Identification and Control of Multivariable Systems – Role of Relay Feedback

Batch and continuous systems are of multivariable in nature. A multivariable system is one in which one input not only affects its own outputs but also one or more other outputs in the plant. Multivariable processes are difficult to control due to the presence of the interactions. Increase in complexity and interactions between inputs and outputs yield degraded process behavior. Such processes are found in process industries as they arise from the design of plants that are subject to rigid product quality specifications, are more energy efficient, have more material integration, and have better environmental performance. Most of the unit operations in process industry require control over product rate and quality by adjusting one/more inputs to the process; thus making multivariable systems. For example, chemical reactors, distillation column, heat exchanger, fermenters are typical multivariable processes in industry. In case of chemical reactor, the output variables are product composition and temperature of reaction mass. The input variables are reactant or feed flow rate and energy added to the system by heating/ cooling through jackets. Product composition can be controlled by manipulating feed rate whereas rate of reaction (thereby temperature) can be controlled by changing addition/ removal rate of energy. But, while controlling product composition, temperature is affected; similarly, while controlling temperature of reaction mass, the composition gets affected, thus, exhibiting interactions between input and output variables. Distillation is widely used for separating components from mixture in refineries. Composition of top and bottom products are controlled by adjusting energy input to the column. A common scheme is to use reflux flow to control top product composition whilst heat input is used to control bottom product composition. However, changes in reflux also affect bottom product composition and component fractions in the top product stream are also affected by changes in heat input. Hence, loop interactions occur in composition control of distillation column. Thus, unless proper precautions are taken in terms of control system design, loop interactions can cause performance degradation and instability. Control system design needs availability of linear models for the multivariable system. The basic and minimum process model for multivariable system is considered here as 2x2 system. The outputs of the loops are given by

Introduction to PID Controllers -Theory, Tuning and Application to Frontier Areas 106 11 1 12 1 2 where y i are system outputs and u i are the system inputs, G is system transfer functions.
Eqn (1) In order to achieve desired quality, specified output characteristics at the cost of spending optimum inputs one needs to design a controller and run the plant under closed loop so that optimal production of product under safe operation. The first thing we need is to select input-output pairs, i.e., which output should be controlled by which input? This needs knowledge in control structure selection or interaction analysis. In the next section, a brief state of art on interaction analysis is presented. Relative gain array (RGA) (Bristol 1966) is the most discussed method for analyzing interactions and it is based on steady state gain information of MIMO processes. Control loops should have input-output pairs which give positive relative gains that have values which are as close as unity as possible. It is dependent on process models, independent of scaling of inputs and outputs and can include all ways of pairing in a single matrix. Niederlinski index (NI) is a useful tool to analyse interactions and stability of the control loop pairings determined using process gain matrix. NI is found by the following formula, where each element of G P is rational and is openloop stable. The values of NI need to be positive. A negative value of NI will imply that the system is un-stable. Ni is used to check if the system (more than 2x2) is unstable or not. NI will detect instability introduced by closing the other control loops. Generally, NI is not used for systems with time delays. Any loop pairing is unacceptable if it leads to a control system configuration for which the NI is negative. But both RGA & NI do not provide dynamic information on the process transients. They do not give information on change in in/op pairing for instances when there is a sudden load disturbance. Singular value decomposition (SVD) is a useful tool to determine whether a system will be prone to control loop interactions resulting in sensitivity problems that rises from model mismatch in process gains. SVD considers directional changes in the disturbances. SVD is applied to steady state gain matrix that is decomposed into product of three matrices, (CN) is defined as ratio between maximum and minimum eigenvalues. Generally if the CN < 50 then the system is not prone to sensitivity problems (a small error in process gain will not cause a large error in the controller's reactions). The greater the CN value, the harder it is for the system in question to be decoupled. An ideal system would have a CN number of one, where each control variable controls a single distinct output variable. CN value tells us how easy it is to decouple a system. Though SVD has good geometric interpretation in terms of selection of measurement and pairing of variables, SVD depends on input-output scaling.

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Moreover, with weak interactions and with large dimensional systems they induce to go for more criteria for selection of pairs. Morari resiliency index (MRI) is also used to select in/out pairs. . These gramian based interaction measures seem to overcome most of the disadvantages of the RGA. One key property of these is that the whole frequency range is taken into account in one single measure. Furthermore, these measures seem to give appropriate suggestions for controller structures selection. The use of the system H 2 norm as a base for an interaction measure has been proposed by Birk and Medvedev (2003) as an alternative to the HIIA. But, dynamic simulation is a powerful tool to be used to test the viability of a control scheme during various process disturbances. Controllers for MIMO systems can be of either multiloop (controllers are designed only for diagonal elements of process TF) or multivariable (controllers are designed for all the elements of the MIMO TF).
Multiloop control scheme has an edge over multivariable as the former can work even if a single loop fails. In presence of interactions between input/output, the process need to be decoupled and then multiloop controllers can be designed. When interaction effects produce a significant deterioration in control system performance, decoupling control should be considered. One of the most powerful and simplest ways of reducing or eliminating interaction is by altering manipulated and / or controlled variables. Improvement of closedloop performance needs proper tuning of controller parameters that requires process model structure and estimation of respective parameters. There are many methods to select input/output pairs or to design control structures, design control strategy (either PID or IMC or predictive or heuristics etc.) and tuning of controller parameters in literature. But because of hazy pictures on above selections, till today, it is difficult to choose correct pairs, carryout interaction analysis and choose tuning rules. Thus the aim of this chapter is to bring out a clear picture of identifying process parameters and designing controller for MIMO systems. The rest of the chapter is carried out as follows: section 2 discusses identification methods of multivariable systems. Interaction analysis is explained in section 3. Control structure selection and determination of input/output pairs are given in section 4. Tuning of controllers is presented in section 5. Stability analysis for multivariable systems is provided in section 6. At the end, conclusion is drawn.

System identification
Most of the chemical and bio-chemical processes are multivariable in nature, having more than one input and outputs. Estimation of process parameters is a key element in  (1994) proposed subspace method of identification that mostly applies to identification of multivariable state space models. This method involves more computational time. Practical industrial plants are easy to identify in closed-loop using relay feedback method (Astrom and Hagguland 1984) and Yu (1999) explains advances in autotuning using sequential identification. System identification is the method of estimating parameters from system's input/output data using numerical techniques:

Transfer function identification
Model structures and parameters of transfer function are constructed from observed plant input output data. Transfer function models are developed using three schemes: (a) Least square (b) subspace and (c) sequential identification method. These approximations made out through each of the methods carry errors that propagate to controller tuning and in turn deteriorates the overall performance.

Least-squares method
Least-squares method, used to reduce the mean square error, is very simple and more numerically stable and can be used to identify the unknown parameters of the 2x2 MIMO transfer function model from the input (u) and output (y) data. Though any type of forcing function (step, pulses or a sequence of positive and negative pulses) can be used, a very popular sequence of inputs, "Pseudo-random binary sequence" (PRBS) is made use of in the present work. Let us consider a process with continuous transfer function The discrete transfer function has three parameters that need to be identified: dead time (D) contained in n k , and other two parameters of the model (k p and ) contained in b 1 , and a 1 .
The discrete output can be represented in the following form: The parameters a 1 and b 1 are calculated using where  is the parameter vector  is state matrix and y is outputs.

State-space model
In the state space form the relationship between the input, noise and output signals are written as a system of first-order differential or difference equations using auxillary state vectors. Transfer function in laplace domain is converted to state space form using a sampling period of 0.1s

Subspace method
The beginning of the 1990s witnesses the birth of a new type of linear system identification algorithms, called subspace method. Subspace identification methods are indeed attractive since a state-space realization can be directly estimated from input/output data without nonlinear optimization. Furthermore, these techniques are characterized by the use of robust numerical tools such as RQ factorization and the singular values decomposition (SVD).
Interesting from numerical point of view, the batch subspace model identification (SMI) algorithms are not usable for online implementation because of the SVD computational complexity. Indeed, in many online identification scenarios, it is important to update the model as time goes on with a reduced computational cost. Linear subspace identification methods are concerned with systems and models of the form The vectors 1 mx k uR  and 1 lx k y R  are the measurements at time instant k of, respectively, the m inputs and l outputs of the process. The vector x k is the state vector of the process at discrete time instant k, are unobserved vector signals, v k is called the measurement noise and w k is called the process noise. It is assumed that they are zero mean, stationary white noise vector sequences and uncorrelated with the inputs u k .
are the covariance matrices of the noise sequences w k and v k. In subspace identification it is typically assumed that the number of available data points goes to infinity, and that the data is ergodic. The main problem of identification is arranged as follows: Given a large number of measurements of the input u k and the output y k generated by the unknown system described by equations (2.7)-(2.9). The task is to determine the order n of the unknown system, the system matrices A, B, C, D up to within a similarity transformation and an estimate of the matrices Q, S and R. Subspace identification algorithms always consist of two steps: Step 1: Make a weighted projection of certain subspace generated from the data, to find an estimate of the extended observability matrix, i  and/or an estimate i X  of the state sequence i X of the unknown system Step 2: Retrieve the system matrices (A, B, C, D and Q, S, R) and from either this extended observability matrix ( i  ) or the estimated states. All the above identification methods involve more computations and many offline methods. These difficulties can be avoided easily by using another method of estimation technique, namely, relay feedback method as explained below:

