K. J. Astrm and B. Wittenmark, Computer Controlled Systems: Theory and Design (Prentice-Hall International, Inc., London, 1990).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the delay operator q 1 and w(t) is the innovation sequence. The observation can thus be expressed as a weighted sum of old observations and an innovation y(t) a 1 y(t 1) a ny( t n) w(t 1) 6. 59) The Wiener Hopf solution for the predictor is equivalent to the steady state Kalman predictor [60]. It is thus convenient to use the state space formulation of (6.58) z(t 1) Az(t) cw(t) y(t) c with A = B B B 1 . n 1 . 010 C C C (6.61) and c = 1 0. 0] The states in (6.60) consists of old observations, z(t) y(t) y(t 1) y(t n 1) The Kalman ....

K.J. Astrom and B. Wittenmark, Computer Controlled Systems: Theory and Design, Prentice Hall, Inc., Englewood Cli#s, NJ, second edition, 1990.

....when, for example, the control signal saturates. The conditioning technique as an anti windup and bumpless transfer scheme was originally formulated by Hanus et al. 14, 15] as an extension of the back calculation strategy of Fertik and Ross [11] to a general class of controllers. Astrom et al. [3, 2] proposed that an observer be introduced into the system to estimate the states of the controller and hence restore consistency between the saturated control signal and the controller states. Walgama and Sternby [20] have very clearly exposed this inherent observer property in several anti windup ....

K. J. Astrom and B. Wittenmark. Computer Controlled Systems Theory and Design. PrenticeHall, Inc., Englewood Cliffs, N.J., 1984.

....(with a linear controller) and may be interpreted in that light. Realistic problems will likely be a hybrid. Additionally, the B 1 ,D 1 terms have been added to allow the analysis of certain problems of interest (the Parallel Projection Operator (Section 8. 3) and the PI anti windup scheme in [1]) 3.3.7 Plant Output Saturation Configuration i oe 6 G(s) oe Figure 3.7: Plant Output Saturation Configuration The plant output saturation configuration is that which arises when the saturation element follows the plant (as shown in Figure 3.7) It should be noted that this ....

K.J. Astrom and B. Wittenmark. Computer controlled systems: Theory and design. Prentice-Hall, Englewood Cliffs, N.J., 1984.

.... BS AR BT t y c For the closed loop system performance to be identical to the desired reference, the closed loop polynomials and the reference model must be identical: B BS AR For the interested reader, this classic formulation of MRC is described in detail by Astrom and Wittenmark [20], including minimum degree solutions, observer polynomials, etc. To tailor the basic MRC approach for the DAC, the driver front steering input will act as a reference command to the MRC of the rear steering model while simultaneously acting as a known output disturbance to the vehicle s yaw rate. ....

....are given as , 20714 s 3727 3 . 155 ) s s s R 48142, 5294 98 . 49 181 . 5 ) s s s s S , 834385 18542 103 ) s s s T . 23 . 90 90 2 ) 0 = s s s A A 0 is a standard observer polynomial utilized in MRC controllers. The reader is referred to [20] for further details. Fig. Error Not a valid link. shows the output of the reference model and the response of the vehicle with DAC for a double lane change. Without DAC, Fig. 8, the system is operating in an open loop. Clearly, with the DAC, the vehicle s yaw degree of freedom behaves much more ....

K.J. Astrom and B. Wittenmark, Computer-Controlled Systems: Theory and Design, 3rd ed. Upper Saddle River, New Jersey: Prentice Hall, 1997.

....forward control packets(FCPs) travelling through it in the forward direction, thereby solving the feedback explosion problem. The essence of the proposed flow control algorithm is the way to compute fair rates. It is based on proportional and integral (PI) control in the feedback control theory [1] [2] and has the following form in discrete time. For each outgoing link of a node, its fair rate f j [k] is updated upon every T epoch by f [k] 1) where Cp 0 and C I 0 are the proportional and the integral control gains respectively, q[k] is the queue length at the link ....

....is given in Figure 2. 2.1 Router Algorithm 2.1.1 Fair Rate Computation Fair rate computation runs independently on each outgoing link to find the max min fair rate among all sessions sharing the link. The proposed fair rate computation is based on PI control in the feedback control theory [1] [2] and has the following form. For each outgoing link j, its fair rate, f j [k] is calculated periodically upon every T epoch by f j [k] j ) 2) where Cp 0 and C I 0 are the proportional and the integral control gains respectively. The choice of Cp and C I determines the ....

K. J. Astrom and B. Wittenmark. Computer Controlled Systems: Theory and Design. Englewood Cli#s: Prentice-Hall, NJ, 1984.

