Which of the following components is an exclusive Component of a closed loop system

4.27.6.3 Problems due to Mineral Scales, Muds, and Sludges in Closed-Loop Water Systems

Closed-loop systems contain a finite volume of water and evaporative processes do not take place, although minor water losses because of leaks will occur. Closed-loop systems will include engines under circumstances where they are employed in emergency/backup electricity generating systems and similar applications (e.g., standby diesel generators for hospitals, commerce, and general industry, and diesel engine utility power generators). Some closed-loop systems are not in fact ‘closed’ and may have, for example, a returned water receiver that channels a number of individual cooling water lines to a single point before being pumped across a heat-exchanger and around another loop.

As a general rule, all closed and semi-closed loop systems tend to suffer to some extent or other from iron and steel corrosion problems because of the prolific use of mixed metals, poor passivation, and limited maintenance. Therefore, one of the most common and difficult foulants found in closed systems is a black mud made up of compacted, fine, black magnetic iron oxide particles, which deposit at heat-transfer surfaces, and clog or block narrow passages in unit heaters, fan coil units, and cooling, reheat, and heating coils in air-handling units. This mud is a result of very fine wet particles of black magnetic iron oxide compacted into a dense adherent mud.

The interior of black iron piping, commonly used for recirculating water, has a natural black iron oxide protective coating ordinarily held intact by oil-based inhibitors used to coat the pipe to prevent corrosion during storage and lay-up. This natural iron oxide protective coating is called mill scale, a very general term which can be applied to any form of pipe scale or filings washed off the interior of the pipe. This mill scale film becomes disturbed and disrupted during construction because of the constant rough handling, cutting, threading, and necessary battering of the pipe. After construction, the recirculating water system is filled and flushed with water, which removes most of the loosened mill scale along with other construction debris. However, very fine particles of magnetic iron oxide will continue to be washed off the metal surface during operation, and in many instances this washing persists for several years before it subsides. Mill scale plugging can be a serious problem. It is best alleviated in a new system by thorough cleaning and flushing with a strong, low-foaming detergent–dispersant cleaner. This, however, does not always solve the problem. Even after a good cleanout, gradual removal of mill scale during ensuing operation can continue.

Lemma 3.2

The closed loop system in Figure 3 is asymptotically stable if and only if (Ap-Bpkp-BpkeCp) has all its eigenvalues in the left half plane. Proof: The closed-loop system can be represented by

(3.30)[X˙mX˙p]=[Am0Bpkm+BpkeCmAp−BpkpCp][XmXp]+[BmBpku]um[ymyp]= [Cm00Cp ][XmXp]

The closed loop system eigenvalues are those of the model Am and those of (Ap-Bpkp-BpkeCp). The reference model is always a stable system and Am has all its eigenvalues in the left half plane. Hence the closed-loop system is asymptotically stable if and only if (Ap-Bpkp-BpkeCp) has all its eigenvalues in the left half plane.

Note that the stability is the most important design criterion and must be maintained at all cost. If CpBp has full row rank and a stabilizing ke can be found such that (Ap-Bpkp-BpkeCp) is stable with kp obtained from equation (3.26), then the closed-loop system is stable and PMF is achieved. However, when such ke does not exist, a trade-off between tracking and stability must be made. In addition, the error dynamics should also be considered to ensure proper transient response. Based on these considerations, a design procedure is proposed as follows:

B THE PREDICTIVE ADAPTIVE CONTROL SCHEME

It has been mentioned earlier that precision high bandwidth control could be achieved using non–colocated tip position feedback, whereas the colocated hub rate feedback tends to stabilize the overall control system. Therefore, by choosing a proper trade–off between these two output feedback gains, one could obtained a desirable stable control, yet provide a satisfactory transient characteristics. The predictive adaptive controller is formulated based on the above observations.

