LTI systems have the extremely important property that if the input to the system is sinusoidal, then the output will alsoīe sinusoidal with the same frequency as the input, but with possibly different magnitude and phase. Representation is used extensively in "modern" control theory. The state-space representation, also referred to as the time-domain representation, can easily handle multi-input/multi-output (MIMO) systems, systems with non-zero initial conditions, and nonlinear systems via Equation (1). Often the case that the outputs do not directly depend on the inputs (only through the state variables), in which case is the zero matrix. The output matrix,, is used to specify which state variables (or combinations thereof) are available for use by the controller. The output equation, Equation (3), is necessary because often there are state variables which are not directly observed orĪre otherwise not of interest. Where is the vector of state variables (nx1), is the time derivative of the state vector (nx1), is the input or control vector (px1), is the output vector (qx1), is the system matrix (nxn), is the input matrix (nxp), is the output matrix (qxn), and is the feedforward matrix (qxp). In fact, the true power of feedback control systems are that they work (are robust) in the presence of the unavoidable modeling uncertainty.įor continuous linear time-invariant (LTI) systems, the standard state-space representation is given below: These results have proven to be remarkably effective and many significant engineering challenges have been solved using LTI Consequently, most of the results of control theory are based on these assumptions. Until the advent of digital computers (and to a large extent thereafter), it was only practical to analyze linear time-invariant (LTI) systems. In this case, the system of first-order differential equations can be represented as a matrix equation, that is. Fortunately, over a sufficiently small operating range (think tangent line near a curve), the dynamics of most The most common in control systems being "saturation" in which an element of the system reaches a hard physical limit to its These nonlinearities arise in many different ways, one of In other words, is typically some complicated function of the state and inputs. In reality, nearly every physical system is nonlinear. The second common assumption concerns the linearity of the system. The state variables,, and control inputs,, however, may still be time dependent. This is often a very reasonable assumption because the underlying physical laws themselves do not typically depend on time.įor time-invariant systems, the parameters or coefficients of the function are constant. , then the system is said to be time invariant. The function does not depend explicitly on time, i.e. There are two common simplifications which make the problem more tractable. The relationship given in Equation (1) is very general and can be used to describe a wide variety of different systems unfortunately, The system order usually corresponds to the number of independent energy storage elements in the system. is referred to as the system order and determines the dimensionality of the state-space. , required in order to capture the "state" of a given system and to be able to predict the system's future behavior (solve Though the state variables themselves are not unique, there is a minimum number of state variables, The state at any future time,, may be determined exactly given knowledge of the initial state,, and the time history of the inputs,, between and by integrating Equation (1). is the vector of external inputs to the system at time, and is a (possibly nonlinear) function producing the time derivative (rate of change) of the state vector,, for a particular instant of time. For instance, in a simple mechanical mass-spring-damper system, the two state variables could be the position and velocity In the above equation, is the state vector, a set of variables representing the configuration of the system at time. For many physical systems, this rule can be stated asĪ set of first-order differential equations: Entering Transfer Function Models into MATLABĭynamic systems are systems that change or evolve in time according to a fixed rule.Entering State-Space Models into MATLAB.
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