Nonlinear valve Results a typical process control loop Without gain scheduling Valve characteristics f() f(u) With gain scheduling A crude approximation! Concentration Control Variable sampling rate System Process model G(s=1+sT where Sample the system with period Performance with changing flow h (a),Outp The sampled model becomes c(kh+h)=ac(kh)+(1-a)u(kh-nh (b), Control signa 6i5 Where a =e-gh/Vm=e-v-d/ (n Vm) g亏11 Notice that the sampled equation does not depend on q!!! e K.J. Astrom and BWittenmarkNonlinear Valve A typical process control loop Σ PI c u f v y Process −1 ˆ f −1 uc G0 (s) Valve characteristics 0 0.5 1 1.5 2 0 10 Time ˆ f(u) f(u) A crude approximation! Results Without gain scheduling 0 10 20 30 40 0.2 0.3 0 10 20 30 40 1.0 1.1 0 10 20 30 40 5.0 5.2 Time Time Time uc y uc y y uc With gain scheduling 0 20 40 60 80 100 0.2 0.3 0 20 40 60 80 100 1.0 1.1 0 20 40 60 80 100 5.0 5.1 Time Time Time uc y uc y uc y Concentration Control System cin Vd Vm c Performance with changing flow 0 5 10 15 20 0.0 0.5 1.0 0 5 10 15 20 0.0 0.5 1.0 1.5 Time Time (a) Output c cr q
0.5 q
0.9 q
1.1 q
2 (b) Control signal cin q
0.5 q
0.9 q
1.1 q
2 Variable Sampling Rate Process model G(s)
1 1 + sT e−sτ where T
Vm q τ
Vd q Sample the system with period h
Vd nq The sampled model becomes c(kh+ h)
a c(kh)+(1 − a)u(kh − nh) where a
e−qh/Vm
e−V−d/(nVm) Notice that the sampled equation does not depend on q!!! c K. J. Åström and B. Wittenmark 7