2024_04_04_Lectrue_Slides_AC.pdf

Automatic Control by Prof. Dr.-Ing. Oliver Nelles University Created 13 April 2015 Prof. Dr.-Ing....

Automatic Control by Prof. Dr.-Ing. Oliver Nelles University Created 13 April 2015 Prof. Dr.-Ing. Page 1 Oliver Nelles of Siegen Contents Note: These slides have been translated from German language. 1. State-Space Control Therefore sometimes German 1.1 Dynamic Systems in State-Space Representation abbreviations are used. 1.2 Solving State-Space Equations Also sometimes a comma is used as a decimal point, i.e. 1,23 instead of 1.23. 1.3 Properties of State-Space Equations 1.4 State-Space Control via Pole Placement 1.5 State-Space Control via Optimization (LQ) 1.6 State Observer 1.7 State-Space Control with Observer (LQG) 1.8 Tracking 1.9 Reference Variable and Disturbance Model Chapters 3, 4, 5 are contained in Predictive Control and Optimization 2. Digital Control 2.1 Introduction 2.2 Brief Overview: Discrete-Time Systems 2.3 Stability of Discrete-Time Systems 2.4 Deadbeat Control 2.5 Adaptive Control University Prof. Dr.-Ing. Page 2 Oliver Nelles of Siegen Contents 3. Optimization: Linear in the Parameters 3.1 Introduction 3.2 Linear Problems 3.3 Quadratic Problems 4. Predictive Control 4.1 Introduction to Predictive Control 4.2 Linear Predictive Control 4.3 Constraints in Predictive Control 4.4 Nonlinear Predictive Control University Prof. Dr.-Ing. Page 3 Oliver Nelles of Siegen Contents 5. Optimization: Nonlinear in the Parameters 5.1 Search Algorithms 5.2 Gradient Method 5.3 Newton’s Method 5.4 Quasi-Newton-Method 5.5 Conjugate Gradient Method 5.6 Line Search 5.7 Nonlinear Problems with Constraints 5.8 Global Search Methods 5.9 Multi-Objective Optimization University Prof. Dr.-Ing. Page 4 Oliver Nelles of Siegen Bibliography State-Space Control: Lunze: „Regelungstechnik 1“, 9. Aufl., Springer 2012 Friedland: „Control System Design: An Introduction to State-Space Methods“, Dover, 2005 Digital Control: Lunze: „Regelungstechnik 2“, 7. Aufl., Springer 2013 Åström und Wittenmark: „Computer-Controlled Systems“, Prentice Hall, 1996 Franklin, Powell, Workman: „Digital Control of Dynamic Systems“, Ellis-Kagle Press; 3. Edition, 2019 Isermann: „Digitale Regelsysteme, Band 1“, 2. Aufl., Springer, 2008 Lutz, Wendt: „Taschenbuch der Regelungstechnik“, 9. Aufl., Deutsch, 2012 Interesting books can be downloaded as PDF under http://link.springer.com/ searching for “State Space” and „Digital Control“. University Prof. Dr.-Ing. Page 5 Oliver Nelles of Siegen 1. State-Space Control University Prof. Dr.-Ing. Page 6 Oliver Nelles of Siegen Content Chapter 1 1. State-Space Control 1.1 Dynamic Systems in State-Space Representation 1.2 Solving State-Space Equations 1.3 Properties of State-Space Equations 1.4 State-Space Control via Pole Placement 1.5 State-Space Control via Optimization (LQ) 1.6 State Observer 1.7 State-Space Control with Observer (LQG) 1.8 Tracking 1.9 Reference Variable and Disturbance Model University 1. State-Space Control Page 7 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation Properties of State-Space Representation Differential equation order n → System ofn differential equations each order 1. Description of the system not only by input/output behavior but by (internal) states by means of state variables allows deeper insights. Design methods predominantly in the time domain. Mathematical model of the controlled system required. Power of linear algebra is fully exploited. Numerically robust methods → M A TLA B. Emerged in the 1960s with groundbreaking work by Kalman. Major technological driver for aerospace (flight to the moon) More recent example of the success of state-space methods is ESP (Bosch). Well extendable to time-varying, nonlinear, and multivariable systems. Complex high-order systems can be displayed more clearly. University 1. State-Space Control Page 8 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation Matrix and Vector Calculus A vector can be interpreted in different ways: The computer science view: As a 1-dimensional array, e.g., for implementing a queue or stack. The engineering view: As a point in an n-dimensional space, or as a pointer from the origin of the coordinate system to that point. For vectors we distinguish between row and column vectors. By default, a vector is always in column form: To obtain a row vector, a column vector must be transposed: University 1. State-Space Control Page 9 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation Matrix and Vector Calculus The inner product or scalar product of two vectors yields a scalar. For this, both vectors must have the same dimension (here: n): The dyadic product of two vectors yields a matrix. For this, both vectors do not have to have the same dimension: University 1. State-Space Control Page 10 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation Matrix and Vector Calculus A matrix can be interpreted in different ways: The computer science view: A matrix is a 2-dimensional field or array. For example, the columns represent x-coordinates and the rows represent y-coordinates. An m×n matrix could represent the pixels of an image with n pixels horizontally and m pixels vertically, e.g., n = 1024, m = 768. Each element (matrix entry) could specify the amount of black in a gray image: 0 (0% black = white), 127 (50% black = gray), 255 (100% black = black). A logical extension of such a matrix notion to higher dimension are tensors, e.g. an order 3 tensor realized by a 3-dimensional field/array describing e.g. voxels in a volume (instead of pixels on a surface). University 1. State-Space Control Page 11 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation Matrix and Vector Calculus The engineering view: A matrix A represents a (linear) mapping of a vector x (input) to another vector y (output): In the general case the vectors can have different dimensions, e.g., dim(x) = n, dim(y) = m. Then the matrix is rectangular: i. input j. output If n > m (more columns than rows), then the matrix is called fat. If n < m (more rows than columns), then the matrix is called skinny. If n = m (equal number of columns and rows), then the matrix is square. University 1. State-Space Control Page 12 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation Matrix and Vector Calculus We are mostly dealing with square matrices. In this case, the mapping A can be understood as a scaling and rotation operation, i.e., the vector x is mapped into a vector y by multiplication with A, by changing the length and the angle of x. Any (linear) transform into another coordinate system can be done by matrix multiplication. Example: Rotation of the Coordinate System The x- und y-axes are each to be rotated counterclockwise by angle 𝜑. The following transformation matrix does this: y y‘ x‘ With it we can transform the old coordinates x and y φ into the new coordinates x‘ and y‘: x University 1. State-Space Control Page 13 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation Matrix and Vector Calculus A matrix-vector multiplication can also be interpreted column-wise: Each column aj of the matrix A is assigned to an input xj : Therefore, the matrix-vector multiplication can also be