% Nise, N.S.
% Control Systems Engineering, 6th ed.
% John Wiley & Sons, Hoboken, NJ, 07030
%
% Control Systems Engineering Toolbox Version 6.0
% Copyright 2011 by John Wiley & Sons, Inc.
%
% Chapter 5: Reduction of Multiple Subsystems
%
% (ch5p1) UFSS Pitch Control System: MATLAB can be used for block diagram
% reduction. Three methods are available: (1) Solution via Series, Parallel, &
% Feedback Commands, (2) Solution via Algebraic Operations, and (3) Solution via
% Append & Connect Commands. Let us look at each of these methods.
%
% (1) Solution via Series, Parallel, & Feedback Commands:
% The closed-loop transfer function is obtained using the following commands
% successively, where the arguments are LTI objects: series(G1,G2) for a cascade
% connection of G1(s) and G2(s); parallel(G1,G2) for a parallel connection of
% G1(s) and G2(s); feedback(G,H,sign) for a closed-loop connection with G(s)
% as the forward path, H(s) as the feedback, and sign is -1 for negative-feedback
% systems or +1 for positive-feedback systems. The sign is optional for
% negative-feedback systems.
%
% (2) Solution via Algebraic Operations:
% Another approach is to use arithmetic operations successively on LTI transfer
% functions as follows: G2*G1 for a cascade connection of G1(s) and G2(s); G1+G2
% for a parallel connection of G1(s) and G2(s); G/(1+G*H) for a closed-loop
% negative-feedback connection with G(s) as the forward path, and H(s) as the
% feedback; G/(1-G*H) for positive-feedback systems. When using division we follow
% with the function minreal(sys) to cancel common terms in the numerator
% and denominator.
%
% (3) Solution via Append & Connect Commands:
% The last method, which defines the topology of the system, may be used effectively
% for complicated systems. First, the subsystems are defined. Second, the subsystems
% are appended, or gathered, into a multiple-input/multiple-output system. Think of
% this system as a single system with an input for each of the subsystems and an
% output for each of the subsystems. Next, the external inputs and outputs are
% specified. Finally, the subsystems are interconnected. Let us elaborate on each
% of these steps.
%
% The subsystems are defined by creating LTI transfer functions for each. The
% subsystems are appended using the command G = append(G1,G2,G3,G4,.....Gn), where
% the Gi are the LTI transfer funtions of the subsystems and G is the appended system.
% Each subsystem is now identified by a number based upon its position in the append
% argument. For example, G3 is 3, based on the fact that it is the third subsystem in
% the append argument (not the fact that we write it as G3).
%
% Now that we have created an appended system, we form the arguments required to
% interconnect their inputs and outputs to form our system. The first step identifies
% which subsystems have the external input signal and which subsystems have the
% external output signal. For example, we use inputs = [1 5 6] and outputs = [3 4] to
% define the external inputs to be the inputs of subsystems 1, 5 and 6 and the external
% outputs to be the outputs of subsystems 3 and 4. For single-input/single-output
% systems, these definitions use scalar quantities. Thus inputs = 5, outputs = 8 define
% the input to subsystem 5 as the external input and the output of subsystem 8 as the
% external output.
%
% At this point we tell the program how all of the subsystems are interconnected.
% We form a Q matrix that has a row for each subsystem whose input comes from another
% subsystem's output. The first column contains the subsystem's number. Subsequent
% columns contain the numbers of the subsystems from which the inputs comes. Thus,
% a typical row might be as follows: [3 6 -7], or subsystem 3's input is formed from
% the sum of the output of subsystem 6 and the negative of the output of subsystem 7.
%
% Finally, all of the interconnection arguments are used in the
% connect(G,Q,inputs,outputs) command, where all of the arguments have been
% previously defined.
%
% Let us demonstrate the three methods for finding the total transfer function by
% looking at the back endpapers and finding the closed-loop transfer function of
% the pitch control loop for the UFSS with K1 = K2 = 1 (Johnson, 1980). The last
% method using append and connect requires that all subsystems be proper (the order
% of the numerator cannot be greater than the order of the denominator). The pitch
% rate sensor violates this requirement. Thus, for the third method, we perform some
% block diagram maneuvers by pushing the pitch rate sensor to the left past the
% summing junction and combining the resulting blocks with the pitch gain and the
% elevator actuator. These changes are reflected in the program. You should
% verify all computer results with hand calculations.
