Block Diagram Manipulations

See systime.

Unless otherwise noted all parameters (input,output) are system data structures.

bddemo (inputs)

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Octave Controls toolbox demo: Block Diagram Manipulations demo.

buildssic (clst ulst, olst, ilst, s1, s2, s3, s4, s5, s6, s7, s8)

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Form an arbitrary complex (open or closed loop) system in state-space form from several systems. buildssic can easily (despite its cryptic syntax) integrate transfer functions from a complex block diagram into a single system with one call. This function is especially useful for building open loop interconnections for H-infinity and H-2 designs or for closing loops with these controllers. Although this function is general purpose the use of sysgroup sysmult sysconnect and the like is recommended for standard operations since they can handle mixed discrete and continuous systems and also the names of inputs outputs, and states. The parameters consist of 4 lists that describe the connections outputs and inputs and up to 8 systems s1-s8. Format of the lists: clst connection list describes the input signal of each system. The maximum number of rows of Clst is equal to the sum of all inputs of s1-s8. Example: [1 2 -1; 2 1 0] means that: new input 1 is old input 1 + output 2 - output 1 and new input 2 is old input 2 + output 1. The order of rows is arbitrary. ulst if not empty the old inputs in vector ulst will be appended to the outputs. You need this if you want to "pull out" the input of a system. Elements are input numbers of s1-s8. olst output list specifiy the outputs of the resulting systems. Elements are output numbers of s1-s8. The numbers are allowed to be negative and may appear in any order. An empty matrix means all outputs. ilst input list specifiy the inputs of the resulting systems. Elements are input numbers of s1-s8. The numbers are allowed to be negative and may appear in any order. An empty matrix means all inputs. Example: Very simple closed loop system. w e +-----+ u +-----+ --->o--*-->| K |--*-->| G |--*---> y ^ | +-----+ | +-----+ | - | | | | | | +----------------> u | | | | +-------------------------|---> e | | +----------------------------+ The closed loop system GW can be optained by GW = buildssic([1 2; 2 -1] 2, [1 2 3], 2, G, K); clst 1st row: connect input 1 (G) with output 2 (K). 2nd row: connect input 2 (K) with negative output 1 (G). ulst Append input of 2 (K) to the number of outputs. olst Outputs are output of 1 (G) 2 (K) and appended output 3 (from ulst). ilst The only input is 2 (K). Here is a real example: +----+ -------------------->| W1 |---> v1 z | +----+ ----|-------------+ | | | +---+ v +----+ *--->| G |--->O--*-->| W2 |---> v2 | +---+ | +----+ | | | v u y min || GW || vz infty The closed loop system GW from [z u]' to [v1, v2, y]' can be obtained by (all SISO systems): GW = buildssic([1 4; 2, 4; 3, 1], 3, [2, 3, 5], [3 4], G, W1, W2, One); where "One" is a unity gain (auxillary) function with order 0. (e.g. One = ugain(1);)

sys = jet707 ()

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Creates a linearized state-space model of a Boeing 707-321 aircraft at v=80 m/s (M = 0.26 Ga0 = -3 deg, alpha0 = 4 deg, kappa = 50 deg). System inputs: (1) thrust and (2) elevator angle. System outputs: (1) airspeed and (2) pitch angle. Reference: R. Brockhaus: Flugregelung (Flight Control) Springer, 1994.

ord2 (nfreq damp, gain)

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Creates a continuous 2nd order system with parameters: Inputs nfreq natural frequency [Hz]. (not in rad/s) damp damping coefficient gain dc-gain This is steady state value only for damp > 0. gain is assumed to be 1.0 if ommitted. Output outsys system data structure has representation with w = 2 * pi * nfreq: / \ | / -2w*damp -w \ / w \ | G = | | | | |, [ 0 gain ], 0 | | \ w 0 / \ 0 / | \ / See also jet707 (MIMO example Boeing 707-321 aircraft model)

sysadd (gsys hsys)

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returns sys = gsys + hsys. Exits with an error if gsys and hsys are not compatibly dimensioned. Prints a warning message is system states have identical names; duplicate names are given a suffix to make them unique. sys input/output names are taken from gsys. ________ ----| gsys |--- u | ---------- +| ----- (_)----> y | ________ +| ----| hsys |--- --------

sys = sysappend (syst b, c, d, outname, inname, yd)

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appends new inputs and/or outputs to a system Inputs syst system data structure b matrix to be appended to sys "B" matrix (empty if none) c matrix to be appended to sys "C" matrix (empty if none) d revised sys d matrix (can be passed as [] if the revised d is all zeros) outname list of names for new outputs inname list of names for new inputs yd binary vector; yd(ii)=0 indicates a continuous output; yd(ii)=1 indicates a discrete output. Outputs sys sys.b := [syst.b b] sys.c := [syst.c ] [ c ] sys.d := [syst.d | D12 ] [ D21 | D22 ] where D12 D21, and D22 are the appropriate dimensioned blocks of the input parameter d. The leading block D11 of d is ignored. If inname and outname are not given as arguments the new inputs and outputs are be assigned default names. yd is a binary vector of length rows(c) that indicates continuous/sampled outputs. Default value for yd is: sys is continuous or mixed yd = zeros(1rows(c)) sys is discrete yd = ones(1rows(c))

clsys = sysconnect (sys out_idx, in_idx, order, tol)

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Close the loop from specified outputs to respective specified inputs Inputs sys System data structure. out_idx in_idx Names or indices of signals to connect (see sysidx). The output specified by out_idx(ii) is connected to the input specified by in_idx(ii). order logical flag (default = 0) 0 Leave inputs and outputs in their original order. 1 Permute inputs and outputs to the order shown in the diagram below. tol Tolerance for singularities in algebraic loops default: 200eps. Outputs clsys Resulting closed loop system. Method sysconnect internally permutes selected inputs outputs as shown below closes the loop, and then permutes inputs and outputs back to their original order -------------------- u_1 ----->| |----> y_1 | sys | old u_2 | | u_2* ---->(+)--->| |----->y_2 (in_idx) ^ -------------------- | (out_idx) | | ------------------------------- The input that has the summing junction added to it has an * added to the end of the input name.