understood as a weighted sum of the column vectors aj of the matrix A with the inputs xj as weights: with University 1. State-Space Control Page 14 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation Matrix and Vector Calculus A matrix-vector multiplication can also be interpreted row by row: One row aiT of the matrix A is assigned to an output yi : Therefore, the matrix-vector multiplication can also be understood as a set of scalar products of the row vectors aiT of the matrix A with the input vectors x: with University 1. State-Space Control Page 15 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation The input/output behavior of an n-th order linear dynamic system with input u(t) and output y(t) can be described by the following transfer function: always achievable by cancellation Without loss of generality, an = 1 was set here. We distinguish 2 cases: m < n: Normal case; no direct feedthrough (= nicht sprungfähig). m = n: Occurs only infrequently; direct feedthrough (= sprungfähig). In the state-space representation, such a system can be described as follows: (State-space differential equations) (Output equation) A: n x n system matrix cT: 1 x n output vector b: n x 1 input vector d: feedthrough scalar (= 0 if m < n, i.e., usually) University 1. State-Space Control Page 16 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation only if system has direct feedthrough : n-dimensional vector : scalar University 1. State-Space Control Page 17 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation How are the Matrices and Vectors of the Equations of State-Space Representation Related to the Coefficients of the DGL and the Transfer Function, respectively? For simplicity, let us first consider the special case where the numerator of the transfer function is equal to 1: Since this is a system of n-th order, we have to define n states. This choice is not distinct or unambiguous. However, each state must be assigned to an energy storage or an integrator of the system. derivative of Xn(s) n states University 1. State-Space Control Page 18 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation Scheme in Frequency Domain...... University 1. State-Space Control Page 19 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation Scheme in Time Domain...... University 1. State-Space Control Page 20 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation From the Scheme One Can Easily Set Up the Equations of State-Space Representation: In matrix/vector notation: University 1. State-Space Control Page 21 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation Let us extend this result to systems with arbitrary numerator polynomials, but initially only for systems without direct feedthrough (m < n): In the previous example with numerator = 1 it was and war. If we keep this definition of the states, the following is still true: CAUTION: Y(s) is now defined differently because of the complete numerator polynomial. But we take over the definition of the states from the simple case. This results in: University 1. State-Space Control Page 22 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation Xm+1 exists, of course, only if m < n. In case of m = n it must be replaced by. For systems without feedthrough (i.e., m < n) this results in the following equations: This special form of the state equations is called controllable canonical form. Among other things, it has the following nice properties: The numerator coefficients of the transfer function appear directly in cT. The denominator coefficients of the transfer function appear directly in A. The system matrix A has a very special structure; of the n x n elements only the n elements of the last row depend on the system. The input vector b is completely independent of the system properties. University 1. State-Space Control Page 23 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation Scheme of the Control Normal Form for Systems without Feedthrough in the Time Domain numerator......... denominator Thus, from each transfer function, the equations of state can be directly set up in controllable canonical form. There is a 1:1 relationship between them. University 1. State-Space Control Page 24 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation For systems with feedthrough m = n holds and the equation transforms to: Eliminating the first term by means of (see scheme in the frequency domain) results in the controllable canonical form for systems with feedthrough: additional terms feedthrough University 1. State-Space Control Page 25 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation Direct Setting Up of the Equations of State Space Representation Plant/process for position control of an electric drive Dynamics of the Dynamics of the electrical part mechanical part KE TE KM TM 1 Input variable: Converter input voltage Output variable: Rotation angle State-space variable: Rotation angle Order/Sequence Speed / angular velocity is arbitrary Current ~ Motor torque University 1. State-Space Control Page 26 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation Source: http://www.rn-wissen.de/index.php/Regelungstechnik University 1. State-Space Control Page 27 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation With this choice of the states the following equations of state result (with another choice of the states other equations of state would result, which would exhibit, however, the same input/output behavior!) (integrator) (mechanical PT1) (electrical PT1) In matrix/vector notation: University 1. State-Space Control Page 28 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation How can we now convert the state equations into, say, controllable canonical form? There are 2 alternative ways: 1. Set up the transfer function. → Create the equationsofstate. 2. Direct transformation in the state space (topic of Section 1.3). Carrying out the 1st way: The numerator and denominator coefficients directly yield the equations of state in controllable canonical form: University 1. State-Space Control Page 29 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation Definition: State of a Dynamic System A vector x is called a state vector (its elements state variables) of a system if the output variable y(te) can be determined solely by means of its initial value x(0) and the course of the input variable u(t), 0 ≤ t ≤ te, and this for any time te. This means, among other things: The state vector must contain all internal information about the system. Otherwise, the calculation of the output variable would not be possible. If we assign one state to each energy store or integrator, we have fulfilled this requirement. The state vector can contain more elements than necessary. If the state vector contains only the necessary number of states, then one speaks of a minimal realization. There are infinitely many choices for the state vector called realizations that lead to the same input/output behavior. Realizations that have certain useful properties are called canonical forms or normal forms. University 1. State-Space Control Page 30 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation Other Examples of the Choice of States