PWAAnalNon.m

clear model pwasys
echo on;
clc
% *************************************************************************
%  ANALYZING A NONLINEAR MODEL FOR STABILITY (GLOBAL LYAPUNOV APPROACH)
% *************************************************************************
% 
% PWATOOL can analyze a nonlinear model, approximated with PWADIs, to 
% check if it is stable at a given point xcl. 
%
% Please hit number 1 and 3 in the main menu of PWATOOL to learn about
% PWADIs.
%
% A nonlinear model which is bounded within a lower and upper envelope is
% stable at a given point xcl if both of the envelopes are stable at that
% given point xcl. 
%
% Reference: B. Samadi and L. Rodrigues. Extension of local linear 
%            controllers to global piecewise affine controllers for uncertain
%            non-linear systems. International Journal of Systems Science,
%            39(9):867-879, 2008.
%
% We explain how a nonlinear system is analyzed for stability.
%
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%              STABILITY ANALYSIS: A GLOBAL LYAPUNOV APPROACH
%
%
% Let a nonlinear model be approximated by PWADIs as follows
%
%  sigma_j:  dx/dt=A{i,j} x+ a{i,j}+ B{i,j} u        j=1,2 and x is in region R_i  (*)
%                                                   
% where regions R_i are defined to be 
%
%            R_i={x | E{i} x + e{i} >= 0 }.    i=1,2,...,NR
%
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%
% (OPTIONAL) Also, let the controller output in each region R_i be
%
%            u=K{i} x +k{i}              if x is in R_i.
%
% NOTE: If controller gains K{i} and k{i} are not defined, the system is
% considered open-loop and controller gains K{i} and k{i} will be zero.
%
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% PWATOOL analyzes the system by searching for a global quadratic Lyapunov
% function for the system; PWATOOL searches for a positive definite matrix
% 'Q' that satisfies the following conditions for a constant positive
% scalar 'alpha'. 
%
%
%    1)  V(x)  = x' Q x         >0                 for x defined by sigma_j
%    2)  dV/dt = d(x' Q x)/dt   <-alpha * V(x)     for x defined by sigma_j
%
% Conditions (1) and (2) amount to the following LMIs 
%
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% Let Z{i} be matrices of proper size with non-negative elements which are
% to be found. Then, the following LMIs are obtained for i=1,2,...,NR and
% j=1,2.
%
%
% (1) Q>0, Z{i}(:) >0
%
% (2) In regions R_i which contain the equilibrium point xcl
%
%     Q*(A{i,j}+B{i,j}*K{i})+(A{i,j}+B{i,j}*K{i})'*Q+alpha*Q <0
%
%
% (3) In regions R_i which do NOT contain the equilibrium point xcl
%
%     [Q*(A{i,j}+B{i,j}*K{i})+(A{i,j}+B{i,j}*K{i})'*Q+alpha*Q+E{i}'*Z{i}*E{i}        (Q*a{i,j}+B{i,j}*k{i})+E{i}'*Z{i}*e{i};
%     (Q*(a{i,j}+B{i,j}*k{i})+E{i}'*Z{i}*e{i})'                                                             e{i}'*Z{i}*e{i}]  < 0
%
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% NOTE 1: PWATOOL uses Yalmip to solve above LMIs. A solution, then, may or
%         may not be found by the solver. Since the previous LMIs are
%         sufficient conditions for stability, we can conclude the system is
%         stable only if we find a Q>0 that satisfies above LMIs. In case a
%         matrix Q>0 is not found we may not be able to comment on the 
%         stability of the system. 
%
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% NOTE 2: The regions can also be approximated with ellipsoids which in the
%         case of slab regions they will be degenerate ellipsoids. This 
%         yields to a different set of LMIs which again are sufficient
%         conditions for stability:
%
% References: (1) L. Rodrigues and S. Boyd. Piecewise-affine state feedback 
%             for piecewise-affine slab systems using convex optimization. 
%             Systems and Control Letters, 54:835-853, 200
%
%             (2) A. Hassibi and S. Boyd. Quadratic stabilization and
%             control of piecewise-linear systems. Proceedings of the
%             American Control Conference, 6:3659-3664, 1998
% 
%             (3) L. Vandenberghe, S. Boyd, and S.-P. Wu. Determinant
%             maximization with linear matrix inequality constraints. SIAM
%             Journal on Matrix Analysis and Applications,9(2):499-533,1998
%
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% To obtain LMIs with ellipsoidal approximation, let 'Elip_i' be the
% ellipsoid that approximates region R_i and is described as follows:
% 
%                Elip_i={x | || EL{i} x + eL{i}|| < 1}
% 
% Also let miu{i} be the negative scalars which are to be found. Then, we
% should satisfy the following LMis.
