State variables and SISOMIMO control systems - PDF document

State variables and SISO/MIMO control systems 1. State variable model for a dynamic system simultaneous first-order differential equations, called the State equation: )(][][tiBxAx to the state vector )(][)(tDito xC . Here we define: state vector system matrix input matrix (N rows x 1 column for a sisystem, (one row x N columns for a SISO system), feedforward matrix (1 x 1 for a SISO system). As shown in the Topic Overview for the ceither the - domain for a system from the state-variable model. For a SISO system, these transfer functions take the forms: ])([)(ssH BAIC , and ])([)(zzH BAIC . is the identity matrix (remember that the elements of A,B, C, and D will not be the same for the continuous and the sampled models, and that in the sampled model they will depend on sampling frequency) Obtaining a state variable model You can get a state variable model for a system in two ways: from detailed knowledge of the internal dynamics of the system, and from a transfer function (which might have been obtained by system identification or by other means not requiring knowledge of the internal dynamics). 2a. from the internal dynamics Suppose we use an electromagnet coil to position permanent bar magnet of mass attached to a wall through a spring of spring constant . We drive this coil with an input voltage, v(t), Frictional forces in this system are represenThe force on the bar magnet is proportional to th. Equating this magnetic force to the sum of inertial, frictional, and -domain) , R sVsXKWsMs)()()  and a transfer function from a RKRWssVsXsG )()()( , and a second-order differential equation, )(tvRKxxd  . red to model it. Choosing a set of states: velocityxpositionx21x the system differential equation becomes )(tvRKx  . This equation, and another obvious one, 21xdtdx , provide enough information to )(10tv434444 , and an output equation, )(001)(titou321 . Important general property: the poles of the transfer function are the eigenvalues of the system matrix, . Setting the denominator of the MKWWsKWsMsr Ÿ  . using the standard matrix method of setting 0]det[ AI yields  M Ks M Ws , the same result. This property can be shown to be true for all linear systems with state-variable models. of the system: another example Signal-flow graphs In order to introduce some key ideas in state-variable system modeling, we need Addition of incoming signals at a node: Here the node is a small a node is the sum of all the signals flowing in. Amplification by a fixed factor, positive or negative: the gain is just written above Integration: This is The state variable model for any linear system isthe outputs of each integrator in a signal-flow graph of a system are the states of that system. For any system, an infinite numb. Let’s look at some proce by means of a specific example. Consider matrix (regardless of what form it is in) are the poles of the transfer function. Also, note that in this form, the coefficients in the equations will generally be complex. This was not the case for the control canonical form earliesfer function coefficients. Here is the general matrix modal form for a fourth-order system: 000000000000 , , 11B 4321AAAA C with Observer canonical form There is one more special form of the state equations that is of interest. In this case the feedback is from the output to the state variables. For this form, we start with a normalized , 3 rd -order transfer function, 32210210)(ssssssGDDDEEE , and draw the following signal-flow graph: which leads to the matrix forms, 001001 , , 2B 001 C , and D contains all zeros. For the control canonical form, we justified the form of the signal-flow graph by solving the differential equation for th the output. For the modal form, we did this by first lookifunction, then adding similar terms. For the observer canonical form, suppose we multiply the normalized transfer function by s raised to that power, thereby creating a rational polynomial in (1/s) as follows: )()()( sssssssIsOsGDDDEEE . This leads to the following LaPlace transform equation relating the input, I(s)O(s) ))()(())()(())()(()( ssOsIssOsIssOsIsODEDEDE . This equation corresponds exactly to the signside gets integrated three times, th Transformation to other state-space representations How are the different state-space representations related, other than in representing the same physical system? If a linear system can be represented by two state , the two vectors must be related through a transformation u=Tv, and v=T -1 u must exist, that is must be non-singular. We can use this relation to transform the state and output equations as well, for example, if with one state vector, QiPuoHiGuu , , then using the transformation, HiGTvu and QiPTvo . Pre-multiplying by the inverse of T gives a new set of state equations, iHTvGTTuTv321321&& and QivPTo . input, and output matrices for the system using the state vector The availability of the transformation, , means that an infinite number of state representations for a system are possible. Only a few of these are interesting. on, calculated using the matrices from the state variable model using ])([)(ssH BAIC is the same as the one at the beginning of this example. Th Pole placement using state feedback Suppose you can measure all the states of a corresponding to each. Such . In the automatic control scheme called , a weighted linear combinati, r(t) To see how this controller works, just substitute the new input,   321)()()(KKKtrtrti , into the state equation. The result is a new state equation for the closed-loop system with )(][trBxBKA . Clearly, the new state matrix for the closed-loop system is [The analytic method for this pole placement is called computer implementation is available. Unfothem is usually insufficient to insure that the closed-loop system’s time- and frequency-domain responses are as you wishcontroller is still a powerful alternative design tool. Multiple input multiple output (MIMO) systems All through this discussion, we have beento analyze systems with multiple inputs and outputs (MIMO systems). The link between the state variable model and the transfer fbelow) which is 1 x 1 for a single-input single output (SISO) system and of higher dimensionality for a MIMO system. . Can we “guarantee” observable states? When using state variables to design a control system, you have to be sure you can sense or measure the states you are using for your model, or you won’t be able to carry out the multiplication, 21322212312111xxKKKKKK in the feedback loop. Regardless of whetheobservable, you can always count on being able to measure the following things about all of its outputs, comprising (t), all of its inputs, comprising the components of the vector, Now, suppose that in addition to these measurements, you have used either a theoretical model or system identificatier function matrix for your plant that relates each output to each input (this example has two inputs and two outputs), )()()()()(sGsGsGsG and that each element of this matrix is in rational polynomial formhave found all of the poles of each element in , and have rewritten each element so that all elements share a common denominator. Further, assume you have normalizedrational polynomial. If the denominators in were 3 rd order, each element in the transfer function matrix would have the following form (generalization to polynomials is straightforward): 3221010)(ssssssGijijDDDEEE . A useful way to interpret G ij is as the transfer function to output from input have either a theoretical model or the results of system identification for all of the coefficients that make up the plant transfer function, cal form of each element in to construct a set of states that are, in principle, guaranteed to er first the transfer 11 . For a 3 rd order system such as our example, the The first subscript of the coefficients refers to the order in the numerator of the rational polynomial. The later pair of subscripts refers to the element in the G-matrix for which we are making the signal-flow graph. To tae transfer function matrix), you If there were more than two elements in the first row of , you would need more sets of coefficients than the ones showcontribution of all the in es in terms of the known coefficients and the measurable inputs and outputs. Note that, no matter how many inputs you might have, the states are 21121111112322121211121211iioxxiioxxoxEEDEED  . Thus, if you can measure and differentiate the this system, you can looks like this: From the signal-flow graph for output 1, you triplet of states, 001001EEEEEE and . 001 From the signal flow graph for output 2, triplet of states, 001001EEEEEE and . 001 For the combined MIMO system, this state-spires six states, and -the state equation: 000001000001000000000001000001EEEEEEEEEEEE , - the output equation: 001000000001 From this discussion with an example having two inputs, two outputs, and a 3 rd common denominator for the plant transfer function matrix (yielding 6 states = denominator order x # outputs), you can see howorder systems and different numbers of outputs asystem that is controllable, but it is guaranteed to yield a set of states that can be calculated from measurements on the inputs polynomial coefficients of the known plant transfer function. Using this final state-space model, you can at least attempt to design a co