travelling crane claw 2nd order (if output = speed) States: armature current 4th order = 2 × 2nd order angular velocity States: position crane 3rd order (if output = rotation angle) velocity crane States: armature current position claw angular velocity velocity claw rotation angle University 1. State-Space Control Page 31 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation Why Do We Speak of the State Space? We can think of the state vector as a point in an n-dimensional space. The state-space differential equation then describes how this point moves in the state space. If the input variable u(t) = 0, then the point evolves according to the proper motion of the system: If the system is stable, then the point x(t) will tend towards the origin of the state space, i.e., x(t → ∞ ) → 0. State vector of a PT2 capable of oscillation The function f (x) = A x maps a vector to a vector. Such a 1.5 function is called a vector field. The point moves in 1 this vector field along the field lines, like a particle in 0.5 a flow field along the flow lines. For 2- and 3-dimensional 0 vector fields a graphical illustration in the plane or in space is -0.5 possible. -1 -2 -1 0 1 2 University 1. State-Space Control Page 32 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation Extension to Multivariable Systems We will deliberately restrict ourselves here to systems with one input and one output, since the treatment of multivariable systems requires many additional considerations that complicate the immediate understanding. However, we will briefly show how easy it is to formally extend the description of dynamical systems in state space to the multivariable case: 1 Input, 1 Output: nu Inputs, ny Outputs: A: n x n , b: n x 1 A: n x n , B: n x nu cT: 1 x n , d: 1 x 1 C: ny x n , D: ny x nu Most equations can be easily extended formally from the single to the multivariable case. In the multivariable case, however, many additional difficulties arise which require a separate and more detailed treatment. University 1. State-Space Control Page 33 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.1 Dynamic Systems in State-Space Representation MATLAB Generate a dynamic system in the state space: System = ss(A,b,c,d); % Generates state-space % equations. [A,b,c,d] = ssdata(System); % Returns matrices and vectors Convert transfer functions to state space and vice versa: [A,b,c,d] = tf2ss(num,den) % Converts transfer function‘s % numerator and denominator % polynomial to state-space form. [num,den] = ss2tf(A,b,c,d); % Conversion the other way % around. Calculation of one minimum realization: System_min = minreal(System) % Calculates one minimum % realization of “system” University 1. State-Space Control Page 34 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.2 Solving State-Space Equations We now want to calculate how the state vector evolves over time; as a function of an initial value x(0) and the time course of the input u(t). Once the state vector x(t) has been calculated, we can use the output equation to determine the output y(t) quite easily. So, let's start with the differential equations of state: with Analogous to the one-dimensional case (see RT), we make an exponential functions approach: Here, k(t) is an n-dimensional vector and the so-called matrix exponential is defined as follows: identity matrix Quite analogous to the series expansion of the scalar exponential function: University 1. State-Space Control Page 35 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.2 Solving State-Space Equations The derivative of the matrix exponential is calculated as follows: Since A can be bracketed out on both the left and right sides, this results in: Substituting (variation of constants) into the differential state-space equations yields: An inversion of the matrix exponential yields:. This leads to: University 1. State-Space Control Page 36 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.2 Solving State-Space Equations Integration provides: with and one obtains: Equation of motion: The following abbreviation is often introduced: This function is called transition matrix or fundamental matrix. It is used to calculate the equation of motion: corresponds to impulse response g(t) in scalar case University 1. State-Space Control Page 37 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.2 Solving State-Space Equations The solution of differential state-space equations is composed of 2 parts: The homogeneous solution xhom(t) describes the proper motion of the system starting from an initial value x0 without input signal, i.e., with u(t) = 0. The inhomogeneous (German: partikulär) solution xpart(t) describes the reaction of the system to a certain course of the input variable u(𝜏), 0 ≤ 𝜏 ≤ t. For stable systems, the homogeneous solution fraction decays exponentially and therefore does not matter for the input/output behavior after a certain time. The output of the system is calculated directly from the state vector (and systems with feedthrough also from the input). This leads to: Output equation of motion: University 1. State-Space Control Page 38 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.2 Solving State-Space Equations Solution of Differential State-Space Equations in Frequency Domain: identity matrix Solving the differential state-space equations according to X(s) provides: The matrix sI always has full rank; therefore, sI – A, is invertible except for single values s = si. The output then results to: Also in the frequency domain, the two solution components show up depending on the initial value and the input Yhom und Ypart. The transfer function is therefore: University 1. State-Space Control Page 39 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.2 Solving State-Space Equations Example: Given the following transfer function: This corresponds in controllable canonical form to the equation of state with From this the transfer function is calculated to: University 1. State-Space Control Page 40 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.2 Solving State-Space Equations Stability of the Equations of State Preliminary considerations: The output equation merely maps the state vector to the output. It is therefore irrelevant for stability considerations. In the state-space differential equations, u(t) = 0 can be set for the stability investigation, because the course of the input signal has nothing to do with the stability. Thus, we study only the homogeneous differential state-space equations: Thus, the stability of the system obviously depends only on the system matrix A ! In the following we will see that the stability depends on the eigenvalues of A. More precisely formulated: Let a dynamic system be described by state equations. This system is stable if all eigenvalues of the system matrix A have a negative real part. University 1. State-Space Control Page 41 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.2 Solving State-Space Equations How are the Eigenvalues and Eigenvectors Related to Stability? Let us recall the geometrical meaning of eigenvalues and eigenvectors. They are determined by the following relationship: An eigenvector of matrix A is mapped by A to a