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18. Synthesis via State Space Methods

We are now in a position to give a state space interpretation to many of the results described earlier in Chapter 7 and 15. In a sense this will duplicate the earlier work. Our reason for doing so, however, is that the alternative state space formulation carries over more naturally to the multivariable case. Results to be presented here include

• pole assignment by state variable feedback
• design of observers to reconstruct missing states from available output measurements
• combining state feedback with an observer
• transfer function interpretation
• dealing with disturbances in state variable feedback
• re-interpretation of the affine parameterization of all stabilizing controllers

• We have shown that controller synthesis via pole placement can also be presented in state space form: Given a model in state space form, and given desired locations of the closed loop poles, it is possible to compute a set of constant gains, one gain for each state, such that feeding back the states through the gains results in a closed loop with poles in the pre-specified locations.
   
• Viewed as a theoretical result, this insight complements the equivalence of transfer function and state space models by an equivalence of achieving pole placement by either synthesizing a controller as transfer function via the Diophantine equation or as constant gain state variable feedback.
   
• Viewed from a practical point of view, implementing this controller would require sensing the value of each state. Due to physical, chemical and economical constraints, however, one hardly ever has actual measurements of all system states available.
   
• This raises the question of alternatives to actual measurements and introduces the notion of so called observers, sometimes also called soft sensors, virtual sensors, filters or calculated data.
   
• The purpose of an observer is to infer the value of an unmeasured state from other states that are correlated with it and that are being measured.
   
• Observers have a number of commonalities with control systems:
  • they are dynamical systems
  • they can be treated in either the frequency or time domain
  • they can be analyzed, synthesized and designed
  • they have performance properties, such as stability, transients, sensitivities, etc.
  • these properties are influenced by the pole/zero patterns of their sensitivities.
     
• State estimates produced by an observer are used for several purposes:
  • constraint monitoring
  • data logging and trending
  • condition and performance monitoring
  • fault detection
  • feedback control
     
• To implement a synthesized state feedback controller as discussed above, one can use state variable estimates from an observer in lieu of unavailable measurements; the emergent closed loop behavior is due to the interaction between the dynamical properties of system, controller and observer.
   
• The interaction is quantified by the third fundamental result presented in the chapter: the nominal poles of the overall closed loop are the union of the observer poles and the closed loop poles induced by the feedback gains if all states could be measured. This result is also known as the separation theorem.
   
• Recall, that controller synthesis is concerned with how to compute a controller that will give the emergent closed loop a particular property, the constructed property.
   
• The main focus of the chapter is on synthesizing controllers that place the closed loop poles in chosen locations; this is a particular constructed property that allows certain design insights to be discussed in the next chapter.
   
• There are, however, other useful constructed properties as well.
   
• Examples of constructed properties for which there exist synthesis solutions include:
  • arrive at a specified system state in minimal time with an energy constraint
  • minimize the weighted square of the control error and energy consumption
  • minimum variance control
     
• One approach to synthesis is to cast the constructed property into a so-called cost-functional, objective function or criterion which is then minimized numerically
   

• This approach is sometimes called optimal control, since one optimizes a criterion.
   
• One must remember, however, that the result can only be as good as the criterion.
   
• Optimization shifts the primary engineering task from explicit controller design to criterion design, which then generates the controller automatically.
   
• Both approaches have benefits, including personal preference and experience.

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