Continuous-time and discrete-time systems Comparison similarities - PowerPoint PPT Presentation

Continuous-time and discrete-time systems Comparison — similarities and differences Continuous-time systems Discrete-time systems signals = functions signals = sequences ODE:s difference equations Operator: p = d qy ( k ) = y ( k + 1) dt � � x = Ax + Bu ˙ qx = Fx + Gu y = Cx y = Hx � � � � A n − 1 B F n − 1 G S = S = B AB . . . G FG . . .     C H  CA   HF      O = O = . .     . .  .   .      CA n − 1 HF n − 1 1 / 13 hans.rosth@it.uu.se Sampling 1

Continuous-time and discrete-time systems Continued Continuous-time systems Discrete-time systems L -transform Z -transform G ( s ) = C ( sI − A ) − 1 B = b ( s ) G ( z ) = H ( zI − F ) − 1 G = d ( z ) a ( s ) c ( z ) Poles: 0 = det( sI − A ) = a ( s ) Poles: 0 = det( zI − F ) = c ( z ) Static gain = G (0) Static gain = G (1) Stability region: Left half plane Interior of the unit circle 2 / 13 hans.rosth@it.uu.se Sampling 1

Sampled data control Continuous-time system and discrete-time controller 2 4 2 1.5 u ( t ) y ( t ) 0 u(t i ) y(t) System 1 −2 Hold ✲ ✲ −4 0.5 −6 u ( t i ) 0 −8 0 2 4 6 0 2 4 6 t t y ( t i ) ✛ 2 4 Sampling Controller ✛ 2 r ( t i ) 1.5 ✛ r ( t ) 0 y(t i ) u(t) 1 −2 −4 0.5 −6 0 −8 0 2 4 6 0 2 4 6 t t The controller operates in discrete time. ◮ Sampling: Convert continuous-time signal to discrete-time signal. ◮ Hold circuit: Convert discrete-time signal to continuous-time. ◮ Notation (mild abuse of): y ( k ) = y ( kh ) = y ( t ) | t = kh , where h = sampling time/interval/period. 3 / 13 hans.rosth@it.uu.se Sampling 1

Sampled data control Continuous- and discrete-time domains — how do we deal with that? The hold circuit: Typically we use zero-order-hold (ZOH), u ( t ) = u ( kh ) for kh ≤ t < kh + h. Two ways to proceed: I: Work in continuous time: Design a continuous-time controller F ( s ) . For implementation (in a computer), approximate F ( s ) by use of a discrete-time controller, F d ( z ) . II: Work in discrete time: Produce a discrete-time model of the system, and design a discrete-time controller F ( z ) directly. 4 / 13 hans.rosth@it.uu.se Sampling 1

Method I Work in continuous time Use a continuous-time model and your favourite control design method ⇒ a continuous-time controller F ( s ) . The approximation of F ( s ) with a discrete-time controller F d ( z ) is called discretization, and there are several approaches: ◮ Impulse-invariant mapping ◮ Pole-zero matching ◮ Frequency response matching (e.g. by least squares) ◮ Zero-order-hold (ZOH) — we will get to this soon ... ◮ Approximation of the time derivative (numerical differentiation and/or integration) 5 / 13 hans.rosth@it.uu.se Sampling 1

Method I: Discretization Approximation of the time derivative Three methods for approximation of the time derivative: ◮ Forward difference (Euler’s method): dty ( kh ) ≈ y ( kh + h ) − y ( kh ) s ← z − 1 d ⇔ h h ◮ Backward difference: dty ( kh ) ≈ y ( kh ) − y ( kh − h ) s ← z − 1 d ⇔ h hz d ◮ Tustin’s approximation: dt y ( kh ) ≈ ∆ y ( kh ) ∆ y ( kh + h ) + ∆ y ( kh ) = y ( kh + h ) − y ( kh ) s ← 2( z − 1) ⇔ 2 h ( z + 1) h (Tustin’s approximation ⇔ trapezoidal rule in numerical integration.) 6 / 13 hans.rosth@it.uu.se Sampling 1

Method I: Performance How well does it work? For sufficiently fast sampling the performance with these techniques will be almost as good as for the continuous-time controller (if it could be implemented). But how fast should the sampling be? ◮ The sampling frequency ω s = 2 π h . ◮ The Nyquist frequency ω n = ω s 2 = π h . ◮ Aliasing: For a sampled signal the frequency components for frequencies ω > ω n will be interpreted as components for frequences in [0 , ω n [ . (Measure at least twice in a period.) Rule of thumb: For acceptable performance, choose ω s ≥ 20 ω B , where ω B is the desired bandwidth of the closed loop system. 7 / 13 hans.rosth@it.uu.se Sampling 1