multiple of itself. This multiple (can also be < 1) is called an eigenvalue. In general, a matrix maps a vector to a vector. For a square matrix, the input vector and output vector are of the same dimension: Such a mapping can be thought of as a rotation (in n-dimensional space) and a scaling of the vector. If no rotation takes place, so the direction of the input vector v remains the same, then v is an eigenvector. The scaling factor 𝜆 is the corresponding eigenvalue. University 1. State-Space Control Page 42 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.2 Solving State-Space Equations An n x n matrix has n eigenvectors and n associated eigenvalues (which need not all be different). Under certain conditions, e.g., if the matrix A has many different real eigenvalues, the eigenvectors are linearly independent. Then the eigenvectors span the entire n-dimensional state space; i.e., every state vector can be written as a linear combination of the eigenvectors Let us return to the stability test: As an approximation we can also write (for very small ∆t): Thus, the state vector x changes per time unit according to what is written on the right side of the equation. The system is stable exactly when a positive state vector becomes smaller in each step, i.e., its change is negative. Since each x can be composed of the eigenvectors of A, we can guarantee stability if all associated eigenvalues are negative. All eigenvectors shrink and therefore also each x composed of these eigenvectors shrinks, and thus tends towards 0. University 1. State-Space Control Page 43 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.2 Solving State-Space Equations Calculation of Eigenvalues and Eigenvectors Starting from the definition equation we get: Since we are not interested in the trivial solution v = 0 here 𝜆 I – A must be singular, i.e.: This is the characteristic equation of the system. The polynomial is called characteristic polynomial and has the degree n. Its n zeros are the eigenvalues of the matrix A and thus poles of the associated transfer function (provided there are no pole/zero truncations). To calculate the eigenvectors, we step-by-step put the eigenvalues in and get for each eigenvalue a system of n equations with n unknowns, namely the n components of the respective eigenvector. These systems of equations are underdetermined because we can only uniquely determine the direction of the eigenvectors, not their magnitude. If v is an eigenvector, so is 𝛼 v. Typically, therefore, the eigenvectors or one of their components are normalized to 1. University 1. State-Space Control Page 44 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.2 Solving State-Space Equations Example: Calculating the eigenvalues and eigenvectors of a 3 × 3 matrix Calculate the characteristic polynomial: This provides the eigenvalues: University 1. State-Space Control Page 45 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.2 Solving State-Space Equations Calculation of the eigenvector to 𝜆1 = 6: Let us solve the first two equations: 2. 1. From this we get the 1st eigenvector: University 1. State-Space Control Page 46 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.2 Solving State-Space Equations Calculation of the eigenvector to 𝜆2 = 3: The last two equations are equivalent. So we only need the first two: 2. 1. From this we get the 2nd eigenvector: University 1. State-Space Control Page 47 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.2 Solving State-Space Equations Calculation of the eigenvector to 𝜆3 = 1: The last two equations are equivalent. So we only need the first two: 2. 1. From this we get the 2nd eigenvector: University 1. State-Space Control Page 48 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.2 Solving State-Space Equations System Matrix A in Diagonal Form A very important canonical form of the state-space equations exists when A has diagonal form (also called modal form). The system matrix then has the following structure: It will be very rare that A has diagonal form right after modeling. Rather, it will usually be necessary to transform the equations of state accordingly. How to do this, is discussed in Section 1.3. Remark: Only if A has exclusively real and unique (not multiple) eigenvalues, it is guaranteed that the state-space equations be transformed to diagonal form. Otherwise, additional conditions must be fulfilled or only a transformation to “approximate” diagonal forms is possible. University 1. State-Space Control Page 49 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.2 Solving State-Space Equations Important Properties of the Diagonal Form The diagonal elements aii of A are the eigenvalues 𝜆i of A und die eigenvectors are the unit vectors: This can be easily shown:... University 1. State-Space Control Page 50 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.2 Solving State-Space Equations This means that the n differential equations of state are decoupled. No state influences another; n PT1 systems run in parallel side by side (this also explains why this cannot simply work the same way for conjugate complex poles).... University 1. State-Space Control Page 51 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.2 Solving State-Space Equations Thus, the diagonal form offers the advantage that one can read the eigenvalues (poles) of the system immediately from A. The diagonal form thus corresponds to the partial fraction decomposition of the associated transfer function. How does the time solution of the differential equations of state simplify when A has diagonal form? The calculation of the matrix exponential function simplifies considerably, because it holds: University 1. State-Space Control Page 52 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.2 Solving State-Space Equations The time solution of the differential state-space equations is generally: With a diagonal matrix A this simplifies to: The functions are also called modes. This is where the name modal form comes from. As soon as state-space equations have been transformed into diagonal form, they can easily be solved as n independent 1st order DGLs. But for such a transformation we have to calculate the eigenvalues and eigenvectors of A (see Section 1.3)! No free lunch! University 1. State-Space Control Page 53 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.2 Solving State-Space Equations Diagonal Form with Complex Eigenvalues If complex eigenvalues occur (and thus complex eigenvectors), the diagonal form is also complex. For example, a diagonal matrix with eigenvalues 𝜆, 𝛼 + i, 𝛼 – i looks like this: (all states are decoupled) Since all signals and parameters of the system are real, one prefers to calculate with purely real quantities. This can be achieved by transforming the above matrix into the following form: (states x2 and x3 are decoupled) The corresponding eigenvectors are now also real. In general, 2 decoupled DGLs (diagonal form!) with conjugate complex parameters can be transformed to 2 coupled combined DGLs with real parameters (PT2). University 1. State-Space Control Page 54 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.2 Solving State-Space Equations Jordan Form with