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# CONTROL SYSTEM DESIGN MEEN 651

Texas A&M

GPA 3.85

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This 9 page Study Guide was uploaded by Kameron Hyatt Sr. on Wednesday October 21, 2015. The Study Guide belongs to MEEN 651 at Texas A&M University taught by Staff in Fall. Since its upload, it has received 34 views. For similar materials see /class/225992/meen-651-texas-a-m-university in Mechanical Engineering at Texas A&M University.

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Date Created: 10/21/15

Texas A 85 M University Department of Mechanical Engineering MEEN 651 Control System Design Dr Alexander G Parlos Fall 2003 Lecture 4A 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 dyt d yt dut dumt t 7 dt 7 7 dtn dt 7 7 dtm 7ut7 07 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 simpli ed as d yt d 71100 Ellt an d MAW 01 dt 1090 2 dmut dm 1ut dut b bm 7 b b t 3 d 1 dtm 1 d ou u Dynamic system 39 Figure 1 Singleeinput7 singleoutput dynamic systemi Figure 2 Mechanical system for example 11 Where an i i i a0 and 17m 1 i 1 b0 are all constant coef cients Again7 m g n Example 1 Derive an inputeoutput 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 Fit is the input and the resulting velocity v1t is the output The system equation of motion is M7111 bv1t Rm 4 So7 here ut and yt v1t with n 1 and m 0 Also7 a0 177 all m and b0 1 Example 2 Derive the inputeoutput equations for the mechanical system show in Figure 3 using the force F1t as the input variable and the displacements x1t and x2t as the output 2 x1 u2g Figure 3 Mechanical system for example 2 var iablesi The equation of motion for mass m1 is m1i1lttgt b1i1lttgtltk1 k2x1t7 1021205 F t The equation of motion for mass m is m2i2t 10212037 k2x1t 0i 6 Combining equations 5 and 6 and eliminating x1t yields the following inputioutput equation for the system 514 71 51 kg 101 kg 51 T4 t HR cm Etata 72 b k d k k k 123 12x2 2 Flat 77117712 dt mlmg m1m2 7 This equation can be solved provided that four initial conditions and the input F1t is known Similarly7 the inputioutput equation relating x1t to F1t can be derived as d4x1 b1 dam kg 101 kg d2x1 W t HWEEEW 1le dam Icle 7 1 sz1 k2 77117712 mlmgxl 7 m1 dt2 m1m2F1t 3 8 ugt gtyi 2 39gt Dynamic gt 3 system 39 u gt gt VP Figure 4 Multiiinput multiioutput dynamic system For multiiinput multiioutput systems equations 1 2 and 3 can be generalized for use Figure 4 depicts the block diagram of such a system StateSpace Models The reason we introduce stateispace models in addition to inputioutput 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 soicalled state vector The models that result from the use of the state vector are called stateispace models Finally we de ne state trajectory as the path over time followed by the state of a system Mathematically the stateispace equations are sets of rstiorder differential equations For a linear system model the statespace equations take the following form Mt a11q1t a12q2t amqna b11u1t b12u2t blmuma 9 int M14105 M21120 a u ananLlttgt bn1u1t bn2u2t a u bnmuralttgt7 10 where q1t qnt are the state variables also denoted by xt and u1t umt are the input variables The system output equations are 9175 61111105 61211205 a u CinlInt d11u1t d12u2t i i i dlmumt 11 4 ypt cn1q1t cn2q2t cmqn t dn1u1t dn2u2t dnmum t7 12 y t7 7 ypt are the output variables These equations can be written in more compact matrix form as W AON 311W 13 and WW CON 911007 14 where A is the n gtlt 71 state matrix7 B is the n gtlt m input7 C is the p gtlt 71 output matrix7 and D is the p gtlt m direct output or feedforward matrix Furthermore7 qt is the state vector7 ut is the input vector7 and yt is the output vector In the case of a general nonlinear system7 the state space equations can be generalized as follows 110 f191t792t7 7Qnt7u1t7u2t7 7umt 15 12t7 39 39 39 71nt7u1t7u2t7 39 39 39 7u mt7 whereas the output equations can be expressed as 91 9191t712t7 7Qnt7u1t7u2t7 7umt 17 12t7 39 39 39 71nt7u1t7u2t7 39 39 39 A block diagram ofthe state space representation of this multi input7 multi output nonlinear system is depicted in Figure 5 InputOutput to Statespace Transition lnput output and state space models are equivalent As a result an input output model can be transformed to a state space model7 and Vice versa though the latter is a bit more cumbersome State Output model model Figure 5 Block diagram of a state model Consider the following simple inputeoutput equation aim dt d t an lu a1 a0yt 17011t7 If we select the following state variables dW dHW 0610 071205 Ta 775 W7 the equivalent set of stateespace equations are 33951 t 12t 5520 13W in1t znt anil mt 7lt gtzllttgtilt gtz2lttgtmilt gtznlttgtlt gtulttgtl an Figure 6 depicts the equivalence between inputeoutput and statespace modelsl Example 3 The inputeoutput equation for a mechanical system is given by 141 t dam t mg W251 W252 W152 WW2 6 19 20 21 22 23 24 25 Slate mode Output u 1 Input output mOdel equation State equation 5 Figure 6 Equivalent input7output and statespace modelsi d2 t m1k1m2k1m2k2b1b2 22 d t b1k2b2k1 3 lama 7 we 26 Where the input is Ft and the output is Derive an equivalent state7space model for this systemi De ne the state variables as dxt 51210 51310 1105 9503711205 dt 711305 W7 I405 Ta 27 and the state7space equations are 1105 12 28 1205 Mt 29 Mt IN 30 Icle 121102 122101 771le 7712101 7nng 121122 7 7 t 7 t 7 t q4ltgt mlmgnm lt mm mm lt W2 gtqaltgt m2171 7712171 W152 1 TWIN WWW 31 The output equation is W 111W 32 Equations 28 through 32 form a state7space model of this mechanical systemi 7 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 Newton7s second law mit bit H9600 Ft7 33 where represent acceleration velocity and displacement vectors and where m b k represent the mass damping and stiffness matrices The vector represents the forcing function of the system Equation 33 can be rewritten as 5500 m 1bzt m 1kzt m 1Ft 34 Now de ne the following state vector l ltgt and the following input vector 1100 Ft 36 Using the state vector 35 the second order system can be written as ml 0 1 qt im lk im lb 1 l ut 37 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 ww01mwomw as Equations 37 and 38 form a state space representation ofthe forced mechanical 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 velocity and acceleration are involved in the equations A similar process of deriving state space equations applies to mechanical systems that include only rotational dynamics and even both translational and rotational dynamics The de nition 8 of the state vector must be augmented to include the rotational degrees of freedom of the system7 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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