DYNAMIC SYST AND CONTROL
DYNAMIC SYST AND CONTROL MEEN 364
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This 10 page Class Notes was uploaded by Kameron Hyatt Sr. on Wednesday October 21, 2015. The Class Notes belongs to MEEN 364 at Texas A&M University taught by Alexander Parlos in Fall. Since its upload, it has received 46 views. For similar materials see /class/225991/meen-364-texas-a-m-university in Mechanical Engineering at Texas A&M University.
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Date Created: 10/21/15
Texas A amp M University Department of Mechanical Engineering MEEN 364 Dynamic Systems and Controls Dr Alexander G Parlos Lecture 7 Statespace Representation of Dynamic Systems The objective of this lecture is to introduce you to the two distinct models used in representing dynamic systems in the time domain namely input output models and state space models The procedure for obtaining the state space representation of an input output model is also presented InputOutput Models In dealing with dynamic systems we de ne inputs and outputs lnputs originate outside the system and are not directly dependent on what happens in the system Outputs are chosen from the set of variables generated by the system as it is subjected to the input variables The choice of the outputs is fairly arbitrary Consider the single input single output dynamic system shown in Figure 1 For most systems we will encounter in this class the relation between the input and the output signal can be represented by the following nth order differential equation fytd37tdzgigtutm 0 1 where m S n for physically realizable systems and where the function f is in general nonlinear For a linear single input single output system equation 1 can be sim Figure l Singleinput7 singleoutput dynamic system Figure 2 Mechanical system for example 1 pli ed as bdeZEt bm1 12161 bout7 3 Where an a0 and bm 7 be are all constant coef cients Again m S 71 Example 1 Derive an input output model for the system shown in Figure 2 The mass m is supported by an oil lm bearing that produces a resisting force proportional to the velocity of the mass For this system the choice of the input and output is rather obvious The force is the input and the resulting velocity 11105 is the output The system equation of motion is mm dt b21105 1w 4 Figure 3 Mechanical system for example 2 So here ut and yt 11115 with n 1 and m 0 Also do b a1mandb01 Example 2 Derive the input output equations for the mechanical system show in Fig ure 3 using the force F1t as the input variable and the displacements 05115 and 05215 as the output variables The equation of motion for mass m1 is miffi b19310 k1 k2fr1t 3295205 F105 5 The equation of motion for mass m2 is mga39c392t k2x2t k2x1t 0 6 Combining equations 5 and 6 and eliminating 05115 yields the following input output equation for the system d42 d32 d22 dt4 m1 dtg m2 m1 m1 dt2 3 Figure 4 Multi input multi output dynamic system bkd kk k 123 12x27 2F1t 7 77117712 dt 77117712 77117712 This equation can be solved provided that four initial conditions and the input F1t is known Similarly the input output equation relating 051t to F1t can be derived as d4x1 b1 dgxl k2 k1 k2 dgxl W t WW E 51 WW b1k2 dxl Irle 1 1 d2F1 k2 Fla 8 7 R1 dtg 77117712 For multi input multi output systems equations 1 2 and 3 can be 05 77117712 dt 77117712 generalized for use Figure 4 depicts the block diagram of such a system StateSpace Models The reason we introduce state space models in addition to input output models is the fact that the former are much more powerful than the latter and they are widely used in modeling complex engineering systems The concept of a state is similar to that de ned in thermodynamics That is state variables constitute the minimum number of variables which if known completely describe the system under consideration When the state variables are grouped together they form the so called state vector The models that result from the use of the state vector are called state space models Finally we de ne state trajectory as the path over time followed by the state of a 4 system Mathematically the state space equations are sets of rst order differential equations For a linear system model the state space equations take the following form 4105 01161105 012 t 0171617105 391111105 391211205 