5.2 Implementation of DRG on the Distillation Process
5.2.1 simulation results of diagonal method
From Figure 5.3(a), it can be seen that the outputs satisfy the constraints, which are shown by yellow and purple dashed lines, for both large and small condition number cases. The results in Figure 5.3(b) indicate that the difference between r(t) and u(t) are smaller in the system with smaller condition number. Figure 5.4 confirms that v(t) is always below r0(t), which is a feature of scalar RGs used in the DRG.
0 10 20 30 40 50 60 70 80 90 100 time 0 0.2 0.4 0.6 0.8 1 1.2 y 0 10 20 30 40 50 60 70 80 90 100 time 0 0.25 0.5 0.75 1 1.25 1.5 u and r
Figure 5.5: Simulation results of identity method. Top plot (a) shows the outputs and the bottom plot (b) is r vs u
5.2.2
simulation results of identity method
In Figure 5.5(a), the constraints are presented by yellow and purple dashes lines. It is clear that the outputs are within the constraints. The results in Figure 5.5(b) show that the gap between r(t) and u(t) is smaller in the system with smaller condition number. Note that u(t) is equal to r(t) during the initial time steps for both small and large condition number cases. The reason is that, during these time steps, no violation is detected for both scalar RGs inside the DRG, and from Figure 5.6, we can see that v(t) is equal to r0(t), which leads to u(t) equals to r(t).
Figure 5.6: Comparison of r0 and v in DRG after applying the identity method. Top plot (a) is for large condition number and bottom plot (b) is for small condition number.
5.2.3
Comparison Between DRG and VRG
In this section, we will compare DRG with VRG in steady-state, and this comparison can help us have a better understanding about Theorem 1.
Figure 5.7 compares u computed by DRG and u computed by VRG in steady state, where the left and right plots are for system with large and small condition number, respectively. r = (1, 1) is the set-point. In both figures, the contour lines represent the level sets of the cost functions for VRG (distance from r). From Figure
Figure 5.7: Comparison between v for DRG and v for VRG in steady state. Left (a) and right (b) plots corresponds to the system with large and small condition number, respectively.
5.7(a), it can be seen that for VRG, the optimal solution is the closest point to r that belongs to the steady-state constraint-admissible set (shaded region). As for DRG, it finds the solution at a different location, which is a sub-optimal solutions with respect to the VRG cost function. Figure 5.7(b) shows that DRG finds the solution at a vertex in the admissible set (as alluded to in Section 4.1), because both r1 and
r2 are inadmissible. In this case, the vertex happens to be the closest point to r, so VRG and DRG find the same solution. In sum, the solutions may or may not be the same, depending on the specific situation.
In this chapter, we introduced the distillation process, and compared the simula- tion results for DRG in both small and large condition number cases. The results show that diagonal and identity DRG both perform better for the systems with smaller con- dition number in the sense of tracking performance. Finally, we compared VRG and DRG in steady-state. The results illustrate that in this specific example, VRG and DRG find the same optimal solution for smaller condition number system. However,
for large condition number case, DRG finds the sub-optimal solution compared to VRG.
Chapter 6
Conclusion and future research
6.1
Conclusion
Reference governor (RG) is an add-on control strategy for constraint management of closed-loop systems. There are several types of reference governors, such as scalar RG (SRG) and vector RG (VRG). SRG has computational benefits but has performance limitation on MIMO systems. Meanwhile, VRG performs better in MIMO systems but requires higher computational effort.
In this work, a method for constraint management of coupled square MIMO systems was studied. The method is referred to as the Decoupled Reference Governor (DRG), which maintains the computational advantages of SRG and, at the same time, performs better than SRG (comparable to VRG). DRG is based on decoupling the input-output dynamics, followed by application of SRGs to each decoupled channel.
In this thesis, we first presented the DRG formulation with two different decou- pling techniques and demonstrated the applicability of the method as a function of the singular values and the condition number of the system. Secondly, we presented
steady state analyses of the DRG and compared the computation time of DRG, SRG, and VRG. It was shown that DRG can run faster than VRG by a factor of 2500 and is similar to SRG in terms of execution time. Finally, a distillation process was used as an illustrative example to compare DRG and VRG with small and large condition numbers.
6.2
Future Research
Future work will explore modifications to DRG to ensure that the inputs to the closed- loop system (i.e. u in Figure 1.5) remain below the references (i.e. r). Moreover, we will explore ways of relaxing the limitations of DRG, specifically, for cases where G(z) is unstable or have non-minimum phase zeros.
