Question

7.58 Consider a system with state matrices A = [ -2 1; 0 -3 ], B = [ 1; 1 ], C = [ 1 3 ]. (a) Use feedback of the form u(t) = -Kx(t) + Nr(t), where N is a nonzero scalar, to move the poles to -3 ± 3j. (b) Choose N so that if r is a constant, the system has zero steady-state error; that is, y(?) = r. (c) Show that if A changes to A + ?A, where ?A is an arbitrary 2 × 2 matrix, then your choice of N in part(b) will no longer make y(?) = r. Therefore, the system is not robust under changes to the system parameters in A. (d) The system steady-state error performance can be made robust by augmenting the system with an integrator and using unity feedback—that is, by setting x?_I = r ? y, where x_I is the state of the integrator. To see this, first use state feedback of the form u = ?Kx ? K_I x_I so that the poles of the augmented system are at -3, -2 ± j?3. (e) Show that the resulting system will yield y(?) = r no matter how the matrices A and B are changed, as long as the closed-loop system remains stable. (f) For part (d), use Matlab (Simulink) software to plot the time response of the system to a constant input. Draw Bode plots of the controller, as well as the sensitivity function (S) and the complementary sensitivity function (T).

          7.58 Consider a system with state matrices

A = [ -2 1; 0 -3 ], B = [ 1; 1 ], C = [ 1 3 ].

(a) Use feedback of the form u(t) = -Kx(t) + Nr(t), where N is a nonzero scalar, to move the poles to -3 ± 3j.
(b) Choose N so that if r is a constant, the system has zero steady-state error; that is, y(?) = r.
(c) Show that if A changes to A + ?A, where ?A is an arbitrary 2 × 2 matrix, then your choice of N in part(b) will no longer make y(?) = r. Therefore, the system is not robust under changes to the system parameters in A.
(d) The system steady-state error performance can be made robust by augmenting the system with an integrator and using unity feedback—that is,

by setting x?_I = r ? y, where x_I is the state of the integrator. To see this, first use state feedback of the form u = ?Kx ? K_I x_I so that the poles of the augmented system are at -3, -2 ± j?3.
(e) Show that the resulting system will yield y(?) = r no matter how the matrices A and B are changed, as long as the closed-loop system remains stable.
(f) For part (d), use Matlab (Simulink) software to plot the time response of the system to a constant input. Draw Bode plots of the controller, as well as the sensitivity function (S) and the complementary sensitivity function (T).

7.58 Consider a system with state matrices

A = [ -2 1; 0 -3 ], B = [ 1; 1 ], C = [ 1 3 ].

(a) Use feedback of the form u(t) = -Kx(t) + Nr(t), where N is a nonzero scalar, to move the poles to -3 ± 3j.
(b) Choose N so that if r is a constant, the system has zero steady-state error; that is, y(?) = r.
(c) Show that if A changes to A + ?A, where ?A is an arbitrary 2 × 2 matrix, then your choice of N in part(b) will no longer make y(?) = r. Therefore, the system is not robust under changes to the system parameters in A.
(d) The system steady-state error performance can be made robust by augmenting the system with an integrator and using unity feedback—that is,

by setting x?I = r ? y, where xI is the state of the integrator. To see this, first use state feedback of the form u = ?Kx ? KI xI so that the poles of the augmented system are at -3, -2 ± j?3.
(e) Show that the resulting system will yield y(?) = r no matter how the matrices A and B are changed, as long as the closed-loop system remains stable.
(f) For part (d), use Matlab (Simulink) software to plot the time response of the system to a constant input. Draw Bode plots of the controller, as well as the sensitivity function (S) and the complementary sensitivity function (T).

Added by Jonathan T.

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