Giorgio Buttazzo, Manel Velasco, Pau Marti, Quality-of-Control Management in Overloaded Real-Time Systems, IEEE Transactions on Computers, v n Martin Törngren, Fundamentals of Implementing Real-Time Control Applicationsin Distributed Computer Systems, Real-Time Systems, v n.3, p. , May. This volume focuses on the design of computer-controlled systems, featuring computational tools that can be applied Karl Johan Åström, Björn Wittenmark.

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Skip to main content. Log In Sign Up. Computer Controlled Systems 3rd ed Astrom solutions. Many of the problems are intentionally made such that ocntrolled students have to use a simulation program to verify the analytical solutions.

This is important since it gives a feeling for the relation between the pulse transfer function and the time domain. Information about macros used to generate the illustrations can be obtained by writing to us. It is also important that a course in digital control includes laboratory exercises. The contents in the awtrom experiments are of course dependent on the avail- able equipment. Examples of experiments are P Illustration of aliasing P Comparison between continuous time and discrete time controllers P State feedback control.

Redesign of continuous time controllers wittenmagk well as controllers based on discrete time synthesis P Controllers based on input-output design P Control of systems subject to stochastic disturbances. Finally we would like to thank wittenmadk and students who have helped us to test the book and the solutions. The pole is real.

For small values of h the pole is close to 1. Using the result from Problem 2. A state space represen- tation is given in Example A. The second pole will move from 1 to the origin when h goes from zero to infinity. Discrete time system is given by cf. This is the same system as in Example 2. In that case there exists a power series expansion and termwise differentiation is allowed. The first step is to translate G s to the corresponding pulse transfer operator Controllde z.

Use the method of page This implies that H3 cannot be obtained by sampling a continuous time system. H4 vomputer be rewritten as 0. The roots are 0. The roots are 2.

Computer Controlled Systems 3rd ed Astrom solutions | Jones Arraes –

There is one of the underlined elements that is zero which indicates that there is at least one root on the stability boundary. The complex conjugate roots are on the unit circle. The following table gives some values for the real and imaginary parts.

This implies dittenmark Wc has rank 2 and the system is thus reachable. Two steps, in general it would take 3 steps since it is a third order system.

Computer-Controlled Systems: Theory and Design, Third Edition

To use the final value theorem in Table 2. For the proportional controller we must then look at the derivative of the error, i. The pulse transfer function is Table 2. There is a pole zero cancellation and only the first order exponential mode is seen at the sampling points.


If the closed loop system is stable then the steady state error is zero if the integral controller is used. This mapping maps the left half plane on the unit circle. These zeros may very well be outside the unit circle. We see that the system settles after two samples and that the constraint on the control signal is fulfilled. Using the results in Problem 4. Wittemark dynamical observer 4.

Using the results from Problem 4. The reduced order observer 4.

In this case we want to find K such that i. In this case is the state but not the disturbance measurable. Compared with the controller in a there is a aetrom in the detection of the disturbance.

The estimate of v is also shown for case c. Systemd the difference wittenmarj scale in the upper left curve. If only the output is measurable then the state and the disturbance can be estimated using an observer of the form 4.

The solutions above have the draw- back that there may be an error in the output due to the disturbance if there are small errors in the model. An witenmark can be incorporated as shown in Section 4. We get the following system of equations computrr determine the state feedback vector.

The parameter Xc will not influence the characteristic equation, but it is a feedforward term from wittenmaro reference signal, see Fig. Compare the discussion in the solution of Problem 4. It is seen that the controller is able to eliminate the disturbance. The closed loop characteristic equation is stable if See Example 3. The stability area is shown in Fig. Wkttenmark that the deadbeat control in a. The closed loop system will be unstable if the feedback from x1 is disconnected i.

To use Algorithm 5. Solution with higher order observer: The solution above is not the only one solving the original problem. To solve the equation for the closed loop characteristic polynomial we must increase the order of R by one. In this case there is a delay of one sample from the measurements y to the control signal u.

In both cases we get the same closed loop pulse-transfer function from uc to y since the observer polynomial is cancelled by T z: You can see the effect of the observer polynomial when regulating a nonzero initial state, but not in the response to a set point change.

The dots corresponds to the zero order controller, and the crosses to the first order controller.

Cancellation of process poles and zeros is handled by Algorithm 5. Now, we can solve the Diophantine equation above, since we have 3 indeterminates r0s0 and s1 and 3 coefficients to set: In this case we do not want to cancel the process zero, so 0. Case a should probably be avoided because of the ringing in the control signal.


Computer-Controlled Systems: Theory and Design, Third Edition

The ringing in the control signal in Case a is due to the cancellation of the process zero on the negative real axis. Further the pole of the process is can- celled. Bm must contain the zeros of B that are outside the unit circle. The first constraint is the same as for the polynomial design discussed in Chapter 5.

By forcing the factor z 1 into R z we thus have obtained integral action in wittenmatk controller. The design problem is solved by using the general Algorithm 5.

The reference value is zero and there is an initial value of the state in the process. The design in Problem 5. The right hand side of the inequality is also shown solid. The right hand side is also shown. Further the desired jj model is stable, i.

If the desired response is slower than the open loop system then the final value is the largest one.

The rule of thumb on p. The fact that we cancel one zero and do not introduce any other zero in Bm causes the delay from the command signal to be one time unit more than the delay of the process. The controller polynomials are now given by 5.

A disturbance in the process will propagate in the direction of the flow. The system is then represented with the block diagram in Fig. Notice the order of the states. The system will remain stable for all positive values of K i. A disturbance will now propagate in the direction opposite the flow. A disturbance in uc will now only influence the first subprocess and will not propagate along with the flow.

The reader is strongly recommended to compare with the case where the disturbance appears at the final product storage instead. Cascade control loops are found for the cooling media witenmark and for the output product flow.

Feedforward is used for the level control loop where the input flow is used as a measurable disturbance. The input flow is also used as feedforward for the cooling of the jacket.

Nonlinear elements are used in the flow control loops of the product output and the coolant flow. Conputer flow is probably measured using differential pres- sure which is proportional to the square of the flow.

The square root device is thus used to remove the nonlinearity of the measurement device. An in- tentional nonlinearity is introduced in the selector.