loopsens

Sensitivity functions of plant-controller feedback loop

Syntax

loops = loopsens(P,C)

Description

loops = loopsens(P,C) computes the multivariable sensitivity, complementary sensitivity, and open-loop transfer functions of the closed-loop system consisting of the controller C in negative feedback with the plant P. To compute the sensitivity functions for the system with positive feedback, use loopsens(P,-C).

Examples

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 PSi.

bodemag(P,'r',loops.PSi,'b')
legend('Plant','Sensitivity*Plant')

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)');

Input Arguments

P — Plant

dynamic system model | control design block | matrix

Plant, specified as a dynamic system model, control design block, or static gain matrix. P can be SISO or MIMO, as long as P*C has the same number of inputs and outputs.

P can be continuous time or discrete time. If P is a generalized model (such as genss or uss) then loopsens uses the current or nominal value of all control design blocks in P.

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 P 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(P,-C).

The 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 reference channels.

Output Arguments

loops — Sensitivity functions

structure

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*P. If P or C is a frequency-response-data model, then the sensitivity functions are frd models.

Field	Description
Si	Input-to-plant sensitivity function.
Ti	Input-to-plant complementary sensitivity function.
Li	Input-to-plant loop transfer function.
So	Output-to-plant sensitivity function.
To	Output-to-plant complementary sensitivity function.
Lo	Output-to-plant loop transfer function.
PSi	Plant times input-to-plant sensitivity function.
CSo	Compensator times output-to-plant sensitivity function.
Poles	Poles of the closed loop feedback(P,C). If either P or C is a frequency-response-data model, then this field is NaN.
Stable	1 if nominal closed loop is stable, 0 otherwise. If either P or C is a frequency-response-data model, then this field is NaN.

More About

Sensitivity functions

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 P and C are MIMO systems.

The following table gives the values of the input and output sensitivity functions for this control structure.

Description	Equation
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