Reduction of Multiple Subsystems Ref: Control System Engineering Norman Nise : Chapter 5 Chapter objectives : How to reduce a block diagram of multiple subsystems to a single block representing the transfer function from input to output How to analyze and design transient response for a system consisting of multiple subsystems How to represent in state space a system consisting of multiple subsystems Multiple subsystems - 2 1
1. Block Diagrams for Dynamic Systems Block diagram an interconnection of blocks representing basic mathematical operations in such a way that the overall diagram is equivalent to the system s mathematical model. In such a diagram, the lines interconnecting the blocks represent the variables describing the system behaviour. x K f A block diagram representing f = Kx Multiple subsystems - 3 Antenna position control Multiple subsystems - 4 2
Block Diagram Reduction Block diagram reduction involves algebraic manipulations of the transfer functions of the subsystems or blocks, which in effect reduce the diagram to a single block. This gives the overall transfer function relating the input r and output c in a block diagram and hence permits, for example, calculation of system transient responses. Multiple subsystems - 5 Summer addition and subtraction of variables x 2 x 1 + + - x 3 y A summer representing y = x 1 + x 2 - x 3 Gain multiplication of a single by a constant (exp. spring) Integrator integration with respect to time u y y y dt dt The block diagram for an integrator Multiple subsystems - 6 3
Constant has no input, and its output never changes c y Combining block diagram Consider the following equation : x fa t Ax Steps (for input output equations) - Solve the given equation for the highest derivative of the unknown output variables - Connect one or more integrator blocks in series to integrate that derivative successively as many times as necessary to produce the output var. - Use the result of step 1 to form the highest output derivatives as the output of a summer and a gain block. Multiple subsystems - 7 Construct block diagrams for the following systems 1. Mx Bx Kx fa t iv 2. M M x M K M K K 1 2 1 1 2 2 1 2 x1 K1K2x1 K2 fa t 3. M x Bx K K x Bx 1 1 1 2 1 2 K2x2 0 Bx K x M x Bx K x f t 1 2 1 Rules for altering diagram structure 2 2 2 2 Transfer functions which are generally the ratio of two polynomials are often denoted by F(s), G(s) or H(s). When the transfer function is a constant, then that block reduces to a gain block. Series combination 2 a X(s) F 1 (s) V(s) F 2 (s) Y(s) Multiple subsystems - 8 4
Parallel combination X(s) F 1 (s) F 2 (s) V 1 (s) V 2 (s) + + Y(s) Example 1 Evaluate the transfer functions Y(s)/U(s) and Z(s)/U(s) for the block diagram below give the results as rational functions of s Multiple subsystems - 9 Equivalent diagrams for the diagram shown in Example 1 Multiple subsystems - 10 5
Moving block to create familiar forms Moving a pick off point a point where an incoming variable in the diagram is directed into more than one block (1) (2) (3) 1 Original diagram, 2 & 3 equivalent diagrams Multiple subsystems - 11 Block diagram algebra for pickoff points - equivalent forms for moving a block a. to the left past a pickoff point; b. to the right past a pickoff point Multiple subsystems - 12 6
Moving a summing junction Ahead of a block After a block Multiple subsystems - 13 Block diagram algebra for summing junctions - equivalent forms for moving a block a. to the left past a summing junction; b. to the right past a summing junction Multiple subsystems - 14 7
Example 2 Modify the bock diagram in (a) to remove the right summing junction, leaving only the left summing junction (a) Original diagram, (b), (c) & (d) equivalent diagrams Multiple subsystems - 15 Reducing diagrams for feedback systems G(s) = Y(s)/E(s) forward transfer function Y(s) H(s) = Z(s)/Y(s) feedback transfer function G(s)H(s) function open-loop transfer Y(s) T(s) = Y(s)/R(s) closed-loop transfer function Z(s) H(s) = 1 system unity feedback Y(s) Multiple subsystems - 16 8
Obtaining the CLTF E()()() s R s YH s Y ()()() s E s G s Y () s G() S R()() s YH s Y () s 1()() GH S GR s Y ()()() s G s1 E s ; R() s 1()() GH1() S R s GH S Y () s R() s is the closed loop TF Multiple subsystems - 17 Unity feedback For a unity feedback, the CLTF is Y ()() s G s R() s 1() G S 1() GH0 S This equation is called the characteristic equation of the closed loop system, giving the root or poles of the TF on the s-plane. Multiple subsystems - 18 9
