Department of Automatic Control LTH, Lund University
What is gain scheduling? How to find schedules? Applications What can go wrong? Some theoretical results LPV design via LMIs Conclusions To read: Leith & Leithead, Survey of Gain-Scheduling Analysis & Design To try out: Matlab -
Controller parameters Gain schedule Operating condition Command signal Controller Control signal Process Output Example of scheduling variables Production rate Machine speed, e.g. DVD player Mach number and dynamic pressure
How to Find Schedules? Select scheduling variables: Variable(s) should reflect changes in system dynamics. Make (linear) control design for different operating conditions: For instance with automatic tuning Use closest control design, or interpolate: Many ad-hoc or theoretically motivated methods exist Verify performance: Simulations. Some methods exist that guarantee performance; usually conservative though
Scheduling on controller output GS Ref FT FIC
A typical process control loop Nonlinear Valve Process u c Σ PI c ˆ f 1 u f v G 0 (s) y 1 Valve characteristics and a crude approximation 10 ˆf(u) f(u) 0 0 0.5 1 1.5 2 Time
Results Without gain scheduling Loop is either too slow or unstable With gain scheduling 0.3 u c y 0.2 0 20 40 60 80 100 Time u c y 1.1 1.0 0 20 40 60 80 100 Time 5.1 u c y 5.0 0 20 40 60 80 100 Time
Schedule on Process Variable GS Ref LIC LT
Schedule on External Variable GS Ref TIC TT FT
Concentration Control
Concentration Control Performance with changing flow V d (a) (b) 1.0 0.5 0.0 0 5 10 15 20 Time 1.5 1.0 0.5 Output c c r q = 0.5 q = 0.9 q = 1.1 q = 2 Control signal c in q = 0.9 q = 0.5 q = 1.1 q = 2 0.0 0 5 10 15 20 Time
Variable Sampling Rate Process model G(s) = 1 1 + st e sτ, T = V m q, V d q Sample system with period h = V d nq Sampled model becomes linear time -invariant c(kh+h) = ac(kh)+(1 a)u(kh nh), a = e qh/vm = e V d/(nv m) Sampled equation does not depend on q!!
Results Digital control with h = 1/(2q). The flows are: (a) q = 0.5; (b) q = 1; (c) q = 2 (a) (b) (c) 1 0 0 5 10 15 20 Time 1 0 0 5 10 15 20 Time 1 c c c 0 0 5 10 15 20 Time 1 0 0 5 10 15 20 Time 1 0 0 5 10 15 20 Time 1 c in c in c in 0 0 5 10 15 20 Time
Flight control Pitch dynamics q = θ θ V α N z δ e Operating conditions 80 Altitude (x1000 ft) 60 40 20 3 4 0 1 2 0 0.4 0.8 1.2 1.6 2.0 2.4 Mach number
The Pitch Control Channel V IAS H Pitch stick V IAS Gear Position Filter A/D K DSE Acceleration Filter A/D H M T 1s 1+ T 1 s H K SG Σ Σ D/A D/A Σ Σ Pitch rate Filter A/D M Filter 1 1+ T 3 s K Q1 Σ K NZ Σ To servos T 2s 1+ T 2 s K QD M H H M V IAS Many scheduling variables
Schedule of K Q wrt airspeed (IAS) and height (H) K QD IAS K QD H 0.5 0.5 0 0 0 1000 0 10 20 V IAS (km/h) H (km)
Surge Tank Control A surge tank is used to smooth flow variations. The is allowed will fluctuate substantially but it is important that the tank does not become empty or that it overflows. LT LIC Surge tank FIC FT
The Igelsta Power Station Controller structure before modification PIC PT TIC Set point Fuel/air TT DH network Boiler Heat exchanger Modified controller structure PIC PT TIC Set point Fuel/air TT XT DH network Boiler Heat exchanger
What can go wrong? Most designs are done for the time-frozen system, i.e. as if scheduling parameter θ is constant. Theory and practice: This will work also when θ(t) is slowly varying. But can go wrong for fast varying parameters. Following example is from Shamma and Athans, ACC 1991
Shamma - Athans Resonant system with varying resonance frequency G θ (s) 1 s = 1 1 s 2 + 0.2s + 1 + 0.5θ(t) s = C(sI A(θ)) 1 B with 1 θ(t) 1 Controller design: LQG + integrator on system input, LQG parameters Q 11 = C T C, Q 22 = 10 8, R 11 = B 2 (θ)b 2 (θ) T, R 22 = 10 2 Gives controller K θ (s) with good robustness margins when θ constant Frozen-system loop-gain G θ (s)k θ (s) is actually independent of θ (using B 2 (θ) T = [1 0 1 + 0.5θ])
Shamma-Athans Step response and GOF for any constant θ look fine. All θ give the same curves 1.2 10 2 10 2 1 S 10 0 PS 10 0 0.8 10-2 10 0 10 2 10-2 10 0 10 2 0.6 0.4 10 2 10 2 0.2 CS 10 0 T 10 0 0 0 0.5 1 1.5 2 2.5 3 10-2 10 0 10 2 w 10-2 10 0 10 2 w
Shamma-Athans But if θ(t) = cos (2t) the system becomes unstable 2.5 2 1.5 1 0.5 0-0.5 0 10 20 30 40 50 60 70 80 90 100 It can be shown that the open loop LTV system is unstable in this case. Several theorems show that if θ varies slowly performance for the frozen-system analysis is maintained for the true system Small-gain theorem Lyapunov theory using V = x T Px or V = x T P(θ)x
Gain-scheduling design Several authors have worked with systematic design methods Shamma-Athans Packard Apkarian-Gahinet Helmersson...
Gain-scheduling for LPV systems by LMIs Apkarian, Gahinet (1995) A Convex Characterization of Gain-Scheduled H Controller Model assumption Controller structure ẋ = A(θ(t))x(t) + B(θ(t))u(t) y = C(θ)x(t) + D(θ(t))u(t) ζ(t) = A K (θ(t))ζ(t) + B K (θ(t))y(t) u(t) = C K (θ(t))ζ(t) + D K (θ(t))y(t) Will assume both process and controller depends on θ via a fractional transformation.
Main Idea ( T ( ) T q θ q θ q y w) = Pa (s) w θ w θ w u ũ
Main Result - a sufficency condition The closed loop system is stable for all θ(t) with θ(t) < 1/γ and the L 2 induced norm from w to q satisfies max T qw < γ θ(t) <1/γ 2 if there is a scaling matrix L commuting with Θ so that ( ) ( ) L 1/2 0 L 1/2 0 F 0 I l (P a, K) < γ 0 I Sufficient, not necessary The condition can be checked by an LMI, also gives the controller.
If Time Permits http://se.mathworks.com/help/control/ug/gain-scheduled-control-of-achemical-reactor.html
Summary - Very useful technique Linearization of nonlinear actuators Surge tank control Control over wide operating ranges Requires good models Issues to consider Choice of scheduling variable(s) Granularity of tables, interpolation Bumpless parameter changes