Sensitivity functions of plant-controller feedback loop
computes the multivariable sensitivity, complementary
sensitivity, and open-loop transfer functions of the closed-loop system consisting
of the controller
loops = loopsens(
C in negative feedback with the plant
P. To compute the sensitivity functions for the system with positive
Single Input, Single Output (SISO) Loop Sensitivities
Consider PI controller for a dominantly first-order plant, with the closed-loop bandwidth of 2.5 rads/sec. Since the problem is SISO, all gains are the same at input and output.
gamma = 2; tau = 1.5; taufast = 0.1; P = tf(gamma,[tau 1])*tf(1,[taufast 1]); tauclp = 0.4; xiclp = 0.8; wnclp = 1/(tauclp*xiclp); KP = (2*xiclp*wnclp*tau - 1)/gamma; KI = wnclp^2*tau/gamma; C = tf([KP KI],[1 0]);
Form the closed-loop (and open-loop) systems with
loopsens, and plot Bode plots of the sensitivity functions at the plant input.
loops = loopsens(P,C); bode(loops.Si,'r',loops.Ti,'b',loops.Li,'g') legend('Sensitivity','Complementary Sensitivity','Loop Transfer')
Finally, compare the open-loop plant gain to the closed-loop value of
Multi Input, Multi Output (MIMO) Loop Sensitivities
Consider an integral controller for a constant-gain, 2-input, 2-output plant. For purposes of illustration, the controller is designed via inversion, with different bandwidths in each rotated channel.
P = ss([2 3;-1 1]); BW = diag([2 5]); [U,S,V] = svd(P.d); % get SVD of Plant Gain Csvd = V*inv(S)*BW*tf(1,[1 0])*U'; % inversion based on SVD loops = loopsens(P,Csvd); bode(loops.So,'g',loops.To,'r',logspace(-1,3,120)) title('Output Sensitivity (green), Output Complementary Sensitivity (red)');
P — Plant
dynamic system model | control design block | matrix
Plant, specified as a dynamic system model, control design block, or static gain
P can be SISO or MIMO, as long as
has the same number of inputs and outputs.
P can be continuous time or discrete time. If
P is a generalized model (such as
loopsens uses the current or
nominal value of all control design blocks in
C — Controller
dynamic system model | control design block | constant matrix
Controller, specified as a dynamic system model, control design block, or static
gain matrix. The controller can be any of the model types that
can be, as long as
P*C has the same number of inputs and outputs.
loopsens computes the sensitivity
functions assuming a negative-feedback closed-loop system. To compute the
sensitivity functions for the system with positive feedback, use
loopsens command assumes one-degree-of-freedom control
architecture. If you have a two-degree-of-freedom architecture, then construct
C to include only the compensator in the feedback path, not any
loops — Sensitivity functions
Sensitivity functions of the feedback loop
feedback(P,C), returned in a structure having the fields shown in
the table below. The sensitivity functions are returned as state-space
ss) models of the same I/O dimensions as
C is a frequency-response-data model,
then the sensitivity functions are
Input-to-plant sensitivity function.
Input-to-plant complementary sensitivity function.
Input-to-plant loop transfer function.
Output-to-plant sensitivity function.
Output-to-plant complementary sensitivity function.
Output-to-plant loop transfer function.
Plant times input-to-plant sensitivity function.
Compensator times output-to-plant sensitivity function.
Poles of the closed loop
1 if nominal closed loop is stable, 0 otherwise. If either
The closed-loop interconnection structure shown below defines the
input/output sensitivity, complementary sensitivity, and loop transfer functions. The
structure includes multivariable systems in which
C are MIMO systems.
The following table gives the values of the input and output sensitivity functions for this control structure.
Input sensitivity Si (closed-loop transfer function from d1 to e1)
|Si = (I + CP)–1|
Input complementary sensitivity Ti (closed-loop transfer function from d1 to e2)
|Ti = CP(I + CP)–1|
Output sensitivity So (closed-loop transfer function from d2 to e2)
|So = (I + PC)–1|
Output complementary sensitivity To (closed-loop transfer function from d2 to e4)
|To = PC(I + PC)–1|
Input loop transfer function Li
|Li = CP|
Output loop transfer function Lo
|Lo = PC|