ICSB2002 Section 1 Slides

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Section 1: Introduction to Feedback Control :

Section 1: Introduction to Feedback Control

Dynamical Models:

Dynamical Models Dynamics State space

Ordinary Differential Equations:

Ordinary Differential Equations Linear: Superposition, limited behaviour Nonlinear: No superposition, wide range of behaviour

Ordinary Differential Equations:

Ordinary Differential Equations Phase portraits Linear Nonlinear

Linearization:

Linearization

Notions of Stability:

Notions of Stability (neutral) stability asymptotic stability

Regions of Stability:

Regions of Stability local stability global stability

Tests for Stability:

Tests for Stability Nonlinear system: stable if there is a dissipative Lyapunov function for the system (a generalized energy) Linear system: asymptotically stable if eigenvalues of A have negative real part (we say A is Hurwitz)

Control Differential Equations:

Control Differential Equations Linear: Nonlinear: The input ‘u’ may include a disturbance acting on the system a reference signal to be tracked a control input to be chosen by the system designer (Here we think of the output as the whole state x)

Feedback Control:

Feedback Control Open loop system: Feedback control: Closed loop system: Linear System:

Allosteric Control:

Allosteric Control Input (u1) X 0 X 1 X i X n … … Output Control Input (u2) Set u2 = - Xn X 0 X 1 X i X n … … Output Feedback Input (u1)

Controllability and Stabilizability:

Controllability and Stabilizability System is stabilizable if for every point x1 there exists an input u(t) which drives the system to an equilibrium System is controllable if for every pair of points x1 and x2 there is an input u(t) which drives the system from x1 to x2

Controllability of Biochemical Networks:

Controllability of Biochemical Networks X 0 X 1 X i X n … … Output (ATP) Control Input (u) Input (glucose) Glycolytic chain Z1 … Zn Output (pentose) Pentose pathway [ATP] is controllable with input u [Pentose] is not controllable with input u [Pentose] is stabilizable with input u

Systems with Outputs:

Systems with Outputs The output y is commonly a subset of the components of the state . The output may represent the ‘part’ of the state which is of interest a measurement of the state

Observability and Detectability:

Observability and Detectability System is observable if the state trajectory x(t) can be determined from the output trajectory y(t) System is detectable if: output tends to equilibrium implies state tends to equilibrium

Observability of Biochemical Networks:

Observability of Biochemical Networks Trivial (static) example: system involving AMP, ADP, ATP, no change in level of adenosine moiety: [AMP] observable through [ADP] and [ATP] Dynamic example: X 0 X 1 X i X n … … Feedback Input (u) Xn+1 Full state observable from y = (X0, X3, X6, Xn) and input u.

Frequency Domain: Fourier Series:

Frequency Domain: Fourier Series

Frequency Domain: Fourier Transform:

Frequency Domain: Fourier Transform

Laplace Transform:

Laplace Transform Laplace Transform: Determines the frequency content of the signal f(t). Crucial feature: Laplace transform of a differential equation is an algebraic equation Differential Equation Algebraic Equation Laplace Transform Transformed Solution Solution Inverse Laplace Transform

Transfer functions:

Transfer functions In frequency space a linear time-invariant (LTI) system acts by multiplication: U(s) Y(s) H(s) Y(s) = H(s) U(s) Satisfies superposition: Y1(s) = H(s) U1(s) and Y2(s) = H(s) U2(s) imply Y1(s) + Y2(s) = H(s) {U1(s) + U2(s)}

Frequency Response: Bode Plots:

Frequency Response: Bode Plots Superposition: a plot of system response versus frequency completely characterizes the system

Feedback provides Robustness:

Feedback provides Robustness No feedback: Feedback: system: -