08-11-2012, 01:53 PM
Discrete State Space Control
[attachment=38564]
Introduction to Direct Digital Design Using State Space Methods
In the previous set of notes we learnt how to design controllers using ‘classical methods’. In this set of
notes we consider ‘modern methods’ of control system design i.e. state space. With state space
design, we remain in the time domain and thus work directly with the differential equation model of our
plant. It is important to realise that whether we work with transfer functions or with differential
equations in state space form, the mathematics describes the same thing and the forms can be
interchanged. The major advantage however of working with a state space model of a system is that
the internal system state is explicitly maintained over time, where as with a transfer function, only the
input output relationship is maintained.
In these notes we consider only direct digital design and do not consider emulation in state space
systems. However, emulation techniques could be equally well applied to state space derived
controllers.
For a direct digital design, it is necessary for us to convert our continuous time plant to a discrete
equivalent that is capable of predicting the output of the plant at the sample instances, given that the
control signal to the plant is updated every sample instance and held using a ZOH circuit in-between
samples. In classical control we developed the formula for G(z) as a function of G(s). However, we
now have a state space model of our plant and would like to derive a similar formula to convert this
model to a discrete equivalent state space model.
Two canonical form of interest for later on
We will often have a continuous plant G(s) that we convert to a discrete equivalent G(z) (by hand say)
and then want to convert G(z) to a particular state space form. Remember the choice of state vector
for a particular system is arbitrary, as long as the vector can fully describe the state of a system. This
means of course that there are an infinite number of state space representations that correspond to
the same transfer function. Two special state space forms called the Control Canonical and Observer
Canonical forms are of particular interest to us when it comes to reducing the calculation complexity
when designing controllers and estimators. This is particularly useful if the controller or estimator
needs to be designed by hand (as in during an exam!).
Estimator Design
Previously we designed controllers using full state feedback. The state vector however is not usually
directly available through measurements. Thus, we need to estimate the state vector given
measurements y(k).
With reference to Figure 7.5 pg 496, the idea behind the estimator is to place a model of the plant in
parallel with the actual plant and to drive them both with the same input. If the model’s initial state
vector is set equal to the plant’s initial state vector then the state estimate (generated by the model)
will track the actual state vector. However, there are always uncertainties in the plant model and in
practice, without feedback, the state estimate would diverge from the true state. The solution is to use
the measurement y(k) and to compare it with the model’s predicted measurement and use the
difference between the two to modify the state estimate in such a way that it converges to the true
state vector.
[attachment=38564]
Introduction to Direct Digital Design Using State Space Methods
In the previous set of notes we learnt how to design controllers using ‘classical methods’. In this set of
notes we consider ‘modern methods’ of control system design i.e. state space. With state space
design, we remain in the time domain and thus work directly with the differential equation model of our
plant. It is important to realise that whether we work with transfer functions or with differential
equations in state space form, the mathematics describes the same thing and the forms can be
interchanged. The major advantage however of working with a state space model of a system is that
the internal system state is explicitly maintained over time, where as with a transfer function, only the
input output relationship is maintained.
In these notes we consider only direct digital design and do not consider emulation in state space
systems. However, emulation techniques could be equally well applied to state space derived
controllers.
For a direct digital design, it is necessary for us to convert our continuous time plant to a discrete
equivalent that is capable of predicting the output of the plant at the sample instances, given that the
control signal to the plant is updated every sample instance and held using a ZOH circuit in-between
samples. In classical control we developed the formula for G(z) as a function of G(s). However, we
now have a state space model of our plant and would like to derive a similar formula to convert this
model to a discrete equivalent state space model.
Two canonical form of interest for later on
We will often have a continuous plant G(s) that we convert to a discrete equivalent G(z) (by hand say)
and then want to convert G(z) to a particular state space form. Remember the choice of state vector
for a particular system is arbitrary, as long as the vector can fully describe the state of a system. This
means of course that there are an infinite number of state space representations that correspond to
the same transfer function. Two special state space forms called the Control Canonical and Observer
Canonical forms are of particular interest to us when it comes to reducing the calculation complexity
when designing controllers and estimators. This is particularly useful if the controller or estimator
needs to be designed by hand (as in during an exam!).
Estimator Design
Previously we designed controllers using full state feedback. The state vector however is not usually
directly available through measurements. Thus, we need to estimate the state vector given
measurements y(k).
With reference to Figure 7.5 pg 496, the idea behind the estimator is to place a model of the plant in
parallel with the actual plant and to drive them both with the same input. If the model’s initial state
vector is set equal to the plant’s initial state vector then the state estimate (generated by the model)
will track the actual state vector. However, there are always uncertainties in the plant model and in
practice, without feedback, the state estimate would diverge from the true state. The solution is to use
the measurement y(k) and to compare it with the model’s predicted measurement and use the
difference between the two to modify the state estimate in such a way that it converges to the true
state vector.