Multiple Eigenvalues If multiple eigenvalues occur, the system matrix may no longer be able to be put into a pure diagonal form. For example, a matrix with a double eigenvalue at 𝜆1 and a single one at 𝜆2 can be put into one of the following forms: or For a matrix with a triple eigenvalue at 𝜆1 and a single one at 𝜆2, one of the following forms holds: 2-dim Jordan block 3-dim Jordan block or or 1-dim Jordan block A matrix that contains at least one Jordan block of size 2 (or more) is in Jordan form. A Jordan block contains “ones” on the first secondary diagonal. University 1. State-Space Control Page 55 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.3 Properties of State-Space Equations Before we can get to the controller design, we need to understand some basic properties of the state-space equations. Cayley-Hamilton Theorem Let P(s) be the characteristic polynomial of a matrix A, then the characteristic equation is: If we insert the matrix A into this (actually scalar) equation, it itself satisfies its own characteristic equation (with ): That is, the matrix An (and all higher powers of A) are linearly dependent on the matrices Ai with i < n: University 1. State-Space Control Page 56 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.3 Properties of State-Space Equations Lyapunov Equation In control engineering Lyapunov equations play an important role in many areas. Therefore, we want to introduce them briefly here. Let the n x n-matrices A, B, und C be given. We are looking for the matrix M which satisfies the following equation: This equation contains n2 unknowns, the elements of the n x n-matrix M. After multiplying out the matrices, a linear system of equations with n2 equations is obtained, which can easily be solved for these unknowns. One can even formulate the Lyapunov equation a bit more generally by allowing different dimensions for A and B: A: n x n, B: m x m → C and M: n x m. In MATLAB such Lyapunov equations can be solved by MATLAB M = lyap(A,B,-C). University 1. State-Space Control Page 57 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.3 Properties of State-Space Equations Example: Solving a 2 x 2-Lyapunov equation  c11 c12   a11 a12  m11 m12   m11 m12  b11 b12  c  =   +    21 c22   a21 a22  m21 m22   m21 m22  b21 b22  a m +a m a11m12 + a12 m22   m11b11 + m12b21 m11b12 + m12b22  =  11 11 12 21  +  a m +  21 11 22 21 21 12 a m a m + a22 22   21 11 m m b + m b 22 21 m b 21 12 + m22 22  b c = L  m → m = L−1  c This system of equations with 4 (2 × 2) equations and 4 unknowns (m11, m12, m21, m22) can be solved easily! University 1. State-Space Control Page 58 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.3 Properties of State-Space Equations Controllability A system is called controllable if it can be transferred from any initial state x0 to any final state x(te) in finite time te by a suitably chosen course of the input variable u(0)... u(te). The connection between the state x and the output variable y does not matter at all. It is only of importance whether one can influence all state variables in x with the input u at will. Therefore, it is clear that the controllability can be checked by means of A and b alone. The final state x(te) is calculated from the initial state x0 , as we already know, with: If we set e.g., x(te) = 0 we get: University 1. State-Space Control Page 59 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.3 Properties of State-Space Equations Reducing leads to: Substituting the series expansion of the matrix exponential yields: I.e., x0 is the sum of infinitely many n-dimensional column vectors. Any initial state x0 can be generated by a suitable course of the input variable only if the vectors on the right-hand side span the entire n-dimensional space. That means, the following matrix (with infinitely many columns) must have rank n: University 1. State-Space Control Page 60 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.3 Properties of State-Space Equations From Cayley-Hamilton theorem we know that An is linearly dependent on the lower powers of A ,i.e. An can be written as a linear combination of I, A, A2,... An–1. The same is true for all higher powers of A. Therefore, only the first n columns of the above matrix contribute to its rank. So, it is sufficient to check whether the matrix, consisting of the first n columns, has rank n. Controllability Criterion The following n x n matrix is called controllability matrix: If the controllability matrix has full rank, then the system (A, b) is controllable: rank Note: In the numerical calculation of controllability, it is necessary due to limited computational accuracy to specify a tolerance limit in order to define from where a rank drop (singularity) is detected. In case of bad conditioning of SS it is quite reasonable to speak of “badly controllable” or “hardly controllable” or similar. University 1. State-Space Control Page 61 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.3 Properties of State-Space Equations What Kind of Systems Are Not Controllable? Intrinsic processes that are not connected to the input cannot be controlled. This can be easily seen in the diagonal form. If there is a bi = 0, then the state xi cannot be influenced by the input. This state and therefore the system are not controllable 2 parallel subsystems with the same dynamic properties are not controllable. Example: Since both subsystems have identical dynamics, the two states can only be influenced linearly dependently. For example, it will always be true that: x1 > 0 → x2 > 0 (if b1 and b2 have the same algebraic sign) and x1 > 0 → x2 < 0 (if b1 and b2 have different algebraic signs). Each zero/pole reduction (in this order) makes zero 𝜆 pole 𝜆 one state each non-controllable. University 1. State-Space Control Page 62 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.3 Properties of State-Space Equations Decomposition of a System Into a Controllable and a Non-Controllable Part Each system can be decomposed into a controllable and non-controllable part. Original system: controllable Decomposed System: non-controllable controllable subsystem (A11, b1) subsystem is controllable non-controllable subsystem Note: If the original system is already controllable, this results in:. University 1. State-Space Control Page 63 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.3 Properties of State-Space Equations Observability A system is called observable if the initial state x0 can be determined from the course of the input variable and output variable known over a finite interval [0, te]. To determine observability, it is only important to know how the state x acts on the output y. Therefore, it is clear that observability can be verified using A and cT alone. (Knowledge of the course of the input is indeed necessary to reconstruct the initial state; however, this is possible with any b, so there is no information about observability in b.) Observability Criterion The following n x n matrix is called observability matrix: If the observability matrix has full rank, then the system (A, cT) is observable: University 1. State-Space Control Page 64 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.3 Properties of State-Space Equations What kind of systems are not observable? Intrinsic processes that are not connected to the output are not observable. This can be easily seen in the diagonal form. If there is a ci = 0, then the state xi cannot influence the output. This state and thus the system are therefore not observable. 