Digital Control Systems State Feedback Control Illustrating the Effects of Closed-Loop Eigenvalue Location and Control Saturation for a Stable Open-Loop System
Continuous-Time System Gs () Y() s 1 = = 2 Us () s + 6. s+ 1, ζ = 3., ω = 1r / n s L NM x x 1 2 O L O. x x QP = 6 1 NM QP L N M O u, y x Q P +L NM O L 1 1 QP = 1 N M 1 2 x 1 2 O QP
Conversion to Discrete Time T =.1 s, zero-order hold x1( k+ 1).937.969 x1( k).969 = + uk ( ) x ( k+ 1).969.9951 x ( k).49 2 2 [ ] yk ( ) 1 x ( k) 1 =, x = x2( k) 1
1 Open-Loop State Response 8 6 x 2 4 2-2 -4 x 1-6 -8 2 4 6 8 1 12 14 16 18 2 Time (s)
Open-Loop Response Open-loop system is asymptotically stable, so both states decay to. States settle to within ±.5 in 17.4 seconds. No control is used in this open-loop configuration.
Closed-Loop Pole Locations disc _11 disc _ k {, } [ ] for each k 1, 1 λ ζ =.7, ω = k, T =.1 s λcont _ k = ω n ζ ± j 1 ζ λ = = e λ cont T n As k increases, closed-loop poles move farther away from open-loop poles and closer to the origin. 2
Closed-Loop Experiments For each set of closed-loop poles, the gain is computed, and closed-loop system is simulated with no control saturation. For the same set of closed-loop poles and gains, the closed-loop system is simulated with control saturation at ±5. State responses and various performance measures are computed and plotted.
State Response with L 1 State Response with L 1 1 1 5 5-5 2 4 6 8 1-5 2 4 6 8 1 Control Signal with L 1 Control Signal with L 1 4 4 3 3 2 1 2 1-1 2 4 6 8 1 Time (s) -1 2 4 6 8 1 Time (s)
State Response with L 2 State Response with L 2 1 1 5 5-5 -5-1 2 4 6 8 1-1 2 4 6 8 1 Control Signal with L 2 Control Signal with L 2 1 6-1 -2 4 2-2 -4-3 2 4 6 8 1 Time (s) -6 2 4 6 8 1 Time (s)
State Response with L 1 State Response with L 1 1 1 5-1 -2-3 -5-4 -1-5 2 4 6 8 1-15 2 4 6 8 1 Control Signal with L 1 Control Signal with L 1 4 6 2 4-2 2-2 -4-4 -6 2 4 6 8 1 Time (s) -6 2 4 6 8 1 Time (s)
State Response with L 11 State Response with L 11 2 1 5-2 -4-6 -5-8 -1-1 2 4 6 8 1-15 2 4 6 8 1 Control Signal with L 11 Control Signal with L 11 1 6 5 4-5 2-2 -1-4 -15 2 4 6 8 1 Time (s) -6 2 4 6 8 1 Time (s)
Closed-Loop Responses As natural frequency increases: settling time decreases; control signals become larger in magnitude; state variable values are larger in magnitude. If control saturation is present: amount of time in saturation increases with gain number; settling time increases with gain number; closed-loop stability is maintained.
1 Eigenvalue Locations.8.6.4.2 Imag Axis -.2 -.4 -.6 -.8-1 -1 -.8 -.6 -.4 -.2.2.4.6.8 1 Real Axis
1 7 Performance Measures Without Control Saturation 1 6 1 5 1 4 Sum of Squares Magnitude 1 3 1 2 Max. Controls 1 1 Gain Norms 1 1-1 Distance from OL 1-2 2 4 6 8 1 12 Gain Number
Performance Measures As distance of closed-loop poles from openloop poles increases: elements in gain matrix increase in magnitude; control signal increases in magnitude because of the larger gain and larger state values; state variables increase in magnitude because of the larger control signals; the sum of squares increases because of the larger state and control magnitudes.
1 7 Comparison of Sums of Squares 1 6 Magnitude 1 5 Without Saturation 1 4 With Saturation 1 3 2 4 6 8 1 12 Gain Number
Sum of Squares J = (x T x+u T u) With control saturation: control signal is limited to ±5, so it is smaller than without saturation for most gains; smaller control signal produces smaller values for state variable magnitudes; J is smaller with saturation than without, except for Gain Number 1 that does not saturate.
1 4 Comparison of Maximum Control Magnitudes 1 3 Magnitude 1 2 Without Saturation and Before Saturation 1 1 With Saturation 1 2 4 6 8 1 12 Gain Number
Maximum Control Magnitudes Control signals computed by the gain matrix have the same maximum magnitudes with and without control saturation. The maximum magnitude control occurs at the initial time for all Gain Numbers > 1. With Gain Number 1, the peak magnitude is less than 5, so there is no saturation. For large Gain Numbers, the control signals before saturation have more oscillations and longer settling times.
1 Comparison of Settling Times 9 8 7 Time Steps 6 5 4 With Saturation 3 2 Without Saturation 1 2 4 6 8 1 12 Gain Number
Settling Times Without saturation, settling time decreases as the closed-loop eigenvalues move toward the origin. Without saturation, when the closed-loop eigenvalues are at the origin, the settling time is 2 time steps, the order of the system. With saturation, settling time increases rather than decreases with the faster eigenvalues due to larger control magnitudes.