The control signals are determined at each sampling instance under the assumption that parameters of the process under control remain constant (and equal to their estimates) during the current control sampling interval. The control is determined so as to minimize a quadratic cost function of the predicted output error and control increment. Note that the following formulation is directly derived for single input, multiple output (SIMO) processes.

Consider a SIMO process modeling a flexible robot arm by a reduced order ARMA process given in Eq. (33). We can rewrite that equation in a state space representation as:

(70)xk+1=Axk+bT˜kzk=Cxk

Here elements of A, b and C depend on the coefficients of the polynomials A2(q− 1), B2(q− 1), A3(q− 1) and B3(q− 1) given in Eq. (33). Since only the hub–rate and tip position are utilized in the feedbacks, the output vector here will contain the non–dimensionalized hub–rate and tip position measurements.

The main steps in the proposed predictive adaptive control algorithm are shown in Fig. 3 where the ith element zi(k) of the output vector z(k) is shown. At time instant k, we may predict output zi(k) at time k + 1 according to information from the past upto k, assuming that the input remains unchanged after time instant k − 1. However, the predicted output may be different from the desired output zd(k). The goal of the predictive adaptive controller is to bring the predicted system output to the desired position as close as possible. Hence, let us define z0(k + 1) as the value of output z(k) when assume:

ΔT˜k=T˜k−T˜k− 1=0

If instead a nonzero ΔT˜k is applied, the corresponding system output z(k + 1) renders the predicted system response to ΔT˜k as follows [32]:

(71)u^k=zk+1−z^0k+1 ΔT˜k

where z^0 k+1 denotes the estimated value of z0(k + 1). Furthermore, in Eq. (71), ûk explicitly gives the estimated system response to a unit step pulse. It has been pointed out [32,38] that because of the application of zero–order hold (every control signal is in the form of a step pulse), ûk will provide enough information for the purpose of control. Consequently, control increment ΔT˜k is determined to minimize the following cost-functional Jk + 1 such that, at sampling interval k + 1, system output z(k + 1) is brought to its desired position zd(k + 1) as close as possible, with a penalty on the magnitude of the control signal T ˜k.

(72)Jk+1=e^Tk+1Pe^k+1+ρ T˜2k

where êk+1 designates the estimated output error vector at time instance k + 1, and ρ is a weighting factor on the control cost. P is a chosen matrix which determines the relative weighting on the predicted tip position error and hub rate error. Definitions of êk+1 and ê0k+1 are given below.

(73)êk+1 =z^0k+1+ΔT˜ku^k−zdk+1 ê0k+1=z^0k+1−zdk+1

Here, êk+1 is the estimated output error assuming ΔT˜k=0.

The optimal solution to Eq. (72), subjected to constraints (70) and (73) can be obtained by choosing ΔT˜k such that:

∂Jk+1 ∂ΔT˜k=0

which yields

(74)ΔT˜k=−ρT˜k−1+ e^0Tk+1Pu^kρ+u^TkPu^k

This algorithm can be applied directly to the control of the flexible one arm robot. The control calculation procedure is as follows: ê0k+1 is derived by prediction (Eq. (70) with ΔT˜k=0), and ûk is approximated by applying Eq. (71) at the k − 1 interval. One assumption has been made in the above procedure that ûk−1, estimate at the previous interval, also applies to the current interval. Consequently, it is required that the system parameters remain constant or change slowly relative to sampling frequency (which is not a conservative assumption considering the availability of micro–computer technology for fast sampling rates). Furthermore, if adequate predictions z^0k+1 and z^k+1 can be achieved, and if z(k) changes slowly from one control interval to the next, the above control scheme will provide an adequate control sequence.

The stability of closed loop system is guaranteed if V(k + 1) = Jk + 1 is a Lyapunov function of system (70) as stated in the following theorem.

Or in a design problem, for a chosen weighting matrix P (symmetric and positive definite), if

(76) Q=−ATCTPCA−CTPC−ATCTPCbbTCTPCAρ+bTCTPCb

is symmetric and positive definite, then the PAC control strategy is optimal and asymptotically stable in the sense of Lyapunov.