2 parallel subsystems with the same dynamic properties are not observable. Example: Although both states influence the output, their influence can no longer be separated, since both subsystems have the same dynamics. It is therefore no longer possible to determine in retrospect which part originates from which state. Each pole/zero reduction (in this order) makes pole 𝜆 zero 𝜆 one state unobservable at a time. University 1. State-Space Control Page 65 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.3 Properties of State-Space Equations Decomposition of a System Into an Observable and a Non-Observable Part Any system can be decomposed into an observable and non-observable part. Original system: observable Decomposed system: non-observable observable subsystem (A11, c1T) subsystem is observable non-observable subsystem Note: If the original system is already controllable, this results in:. University 1. State-Space Control Page 66 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.3 Properties of State-Space Equations Kalman Decomposition If we summarize the decompositions regarding controllability and observability, we can decompose a system into a total of 4 parts: Part 1: controllable and observable Part 2: controllable and non-observable Part 3: non-controllable and observable Part 4: non-controllable and non-observable University 1. State-Space Control Page 67 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.3 Properties of State-Space Equations Duality between Controllability and Observability The controllability and observability criteria are very closely related. We note that the controllability criterion merges into the observability criterion and vice versa when und Therefore, the two properties (controllability and observability) are said to be dual to each other. Besides the “normal” (“primal”) system, one can consider the following dual system: Primal System: Dual System: If the primal system (A, b, cT) is controllable, then the dual system (AT, c, bT) is observable. If the primal system (A, b, cT) is observable, then the dual system (AT, c, bT) is controllable. University 1. State-Space Control Page 68 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.3 Properties of State-Space Equations Transformation of the State-Space Equations We already know that there are infinitely many possible realizations of a transfer function in the state space, even if we restrict ourselves to the minimal realizations, i.e., those which contain only the minimal number of states. The different realizations differ only in the choice of the state vector; they are identical with respect to their input/output behavior. How to transform state-space equations into other realizations? First, a regular (i.e. a full rank) n x n transformation matrix T is chosen. Between the old state vector x and the new (transformed) state vector xT it is converted like this: How exactly the transformation matrix T has to be chosen in order to be able to transform into the most important canonical forms is discussed further below. University 1. State-Space Control Page 69 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.3 Properties of State-Space Equations This transformation results in the following new state-space equations: → With the definition of a new (transformed) system matrix and input and output vectors, the transformed state-space equations result in: with University 1. State-Space Control Page 70 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.3 Properties of State-Space Equations Transformation Into the Diagonal Form To transform the equations of state into the diagonal form, the transformation matrix must be constructed with the eigenvectors vi of the system matrix A: This can be easily shown by calculating the transformed system matrix: Since vi are the eigenvectors of A, it follows that and therefore: It is indeed diagonal! University 1. State-Space Control Page 71 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.3 Properties of State-Space Equations Transformation Into the Controllable Canonical Form In order to transform the equations of state into the controllable canonical form, a transform- ation matrix must be constructed which depends on the inverse of the controllability matrix SS. This inverse exists only if SS has full rank, i.e., the system is controllable. For non-controllable systems the controllable canonical form does not exist, only for the controllable part! The controllability matrix is defined as follows: From its inverse the last row is needed: From this, the transformation matrix is calculated to be: University 1. State-Space Control Page 72 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.3 Properties of State-Space Equations Transformation Into the Observable Canonical Form In order to transform the equations of state into the observable canonical form, a transform- ation matrix, depending on the inverse of the observability matrix SB , has to be constructed. This inverse exists only if SB has full rank, i.e., the system is observable. For non-observable systems the observable canonical form does not exist, only for the observable part. The observability matrix From its inverse is defined as follows: the last column is needed: From this, the transformation matrix is calculated to be: University 1. State-Space Control Page 73 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.3 Properties of State-Space Equations The observable canonical form, like the controllable canonical form, can be determined directly from the numerator and denominator coefficients of the associated transfer function. For systems with feedthrough, it results from the following equations: For systems without feedthrough the equations simplify because bn = 0. The input vector b has then only the elements bi (i = 0,..., n–1) and the feedthrough is omitted. There exists a duality between the controllable and the observable canonical forms. The following is true (the indices C and O stand for the respective canonical forms): University 1. State-Space Control Page 74 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.3 Properties of State-Space Equations MATLAB Calculation of eigenvectors and eigenvalues: A = diag([2 3 4]); % Generates a diagonal matrix [EV, EW] = eig(A); % Calculates the eigenvectors and eigenvalues of % A. The eigenvectors are in the columns of the % matrix EV. The eigenvalues are in the diagonal % of the matrix EW. Calculation of controllability and observability: System = ss(A,b,c,d); % Generates state-space equations S_S = ctrb(System); % Generates controllability matrix S_B = obsv(System); % Generates observability matrix r = rank(S_S); % Calculates rank of a matrix Transformation to diagonal form and observable canonical form: [System_T,T] = canon(System,'modal'); % Transformation to diagonal % form (modal form). % T = transformation matrix [System_T,T] = canon(System,'companion'); % Transformation to % observable canonical form University 1. State-Space Control Page 75 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.4 State-Space Control via Pole Placement The basic idea of a state-space controller is to feed back the states − i.e.,internal information from the process. This is in contrast to the standard control loop, where only the controlled variable, i.e., the output of the process, is fed back. Thus, the state-space controller can be understood as a kind of systematic extension of the idea of cascade control loop. Advantages of State-Space Controllers: Small deviations are detected and corrected very quickly because a lot of information from inside the process is used (same idea as cascade control). Very elegant and sophisticated methods for controller design exist. The achievable control quality is usually very high. Drawbacks of State-Space Controllers: State-space methods require good models and computer support. Simple interpretability of results (e.g., of controller parameters) not given. Lack of experience of staff; no direct connection to the PID controller. University 1. State-Space Control Page 76 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.4 State-Space Control via Pole Placement The block diagram shows that not the output y(t) is fed back, but the entire n-dimensional state vector x(t). The controller sums up the individual states weighted (scalar product) and leads the (scalar) result to the reference value / actual value comparison. The manipulated variable is thus calculated by: Many books focus on the case: Initial values x0 are compensated and set w(t) = 0. We continue to calculate with w(t), because of the more direct reference to the standard control loop. Process Note: For an easier understanding, we will only deal with systems without direct feedthrough in the following. However, an extension to systems with direct feedthrough Controller is easily possible. University 1. State-Space Control Page 77 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.4 State-Space Control via Pole Placement The state-space controller therefore acts proportionally with the factors ki on the states. Is it therefore also a P-controller? No! To make the relationships as clear as possible, we assume the following process with numerator polynomial = 1: In the state space the corresponding controllable canonical form is: I.e., y(t) = x1(t). University 1. State-Space Control Page 78 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.4 State-Space Control via Pole Placement Written in the Laplace domain, the initial equation and the first n-1 differential equations of state after an exchange of the right-hand side with the left-hand side become:... The state-space controller returns the states x1(t), x2(t),..., xn(t). Thus, it (implicitly) returns the output, the derived output, the twice derived output, etc.: P D D2 Dn–1 It is therefore a PDn–1-controller. However, with a decisive difference to the standard controller: The derivatives are not obtained by differentiation of the output signal. Instead, the states are measured which already correspond to these differentiated quantities! So, there is no amplification of noise here! University 1. State-Space Control Page 79 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.4 State-Space Control via Pole Placement Note: If you have paid close attention, you will have recognized another difference to the standard controller: The derivatives refer exclusively to the output signal y(t). For the standard control loop, they would refer to the control deviation e(t) = w(t) – y(t). I.e., PDn–1 controller in the standard control loop would (in Laplace domain) look like this: But for constant reference variables w(t) = constant all derivatives of w(t) are zero anyway: sW(s) = 0, s2W(s) = 0, etc., and then both equations are (almost) identical. For the design of a state-space controller, i.e., the systematic determination of the values for ki, there are two main methods: Pole placement: There are n control parameters available (k1, k2,..., kn), with which it is possible to shift the n poles of the characteristic polynomial to desired positions. Optimization: Minimizing a quadratic loss function leads to a globally optimal, simple, easily computable solution for the state-space controller (in contrast to the standard controller). This is discussed in Section 1.5. University 1. State-Space Control Page 80 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.4 State-Space Control via Pole Placement Our goal is now to derive an equation for determining the controller parameters. To do this, we need to find the relationship between the controller parameters and the characteristic equation. The equations of state with controller are as follows: I.e., the system matrix A is changed by the state feedback and the reference variable w takes the place of the manipulated variable u. Everything else remains as it was. Since A contains all information about the poles of the system (all denominator coefficients ai of the transfer function), we can expect to be able to change these poles specifically with the help of the state feedback; but also, only these (i.e., no zeros). We have n degrees of freedom with k ! Note: The dyadic product b kT is calculated as follows: University 1. State-Space Control Page 81 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.4 State-Space Control via Pole Placement Now, if we would like to specify the n poles 𝜆1, 𝜆2,..., 𝜆n this is equivalent to: we would like the following characteristic equation: Desire: For the characteristic equation of the closed loop, we have to calculate the determinant sI – (A – b kT). Equating the characteristic equation of the closed loop with the desired polynomial N(s) leads to the following condition: Calculating the determinant also leads to a polynomial of n-th degree and coefficient comparison then yields the sought-after controller parameters ki. However, the state-space controller is able to move all poles arbitrarily only if the system (A, b) is controllable. Because we know: For non-controllable systems, the manipulated variable has no influence on certain states and no controller can impose a desired dynamic on these states. University 1. State-Space Control Page 82 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.4 State-Space Control via Pole Placement The calculation of the determinant in the pole placement equation turns out to be quite tedious, especially for high order systems (large n). This calculation is extremely simplified if the state-space equations are available in controllable canonical form: If this form exists then we also know: The system is controllable! University 1. State-Space Control Page 83 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.4 State-Space Control via Pole Placement This means: Also the closed-loop control can be written in controllable canonical form. Thus, we get from the system matrix of the open loop to that of the closed loop control by the following simple operation: open loop ➝ closed loop Because the system matrix of the closed loop control occurs again in controllable canonical form, we can read off the characteristic polynomial directly: Equating with the desired polynomial yields: University 1. State-Space Control Page 84 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.4 State-Space Control via Pole Placement This directly results in the controller parameters: Or in vectorial form: Thus, the controller parameters just correspond to the differences between the coefficients of the desired characteristic polynomial and those of the open loop. The more they deviate from each other, the more the feedback is amplified (ki is greater). In controllable canonical form, the controller design by pole specification is therefore almost trivial. If a system is not in controllable canonical form (this is the normal case), then we have two alternatives: 1. Calculation of the determinant of sI – (A – b kT) and subsequent coefficient comparison with the desired polynomial. 