....state feedback, but lateral velocity states are inherently difficult to determine. Relatively few of the previous investigations attempt to actually make the vehicle behave as if it had a different set of dynamics through feedback. To construct the DAC, a model reference controller (MRC) strategy [24] will be used along with a modification based on rejection of known disturbance dynamics. Model reference (model following) controller design is a method by which desired closed loop characteristics can be introduced into a system, i.e. a pole placement method. The basic MRC approach has a ....

....polynomial matching to (31) 32) where . As long as and are relatively prime, 31) has a solution. For , and to be causal, the following conditions must be satisfied: degree degree degree degree (33) A more detailed discussion of this topic, including minimum degree solutions, is given in [24]. To tailor the basic MRC approach for the DAC, the driver front steer input will act as a reference command to the MRC of the rear steer model while simultaneously acting as a known output disturbance to the vehicle s yaw rate. Assume the same reference model from (26) but the plant dynamics of ....

K. J. Astrom and B. Wittenmark, Computer Controlled Systems: Theory and Design. Englewood Cliffs, NJ: Prentice-Hall, 1997.

....a collection of N subsystems, each with their own controller: # # ###### # ## # # ### # #### # ######## (9) where the round trip delays have been ordered such that # # # # ### # # without loss of generality. Our control aim now is to minimize ### # ####. This is equivalent (refer to [15]) to requiring # # # #####for all k. Taking the expectation of (9) and setting # # # #####,we have the following general control law for each subsystem: # # #### # # # ## # # # # #### # # ## (10) Thus, we require the allowed rates of the individual source rates # # ### to be equal to the ....

K. Astrom and B. Wittenmark, Computer-Controlled Systems Theory and Design. Prentice Hall, New Jersey, 3rd ed., 1997.

....the available bandwidth for a particular elastic source (3) The system now becomes a collection of subsystems, each with their own controller (4) where the roundtrip delays have been ordered such that without loss of generality. Our control aim now is to minimize . This is equivalent (refer to [12]) to requiring for all . Taking the expectation of (4) and setting , we have the following general control law for each subsystem: 5) Thus, we require the allowed rates of the individual source rates to be equal to the predicted values of the system parameters , which are determined by the ....

K. strm and B. Wittenmark, Computer-Controlled Systems Theory and Design, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1997.

....the response for the discrete time design becomes even more oscillatory, and shows a very high peak in the frequency response. The details may be found in [20] 8 Bibliographical Notes For classical treatments of sampled data control, it is instructive to consult [21, 28, 34] The textbooks [1, 35] covers both classical and modern aspects of digital control. For discrete time design methods and other related topics, the reader is also referred to the Handbook [27] As noted in the main text, discrete time lifting has been introduced and re discovered by several authors. See, for example, ....

K. J. Astrom and B. Wittenmark, Computer Controlled Systems---Theory and Design, 3rd Edition, Prentice Hall, Upper Saddle River, 1996.

....(21) is not much bigger than t ott f fl flT(t) Gamma T fl ; 22) t) is a vector with all its entries equal to T (t) 5. 1 Discretization in time In an RTP system under automatic temperature control it is almost certain that the lamps will be controlled by a digital computer [FPW90, AW90] The 24 lamp power signals will most likely be zero order hold signals, that is, signals that are constant between sample instants. In this section Deltat will denote the sampling period; accordingly, 1 = Deltat is the sample rate. It will be assumed that the length of the interval [t o ; t f ....

K. J. Astrom and B. Wittenmark. Computer Controlled Systems: Theory and Design. Prentice-Hall, 2nd edition, 1990.

....places. A Place changes its status as token = AnyTrue(inp.set) or (pre(token) and not AnyTrue(out.reset) The reporting of the state about accessible token to output transitions takes care of giving priority to transitions. The first transition gets information if the place has a token out[1].available = pre(token) The token is hidden to the second transition if the first transition decides to fire and sends a reset condition: out[2] available = out[1] available and not out[1] reset; The general case can be written in Modelica as if i= 1 then pre(token) else ....

....output transitions takes care of giving priority to transitions. The first transition gets information if the place has a token out[1] available = pre(token) The token is hidden to the second transition if the first transition decides to fire and sends a reset condition: out[2] available = out[1].available and not out[1] reset; The general case can be written in Modelica as if i= 1 then pre(token) else out[i 1] available and not out[i 1] reset; A Place needs to signal if it can accept a token by telling if it has a token or is about to receive one via transitions with higher ....

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K.J. strm and B. Wittenmark. Computer-Controlled Systems --- Theory and Design. Prentice Hall, 3 edition, 1996.

