Example of a state-feedback control system

For a nd-order plant, described by

and

with the disturbance

and the initial condition

a state-feedback controller with and without observer is to be designed so that the poles of the closed-loop system are . The eigenvalues of the observer are given as .

The plant is unstable and has the eigenvalues at 1.0 and -0.5. The desired characteristic polynomial with the zeros as the given poles is from Eq. (13.29)

The controllability matrix is

and its inverse

The last row of the inverse is

With this data, the feedback vector using Ackermann's formula Eq. (13.51) is

The feedforward gain according to Eq. (13.19) is

For the observer design the dual system

is used. The desired characteristic polynomial with the zeros as the given eigenvalues of the observer is from Eq. (13.29)

The controllability matrix of the dual system is

and its inverse

The last row of the inverse is

With this data, the feedback vector using Ackermann's formula Eq. (13.51) is

The observer system matrix is

and, finally, the following observer equation is obtained:

Figure 13.5 shows the time responses of the state-feedback control system with and without observer. The responses are divided into three periods. The control system is started at s and until s the decay of the observing error is demonstrated. At s one can see the behaviour of the system following the change in the set point . Then from s the disturbance is active and one can see the disturbance behaviour. The observer is started at s with zero initial conditions, whereas the plant state is not zero. The estimated states in Figure 13.5c and 13.5d converge asymptotically to the real states and the value of is reached by the controlled variable as shown in Figure 13.5b. The control system follows the set-point change from from 1 to 4 applied at s. However, due to the proportional behaviour of the open-loop system the controlled variable shows a large steady-state error after the disturbance is applied at s.

The control structure used does not show the desired static behaviour with a vanishing control error under disturbances. Therefore, a state-feedback controller with an integrator is required to cope with the disturbance problem. For the modified control structure according to Figure 13.2 one has to design a state-feedback controller for the extended system according to Eqs. (13.24) and (13.25):

The initial condition is now

For the eigenvalues of the closed-loop system a third value must be given, as the additional state of the integrator is introduced. For simplicity the choice is taken and the desired characteristic polynomial is

The controllability matrix is

and its inverse

The last row of the inverse is

With this data, the feedback vector using Ackermann's formula Eq. (13.51) is

From this the feedback vector of the state-feedback is

and the feedforward gain

As the state of the integrator is known, the same observer can be used as in the case without integrator.

Figure 13.6 shows the time responses of the same experiment as in Figure 13.5. The responses in the first and second period are similar, the amplitudes are somewhat larger and due to the additional integrator the response on set-point changes is slower, but there is no steady-state error on disturbances. This is for the case with and without observer, though the observer shows a steady-state error when the disturbance is active.

Figure 13.5: Time responses of the state-feedback control system without integrator,
(a) manipulated variable ,
(b) controlled value , set point and disturbance ,
(c) state and its estimate ,
(d) state and its estimate

Figure 13.6: Time responses of the state-feedback control system with integrator,
(a) manipulated variable ,
(b) controlled value , set point and disturbance ,
(c) state and its estimate ,
(d) state and its estimate

Demonstration Example 13.1   A virtual experiment using state-feedback control of a tank system

Demonstration Example 13.2   A virtual experiment using state-feedback control of a VTOL