Sequential identification
Based on the concept of sequential auto tuning (Shen & Yu, 1994) method each controller is designed in sequence. Let's consider a 2-by-2 MIMO system with a known pairing   11 y u  and   22 y u  under decentralized PI control ( Figure 1). Initially, an ideal / biased relay is placed between 1 y and 1 u , while loop 2 is on manual (Figure 2a). Following the relayfeedback test, a controller can be designed from the ultimate gain and ultimate frequency. The next step is to perform relay-feedback test between 2 y and 2 u while loop 1 is on automatic (Figure 2b). A controller can also be designed for loop 2 following the relayfeedback test. Once the controller on the loop 2 is put on automatic, another relay-feedback experiment is performed between 1 y and 1 u , (Figure 2c). Generally, a new set of tuning constants is found for the controller in loop 1. This procedure is repeated until the controller parameters converge. Typically, the controller parameters converge in 3 -4 relay-feedback tests for 2 x 2 systems. In order to proceed with sequential identification, it is necessary to derive closed-loop transfer functions for the above mentioned schemes. The following notations will be used for 2-by-   Table 1.The output (y) and input data (to original WB plant transfer function) are used to form matrix. The parameters a 1 and b 1 were calculated using Eq.(2.6). On applying subspace algorithms to an unknown 2x2 MIMO process (WB column) the following steps are followed Step 1: From the transfer function matrix State space representation matrices are calculated.
Step 2: A, B, C and D matrices are simulate to get output data for a random input signal.

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Step 3: From the output and input data Henkel matrix are formed and LQ decomposition method is used to spilt the matrix Step 4: Then Singular value decomposition method is used to estimate A, B, C and D matrices.
Step 5: From estimated matrices the transfer function were found.  Mostly, the purpose of identification of transfer functions is to design controller for the system in order to achieve desired performance. Three methods of identifications (two in openloop mode and the other in closed-loop mode) are used to identify the two-input-twooutput process, WB column. Least square and subspace methods have been used to identify the process in openloop and sequential identification technique is used to estimate the process in closedloop.

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The identified models and actual plant model are compared (Table-2  After identifying the model structures and estimating process parameters of the models, next work is to select a suitable control strategy for the process.

Different control strategies
MIMO systems came into use in chemical industries as the processes were redesigned to improve efficiency. Multivariable control involves the objective of maintaining several controlled variables at independent set points. Interaction between inputs and output cause a manipulated variable to affect more than one controlled variable. The various control schemes studied here are the decentralized, centralized and decoupled systems. In decentralized structure, diagonal controllers are used. Hence they result in systems having n controllers. The centralized control systems have n x n controllers. In decoupled systems the process interactions are decoupled before they can actually reach and affect the processes.

Centralized structure
Centralized control scheme is a full multivariable controller where the controller matrix is not a diagonal one. The decentralized control scheme is preferred over the centralized control scheme mainly because the control system has only n controlling n output variables, and the operator can easily understand the control loops. However, the design methods of such decentralized controllers require first pairing of input-output variables, and tuning of controllers requires trial and error steps. The centralized control system requires n x n controllers for controlling n output variables using n manipulated variables. But if we are calculating the control action using a computer, then this problem of requiring n x n controllers does not exist. The advantage of the centralized controller is easy to tune even with the knowledge of the steady state gain matrix alone, multivariable PI controllers can be easily designed. , the PI controller settings are as follows: These tuning relations are derived by comparing IMC control with the conventional PID controller and solving to determine the proportional gain and integral time.