Proof. In an optimal control problem, even though the quadratic performance index is minimized by a physically realizable system trajectory, the stability is not guaranteed [39,40]. The system is asymptotically stable, however, if the cost function is also a suitable Lyapunov function. Hence, we chose a Lyapunov function for system (70) as:

(77)Vk+1=e^Tk+1Pe^k+1+ ρT˜2k

Obviously, V(k + 1) is continuous in the state variable x(k) and the driving torque T˜ k, and is positive definite. Without loss of generality, it is assumed that zd = 0. Therefore, V(k + 1) has the following increment.

(78)ΔVk+1=Vk+1−Vk=xTkATCTPCA− CTPCxk+2xTkATCTPCbT˜k+T˜TkbTCTPCbT˜k+ρT˜2k−ρT˜2k−1

Furthermore, ΔV(k + 1) has a minimum value with respect to T˜ k because it is in a quadratic form of ΔT˜k. ΔV(k + 1) is minimized if we choose T˜k in such a way that

∂ΔVk+1∂T ˜k=0

which renders the following solution:

(79)T˜k= xTkATCTPCbρ+bTCTPCb

Substituting Eq. (79) into Eq. (78) gives:

(80)ΔVk+1=−xTkQx−ρ T˜2k−1

where Q is given in Eq. (75). As a result, ΔV(k + 1) is negative if Q is positive definite, guaranteeing an asymptotically stable control. Note that, for positive ρ (ρ > 0), Q can be a symmetric positive semidefinite matrix for asymptotic stability.

To illustrate that the control sequence from Eq. (79) is truly the PAC given by Eq. (74), keep in mind that:

T˜k=T˜k−1+ΔT˜k,e0k+1=C Axk+bT˜Tk−1 u^k=zk+1− z0k+1ΔT˜k=C b

Then we have Eq. (79) as:

(81)ΔT˜k=−ρ T˜k−1+e^0Tk+1Pu^kρ+u^TkPu^k

And it agrees with Eq.(74).

To summarize, the above shows that, if Eq.(75) holds and the control increment is determined according to the PAC algorithm Eq.(74), then the increment of the Lyapunov function Eq. (78) is negative definite. In other words, the control system is asymptotically stable in the sense of Lyapunov, and the control is also optimal in the sense that cost function (72) is minimized.

Furthermore, the increment of the Lyapunov function can be utilized as a measure of the transient system behavior, because it can be interpreted as a “distance” from an arbitrary state to the origin of the state plane. If this increment is minimized, the system has fast speed of response. We may therefore select proper value for ρ for the above argument. However, ρ must remain non–negative.

The weighting matrix P plays also a important role in the system transient behavior through Eq. (80). This equation reveals that, if Eq. (75)holds, the PAC will be stable. Moreover, the speed of response, measured by ΔV(k + 1), depends on the eigenvalues λi of Q. The bigger λi, the faster the response. The physical interpretation of the above argument in the control of a flexible robot is that increasing the weighting on the hub–rate, while decreasing the penalty on the tip position error, yields a stable control but with a slower transient response. If P is changed in the opposite way, one may obtain a higher closed–loop bandwidth (faster transient response), whereas the stability margin will be reduced. A good relative weighting between hub–rate and tip position error feedbacks should be such that all eigenvalues of Q in Eq. (75) are positive with bigger magnitude. In the following sections, the predictive adaptive output feedback control combined with lattice filter parameter estimator is applied to a specific one–arm flexible robot.

Theorem 2.1

The asymptotically stable closed-loop system (1),(3) admits implicit models if and only if the following conditions hold:

the closed-loop characteristic polynomial has the structure

(7)AR−BP=QC

(8)gcd( R,P)=gcd(R,P,Q)

polynomial T can be factorized as

(9)T=SC∘

(B0,C0)≜(B​,C)gcd(B,C)

at least in correspondence with one pair (A, B) solving the Bezoutian equation

(10)AR−BP=Q

(11)S(B°−C°B)is a multiple of R

Besides, condition (iii) can be substituted by the following

polynomial T(q−1) can be factorized as

(12)T =S¯C¯°

(A°,C°)≜ (A,C)gcd(A,C)

Similarly condition iv) can be substituted by

at least in correspondence with one pair (A, B) solving the Bezoutian equation (10),

(13) S(A°−C°A) is a multiple of P

Moreover, in case the above necessary and sufficient conditions hold, every triple (A, B, D), where (A, B) satisfy (10) and (11), or (13), and D is given by

(14)D=S(B°−C°B)R

or, equivalently, by

(15) D=S(A°−C°A)P

is an implicit model, and no other implicit model exists.

B DYNAMIC FEEDBACK CASE

We Consider the fixed order dynamic controller

(4.21)xck+1 =Acxck+Bczkuk=Ccxck+Hczk.

Let v = 0. Then the closed loop system has the following form

(4.22)xk+1=A^+B^GM^xk+D^wpk+ D^vvkyk= C^TM^TGTTxk+H^vk.

where x(∙), y(∙), Â, B^, D^, Ĉ and M^ are defined in (3.11). The fixed order dynamic controller design yields the standard static measurement feedback problem, as a special case.

Special Case: Full Order Dvnamic Controller

Consider the following strictly proper full order dynamic controller

(4.23)xck+1=Acxck+Bczkuk=Ccxck

By defining the closed loop system state vector x, output vector y and input vector w as (3.17), and defining the closed loop system matrix A, D and C as (3.18). The discrète time EOL∞ problem with full order dynamic controller can be stated as follows

(4.24)minAc,Bc, Cc.W>0traceNW−1.traceRCuX CuT;Cu=0Ccsubjectto2.1andσ¯[CiXCiT⋅traceNW−1≤εi2,i=1,2,…,m.

The proof is similar to that of Theorem 3.4.

7.2.4 The State Feedback Regulator (SFR)1

Assuming that all system states are available (measurable), a state feedback regulator may be designed to control the output around the zero state:

(7.84)uk=−Kxk

where K is the state feedback regulator (SFR) gain matrix. In order to determine the elements of K, uk is substituted in the state equation and z-transformed.

(7.85)xk+1=Axk−BKxk=A−BKxk

(7.86)xz=zI−A+BK−1zx0

The stability of the closed-loop system depends on the roots of determinant zI−A+BK=0 which is the closed-loop characteristic equation of the system. Note that for n states and m manipulated variables, the state feedback gain matrix K will have n×m unknown elements. One can specify n desired closed-loop poles and have n×m−n degrees of freedom. There are techniques to reduce the number of the degrees of freedom to zero.

Example 7.10

Design a state feedback regulator for the system given by

(7.87)xk+1=310.9−2xk+21ukyk=10xk

Which places two closed-loop poles at +0.5 and +0.6

(7.88)zI−A+BK=z−0.5z −0.6

(7.89)z−3−1−0.9z+2+2k12k2k1k2=z2+2k1+k2−1z+5k1−1.2k2−6.9

K=2.72−5.54

5.5.2 Direct Substitution Method

For a marginally stable closed-loop system, the closed-loop CLCE has a pair of pure imaginary roots. Therefore if s is replaced by jω in the CLCE shown in Fig. 5.10, the corresponding controller proportional gain will be the maximum or the ultimate gain Kc,max=Kult.

Example 5.4

Using the direct substitution method, find the maximum or the ultimate controller gain for the system whose CLCE is given by

(5.25)CLCE=s3+3s2+3s+1+30Kc=0

CLCE=jω3+3jω2+3jω +1+30Kc=−jω3−3ω2+3 jω+1+30Kc=0+0j

The imaginary part of the previous polynomial identity yields the value of ω=±3, once substituted in the real part of the equation, it renders the maximum controller gain.

