Pole Placement Pole Placement A majority of the

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Pole Placement

Pole Placement

Pole Placement A majority of the design techniques in modern control theory is based

Pole Placement A majority of the design techniques in modern control theory is based on the state-feedback configuration. That is, instead of using controllers with fixed configurations in the forward or feedback path, control is achieved by feeding back the state variables through real constant gains. 2

Definition of Pole Placement • (Pole Assignment, Pole Allocation) Placing the poles or eigenvalues

Definition of Pole Placement • (Pole Assignment, Pole Allocation) Placing the poles or eigenvalues of the closed-loop system at specified locations. • Poles can be arbitrarily placed if and only if the system is controllable. • Pole placement is easier if the system is given in controllable form. 3

Pole Placement 4

Pole Placement 4

Pole Placement 5

Pole Placement 5

State Feedback: involves the use of the state vector to compute the control action

State Feedback: involves the use of the state vector to compute the control action for specified system dynamics. 6

State Feedback X(k+1)=Ax(k)+Bu(k) Y(k)=C x(k) U(k)=-K x(k)+v(k) X(k+1)=[A-BK]x(k)+B v(k) Let Ad= A-BK The closed-loop

State Feedback X(k+1)=Ax(k)+Bu(k) Y(k)=C x(k) U(k)=-K x(k)+v(k) X(k+1)=[A-BK]x(k)+B v(k) Let Ad= A-BK The closed-loop system state space: X(k+1)= Ad x(k)+B v(k) Y(k)=C x(k) 7

Output Feedback 8

Output Feedback 8

Output Feedback U(k)=-K y(k)+v(k)= -K C x(k)+ v(k) X(k+1)=[A-BKC]x(k)+B v(k) 9

Output Feedback U(k)=-K y(k)+v(k)= -K C x(k)+ v(k) X(k+1)=[A-BKC]x(k)+B v(k) 9

Pole Placement Theorem 9. 1: State Feedback. If the pair (A, B) is controllable,

Pole Placement Theorem 9. 1: State Feedback. If the pair (A, B) is controllable, then there exists a feedback gain matrix K that arbitrarily assigns the system poles to any set [λi i=1, …. n]. Furthermore, if the pair (A, B) is stabilizable, then the controllable modes can all be arbitrarily assigned. 10

Procedure 9. 1: Pole Placement by Equating Coefficients 11

Procedure 9. 1: Pole Placement by Equating Coefficients 11

Example 9. 1 • 12

Example 9. 1 • 12

MATLAB >> A = [0, 1; 3, 4]; >>B = [0; 1]; >> poles

MATLAB >> A = [0, 1; 3, 4]; >>B = [0; 1]; >> poles = [0. 3 + j*. 2, 0. 3 – j*0. 2]; >> K = place(A, B, poles) K = 3. 1300 3. 4000 13

Pole Placement by Transformation to Controllable Form Any controllable single-input-single-output (SISO) system can be

Pole Placement by Transformation to Controllable Form Any controllable single-input-single-output (SISO) system can be transformed into controllable form using the transformation 14

Procedure 9. 2 • 15

Procedure 9. 2 • 15

Example 9. 2 16

Example 9. 2 16

Solution 17

Solution 17

Servo Problem The schemes shown in Figures 9. 1 and 9. 2 are regulators

Servo Problem The schemes shown in Figures 9. 1 and 9. 2 are regulators that drive the system state to zero starting from any initial condition capable of rejecting impulse disturbances. In practice, it is often necessary to track a constant reference input r with zero steadystate error. 18

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Servo Problem 20

Servo Problem 20

Servo Problem The reference input of (9. 2) becomes v(k) = F r(k), and

Servo Problem The reference input of (9. 2) becomes v(k) = F r(k), and the control law is chosen as with r(k) the reference input to be tracked. The corresponding closed-loop system equations are 21

where the closed-loop state matrix is The z-transform of the corresponding output is given

where the closed-loop state matrix is The z-transform of the corresponding output is given by The steady-state tracking error for a unit step input is given by 22

Servo Problem For zero steady-state error, we require the condition If the system is

Servo Problem For zero steady-state error, we require the condition If the system is square (m = l) and Acl is stable (no unity eigenvalues), we solve for the reference gain 23

Example 9. 5 Design a state–space controller for the discretized state–space model of the

Example 9. 5 Design a state–space controller for the discretized state–space model of the DC motor speed control system described in Example 6. 8 (with T = 0. 02) to obtain (1) zero steady-state error due to a unit step, (2) a damping ratio of 0. 7, and (3) a settling time of about 1 s. 24

Solution The discretized transfer function of the system with digital-to-analog converter (DAC) and analog-todigital

Solution The discretized transfer function of the system with digital-to-analog converter (DAC) and analog-todigital converter (ADC) is The corresponding state–space model, computed with MATLAB, is 25

Solution The desired eigenvalues of the closed-loop system are selected as {0. 9 ±

Solution The desired eigenvalues of the closed-loop system are selected as {0. 9 ± j 0. 09} (see Example 6. 8). This yields the feedback gain vector and the closed-loop state matrix 26