....in the control community to derive the linear gain vector L. L is found by minimizing a loss function J, which is a performance index for a controller: the lower the better and when it means that the closed loop system becomes unstable. A commonly used and general loss function is found in e.g. [strm and Wittenmark, 1997] (see table 1 for the notation) 5) for continuous time and with a unique conversion: 6) for discrete time control design, where Q N and Q are defined by the other weight matrices. The minimum value of the loss function is given by: 7) Those equations are general and are sometimes modified. ....

....loop. In the control design of the closed loop, the open loop dynamics and the disturbance dynamics are taken into account. The choice of control period can e.g. be guided by a rule of thumb: for a first order process the number of sampling instances per rise time should be between 4 and 10, [strm and Wittenmark, 1997] When moving from control system design to implementation in a computer system the choice of control period h c becomes the choice of task period P i . With a liberal choice of h c , exemplified by the rule of thumb above, the trivial upper limit on the utilization U, 10) of a processing unit, ....

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K.-J. strm, B. Wittenmark, "Computer Controlled Systems - Theory and Design", 3rd edition, ISBN 0-13-314899-8, 1997.

....for a practical application. A bad choice of sampling period can lead to loss of controllability or observability. Those requirement are so vital that they can not be negotiated. Properties of a closed loop which could serve as quality measures are based on traditional control design, see e.g. [11]: Control performance. Control performance can be specified in the time or frequency domain. There are several ways this can be done. One way is to construct a function to optimize. Sensitivity. The sensitivity is a measure of how much the closed loop is affected by a perturbation. Bad ....

....where the three processes in equation (4) for convenience have been used to illustrate the relationship. The selected control periods h P,i T i P c i a i d i p i , J P i , 1 h p i , E y i 2 t ( t d kh k 1 ( h = 5 for each process follow the rule of thumb in [11] and are marked with dots in the figure. The loss function is interpreted in the following way: the smaller the loss, the better the performance, i.e. the smaller is the variance of the control error. When the loss goes to infinity this means that the control loop becomes unstable. Figure 2. The ....

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K.-J. strm, B. Wittenmark, "Computer Controlled Systems - Theory and Design", 2nd edition, ISBN 013 -172784-2, 1990.

....system equations can be written as # ( xt Axt But t kh k h yt Cxt kk = 1 1 , ut Kxt t kh k kk = 012# (8) where ut( is piecewise continuous and only changes value at kh k . Sampling the system with period hwe obtain [14] xk h xkh ukh uk h ykh kk ( 11 01 = Cx kh ( where = e eBds eBds Ah k As h k As h h k k , 0 0 1 Defining zkh x khu k h TT T ( 1 as the augmented state vector, the augmented closed loop ....

....period (say, 0 k lh, l 1 ) one may receive zero, one, or more than one (up to l) control sample(s) in a single sampling period. In the special case where ( lh lh k 1 for all k, one control sample is received every sample period for kl . In this case, the analysis follows that in [14], resulting in ( k I K kk = 10 0 00 0 000 # # ##### # , 10) where kklh = 1 and the augmented state vector is zkhxkhuklh ukh TT T T ( # 1 . In the more general case, tedious bookkeeping must be ....

K.J. strm and B. Wittenmark, Computer-Controlled Systems: Theory and Design, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1997.

....A distributed system has a natural potential of hardware to detect and recover from errors. 2.2 Characteristics of Distributed Computer Control Systems 2.2. 1 Control application The sequence of actions in a regular control loop is: read A D, calculate output, write D A and update state, [CCS90]. These are partitioned into several tasks depending on the degree of distribution and timing requirements. Two instances are generally not executed or transmitted at the same time. There are different kinds of tasks and messages in a regular control system: 1) mode change requests and other ....

....and latency that satisfy these requirements are good for a feasible schedule. The deviation of poles from their nominal locations, as a performance index, has also been proposed, SHI95] 3. 2 Control Delay A constant control delay can be incorporated in the model for the controller design, [CCS90]. This will counteract performance degradation imposed by the pure phase delay. SHI95] investigates the effect on constant control delay from both performance and stability aspects. When scheduling a set of tasks and messages it is important to make the latency as small as possible. Some ....

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K.-J. strm, B. Wittenmark, "Computer Controlled Systems - Theory and Design", 2nd edition, ISBN 0-13-172784-2, 1990.

....the continuous like performance under this discrete control. From Nyquist sampling theorem and from control theory it is known that, in order to have a continuous like performance of the system, the ratio of the rise time of the system over the sampling time must fall into the interval (2, 4) [24]. Consider the principal root, of the normalized system (22) If , the system in the asymptotic region is nonoscillatory and hence the rise time of the system, denoted by , is equal to the time constant . Otherwise, the system in the asymptotic region is oscillatory and the rise time of the ....