%
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%
% For i=1,2,...,NR and j=1,2 
%
% (1) Q>0 , miu{i} <0
%
% (2) In regions R_i which contain the equilibrium point xcl
%
%     Q*(A{i,j}+B{i,j}*K{i})+(A{i,j}+B{i,j}*K{i})'*Q+alpha*Q <0
%
%
% (3) In regions R_i which do NOT contain the equilibrium point xcl
%
%     [Q*(A{i,j}+B{i,j}*K{i})+(A{i,j}+B{i,j}*K{i})'*Q+alpha*Q+miu{i}*EL{i}'*EL{i}        Q*(a{i,j}+B{i,j}*k{i})+miu{i}*EL{i}'*eL{i};
%     (Q*(a{i,j}+B{i,j}*k{i})+miu{i}*EL{i}'*eL{i})'                                                      -miu{i}*(1-eL{i}'*eL{i});]<0
%
%
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%
%          STABILITY ANALYSIS OF A NONLINEAR MODEL USING PWATOOL
%
% We use the inverted pendulum example (L. Rodrigues and J.P. How, 2002)
% to show how PWATOOL analyzes a nonlinear model.
%
% |-----------------------------------------------------------------------|
% | STEP 0: APPROXIMATING THE NONLINEAR MODEL WITH PWADI
% |-----------------------------------------------------------------------|
%
% Please hit number 3 in the main menu of PWATOOL for "CREATING A PWADI
% MODEL"
%
% For the inverted pendulum example we load the variable 'inv_pend_pwadi'
% 
         load inv_pend_pwadi
%
% by doing so, a variable called 'pwainc' will contain the PWADI
% approximation of the inverted pendulum.
% 
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% |-----------------------------------------------------------------------|
% | STEP 1: CHECKING THE STABILITY OF A PWADI model
% |-----------------------------------------------------------------------|
%
% To check the stability of the the PWADI model 'pwainc' at the equilibrium
% point 'xcl', type 
%
%           pwaanalysis(pwainc, setting);         
%
% where setting is a structured variable with the following fields:
%
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%
% setting.Lyapunov: shows whether 'global' or 'pwq' (piecewise quadratic)
%                   Lyapunov function should be used in the analysis
%
% For the inverted pendulum example we set 
%
       setting.Lyapunov='global';
%
% This means that PWATOOL uses only global Lyapunov function in analysis.
%
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% setting.ApxMeth:  shows whether 'ellipsoidal' or 'quadratic' curve
%                   method should be used to approximate the regions.
%
% For the inverted pendulum example we don't set this field so that it
% takes the default value {'quadratic', 'ellipsoidal'}.
%
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%
% setting.alpha :  specifies the decay rate in finding the Lyapunov function.
%                  Generally, bigger values of alpha make the convergence 
%                  harder.
%
% For the inverted pendulum example, we set
%
       setting.alpha=.1;
%
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%
% setting.xcl   :  shows the desired equilibrium point which user sets.
%                  If it is not set by the user, PWATOOL uses the
%                  pwainc.xcl (if exists) as the equilibrium point.
%                  PWATOOL issues an error and stops running if xcl is not
%                  defined or if at xcl we cannot satisfy the equilibrium
%                  point equations analytically.
%
% For the inverted pendulum example, we set
%
       setting.xcl=[0 pi 0 0]';
%
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% We call 'pwaanalysis' to analyze the stability of the model at xcl
%
pause; %strike any key to continue
%
        pwaanalysis(pwainc, setting);
 
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% |-----------------------------------------------------------------------|
% | STEP 2: RESULTS AND DUISCUSSIONS
% |-----------------------------------------------------------------------|
%
% We can only say the system is stable if a positive definite matrix Q is 
% found; otherwise we cannot comment on the stability of the nonlinear 
% system.
%
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%
% In the inverted pendulum example the message
% "I could not verify if the open-loop nonlinear system is stable at xcl."
% implies that LMIs have not been able to find a matrix Q>0 that solves
% the LMIs. Therefore, xcl may or may not be an equilibrium point for the 
% system.
%
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% |-----------------------------------------------------------------------|
% | STEP 2: FORMATS OF CALLING PWAANALYSIS
% |-----------------------------------------------------------------------|
%
% pwaanalysis can also be called by three inputs as follows.
%
%        pwaanalysis(pwainc, setting, gain);
% 
% In this format, 'gain' should be the control gain Kbar (please hit
% number 1 in the main menu of PWATOOL to learn about Kbar).
%
% We use a stabilizing gain Kbar to check this option.
%
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%
       load Kbar_pendul
%
% Now, let's analyze the inverted pendulum around xcl one more time; this
% time the loop is closed through the gain Kbar
%
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%
        pwaanalysis(pwainc, setting, Kbar_pendul);
% The message "The closed-loop PWA system is stable at xcl." shows PWATOOL
% has been successful in finding a matrix Q>0 for solving the LIMs and
% hance the closed-loop system is stable around xcl.
%
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% 'pwanalysis' also produces an output which is 1, whenever it has verified
% the stability of the system at a given point xcl. Therefore, the
% following formats in calling pwaanalysis are valid:
%
%  Y = pwaanalysis(pwainc, setting); 
%  Y = pwaanalysis(pwainc, setting, gain); 
%
pause; %strike any key to return to the main menu
echo off