'(ch5p1) UFSS Pitch Control System'
'Solution via Series, Parallel, & Feedback Commands'
% Display labels.
numg1=[-1]; % Define numerator of G1(s).
deng1=[1]; % Define denominator of G1(s).
numg2=[0 2]; % Define numerator of G2(s).
deng2=[1 2]; % Define denominator of G2(s).
numg3=-0.125*[1 0.435]; % Define numerator of G3(s).
deng3=conv([1 1.23],[1 0.226 0.0169]);
% Define denominator of G3(s).
numh1=[-1 0]; % Define numerator of H1(s).
denh1=[0 1]; % Define denominator of H1(s).
G1=tf(numg1,deng1); % Create LTI transfer function,
% G1(s).
G2=tf(numg2,deng2); % Create LTI transfer function,
% G2(s).
G3=tf(numg3,deng3); % Create LTI transfer function,
% G3(s).
H1=tf(numh1,denh1); % Create LTI transfer function,
% H1(s).
G4=series(G2,G3); % Calculate product of elevator and
% vehicle dynamics.
G5=feedback(G4,H1); % Calculate closed-loop transfer
% function of inner loop.
Ge=series(G1,G5); % Multiply inner-loop transfer
% function and pitch gain.
'T(s) via Series, Parallel, & Feedback Commands'
% Display label.
T=feedback(Ge,1) % Find closed-loop transfer function.
'Solution via Algebraic Operations' % Display label.
clear % Clear session.
numg1=[-1]; % Define numerator of G1(s).
deng1=[1]; % Define denominator of G1(s).
numg2=[0 2]; % Define numerator of G2(s).
deng2=[1 2]; % Define denominator of G2(s).
numg3=-0.125*[1 0.435]; % Define numerator of G3(s).
deng3=conv([1 1.23],[1 0.226 0.0169]);
% Define denominator of G3(s).
numh1=[-1 0]; % Define numerator of H1(s).
denh1=[0 1]; % Define denominator of H1(s).
G1=tf(numg1,deng1); % Create LTI transfer function,
% G1(s).
G2=tf(numg2,deng2); % Create LTI transfer function,
% G2(s).
G3=tf(numg3,deng3); % Create LTI transfer function,
% G3(s).
H1=tf(numh1,denh1); % Create LTI transfer function,
% H1(s).
G4=G3*G2; % Calculate product of elevator and
% vehicle dynamics.
G5=G4/(1+G4*H1); % Calculate closed-loop transfer
% function of inner loop.
G5=minreal(G5); % Cancel common terms.
Ge=G5*G1; % Multiply inner-loop transfer
% functions.
'T(s) via Algebraic Operations' % Display label.
T=Ge/(1+Ge); % Find closed-loop transfer function.
T=minreal(T) % Cancel common terms.
'Solution via Append & Connect Commands'
% Display label.
'G1(s) = (-K1)*(1/(-K2s)) = 1/s' % Display label.
numg1=[1]; % Define numerator of G1(s).
deng1=[1 0]; % Define denominator of G1(s).
G1=tf(numg1,deng1) % Create LTI transfer function,
% G1(s) = pitch gain*(1/pitch rate sensor).
'G2(s) = (-K2s)*(2/(s+2)' % Display label.
numg2=[-2 0]; % Define numerator of G2(s).
deng2=[1 2]; % Define denominator of G2(s).
G2=tf(numg2,deng2) % Create LTI transfer function,
% G2(s) = pitch rate sensor* vehicle dynamics.
'G3(s) = -0.125(s+0.435)/((s+1.23)(s^2+0.226s+0.0169))'
% Display label.
numg3=-0.125*[1 0.435]; % Define numerator of G3(s).
deng3=conv([1 1.23],[1 0.226 0.0169]);
% Define denominator of G3(s).
G3=tf(numg3,deng3) % Create LTI transfer function,
% G3(s) = vehicle dynamics.
System=append(G1,G2,G3); % Gather all subsystems
input=1; % Input is at first subsystem, G1(s).
output=3; % Output is output of third subsystem, G3(s).
Q=[1 -3 0 % Subsystem 1, G1(s), gets its input from the
% negative of the output of subsystem 3, G3(s).
2 1 -3 % Subsystem 2, G2(s), gets its input from subsystem
% 1, G1(s), and the negative of the output of
% subsystem 3, G3(s).
3 2 0]; % Subsystem 3, G3(s), gets its input from subsystem
% 2, G2(s).
T=connect(System,Q,input,output); % Connect the subsystems.
'T(s) via Append & Connect Commands'% Display label.
T=tf(T); % Create LTI closed-loop transfer function,
T=minreal(T) % Cancel common terms.