To make DRG be more applicable for practical systems, we will also explore the inclusion of external disturbances, plant model mismatch, and observers in the DRG design.
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Appendix A
Matlab
Code
for
Distillation
Process
In the distillation process example, we test two cases, one with large condition number, another with small condition number. We will only illustrate the code for large condition number system, the code for small condition number case is similar to that of large condition number case.
% Assume we know the state-space of F system: A_f, B_f, C_f, D_f, % State-space of F^(-1): A_finv, B_finv, C_finv, D_finv and
% State-space of two decoupled systems: A1,B1, C1, D1 % and A2, B2, C2, D2
% Sample time is 10ms. Simulation length is 10,000 timesteps. % We assume all states are known.
time=0.01 * [0:10000]'; r1 = 1*stepfun(time,0.2); r2 = 1*stepfun(time,0.2); r = [r1, r2];
% build O_inf for two decoupled system
[Hx1, Hv1, h1, jstar1] = Oinf_builder(A1, B1, C1, D1 ,S1, s1, e1);
[Hx2, Hv2, h2, jstar2] = Oinf_builder(A2, B2, C2, D2 ,S2, s2, e2);
% implement SRG to decoupled system x_finv=zeros(size(A_finv,1),1);
v1 = zeros(10000,1);x1 = zeros(size(Hx1,2),1); v2 = zeros(10000,1);x2 = zeros(size(Hx2,2),1); for t=1:10000
% get r1' and r2'
xfinv_later = A_finv*x_finv + B_finv*r(t,:)'; r_new(t+1,:) = C_finv * x_finv + D_finv * r(t,:)'; x_finv=xfinv_later;
r_new(t+1,1) = r1_new(t+1); r_new(t+1,2) = r2_new(t+1);
% find kapa through Hx, Hv, h, jstar, %A, B, C, D, r' and x
kapa1 = findkapa(Hx1, Hv1, h1, jstar1, r_new(t+1), x1, v1(t)); kapa2 = findkapa(Hx2, Hv2, h2, jstar2,
r_new(t+2), x2, v2(t)); % using the equation: v(t+1) = v(t) + % kapa * (r(t+1)-v(t)) to
% get governed input to closed-loop system v1(t+1) = v1(t) + kapa1*(r_new(t+1,1)-v1(t)); v2(t+1) = v2(t) + kapa2*(r_new(t+1,2)-v2(t)); % observer code (if needed) goes here
x1_later = A1 * x1 + B1* v1(t+1); y1(t+1,:) = C1 * x1 + D1 * v1(t+1); x1=x1_later; x2_later = A2 * x2 + B2* v2(t+1); y2(t+1,:) = C2 * x2 + D2* v2(t+1); x2 = x2_later; end
% get the inputs to the closed-loop system
x_f=zeros(size(A_f,1),1);xf_later=zeros(size(A_f,1),1); for t=1:10000
xf_later = A_f*x_f + B_f*v; u(t,:) = C_f*x_f + D_f*v; x_f = xf_later;
end
%--- function2: build O_inf ---% function [Hx, Hv, h,jstar] = Oinf_builder(A,B,C,D,S,s,e) nu = size(A,1)
I=eye(nu);
% first two rows of Hx Hx=[0*S*C;S*C];
% first two rows of Hv
Hv=[S*(C*(I-A)^(-1)*B+D);S*D]; % first two rows of h
h=[(1-e)*s;s]; a=[Hx,Hv]; b=h;
% we assume upper bound of 400 timesteps for m=1:400 Hx=[Hx;S*C*A^(m)]; Hv=[Hv;S*(C*(I-A^(m))*(I-A)^(-1)*B+D)]; h=[h;s]; num1_1 = size(a,1); num2_1 = size(b,1); fun=[-(Hx(num1_1+1,:)),-(Hv(num1_1+1,:))]; [x,FVAL] = linprog(fun,[a;-fun],[b;h(num2_1+1)+100]); if ((-FVAL)<=h(num2_1+1)) break; else a=[a;Hx(num1_1+1,:),Hv(num1_1+1,:)]; b = [b;h(num2_1+1,:)]; end
end
jstar = m; end
%--- function3: find kapa ---% function kapa = findkapa(Hx, Hv, h, jstar, r, x, v)
% solve linear program: kapa*Hv*(r(t)-v(t)) <= h-Hx*x-Hv*v(t) % to find kapa.
a = Hv*(r-v); b = h-Hx*x-Hv*v;
k=1; % Algorithm 1 to find kapa. for i=1:jstar +2 if a(i) > 0 k=min(k,b(i)/a(i)); end end k=max(k,0); kapa = k; end