Decentralized structure
In spite of developments of advanced controller synthesis for multivariable controllers, decentralized controller remain popular in industries because of the following: 1. Decentralized controllers are easy to implement. 2. They are easy for operators to understand. 3. The operators can easily retune the controllers to take into account the change in process conditions. 4. Some manipulated variables may fail. Tolerances to such failures are more easily incorporated into the design of decentralized controllers than full controllers. 5. The control system can be bought gradually into service during process start up and taken gradually out of service during shut down. The design of a decentralized control system consists of two main steps: Step 1 is control structure selection and step 2 is the design of a SISO controller for each loop. In decentralized control of multivariable systems, the system is decomposed into a number of subsystems and individual controllers are designed for each subsystem. For tuning the controller, Biggest Log Modulus Tuning (BLT) method (Lubed 1986) is used, which is an extension of the Multivariable Nyquist Criterion and gives a satisfactory response. A detuning factor F (typical values are said to vary between 2 and 5) is chosen so that closed-loop log modulus, L cm max >= 2n, 20 log 1 where G c is an n x n diagonal matrix of PI controller transfer functions, G p is an n x n matrix containing the process transfer functions relating the n controlled variables to n manipulated variables. Now the PI controller parameters are given as, where ciZ N k  and IiZ N   are Zeigler-Nichols tuning parameters which are calculated from the system perturbed in closed loop by a relay of amplitude h, reaches a limit cycle whose amplitude a and period of oscillation P, are correlated with the ultimate gain (k u ) and frequency (w u ) by the following relationships: Detuning factor F determines the stability of each loop. The larger the value of F, more stable the system is but set point and load responses are sluggish. This method yields settings that give a reasonable compromise between stability and performance in multivariable systems. The decentralized scheme is more advantageous in the fact that the system remains stable even when one controller goes down and is easier to tune because of the less number of tuning parameters. But however pairing (interaction) analysis needs to be done as n! pairings between input/output are possible.

Decoupled structure
This structure has additional elements called decouplers to compensate for the interaction phenomenon. When Relative gain Array shows strong interaction then a decoupler is designed. But however decouplers are designed only for orders less than 3 as the design procedure becomes more complex as order increases. The BLT (Luyben 1986) procedure of tuning the decentralized structure follows the generalized way for all n x n systems as mentioned above. The centralized controllers are tuned using the IMC-PI tuning relations which are appropriately selected for first order and second order systems. The decoupled structure adopts the various methods like partial, static and dynamic decoupling to procedure the best results. The design equations for a general decoupler for n x n systems are conveniently summarized using matrix notations defined as follows: which defines the decoupler For a 2 x 2 system, equations are derived for decouplers, taking that loop and the other interacting loops into account.

Decentralized controller
The wood and berry distillation column process whose transfer function

Decoupled PID controller
The Wood and Berry binary distillation column is a multivariable system that has been studied extensively. The process has transfer function

Input-output pairing
Many control systems are multivariable in nature. In such systems, each manipulated variable (input signal) may affect several controlled variables (output signals) causing interaction between the input/output loops. Due to these interactions, the system becomes more complex as well as the control of multivariable systems is typically much more difficult compared to the single-input single-output case.

The Relative Gain Array analysis
The RGA is a matrix of numbers. The i jth element in the array is called ij The gain between y 1 and m 1 when y 2 is constant (y 2 = 0) is found from solving the equations 11  where K is an n x m matrix. U is an n x n orthonormal matrix, the columns of which are called the 'left singular vectors'. V is an m x m orthonormal matrix, the columns of which are called the 'right singular vectors'.  is an n x m diagonal matrix of scalars called the "singular values" SVD is designed to determine the rank and the condition of a matrix and to show geometrically the strengths and weaknesses of a set of equations so that the errors during computation can be avoided.

Example
Consider a very simple mixing example, a multivariable process whose gain matrix is as follows: At this point these singular values and vectors are merely numbers; however, consider the relationship between these values and an experimental procedure that could be applied to measure the steady-state process characteristics.

Niederlinski index
A fairly useful stability analysis method is the Niederlinski index. It can eliminate unworkable pairings of variables at an early stage in the design. The controller settings need not be known, but it applies only when integral action is used in all the loops. It utilizes only the steady state gains of the process transfer function matrix. The method is necessary but not the sufficient condition for stability of a closed loop system with integral action. If the index is negative, the system will be unstable for any controller settings. If the index is positive, the system may or may not be stable. Further analysis is necessary.
where, k p is a matrix of steady state gains from the process openloop transfer function k pjj is the diagonal elements in steady state gain matrix Example: Calculate the Niederlinski index for the wood and berry column: Since NI is positive, the closed loop system with the specified pairing may be stable.

INA and DNA methods
Rosenbrock extended the nyquist stability and design concepts to MIMO systems containing significant interaction. The methods are known as the inverse and direct Nyquist array (INA and DNA) methods. As an extension from the SISO nyquist stability and design concepts, these methods use frequency response approach. These techniques are used because of their simplicity, high stability, and low noise sensitivity. In actual applications, there will be a region of uncertainty for interaction, as the process transfer function can be different from what was used in the controller design (due to modeling errors and process variations).

INA design methodology
The following is the design procedure for the INA technique: