Control System Design


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1 ELEC4410 Control System Design Lecture 19: Feedback from Estimated States and DiscreteTime Control Design Julio H. Braslavsky School of Electrical Engineering and Computer Science Lecture 19: Feedback from Estimated States p. 1
2 Outline Feedback from Estimated States DiscreteTime Control Design DeadBeat Control Lecture 19: Feedback from Estimated States p. 2
3 Feedback from Estimated States In the last lecture we discussed the general scheme for feedback control from estimated states, involving Lecture 19: Feedback from Estimated States p. 3
4 Feedback from Estimated States In the last lecture we discussed the general scheme for feedback control from estimated states, involving a state feedback gain K Plant Lecture 19: Feedback from Estimated States p. 3
5 Feedback from Estimated States In the last lecture we discussed the general scheme for feedback control from estimated states, involving a state feedback gain K a state estimator (observer), with gain L Plant Observer Lecture 19: Feedback from Estimated States p. 3
6 Feedback from Estimated States In the last lecture we discussed the general scheme for feedback control from estimated states, involving a state feedback gain K a state estimator (observer), with gain L To design we used the BassGura formula to make Plant Hurwitz To design in the observer, we used duality and the same BassGura formula to make Observer Hurwitz Lecture 19: Feedback from Estimated States p. 3
7 Feedback from Estimated States By designing K such that (A BK) is Hurwitz, with the desired eigenvalues, we can guarantee that the closedloop system will be asymptotically and BIBO stable, and will have the specified dynamic response. Lecture 19: Feedback from Estimated States p. 4
8 Feedback from Estimated States By designing K such that (A BK) is Hurwitz, with the desired eigenvalues, we can guarantee that the closedloop system will be asymptotically and BIBO stable, and will have the specified dynamic response. By designing L such that (A LC) is Hurwitz, we guarantee that the observer will be asymptotically stable, and the estimate of the states ^x(t) will converge to the real states x(t) as t. Lecture 19: Feedback from Estimated States p. 4
9 Feedback from Estimated States By designing K such that (A BK) is Hurwitz, with the desired eigenvalues, we can guarantee that the closedloop system will be asymptotically and BIBO stable, and will have the specified dynamic response. By designing L such that (A LC) is Hurwitz, we guarantee that the observer will be asymptotically stable, and the estimate of the states ^x(t) will converge to the real states x(t) as t. But K and the observer are designed independently... Will they work the same when we put them together in a feedback from estimated states scheme? Lecture 19: Feedback from Estimated States p. 4
10 Feedback from Estimated States Three basic questions arise regarding feedback from estimated states: The closedloop eigenvalues were set as those of (A BK) by using state feedback u = Kx. Would we still have the same eigenvalues if we do feedback from estimated states u = K^x? Lecture 19: Feedback from Estimated States p. 5
11 Feedback from Estimated States Three basic questions arise regarding feedback from estimated states: The closedloop eigenvalues were set as those of (A BK) by using state feedback u = Kx. Would we still have the same eigenvalues if we do feedback from estimated states u = K^x? Would the interconnection affect the observer eigenvalues, those of (A LC)? Lecture 19: Feedback from Estimated States p. 5
12 Feedback from Estimated States Three basic questions arise regarding feedback from estimated states: The closedloop eigenvalues were set as those of (A BK) by using state feedback u = Kx. Would we still have the same eigenvalues if we do feedback from estimated states u = K^x? Would the interconnection affect the observer eigenvalues, those of (A LC)? What would be the effect of the observer in the closedloop transfer function? Lecture 19: Feedback from Estimated States p. 5
13 Feedback from Estimated States To answer these questions we take a look at the state equations of the full system, putting together plant and observer, that is, ẋ = Ax+Bu ^x = (A LC)^x + LCx+Bu plant observer Lecture 19: Feedback from Estimated States p. 6
14 Feedback from Estimated States To answer these questions we take a look at the state equations of the full system, putting together plant and observer, that is, ẋ = Ax BK^x + BNr ^x = (A LC)^x + LCx BK^x + BNr plant observer after replacing u = Nr K^x. Lecture 19: Feedback from Estimated States p. 6
15 Feedback from Estimated States To answer these questions we take a look at the state equations of the full system, putting together plant and observer, that is, ẋ = Ax BK^x + BNr ^x = (A LC)^x + LCx BK^x + BNr plant observer after replacing u = Nr K^x. Packaging these equations in a more compact form we have ẋ = A BK x^x + BN r ^x LC (A LC BK) BN [ ] y = C 0 x ^x Lecture 19: Feedback from Estimated States p. 6
16 Feedback from Estimated States Let s make a change of coordinates, so that the new coordinates are the plant state x and the estimation error ε = x ^x, = x = I 0 x ε x ^x Note that P 1 = [ ] I 0 1 [ I I = I 0 ] I I = P. I I } {{ } P x^x Lecture 19: Feedback from Estimated States p. 7
17 Feedback from Estimated States Let s make a change of coordinates, so that the new coordinates are the plant state x and the estimation error ε = x ^x, = x = I 0 x ε x ^x I I } {{ } P x^x BK Note that P 1 = [ ] I 0 1 [ I I = I 0 ] I I = P. With this equivalence transformation we get the matrices in the new coordinates as Ā KL = PA KL P 1 (A BK) =, B KL = PB KL = BN 0 C KL = C KL P 1 = [ ] C 0 0 (A LC) Lecture 19: Feedback from Estimated States p. 7
18 Feedback from Estimated States The full system in the new coordinates is thus represented as (A BK) ẋ = BK x + BN r ε 0 (A LC) ε 0 [ ] y = C 0 x ε Because Ā KL is triangular, its eigenvalues are the union of those of (A BK) and (A LC). Controller and observer do not affect each other in the interconnection. Lecture 19: Feedback from Estimated States p. 8
19 Feedback from Estimated States Note that the estimation error is uncontrollable, hence the observer eigenvalues will not appear in the closed loop transfer function. Lecture 19: Feedback from Estimated States p. 9
20 Feedback from Estimated States The property of independence between control and state estimation is called the Separation Principle Separation Principle: The design of the state feedback and the design of the state estimator can be carried out independently. The eigenvalues of the closedloop system are as designed by the state feedback law, unaffected by the use of a state estimator. Lecture 19: Feedback from Estimated States p. 10
21 Feedback from Estimated States The property of independence between control and state estimation is called the Separation Principle Separation Principle: The design of the state feedback and the design of the state estimator can be carried out independently. The eigenvalues of the closedloop system are as designed by the state feedback law, unaffected by the use of a state estimator. The eigenvalues of the observer are unaffected by the state feedback law. Lecture 19: Feedback from Estimated States p. 10
22 Feedback from Estimated States The closedloop transfer function will only have the eigenvalues arising from (A BK), since the estimation error is uncontrollable, G cl (s) = C(sI A + BK) 1 BN. Transients in state estimation, however, will be seen at the output, since the estimation error is observable. Lecture 19: Feedback from Estimated States p. 11
23 Outline Feedback from Estimated States DiscreteTime Control Design DeadBeat Control Lecture 19: Feedback from Estimated States p. 12
24 DiscreteTime Control Design For discretetime state equations x[k + 1] = Ax[k] + Bu[k], y[k] = Cx[k], the design procedure for a state feedback law u[k] = Kx[k] is the same as for continuoustime systems. The same goes for a discretetime state observer, ^x[k + 1] = (A LC)^x[k] + Bu[k] + Ly[k]. One difference is the location of the desired eigenvalues. E.g., for asymptotic stability, they should be inside the unit circle. Lecture 19: Feedback from Estimated States p. 13
25 DiscreteTime Control Design For discretetime state equations x[k + 1] = Ax[k] + Bu[k], y[k] = Cx[k], the design procedure for a state feedback law u[k] = Kx[k] is the same as for continuoustime systems. The same goes for a discretetime state observer, ^x[k + 1] = (A LC)^x[k] + Bu[k] + Ly[k]. One difference is the location of the desired eigenvalues. E.g., for asymptotic stability, they should be inside the unit circle. A practical rule to choose desired discretetime eigenvalues is 1. choose a desired location in continuoustime, say p i 2. translate it to discretetime using the relation λ i = e p it. Lecture 19: Feedback from Estimated States p. 13
26 DiscreteTime Control Design 0.3 Imaginary j Loci with constant damping and constant frequencies the discrete in complex plane ωt 2π Real In MATLAB this grid may be obtained with the zgrid command ζ j Lecture 19: Feedback from Estimated States p. 14
27 DiscreteTime Control Design: Example Example (Discretetime speed control of a DC motor). We return to the DC motor we considered in the examples of the last lecture. We will suppose that the motor is to be controlled with a PC. Hence, the controller has to be discretetime. Lecture 19: Feedback from Estimated States p. 15
28 DiscreteTime Control Design: Example Example (Discretetime speed control of a DC motor). We return to the DC motor we considered in the examples of the last lecture. We will suppose that the motor is to be controlled with a PC. Hence, the controller has to be discretetime. This time, with a view to design a discretetime control law, we first discretise the continuoustime model h 5 i The first design parameter to define is the sampling period. Lecture 19: Feedback from Estimated States p. 15
29 DiscreteTime Control Design: Example Example (Discrete design, step 1: choice of sampling period). From Shannon s Sampling Theorem, the sampling frequency ω s = 2π/T should be at least twice the bandwidth of the closedloop system (because we will change the system bandwidth with the control action). Lecture 19: Feedback from Estimated States p. 16
30 DiscreteTime Control Design: Example Example (Discrete design, step 1: choice of sampling period). From Shannon s Sampling Theorem, the sampling frequency ω s = 2π/T should be at least twice the bandwidth of the closedloop system (because we will change the system bandwidth with the control action). The specification that we had for the previous continuoustime design was a settling time t s of about 1s. The rule based on Shannon s Theorem would then give a sampling time of less than T = 0.5s. In practice T is chosen at least 10 to 20 times faster than the desired closedloop settling time. Here we choose T = 0.1s Lecture 19: Feedback from Estimated States p. 16
31 DiscreteTime Control Design: Example Example (Discrete design, step 1: choice of sampling period). From Shannon s Sampling Theorem, the sampling frequency ω s = 2π/T should be at least twice the bandwidth of the closedloop system (because we will change the system bandwidth with the control action). The specification that we had for the previous continuoustime design was a settling time t s of about 1s. The rule based on Shannon s Theorem would then give a sampling time of less than T = 0.5s. In practice T is chosen at least 10 to 20 times faster than the desired closedloop settling time. Here we choose T = 0.1s Note that the maximum sampling speed is limited by the available computer clock frequency, and thus the time required for computations and signal processing operations. Lecture 19: Feedback from Estimated States p. 16
32 DiscreteTime Control Design: Example Example (Discrete design, step 2: discretisation of the plant). Having defined the samplingtime T, we compute the discretetime state matrices A d = e At, and B d = T 0 e Aτ B dτ From MATLAB, we do [Ad,Bd]=c2d(A,B,0.1) and obtain A d =, B d = As we can verify, the open loop discretetime eigenvalues are = e ( ) and = e ( ) Lecture 19: Feedback from Estimated States p. 17
33 DiscreteTime Control Design: Example Example (Discrete design, step 3: design of discrete feedback gain). The discretetime characteristic polynomial is Lecture 19: Feedback from Estimated States p. 18
34 DiscreteTime Control Design: Example Example (Discrete design, step 3: design of discrete feedback gain). The discretetime characteristic polynomial is For the desired discrete characteristic polynomial, we first obtain the discrete mapping for the continuoustime system, i.e., of the eigenvalues specified which yield the desired discrete characteristic polynomial Lecture 19: Feedback from Estimated States p. 18
35 DiscreteTime Control Design: Example Example (Discrete design, step 3: design of discrete feedback gain). The discretetime characteristic polynomial is For the desired discrete characteristic polynomial, we first obtain the discrete mapping for the continuoustime system, i.e., of the eigenvalues specified which yield the desired discrete characteristic polynomial From the coefficients of feedback gain h and we get the discrete state i h i Lecture 19: Feedback from Estimated States p. 18
36 DiscreteTime Control Design: Example Example (Discrete design, step 3, continuation). We then compute, just following the same procedure used in continuoustime, from the discrete matrices A d and B d, the discrete controllability matrices C d and C d, C d = , and C d = We thus obtain, in the original coordinates, the discretetime feedback gain ] K d = K d C d C 1 d [ = As can be verified with MATLAB, (A d B d K d ) will have the desired discrete eigenvalues. 1 Lecture 19: Feedback from Estimated States p. 19
37 DiscreteTime Control Design: Example Example (Discrete design: continuation). In a similar fashion, we can carry out the design of the discretetime observer, based on the discretetime model of the plant. The output feedback design is finally implemented on the continuoustime plant through a Zero Order Hold and a Sampler. Discrete equivalent plant Continuous time plant ZOH T Discretetime Observer Lecture 19: Feedback from Estimated States p. 20
38 DiscreteTime Control Design: Example Example (Discrete design: MATLAB script). We used the following MATLAB script to compute the gains and run the simulations. % Continuoustime matrices A=[10 1; ];B=[0;2];C=[1 0];D=0; G=ss(a,B,C,D); % state space system definition T=0.1; % Sampling time Gd=c2d(G,T, zoh ) % discretisation % Discretetime feedback gain Kd=place(Gd.a,Gd.b,exp([5i,5+i]*T)) % Discretetime observer gain Ld=place(Gd.a,Gd.c,exp([6i,6+i]*T)) % steady state error compensation N=inv(Gd.c*inv(eye(2)Gd.a+Gd.b*Kd)*Gd.b) % Run simulink diagram sim( dmotorofbk ) % Plots (after simulations have been run) subplot(211),plot(y(:,1),y(:,2));grid subplot(212),stairs(u(:,1),u(:,2));grid Lecture 19: Feedback from Estimated States p. 21
39 DiscreteTime Control Design Example (Discrete design: SIMULINK diagram). We used the following SIMULINK diagram to run the simulations. N G.b* u 1 s G.c* u r Gain Zero Order Hold Matrix Gain1 Integrator Matrix Gain2 y(t) G.a* u Matrix Gain Gd.b* u u[k] Kd* u 1 z Unit Delay Ld* u Gd.a Ld*Gd.c* u Notice the sampled signals coloured in red (check the option Sample Time Colors in the Format menu). All blocks with discretetime signals include a sampler at their input. Lecture 19: Feedback from Estimated States p. 22
40 DiscreteTime Control Design: Example Example (Discrete design: simulations). The following plots show the response of the closedloop sampleddata controlled system: continuoustime output 1.2 and discretetime control signal y(t) time [s] u[k] kt [s] Lecture 19: Feedback from Estimated States p. 23
41 DiscreteTime Design: Peculiarities Two special differences in the discretetime design procedure: The gain for steady state error compensation is, as in continuoustime, the inverse of the steadystate DC gain of the closedloop transfer function. Notice though, that in discretetime Lecture 19: Feedback from Estimated States p. 24
42 DiscreteTime Design: Peculiarities Two special differences in the discretetime design procedure: The gain for steady state error compensation is, as in continuoustime, the inverse of the steadystate DC gain of the closedloop transfer function. Notice though, that in discretetime If we implement robust tracking by adding Integral Action into the state feedback design, notice that the plant augmentation is different in discretetime, since the discreteintegration of the tracking error has to be implemented as Lecture 19: Feedback from Estimated States p. 24
43 DiscreteTime Design: Peculiarities The implementation of the discretetime integral action in the diagram should be consistent, i.e. discreteintegration of the tracking error. ZOH Apart from these two differences, and the locations for the eigenvalues/poles, the discretetime design is obtained by performing the same computations for continuoustime state feedback and observer design. Lecture 19: Feedback from Estimated States p. 25
44 Outline Feedback from Estimated States DiscreteTime Control Design DeadBeat Control Lecture 19: Feedback from Estimated States p. 26
45 DiscreteTime Design: DeadBeat Control A special discretetime design that has no correlate in continuoustime is deadbeat control. A deadbeat response is a response that settles in its final value at a finite time. This feature arises in a discretetime system which has all its poles at z = 0, e.g., 1.5 Step Response 1 G(z) = (7z 1)(5z 1) 24z 3, Amplitude which will settle in 3 sampling periods (the dynamics is just a delay of 3 sample periods). Time (sec) Lecture 19: Feedback from Estimated States p. 27