....and hence the rise time of the system, denoted by , is equal to the time constant . Otherwise, the system in the asymptotic region is oscillatory and the rise time of the system is given by where and , satisfying and , are the natural frequency and the damping ratio, respectively, and [24]. Therefore, in order for the closed loop dynamics with discrete time control to have the continuous like performance, must be satisfied. VII. ESTIMATION In Section IV, we showed that normalization of the controller gains by makes the closed loop dynamics to be virtually independent of . ....

K. J. strm and B. Wittenmark, Computer Controlled Systems: Theory and Design. Englewood Cliffs, NJ: Prentice-Hall, 1984.

....k e = is the plan output error in the k th discrete time step, t t N t D = max the maximal number of time steps (250 0.2) and t D =0.2 min is the sampling time. Among the conventional solutions, the proposed control algorithm was compared with fixed gain discrete PI controller formulated as [2]: 1 ( 1 ( 1 0 = k e q k e q k u k u (47) D = i T t K q 1 0 , K q = 1 (48) The parameters of the optimal PI controller were determined by optimization with Sequential Quadratic Programming method with MATLAB Optimization Toolbox s constr function [7] The ....

K. J. Astrm and B. Wittenmark, Computer Controlled Systems: Theory and Design (Prentice-Hall International, Inc., London, 1990).

....in discretizing a continuous time system be adequate to capture the dynamic behaviour of such a system. Sampling Frequency For identification purposes, the rate at which the plant is sampled is selected in order to satisfy the following consideration: Theorem 4.1. Shannon s Sampling theorem [2]) A continuous time signal with a Fourier transform that is zero outside the interval ( Gammaw 0 ; w 0 ) is determined uniquely by its value at equidistant points if the sampling frequency is higher than 2w 0 . In practice, we should sample a continuous time system at least twice as quickly as ....

....n 1) u(k Gamma 1) u(k Gamma m) Delta u(k) 4.3) The functions in (4.3) are written in short form as f ( Delta; w) and g( Delta; v) where (w; v) represent the free parameters (in this case synaptic weights and or biases of the neural networks) of the identification routine. As in [2], the system identification will proceed as follows. ffl Experimental planning: determine how input and output data will be generated to train the networks ffl model structure selection: select a skeleton model structure that the free parameters will be adjusted to fit. ffl parameter ....

K.J. Anstrom and B. Wittenmark. Computer-controlled systems - theory and design. Prentice Hall, NJ, USA, 1990.

....in discretizing a continuous time system be adequate to capture the dynamic behaviour of such a system. Sampling Frequency For identification purposes, the rate at which the plant is sampled is selected in order to satisfy the following consideration: Theorem 4.1. Shannon s Sampling theorem [2]) A continuous time signal with a Fourier transform that is zero outside the interval ( w 0 , w 0 ) is determined uniquely by its value at equidistant points if the sampling frequency is higher than 2w 0 . In practice, we should sample a continuous time system at least twice as quickly as the ....

....g[ y(k) y(k n 1) u(k 1) u(k m) u(k) 4.3) The functions in (4.3) are written in short form as f( w) and g( v) where (w, v) represent the free parameters (in this case synaptic weights and or biases of the neural networks) of the identification routine. As in [2], the system identification will proceed as follows. Experimental planning: determine how input and output data will be generated to train the networks . model structure selection: select a skeleton model structure that the free parameters will be adjusted to fit. parameter estimation: ....

K.J. Anstrom and B. Wittenmark. Computer-controlled systems - theory and design. Prentice Hall, NJ, USA, 1990.

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K. J. Astrm and B. Wittenmark, Computer Controlled Systems: Theory and Design (Prentice-Hall International, Inc., London, 1990).

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strm, K. J.,& Wittenmark, B. (1997). Computer-Controlled System: Theory and Design (3rd ed.). NJ: Prentice Hall.

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K. J. Astr om and B. Wittenmark. Computer-controlled Systems: Theory and Design. Prentice-Hall International Editions, 1990.

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K. J. Astrom and B. Wittenmark, Computer Controlled Systems: Theory and Design. NJ: Englewood Cliffs: Prentice-Hall, 1984. 26

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K. J. Astrom and B. Wittenmark. Computer Controlled Systems Theory and Design. Prentice-Hall, Inc., Englewood Cliffs, NJ, 1984.

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AstrSm, K. J., Wittenmark, B., "Computer Controlled Systems: Theory and Design", Prentice Ha]J, 198d.

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