[csys acd, ccd] = syscont (sys)

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Extract the purely continuous subsystem of an input system. Input sys system data structure. Outputs csys is the purely continuous input/output connections of sys acd ccd connections from discrete states to continuous states discrete states to continuous outputs respectively. returns csys empty if no continuous/continous path exists

[dsys adc, cdc] = sysdisc (sys)

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Input sys System data structure. Outputs dsys Purely discrete portion of sys (returned empty if there is no purely discrete path from inputs to outputs). adc cdc Connections from continuous states to discrete states and discrete. outputs respectively.

retsys = sysdup (asys out_idx, in_idx)

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Duplicate specified input/output connections of a system Inputs asys system data structure out_idx in_idx indices or names of desired signals (see sigidx). duplicates are made of y(out_idx(ii)) and u(in_idx(ii)). Output retsys Resulting closed loop system: duplicated i/o names are appended with a "+" suffix. Method sysdup creates copies of selected inputs and outputs as shown below. u1 y1 is the set of original inputs/outputs, and u2 y2 is the set of duplicated inputs/outputs in the order specified in in_idx out_idx, respectively ____________________ u1 ----->| |----> y1 | asys | u2 ------>| |----->y2 (in_idx) -------------------- (out_idx)

sys = sysgroup (asys bsys)

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Combines two systems into a single system. Inputs asys bsys System data structures. Output sys sys = block diag(asysbsys) __________________ | ________ | u1 ----->|--> | asys |--->|----> y1 | -------- | | ________ | u2 ----->|--> | bsys |--->|----> y2 | -------- | ------------------ Ksys The function also rearranges the internal state-space realization of sys so that the continuous states come first and the discrete states come last. If there are duplicate names the second name has a unique suffix appended on to the end of the name.

sys = sysmult (Asys Bsys)

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Compute sys = Asys*Bsys (series connection): u ---------- ---------- --->| Bsys |---->| Asys |---> ---------- ---------- A warning occurs if there is direct feed-through from an input or a continuous state of Bsys through a discrete output of Bsys to a continuous state or output in Asys (system data structure does not recognize discrete inputs).

retsys = sysprune (asys out_idx, in_idx)

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Extract specified inputs/outputs from a system Inputs asys system data structure out_idx in_idx Indices or signal names of the outputs and inputs to be kept in the returned system; remaining connections are "pruned" off. May select as [] (empty matrix) to specify all outputs/inputs. retsys = sysprune (Asys [1:3,4], "u_1"); retsys = sysprune (Asys {"tx", "ty", "tz"}, 4); Output retsys Resulting system. ____________________ u1 ------->| |----> y1 (in_idx) | Asys | (out_idx) u2 ------->| |----| y2 (deleted)-------------------- (deleted)

pv = sysreorder (vlen list)

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Inputs vlen Vector length. list A subset of [1:vlen]. Output pv A permutation vector to order elements of [1:vlen] in list to the end of a vector. Used internally by sysconnect to permute vector elements to their desired locations.

retsys = sysscale (sys outscale, inscale, outname, inname)

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scale inputs/outputs of a system. Inputs sys Structured system. outscale inscale Constant matrices of appropriate dimension. outname inname Lists of strings with the names of respectively outputs and inputs. Output retsys resulting open loop system: ----------- ------- ----------- u --->| inscale |--->| sys |--->| outscale |---> y ----------- ------- ----------- If the input names and output names (each a list of strings) are not given and the scaling matrices are not square then default names will be given to the inputs and/or outputs. A warning message is printed if outscale attempts to add continuous system outputs to discrete system outputs; otherwise yd is set appropriately in the returned value of sys.

sys = syssub (Gsys Hsys)

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Return sys = Gsys - Hsys. Method Gsys and Hsys are connected in parallel. The input vector is connected to both systems; the outputs are subtracted. Returned system names are those of Gsys. +--------+ +--->| Gsys |---+ | +--------+ | | +| u --+ (_)--> y | -| | +--------+ | +--->| Hsys |---+ +--------+

ugain (n)

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Creates a system with unity gain no states. This trivial system is sometimes needed to create arbitrary complex systems from simple systems with buildssic. Watch out if you are forming sampled systems since ugain does not contain a sampling period.

W = wgt1o (vl vh, fc)

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State space description of a first order weighting function. Weighting function are needed by the H-2/H-infinity design procedure. These function are part of the augmented plant P (see hinfdemo for an application example). Inputs vl Gain at low frequencies. vh Gain at high frequencies. fc Corner frequency (in Hz not in rad/sec) Output W Weighting function given in form of a system data structure.

ksys = parallel (asys bsys)

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Forms the parallel connection of two systems. -------------------- | -------- | u ----->|----> | asys |--->|----> y1 | | -------- | | | -------- | |--->|----> | bsys |--->|----> y2 | -------- | -------------------- ksys

[retsys nc, no] = sysmin (sys, flg)

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Returns a minimal (or reduced order) system Inputs sys System data structure flg When equal to 0 (default value) returns minimal system, in which state names are lost; when equal to 1 returns system with physical states removed that are either uncontrollable or unobservable (cannot reduce further without discarding physical meaning of states). Outputs retsys Returned system. nc Number of controllable states in the returned system. no Number of observable states in the returned system. cflg is_controllable(retsys). oflg is_observable(retsys).