517211172105 9 Q7705 an1q1t an2q2t amqnt bn1u1t bn2u2t bnmumt7 10 Where q1t7 7 17705 are the state variables also denoted by xt7 and u1t7 7 umt7 are the input variables The system output equations are 3105 01161105 012 t 0171617105 611111105 611211205 6117211172105 11 Cn1Q1Cn2CI2t dnlul dn2u2 12 y1t7 7 3100 are the output variables These equations can be written in more compact matrix form as 3910 40105 51107 13 and 33905 30105 1911037 14 Where A is the n x 71 state matriX7 B is the n x m input7 C is the p x 71 output matriX7 and D is the p x m direct output or feedforward matrix Furthermore7 qt is the state vector7 ut is the input vector7 and yt is the output vector Figure 5 Block diagram of a state model In the case of a general nonlinear system the state space equations can be generalized as follows 1115 f1q1tq2t qntu1tu2t umt l5 17125 fnq1tq2t qnt u1tu2t umt 16 Whereas the output equations can be expressed as 3105 91q1 t 1205 qntu1 t U205 umt l7 ypt gnql 07 q2t7 39 7qnt7 1 07 u2t7 7umt A block diagram of the state space representation of this multi input multi output nonlinear system is depicted in Figure 5 InputOutput to Statespace Transition Input output and state space models are equivalent As a result an input output model can be transformed to a state space model and Vice versa though the latter is a bit more cumbersome 6 Figure 6 Equivalent inputoutput and statespace modelsi Consider the following simple input output equation dny 61105 an dtn a1 dt dogt b01105 19 If we select the following state variables ellt 617143105 95115 0795205 W7 7957105 W7 20 the equivalent set of state space equations are 210 95205 21 x20 x30 22 lt23 271105 i xnt 24 mt ltgtx1lttgt Ema C n 1gtxnlttgt Ema 25gt an an an an Figure 6 depicts the equivalence between input output and state space mod els Example 3 The input output equation for a mechanical system is given by d4x t d3xt 1154 1153 quot117712 7712391 m2b2 m1b2 7 d2 t m1k1 m2k1 m2k2 bib2 ml d12 dxt b k b k 7 1 2 2 1 dt k1k2lttgt k2Ft where the input is F t and the output is Derive an equivalent state space model for this system De ne the state variables as i i dxt i d2xt i d3xt 1115 95t7Q2t 771305 W440 W7 27 and the state space equations are i105 120 i205 61307 i305 14037 k1k2 bik2 b2k1 m1k2 m2k1 m2k2 51392 t t t qio mmmo lt mm M lt mm mgbl mgbl 771le kg t F t lt mm m gt ltm1m2gt o The output equation is 105 110 32 Equations 28 through 32 form a state space model of this mechanical system Let us now consider the transformation of a model to state space form from the standard second order form of mechanical systems Consider the following mechanical system model obtained following repeated application of Newton s second law Inf3505 bf19 t 149505 1057 33 where represent acceleration velocity and displace ment vectors and where m b k represent the mass damping and stiffness 8 Q3t matrices The vector F represents the forcing function of the system Equation 33 can be rewritten as mwnrugmnnruqmwnquan Bb Now de ne the following state vector B and the following input vector MUF BQ Using the state vector 35 the second order system can be written as 0 I qt m 1k m 1b 0 u BU What about the system outputs As mentioned before one can arbitrarily de ne any linear combination of states as the outputs for this system For example let us de ne all of the velocities as the outputs of this system ie de ne the output vector yt as the vector Then we can express is in terms of the state vector as follows 01lqnlobw Equations 37 and 38 form a state space representation of the forced me chanical system 34 This representation is appropriate for any M degree of freedom MDOF mechanical system Furthermore the mechanical system 34 contains only translational dynamics ie only linear displacement ve locity and acceleration are involved in the equations A similar process of de riving state space equations applies to mechanical systems that include only rotational dynamics and even both translational and rotational dynamics The de nition of the state vector must be augmented to include the rota tional degrees of freedom of the system that is the angular displacements and angular velocities Reading Assignment Read pages 41 45 the textbook Read Handout A3 and examples Handout E7 posted on the course web page
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