2. a) Transformation of the system into controllable canonical form (x → xT). b) Control design in controllable canonical form by means of ki+1 = ni – ai. c) Back transformation of the controller to the original states of the system (xT → x). University 1. State-Space Control Page 85 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.4 State-Space Control via Pole Placement Pole Specification for Equations of State that Are Not in Controllable Canonical Form In order to be able to use the simple formula for the pole placement in the controllable canonical form, it makes sense to choose the 2nd alternative. To do this, the state-space equations must first be converted to controllable canonical form: controllable canonical form original system This is done with the help of the following transformation matrix, in which the last row of the inverse controllability matrix sST occurs: After the controller design in controllable canonical form is carried out, the system is transformed back to the state space of the original system: University 1. State-Space Control Page 86 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.4 State-Space Control via Pole Placement The information how the controller kT must be transformed is still lacking. The controlled original system follows the state differential equations: The transformed system then becomes: → From this follows: controllable canonical form original system University 1. State-Space Control Page 87 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.4 State-Space Control via Pole Placement Instead of performing above transformations in sequence, it is possible to simplify the equations first. If the controller is designed in the controllable canonical form, it should be transformed back to the original system with the transformation given on the last slide: Substituting the transformation matrix yields: Expanding the right product results in: According to the Cayley-Hamilton theorem, the expression in the parenthesis is equal to –An. Substituting this, we obtain a simplified equation for controller design by pole placement for systems in arbitrary state-space form: University 1. State-Space Control Page 88 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.4 State-Space Control via Pole Placement This term can also be incorporated into the vector by multiplying it with 1 Formula According to Ackermann The state-space controller by pole placement for arbitrary state space forms is calculated by: sST is the last row of the inverse controllability matrix of the original system. University 1. State-Space Control Page 89 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.4 State-Space Control via Pole Placement Design of a State-Space Controller by Means of the Lyapunov Equation. A very sophisticated and easy to implement method of pole placement design is available via the solution of a Lyapunov equation. The method works like this: 1. Choose a matrix N which has eigenvalues equal to the desired poles of the closed loop. Restriction: These eigenvalues must all be different from the eigenvalues of A, i.e., all open-loop poles must be moved (which is standard anyway). So, we can either directly specify the desired poles by choosing a diagonal form (then the eigenvalues are directly equal to the diagonal elements) or we specify a characteristic polynomial by choosing a system matrix in controllable canonical form: or identical eigenvalues University 1. State-Space Control Page 90 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.4 State-Space Control via Pole Placement 2. Choose a 1×n vector , so that is observable. This vector can be interpreted as a transformed feedback vector / control vector. A simple possible choice that ensures observability is: 3. Solve the following Lyapunov equation in order to obtain T : 4. The feedback vector of the state-space controller is then calculated from: The operation of this method can be verified as follows: different realizations of the same system If we interpret T as transformation matrix, then and N are system matrices of different realizations of the same system. Thus, both have also the same eigenvalues! So, the closed loop has the same eigenvalues as N, which possesses the desired eigenvalues/poles. University 1. State-Space Control Page 91 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.4 State-Space Control via Pole Placement Plant, Open-Loop, and Closed-Loop Control in Frequency Domain It is often useful and conducive for comprehension to describe important input/output behaviors not only in state-space but also as a transfer function. The transfer function of the control plant u → y is: The transfer function of the open-loop control runs back via kT instead via cT to the output and therefore is: The transfer function of the closed-loop control w → y must take the state feedback into account as well: University 1. State-Space Control Page 92 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.4 State-Space Control via Pole Placement MATLAB State-space Control via Pole Placement: System = ss(A,b,c,d); % Define a system in state space rank(ctrb(System)) % Check its controllability (result == n ?) EW = [-3 -2+i -2-i]; % Desired eigenvalues k = place(A,b,EW); % State-space control via pole placement University 1. State-Space Control Page 93 Prof. Dr.-Ing. Oliver Nelles of Siegen 1.5 State-Space Control via Optimization (LQ) From Section 11.1 in RT we know the following quadratic loss function: It penalizes the control deviation and the actuating effort. Let us first assume a constant reference variable, i.e., a fixed command control (regulation problem). Then we can set the reference (or command) variable w = 0 (another constants are possible and can easily be dealt with by shifting the operating point of y). Then the loss function simplifies to: So, we want to regulate a disturbance within a time te “as good as possible” (y should go back to 0). The above loss function indicates what exactly is meant by “as good as possible”. If the end time te is chosen small, it may be that y is still far away from 0. Therefore, it makes sense to include a term for the deviation of y from the desired final value (= 0) at time te in the loss function. University 1. State-Space Control Page 94 Prof. Dr.-Ing. Oliver Nelles