Consider the cascade or series connection of two blocks in the Fig. By definition, C = G 2 M M = G 1 R Substituting the second into the first yields C = GR G = G 1 G 2 By direct extension it follows that The overall transfer function of a series of blocks equals the product of the individual transfer functions. Multiple subsystems - 19 Multiple subsystems - 20 10
Multiple subsystems - 21 Loop Gain In words, and in somewhat generalized form, this may be stated as follows: The closed-loop transfer function of the standard loop equals the product of the transfer functions in the forward path divided by the sum of 1 and the loop gain function. The loop gain function is defined as the product of the transfer functions around the loop. Multiple subsystems - 22 11
For the present system, the Loop gain function = G 1 G 2 H(s) If H = 1, then E = R - C is the system error, then E/R is the input-to-error transfer function. This will permit the error response for a given input r(t) to be found directly. Multiple subsystems - 23 Error to input TF Since C = G 1 G 2 E(s), then the error to input TF is E() s 1 R() s 1() G G H S 1 2 If the feedback H = 1 and G = G1G2 E() s 1 R() s 1 G Note that the characteristic equation here is 1+G = 0. Multiple subsystems - 24 12
Reducing TF blocks Multiple subsystems - 25 Multiple subsystems - 26 13
Multiple subsystems - 27 Example 3 Multiple subsystems - 28 14
Multiple subsystems - 29 Multiple subsystems - 30 15
Multiple subsystems - 31 Block diagram reduction via familiar form Example 4 reduce the block diagram shown below to a single transfer function Multiple subsystems - 32 16
Steps in solving Example 4: a. collapse summing junctions; b. form equivalent cascaded system in the forward path and equivalent parallel system in the feedback path; c. form equivalent feedback system and multiply by cascaded G1(s) Multiple subsystems - 33 Block diagram reduction by moving blocks Example 5 reduce the block diagram shown below to a single transfer function Multiple subsystems - 34 17
Steps in the block diagram reduction for Example 5 Multiple subsystems - 35 Example 6 reduce the block diagram shown below to a single transfer function Multiple subsystems - 36 18
Example 7 find the equivalent transfer function T(s)=C(s)/R(s) Multiple subsystems - 37 Example 8 Find the closed-loop transfer function for the feedback system below. Compare the locations of the poles of the open-loop and closed-loop transfer function in s-plane. Multiple subsystems - 38 19
Example 9 Find the closed-loop transfer function of the two-loop feedback system in Fig 1. Also express the damping ratio and the un-damped natural frequency of the closedloop system in terms of the gains a 0 and a 1. Figure 1 Equivalent block diagrams Multiple subsystems - 39 Second order system Example 10 construct the block diagram for the system described by the differential equation a y a y a y 2 1 0 f t which involves no input derivatives in its input function. Then use the block diagram to find a state-variable model for the system Multiple subsystems - 40 20
2. Analysis and Design of Feedback System Immediate application of the principles of block diagram. Example 11 find the peak time, percent overshoot and settling time. Example 12 design the value gain K for the system below so that the system will respond with a 10 % overshoot Multiple subsystems - 41 Two or more inputs Multiple subsystems - 42 21
Since the plant is assumed linear, the total output c(t) can be independently evaluated due to the disturbance or load D the input R and then added together The same concept can be applied for transfer functions Multiple subsystems - 43 First set D(s) = 0 to evaluate C 1 /R C1 () s G1G 2 R() s 1() G G H S 1 2 Second set R(s) = 0 to evaluate C2/D C 2 () s 2 G L D() s 1() G G H S 1 2 The characteristic equations are the same Multiple subsystems - 44 22
3. Signal-Flow Graphs Signal flow graphs are alternative to block diagram. A signal flow graph consists only of branches, which represent systems, and nodes, which represent signals. Signal-flow graph components: a. system; b. signal; c. interconnection of systems and signals Multiple subsystems - 45 Converting common block diagrams to signal-flow graphs a. cascaded system nodes; b. cascaded system signal-flow graph; Multiple subsystems - 46 23
c. parallel system nodes; d. parallel system signal-flow graph; Multiple subsystems - 47 e. feedback system nodes; f. feedback system signal-flow graph; Multiple subsystems - 48 24
Example 13 Convert the block diagram in Example 4 to signal-flow graph. Signal-flow graph development: a. signal nodes; b. signal-flow graph; c. simplified signal-flow graph Multiple subsystems - 49 Example 14 Convert the block diagram below to signal-flow graph Multiple subsystems - 50 25