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Chapter 6 Classic Design. 6.1 Introduction. Figure 6.1 Typical CCS. Figure 6.2 Simplified CCS framework. There are two kinds of design methods for the design of CCS, i.e., indirect design method and direct design method. 6.2 Indirect design method. Discretize.
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Chapter 6 Classic Design
6.1 Introduction Figure 6.1 Typical CCS Figure 6.2 Simplified CCS framework There are two kinds of design methods for the design of CCS, i.e., indirect design method and direct design method.
6.2 Indirect design method Discretize Figure 6.3 Indirect design method Advantage: Simple; Abundant experience in continuous design Disadvantage: Have a larger phase delay than that of continuous-time system
6.2 Indirect design method Concrete procedure: 1.Choose sampling period, and design pre-antialiasing filter 2.Design or discretize continuous controller 3.Test the performance of the discrete system 4. If the system cannot meet the specification, there are three ways to improve it. a. Choose the more appropriate discretized method b. Increase sampling frequency c. Modify the original continuous controller
6.2 Indirect design method 6.2.1 Approximations based on transfer functions The system’s transfer function is equivalent to the differential equation (6.1) where a=1/(RC) Figure 6.4 Continuous time filter
6.2 Indirect design method 1.Numerical integration • If we write Eq. (6.1) in integral form as following • Many rules have been developed based on how the incremental area term is approximated. • Three possibilities are sketched in Fig. 6.5 which is called forward rectangular rule, backward rectangular rule and trapezoid rule respectively.
6.2 Indirect design method Figure 6.5 Sketches of three ways the area under the curve from kT to kT+T can be approximated
6.2 Indirect design method (1) forward rectangular rule (also known as Euler’s rule) • We approximate the area by the rectangle looking forward from kT-T and take the amplitude of the rectangle to be the value of the integrand at kT-T. The width of the rectangle is T. The result is an equation in the first approximation, u1 • The transfer function corresponding to the forward rectangular rule in this case is
6.2 Indirect design method • Remarks • 1. Very simple even for large system • 2. Stable perhaps will obtain unstable, so the former rectangular • rule can’t be used in the real application • 3. Steady-state gain is invariant, i.e., H(s)|s=0=H(z)|z=1
6.2 Indirect design method (2) backward rectangular rule • we approximate the area by the rectangle looking backward from kT toward kT-T. The equation for u2 is • The corresponding transfer function is
6.2 Indirect design method • Remarks • 1. Stable will obtain stable • 2. There are large distortions on dynamic response and frequency • response properties. • 3. T should be smaller. • 4. Steady-state gain is invariant, i.e., H(s)|s=0=H(z)|z=1
6.2 Indirect design method (3) trapezoid rule • Taking the area to be that of the trapezoid formed by the average of the previously selected rectangles. The approximating difference equation is • The corresponding transfer function from the trapezoid rule is
6.2 Indirect design method • Remarks • 1. Stable will obtain stable and is not same as z transform. • 2. Have frequency distortion or warping • 3. Frequency pre-warping can decrease the distortion on frequency • response. • 4. Steady-state gain is invariant, i.e., H(s)|s=0=H(z)|z=1
6.2 Indirect design method (4) Frequency pre-warping set Then If is small ; ,
6.2 Indirect design method Example 6.1
6.2 Indirect design method 2.Input response invariance--Step invariance • If the input signal is constant over the sampling intervals, equation H(z) = (1 – z-1) Z{G(s)/s} gives an appropriate pulse-transfer function H(z) for a given transfer function G(s).
6.2 Indirect design method Remarks: 1. Stable obtain stable 2. Frequency folding phenomena, but thanks to the low-pass characteristics of the ZOH, it is a little better. 3. Complex computation for large-scale systems 4. Steady-state value is invariant, i.e., G(s)|s=0=H(z)|z=1
6.2 Indirect design method Example: bode(1,[1 1]) hold on dbode([1 1],[3 -1],1)
6.2 Indirect design method Example: sysc=tf([1 1 9],[1 2 9]); sysdt=c2d(sysc,1,'tustin') Transfer function: 0.8824 z^2 + 0.5882 z + 0.6471 ------------------------------ z^2 + 0.5882 z + 0.5294 Sampling time: 1 >> bode(sysc) >> hold on >> dbode([0.8824 .5882 .6471],[1 .5882 .5294],1)
6.2 Indirect design method Example: sysc=tf([1 1 9],[1 2 9]); sysdp=c2d(sysc,1,'prewarp',3) Transfer function: 0.9775 z^2 + 1.891 z + 0.9326 ----------------------------- z^2 + 1.891 z + 0.9101 Sampling time: 1 >> dbode([.9775 1.891 .9326],[1 1.891 .9101],1) >> bode(sysc) >> hold on >> dbode([0.8824 .5882 .6471],[1 .5882 .5294],1)
6.2 Indirect design method Comparison of discretization methods
6.2 Indirect design method 6.2.2 Approximations based on state models (1) Forward rectangular rule is to replace s with (z-1)/T is a state-space formula for the forward rule equivalent.
