Reference tracking

Overview

The controllers we have designed so far make the state converge to zero — that is, they make

Suppose we want the state to converge to something else — that is, to make

We will see that, under certain conditions, this is easy to do.

Reference tracking with full state feedback

Consider the dynamic model

where . Suppose we linearize this model about some equilibrium point to produce the state-space model

where

Suppose we design linear state feedback

that would make the closed-loop system

asymptotically stable — that is, that would make

Denote the standard basis for by

Suppose there is some index that satisfies

That is, suppose the function is constant in — or, does not vary with — the ‘th element of . Then, the following three things are true:

Invariance of equilibrium point. Since

then is also an equilibrium point for any .

Invariance of error in approximation of the dynamic model. The linear model

is an approximation to the nonlinear model

The amount of error in this approximation is

We will show that this approximation error is constant in — or, does not vary with — the ‘th element of . Before we do so, we will prove that

First, we show that the ‘th column of is zero:

Next, denote the columns of by

Then, we compute

Now, for the approximation error:

What this means is that our state-space model is just as accurate near as it is near the equilibrium point .

Invariance of control. Suppose we implement linear state feedback with reference tracking:

where

for any . Let’s assume (for now) that is constant, and so is also constant. What will converge to in this case? Let’s find out. First, we define the error

and note that

Second, we derive an expression for the closed-loop system in terms of this error:

This means that

or equivalently that

so long as all eigenvalues of have negative real part — exactly the same conditions under which the closed-loop system without reference tracking would have been asymptotically stable.

Reference tracking with partial state feedback

Consider the system

where . Suppose we linearize this model about some equilibrium point to produce the state-space model

where

Suppose we design a controller

and observer

that would make the closed-loop system

asymptotically stable — that is, that would make

where

Suppose, as for reference tracking with full state feedback, that there is some index for which

This implies invariance of equilibrium point and invariance of error in approximation of the dynamic model, just like before. Suppose it is also true that, for some constant vector , the sensor model satisfies

Then, the following two more things are true:

Invariance of error in approximation of the sensor model. The linear model

is an approximation to the nonlinear model

The amount of error in this approximation is

We will show that this approximation error is constant in — or, does not vary with — the ‘th element of . First, we show that the ‘th column of is :

Next, denote the columns of by

Then, we compute

Now, for the approximation error:

What this means is that our state-space model is just as accurate near as it is near the equilibrium point .

Invariance of control. Suppose we implement linear state feedback with reference tracking:

where

for any . Let’s assume (for now) that is constant, and so is also constant. What will converge to in this case? Let’s find out. First, we define the state error

and the state estimate error

Second, we derive an expression for the closed-loop system in terms of these errors. Let’s start with the state error:

Now, for the state estimate error:

Putting these together, we have

This means that

or equivalently that

so long as all eigenvalues of and all eigenvalues of have negative real part — exactly the same conditions under which the closed-loop system without reference tracking would have been asymptotically stable.

Tracking more than one element of the state

Our discussion of reference tracking with full state feedback and with partial state feedback has assumed that we want to track desired values of exactly one element of the nonlinear state . All of this generalizes immediately to the case where we want to track desired values of more than one element of .

In particular, suppose there are two indices that satisfy

and, in the case of partial state feedback, that also satisfy

for constant vectors and . Then, choosing

would produce the same results that were derived previously.

Choosing the desired state to keep errors small

Our proof that tracking “works” relies largely on having shown that our state-space model is just as accurate near as it is near the equilibrium point . Equivalently, it relies on having shown that this model is just as accurate near as it is near .

Despite this fact, it is still important to keep the state error

small. The reason is that the input is proportional to the error — for full state feedback as

and for partial state feedback as

So, if error is large, the input may exceed bounds (e.g., limits on actuator torque). Since our state-space model does not include these bounds, it may be inaccurate when inputs are large.

As a consequence, it is important in practice to choose so that the state error

remains small. Here is one common way to do this, both in the case of full state feedback and partial state feedback.