The secondД®quation is called the output equation and it has the output on the left and Variable(s) and input(s), multiplied by matrices, on the right. The first equation is called the state equation and it hasДЄ first order derivative of the state variable(s) on the left, and the state State space approach as a single first order matrix differential equation: Key Concept: Defining a State Space RepresentationДЄ n th order linear physical system can be represented using a ( ItД¬an be shown that the systems are identical by transforming the state space representation One, and it is not at all obvious that they represent the same system. This new state space system is quite different from the original They are given here if you are interested): In this case the new state space variables are given by ( theДetails of how these matrices are determined are not important for this discussion. New state variables by forming linear combination of the original state variables System reveals that they represent the same system. In the previous case careful examination of the original and modified state space Results in a new state space representationД¬ase 2: Alternate State Space Representations (the new state variables are labeled q new). Is not unique Case 1: Alternate State Space RepresentationДЄnother important point is that the state space representation is not unique.ДЄs a simple example we could simply reorder the variables from the example above Of transformations between system representations. Such cases are explained in the discussion In manyД¬ases (e.g., if there are derivatives on the right side of the differential equation) This is compactly written in state space format asДЇor this problem a state space representation was easy to find. We can now rewrite the 4 th order differential equation We can define 4 new variables, q1 through q4. Order differential equation with input x and output z. Computers easily simulate first order equations.Д¬onsider an 4 th order system represented by a single 4 th.To develop general techniques to solve these systems. Because all systems are represented by the same notation, it is very easy.We will consider) these variables become (with r=1 and m=1):ДЄdvantages of this representation include: There are no derivatives on the right hand side.ДЇor systems with a single input and single output (i.e., most of the systems State vector, q(t), and the input u(t) on The output equation has the output on the left, and the The state vector on the left, and the state vector, q(t), and The state equation has a single first order derivative of (n rows by 1 column) q is called the state vector, it is aД is the direct transition (or feedthrough) matrix, a constant R inputs and m outputs the size of each of the matrices is as (i.e., it can be represented by an n th order differential equation) with The first equation is called the state equation, the secondД®quation is called the output equation. Variable q will be used here since we will often use x Is the variable q when state variables are discussed. X is more commonly used in textbooks and other references than Note: Bold face characters denote a vector or matrix.The variable Of a system replaces an n th order differential equation with a singleДЇirst order matrix differential equation. Method which largely alleviates this problem. Or transfer functions becomes cumbersome. State Space Representations of Linear Physical SystemsДЄs systems become more complex, representing them with differential equations

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