Method II Work in discrete time Find a discrete-time model representation of the system. Then use your favourite control design method to obtain a discrete-time controller, which can be implemented directly as it is. We have the continuous-time model � x ( t ) = Ax ( t ) + Bu ( t ) , ˙ ZOH sampling ⇒ y ( t ) = Cx ( t ) u ( t ) is piecewise constant We want to obtain a discrete-time model � x ( k + 1) = Fx ( k ) + Gu ( k ) , ( x ( k ) = x ( kh ) etc ) y ( k ) = Hx ( k ) If possible, we also want x ( k ) = x ( t ) | t = kh , k = 0 , 1 , . . . 8 / 13 hans.rosth@it.uu.se Sampling 1

System sampling A.k.a. ZOH sampling Use the solution of the continuous-time state equation (eq. (3.2)): � t x ( t ) = e A ( t − t 0 ) x ( t 0 ) + e A ( t − τ ) Bu ( τ ) dτ t 0 Here we have u ( t ) = u ( kh ) for kh ≤ t < kh + h . Set t 0 = kh and t = kh + h ⇒ � kh + h x ( k + 1) = x ( kh + h ) = e Ah x ( kh ) + e A ( kh + h − τ ) Bu ( τ ) dτ kh � 0 = [ s = kh + h − τ ] = e Ah x ( kh ) + e As Bu ( kh )( − ds ) h �� h � = e Ah x ( kh ) + e As Bds u ( kh ) 0 9 / 13 hans.rosth@it.uu.se Sampling 1

ZOH sampling Theorem 4.1 Consider the continuous-time system � x ( t ) = Ax ( t ) + Bu ( t ) , ˙ with u ( t ) = u ( kh ) y ( t ) = Cx ( t ) , for kh ≤ t < kh + h. Then x ( k ) = x ( t ) | t = kh , ∀ k ∈ Z , where x ( k ) is given by the discrete-time system � x ( k + 1) = Fx ( k ) + Gu ( k ) , y ( k ) = Hx ( k ) , where � h F = e Ah , e At Bdt G = and H = C. 0 10 / 13 hans.rosth@it.uu.se Sampling 1

Examples of sampled systems The harmonic oscillator ◮ Continuous-time system: � � � �  0 − 1 1  x = ˙ x + u, 1   1 0 0 Y ( s ) = s 2 + 1 U ( s ) ⇔ � �  y =  0 1 x  ◮ Discrete-time state space model: � � � �  cos( h ) − sin( h ) sin( h )  x ( k + 1) = x ( k ) + u ( k ) ,   sin( h ) cos( h ) 1 − cos( h ) � �   y ( k ) = 0 1 x ( k ) .  ◮ Discrete-time transfer function: G ( z ) = H ( zI − F ) − 1 G = (1 − cos( h ))( z + 1) z 2 − 2 cos( h ) z + 1 11 / 13 hans.rosth@it.uu.se Sampling 1

Examples of sampled systems The double integrator ◮ Continuous-time system: � � � �  0 0 1  x = ˙ x + u,  Y ( s ) = 1  1 0 0 s 2 U ( s ) ⇔ � �   y = 0 1 x  ◮ Discrete-time state space model: � � � �  1 0 h  x ( k + 1) = x ( k ) + u ( k ) ,   0 . 5 h 2 h 1 � �   y ( k ) = 0 1 x ( k ) .  ◮ Discrete-time transfer function: � − 1 � = 0 . 5 h 2 ( z + 1) � � z − 1 0 � h � G ( z ) = 0 1 0 . 5 h 2 − h z − 1 ( z − 1) 2 12 / 13 hans.rosth@it.uu.se Sampling 1

Examples of sampled systems DC motor ◮ Continuous-time system: � � � �  0 0 1  x = ˙ x + u,  1  0 − 1 1 Y ( s ) = s ( s + 1) U ( s ) ⇔ � �   y = 1 − 1 x  ◮ Discrete-time state space model: � � � �  1 0 h  x ( k + 1) = x ( k ) + u ( k ) ,   e − h 1 − e − h 0 � �  y ( k ) = x ( k ) .  1 − 1  ◮ Discrete-time transfer function: G ( z ) = H ( zI − F ) − 1 G = ( h − 1 + e − h ) z + 1 − (1 + h ) e − h ( z − 1)( z − e − h ) ◮ Zero in z ′ = − 1 − (1+ h ) e − h h − 1+ e − h , and − 1 < z ′ < 0 for h > 0 . 13 / 13 hans.rosth@it.uu.se Sampling 1