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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
- combining state feedback with an observer
- transfer function interpretation
- dealing with disturbances in state variable feedback
- re-interpretation of the affine parameterization of all
- 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
- 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
- State estimates produced by an observer are used for several
- 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
- arrive at a specified system state in minimal time with an
- minimize the weighted square of the control error and energy
- 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
- 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