Solution The feedforward gain is The response of the system to a step reference

Solution The feedforward gain is The response of the system to a step reference input r is shown in Figure 9. 8. The system has a settling time of about 0. 84 s and percentage overshoot of about 4% with a peak time of about 1 s. All design specifications are met. 27

Solution 28

Solution 28

Integral Control The control law (9. 21) is equivalent to a feedforward action determined

Integral Control The control law (9. 21) is equivalent to a feedforward action determined by F to yield zero steady-state error for a constant reference input r. Because the forward action does not include any form of feedback, this approach is not robust to modeling uncertainties. Thus, modeling errors (which always occur in practice) will result in nonzero steady-state error. To eliminate such errors, we introduce the integral control shown in Figure 9. 9, with a new state added for each control error integrated. 29

Integral Control 30

Integral Control 30

The resulting state–space equations are 31

The resulting state–space equations are 31

Integral Control The eigenvalues of the closed-loop system state matrix can be arbitrarily assigned

Integral Control The eigenvalues of the closed-loop system state matrix can be arbitrarily assigned by computing the gain matrix K using any of the procedures for the regulator problem as described in Sections 9. 1 and 9. 2. 32

Example 9. 6 Solve the design problem presented in Example 9. 5 using integral

Example 9. 6 Solve the design problem presented in Example 9. 5 using integral control. 33

Solution The state–space matrices of the system are In Example 9. 5, the eigenvalues

Solution The state–space matrices of the system are In Example 9. 5, the eigenvalues were selected as {0. 9 ± j 0. 09}. Using integral control increases the order of the system by one, and an additional eigenvalue must be selected. The desired eigenvalues are selected as {0. 9 ± j 0. 09, 0. 2}, and the additional eigenvalue at 0. 2 is chosen for its negligible effect on the overall dynamics. This yields the feedback gain vector 34

Solution The response of the system to a unit step reference signal r is

Solution The response of the system to a unit step reference signal r is shown in Figure 9. 10. The figure shows that the control specifications are satisfied. The settling time of 0. 87 is well below the specified value of 1 s, and the percentage overshoot is about 4. 2%, which is less than the value corresponding to ζ = 0. 7 for the dominant pair. 35

Solution 36

Solution 36

State Estimation In most applications, measuring the entire state vector is impossible or prohibitively

State Estimation In most applications, measuring the entire state vector is impossible or prohibitively expensive. To implement state feedback control, an estimate xˆ (k) of the state vector can be used. The state vector can be estimated from the input and output measurements by using a state estimator or observer. 37

Estimators In the previous section, we have designed a state feedback controller, assuming that

Estimators In the previous section, we have designed a state feedback controller, assuming that all states are measured. We also know, however, that not all states can be measured, for a detailed discussion on this topic. Does this mean that the state feedback controllers cannot be implemented? The answer is that we can obtain a reasonably equivalent state feedback controller through estimated state measurements. A state estimator, or simply an estimator, is a mathematical construct that helps estimate the states of a system. 38

Full-Order Observer To estimate all the states of the system, one could in theory

Full-Order Observer To estimate all the states of the system, one could in theory use a system with the same state equation as the plant to be observed. In other words, one could use the open-loop system xˆ (k +1) = Axˆ (k) + Bu(k) However, this open-loop estimator assumes perfect knowledge of the system dynamics and lacks the feedback needed to correct the errors that are inevitable in any implementation. The limitations of this observer become obvious on examining its error dynamics. We obtain the error dynamics by subtracting the open-loop observer dynamics from the system dynamics (9. 1). Where 39

Full-Order Observer The error dynamics are determined by the state matrix of the system

Full-Order Observer The error dynamics are determined by the state matrix of the system and cannot be chosen arbitrarily. For an unstable system, the observer will be unstable and cannot track the state of the system. A practical alternative is to feed back the difference between the measured and the estimated output of the system, as shown in Figure 9. 12. This yields to the following observer: 40

Full-Order Observer 41

Full-Order Observer 41

Full-Order Observer Subtracting the observer state equation from the system dynamics yields the estimation

Full-Order Observer Subtracting the observer state equation from the system dynamics yields the estimation error dynamics The error dynamics are governed by the eigenvalues of the observer matrix. We transpose the matrix to obtain 42

Full-Order Observer Theorem 9. 2: State Estimation. If the pair (A, C) is observable,

Full-Order Observer Theorem 9. 2: State Estimation. If the pair (A, C) is observable, then there exists a feedback gain matrix L that arbitrarily assigns the observer poles to any set {l , i = 1, . . . , n}. Furthermore, if the pair (A, C) is detectable, then the observable modes can all be arbitrarily assigned. i 43

Example 9. 8 Determine the observer gain matrix L for the discretized state–space model

Example 9. 8 Determine the observer gain matrix L for the discretized state–space model of the armature controlled DC motor described in Example 7. 15 with the observer eigenvalues selected as {0. 1, 0. 2 ± j 0. 2}. 44

Solution Recall that the system matrices are 45

Solution Recall that the system matrices are 45

HW 7 9. 11 9. 15 46

HW 7 9. 11 9. 15 46