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7. Synthesis of SISO Controllers

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In Chapter 5 it was shown how feedback control loop performance and stability can be characterized using a set of four sensitivity functions. A key feature of these functions is that all of them have poles belonging to the same set, the set of closed loop poles. Hence, this set determines stability and the natural modes of the closed loop. A key synthesis question is therefore: given a model, can one systematically synthesize a controller such that the closed loop poles are in predefined locations? This chapter will show that this is indeed possible. We call this pole assignment, which is a fundamental idea in control synthesis.

In this chapter we will use a polynomial description. This evolves naturally from the analysis in Chapter 5. In a later chapter we will use a state space description. This will provide a natural transition from SISO to MIMO control systems in later chapters.

It is also shown how both approaches can accommodate other design requirements such as zero steady state tracking errors and disturbance rejection.

Finally, PID controllers are placed into this general framework and it is shown how they may be synthesized using pole assignment methods.

The question of how to choose a set of values for the closed loop poles to meet the required performance specifications is, of course, a crucial issue. This is actually a non-trivial question which needs to be considered as part of the intricate web of design trade-offs associated with all feedback loops. The latter will be taken up in the next chapter, where the design problem will be analyzed.

Summary
  • This chapter addresses the question of synthesis and asks:
    Given the model $G_0(s)=\frac{B_o(s)}{A_o(s)}$, how can one synthesize a controller, $C(s)=\frac{P(s)}{L(s)}$ such that the closed loop has a particular property.
     
  • Recall:
    • the poles have a profound impact on the dynamics of a transfer function;
    • The poles of the four sensitivities governing the closed loop belong to the same set, namely the roots of the characteristic equation Ao(s)L(s) + Bo(s)P(s) = 0.
       
  • Therefore, a key synthesis question is: Given a model, can one synthesize a controller such that the closed loop poles (i.e. sensitivity poles) are in pre-defined locations?
     
  • Stated mathematically:
    Given polynomials Ao(s),Bo(s) (defining the model) and given a polynomial Acl(s) (defining the desired location of closed loop poles), is it possible to find polynomials P(s) and L(s) such that Ao(s)L(s) + Bo(s)P(s) = Acl(s)? This chapter shows that this is indeed possible.
     
  • The equation Ao(s)L(s) + Bo(s)P(s) = Acl(s) is known as a Diophantine equation.
     
  • Controller synthesis by solving the Diophantine equation is known as pole placement. There are several efficient algorithms as well as commercial software to do so.
     
  • Synthesis ensures that the emergent closed loop has particular constructed properties.
    • However, the overall system performance is determined by a number of further properties which are consequences of the constructed property.
    • The coupling of constructed and consequential properties generates trade-offs.
       
  • Design is concerned with
    • efficiently detecting if there is no solution that meets the design specifications adequately and what the inhibiting factors are
    • choosing the constructed properties such that, whenever possible, the overall behavior emerging from the interacting constructed and the consequential properties meets the design specifications adequately.
       
  • This is the topic of the next chapter.