matlab的mpt工具箱（含voronoi函数）

User Interface for Modeling and Control Synthesis for MPT 3.0 ============================================================= The proposed user interface (UI) aims to fix the design flaws of the old MPT 2.6.x interface. The main points behind the new UI are: * consistency with respect to working with various prediction models (LTI, PWA, MLD, compositions of various subsystems) * consistency of mathematical formulation between on-line and explicit controllers * easy extensibility without the need to maintain a single monolithic implementation * ability to seamlessly import MPT 2.6.x formulations * promotion of the "correct" workflow where tuning is done on on-line controllers first, only converting them to the explicit form when the tuning is complete (avoiding frequent re-computation of the explicit solution) The UI consists of three primary layers, described in more details below. 1) Classes representing prediction models 2) Classes representing controllers 3) Class representing a generic closed-loop system 1. Representation of prediction models -------------------------------------- The UI allows to define four basic types of prediction models: 1) Linear Time Invariant (LTI) models 2) Piecewise Affine (PWA) models 3) Mixed-Logical Dynamical (MLD) models 4) Compositions consisting of several interconnected LTI/PWA/MLD models 1.1 LTI models -------------- LTI models describe linear (in fact, affine) time-invariant systems of the form x^+ = A x + B u + f y = C x + D u + g Such models, represented by the @LTISystem class, can be constructed in three ways: 1) By providing the matrices: lti = LTISystem('A', A, 'B', B, 'C', C, 'D', D, 'f', f, 'g', g) The 'f' and 'g' parameters can be omitted. 2) By importing from state-space models of the Control Toolbox: ss = ss(A, B, C, D, Ts) lti = LTISystem(ss) Note that only discrete-time state-space objects can be imported. 3) By importing from MPT2 sysStruct: lti = LTISystem(sysStruct) The returned object contains: 1) matrices A, B, C, D, f, g of the state-update and output equations (read-only) 2) dimensions nx, nu, ny (read-only) 3) sampling time Ts (read-only) 4) signals representing states (lti.x), inputs (lti.u) and outputs (lti.y) of the system (read/write) 5) domain of the system in the state-input space (can be set using the setDomain method) Properties of these signals (e.g. constraints and penalties) can be defined as described in Section 1.4. The class implements following methods: * initialize(x0): Sets the current state of the system * getStates(): Returns the current state of the system * update(u): Performs the state update using the control action "u" * output(u): Computes the output (note that providing the control action is only needed if the system contains direct feed-through) * setDomain(type, D): sets the domain of the system either in the state space (type='x'), input space (type='u') or state-input space (type='xu') * plot(): to visualize the optimized open-loop profiles of states, inputs and outputs (requires prior call to an optimization routine, explained in Section 2.x) The object remembers the value of its state and updates it whenever update() is called. This means that evolution of the system can be obtained in the following way: lti.initialize(x0) lti.update(u0) lti.update(u1) lti.update(u2) lti.getStates() See mpt3_lti_demo1.m, mpt3_lti_demo2.m, mpt3_lti_demo3.m, mpt3_lti_demo4.m, mpt3_lti_demo5.m for examples. 1.2 PWA models -------------- PWA models encode the PWA dynamics x^+ = A_i*x + B_i*u + f_i IF [x; u] \in Dom_i y = C_i*x + D_i*u + g_i IF [x; u] \in Dom_i PWA models can be created in three ways: 1) By providing an array of LTI systems, each of which represents one mode of the PWA system: dyn1 = LTISystem(...) dyn1.setDomain('x', Polyhedron(...)) % Region of validity dyn2 = LTISystem(...) dyn2.setDomain('x', Polyhedron(...)) % Region of validity pwa = PWASystem([dyn1, dyn2]) 2) By converting an MLD description to a PWA form: mld = MLDSystem(...) pwa = PWASystem(mld) 3) By importing from MPT2's sysStruct: opt_sincos pwa = PWASystem(sysStruct) Internally, PWA models are represented by the @PWASystem class, which contains following fields (all read-only): * A, B, C, D, f, g: cell arrays of matrices * domain: array of polyhedra denoting region of validity of corresponding local affine models * nx, nu, ny, ndyn: number of states, inputs, outputs and dynamics * Ts: sampling time * x, u, y, d: signals representing states, inputs, outputs and binary mode selectors Properties of signals 'x', 'u', 'y', 'd' (e.g. constraints and penalties) can be defined as described in Section 1.5. The class implements the same methods as in the LTI case (Section 1.1). 1.3 MLD models -------------- MLD models are represented by the @MLDSystem class and can be imported from a HYSDEL source file by calling mld = MLDSystem('filename.hys') If the source file contains symbolic parameters, their values can be provided as a structure in the second input argument: mld = MLDSystem('filename.hys', struct('param1', value1)) MLD models can be converted to a PWA form using the toPWA() method: pwa = mld.toPWA(); The class implements the same methods as in the LTI case (Section 1.1). 1.4 Signals ----------- As mentioned in Section 1.1-1.3, each prediction model contains "signals", which represent the state, input and output variables of the model. A "signal", represented by the @SystemSignal class, is a basic primitive which does two things: 1) it represents the prediction of a given quantity over a prediction horizon as a YALMIP variable 2) allows to define basic properties of the quantity A "signal" is created by calling the s=SystemSignal(n) constructor where "n" is the dimension of the signal. (NOTE: the user never has to create signals manually.) Once the signal is created, following basic properties can be set by the user: * lower bound (s.min = [...]) * upper bound (s.max = [...]) * penalty function (s.penalty = ...) As a specific example, consider the following LTI model of a double integrator: Ts = 1 s = c2d(ss(tf(1, [1 0 0])), Ts) lti = LTISystem(s) Then the "lti" object contains following signals: * lti.x: the 2x1 state vector * lti.u: the 1x1 input * lti.y: the 2x1 output vector To set the lower/upper bound on the input, one would call lti.u.min = -1; lti.u.max = 1; To set a quadratic state penalty of the form x'*Q*x on the state with Q=eye(2), call lti.x.penalty = Penalty(eye(2), 2) NOTE: Definition of penalties is described in Section 1.5. Besides the minimal set of properties described above, arbitrary additional properties can be added on-the-fly by the concept of custom properties, called "filters". Examples include polyhedral constraints, soft constraints, or move blocking. These custom properties are described by separate methods of the SystemSignal class, prefixed by "filter_" (see "methods SystemSignal"). Examples: * polyhedral set constraint (for states, inputs, outputs): lti.x.with('setConstraint') % enable the property lti.x.setConstraint = Polyhedron(...); % specification of the property * marking variables as binary (for states, inputs, outputs): lti.u.with('binary') lti.u.binary = true; * move-blocking constraint u(1)=u(2)=u(3): lti.u.with('block') lti.u.block.from = 1 lti.u.block.to = 3 * soft constraints (for states, inputs, outputs): lti.x.with('softMin') % soft lower bound lti.x.with('softMax') % soft upper bound * penalty on the final predicted state: lti.x.with('terminalPenalty') lti.x.terminalPenalty = Penalty(...) * polyhedral constraints on the final predicted state: lti.x.with('terminalSet') lti.x.terminalSet = Polyhedron(...) * adding a reference trajectory for MPC (for