model is then: ˙x = A x . (6) Recall that matrix A can be diagonalized by using S 1 AS = Λ = λ 1 0 0 0 0 λ 2 0 0 0 0 . . . . . . 0 0 ... λ
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unsw ceic3000 w4 process modeling and analysis - Process...

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