Robust Control Toolbox™

Sensitivity functions of plant-controller feedback loop

Syntax

• loops = loopsens(P,C) 
  

Description

loops = loopsens(P,C) creates a struct, loops, whose fields contain the multivariable sensitivity, complementary and open-loop transfer functions. The closed-loop system consists of the controller C in negative feedback with the plant P. C should only be the compensator in the feedback path, not any reference channels, if it is a 2-dof controller as seen in the figure below. The plant and compensator P and C can be constant matrices, double, lti objects, frd/ss/tf/zpk, or uncertain objects umat/ufrd/uss.

The loops returned variable is a structure with fields:

Field	Description
Poles	Closed-loop poles. NaN for frd/ufrd objects
Stable	1 if nominal closed loop is stable, 0 otherwise. NaN for frd/ufrd objects
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

The multivariable closed-loop interconnection structure, shown below, defines the input/output sensitivity, complementary sensitivity, and loop transfer functions.

Description	Equation
Input sensitivity ()	(I+CP)-1
Input complementary sensitivity ()	CP(I+CP)-1
Output sensitivity ()	(I+PC)-1
Output complementary sensitivity ()	PC(I+PC)-1
Input loop transfer function	CP
Output loop transfer function	PC

Example

Single Input, Single Output (SISO)

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') 
  
  
  

Multi Input, Multi Output (MIMO)

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)) 
  
  
  

See Also
loopmargin  Performs a comprehensive analysis of feedback loop

robuststab  Calculate stability margins of uncertain systems

wcsens      Calculate worst-case sensitivities for feedback loop

wcmargin    Calculate worst-case margins for feedback loop

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