9.1.1 Ackermann's Formula
According to Ackermann's formula, the feedback matrix K that places the poles of a single-input system into the positions can be found as
is the unit vector of length ,
is the controllability matrix, and
is the characteristic polynomial of matrix . Here is the identity matrix, and the coefficients are such that , , , are the roots of the polynomial:
The function StateFeedbackGains with the option Method Ackermann implements this algorithm.
State feedback design using Ackermann's formula.
The Ackermann method, besides being useful for single-input systems, may also find application if an attempt is to be made to control a multi-input system through a single input. The option ControlInput Automatic is used in such cases to find the "best" control using the condition number of the corresponding controllability matrix as a criterion. It is also possible to specify the control input explicitly.
Option specific to the Ackermann method.
Figure 9.2. F-8 aircraft in flight. Photograph by Dryden Flight Research Center, NASA.
As an example we consider an approximate model of the lateral dynamics of an F-8 aircraft (Figure 9.2) linearized about a particular set of flight conditions and reproduced after Brogan (1991). The state and input vectors in the model are
where , , , and are the roll and yaw rates and the sideslip and roll angles, respectively, and and are the aileron and rudder deflections. Figure 9.3 introduces the nomenclature.
Figure 9.3. Aircraft schematic.
This is the state-space model of the aircraft.
Here are the closed-loop poles we wish the system to have.
StateFeedbackGains may be asked to determine whether it is possible to control the aircraft using only one of the inputs if, say, a malfunction prevents manipulation of the other. If such input exists, the feedback gain matrix will contain a nonzero row corresponding to this input.
StateFeedbackGains finds that the system is better controlled through the second input (i.e., the rudder deflection) and returns the corresponding feedback gains.
The attempt to control the aircraft from only the first input fails, and we are presented with messages suggesting that the trouble possibly stems from the system being uncontrollable.
Indeed, the system is not controllable from its first input; in other words, the aircraft cannot be controlled by only the aileron deflections (at least not within the linearized model).