We saw in the previous subsection how
can play the role of a disturbance input, in contrast with the standard LQR setting of Chapter 6 where, as in the rest of this book, it plays the role of a control input. Now, let us make the situation more interesting by allowing both types of inputs to be present in the system, the task of the control being to stabilize the system and attenuate the unknown disturbance in some sense. Such control problems fall into the general framework of *robust control theory*, which deals with control design methods for providing a desired behavior in the presence of uncertainty. (We can also think of the control and the disturbance as two opposing players in a differential game.) In the specific problem considered in this subsection, the level of disturbance attenuation will be measured by the
gain (or
norm) of the closed-loop system.

The control problem concerns itself with a system of the form

where is the control input, is the disturbance input, is the measured output (the quantity available for feedback), and is the controlled output (the quantity to be regulated); for simplicity, we assume here that there are no feedthrough terms from and to and . The corresponding feedback control diagram is shown in Figure 7.4. We will first consider the simpler case of

The control objective is to stabilize the internal dynamics and attenuate in the sense. More specifically, we want to design the feedback gain matrix so that the following two properties hold:

- The closed-loop system matrix is Hurwitz.
- The gain of the closed-loop system from to (or, what is the same, the norm of the closed-loop transfer matrix ) does not exceed a prespecified value .

Note that this is not an optimal control problem because we are not asking to minimize the gain (although ideally of course we would like to be as small as possible). Controls solving problems of this kind are known as

It follows from the results of the previous subsection--applied with the change of notation from to , respectively--that to have property 2 (the bound on the gain) it is sufficient to find a positive semidefinite solution of the Riccati inequality

To also guarantee property 1 (internal stability of the closed-loop system) requires slightly stronger conditions: needs to be positive definite and the inequality in (7.27) needs to be strict. The latter condition can be encoded via the Riccati equation

where and . (This implies which is the well-known Lyapunov condition for to be a Hurwitz matrix.) Next, we need to convert (7.28) into a condition that is verifiable in terms of the original open-loop system data. Introducing a matrix as another design parameter (in addition to and ), suppose that there exists a solution to the Riccati equation

Then, if we let , a straightforward calculation shows that the feedback law enforces (7.28) and thus achieves both of our control objectives. Conversely, it can be shown that if the system is stabilizable with an gain less than , then (7.29) is solvable for .

The general case--when the full state is not measured and the controller is a dynamic output feedback--is more interesting and more complicated. Without going into details, we mention that a complete solution to this problem, in the form of necessary and sufficient conditions for the existence of a controller achieving an gain less than , is available and consists of the following ingredients:

- Finding a solution
of a Riccati equation from
the state feedback case.
- Finding a solution
of another Riccati equation obtained from the first one by the substitutions
and
.
- Checking that the largest singular value of the product
is less than
.

These elegant conditions in terms of two coupled Riccati equations yield a controller that can be interpreted as a state feedback law combined with an estimator (observer).