Foreword: In biological system modelling, often we are faced with a (huge) set of differential equations. They describe physical phenomena and are therefore relatively easy to construct and inspect if the system was constructed in the right way. However, solving these are computationally taxing and there is no way to predict the outcome just by looking at the entire set of differential equations. Since being introduced to System Engineering, I have always wanted to use the control theories to simplify these massive (and messy) systems, and thus be able to anticipate and adjust the system behaviour according to the need of the model.
System Modelling
Here we have a very simple process model: an input is manipulated by a to produce an output .
A set of differential equations are commonly used to model the system, which then gives the response in time domain. However this method gives very limited information on the predictable system behaviours.
1. Laplace Transform
 Transform statespace equations in time domain into frequency domain
 Definition:
 Similar to integration, we can look up the Laplace Transform of a specific function:
(only some common functions are listed)
 With the following theorems, system analysis in the frequency domain is greatly simplified:
Theorem  Name 

Frequency Shift Theorem  
Time Shift Theorem  
First Order Differentiation Theorem  
Second Order Differentiation Theorem  
Integration Theorem  
Final Value Theorem  
Initial Value Theorem 
 An example of how Laplace Transform is used to solve ordinary differential equations is shown below:
 Equation:
 Given
 When we take the Laplace transform (from the lookup table):
 also the input
 Now we take the partial fraction (and that ):
 Take the inverse Laplace (to get back the timedomain representation):
 This is to prove that Laplace Transform can solve ODEs as with the usual explicit method. The benefit of this approach is the simplicity of the transformed system, and the underlying information (including the speed and stability of responses) in the system is easily accessible using simple tools.
2. Block Diagram
 Rationale: Compartmentalise different components of the system
 Processes = System/Plant
 Inputoutput = measurable variables eg. molecules production, genes transcription/translation events or any physical quantities
 In a closedloop system, a sensor for the output signal sends back information to determine the error between the actual and desired output.
In the open loop system presented here, is the input to the plant to give the output . This gives the relationship of . Hence, the plant is
In the closed loop system: is the reference (desirable) output; is the error between the actual and desirable outputs, which in this case acts as the input to the system .
System analysis rules the following:
By this the error term is eliminated to establish the process based on the information of the actual and desirable outputs.
Alternatively, system reduction according to Mason’s rule (as shown on the right of the diagram) gives the following equation for the plant:
The Laplace transform of the plant can be implemented in MATLAB with the following codes:
Assume
syms s t;g=t*exp(t); % Enter equation in g(t) GL=laplace(g,s); % Laplace transform [num,den]=numden(GL); % Extract numerator and denominator num=sym2poly(num); den=sym2poly(den); % Get coefficients G=tf(num,den);% Set up Transfer function
3. System Analysis
 Rootfinding:
 First order polynomial: then
 Second order polynomial: then
 Third order polynomial with imaginary roots: where then
 Inspect convergence from PoleZero Map:
 Usage: Inspect stability and convergence
 Pole = root(s) of the denominator polynomial (‘x’)
 Zero = root(s) of the numerator polynomial (‘o’)
 MATLAB command
pzmap(sys)
 Example:
 has poles (‘x’) at and and zeros (‘o’) at
 Properties:
 Righthalf plane (RHP) refers to the real part of the root > 0;
 Lefthalf plane (LHP) refers to the real part of the root < 0.
 We only consider the denominator polynomial (poles) for convergence evaluation:

1st Order Polynomial 2nd Order Polynomial Attribute s+a All additive terms Root in LHP Convergent sb Contain subtractive terms Root in RHP Divergent  For third order polynomial: according the Routh array, for , has all roots in LHP only if

Pole position gives an insight on the system behaviour. This diagram roughly defines the regions of pole positions with predicted response. RHP is not considered since the systems do not converge (unlike genetic circuits, in engineering, we strive to model stabilisation and convergence!)
 Now we focus on the oscillatory responses by inspecting the characteristics of second order systems:
 We first transform the system into monic form:
 Damping : ratio of exponential decay constant to natural frequency: (hence gives system poles at )
 Undamped natural frequency:
 Here are four types systems responses:
 Some other specific characteristics which are interesting to look at in Underdamped system ($latex 0<\zeta
 Damped natural frequency:
 Rise time : time for the response to go from 10% to the first 90% point of the final value
 Peak time : time to the first peak
 Percentage overshoot : highest peak to steadystate value in percentage
 Settling time : time for oscillations to reach and stay within 2% (sometimes 5%) of steadystate value
 Ultimately, with all the system analysis tools, we would like to manipulate different blocks of the system to attaindesirable outputs. Given the existing plant, how can we manipulate the system behaviour? What if there are disturbances in the system – how do we reject these disturbances?
 A typical engineering scenario is that, when an aircraft during automatic cruising faces changes in air currents around it, how do different components (engine speed, direction etc.) coordinate to respond and provide a stable flight experience?
4. Controller design
 Controllers are inserted into the system to enhance the plant performance and/or reject disturbances. Here is an overview of the two traditional controller design strategies:
 PID control:

Control Action Effects on System Response Proportional Scaling error signal direct effect on steadystate error
Transient response (change )
Increases oscillationsIntegral Removes steadystate error (operates even when error is 0)
Introduces oscillations
Introduces critically stable poleDerivative Damp down oscillations
Provides prediction of future error
 Classical control:
 Stability criteria in the frequency domain step up to another level of complexity. In circuits, these controller designs are determined by Rootlocus diagram, bode plot (gain and phase margins) and Nyquist criterion. With these indicators, lead compensator and lag compensator are chosen to bring about desirable system response in terms of the bandwidth, high and low frequency gains (final steady states) and so on.
 Controller design in MATLAB is simple with the toolbox:
simulink
 Outlook: Designing compensators to drive desirable system response can give us an insight into how other components in the biological system (which had not been considered in the main system plant ) are contributing to the system. For instance, the need to introduce a lead compensator (which speeds up response) might translate to the fact that some biological factors unknown to the system (ie. not represented in the system) are acting to speed up the responses from known compensators.
Footnote: Technical parts are reliant on general knowledge which could be obtained from standard control engineering textbooks; views are my own.
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