6.2 Indirect design method (2) For the backward rule, substitute with the result which corresponds to the time domain equation
6.2 Indirect design method (3) Finally, for the trapezoid or bilinear rule, the z-transform equivalent is obtained
6.2 Indirect design method • These results can be tabulated for convenient reference. Suppose we have a continuous system described by • Then a discrete equivalent at sampling period T will be described by the equations where , , and J are given as follows:
6.2 Indirect design method 6.2.3 Digital PID controllers (1) Analog PID controllers
e 1 t t0 u Kp u0 t0 t 6.2 Indirect design method • P control
e 1 t t0 Kp u Kp u0 Ti t0 t 6.2 Indirect design method • PI control
e 1 t0 t Kp u Kp u0 Ti t0 t 6.2 Indirect design method • PID control
6.2 Indirect design method (2) Digital PID controllers • Position algorithm Suppose t= kT, k= 0,1,2,… So e(t) = e(kT) We call it position algorithm or an absolute algorithm.
6.2 Indirect design method The output of the controller is the absolute value of the control signal, for instance, a valve position.
6.2 Indirect design method • Incremental algorithm we call it incremental algorithm. or
6.2 Indirect design method In an incremental algorithm, the output of the controller should then represent the increments of the control signal. • Advantages of the incremental algorithm • When there exists failure in computer, the influence on the • system is smaller. • 2. Impact of the control is smaller when changing between manual and automatic mode.
6.2 Indirect design method • Pulse-transfer function for PID controllers which is equivalent to the following equation
6.2 Indirect design method P control PI control PD control
6.2 Indirect design method Remarks: 1. Integrator windup b y a r* u a b umax τ
6.2 Indirect design method 2. Modification of D control
k 6.2 Indirect design method e D control 1 Modification of D control D control Modification of D control
6.2 Indirect design method • Tuning of PID control parameters Step-response method a=RL
6.2 Indirect design method Exercise: 1. , T=1, using step invariance, backward rectangle rule, bilinear transformation, prewarping bilinear transformation method to discretize D(s). Where have prewarping bilinear at , at , . 2. PID controller a. Using backward rectangle rule to discretize D(s). T=0.1. b. Suppose the input signal is e(t) and output signal is u(t), determine the difference expression of the u(k) when realizing on computer. c. Determine the analytical solution of u(k) when e(k) is unit step.
_ The controller system with time delay 6.2 Indirect design method 6.2.4 Smith predictor - controllers for system with time delay (1) The problem of the control of the systems with time delay • The time delay is aroused by the delay of transition of the materials and energies • In 1950s, Smith proposed a compensation model for the systems with time delay, which is difficult to be realized with analog controller
6.2 Indirect design method (2) Traditional methods and its characteristic equation is
6.2 Indirect design method (3) Smith predictor Figure 6.6 Block diagram of a Smith-predictor Smith predictor
6.2 Indirect design method and its characteristic equation is
6.2 Indirect design method • The controller consists of a controller D(s) and a loop around it that contains a predictor model. The controller D(s) is designed as if the time delay T in the process was absent and the feedback around the controller ensures that the system with the time delay will be well behaved. • So procedure for smith predictor is as follows: 1. Design D(s) for Gp firstly. 2. Realize the controller as the figure shown in Fig. 6.6.
6.2 Indirect design method Example 6.2 • A time-delay process is described by the model • If there were no time-delays, a PI-controller with gain 0.4 and integration time Ti = 0.4 would give good control. • This PI-controller will not give good control if the process has a time delay. To obtain good PI-regulation, it is necessary to have a gain of 0.1 and Ti = 0.5.
