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loopsens

Sensitivity functions of plant-controller feedback loop

Description

example

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

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Consider PI controller for a dominantly 1st-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 using the gains at the plant input.

loops = loopsens(P,C); 
bode(loops.Si,'r',loops.Ti,'b',loops.Li,'g')

Finally, compare the open-loop plant gain to the closed-loop value of PSi.

bodemag(P,'r',loops.PSi,'b')

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

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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.

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

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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

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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.

Control structure feedback(P,C) for computing sensitivity functions. The structure includes disturbance inputs d1 (plant input) and d2 (controller input), and measurement outputs e1 (plant input), e2 (controller output), e3 (controller input), and e4 (plant output).

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
Introduced before R2006a