Digital control system

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EEE 188 Review Problems Date: 05/01/2018

Problem 1: Controllability of a two-tank system

The hydraulic system shown in figure 1 consists of two tanks. It is obvious that the input cannot change level x1. Without any calculations, is the system (two tanks) controllable? If your answer is yes, do the following

1) Use the Ziegler-Nichols method to design a PID controller so that r1 = 10cm,r2 = 12cm 2) Design state feedback with integral action to place the poles at 0.15,0.11,0.55.

Problem 2: Observability without the output matrix

For a given system x(k + 1) = −0.25x(k) + αu(k) (1)

Are there any restrictions on the values of α for the system to be observable? If the answer is yes, discuss these restrictions and design an observer for the system.

Problem 3: State feedback for cruise control

We want to design a cruise control system to keep constant speed at 40 (the unit is miles/ hour). The schematics and block diagrams are shown in figure 2. In this problem we can measure the speed of the car. Newton’s law gives

v̇ = − b

m v +

1

m u (2)

where m is the car’s mass and b is damping coefficient. It is assumed that the parameters of the system are:

m = 1000kg, b = 50N.s/m (3)

Therefore, the continuous time system is given by

v̇ = −0.05v + 10−3u (4)

The discrete time state space model is v(k + 1) = 0.95v(k) + 10−3u(k) (5)

The sampling time for this problem is T = 1s. The output equation is given by

y(k) = v(k) (6)

Fig. 1. A system of two tanks

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Fig. 2. Cruise Control system

We want to design state feedback with integral action. Under this formulation, the augmented system is given by

x(k + 1) =

[ v(k + 1) z(k + 1)

] =

[ 0.95 0 −1 1

][ v(k) z(k)

] +

[ 10−3

0

] u(k) +

[ 0 1

] r(k) (7)

where the augmented state space vector is x = [ v z

]T 1) What is the numerical value of C (output matrix)? 2) What is the size of the gain matrix K under the proposed control law? 3) Design state feedback with integral control to place the eigenvalues of the closed loop system at 0.3 and 0.6. Note

that:

det

( λI −

([ 0.95 0 −1 1

] − [ 10−3

0

][ K1 K2

])) = λ2 − K2/1000 − (39λ)/20 − K1/1000 + (K1λ)/1000 + 19/20

4) Write the numerical values of the state transition matrix of the closed loop system Acl = A − BK 5) Find the numerical values of the eigenvalues of the closed loop system (Acl). 6) How are these closed loop system eigenvalues related to the desired eigenvalues? 7) What does the second variable in the augmented system (variable z) represent (error, square of the error, integral

of the error . . .)?

Problem 4: Controllability and observability

In this problem we want to control the liquid levels h1 and h2 in the coupled tank system of figure 3. Assuming the cross sectional area of the tanks is equal to 1 (SI unit), the liquid levels can be described by the following equations in the continuous state space domain

ḣ1(t) = qi(t) − qb(t) (8) ḣ2(t) = qb(t) − qc(t)

where qi and qc are control variables. The flow qb between Tank 1 and Tank 2 cannot be used to change the liquid levels in a controlled way, therefore it is treated as a disturbance. The desired values for the liquid levels are r1 = 2cm and r2 = 3cm for Tank 1 and Tank 2, respectively.

1) Write the discrete time model corresponding to system (8). For simplicity, we take a sampling time of 1 second (T = 1s).

2) Assuming the discrete time model is given by

h1(k + 1) = h1(k) + qi(k) − qb(k) (9) h2(k + 1) = h2(k) + qb(k) − qc(k) (10)

y1(k) = h1(k) (11)

y2(k) = h2(k) (12)

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Fig. 3. Two Tank system for problem

where y1,y2 represent the outputs. Write matrices A,B and C for the system. Recall that qb is a disturbance, therefore, its is not part of the input space (you can assume that qb = 0 when writing the input matrix).

3) Build the controllability matrix 4) Is the system controllable?Explain. 5) Build the observability matrix 6) Is the system observable?Explain 7) Write simple equations for the errors e1(k),e2(k) in terms of the liquid levels and the desired values.

Problem 5 PI controller design

This problem can be seen as a continuation of the previous problem. We want to design a proportional and integral controller for Tank 1 and just a proportional controller for Tank 2. Based on this, the expression in the z-domain of the control inputs are given by

Qi(z) = K1E1(z) + K2 z

z − 1 E1(z) (13)

Qc(z) = K3E2(z) (14)

where K1,K2 and K3 are the controllers gains and E1(z) and E2(z) are the errors for Tank 1 and Tank 2, respectively.

1) After closing the loop for Tank 2, we obtain

H2(z) = −K3R2(z) + Qb(z)

z − 1 − K3 (15)

What is the interval of K3 to keep a stable closed loop system? 2) Design a proportional controller to place the pole at 0.5. 3) After closing the loop for Tank 1, we obtain

H1(z) = (K1z + K2z − K1)R2(z) − (z − 1)Qb(z)

z2 + (K1 + K2 − 2)z + 1 − K1 (16)

Design a proportional and integral controller to place the poles at 0.5,0.4. Use pole placement technique. 4) The closed loop response for the liquid levels is shown in figure 4 (top corresponds to h1(k) and bottom to h2(k))

What is the steady state error for h1 from the graph?

4

0 2 4 6 8 10 12 14 16 18 20 0

0.5

1

1.5

2

2.5

L e v e l h 1

time (s)

h1(k)

r1 = 2cm

0 2 4 6 8 10 12 14 16 18 20 0

1

2

3

4

L e v e l h 2

time (s)

r2 = 3cm

h2(k)

Fig. 4. Evolution of h1 and h2. The desired values are r1 = 2cm and r2 = 3cm for Tank 1 and Tank 2, respectively.

5) What is the steady state error for h2 from the graph? 6) Why is the steady state error zero for one tank but not for the other one? 7) Propose a simple way to achieve zero steady state error for tank 2. No calculations needed. 8) Under the proposed control laws, the controllers’ actions are given by

qi(k) = K1e1(k) + K2

k∑ n=0

e1(k) (17)

qc(k) = K3e2(k) (18)

Knowing that at time k = 6, the liquid levels are h1(6) = 2.38 and h2(6) = 2.58, calculate the control actions at time k = 7. Note that: qb = 0.1,K1 = 0.5,K2 = 0.15,K3 = −0.2,r1 = 2,r2 = 3. Note that the area between the curve representing h1 and r1 in the time interval [0,6] is 1.33.