Graph-theory description of the model can intuitively get the nature of influence of the input variables on the state and the output ones of a dynamic model. To have a deeper understanding of the dynamic models, the structure-matrix method is employed for the model analysis to provide important guidance for the feasibility analysis of system control scheme.
4.3.1 Model Structural Matrix
Consider a linear time-invariant system:
_
xðsị ẳAxðsị ỵBuðsị _
yðsị ẳCxðsị ỵDuðsị
ð4:21ị
whereA,B,C, and Dare quantity matrices.
For such kind of mathematical description of (A, B, C, D), there must be a corresponding structural description of ðA;B;C;Dị i.e., structural matrix among whichA;B;C;D are corresponding matrix parameters with zero elements‘0’and other nonzero elements marked by ‘’. It can be seen that suppose two linear systemsðA1;B1;C1;D1ịand ðA2;B2;C2;D2ịhave different mathematical descrip- tions but corresponding position of the zero elements in the matrix is the same, then they have the same structural system ðA;B;C;Dị. In such case, the two systems have structural equivalence.
Concerning a structural matrix, since there is no quantitative description of its elements, we cannot identify its rank. But the rank of a structural matrix can be obtained by maximum rank of a numerical matrix corresponding to the structural matrix. In order to differentiate the rank of the numerical matrix, we name the rank of a structural matrix as‘generic rank’as below:
grðSị ẳmax rankð ðS)ị ð4:22ị For details on calculation of generic rank of a structural matrix, please refer to Johnson [4].
4.3.2 Reachability Analysis of Model Input – Output
In a dynamic system model, there are basically three types of variables: i.e., the stateðxị, the inputðuị, and the outputðyị. To fully understand the mutual influence among them, especially whether or not the state variables can be reachable by the
input ones, or whether or not the state variables can affect the output ones. This is extremely important during the structural analysis of a dynamic model. Suppose every state variable of the system modelð ịxi can be reachable by at least one input variable uj , then the system model is of input reachability. Suppose every state variable of the system modelð ịxi can reach at least one of the output onesð ịyk , then the system model is of output reachability. The input reachability and output reachability reflect whether or not the state variables of the model system can be changed by the input ones and whether or not they can be reflected by the output ones of the model system.
Introducing adjacency matrix to describe the logical relationship of the system model [5] in accordance with the dynamic model described by Eq. (4.21), the basic elements are the state, the input, and the output variables of the model. Suppose these variables are the basic units and are arranged in form of block combination, then the adjacency matrix of the system can be written as follows:
Aaẳ
x u y
x u y
AT 0 CT BT 0 DT
0 0 0
2 4
3
5 ð4:23ị
From Eq. (4.23), we can easily see the structural matrix that appears inAai.e., A;B;C;Dare all transposed. Taking the elements in matrixA, for example, suppose an element in the matrixA a ij= ‘×’suggests thatxiwould be affected byxj, then such kind of influence means that there should beaji =‘×’inAa. In the adjacency matrix, the all-zero column corresponds to the input vector of system model, and the all-zero row corresponds to the output vector of system model.
The adjacency matrix only shows the direct influence among system units. In practice, our main concern is whether or not some system units can impact other units within the system through others (i.e., indirect influence). In such case, we can use kth power of Aa (Aak
) to represent the kth heavy indirect relationship among system units, e.g., one elementakij=‘×’inAka means the unitSiaffects the unitSj through the other (k− 1) units. The power operation here can be made based on the general rules of matrix multiplication. SinceAais structural matrix, the addition and multiplication operation between the elements should be the logic addition and multiplication.
For a system consisting of N elements, suppose there exists reachable rela- tionship between any of its two elementsSiandSj, we can always get a reflection in the corresponding unitsAa;A2a;. . .;ANa. If each unit can reach itself, we can get the following matrix to describe every reachable relationship:
RẳI[Aa[A2a[. . .[ANa ð4:24ị The matrixRin Eq. (4.24) is called system reachability matrix. It is also a form of expression of structural matrix in which the elementrij= ‘×’means that the unit
Sican directly or indirectly reach the unitSj, and the elementrij=‘0’means thatSi
can not reach the unitSj. According to the logic algorithm, formula (4.24) can be rewritten as follows:
RẳðI[AaịN
ẳ
ðATỵIịN 0 ðATỵIịN1CT BTðATỵIịN1 I BTðATỵIịN2CT
0 0 I
2 64
3
75 ð4:25ị
Meanwhile,
Rẳ
x u y
x u y
Rxx Rxu Rxy Rux Ruu Ruy Ryx Ryu Ryy 2
4
3
5 ð4:26ị
Thus, the input reachability and output reachability can be described by use of structural matricesRux andRxy, respectively, as below:
RuxẳBTðATỵIịn1 ð4:27ị Rxyẳ ðATỵIịn1CT ð4:28ị If there does not exist all-zero column in the matrixRux, then the system is of input reachability; and if there does not exist all-zero row in the matrixRxy, then the system is of output reachability. If there exsits all-zero row in the matrixRux, then it means the input variables corresponding to the all-zero row do not affect any of the state variables of model system. By using the method of structure-matrix analysis, the input and output reachability of dynamic component models (state-space model) can be investigated.
4.3.3 Controllability/Observability Analysis of Model
In order to get/find a feasible control scheme, we must discuss the system’s con- trollability and observability in advance. Using the previously described system structural matrix, we can carry out structural analysis of system controllability. This is very meaningful since it helps to understand the influence of each control action on the system and provide us global guiding information prior to the design of control system. Although the study of system structural controllability cannot characterize it quantitatively, and so cannot make a detailed analysis of strength of
controllability, etc., but we can carry out qualitative analysis of the system con- trollability based on logic.
Likewise, the linear time-invariant system (Eq.4.21) is considered for the dis- cussion. IfðA;Bịhas an equivalent system of controllable structure in the common sense, then ðA;Bị is structurally controllable. Otherwise, ðA;Bị is structurally uncontrollable.
Davidson and Morari started from system input reachability and combined it with judgment of the generic rank to get the necessary and sufficient conditions for controllability of a system structure as below [5]:
(a) The system is input reachable;
(b) grđơA;Bỡ Ửn:
The condition (a) points out that all system states can be affected by changes of input; and the condition (b) points out that inputs have sufficient freedom to change the system state to a desired state.
In the control of a dynamic system, there often exist many several possibilities for variable selection. In system design, we should decide which control actions are necessary, and the rest of control actions can be omitted or may be used to implement an auxiliary control target. For a feasible control scheme, apart from need to consider system constraint conditions given by process conditions, the most important point is that all the selected control variables must guarantee system controllability. Normally, with regard to system structural matrix ðA;Bị, suppose the generic rank of the structural matrix does not change after deleting a column (which corresponds to a control variable) in the matrix, then it is considered that the control variable corresponding to the deleted column does not affect the system controllability. Otherwise, the control variable corresponding to the deleted column does affect the system controllability, and it must be considered when choosing controllable variables. It is admitted that the controllability usually describes the ability of input variables to control the state of system.
The observability means to what degree the state of system can be reflected by the input and the output variables, i.e., provided the system model is fully known, the system states can be got by the input and the output information from mea- surements. Like the controllability analysis, the observability analysis is of equally important significance to the state feedback control design. The necessary and sufficient conditions for system structural observability are as follows [5]:
(a) Output reachable system (b) gr A
C
ẳn: