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5. Analysis of SISO Control Loops

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Control system design makes use of the two key enabling techniques: analysis and synthesis. Analysis concerns itself with the impact that a given controller has on a given system when they interact in feedback while synthesis asks how to construct controllers with certain properties. This chapter covers analysis. For a given controller and plant connected in feedback it asks (and provides answers for) the following:

  • Is the loop stable?
  • What are the sensitivities to various disturbances?
  • What is the impact of linear modeling errors?
  • How do small nonlinearities impact on the loop?

We also introduce several analysis tools; specifically

  • Root locus
  • Nyquist stability analysis

Summary
  • This chapter introduced the fundamentals of SISO feedback control loop analysis.
     
  • Feedback introduces a cyclical dependence between controller and system:
    • the controller action affects the systems outputs,
    • and the system outputs affect the controller action.
       
  • For better or worse, this has a remarkably complex effect on the emergent closed loop.
     
  • Well designed, feedback can
    • make an unstable system stable;
    • increase the response speed;
    • decrease the effects of disturbances
    • decrease the effects of system parameter uncertainties, and more.
       
  • Poorly designed, feedback can
    • introduce instabilities into a previously stable system;
    • add oscillations into a previously smooth response;
    • result in high sensitivity to measurement noise;
    • result in sensitivity to structural modeling errors, and more.
       
  • Individual aspects of the overall behavior of a dynamic system include
    • time domain: stability, rise time, overshoot, settling time, steady state errors, etc.
    • frequency domain: bandwidth, cut off frequencies, gain and phase margins, etc.
       
  • Some of these properties have rigorous definitions, others tend to be qualitative.
     
  • Any property or analysis can further be prefixed with the term nominal or robust;
    • nominal indicates a temporarily idealized assumption that the model is perfect;
    • robust indicates an explicit investigation of the effect of modeling errors.
       
  • The effect of the controller $C(s)=\frac{P(s)}{L(s)}$ on the nominal model $G_o(s)=% \frac{B_o(s)}{A_o(s)}$ in the feedback loop shown in Figure fig:fb1x is
\begin{align}Y&=\frac{G_oC}{1+G_oC}(R-D_m)+\frac 1{1+G_oC}D_o+\frac{G_o}{1+G_oC... ...ac{A_oP}{A_oL+B_oP}(R-D_o-D_m)-\frac{B_oP}{A_oL+B_oP}D_i \nonumber \end{align}
  • Interpretation, definitions and remarks:
    • The nominal response is determined by four transfer functions.
    • Due to their fundamental importance, they have individual names and symbols:
      $S_o:=\dfrac 1{1+G_oC}=\dfrac{A_oL}{A_oL+B_oP},$ the nominal sensitivity function;
      $T_o:=\dfrac{G_oC}{1+G_oC}=\dfrac{B_oP}{A_oL+B_oP},$ the nominal complementary sensitivity;
      $S_{io}:=\dfrac{G_o}{1+G_oC}=\dfrac{B_oL}{A_oL+B_oP},$ the nominal input sensitivity;
      $S_{uo}:=\dfrac C{1+G_oC}=\dfrac{A_oP}{A_oL+B_oP},$ the nominal control sensitivity
      and are collectively called the nominal sensitivities.
    • All four sensitivity functions have the same poles, the roots of AoL+BoP.
    • The polynomial AoL+BoP is also called the nominal characteristic polynomial.
    • Recall that stability of a transfer function is determined by the roots only.
    • Hence, the nominal loop is stable if and only if the real parts of the roots of AoL+BoP are all strictly negative. These roots have an intricate relation to the controller and system poles and zeros.
    • The properties of the four sensitivities, and therefore the properties of the nominal closed loop, depend on the interlacing of the poles of the characteristic polynomial (the common denominator) and the zeros of AoL,BoP,BoL, and AoP, respectively.
       
  • Linear modeling errors:
    • If the same controller is applied to a linear system, $G\left( s\right) $, that differs from the model by $G(s)=G_o(s)G_\Delta (s),$ then the resulting loop remains stable provided that $\left\vert T_o(j\omega )\right\vert \left\vert G_\Delta (j\omega )\right\vert <1$, $\forall \omega$.
    • This condition is also known as the small-gain theorem.
    • Obviously it cannot be easily tested, as the multiplicative modeling error, $G_\Delta $, is typically unknown. Usually bounds on $G_\Delta (j\omega)$ are used instead.
    • Nevertheless, it gives valuable insight. For example, we see that the closed loop bandwidth must be tuned to be less than the frequencies where one expects significant modeling errors.
       
  • Non-linear modeling errors: If the same controller is applied to a system, $G_{nl}\langle\circ\rangle$, that not only differs from the model linearly but that is nonlinear (as real systems will always be to some extent), then rigorous analysis becomes very hard in general but qualitative insights can be obtained into the operation of the system by considering the impact of model errors.