46 DiscreteTime Design: DeadBeat Control To design a deadbeat controller, we just have to find the closedloop poles at designed deadbeat, with to place all. The discretetime observer can also be to place all the observer poles at. The plot shows the response of the DC motor of the example controlled to have deadbeat response (state feedback no observer). There is not much flexibility in deadbeat control, the only parameter to change the response is the sampling period. y(t) u[k] time [s] Deadbeat usually requires large control action, which may saturate actuators kt [s] Lecture 19: Feedback from Estimated States p. 28
47 Conclusions We have reviewed the state feedback and state estimator design procedure, and showed that the Separation Principle holds: the two designs can be carried out independently Lecture 19: Feedback from Estimated States p. 29
48 Conclusions We have reviewed the state feedback and state estimator design procedure, and showed that the Separation Principle holds: the two designs can be carried out independently We have discussed the design of discretetime state feedback controllers and observers. The procedures are the same as in continuoustime, save for the choice of desired eigenvalues (which have to be in the unit circle for stability) the compensation of steady state tracking error via the feedforward gain N (be careful with the formula used) the implementation of integral action to achieve robust tracking and disturbance rejection (be careful, use discrete integration) Lecture 19: Feedback from Estimated States p. 29
49 Conclusions We have reviewed the state feedback and state estimator design procedure, and showed that the Separation Principle holds: the two designs can be carried out independently We have discussed the design of discretetime state feedback controllers and observers. The procedures are the same as in continuoustime, save for the choice of desired eigenvalues (which have to be in the unit circle for stability) the compensation of steady state tracking error via the feedforward gain N (be careful with the formula used) the implementation of integral action to achieve robust tracking and disturbance rejection (be careful, use discrete integration) We presented a special discrete design: deadbeat control. Lecture 19: Feedback from Estimated States p. 29
50 Conclusions We have reviewed the state feedback and state estimator design procedure, and showed that the Separation Principle holds: the two designs can be carried out independently We have discussed the design of discretetime state feedback controllers and observers. The procedures are the same as in continuoustime, save for the choice of desired eigenvalues (which have to be in the unit circle for stability) the compensation of steady state tracking error via the feedforward gain N (be careful with the formula used) the implementation of integral action to achieve robust tracking and disturbance rejection (be careful, use discrete integration) We presented a special discrete design: deadbeat control. Lecture 19: Feedback from Estimated States p. 29
51 Conclusions We have reviewed the state feedback and state estimator design procedure, and showed that the Separation Principle holds: the two designs can be carried out independently We have discussed the design of discretetime state feedback controllers and observers. The procedures are the same as in continuoustime, save for the choice of desired eigenvalues (which have to be in the unit circle for stability) the compensation of steady state tracking error via the feedforward gain N (be careful with the formula used) the implementation of integral action to achieve robust tracking and disturbance rejection (be careful, use discrete integration) We presented a special discrete design: deadbeat control. Lecture 19: Feedback from Estimated States p. 29
52 Conclusions We have reviewed the state feedback and state estimator design procedure, and showed that the Separation Principle holds: the two designs can be carried out independently We have discussed the design of discretetime state feedback controllers and observers. The procedures are the same as in continuoustime, save for the choice of desired eigenvalues (which have to be in the unit circle for stability) the compensation of steady state tracking error via the feedforward gain N (be careful with the formula used) the implementation of integral action to achieve robust tracking and disturbance rejection (be careful, use discrete integration) We presented a special discrete design: deadbeat control. Lecture 19: Feedback from Estimated States p. 29
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