(5.26)−33+1+30Kc,max=0Kc,max =Kult=830

As expected, the result is the same obtained by the Routh test. The advantage of this method is that it can be used in a CLCE which has a “time delay,” without having to approximate the delay term with a Padé or Taylor series approximation.

Example 5.5

Using the direct substitution method, find the maximum or ultimate controller gain for the system that has a time delay given as follows:

(5.27)Gps= 2e−4s3s+1;Gvs=Gms=1;Gcs=Kc

(5.28)CLCE=1+2e−4s3s+1Kc=03s+1+2Kce− 4s=0

Use the direct substitution method:

(5.29)3jω+1+2Kc,maxe−4jω=0

Using the Euler equation, e− jθ=cosθ−jsinθ, we have

(5.30)3jω+1+2Kc,maxcos4ω−jsin4ω=0+0j

Real→1+2Kc, maxcos4ω=0Imag→3ω−2Kc,max sin4ω=0

Divide the imaginary part by the real part,

(5.31)sin4ω cos4ω=−3ω

Use trial and error to find ω=0.532rad/s. If we substitute this value in the real part of the CLCE,

(5.32)2Kc,maxcos4ω=2Kc,maxcos4×0.532=−2Kc,max −0.5288=−1

Kc,max=9.45

B NECESSARY CONDITIONS FOR PERIODIC OCC PROBLEMS

Consider the stochastic OCC problem. Let v(k,n) and wp(k,n) be zero mean cyclo-stationary processes with periodic static covariance matrices

(3.19a)limk→∞ℰwpk1n1wpTkn=Wpnδk1−kδn1−n;

(3.19b)limk→∞ℰvk1n1vTkn=Vpnδk1−kδn1−n.

Firstly, we consider the measurement feedback case by setting v(k,n) = 0. Let the periodic time-varying controller be

(3.20)u kn=Gnzkn;zkn=Mp nxpkn.

In this case the closed loop system matrices in (3.5) are defined by

(3.21a)An =Apn+BpnGnMpn;Dn=Dpn;

(3.21b)Cyn =Cpn;Cun=GnMpn.

❑

The proof is similar to the continuous case in chapter 3 of [9].

Publisher Summary

This chapter focuses on automated closed-loop systems, indirect control systems, and penetration control techniques for critical root beads. The range of control options for welding varies from the traditional open-loop manual systems based on welding procedures to complex closed-loop automated techniques. Improved monitoring techniques and a wide range of sensors make it possible to measure process performance in both manual and automated applications and should enable more consistent weld quality to be achieved. The application of the appropriate control approach should result in improved productivity and lower total cost. The application of all monitoring techniques and in particular computer-based systems is increasingly justified by economic and quality requirements. The sophisticated adaptive control systems should not, however, be used to compensate for avoidable deficiencies in joint preparation and component quality; this will often prove costly and ineffective.

3 Controller design

To assist controller development and subsequent stability analysis, the matrix sech⋅∈Rn×n,cosh⋅∈Rn ×nand vector tanh⋅∈Rnare defined as follows:

(3.3)Sechx=diagsechx1…sechxn

(3.4) Coshx=diagcoshx1…coshxn

(3.5)Tanh(x)=tanhx1…tanhxnT

where x = [x1, …, xn]T ∈ ℜn, and in this study, xi denotes the coordinates of cell in x, y, and z (namely qx,qy or qz). By utilizing the properties of hyperbolic secant, cosine, and tangent functions, the following inequalities can be derived:

(3.6)x≥Tanhx

(3.7)x2≥∑i=1nlncoshxi≥lncosh‖x‖≥12tanh2‖x‖

(3.8) xTTanhx≥TanhTxTanhx

(3.9) λmaxSech2x=1

where x=xTx, and λmax(⋅) denotes the maximum eigenvalue of a matrix.